Grade 6 2023 Virginia SOL - Teacher Edition by Mathspace

Grade 6 Teacher’s Edition

Built for the

The front matter for our print textbooks will be ready soon. Scan the QR code below to register for updates when it’s available.

© 2024 Mathspace Pty Ltd Copyright Notice This Work is copyright. All rights are reserved. No part of this Work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without prior written permission of the Publisher. For permission to use material from this text or product, please email hello@mathspace.co Mathspace Inc. 228 Park Ave S #15992 New York NY 10003-1502 For our full digital offering, visit mathspace.co Mathspace Virginia: Grade 6 for 2023 SOLs - Print Teacher Edition ISBN: 978-1-963022-15-5

Contents

1 2 3 4

Operations with Fractions

1.01

Review: Simplify fractions and mixed numbers

1.02

Add and subtract fractions and mixed numbers (6.CE.1)

1.03

Multiply fractions and mixed numbers (6.CE.1)

1.04

Divide fractions and whole numbers (6.CE.1)

1.05

Divide fractions and mixed numbers (6.CE.1)

1.06

Solve problems with fraction operations (6.CE.1)

Fractions, Decimals, & Percents

112

2.01

Percents as fractions (6.NS.1)

116

2.02

Percents as decimals (6.NS.1)

133

2.03

Convert between fractions, decimals, and percents (6.NS.1)

142

2.04

Compare and order fractions, decimals, and percents (6.NS.1)

153

Integers & Exponents

172

3.01

Identify and represent integers (6.NS.2)

176

3.02

Compare and order integers (6.NS.2)

193

3.03

Introduction to exponents (6.NS.2)

205

3.04

Patterns with perfect squares (6.NS.3)

217

3.05

Powers of 10 and place value (6.NS.2)

229

Operations with Integers

244

4.01

Add and subtract integers (6.CE.2)

248

4.02

Multiply and divide integers (6.CE.2)

266

4.03

Absolute value of integers (6.CE.2)

280

4.04

Real-world problems with integers (6.CE.2)

292

4.05

Integers in the coordinate plane (6.MG.3)

304

Contents mathspace.co

iii

5 6 7 8

Ratios & Proportional Relationships

332

5.01

Introduction to ratios (6.PFA.1)

336

5.02

Equivalent ratios and ratio tables (6.PFA.1)

355

5.03

Unit rates (6.PFA.2)

372

5.04

Proportional relationships (6.PFA.2)

390

Equations & Inequalities

418

6.01

Algebraic expressions (6.PFA.3)

422

6.02

Properties of real numbers

436

6.03

One-step equations with addition and subtraction (6.PFA.3)

453

6.04

One-step equations with multiplication and division (6.PFA.3)

476

6.05

Write inequality statements (6.PFA.3)

493

6.06

Solutions to inequalities (6.PFA.4)

505

Polygons 526 7.01

Congruent figures (6.MG.4)

530

7.02

Regular polygons and symmetry (6.MG.4)

552

7.03

Perimeter of triangles and parallelograms (6.MG.2)

567

7.04

Area of parallelograms (6.MG.2)

583

7.05

Area of triangles (6.MG.2)

596

7.06

Polygons in the coordinate plane (6.MG.3)

608

Circles 630 8.01

Characteristics of circles (6.MG.1)

634

8.02

Circumference and pi (6.MG.1)

646

8.03

Area of a circle (6.MG.1)

664

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Statistics 678 9.01

Formulate questions and collect data (6.PS.1)

682

9.02

Create and interpret circle graphs (6.PS.1)

704

9.03

Compare representations of data (6.PS.1)

730

9.04

Review: measures of center and spread

755

9.05

Mean as a balance point (6.PS.2)

771

9.06

Changing data values and measures of center (6.PS.2)

788

9.07

Outliers (6.PS.2)

806

Contents mathspace.co

1 Operations with Fractions Big ideas • The properties of real numbers can be applied to many types of expressions. • Real numbers are either rational or irrational. • Expressions are the building blocks of algebra. They can be used to represent and interpret realworld situations.

Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06

Review: Simplify fractions and mixed numbers Add and subtract fractions and mixed numbers (6.CE.1) Multiply fractions and mixed numbers (6.CE.1) Divide fractions and whole numbers (6.CE.1) Divide fractions and mixed numbers (6.CE.1) Solve problems with fraction operations (6.CE.1)

6 23 40 60 79 95

Dividing fractions is like sharing a chocolate bar equally among friends. Everyone gets a fair share!

1. Operations with Fractions Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 5.CE.1 — The student will estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition, subtraction, multiplication, and division with whole numbers.

5.CE.2 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of fractions with like and unlike denominators (with and without models), and solve single-step contextual problems involving multiplication of a whole number and a proper fraction, with models.

Big ideas and essential understanding The properties of real numbers can be applied to many types of expressions. 1.02 — Adding and subtracting rational numbers in different forms can be easily done by converting them first to the same form. 1.03 — Multiplying fractions and mixed numbers can be efficiently done by converting them to improper fractions and applying the multiplication operation. 1.04, 1.05 — Dividing by a fraction is the same as multiplying by its reciprocal.

Real numbers are either rational or irrational. 1.01 — The same fraction can be represented by an infinite set of different but equivalent fractions. A fraction can be expressed in its simplest form by dividing the numerator and denominator by common factors until there are no common factors other than 1. Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 1.06 — Determining the correct operation is essential to writing expressions to solve contextual problems with real numbers.

Standards 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context. 6.CE.1a — Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.* 1.03 Multiply fractions and mixed numbers

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

1.04 Divide fractions and whole numbers 1.05 Divide fractions and mixed numbers 6.CE.1b — Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.* 1.03 Multiply fractions and mixed numbers 1.04 Divide fractions and whole numbers 1.05 Divide fractions and mixed numbers

6.CE.1c — Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.* 1.03 Multiply fractions and mixed numbers 1.04 Divide fractions and whole numbers 1.05 Divide fractions and mixed numbers

6.CE.1e — Estimate, determine, and justify the solution to single-step and multistep problems in context that involve multiplication and division with fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form. 1.06 Solve problems with fraction

6.CE.1d — Estimate, determine, and justify the solution to single-step and multistep problems in context that involve addition and subtraction with fractions (proper or improper) and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form. 1.02 Add and subtract fractions and mixed numbers 1.06 Solve problems with fraction

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Continuous Assessment Measure standards proficiency with check-ins Before starting a new topic, it’s a great time to go online and have students complete a Skills Check-in to measure their readiness for the topic.

1. Operations with Fractions mathspace.co

1.01 Review: Simplify fractions and mixed numbers Subtopic overview Lesson narrative In this lesson, students will review and practice simplifying fractions and mixed numbers. The lesson begins with a focus on identifying and converting between improper fractions and mixed numbers. Students will learn to find equivalent fractions and simplify fractions by dividing the numerator and denominator by their greatest common factor. The lesson includes visual representations, number line examples, and multiple step-by-step practice problems to reinforce these concepts. An exploration with an interactive applet allows students to investigate equivalent fractions by manipulating numerators and denominators. By the end of the lesson, students should confidently simplify fractions and convert between improper fractions and mixed numbers.

1.01 Review: Simplify fractions and Learning objectives mixed numbers Students: Page 4

After this lesson, you will be able to... • rewrite fractions and mixed numbers in simplest form.

Improper fractions and mixed numbers

Key vocabulary Fractions like make up less than one whole. We know this because the numerator is less than the denominator. We call these proper fractions.  equivalent fractions  greatest common factor  improper fraction common factor What about a fraction like ? Notice the numerator is greater thanform the denominator. This means that the fraction  mixed  simplest number properthat fraction 

is greater than a whole. We call these improper fractions.

Essential understanding

Each rectangle in this image has been split into ﬁve equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator.

The same fraction can be represented by an infinite set of different but equivalent fractions. A fraction can be Since the number of shaded is more than until the number of parts expressed in its simplest form by dividing the numerator and denominator byparts common factors there are no in one whole, a complete rectangle and three more parts have common factors other than 1. been shaded. We can write this number as 1 , which we call one and three ﬁfths.

Standards Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number This subtopic addresses theline. following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

For

or 1 , each whole number on the number line is split into

5 equal parts.

0 1 2 For the improper fraction, we would count 8 tick marks from 0, and Teachers can integrate this goal into their instruction by posing real-world problems that require simplifying fractions place the point there. and mixed numbers. For example, they can ask students to determine the total amount of an ingredient needed the mixed number, we would countThis 3 tickwould marksrequire from 1. students to when combining two recipes that require differentFor fraction amounts of that ingredient. add the fractions together and simplify the result.

Example 1 6

Mathspace

Virginia SOL Grade 6 Teacher Edition

mathspace.co What number is plotted on the number line? Give your answer as a mixed number and as an improper fraction.

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can help students connect the current lesson to previous ones by reminding them of the skills they learned in 5.CE.2c and how this lesson builds on those skills. For example, teachers can remind students of the process of finding the GCD and how it’s used to simplify fractions. Teachers can also connect mathematical concepts to other disciplines, such as cooking or woodworking when discussing real-world examples of simplifying fractions and mixed numbers.

Teachers can incorporate visual models, such as fraction bars/strips, visual models, and number lines, to represent fractions and mixed numbers. For example, they can use fraction bars to demonstrate how to simplify fractions or use number lines to show how fractions and mixed numbers are related. Teachers can also ask students to create their own visual representations of fractions and mixed numbers and use these to identify the simplest form.

Prior connections 5.CE.2 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of fractions with like and unlike denominators (with and without models), and solve single-step contextual problems involving multiplication of a whole number and a proper fraction, with models.

Future connections 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

Lesson Preparation Tools You may ﬁnd these tools helpful: • Fraction bars or fraction circles

1.01 Review: Simplify fractions and mixed numbers mathspace.co

Student lesson & teacher guide Improper fractions and mixed numbers Students will revisit proper fractions, improper fractions and mixed numbers, looking at how to visually represent these concepts using fraction bar and a number line.

Students: Page 4

1.01 Review: Simplify fractions and mixed numbers After this lesson, you will be able to... • rewrite fractions and mixed numbers in simplest form.

Improper fractions and mixed numbers Fractions like

make up less than one whole. We know this because the numerator is less than the denominator.

We call these proper fractions. What about a fraction like

? Notice that the numerator is greater than the denominator. This means that the fraction

is greater than a whole. We call these improper fractions. Each rectangle in this image has been split into ﬁve equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator. Since the number of shaded parts is more than the number of parts in one whole, a complete rectangle and three more parts have been shaded. We can write this number as 1 , which we call one and three ﬁfths. Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number line. For

or 1 , each whole number on the number line is split into

5 equal parts. 0

For the improper fraction, we would count 8 tick marks from 0, and place the point there. For the mixed number, we would count 3 tick marks from 1.

Example 1 Step-by-step checklist for converting fractions Student withisdisabilities What number plotted on thesupport number line? Give your answer as a mixed number and as an improper fraction. To assist students who struggle with organization and memory, provide them with a step-by-step checklist for 3 converting between improper fractions and mixed numbers. Create a structured4 guide that outlines each step of the process. Encourage students to refer to this checklist as they work through problems, checking off each Create a strategy step as they complete it. This approach helps students keep track of multiple parts of a problem and ensures For the fraction part, count the number of equal spaces between the two whole numbers, then count how many they don’t skip crucial steps. spaces after the previous whole number the point is located.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 4 Mathspace Virginia SOL Grade 6 mathspace.co

Improper fractions and mixed numbers Fractions like

make up less than one whole. We know this because the numerator is less than the denominator.

We call these proper fractions.

For example:

What about a fraction like

? Notice that the numerator is greater than the denominator. This means that the fraction

Example

is greater than a whole. We call these improper fractions.

Step 1

Each rectangle in this image has been split into ﬁve equal parts, Divide the numerator by the denominator

Step 2 Step 3

The whole number is the quotientnumerator. 4 The numerator is the remainder Since the number of shaded parts 42 is more than the number of parts

Step 4

The denominator stays the same been shaded.

so 5 is the denominator. Eight parts have been shaded, so 8 is the

in one whole, a complete rectangle and three more parts have

We can write this number as 1 , which we call one and three ﬁfths.

Additionally, visually organize the checklist with clear headings and bullet points, and consider using symbols Numbers to likehighlight this are called mixed numbers mixedstrategy numerals.will Mixed numbers and improper cantasks also be or color-coding important actions.orThis help students managefractions complex more represented a number line.in their ability to simplify fractions successfully. efficiently and buildonconfidence For

or 1 , each whole number on the number line is split into

5 equal parts.

Examples 0

For the improper fraction, we would count 8 tick marks from 0, and place the point there.

Students: Pages 4–5

For the mixed number, we would count 3 tick marks from 1.

Example 1 What number is plotted on the number line? Give your answer as a mixed number and as an improper fraction. 4

Create a strategy For the fraction part, count the number of equal spaces between the two whole numbers, then count how many spaces after the previous whole number the point is located.

Apply the idea The point is located between 3 and 4. It is greater than 3 but less than 4. So, the whole number part is 3. There are 10 equal spaces between 3 and 4, so each space represents the fraction is

. The point is 9 spaces to the right of 3, so

Mathspace Virginia SOL Grade 6 The number plotted on the number line is 3 mathspace.co 4

This can also be written as

Example 2 Purpose Show students to improper accurately determine the value of a number represented on a number line, and express Rewrite 4how as an fraction. that number as a mixed number and an improper fraction. Create a strategy

Expected mistakes To ﬁnd the improper fraction, multiply the whole number by the denominator, then add the numerator. Students might incorrectly label the tick marks between the whole numbers on the number line. Encourage studentsApply to count the number of spaces between 3 and 4; this will be the denominator. Then, they can label the idea the ticks with mixed numbers, or convert 3 and 4 to and respectively and label the ticks with improper Multiply the denominator and whole number fractions. Evaluate the multiplication Evaluate the addition

Reﬂect and check We could also draw an array for 4

where each part represents .

We can see that there are 9 shaded parts. So, the improper fraction is .

1.01 Review: Simplify fractions and mixed numbers mathspace.co

There are 10 equal spaces between 3 and 4, so each space represents the fraction is

. The point is 9 spaces to the right of 3, so

The number plotted on the number line is 3 This can also be written as

Students: Page 5

Example 2 Apply the Rewrite 4 idea as an improper fraction. The point is located between 3 and 4. It is greater than 3 but less than 4. So, the whole number part is 3. Create a 10 strategy There are equal spaces between 3 and 4, so each space represents

. The point is 9 spaces to the right of 3, so

To the improper fraction, multiply the whole number by the denominator, then add the numerator. theﬁnd fraction is . The number plotted on the number line is 3

Apply the idea

This can also be written as

Multiply the denominator and whole number Evaluate the multiplication

Example 2

Evaluate the addition

Rewrite 4andascheck an improper fraction. Reﬂect

Create a strategy

We could also draw an array for 4

where each part represents .

We can see that there are 9 shaded parts. To ﬁnd the improper fraction, multiply the whole number by the denominator, then add the numerator. So, the improper fraction is .

Apply the idea Multiply the denominator and whole number Evaluate the multiplication Evaluate the addition

Example 3 PurposeReﬂect and check Show students convert a mixed number into an improper fraction. Rewrite how as a to mixed number. We could also draw an array for 4

where each part represents .

Expected mistakes We can see that there are 9Apply shaded Create a strategy theparts. idea StudentsDivide might multiply 4 by 1, then add 2. To address this misconception, encourage them to draw a picture So, the improper fraction is . the numerator by the denominator. The remainder 11 divided by 3 is 3 remainder 2. of

. Have separate of the 4 wholes in 2 equal-sized pieces, then count the total number of halves will bethem the numerator of each the mixed fraction. So, is made up of 3 wholes and 2 out of 3 remaining.

they see.

Students: Page 5 Example 3 Rewrite

1.01 Review: Simplify fractions and mixed numbers mathspace.co

as a mixed number.

Create a strategy

Apply the idea

Divide the numerator by the denominator. The remainder will be the numerator of the mixed fraction.

11 divided by 3 is 3 remainder 2. So,

is made up of 3 wholes and 2 out of 3 remaining.

Purpose 1.01 Review: Simplify fractions and mixed numbers mathspace.co Show students how to convert an improper fraction into a mixed number

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Reflecting with students Aski students to describe how

and

are the same, and encourage them to use precise mathematical

language. A precise response might be: “

equals

because when we divide 11 parts into groups of 3, the

result is 3 whole groups with a remainder of 2 parts. This means we have 3 whole groups and

of a group, so

.” Encourage them to verify their work by showing how both forms represent the same point on a number line or through visual models, thereby reinforcing accurate mathematical communication and understanding.

Idea summary Students: Page 6 Fractions where the numerator is greater than the denominator are called improper fractions. Improper fractions can be rewritten as mixed numbers.

Idea summary

Equivalent fractions Fractions where the numerator is greater than the denominator are called improper fractions. Sometimes different fractions representasthe samenumbers. amount. Consider the shaded area of the hexagon below. Improper fractions can can be rewritten mixed We can see that 3 of the 6 parts have been shaded in, so the area represents

of the whole shape.

Equivalent fractions

But we can also see that of the shape has been shaded in. Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below. Since the same area is shaded for both and , these two fractions We can that 3 of the 6 parts have been shaded in, so the area must besee equal. Students will explore equivalent fractions, understanding their representation, and applying ofvisual the whole shape. These are called equivalent fractions, since theyrepresents have different numerators and different denominators butthese are stillconcepts to

Equivalent fractions Idea summary

rewrite fractions equal. with different denominators.

of theimproper shape has been shaded in. also see that Fractions where the numerator is greater But thanwe thecan denominator are called fractions.

Improper be rewritten as mixed numbers. Students: Pages 6–7fractions can Interactive exploration Since the same area is shaded for both

and , these two fractions

Explore online to answer the questionsmust be equal.

These are called equivalent fractions, since they have different numerators and different denominators but are still mathspace.co equal. Equivalent fractions Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below. Use theInteractive interactive exploration in 1.01 to answer these questions. exploration We can see that 3 of the 6 parts have been shaded in, so the area online to answer the questions 1. Explore When multiplying the numerator and denominator of the original fraction by 2, how many pieces is the represents of the whole shape. shaded part of the new fraction separated into? 2.

mathspace.co When multiplying the numerator and denominator thesee original by 3, how pieces isin.the of the shape has many been shaded But we can of also that fraction

shaded part of the new fraction separated into? Use the interactive exploration in 1.01 to answer these Since thequestions. same area is shaded for both and , these two fractions 3. Do you think this is the case for any multiple of the numerator and denominator of any fraction? must be equal. 1. When multiplying the numerator and denominator of the original fraction by by 2, how manynumber? pieces is the 4. Why do you think we have to multiply both the numerator and denominator the same These areshaded called equivalent since they have different numerators and different denominators but are still part of the fractions, new fraction separated into? equal.2. When multiplying the numerator and denominator of the original fraction by 3, how many pieces is the We can see equivalent fractions action by representing shaded part of the newinfraction separated into? them visually and seeing how we can create them. Consider grid:think this exploration Interactive 3. this Do you is the case for any multiple of the numerator and denominator of any fraction? 4.

Explore to answer the This the gridnumerator is divided and into denominator 6 equal partsby as the shown bynumber? the solid grid Why do online you think we have toquestions multiply both same

lines. 1 out of 6 parts is shaded, so we can represent this area as of the grid. We can see equivalent fractions in action by representing them visually and seeing how we can create them. If we separate each of the 6 parts into 3 more equal parts, as shown Use thethis interactive exploration in 1.01 to answer these questions. Consider grid: by the dashed lines, we can create an equivalent fraction. 1. When multiplying the numerator and denominator of the original fraction by 2, many is grid the This grid is divided into 6 equal parts ashow shown by pieces the solid shaded part of the new fraction separated into? Notice nowis18shaded, grid squares, of them are lines. 1 there out ofare 6 parts so we and can 3represent thisshaded. area as

mathspace.co

This that can represent the3,shaded areapieces with the When multiplying the numerator and denominator thewe original fraction by how many is the of themeans grid. of shaded part of the new fraction separated into? . fraction If we separate each of the 6 parts into 3 more equal parts, as shown 3. Do you think this is the case for any multiple the numerator of any fraction? by theofdashed lines, weand candenominator create an equivalent fraction. 4. Why do you think we have to multiply both the numerator and denominator by the same number? Notice there are now 18 grid squares, and 3 of them are shaded. This means that we can represent the shaded area with the We can see equivalent fractions in action by representing visually and seeing how we can create them. fraction them . Consider this grid: 2.

Mathspace Virginia SOL Grade 6 mathspace.co

This grid is divided into 6 equal parts as shown by the solid grid lines. 1 out of 6 parts is shaded, so we can represent this area as of the grid. 1.01 Review: Simplify fractions and mixed numbers If we separate each of the 6 parts into 3 more equal parts, asmathspace.co shown by the dashed lines, we can create an equivalent fraction. Notice there are now 18 grid squares, and 3 of them are shaded.

This shows that

and

are equivalent fractions, which we can express using an equal sign:

By separating each of the original parts into 3 more equal parts, we are multiplying the numerator (the shaded part) and denominator (the entire grid) by 3:

We can always ﬁnd an equivalent fraction by multiplying both the numerator and denominator by the same number.

Example 4 Rewrite with a denominator of 40. Fraction bingo

Targeted instructional strategies Create a strategy

FractionMultiply Bingo the is anumerator fun game can be used teachofequivalent fractions. andthat denominator by theto quotient 40 divided by 8. Materials needed: Apply the idea • Bingo cards with fractions written in each square (students may create their own) Dividing 40 by 8 gives 40 ÷ 8 = 5. • Fraction chips or markers • Bingo caller Multiply the numerator and denominator by 5 Instructions:

Evaluate

1. Give each student a bingo card with different fractions in each square. 2. Have students place fraction chips or markers on the fractions called out by the bingo caller. 3. The first student to get a full line of fractions (horizontal, vertical, or diagonal) shouts “Bingo!” and is the winner of that Idearound. summary two fractions represent the samefractions amount ofas a whole, they are fractions. 4. EncourageWhen students to identify equivalent they play theequivalent game. For example, if a fraction of

We can create an equivalent fraction by multiplying the numerator and denominator by the same number.

called, students can look for equivalent fractions such as marker.

on their bingo cards and cover them with a

5. After each round, discuss the equivalent fractions found by the students and how they relate to each other.

Simplify fractions

We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this Collect and display simplifying the fraction.

English language learner support Previously, we saw that As students are working on simplifying fractions and converting between improper fractions and mixed numbers, note how they describe the concepts of “mixed number,” “improper fraction,” “proper fraction,” and From here, we can remove the common factor of 3: “equivalent fraction.” Collect the different ways that students understand and explain these concepts and display them in a common place for all students to access. If students do not come up with alternative ways to express these concepts or seem confused, suggest some of This shows: your own. For example: • Proper fraction • A The fraction where numerator is between less than denominator fraction hasthe no common factors thethe numerator and denominator (other than 1), so it is in simplest • Represents a quantity less than one form. This also means that 3 was the greatest common factor of 3 and 18. • Mixed number • A whole number combined with a fraction 1.01 Review: Simplify fractions and mixed numbers 7 • Represents a quantity greater than one mathspace.co • Can be converted to an improper fraction

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

• Improper fraction • A fraction where the numerator is greater than or equal to the denominator • Represents a quantity greater than or equal to one • Can be converted to a mixed number Idea summary • Equivalent fraction Fractions where the numerator is greater than the denominator are called improper fractions. • Different fractions that represent the same value Improper fractions can be rewritten as mixed numbers. • Have different numerators and denominators but are equal when simplified • Can be found by multiplying or dividing the numerator and denominator by the same number Take care to address any misunderstandings or misconceptions that may arise, such as confusing improper Equivalent fractions fractions with proper fractions or not recognizing that mixed numbers and improper fractions can represent Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below. the same quantity. Encourage students to make connections between these concepts and provide visual We can see that 3 of the 6 parts have been shaded in, so the area representations, like fraction bars or number lines, to help clarify their understanding. represents

of the whole shape.

But we can also see that

Exploration

of the shape has been shaded in.

Since the same area is shaded for both must be equal.

and , these two fractions

Students:These Page are6 called equivalent fractions, since they have different numerators and different denominators but are still equal.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.01 to answer these questions. 1.

When multiplying the numerator and denominator of the original fraction by 2, how many pieces is the shaded part of the new fraction separated into?

When multiplying the numerator and denominator of the original fraction by 3, how many pieces is the shaded part of the new fraction separated into?

Do you think this is the case for any multiple of the numerator and denominator of any fraction?

Why do you think we have to multiply both the numerator and denominator by the same number?

We can see equivalent fractions in action by representing them visually and seeing how we can create them. Consider this grid:

Suggested student grouping: In pairs

This grid is divided into 6 equal parts as shown by the solid grid

In this exploration, students will use a GeoGebralines. applet equivalent fractions. willasexplore 1 outto ofinvestigate 6 parts is shaded, so we can represent They this area how multiplying the numerator and denominatorofofthe a grid. fraction by the same number results in an equivalent fraction because the shaded parts of the two fractions are the same. If we separate each of the 6 parts into 3 more equal parts, as shown by the dashed lines, we can create an equivalent fraction.

Ideal student responses Notice there are now 18 grid squares, and 3 of them are shaded. These ideal responses may differ from other correct student responses. Less formal responses can be This means that we can represent the shaded area with the connected with the more precise mathematical language presented here. fraction

1. When multiplying the numerator and denominator of the original fraction by 2, how many pieces is the shaded part of the new fraction separated into? When you multiply both the numerator and the denominator by 2, the shaded part of the new fraction is separated into twice as many pieces as the original fraction. 2. When multiplying the numerator and denominator of the original fraction by 3, how many pieces is the Mathspace Grade 6separated into? 6 shaded part of theVirginia new SOL fraction mathspace.co When you multiply both the numerator and the denominator by 3, the shaded part of the new fraction is separated into three times as many pieces as the original fraction.

1.01 Review: Simplify fractions and mixed numbers mathspace.co

3. Do you think this is the case for any multiple of any fraction? Yes, this is the case for any multiple of any fraction. When you multiply both the numerator and the denominator of a fraction by the same number, the shaded part will be separated into that many times more pieces while still representing the same value as the original fraction. 4. Why do you think we have to multiply both the numerator and denominator by the same number? We have to multiply both the numerator and denominator by the same number to maintain the proportion or relationship between the two parts of the fraction. Purposeful questions • Try dividing the numerator and denominator of a fraction by the same number (make sure the fraction is divisible by that number). Can you draw a picture of the two fractions? How does the new fraction relate to the original fraction? • Choose a fraction and multiply its numerator and denominator by different numbers. What happens to the value of the fraction? Does it still represent the same part of the whole? Possible misunderstandings • Students may struggle to visualize the equivalent fractions using the applet. They may have difficulty understanding how the shaded part of the new fraction is separated into different pieces when the numerator and denominator are multiplied by the same number. This shows that and will are learn equivalent fractions,represent which we can express using an equal sign: a grid, create them by After the exploration, students to visually equivalent fractions using dividing the grid further, and solve problems. By separating each of the original parts into 3 more equal parts, we are multiplying the numerator (the shaded part) and denominator (the entire grid) by 3:

Examples

Students: Page 7 We can always ﬁnd an equivalent fraction by multiplying both the numerator and denominator by the same number.

Example 4 Rewrite

with a denominator of 40.

Create a strategy Multiply the numerator and denominator by the quotient of 40 divided by 8.

Apply the idea Dividing 40 by 8 gives 40 ÷ 8 = 5. Multiply the numerator and denominator by 5 Evaluate

Idea summary Purpose When two fractions represent the same amount of a whole, they are equivalent fractions. Show studentsWethat rewrite fractions different denominators, maintaining the value of the fraction. canthey createcan an equivalent fraction bywith multiplying the numerator and denominator by the same number. Expected mistakes Students might simply change the denominator to 40, while maintaining a numerator of 3. Ask students whether Simplify fractions the fractions are equivalent. Explain that the goal is to create an equivalent fraction, not a different fraction We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can altogether. remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction. Previously, we saw that

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co From here, we can remove the common factor of 3:

Dividing 40 by 8 gives 40 ÷ 8 = 5.

Example 4 Rewrite

Multiply the numerator and denominator by 5

with a denominator of 40. Evaluate

Students:Create Pagea 7strategy Multiply the numerator and denominator by the quotient of 40 divided by 8.

Apply the idea

Idea summary

Dividing 40 by 8 gives 40 ÷ 8 = 5. When two fractions represent the same amount of a whole, they are equivalent fractions. Multiply by denominator 5 We can create an equivalent fractionthe bynumerator multiplyingand thedenominator numerator and by the same number. Evaluate

Simplify fractions We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can Simplify fractions remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction. Students recall the concept of simplifying fractions by cancelling common factors in both the numerator and Idea summary Previously, we saw that denominator. When two fractions represent the same amount of a whole, they are equivalent fractions. We can create an equivalent fraction by multiplying the numerator and denominator by the same number.

Students: Page 7

From here, we can remove the common factor of 3:

Simplify fractions We also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can Thiscan shows: remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction. Previously, we saw that The fraction has no common factors between the numerator and denominator (other than 1), so it is in simplest form. This also means that 3 was the greatest common factor of 3 and 18. From here, we can remove the common factor of 3: 1.01 Review: Simplify fractions and mixed numbers mathspace.co

This shows:

The fraction

has no common factors between the numerator and denominator (other than 1), so it is in simplest

form. This also means that 3 was the greatest common factor of 3 and 18.

Equivalent fractions and simplification

1.01 Review: Simplify fractions and mixed numbers mathspace.co

Address student misconceptions Some students may think all equivalent fractions are simplified to their lowest terms. They may believe that two fractions are not equivalent if they are not simplified to their lowest terms. Clarify to students that equivalent fractions have the same value but may not always be in their simplest form. Provide examples of equivalent fractions that are not in their simplest form to help students understand the concept. Encourage them to practice finding equivalent fractions with different denominators and simplify fractions by canceling common factors.

1.01 Review: Simplify fractions and mixed numbers mathspace.co

Examples Students: Page 8

Purpose Make students aware that fractions can be simplified using a visual representation, such as a grid. Reflecting with students Challenge advanced learners, or any student that is ready, to design their own grids, shading a portion to represent a fraction of their choice. Then, have them swap grids with a partner. Each student will determine and simplify the fraction represented by their partner’s grid. Ask students to explain the steps they took to simplify the fraction and discuss any patterns or relationships they observed between the grid and the simplified fraction. This promotes critical thinking and helps students make connections between visual models and numerical expressions.

Students: Pages 8–9

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Now, simplify each of the fractions in the answer choices and compare them to : • Option A Factor out the GCF of 10 Divide The fraction

is equivalent to

because they both simplify to .

• Option B The fraction

is already in its simplest form because 7 and 3 have no common factors other than 1. The fraction

is not equivalent to

because it is greater than

and not equal to it.

• Option C Factor out the GCF of 2 Divide The fraction

is not equivalent to

because it is smaller than

and not equal to it.

• Option D The fraction

is already in its simplest form and is equivalent to

which we already found simpliﬁes to .

So, options A and D are the correct answers.

Reﬂect and check Simplifying creates equivalent fractions by making the numerator and denominator smaller. We can also make equivalent fractions by making numerators and denominators larger. is simpliﬁed to . We can also scale up

to form another equivalent fraction:

Multiply the numerator and denominator by 10 Evaluate This conﬁrms our answer, showing that we have correctly identiﬁed the equivalent fractions

and

Idea summary Purpose When we ﬁnd an equivalent fraction by removing common factors from the numerator and denominator, we are simplifying the fraction. Challenge students to determine if fractions are equivalent by simplifying the fractions to simplest form and comparing them. When the fraction has no common factors between the numerator and denominator (other than 1), it is in simplest form.

Expected mistakes To write a fraction in simplest form, divide both the numerator and denominator by their greatest common Students might not choose Option A because there is no whole number that they can multiply to the numerator factor. Two fractions when they can be simpliﬁed to the same fraction. and denominator of toare getequivalent .

Remind students that two fractions are equivalent when they represent the same portion of a whole. In this case, both fractions represent two-thirds of a whole, which makes them equivalent. Draw pictures of both fractions to reinfornce this concept, if necessary. 1.01 Review: Simplify fractions and mixed numbers 9 mathspace.co

1.01 Review: Simplify fractions and mixed numbers mathspace.co

is simpliﬁed to . We can also scale up

to form another equivalent fraction:

Multiply the numerator and denominator by 10 Evaluate

Students: Page 9

This conﬁrms our answer, showing that we have correctly identiﬁed the equivalent fractions

and

Idea summary When we ﬁnd an equivalent fraction by removing common factors from the numerator and denominator, we are simplifying the fraction. When the fraction has no common factors between the numerator and denominator (other than 1), it is in simplest form. To write a fraction in simplest form, divide both the numerator and denominator by their greatest common factor. Two fractions are equivalent when they can be simpliﬁed to the same fraction.

1.01 Review: Simplify fractions and mixed numbers mathspace.co

Practice Students: Pages 10–12

What do you remember? 1

What does it means when two fractions are equivalent? Use an example to support your answer.

Which of these statements are correct? Select all that apply.

A fraction is in its simplest form if the numerator and denominator have common factors other than 1.

A fraction is in its simplest form if the numerator and denominator have no common multiple.

A fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

A fraction is in its simplest form if the numerator is 1.

A mixed number is in its simplest form if the whole number is equal to 1.

An improper fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

Which of these fractions are fully simpliﬁed? A

Complete the solution to simplify the fraction

Let’s practice 5

Complete each equivalent fraction statement: a

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Use the diagrams provided to simplify these fractions: a

Which fraction is eqivalent to A

in simplest form?

E 8

Which fraction is equivalent to A

in simplest form?

1.01 Review: Simplify fractions and mixed numbers mathspace.co

Complete the statement using the fractions from the list: ⬚ is equal to ⬚

Complete the statement using the mixed numbers from the list: ⬚ is equal to ⬚

Select the equivalent fractions from each list: a

Simplify:

Are

Which of these pairs of fractions are equivalent? Justify your reasoning.

and

equivalent fractions? Explain your reasoning.

and , and , and 15

Which of these pairs of fractions are equivalent? Justify your reasoning. and ,

Jane and Timothy disagree on the simplest form of

and

. Determine who is correct and provide an explanation. • Timothy’s Answer:

• Judy’s Answer:

Let’s extend our thinking 17

Determine an equivalent fraction to

If the numerator and denominator of a fraction are both prime numbers, is the fraction in fully simpliﬁed form? Explain your answer.

Maria has a recipe that calls for

that has a denominator of 12. Justify your reasoning.

cup of sugar.

If she wants to double the recipe, how much sugar will she need?

Write a fraction that is equivalent to your answer in part (a).

A pizza has been cut into 8 pieces, and 3 pieces have been eaten. a

What fraction of the pizza remains?

Write a fraction with a denominator of 16 that is equivalent to your answer in part (a).

Adityah is collecting data on favorite colors. They found that 5 out of 20 students in their class like blue. The class next door has 28 students. If the fraction of students who like blue in Adityah’s class is equivalent to the fraction of students who like blue in the class next door, how many of the students in the classroom next door like blue?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

14 Answer may look like this: To determine if the pairs of fractions are equivalent, we need to compare them by simplifying or ﬁnding a common denominator.

1.01 Review: Simplify fractions and mixed numbers

• and :

What do you remember? 1 Two fractions are considered equivalent when they represent the same portion or part of a whole, even though they may appear different. This happens when the numerator and denominator of one fraction can be multiplied or divided by the same number to yield the numerator and denominator of the other fraction. and

For example,

are equivalent fractions because

The fractions

you can simplify

or ﬁnd a common denominator for

3 and 9 is 9. Converting

to a denominator of 9 gives

, which is not equal to . • and : and

are not equivalent. Their

denominators are different, and there is no way to

3 B, D

simplify either fraction to make them have the same numerator and denominator. Also, ﬁnding a common denominator and comparing the converted fractions will show that they are not equal.

4 Let’s practice

• and

5 a

6 a

The fractions

and

are equivalent. This can be simpliﬁes to ,

seen by simplifying both fractions. and

are not equivalent. To see this,

both fractions. The simplest common denominator for

The fractions 2 C, D, F

and

also simpliﬁes to . Since both simpliﬁed forms

are equal, the original fractions are equivalent. and

Therefore, out of the given pairs, only equivalent fractions.

are

15 Answer may look like this: To determine if the pairs of fractions are equivalent, let’s ﬁrst convert any mixed numbers to improper fractions. Then, we can compare the fractions either by simplifying or ﬁnding a common denominator.

7 D 8 A 9 The equivalent fractions from the list are

and

• and :

10 The equivalent mixed numbers from the list are: Convert

and

to an improper fraction:

Now, compare and . These fractions are not equivalent as their numerators and denominators are different, and they cannot be simpliﬁed to the same value.

11 a

12 a

• and

Convert

to an improper fraction:

Compare

and

13 Answers may look like: Yes,

and

are equivalent fractions. To see why, we

can simplify each fraction to lowest terms. lowest terms, while

is already in

can be simpliﬁed by dividing both

. We can simplify

by dividing

both the numerator and the denominator by 2, which gives us

. So, these fractions are equivalent.

the numerator and denominator by 4 to get . Therefore, the two fractions are equivalent.

Answers mathspace.co

•

and

Convert

18 Answers may look like:

Yes a fraction with prime numbers in numerator and denominator is in fully simpliﬁed form as there is no common factor that the numerator and denominator can be divided by, other than 1.

to an improper fraction: .

and . These fractions are not Now, compare equivalent because their numerators are different while they have the same denominator. and

In summary, among the given pairs, only equivalent fractions.

are

To ﬁnd the simplest form of the fraction , we need to reduce it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of 12 and 72 is 12. So, we divide both the numerator and the denominator by 12:

is . This means that

Jane’s answer, , is the correct one. Timothy’s answer of simplest form, as

is a reduced form but not the can be further simpliﬁed to .

Let’s Extend our Thinking 17 Answers may look like: To ﬁnd an equivalent fraction to with a denominator of 12, we need to multiply both the numerator and denominator by the same factor. We can start by multiplying the denominator of , which is 3, by 4 to get 12. To keep the fraction equivalent, we also need to multiply the numerator by 4. Therefore, an equivalent fraction to

with a denominator of 12 is

b Answers may vary:

20 a

21 Answers may look like:

16 Answers may look like:

Therefore, the simplest form of

19 a cup

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

has the simpliﬁed fraction

(divide the numerator and

denominator by 5). Multiply both numerator and denominator by 7 to have (equivalent fraction). So, 7 students will like blue in the class next door

1.02 Add and subtract fractions and mixed numbers Subtopic overview Lesson narrative In this lesson, students will learn to add and subtract fractions and mixed numbers. They begin with adding and subtracting fractions with like denominators, understanding that this involves adding or subtracting the numerators while keeping the denominator the same. The lesson progresses to handling fractions with unlike denominators, where students find a common denominator before performing the operations. An interactive applet is included to help students explore adding fractions with area models. Types of problems students will solve include adding and subtracting simple fractions, finding common denominators, converting improper fractions to mixed numbers and vice versa, and solving word problems that require adding or subtracting mixed numbers. By the end of the lesson, students should be able to confidently add and subtract both fractions and mixed numbers, regardless of the denominators.

1.02 Add and subtract fractions and Learning objectives mixed numbers Students: Page 13

After this lesson, you will be able to... • add and subtract fractions with like and unlike denominators. • add and subtract mixed numbers.

Add and subtract fractions with the same denominator

Key vocabulary Suppose we want to ﬁnd 

. least common denominator (LCD)



unlike denominators



like denominators

Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded. Notice that the parts of each circle are the same size.

Essential understanding Since the parts areeasily the same we can place onefirst circle of Adding and subtracting rational numbers in different forms can be donesize, by converting them to on thetop same the other. Now, we can see 5 sevenths of the circle are shaded in. form. So, we can conclude that

Standards When the denominators are the same (called like denominators), we are adding quantities of the same size. This means we can add the the number of shaded pieces, without needing Standards to change the number of parts in the whole. This subtopic addresses following Virginia 2023 Mathematics of Learning standards. Mathematically, we are adding the numerators but keeping the same denominator. Suppose we want to ﬁnd

. Using the same circles, we can take 2 sevenths away from 3 sevenths. The part that remains is 1 seventh of the circle. So, we can conclude that

1.02 Add and subtract fractions and mixed numbers

When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded mathspace.co pieces is changing, but the number of parts in a whole stays the same. Mathematically, we are subtracting the numerators but keeping the same denominator.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can incorporate this goal by presenting contextual problems involving the addition and subtraction of fractions and mixed numbers. They should model effective practices, such as finding common denominators, converting mixed numbers to improper fractions, and validating solutions. They can also encourage students to create their own problems based on real-world situations and find solutions.By guiding students to understand, plan, execute, and review their problem-solving processes, teachers can develop students’ abilities to estimate, justify, and arrive at reasonable solutions in real-world contexts.

Teachers can foster mathematical reasoning by guiding students through contextual problems involving fractions and mixed numbers. This lesson helps students make and test predictions, justify their problem-solving steps, and evaluate their conclusions. Teachers should encourage students to think about the operations involved, such as how the meaning of addition or subtraction applies to the problem. By explaining their reasoning and discussing their thought processes, students gain a deeper understanding of operations and improve their strategic problem-solving skills, moving beyond reliance on key words.

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can help students make mathematical connections by recalling multiple problem types learned in earlier grades and applying them to more complex problems. They can introduce solution methods with worked-out examples, explaining each step and the decisions made during the problem-solving process. By demonstrating how to solve similar problems, discussing each decision, and engaging students with guiding questions, teachers can effectively connect prior knowledge to new, contextual problem-solving methods.

Teachers can help students develop mathematical connections by guiding them to use their fluency with fractions and mixed numbers to solve contextual problems. This lesson on adding and subtracting fractions and mixed numbers requires students to draw on their prior knowledge of computation with fractions to find exact solutions. Teachers can support students in identifying essential vocabulary, determining the appropriate operations, and planning their solutions. Encouraging students to explain their thought processes and justify their reasoning helps them check for reasonableness and strengthens their problem-solving skills.

MPG5 — Mathematical Representations Teachers can integrate this goal by using various models and manipulatives to represent the addition and subtraction of fractions and mixed numbers. For example, teachers can use fraction bars/strips, number lines, area models, and manipulatives like fraction tiles or pattern blocks to visually represent these operations. Teachers can also encourage students to use these models and manipulatives to help them solve problems and understand concepts. When presenting real-world examples and contextual problems, teachers can encourage students to create their own representations to solve the problems.

Content standards 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.01 Review: Simplify fractions and mixed numbers

Student lesson & teacher guide Add and subtract fractions with the same denominator Students will explore the concepts of addition and subtraction with fractions that have the same denominator. Through annotated images of shaded circles, students will learn that the pieces are the same size when the denominators are the same, which is why they can focus only on the numerators while keeping the denominators unchanged when adding and subtracting fractions.

Students: Page 13

1.02 Add and subtract fractions and mixed numbers After this lesson, you will be able to... • add and subtract fractions with like and unlike denominators. • add and subtract mixed numbers.

Add and subtract fractions with the same denominator Suppose we want to ﬁnd

. Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded. Notice that the parts of each circle are the same size.

Since the parts are the same size, we can place one circle on top of the other. Now, we can see 5 sevenths of the circle are shaded in. 1.02 Add and subtract fractions and mixed numbers . So, we can conclude that mathspace.co When the denominators are the same (called like denominators), we are adding quantities of the same size. This

• add and fractions with like and unlike denominators. Suppose we want tosubtract ﬁnd . • add and subtract mixed numbers. Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded. Notice that the parts of each circle are the same size.

Add and subtract fractions with the same denominator Suppose we want to ﬁnd

. Since the parts are the same size, we can place one circle on top of Here we have circle and aare circle with in. the other. Now,awe canwith see25sevenths seventhsshaded of the circle shaded 3 sevenths shaded. . So, we can conclude that Notice that the parts of each circle are the same size.

When the denominators are the same (called like denominators), we are adding quantities of the same size. This means we can add the number of shaded pieces,Since without change number partsone in the whole. theneeding parts aretothe samethe size, we canofplace circle on top of the other. Now, we can see 5 sevenths of the circle are shaded in. Mathematically, we are adding the numerators but keeping the same denominator. Suppose we want to ﬁnd

So, we can conclude that

Using the same circles, we can take 2 sevenths away from sevenths. The part thatadding remains is 1 seventh of the circle. When the denominators are the same (called like3denominators), we are quantities of the same size. This means we can add the number of shaded pieces, without needing to change the number of parts in the whole. . So, we can conclude that Mathematically, we are adding the numerators but keeping the same denominator. Suppose want to ﬁndare the. same, we are subtracting quantities of the same size. Again, the number of shaded When thewe denominators pieces is changing, but the number of parts in a whole the same. Using stays the same circles, we can take 2 sevenths away from 3 sevenths. Thethe part thatdenominator. remains is 1 seventh of the circle. Mathematically, we are subtracting the numerators but keeping same So, we can conclude that

Example 1

Examples When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded Evaluate

and simplify your answer.

is changing, but the number of parts in a whole stays the same. Students:pieces Pages 13–14

Mathematically, we are subtracting the numerators but keeping the same denominator.

Create a strategy Add the numerators over the same denominator and simplify. Example 1 Evaluate

and simplify your answer.

Create a strategy Add the numerators over the same denominator and simplify.

1.02 Add and subtract fractions and mixed numbers mathspace.co

Apply the idea Add the numerators Rewrite with a common factor of1.02 2 Add and subtract fractions and mixed numbers

mathspace.co

Evaluate

Idea summary

Purpose When two fractions have the same denominator, we can add or subtract them by adding or subtracting the Show studentsnumerators, how to add two fractions with the same denominator, then simplify the result. while keeping the denominator the same. Apply the idea

Expected mistakes

Add the numerators

and state with that asdifferent their final answer. Remind students that answers must be in StudentsAdd may and forgetsubtract to reduce fractions denominators simplest form to be correct. Rewrite with a common factor of 2

If the denominators of two fractions are different (called unlike denominators), then we are not adding quantities of the same size. Evaluate Students: Page 14

Interactive exploration Explore online to answer the questions

Idea summary mathspace.co

When two fractions have the same denominator, we can add or subtract them by adding or subtracting the numerators, while keeping the denominator the same. Use the interactive exploration in 1.02 to answer these questions. 1.

Type the fractions

and

into the fraction boxes. Then, press enter on your keyboard. Why can’t we

add the fractions the way they are?with different denominators Add and subtract fractions

2. Virginia Click “Show common units.” How did the area models change? Mathspace SOLofGrade 6 Teacher If the denominators two fractions are Edition different (called unlike denominators), then we are not adding quantities of 3. Explain how we can add the fractions now. What is the sum? mathspace.co the same size. 4. Does this work for other fractions with different denominators?

Interactive exploration

Add and subtract fractions with different denominators Apply the idea If are different (called If the the denominators denominators of of two two fractions fractionsAdd arethe different (called unlike unlike denominators), denominators), then then we we are are not not adding adding quantities quantities of of numerators the same the same size. size. Rewrite with a common factor of 2

Add and subtract Interactivefractions explorationwith different denominators Explore the Explore online online to to answer answerEvaluate the questions questions

Students will explore how to add and subtract fractions with different denominators by converting them into equivalent fractionsmathspace.co with a common denominator. Using area models and interactive applets, students will visually manipulate fractions to understand the necessity of having the same denominator for straightforward addition or summary the exploration subtraction. Use Use Idea the interactive interactive exploration in in 1.02 1.02 to to answer answer these these questions. questions. 1. 1.

When two fractions have the same denominator, we can add or subtract them by adding or subtracting the Type and into into the the fraction fraction boxes. boxes. Then, Then, press press enter enter on on your your keyboard. keyboard. Why Why can’t can’t we we Type the the fractions fractions and

numerators, Students: Pages 14–15while keeping the denominator the same. add the fractions the way they are? 2. 2. 3. 3.

add the fractions the way they are? Click Click “Show “Show common common units.” units.” How How did did the the area area models models change? change?

Explain how add now. What Explain how we we can canfractions add the the fractions fractions now.different What is is the the sum? sum? Add and subtract with denominators

4. Does for other with denominators? Does this this work work other fractions fractions with different different denominators? If the 4. denominators of twofor fractions are different (called unlike denominators), then we are not adding quantities of the same size. We We can can use use area area models models to to change change fractions fractions to to equivalent equivalent fractions, fractions, so so the the denominators denominators are are the the same. same. Then, Then, we we can can Interactive exploration add the equivalent fractions. add the equivalent fractions. Consider Consider

Explore online to answer the questions

.. These These two two fractions fractions look look like like this: this:

mathspace.co

If If we we tried tried to to add add the the pieces pieces together, together, how how could could we we write write the the denominator? denominator?

We We cannot cannot place place one one circle circle on on top top of of the the other other because because the the number number of of parts parts Use the interactive exploration in 1.02 to answer these questions. in in a a whole whole are are different. different. 1. Type the fractions and into the fraction boxes. Then, press enter on your keyboard. Why can’t we Before we weadd canthe addfractions these two two fractions, we need to to rewrite rewrite them them with with the the same same denominator. denominator. To To do do this, this, we we create create Before can add these need thefractions, way theywe are? equivalent fractions. equivalent fractions. 2. Click “Show common units.” How did the area models change? The denominators denominators herewe arecan 9 and and 4, sofractions the least leastnow. common multiple of the the denominators denominators is is 4 4 ⋅⋅ 9 9= = 36. 36. This This is is The here are 9 so the common of 3. Explain how add4, the Whatmultiple is the sum? sometimes called called the the least least common common denominator. denominator. sometimes 4. Does this work for other fractions with different denominators? Now, Now, we we can can rewrite rewrite the the fractions fractions with with a a denominator denominator of of 36. 36. We can use area models to change fractions to equivalent fractions, so the denominators are the same. Then, we can add the equivalent fractions. Consider 14 14

. These two fractions look like this:

Mathspace Mathspace Virginia Virginia SOL SOL Grade Grade 6 6 mathspace.co mathspace.co

fractions can now be divided into the of parts. If we triedThe to add the pieces together, how could wesame writenumber the denominator? After shading the correct amount of pieces, the fractions look like We cannot place one circle on top of the other because the number of parts this. in a whole are different.

Before we can add these two fractions, we need to rewrite them with the same denominator. To do this, we create Since the sizes of the parts are the same, we can add the fractions equivalent fractions. together. The denominators here are 9 and 4, so the least common multiple of the denominators is 4 ⋅ 9 = 36. This is sometimes called the least common denominator. Now, we can rewrite the fractions with a denominator of 36. This shows that

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract the numerators. Mathspace Virginia SOL Grade 6 14 mathspace.co

Example 2 Evaluate

Create a strategy Find the least common multiple of the two denominators.

Apply the idea The least common multiple of 4 and 8 is 8. Since the denominator of of 8.

is already 8 we only need to ﬁnd the equivalent fraction to

that has a denominator

Multiply the numerator and denominator by 2 Evaluate Substitute the equivalent fraction

1.02 Add and subtract fractions and mixed numbers mathspace.co

Idea summary When two fractions have the same denominator, we can add or subtract them by adding or subtracting the numerators, while keeping the denominator the same.

Exploration

Add and subtract fractions with different denominators

Students:If Page 14 the denominators of two fractions are different (called unlike denominators), then we are not adding quantities of the same size.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.02 to answer these questions. 1.

Type the fractions

and

into the fraction boxes. Then, press enter on your keyboard. Why can’t we

add the fractions the way they are? 2.

Click “Show common units.” How did the area models change?

Explain how we can add the fractions now. What is the sum?

Does this work for other fractions with different denominators?

We can use area models to change fractions to equivalent fractions, so the denominators are the same. Then, we can add the equivalent fractions.

Suggested student .grouping: In pairslook like this: Consider These two fractions The interactive exploration utilizes a Geogebra applet to demonstrate the process of adding fractions with If we tried to add the pieces together, how could we write the denominator? different denominators using area models. By the end of the exploration, students should have a deeper We cannot place one circle on top of the other because the number of parts understanding of why common denominators are needed for addition and subtraction. in a whole are different.

Ideal student responses

Before we can add these two fractions, we need to rewrite them with the same denominator. To do this, we create

These ideal responses may differ from other correct student responses. Less formal responses can be equivalent fractions. connected with the more precise mathematical language presented here. 1.

The denominators here are 9 and 4, so the least common multiple of the denominators is 4 ⋅ 9 = 36. This is called theand least common denominator. Typesometimes the fractions into the fraction boxes. Then, press enter on your keyboard. Why can’t we we can rewrite fractions with a denominator of 36. add Now, the fractions the the way they are?

We cannot directly add the fractions and because they have different denominators. Since the denominators represent different divisions of a whole, the pieces are not the same size, so we cannot add the pieces directly. 2. Click14“Show common units.” How Mathspace Virginia SOL Grade 6 did the area models change? mathspace.co When “Show common units” is clicked, the area models adjust so that both fractions are represented with the same number of equal parts, so they share a common denominator. 3. Explain how we can add the fractions now. What is the sum? Once the fractions are expressed with a common denominator, you can simply add their numerators (which represents the number of shaded parts) while keeping the denominator the same (since the number of parts in a whole remains the same once combined). For of 15, results in

and

. Adding these gives

and , converting them to have a common denominator

4. Does this work for other fractions with different denominators? Yes, this method works for any set of fractions with different denominators. The key is to convert all fractions involved to equivalent fractions with a common denominator (often the least common multiple of the original denominators). This allows us to easily add or subtract the numerators.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purposeful questions • After entering your fractions into the applet and clicking “Show common units”, how did the number of pieces in each model change? • Why is finding a common denominator essential before adding or subtracting fractions? The fractions can now be divided into the same number of parts. • How does the visual method in the applet compare to the numerical method you use when finding After shading the correct amount of pieces, the fractions look like equivalent fractions? Which method do you prefer and why? this. Possible misunderstandings • Students may confuse the process of findingSince a common the actual or subtraction the sizesdenominator of the parts areand the same, we canaddition add the fractions of fractions. They might not realize that the fractions together. are equivalent fractions or that the new denominators are the least common multiple of the original denominators. This shows that

Examples the 15 denominators are different, we create equivalent fractions with the same denominator, then add or subtract Students:When Page the numerators.

Example 2 Evaluate

Create a strategy Find the least common multiple of the two denominators.

Apply the idea The least common multiple of 4 and 8 is 8. Since the denominator of of 8.

is already 8 we only need to ﬁnd the equivalent fraction to

that has a denominator

Multiply the numerator and denominator by 2 Evaluate Substitute the equivalent fraction Evaluate

Idea summary Purpose When two fractions have different denominators, we ﬁrst rewrite the fractions with the same denominator. Show studentsThen, howwetocan subtract fractions finding the leastthe common multiple of the denominators and add or subtract theby numerators and keep denominator the same. converting the fractions to equivalent fractions with the same denominator. Expected mistakes Students might assume that the answer is supposed to be positive rather than negative. Point out that the second term in the expression is almost 2 wholes, so it has a bigger value than the first term. Any time we subtract a large positive value from a smaller positive value, the answer will be negative.

1.02 Add and subtract fractions and mixed numbers mathspace.co

This shows that

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract

Subtract fractions with different denominators using number lines the numerators.

use with Example 2

Student with disabilities support

2 conceptual processing difficulties, use number lines to illustrate the subtraction of To help Example students with fractions with unlike denominators. Begin by drawing two number lines: one divided into fourths and the other Evaluate

into eighths. Show students how to represent

on the first number line and how to identify an equivalent

fraction,Create on the second. a strategy Find the least common multiple of the two denominators.

−1

−2

Apply the idea

The least common multiple of 4 and 8 is 8.

−2of Since the denominator of 8.

0 −1 need to ﬁnd the equivalent is already 8 we only fraction to

1 a denominator that has

Next, encourage students to use theMultiply eighths number line to perform the subtraction step by step. Students the numerator and denominator by 2

should subtract, or move left, 15 steps from . By engaging with the number lines, students can better grasp the Evaluate abstract concepts involved in fraction operations. Substitute the equivalent fraction

Students: Page 15

Evaluate

Idea summary When two fractions have different denominators, we ﬁrst rewrite the fractions with the same denominator. Then, we can add or subtract the numerators and keep the denominator the same.

Add and subtract mixed numbers Students will learn that there are two methods for adding and subtracting mixed numbers. The first method involves separately adding or subtracting the whole numbers and the fractional parts, requiring a common denominator for the fractions when necessary. The second method converts mixed numbers into improper fractions, simplifying the arithmetic process, though it also may require finding a common denominator for fractions different parts.15 1.02 Add and subtract andfractional mixed numbers mathspace.co

Students: Page 16

Add and subtract mixed numbers Mixed numbers have a whole number part and a fractional part. There are two methods for adding or subtracting mixed numbers: 1. Add or subtract the whole parts, then add or subtract the fractional parts. • If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator before adding or subtracting. 2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions. • If the denominators are different, we need to rewrite them with the same denominator before adding or subtracting.

Example 3 Fill in the boxes to show the work for:

a Rewrite as improper fractions.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Create a strategy Multiply the whole number by the denominator, then add the numerator.

Mixed numbers have a whole number part and a fractional part. There are two methods for adding or subtracting mixed numbers: 1. Add or subtract the whole parts, then add or subtract the fractional parts. • If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator before adding or subtracting.

Examples 2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions. Add and subtract mixed numbers

• If the16 denominators are different, we need to rewrite them with the same denominator before adding or Students:Mixed Page numbers have a whole number part and a fractional part. There are two methods for adding or subtracting subtracting. mixed numbers:

1. Add or subtract the whole parts, then add or subtract the fractional parts.

Example 3

• If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator

before adding or subtracting. Fill in the boxes to show the work for: 2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions. • If the denominators are different, we need to rewrite them with the same denominator before adding or subtracting. a Rewrite as improper fractions.

Example 3 Create strategy Fill in theaboxes to show the work for: Multiply the whole number by the denominator, then add the numerator.

Apply the idea

a Rewrite as improper fractions. Multiply the whole number by the denominator Evaluate the multiplication

Create a strategy

Evaluate the addition Multiply the whole number by the denominator, then add the numerator.

Apply the idea

b Rewrite with a common denominator. Multiply the whole number by the denominator

Purpose Evaluate the multiplication Students demostrate that they can convert mixed numbers to improper fractions. Create a strategy

Evaluate the addition

Students:Find Page 16 common multiple (LCM) of the denominators, and multiply each numerator and denominator by the the least quotient obtained by dividing the LCM by their respective denominators. b Rewrite with a common denominator.

Apply the idea

The LCM of 3 and 4 is 12. So, we need to multiply both parts of

by 12 ÷ 3 = 4, and multiply both parts of

by 12 ÷ 4 = 3.

Create a strategy

Multiply by and by Find the least common multiple (LCM) of the denominators, and multiply each numerator and denominator by the quotient obtained by dividing the LCM by their respective denominators. Evaluate

Apply the idea The LCM of 3 and 4 is 12. So, weMathspace need to multiply both parts of Virginia SOL Grade 6 16 mathspace.co

by 12 ÷ 3 = 4, and multiply both parts of Multiply

and

by 12 ÷ 4 = 3.

Evaluate

Mathspace

Virginia SOL Grade 6

Purpose mathspace.co Students demonstrate that they can rewrite fractions with different denominators by finding a common denominator.

1.02 Add and subtract fractions and mixed numbers mathspace.co

Expected mistakes Students might try to multiply both fractions by the same number, or they may multiply a number by the denominators but not the numerators. Instruct students to organize their work by creating each equivalent fraction on the side, rather than trying to write it on the same line of work.

Students: Page 17 c Evaluate the difference.

Create a strategy Evaluate the answer from part (b).

Apply the idea Subtract the numerators

Example 4 Purposec Evaluate the difference. Show students how fair, to subtract fractions withofthe subtracting For the school Aimee is making a batch icedsame tea bydenominator mixing liters by of water with ofthe a numerators. liter of tea concentrate. What is the total volume of the iced tea mixture in liters?

Reflecting with students Createtoa classify strategy Ask students

as a proper fraction, improper fraction, or mixed number. Then, encourage them to

Evaluate the answer from part (b).

rewrite their answer as a mixed number instead. Inform students that

and

are both correct answers, so

the to idea they canApply choose write the answer in either form.

Students:Create Pagea 17 strategy

Subtract the numerators

Add the volume of water to the volume of tea concentrate.

Example 4 Apply the idea Write the water and tea concentrate volumes For the school fair, Aimee is making a batch of iced tea by mixing liters of water with

of a

Convert number to an in improper liter of tea concentrate. What is the total volumethe of mixed the iced tea mixture liters? fraction Find a common denominator Add the fractions Convert back to a mixed number The totalavolume of the iced tea mixture is Create strategy

liters.

Add the volume of water to the volume of tea concentrate.

Apply the idea

Idea summary

Write the water and tea concentrate volumes To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with the same denominator, then add or subtract them. Convert the mixed number to an improper fraction Another method of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same denominator, then add or subtract the whole parts and add or subtract the fractional parts. Find a common denominator Add the fractions Convert back to a mixed number The total volume of the iced tea mixture is

liters.

1.02 Add and subtract fractions and mixed numbers mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition Idea summary mathspace.co To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with the same denominator, then add or subtract them. Another method of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same

Create a strategy Add the volume of water to the volume of tea concentrate.

Apply the idea Write the water and tea concentrate volumes Convert the mixed number to an improper fraction Find a common denominator Add the fractions Convert back to a mixed number The total volume of the iced tea mixture is

liters.

Idea summary

Purpose To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with Show studentsthehow todenominator, add a mixed number and a them. fraction by converting the mixed number to an improper same then add or subtract fraction and finding common denominator. Anotheramethod of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same denominator, then add or subtract the whole parts and add or subtract the fractional parts.

Reflecting with students Encourage advanced learners, or any student that is ready, to add the mixed number and fraction directly without converting to an improper fraction. Students can add the fractional parts by finding a common denominator. This approach allows students to practice combining mixed numbers more efficiently and helps them develop a deeper number sense. 1.02 Add and subtract fractions and mixed numbers 17 mathspace.co

Three reads

use with Example 4

English language learner support Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students, “What do you think is happening in this question?” or “Can you explain what this question is about?” On the second read, students should aim to interpret the problem by answering questions like, “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: • Aimee has

liters of water.

• Aimee has

liters of tea.

• Aimee is adding the tea and water together. Students can be prompted by framing these as questions like “How much of each liquid does Aimee have?” or “What does Aimee need to do with the two different liquids?”

1.02 Add and subtract fractions and mixed numbers mathspace.co

Find a common denominator Add the fractions Convert back to a mixed number total volume Students:The Page 17 of the iced tea mixture is

liters.

Idea summary To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with the same denominator, then add or subtract them. Another method of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same denominator, then add or subtract the whole parts and add or subtract the fractional parts.

1.02 Add and subtract fractions and mixed numbers mathspace.co

Practice Students: Pages 18–20

What do you remember? 1

Find the least common denominator for each set of fractions: b

a 2

The image shows

of the rectangle shaded.

What is the equivalent fraction in tenths?

We now want to take away What is the answer to

. ?

Write your answer in simplest form. 3

The image shows

in blue and

Write the equation that describes the image if the blue and green tiles would be added.

What is the total shaded? Express your answer in simplest form.

Complete each statement: a

in green.

Rewrite with a common denominator

Evaluate

Rewrite as a mixed number

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Rewrite with a common denominator

Evaluate

Rewrite as a mixed number

Fill in the boxes to finish showing the work:

Group the whole number parts and fraction parts

Add the whole number parts and fraction parts

Evaluate

For

Rewrite as improper fractions.

Using part (a), rewrite the improper fractions with the same denominator.

Simplify the expression.

Let’s practice 7

Calculate the following and express in simplest form: a

Jamie solved the following subtraction problem incorrectly: a

Identify the error in Jamie’s calculation.

Correct the error.

Explain a strategy that could help Jamie get the correct answer.

Which value is closest to the sum of A

and ?

Calculate the following and express in simplest form: a

1.02 Add and subtract fractions and mixed numbers mathspace.co

Find the value and express in simplest form: a

Emma adds the mixed numbers understand the process.

Which value is closest to the difference between A

. Show the correct ﬁnal answer and explain the steps to help Emma

and

What is the closest value to the difference between

and

At a party, Bill makes fruit punch by combining

? C

A 16

and

liters of water with

liters of juice

concentrate. Find the total amount of liters in the punch.

During a blizzard, it snowed 10 snow was left on the ground?

Katrina and Luigi ordered a pizza to share. Katrina ate

cm. When the sun came out the next day,

of the pizza while Luigi ate .

Use a model to estimate what fraction of the pizza Katrina and Luigi ate altogether.

What fraction of the pizza did they actually eat altogether?

What fraction of pizza is left?

Justify your solutions using words.

Alex threw a football

yards. Morgan threw the football

Which statement is true?

cm of snow melted. How much

Alex threw the football

Morgan threw the football

Alex threw the football

Morgan threw the football

yards farther than Morgan. yards farther than Alex. yards farther than Morgan. yards farther than Alex.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

yards.

Let’s extend our thinking 20

Skye wants to decorate her window sill with fairy lights. The window sill is lights, how much of the window sill will not be decorated? a

Estimate your solution with a model.

Determine how much of the window sill won’t be decorated.

ft wide. If she only has

A rectangular cake measures 9 cm by 8 cm. You cut a piece that measures 4 cm by 3 cm. What fraction of the cake is left after cutting this piece? Explain your reasoning using a diagram.

Danielle is studying for three tests. She studies Science for for

For

hours, English for

ft of fairy

of an hour and Visual Arts

of an hour. How many hours does she spend studying? Justify your solution using a model. :

Calculate the answer by ﬁrst converting both mixed numbers to improper fractions.

Calculate the answer by ﬁrst subtracting the whole numbers and then subtracting the fraction parts.

Which method do you prefer? Explain.

1.02 Add and subtract fractions and mixed numbers mathspace.co

Answers

10 2

1.02 Add and subtract fractions and mixed numbers

11 a

12 a

What do you remember? 1 a 6

b 20

c 12

2 a

13 The correct ﬁnal answer is

3 a

Emma added the numbers correctly, but she forgot to simplify the ﬁnal result. She could either simplify the improper fraction

4 a

Rewrite with a common denominator

Evaluate the sum

Rewrite as a mixed number

14 3

Evaluate the difference

Rewrite as a mixed number

Group the whole number parts and fraction parts

Add the whole number parts and

or simplify the mixed number

. In

either case, the numerator and denominator have a common factor of 3.

Rewrite with a common denominator

liters

17 18 a

fraction parts

Evaluate the sum

Luigi ate

6 a

Katrina ate

b Example answer:

c or d Katrina ate , and Luigi ate . To ﬁnd the total fraction Let’s practice

they ate together, we add these fractions.

7 a

We then subtract the sum from 1 to determine the fraction of pizza that remains.

19 B

8 a

Let’s extend our thinking

20 a Available Undecorated

9 a J amie did not correctly ﬁnd a common denominator before subtracting the fractions. b

c A strategy for Jamie is to ﬁrst ﬁnd a common denominator, which in this case is 66, and then perform the subtraction using the equivalent fractions.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

fairy lights

part

about 1.3 0

1 2 3 window sill width

21 Answers may look like: To determine the fraction of the cake that is left after cutting a piece, we need to compare the area of the piece that was cut to the area of the original cake.

hours

The original cake measures 9 cm by 8 cm, so its area is:

9 cm × 8 cm = 72 cm

The piece that was cut measures 4 cm by 3 cm, so its area is:

23 a

4 cm × 3 cm = 12 cm2 12 cm2 is

of the cake. 9 cm

c Answers may vary. Example answer: 4 cm

8 cm

I prefer the ﬁrst method because there are no negative fractions.

3 cm

After cutting this piece, what was left of the cake is:

Rewrite with a common denminator

Evaluate

of the cake will be left.

Answers mathspace.co

1.03 Multiply fractions and mixed numbers Subtopic overview Lesson narrative In this lesson, students will learn to multiply fractions and mixed numbers. They start with multiplying fractions by whole numbers and progress to multiplying fractions by fractions. Students will explore these concepts using area models and number lines. They will solve various types of problems, including finding fractions of quantities, multiplying improper fractions, and solving real-world scenarios involving time and measurements. Interactive applets allow students to model multiplication of whole numbers by fractions and fractions by fractions to observe patterns. By the end of the lesson, students should be proficient in multiplying fractions and mixed numbers.

Learning objectives Students: Page 21

Key vocabulary 

unit fraction

Essential understanding Multiplying fractions and mixed numbers can be efficiently done by converting them to improper fractions and applying the multiplication operation.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can incorporate this goal by presenting contextual problems involving the multiplication of fractions and mixed numbers. For example, they can present a problem that involves dividing resources or scaling a recipe. Students will need to apply their understanding of fraction multiplication to solve these problems. Teachers should model effective practices, such as converting mixed numbers to improper fractions and validating solutions. They can also encourage students to create their own problems based on realworld situations and find solutions. By guiding students to understand, plan, execute, and review their problemsolving processes, teachers can develop students’ abilities to estimate, justify, and arrive at reasonable solutions in real-world contexts.

Teachers can foster mathematical reasoning by guiding students through contextual problems involving fractions and mixed numbers. This lesson helps students make and test predictions, justify their problem-solving steps, and evaluate their conclusions. Teachers should encourage students to think about the operations involved, such as how the meaning of multiplication applies to the problem. By explaining their reasoning and discussing their thought processes, students gain a deeper understanding of operations and improve their strategic problem-solving skills, moving beyond reliance on key words. MPG4 — Mathematical Connections

Teachers can help students develop mathematical connections by guiding them to use their fluency MPG2 — Mathematical Communication with fractions and mixed numbers to solve contextual problems. This lesson on multiplying fractions and Teachers can help students recall multiple problem mixed numbers requires students to draw on their prior types learned in earlier grades and applying them to knowledge of computation with fractions to find exact more complex problems. They can introduce solution solutions. Teachers can support students in identifying methods with worked-out examples, explaining each step and the decisions made during the problem-solving essential vocabulary, determining the appropriate process. By demonstrating how to solve similar problems, operations, and planning their solutions. Encouraging students to explain their thought processes and justify discussing each decision, and engaging students with guiding questions, teachers can effectively connect prior their reasoning helps them check for reasonableness and knowledge to new, contextual problem-solving methods. strengthens their problem-solving skills. MPG5 — Mathematical Representations Teachers can integrate this goal by using various models and manipulatives to represent the multiplication of fractions and mixed numbers. For example, teachers can use fraction bars/strips, number lines, area models, and manipulatives like fraction tiles or pattern blocks to visually represent these operations.Teachers can encourage students to use these models and manipulatives to help them solve problems and understand concepts. Teachers can also encourage students to draw their own representations or use physical objects to represent the multiplication process in real-world problems.

Content standards 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1b — Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.*

6.CE.1a — Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.*

6.CE.1c — Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.*

1.03 Multiply fractions and mixed numbers mathspace.co

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.01 Review: Simplify fractions and mixed numbers

Tools You may ﬁnd these tools helpful: • Grid paper • Colored pencils

Student lesson & teacher guide Multiply fractions and whole numbers Students review the concept of fractions as parts of a quantity and relate that to multiplying a fraction by a whole number, both visually and through arithmetic. The concept is reinforced through an interactive exploration activity.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 21

1.03 Multiply fractions and mixed numbers mathspace.co

Exploration Students: Page 22

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.03 to answer these questions. 1.

What is happening when you slide the slider? How does this represent the multiplication?

What does checking the ‘Arrange’ checkbox do? How might this help with understanding the result?

Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?

What can you say about multiplying a whole number by a fraction between 0 and 1?

When a whole number is multiplied by a fraction between 0 and 1, the result is smaller than the original whole number.

Example 1 Suggested pairs Eachstudent rectangle grouping: represents 1 In whole. In this exploration, students will use a GeoGebra applet to understand the concept of multiplication involving a whole number and a fraction. Through this exploration, they will observe and analyze the process of multiplication and outcome, thus gaining a better a Write the its product that is represented by the model.understanding of the concept. Ideal student Createresponses a strategy

Apply the idea

These ideal differ from other correct student responses. Lessare formal responses be Look responses at the shadingmay in each rectangle to ﬁgure out what In each rectangle, there 6 equal parts and 4can of them fraction shaded. Multiply the shaded fraction oflanguage a are shaded. We could say “4 out of 6 parts of each connected withis the more precise mathematical presented here. rectangle by the total number of large rectangles to ﬁnd

rectangle are shaded, of each rectangle is shaded. This 1. Whattheisproduct happening when you slide the slider? How does this represent the multiplication? represented by the model. simpliﬁes to . There are 3 rectangles, so the product When sliding the slider, the two fraction models are combined to show the product. One fraction is shaded in represented theshading model is: overlaps. columns and the other is shaded in rows, and the result is wherebythe 3 ⋅ or 3⋅ 2. What does checking the ‘Arrange’ checkbox do? How might this help with understanding the result? Checking the ‘Arrange’ checkbox groups the fractional parts together. This provides a different visual representation of the fractional b Evaluate the product using theanswer, model. making it easier to determine how many parts of one whole are shaded.

the idea 3. Use Apply the applet with a few more multiplication problems. What patterns do you notice between the Rewrite the whole number as a fraction numbers being multiplied and the result? The number of parts in the resulting fraction is always the same as the number of parts in the original Multiply the numerators and denominators fraction. The number of pieces shaded is the product of the whole number and the numerator of the original Divide fraction. Another way to evaluate this is we divide out the common factors in the numerator and denominator:

4. What can you say about multiplying a whole number by a fraction between 0 and 1? thewhole whole number as This a fraction The result is always less than theRewrite original number. is because multiplying by a fraction is the same as finding a part of each whole, which results is a smaller value. Divide out the common factors

Purposeful questions • When you multiply a fraction by aSimplify whole number, how does the product compare to the original fraction? productwhy represented by the model is 2. Can So, youthe explain this is the case? • Can you think of a real-life scenerio that would require you to multiply a fraction by a whole number? Possible22misunderstandings Mathspace Virginia SOL Grade 6 mathspace.co

• Students may assume that because the quantities are being multiplied, the product will be a larger number. Use the visual models in the applet to reinforce the concept of multiplying fractions by a whole number.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

1. 2. 3.

What is happening when you slide the slider? How does this represent the multiplication? What does checking the ‘Arrange’ checkbox do? How might this help with understanding the result?

Interactive exploration Use the applet with a few more multiplication problems. What patterns do you notice between the Explore answer the numbersonline beingtomultiplied andquestions the result?

What can you say about multiplying a whole number by a fraction between 0 and 1? Examples mathspace.co 4.

Students: Page 22

Useathe interactive to answer these0questions. When whole numberexploration is multipliedinby1.03 a fraction between and 1, the result is smaller than the original whole number. 1.

What is happening when you slide the slider? How does this represent the multiplication?

2. What Example 1 does checking the ‘Arrange’ checkbox do? How might this help with understanding the result? 3. Use the applet with a few more multiplication problems. What patterns do you notice between the Each rectangle represents 1 whole. and the result? numbers being multiplied 4.

What can you say about multiplying a whole number by a fraction between 0 and 1?

a Write the product that is represented by the model. When a whole number is multiplied by a fraction between 0 and 1, the result is smaller than the original whole number.

Create a strategy

Apply the idea

Example 1 Look at the shading in each rectangle to ﬁgure out what

In each rectangle, there are 6 equal parts and 4 of them are shaded. We could say “4 out of 6 parts of each

fraction is shaded. Multiply the shaded fraction of a Each rectangle 1 whole. rectangle by therepresents total number of large rectangles to ﬁnd the product represented by the model. a Write the product that is represented by the model.

Create a strategy

rectangle are shaded,

of each rectangle is shaded. This

simpliﬁes to . There are 3 rectangles, so the product represented by the model is: 3⋅

or 3 ⋅

Apply the idea

Look at the shading in each rectangle to figure out what In each rectangle, there are 6 equal parts and 4 of them b Evaluate the product using the model. fraction is shaded. Multiply the shaded fraction of a are shaded. We could say “4 out of 6 parts of each Purposerectangle by the total number of large rectangles to find rectangle are shaded, of each rectangle is shaded. This Apply the idea product represented the model. Studentstheshow how to use a by visual model to represent multiplication a whole number with a product fraction and how simplifies to . of There are 3 rectangles, so the Rewrite the whole number as a fraction to calculate the product from the model. represented by the model is: Multiply the numerators and denominators 3 ⋅ or 3 ⋅ Students: Pages 22–23 Divide Another way to evaluate this is we divide out the common factors in the numerator and denominator: b Evaluate the product using the model. Rewrite the whole number as a fraction

Apply the idea Divide out the common factors Rewrite the whole number as a fraction Simplify Multiply the numerators and denominators So, the product represented by the model is 2. Divide Another way to evaluate this is we divide out the common factors in the numerator and denominator: 22

Mathspace Virginia SOL Grade 6Rewrite the whole number as a fraction mathspace.co

Divide out the common factors Simplify So, the product represented by the model is 2.

Reﬂect and check 22 Mathspace Virginia SOL Grade 6by rearranging the shaded boxes into smaller rectangles and comparing them with We can further visualize the result mathspace.co two whole rectangles that have 6 parts each.

In the image, we can see that the 12 shaded boxes from the original three rectangles can be rearranged to ﬁll two whole rectangles with 6 parts each. There is one rectangle without any shaded parts. This shows that the shading in the model represents the product 2, as the shaded boxes are equivalent to two whole rectangles.

Example 2 Evaluate

⋅ 35

Create a strategy

1.03 Multiply fractions and mixed numbers mathspace.co

Purpose Students use visual models to evaluate and simplify the product of a whole number and a fraction. Expected mistakes Students might mistakenly multiply the whole number 3 by both the numerator and denominator of the fraction and arrive at the answer of . To address this, remind students that 3 is equivalent to , which should be used for the multiplication.

Connecting visual models to repeated addition and multiplication

use with Example 1

Student with disabilities support If students struggle to write a product, encourage them to describe the shading in the model as an addition statement instead. They should recognize that each rectangle has shaded, and there are three rectangles. Reﬂect and check

and rectangles verbal representation helpswith students Guide them to further express the total shaded area as the shaded. This We can visualize the result by rearranging boxesvisual into smaller and comparing them rectangles that have 6 parts each. see howtwo thewhole model represents adding the same fraction multiple times.

Once they are comfortable with the addition statement, remind them that multiplication is a shortcut for repeated addition. Show how

is another way to represent the same total shaded area. This approach

reinforces their understanding by linking the visual model to the concepts of addition and multiplication, making the abstract idea of multiplying fractions more concrete. In the image, we can see that the 12 shaded boxes from the original three rectangles can be rearranged to ﬁll two whole rectangles with 6 parts each. There is one rectangle without any shaded parts. This shows that the shading in model23 represents the product 2, as the shaded boxes are equivalent to two whole rectangles. Students:thePage

Example 2 Evaluate

⋅ 35

Create a strategy Multiply numerators and denominators separately. The denominator of a whole number is always 1.

Apply the idea Multiply numerators and denominators Evaluate Simplify

Idea summary

Purpose Finding a fraction of a quantity is the same as multiplying a fraction by a whole number. Students demonstrate that they can multiply a fraction by a whole number. To multiply a fraction by a whole number, multiply the numerator by the whole number.

When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole Reflecting with students number. Ask the students if there is another way they can evaluate the given problem above. Lead them to consider dividing 35 by 5 first before multiplying by 2.

Multiply fractions by fractions To multiply two fractions together, we’ll start by thinking of the fractions as multiples of unit fractions, and work towards a more eﬃcient strategy. Let’s take an example of

. We can rewrite these fractions as and

We can then multiply the whole parts together:

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

1.03 Multiply fractions and mixed numbers mathspace.co

the model represents the product 2, as the shaded boxes are equivalent to two whole rectangles.

Apply the idea

Multiply numerators and denominators

Example 2

Students:Evaluate Page 23 ⋅ 35

Evaluate Simplify

Create a strategy MultiplyIdea numerators and denominators separately. The denominator of a whole number is always 1. summary Finding a fraction of a quantity is the same as multiplying a fraction by a whole number.

Apply the idea

To multiply a fraction by a whole number, multiply the numerator by the whole number. Multiply numerators and denominators When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole number. Evaluate Simplify

Multiply fractions by fractions To multiply two fractions together, we’ll start by thinking of the fractions as multiples of unit fractions, and work Multiply fractions by fractions summary towardsIdea a more eﬃcient strategy. Finding a fraction of a quantity is the same as multiplying a fraction by a whole number. Students are introduced to the of multiplying . We can rewrite these fractions fractions asby understanding them as multiples of unit fractions. Let’s take an example of process multiply a fraction by a whole number, multiply the numerator by the whole number. They will learn toTomultiply the numerators and denominators and also apply this concept to mixed numbers. and When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole number. can then23–24 multiply the whole parts together: Students:WePages

mathspace.co

. We can rewrite these fractions as and

We can then multiply the whole parts together:

What can we do with the product of the unit fractions

and ?

1.03 Multiply fractions and mixed numbers mathspace.co

Well, this is like taking one whole, dividing it into 3 pieces to get thirds.

Then dividing each of those thirds into 5 pieces. The result is that the whole has been divided into 15 pieces where we only want 1 piece. This image represents the fraction

We can now ﬁnish our multiplication:

Do you notice the pattern that has happened here? In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are. That is:

1.03 Multiply fractions and mixed numbers mathspace.co

The result is that the whole has been divided into 15 pieces where we only want 1 piece. This image represents the fraction

We can now ﬁnish our multiplication:

We can multiply with mixed numbers as well, because they’re really just fractions. We just have the added step of converting the mixed number to a fraction ﬁrst.

Interactive exploration Explore online to answer the questions

Concrete-Representational-Abstract (CRA) approach mathspace.co Targeted instructional strategies

the by interactive exploration in 1.03 to answer these questions. Concrete: Use Begin engaging students with physical manipulatives using grids and overlays to explore 1. What by is happening slide theby slider? How does this represent the multiplication? multiplying fractions fractionswhen and you fractions mixed numbers. Provide students with grid paper or 2. Use the applet with amultiplying few more multiplication problems.by What patterns do you notice between the or mixed transparent grid overlays. When a proper fraction another fraction (proper, improper, numbers being multiplied and result? or mixed number using vertical columns on the grid. For number), have students first represent thethe improper What can you say about a fraction by a fraction between 0 andequal 1? example, to 3. represent , they shademultiplying one whole rectangle divided into four columns and three out of

four equal columns of the next rectangle. Next, represent the proper fraction by dividing the grid into horizontal When a fraction is multiplied by a fraction between 0 and 1, the result is smaller than the original fraction. rows and shading the appropriate number of rows. If the proper fraction is , divide the grid into three equal

horizontal rows and shade two of them. The overlapping area where the shaded columns and shaded rows Mathspace Virginia SOL 6 24 meet represents the product ofGrade the two fractions. This hands-on activity allows students to physically see how mathspace.co the fractions combine, making the concept of multiplying fractions more concrete. Representational: Transition to visual representations by having students draw models of the multiplication problems. Teach them how to create area models on paper, dividing rectangles into equal parts to represent the fractions. They can draw a rectangle divided into vertical columns for the improper or mixed number and horizontal rows for the proper fraction. By shading the appropriate columns and rows, the overlapping shaded area represents the product. Encourage students to label their diagrams with the corresponding fractions and write brief explanations of each step. Use graph paper to help with precise drawings. Incorporate interactive digital tools or apps that allow students to manipulate virtual grids and see how changing the fractions affects the overlapping area. Abstract: Move to the abstract stage by focusing on numerical calculations without visual aids. Teach students the standard algorithm for multiplying fractions: multiply the numerators to find the new numerator and the denominators to find the new denominator. Show them how to convert mixed numbers into improper fractions before multiplying. Provide practice problems that include multiplying fractions by fractions and fractions by mixed numbers. Discuss the effect of multiplying by a fraction between 0 and 1, highlighting that the product will be smaller than at least one of the original numbers. Encourage students to solve problems and explain their reasoning using proper mathematical notation and vocabulary.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are. That is:

Exploration Students:WePage 24 with mixed numbers as well, because they’re really just fractions. We just have the added step of can multiply converting the mixed number to a fraction ﬁrst.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.03 to answer these questions. 1.

What is happening when you slide the slider? How does this represent the multiplication?

Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?

What can you say about multiplying a fraction by a fraction between 0 and 1?

When a fraction is multiplied by a fraction between 0 and 1, the result is smaller than the original fraction.

Mathspace Virginia SOL Grade 6 mathspace.co

Suggested student grouping: In pairs Students will use a GeoGebra applet to explore the concept of multiplying fractions. They will choose different fractions to multiply and then use a slider to combine the models. This exploration will not only help them visualize the process of multiplication but also discover underlying patterns between the multiplicands and the product. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What is happening when you slide the slider? How does this represent the multiplication? When sliding the slider, the two fraction models are combined to show the product. One fraction is shaded in columns and the other is shaded in rows, and the result is where the shading overlaps. 2. Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result? The numerator of the result is the product of the original numerators, and the denominator of the result is the product of the original denominators. 3. What can you say about multiplying a fraction by a fraction between 0 and 1? When the second fractions is less than 1, the product is always smaller than the first original fraction. This is because when we multiply fractions, we’re finding a part of a part, which results in a smaller number. Purposeful questions • How are the original numerators related to the numerator of the final result? Is the same true for the denominators? • When you multiply by a fraction less than 1, is the product smaller or larger than the original fraction (the fraction on the left)? Can you explain why this is the case? • Can you think of a real-life scenario that would require you to multiply two fractions? Possible misunderstandings • Students may think that they should add the numerators and the denominators instead of multiplying them. Encourage students to record the numerators and denominators of both the factors and the products, organizing the data in a table. Guide them to analyze this data to recognize that multiplying the numerators together and the denominators together consistently yields the product.

1.03 Multiply fractions and mixed numbers mathspace.co

Examples The following supports may be useful for the examples in this section.

Stronger and clearer each time English language learner support After working through Example 2, Example 3 and Example 4 part (a), ask students to individually write an explanation of what happens when multiplying a fraction, whole number, or mixed number by a fraction between 0 and 1. Encourage them to use mathematical vocabulary such as “product,” “fraction,” “less than,” and “original number.” Then, have students pair up to share their explanations, listening carefully and asking clarifying questions. After the discussion, prompt students to revise their explanations, incorporating new ideas or language from their partner. Repeat this process with a new partner for further refinement. As they work, encourage students to use phrases like “multiplying by a fraction between 0 and 1 reduces the original number” or “the product is less than the original number.” Circulate to support their use of precise language and to address misconceptions. Finally, have students write a final version of their explanation, which should now be stronger and clearer, demonstrating enhanced understanding and vocabulary.

Students: Page 25 Example 3 Demonstrate how to multiply

and

using a number line. Then ﬁnd the product.

Create a strategy To multiply

and

Apply the idea using a number line, start by

drawing a number line from 0 to each part between 0 and

Draw a number line from 0 to

. Then, divide

into 3 equal parts.

Finally, shade 2 of the 3 parts in each of

Divide each part between 0 and

into 3 equal parts.

the sections. 0

Shade 2 of the 3 parts in each of the sections. 0

Since 10 pieces are shaded and 12 pieces make up one whole,

Example 4 Purpose Find the value following: Show students how of tothe use a number line to visually represent and solve multiplication problems involving fractions. a Create a strategy

Apply the idea

Multiply the numerators and denominators together.

Multiply the numerators and denominators Evaluate

Reﬂect and check 50

Mathspace Virginia SOLmodel Gradeto6verify Teacher We can use an area our Edition answer. mathspace.co For an area model, we can think of the product as the area of a rectangle with a width of

and a length of

The ﬁrst row of the rectangle represents

of its total area and is shaded blue. We

Shade 2 of the 3 parts in each of the sections. 0

Since 10 pieces are shaded and 12 pieces make up one .

whole,

Students: Page 25

Example 4 Find the value of the following: a

Create a strategy

Apply the idea

Multiply the numerators and denominators together.

Multiply the numerators and denominators Evaluate

Reﬂect and check We can use an area model to verify our answer. For an area model, we can think of the product as the area of a rectangle with a width of

and a length of

The ﬁrst row of the rectangle represents

of its total area and is shaded blue. We

further divide each of these thirds into 10 equal pieces, resulting in a total of 30 small rectangles. We then shade 7 columns, each containing three small rectangles, in red. The overlapping purple-shaded area represents the product of

and

There are 7 purple small rectangles out of 30 small rectangles, which shows that the product of

Purpose Students demonstrate that they can multiply two proper fractions.

and

1.03 Multiply fractions and mixed numbers mathspace.co

Students: Page 26

Create a strategy Multiply numerators and denominators separately.

Apply the idea Multiply numerators and denominators Evaluate Simplify

Purpose the idea that they can multiply two proper fractions. StudentsApply demonstrate Rewrite mixed numbers as improper fractions

Reflecting with students Ask students if they can see a way to simplify the given fractions before multiplying, and to verify that Simplify multiplying the simplified fractions would give the same result. Simplify Evaluate

Reﬂect and check

1.03 Multiply fractions and mixed numbers mathspace.co

We can use estimation as a quick way to check if our answer is reasonable. Start by rounding the mixed numbers to numbers that are more familiar and easier to work with.

Multiply numerators and denominators Evaluate Simplify

Students: Page 26

Apply the idea Rewrite mixed numbers as improper fractions Simplify Simplify Evaluate

Reﬂect and check We can use estimation as a quick way to check if our answer is reasonable. Start by rounding the mixed numbers to numbers that are more familiar and easier to work with. is close to

because

Next, we can approximate

is close to

as 5 because it is just

For our estimation we are calculating:

less than 5.

We can easily ﬁnd that 5 ⋅ 5 = 25 and

so 5

⋅ 5 = 25 + 2.5 = 27.5

This is close to our actual answer of 26 so we know our answer is reasonable. However, our estimation is a little too high. This is because we rounded

up to 5.

Purpose Students demonstrate that they can multiply two mixed numbers. Reflecting with students Challenge advanced learners to multiply the mixed numbers without converting them to improper fractions first. If students are stuck, let them know that they would need to multiply 5 by Mathspace Virginia SOL Grade 6 26 product. to find the mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

and

, then add the results

Students: Page 27 Example 5 Danielle takes traﬃc lights.

minutes to drive from her home to the local shopping center. She spends

of this time waiting at

Find the number of minutes she spends waiting.

Create a strategy Multiply the given numbers.

Apply the idea

Example 5 Danielle takes traﬃc lights.

Multiply

Rewrite

as improper fraction

Evaluate minutes to drive from her home to the local shopping center. She spends

of this time waiting at

Find the number of minutes she spends waiting.

Idea summary

Purpose To multiply two fractions, multiply the numerators and the denominators separately. a strategy they can perform multiplication in a real-world context. StudentsCreate demonstrate To multiply that mixed numbers by a fraction or by another mixed number, convert the mixed number to an Multiply the given numbers. improper fraction ﬁrst. Then, multiply the numerators and denominators separately.

Expected mistakes When multiplying any number by a fraction between 0 and 1 the result is smaller than the original number. the idea StudentsApply might not convert the mixed number to an improper fraction before multiplying. Inform them that, if Multiply by they do not convert the mixed number, they would need to multiply 4 by find the product.

Practice

Rewrite

Students: Page 27

Evaluate

and

by , then add the results to

as improper fraction

What do you remember? SOL

Each circle represents 1 whole.

Ideaproduct summary Which is best represented by the shading shown on this model? To multiply two fractions, multiply the numerators and the denominators separately. To multiply mixed numbers by a fraction or by another mixed number, convert the mixed number to an improper fraction ﬁrst. Then, multiply the numerators and denominators separately. When multiplying any number by a fraction between 0 and 1 the result is smaller than the original number.

Practice A

Find the value of:

Each circle represents 1 whole. a b

What do you remember? SOL

Which product is best represented by the shading shown on this model?

1.03 Multiply fractions and mixed numbers mathspace.co

A 2

Find the value of: a

1.03 Multiply fractions and mixed numbers d mathspace.co

Practice Students: Pages 27–30

What do you remember? 1

Each circle represents 1 whole. Which product is best represented by the shading shown on this model?

A 2

Find the value of: a

What is a reciprocal? A

A fraction where the numerator and denominator are the same.

A fraction that represents the square root of the original fraction.

A fraction obtained by interchanging the numerator and the denominator of the original fraction.

A fraction that represents the sum of the numerator and denominator of the original fraction.

Complete the statement: We can multiply two fractions by multiplying the ﬁrst numerator by the second ⬚ and the ﬁrst ⬚ by the second denominator.

Which expression is best represented by this model? A

The rectangle of blocks represents what happens when by . Use the blocks to evaluate

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

is multiplied

The rectangle of blocks represents what happens when Use the blocks to evaluate

is multiplied by .

Let’s practice 8

For each value, state whether the product is greater than, less than, or equivalent to the whole number. b

The area model shows . Use the model to ﬁnd the product of

The model shows

emonstrate how to multiply D the product.

Is the product of fractions?

Demonstrate how to multiply

Is the product of

. Use the model to ﬁnd the product of

and

using the fraction bars. Then ﬁnd

greater than or less than each of the original

and

using a number line. Then ﬁnd the product.

greater than or less than

1.03 Multiply fractions and mixed numbers mathspace.co

Demonstrate how to multiply

and

using pattern blocks. Then ﬁnd the product. = 1 whole

In a math problem, Trixie tries to multiply

using a number line. She draws a number line from 0 to 2 and

marks the point

on it. Then, she divides the segment from 0 to

to represent

. Based on this method, Trixie concludes that

into 3 equal parts and takes 2 of these parts .

Identify and explain the mistake in Trixie’s method. Then, correctly use a number line to ﬁnd the product. 15

Evaluate and express your answer in simplest form: a

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s extend our thinking 21

Consider the expression

Without calculating, determine if the product will be greater than or less than

Will the product be greater than or less than ? Explain your reasoning.

Find a fraction that, when multiplied by , results in a product that is less than

. Explain your reasoning.

. What conclusion can you draw

about multiplying two proper fractions based off your investigation? Explain your thinking. 23

Stephen is Sydney?

In Wings Station, one batch of chicken wings requires

years old. He has spent

of his life living in Sydney. How many years has he spent living in

cups of ﬂour. If the Wings Station is making

batches this Sunday, how much ﬂour will they need? Draw a picture to justify your thinking. 25

Georgia and Chenny need to follow a recipe which requires

of a cup of ﬂour. However, they only want half

of the quantity that the recipe is for. Georgia thinks they should divide multiply 26

. Who is correct? Use a number line to model your thinking.

Every day of the week, Mohamad walks to his school which is he is

, but Chenny thinks they should

of a mile away from his house. On Thursday,

of the way to school when he realizes that he forgot his English textbook at home and turns back to get

it. Explain how to ﬁnd how far Mohamad has traveled in miles when he turns back. 27

Jenna is asked to multiply

and she answered

right away by canceling the 5s in the numerator and

denominator. Explain why this method can be used.

1.03 Multiply fractions and mixed numbers mathspace.co

Answers

Then, we look for a piece that has the same size as the shaded pieces. We can see that the shaded pieces are

1.03 Multiply fractions and mixed numbers

the same size as the

th rectangle, so

b Less than

What do you remember?

12 a To multiply

1 A 2 a 14 b 40

d 7

and

using a number line:

We can draw a number line from 0 to 0

3 C 4 We can multiply two fractions by multiplying the ﬁrst numerator by the second numerator and the ﬁrst denominator by the second denominator.

Then, divide each part between 0 and parts.

5 D

into 5 equal

Finally, we shade 3 of the 5 parts in each of the eighths.

Let’s practice

Since 33 pieces are shaded and 40 pieces make up

8 a Less than 9

b Less than 12

c Equivalent to 2

d Greater than 4

one whole,

b Less than 13 To multiply of

, we can ﬁnd 1 group of

and then

10 Then, count the total number of colored blocks and the total number of the parts of the pattern. Since we have 3 wholes and 3 sixths, or 11 a To multiply

and

using a fraction bar:

We can split one of the th rectangles into 3 equal pieces, then shade in two of the new parts.

14 Trixie’s method would have worked, but it is diﬃcult to ﬁnd the resulting fraction from this method, which is why her answer of

is incorrect.

A better method is to ﬁrst identify the point number line, then ﬁnd

on the

of each fourth. This can be

visualized by ﬁrst marking on the number line, then dividing each fourth into 3 equal segments and taking 2 of those segments to ﬁnd the new product. 0

Using this method, we can see there are 12 pieces in one whole, and 14 total pieces are shaded. Therefore, .

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

blocks and 1 quarter block, and 1 half of a quarter block.

15 a

16 a 3

17 a 1

18 a

c 1

19 a

20 a

3. Total Blocks Count: There are 30 whole blocks, 7 quarter blocks, and 1 half of a quarter block. Since 4 quarters make one whole, there are now 31 whole blocks. Since 1 half of a quarter block is 1 eighth, we can say there are 7 eighths blocks.

Wings Station will need batches of chicken wings.

cups of ﬂour to make

25 Model the division or multiplication: Let’s extend our thinking 21 a

is less than

• Georgia’s suggestion because multiplying any number

by a fraction between 0 and 1 results in a product that is less than the original number. b I t is greater than because multiplying any number by a number that is greater than 1 results in a product greater than the original number.

is divided into 2 equal groups, where the size of each group is . :

years

is split into 4 equal parts, and 3 of the 4 parts is .

24 To justify this with a model: 1. Visualizing the amount of ﬂour per batch: Each batch cups of ﬂour. This can be visualized as four requires full blocks and one quarter block. Each full block represents 1 cup, and the quarter block represents a cup.

• Chenny’s suggestion

22 The product of two proper fractions is always less than each of the original fractions. 23

2. Replicating for multiple batches: For batches, replicate this set of blocks 7 times fully and once more at half its size. This means you’ll have 7 full sets of 4 full

The result is . Therefore, both Georgia and Chenny are correct. When they either divide

by 2 or multiply

by ,

they will end up with of a cup of flour, which is exactly half of the original quantity required by the recipe. 26 The first fraction is the total distance from Mohamad’s school to his house, while the second fraction is the part of the total distance. This means we are asked to find the fraction of a fraction, so we need to use multiplication. .

Multiplying the fractions gives So Mohamad has traveled 27 The

is the same as

miles when he turns back. where

, so

This means if there are same numbers in the numerator and denominator, we can multiply the remaining numerator and denominator by 1 to get the answer right away.

Answers mathspace.co

1.04 Divide fractions and whole numbers Subtopic overview Lesson narrative In this lesson, students will learn to divide fractions and whole numbers. They start by understanding how to divide whole numbers by fractions and then explore dividing fractions by whole numbers. The lesson includes detailed explorations using interactive applets that allow students to model division problems visually and conceptually. These explorations help students understand the relationship between multiplication and division of fractions. Problems include using number lines, finding reciprocals, and converting between mixed numbers and improper fractions. By the end, students should be able to confidently perform division operations involving fractions and whole numbers.

Learning objectives Students: Page 31

Key vocabulary 

reciprocal

Essential understanding Dividing by a fraction is the same as multiplying by its reciprocal.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate this goal by presenting contextual problems involving the division of whole numbers by fractions and fractions by whole numbers. For instance, asking students how to equally distribute a certain number of resources among a group of people. Teachers can also encourage students to create and solve their own problems based on real-world scenarios. Teachers should model effective practices, such as converting mixed numbers to improper fractions and validating solutions. By guiding students to understand, plan,execute, and review their problem-solving processes, teachers can develop students’ abilities to estimate, justify, and arrive at reasonable solutions in real-world contexts. 60

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can help students recall multiple problem types learned in earlier grades and applying them to more complex problems. They can introduce solution methods with worked-out examples, explaining each step and the decisions made during the problem-solving process. By demonstrating how to solve similar problems, discussing each decision, and engaging students with guiding questions, teachers can effectively connect prior knowledge to new, contextual problem-solving methods.

Teachers can help students make mathematical connections by linking the lesson’s content to prior knowledge. For instance, teachers could remind students of the process for dividing whole numbers, which they learned in 5.CE.1b, and explain how this process is related to dividing whole numbers by fractions.

MPG3 — Mathematical Reasoning Teachers can foster mathematical reasoning by guiding students through contextual problems involving the division of fractions by whole numbers and whole numbers by fractions. This lesson helps students make and test predictions, justify their problem-solving steps, and evaluate their conclusions. Teachers should encourage students to think about the operations involved, such as how the meaning of division applies to the problem. By explaining their reasoning and discussing their thought processes, students gain a deeper understanding of operations and improve their strategic problem-solving skills, moving beyond reliance on key words.

MPG5 — Mathematical Representations Teachers can integrate mathematical representations by using various models to represent the division of whole numbers and fractions. They can use fraction bars/strips, area models, number lines, and manipulatives. This can also be expanded to include students creating their own models or drawings to represent their solutions. Utilizing interactive technology, such as online fraction tools or virtual manipulatives, can further enhance students’ understanding of mathematical representations.

Content standards 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1b — Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.*

6.CE.1a — Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.*

6.CE.1c — Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.*

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

1.04 Divide fractions and whole numbers mathspace.co

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.03 Multiply fractions and mixed numbers

Student lesson & teacher guide Divide whole numbers by fractions Students revisit the idea that dividing a whole number by a unit fraction is equivalent to multiplying the reciprocal of the unit fraction, which is a whole number. This idea is extended to dividing a whole number by a fraction that is not a unit fraction, and an example is shown on a number line.

Students: Pages 31–32

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Consider

. We cannot ask, “What is 6 divided into

are in 6?”

sense. Instead, we can say, “How many groups of

groups of the same size” because that does not make

To answer this question, we can begin by dividing the 6 wholes into thirds. Then, we can make groups of 2 thirds. If we count, we can see there are 9 equal groups of

in 6.

This shows 6 ÷ = 9. . We cannot ask, “What is 6 divided into groups of the same size” because that does not make

Consider

Again, instead by“How a fraction, can multiply the reciprocal of the fraction: in 6?” sense. Instead,of wedividing can say, many we groups of areby

Interactive exploration Explore online to answer the questions

Exploration

To answer this question, we can begin by dividing the 6 wholes into thirds. Then, we can make groups of 2 thirds. If we count, we can see there are 9 equal groups of This shows 6 ÷

in 6.

= 9.

mathspace.co Again, instead of dividing by a fraction, we can multiply by the reciprocal of the fraction: Students: Page 32 Use the interactive exploration in 1.04 to answer these questions. 1.

Explain how the model is showing the division.

Interactive exploration How does the model relate to the final fraction?

Explore online to answer the questions

What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

mathspace.co

What do you notice about dividing a whole number by a fraction between 0 and 1?

Use the interactive exploration in 1.04 to answer these questions. When1.a whole number is divided byshowing a fraction between Explain how the model is the division.0 and 1, the result is larger than the original whole number. 2.

How does the model relate to the final fraction?

3. What Example 1 is the relationship between the numbers in the original division expression and the numbers in the final fraction? The number linedo below showsabout 4 wholes split a into sized parts. 4. What you notice dividing whole number by a fraction between 0 and 1?

When a whole number is divided by a fraction between 0 and 1, the result is larger than the original whole number. 0

Example 1 a Use the model to evaluate . The number line below shows 4 wholes split into

sized parts.

Create a strategy To evaluate

, we can use the number line model to count the number of one-third sized parts that make up the

whole sections.

Apply the idea

a Use the model to evaluate . There are 4 whole sections, and each section is divided into 3 parts

Create strategy in the a diagram so 4 ÷

= 12.

1.04 Divide fractions and whole numbers mathspace.co . There are 12 parts of size

Suggested student grouping: Small groups In this exploration, students will be using a GeoGebra applet to explore the concept of dividing a whole number by a fraction. Students will create different division expressions and observe how the fraction bars overlap to visually represent the division operation. They will then relate their observations to the final fraction result. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Explain how the model is showing the division. Each fraction bar represents one whole. When we divide, the fraction bars representing the whole number is divided into the same size of pieces as the fraction. 2. How does the model relate to the final fraction? The model visually demonstrates how many fraction pieces fit into the whole number. 3. What is the relationship between the numbers in the original division expression and the numbers in the finalConsider fraction? . We cannot ask, “What is 6 divided into groups of the same size” because that does not make aremultiplying in 6?” Instead, say,fraction “How many groups of of The sense. numerator ofwe thecan final is the result the numerator of the whole number by the fraction’s denominator. If the whole number is written as a fraction out of 1, the denominator of the final To answer this question, we can begin by dividing the 6 wholes into fraction0is the1 product of3 the denominator the first fraction numerator of the second fraction. 2 4 5 6 of thirds. Then, we canand makethe groups of 2 thirds. 4. What do you notice about dividing a whole Ifnumber fraction between 0 and 1?of in 6. we count,by weacan see there are 9 equal groups The result is always a whole number that is larger than the original whole number. This shows 6 ÷

= 9.

Purposeful questions Again, instead of dividing by a fraction, we can multiply by the reciprocal of the fraction: • Can you explain how dividing a whole number by a fraction is similar to multiplying by the reciprocal of the fraction? Why does this method work? • How would you represent the division of a whole number by a fraction using a visual model? How does this help in understanding the division process? Interactive exploration Explore online to answer the questions • Is the quotient the same, smaller, or larger than the original whole number?

mathspace.co Possible misunderstandings • Students may think that dividing by a fraction will make the number smaller. Help students see that dividing Use the interactive exploration in 1.04 to answer these questions. by a fraction (less than one) increases the original number. Use visual models and concrete examples to how the model is showing the division. reinforce1.thisExplain concept. 2.

How does the model relate to the final fraction?

What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

Examples4. What do you notice about dividing a whole number by a fraction between 0 and 1? Students: Page 32

When a whole number is divided by a fraction between 0 and 1, the result is larger than the original whole number.

Example 1 The number line below shows 4 wholes split into

a Use the model to evaluate

sized parts.

Create a strategy To evaluate

, we can use the number line model to count the number of one-third sized parts that make up the

whole sections.

Apply the idea 64

There are 4 whole sections, each section Mathspace Virginia SOL Grade and 6 Teacher Editionis divided into 3 parts mathspace.co in the diagram so 4 ÷ = 12.

. There are 12 parts of size

The number line below shows 4 wholes split into

a Use the model to evaluate

sized parts.

Create a strategy To evaluate

, we can use the number line model to count the number of one-third sized parts that make up the

whole sections.

Apply the idea There are 4 whole sections, and each section is divided into 3 parts in the diagram so 4 ÷

. There are 12 parts of size

= 12.

Purpose32 Mathspace Virginia SOL Grade 6 mathspace.co Make students aware that a number line can be used to determine how many parts are created when a whole is subdivided.

Students: Page 33 b If 4 is divided into parts that are

of a whole each, how many parts are there in total?

Create a strategy We can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal of

or 3.

Apply the idea Divide the whole number by the unit fraction Multiply by the reciprocal b If 4 is divided into parts that are

Evaluate of a whole each, how many parts are there in total?

Create a strategy

Example 2 PurposeWe can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal is or 3. Studentsof demonstrate of whole numbers by unit fractions. Rewrite usingdivision multiplication. Apply the idea Expected mistakes Create a strategy Apply the idea Students might think this is asking a different question and trybytothefind answer that is different to part (a). Divide the whole number unitan fraction To rewrite the expression using multiplication, remember

Multiply by the reciprocal

Help them howbythis question theas same as thatsee dividing a fraction is theis same multiplying its for the result of 4 divided by . Multiply byasking thebyreciprocal reciprocal. The reciprocal of

Students: Page 33

is .

Evaluate

Idea2 summary Example

Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that Rewritefraction. using multiplication.

Create a strategy

Apply the idea

To rewrite the expression using multiplication, remember Multiply by the reciprocal Divide fractions by whole numbers that dividing by a fraction is the same as multiplying by its When we divided a whole number by a fraction, such as 2 ÷ , we asked the question “how many parts of size ﬁt reciprocal. The reciprocal of is . into 2 wholes?” Dividing a fraction by a whole number is the reverse of this. Let’s look at

÷ 2 as an example:

Idea summary

We start with of a whole, shown as the shaded area in the image. Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that 1.04 Divide fractions and whole numbers fraction. mathspace.co

Purpose Students demonstrate the concept of reciprocal to convert a division into a multiplication operation. Reflecting with students Encourage students to complete the problem by evaluating the multiplication. Then, have them consider whether the result is larger or smaller than the original number. Students should recognize that when a whole number is divided by a fraction less than 1, the result is larger than the original whole number. b If 4 is divided into parts that are

of a whole each, how many parts are there in total?

Critique, correct, and clarify

use with Example 2

Create a strategy

English language learner support

We can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal

Provide of students the incorrect statement: “9 divided by is or with 3.

equals

.” Write this statement on the board

and ask students to work in pairs to critique it. Encourage them to identify any errors and discuss why the Apply statement maythe beidea incorrect. Divide whole number of by fractions the unit fraction Prompt students to consider how division andthe multiplication work, and to use mathematical b If 4 is divided into parts that are of a whole each, how many parts are there in total? Multiply the reciprocal in their explanations. Ask guiding questions vocabulary such as “reciprocal,” “division,” andby “multiplication” Evaluate like, “What does it mean to divide a whole number by a fraction?” and “How can we rewrite division as a Create a strategy multiplication statement?” We can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal

After students of is have or 3. identified that dividing by a fraction is the same as multiplying by its reciprocal, have them correct the statement. Encourage students to clarify their understanding by explaining each step of the Example 2 Apply the idea correction. Rewrite

using multiplication.

the whole number by the unit This activity helps students deepen theirDivide comprehension of dividing byfraction fractions and reinforces precise mathematical language, while also addressing any language barriers with terms like “reciprocal.” Multiply by the reciprocal Create a strategy Apply the idea Evaluateremember To rewrite the expression using multiplication, that dividing by a fraction is the same as multiplying by its

Students:reciprocal. Page 33 The reciprocal of

Multiply by the reciprocal

is .

Example 2 RewriteIdea using multiplication. summary Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that

Createfraction. a strategy

Apply the idea

To rewrite the expression using multiplication, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of

Multiply by the reciprocal

is .

Divide fractions by whole numbers we divided a whole number by a fraction, such as 2 ÷ , we asked the question “how many parts of size ﬁt DivideWhen fractions by whole numbers into 2 wholes?”

Idea summary Students visualize fractions byis whole numbers exploration. Dividing adividing fraction byunit a whole number the reverse of this. before Let’s lookengaging at ÷ 2 as in an an example:

Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that fraction. Students: Pages 33–34 We start with of a whole, shown as the shaded area in the image.

Divide fractions by whole numbers When we divided a whole number by a fraction, such as 2 ÷ , we asked the question “how many parts of size ﬁt into 2 wholes?” We then divide each of these thirds into 2 parts. Dividing a fraction by a whole number is the reverse this. look at shaded ÷ 2 as an example: Howofbig is Let’s the remaining area? Well, there are now 6 parts of equal area and 1 of them is shaded, so this is equal to of the We start with of a whole, shown as the shaded area in the image. whole.

1.04 Divide fractions and whole numbers mathspace.co

We then divide each of these thirds into 2 parts.

How big is the remaining shaded area? Well, there are now 6 parts

of equal area and 1 of them is shaded, so this is equal to Mathspace Virginia SOL Grade 6 Teacher Edition whole. mathspace.co

of the

We start with

of a whole, shown as the shaded area in the image.

We then divide each of these thirds into 2 parts. How big is the remaining shaded area? Well, there are now 6 parts of equal area and 1 of them is shaded, so this is equal to whole.

of the

1.04 Divide fractions and whole numbers mathspace.co

Physical division Targeted instructional strategies Have students take a sheet of paper and divide it in two (fold in half). This is obviously two halves. Now have them repeat this process with one of the halves and they can see it is now a quarter of the original shape.

If they take the remaining half, they could divide it into four equal parts and see that is an eighth of the original shape.

Exploration Students: Page 34

1.04 Divide fractions and whole numbers mathspace.co

Suggested student grouping: Small groups Students are asked to devise a method for dividing fractions by whole numbers, and then apply this method to a series of examples. They will then compare the results obtained with their method to a model of the division process. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What is the method you used to divide the fractions by whole numbers? If the numerator of the fraction is a multiple of the whole number, the numerator is divided by the whole number to get the new numerator. The denominator remains the same. Alternatively, we multiply the whole number by the denominator of the fraction and keep the numerator the same. 2. Evaluate each example using a model. How does this compare to the method you developed? Students can use fraction bars, fraction circles or number lines to model each division. To find the result of each expression using a model, students should shade a whole according to the first fraction, divide each shaded piece into the whole number, and determine how many parts of the whole are now shaded. 3. Did you get the same result using both your method and a model? Why or why not? Yes, the same results are observed with both methods. This is because the method is basically performing the same operation as the model, just in a more abstract way. 4. Can your method be applied to all fractions and whole numbers? Why or why not? Yes, my method can be applied to all fractions and whole numbers. This is because we can always find the reciprical of a whole number or fraction and multiply it to another whole number or fraction. Purposeful questions • Consider the two example problems given. How do the results compare to the original numbers? What operation is used? • Why does dividing a unit fraction by a whole number result in a smaller fraction? • How can you relate the division of a unit fraction by a whole number to a real-life situation? How does this help you understand the concept better? Possible misunderstandings • Students may think that dividing a unit fraction by a whole number increases the size of the fraction. For example, students may not understand that dividing

by 2 results in a smaller fraction because they

are splitting a small part into even smaller parts. This can be clarified through use of visual aids.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examples Students: Page 34

Purpose Show students how to use a number line to represent division of unit fractions by wholes.

Students: Page 35 b What is the size of the part created when

is divided by 4?

Create a strategy We can also count the number of spaces between 0 and 1 on the number line.

Apply the idea 0

We can see that there are 12 spaces, so each space (or part) has a size of

Reﬂect and check We can also divide the unit fraction by the whole number. Divide the unit fraction by the whole number Multiply the denominator by the whole number Evaluate

Example 4 Evaluate the following:

1.04 Divide fractions and whole numbers mathspace.co

Apply the idea 0

We can see that there are 12 spaces, so each space (or part) has a size of

Reﬂect and check We can also divide the unit fraction by the whole number. b What is the size of the part created when is divided by 4? Divide the unit fraction by the whole number

Create a strategy

Multiply the denominator by the whole number We can also count the number of spaces between 0 and 1 on the number line.

Apply the idea

Evaluate

4 PurposeExample We can see that there are 12 spaces, so each space (or part) has a size of . StudentsEvaluate demonstrate that they can divide a unit fraction buy wholes arithmetically. the following: Reﬂect check a Reflecting withand students We can also divide the unit fraction by the whole number. Ask students to consider whether this method works if the fraction is not a unit fraction. How many pieces Create a strategy would we consider then? Dividing byDivide a whole number affects denominator, so the number of pieces the unit fractiononly by the whole the number Rewrite the whole number a fraction. multiply thefor ﬁrsta fraction bylike the reciprocal of the being considered, whether it’sas1 for a unitThen fraction or 2 fraction , will stay thesecond same.fraction. The only thing that Multiply the denominator by the whole number

changesApply is thethe denominator, since we are dividing each piece into smaller pieces. idea

Students: Page 35

Evaluate Rewrite the whole number as fraction Multiply by the reciprocal of

Example 4

Multiply the numerators and denominators

Evaluate the following: a

Create a strategy Rewrite the whole number as a fraction. Then multiply the ﬁrst fraction by the reciprocal of the second fraction.

Apply the idea Rewrite the whole number as fraction Multiply by the reciprocal of Multiply the numerators and denominators1.04 Divide fractions and whole numbers

mathspace.co

Purpose Show students how to divide a fraction by a whole number and multiplying by its reciprocal. Expected mistakes Students might forget to convert the whole number 5 into a fraction and find its reciprocal before multiplying. Alternatively, they may find the reciprocal of the first fraction. Remind students that whole numbers can be written as fractions out of 1 since a number divided by 1 is the number itself, and that we always find the reciprocal of the second fraction (never the first). 1.04 Divide fractions and whole numbers mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 36 b

Create a strategy Convert the mixed number into an improper and the whole number into a fraction. Then rewrite as multiplication by the reciprocal.

Apply the idea First, convert the mixed number

into an improper fraction. Rewrite the mixed fraction Evaluate the multiplication Evaluate

Then convert the whole number 4 into a fraction. Rewrite the whole number Rewrite the division as multiplication by the reciprocal. Divide improper fractions Multiply by the reciprocal Multiply the numerators and denominators Evaluate Simplify

Idea summary

Purpose Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of the whole Show studentsnumber. the process of dividing a mixed number by a whole number. The reciprocal of a whole number is: Reflecting with students Challenge advanced learners to evaluate the expression without converting the mixed number to an improper number first. Students should divide the whole and the fraction by 4 individually, then combine the results.

Mathspace Virginia SOL Grade 6 mathspace.co

1.04 Divide fractions and whole numbers mathspace.co

Apply the idea First, convert the mixed number

into an improper fraction. Rewrite the mixed fraction

Introduce the keep, change, flipthephrase to support memory Evaluate multiplication

use with Example 4

Student with disabilities support Evaluate

Introduce students to the phrase ‘Keep, change, flip’ to help them remember the process of dividing fractions Then convert the whole number 4 into a fraction. and whole numbers. Keep the first fraction or whole number as it is, change the division sign to multiplication, Rewrite the whole number and flip the second fraction to its reciprocal. Rewrite the to division as multiplication reciprocal. Use color coding highlight each partbyofthe the process—for instance, green for ‘keep,’ blue for ‘change,’ and red for ‘flip’—which can aid visual-spatial Encourage students to verbalize each step as they work Divideprocessing. improper fractions through problems to strengthen language and memory skills. Multiply by the reciprocal

Provide ample guided practice with step-by-step instructions before moving on to independent work to help Multiply the numerators and denominators students build confidence and mastery. Evaluate

Students: Page 36

Simplify

Idea summary Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of the whole number. The reciprocal of a whole number is:

Practice Students: Pages 37–40

What do you remember? 1

This rectangle is broken into two halves: 36

Which rectangle shows that each half has been divided into 3 parts? A

Mathspace Virginia SOL Grade 6 mathspace.co

Find the size of one piece created when

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

is divided by 3.

This rectangle shows 1 whole split into 4 parts of size :

Which rectangle shows that each quarter has been divided into 3 parts? A

b 3

Find the size of the piece created when

are in 6?

Consider the squares shown:

Divide 10 into 2 groups of equal size.

Find the size of each group when 10 is divided by 2.

Consider the fraction bars shown:

Divide 1 whole into 2 groups of equal size.

Find the size of each group when 1 is divided by 2.

iii Divide

into 2 groups of equal size.

iv Find the size of each group when

is divided by 3.

This picture represents 6 pizzas. Exactly how many

is divided by 2.

Rewrite each expression using multiplication: a

Let’s practice 6

Use the given models to evaluate the expressions. a

1.04 Divide fractions and whole numbers mathspace.co

Consider the expression a

Use a model to show 2 wholes split into

How many pieces of size

Is the quotient of 2 ÷

greater than, less than or equivalent to the original whole number?

Find the number of

sized groups in 2 wholes.

Is the quotient of 2 ÷

greater than, less than or equivalent to the original whole number?

are there in total?

Consider the expression a

Use the model to show

wholes split into 2 groups of equal size.

How many pieces of size

are in each group?

Which expression results in a quotient that is larger than 5? A

size pieces.

Use the number line to ﬁnd the value of each expression. Express your answer in simplest form. a 0

b 0

c 0

d 0

e 1

f 0

5÷2

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Use the fraction bars to ﬁnd the value of each expression. Express your answer in simplest form. a 1 WHOLE

1 WHOLE

b 1 1 1 1 1 1 1 1 1 WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE

c 1 WHOLE

The fraction bar shows groups?

A 13

1 WHOLE

divided into 2 equal groups. What fraction of the whole strip makes up one of those

The fraction bar shows 3 wholes, and

is shaded.

1 WHOLE

Divide the fraction bar into 4 groups of equal size.

Use your model from part (a) to evaluate

1 WHOLE

= 1 whole

Use the pattern blocks to show how to divide 3 by , and write the quotient.

Use the pattern blocks to show how to divide

by 3, and write the quotient.

Divide each expression. Express your answer in simplest form. a

1.04 Divide fractions and whole numbers mathspace.co

Divide each expression. Express your answer in simplest form. a

Divide each expression, giving your answer as a mixed number: a

Let’s extend our thinking 18

Jordan has

gallons of paint. Each wall requires 2 gallons of paint. How many walls can Jordan paint with the

available paint? Use a model to solve. 19

A container holds

Sophia is preparing gift bags for a party. She has 48 chocolate bars and wants to distribute them equally

liters of juice. How many containers can be ﬁlled with 18 liters of juice?

among gift bags. If each gift bag is supposed to contain prepare? Use a model to explain your thinking.

chocolate bars, how many gift bags can Sophia

When 24 is divided by a number, the quotient is 21. Explain how to ﬁnd the number.

Mr. Marge gives the following expressions to the class:

Which expression should result in the larger number? Explain your thinking. 23

A bag contains 45 cups of dog food. Robin’s dog eats

cups of dog food each day.

How many days does the bag of dog food last? Explain your answer.

How many cups of dog food does Robin’s dog eat in

If a bag of dog food costs $12, how much should she pay for dog food that will last for

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

months? month?

Answers

8 a

b 3

1.04 Divide fractions and whole numbers What do you remember? 1 a C

9 The expression that results in a quotient larger than 5 is

2 a B

. This expression evaluates to 17.5, which is greater than 5.

3 C

10 a 3

4 a i

b 0

ii The size of each group is 5. b i

c 5

d ii The size of each group is . iii

e 0

iv The size of each group is . 5 a

Let’s practice 6 a

1 3

2 3

1 2 1 3 3

1 2 2 3 3

1 2 3 3 3

1 2 4 3 3

1 3

11 a 8 1 WHOLE 1 WHOLE 1 WHOLE 1 WHOLE 1 WHOLE 1 WHOLE b 4

7 a Models may vary.

1 1 1 1 1 1 1 1 1 WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE WHOLE

c b 6

c Greater than

d 3

e Greater than

1 WHOLE

12 D

Answers mathspace.co

13 a

20 This shows 48 squares: 1

14 a

We can divide the 48 squares into 12 groups of b

15 a

b 14

16 a

g 28

17 a

to start:

This shows the remaining from the previous image.

There is 1 more group of

pieces There are remaining, which is not enough to make another whole group.

So, we conclude that Sophia can prepare 13 complete gift bags for the party.

Let’s extend our thinking

21 Finding the number is the same as dividing 24 by 21.

So we have

. Now, the number is .

22 Splitting the wholes into fractional parts will result in a larger answer than splitting a fraction into smaller fractional parts. while

Observe that . Now, 23 a Notice that Jordan can paint 3 walls with the available paint. The calculation is based on dividing the total gallons of paint Jordan has by the gallons required per wall, resulting in . Since Jordan cannot paint a fraction of a wall, the maximum whole number of walls that can be painted is 3. 19 8 containers

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

. This means the bag of dog food or 20 days.

will last for b

is the larger number.

cups

c (90 ÷ 45) ⋅ 12 = $24

1.05 Divide fractions and mixed numbers Subtopic overview Lesson narrative In this lesson, students will learn to divide fractions and mixed numbers. They begin by understanding the division of fractions and progress to dividing mixed numbers by converting them into improper fractions. The lesson includes detailed explorations with interactive applets, where students create division expressions and observe models of the division process. These models help visualize and understand the division concept. Types of problems include dividing fractions by fractions, mixed numbers, and solving real-world problems involving measurements. By the end, students should confidently divide fractions and mixed numbers using models and mathematical operations.

1.05 Divide fractions and mixed Learning objectives numbers Students: Page 41

After this lesson, you will be able to... • divide fractions. • divide mixed numbers. • explain the effect of dividing a fraction by a number between 0 and 1.

Divide fractions and mixed numbers We’ve divided fractions and whole numbers, but what does it look like to divide a fraction by another fraction? Key vocabulary 

Let’s look at reciprocal

. Let’s start by asking, “How many groups of

make up .

Essential understanding

First we can draw a diagram of . Dividing by a fraction is the same as multiplying by its reciprocal.

Standards Then we can splitStandards it into groups of . This subtopic addresses the following Virginia 2023 Mathematics of Learning standards. We can see that there are 4 equal groups of

in . So

We can also apply the method of rewriting division as multiplication by the reciprocal like we did when dividing with whole numbers. Multiply

by the reciprocal of

Multiply the numerators together and denominators together Evaluate the multiplication Divide

1.05 Divide fractions and mixed numbers mathspace.co

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can incorporate this goal by giving students a variety of practice problems that involve dividing fractions, mixed numbers, and fractions by fractions with different denominators up to 12. Teachers can also provide real-world examples where these concepts are applied, such as dividing resources, determining the number of equal parts in a whole, or finding unit rates. Allowing students to work through these problems can help develop their problem-solving skills. Teachers can also encourage students to create and solve their own problems based on real-world scenarios. Teachers should model effective practices, such as multiplying by the reciprical of the fraction, converting mixed numbers to improper fractions and validating solutions. By guiding students to understand, plan,execute, and review their problem-solving processes, teachers can develop students’ abilities to estimate, justify, and arrive at reasonable solutions in real-world contexts.

Teachers can foster mathematical reasoning by guiding students through contextual problems involving the division of fractions and mixed numbers. This lesson helps students make and test predictions, justify their problem-solving steps, and evaluate their conclusions. Teachers should encourage students to think about the operations involved, such as how the meaning of division applies to the problem. By explaining their reasoning and discussing their thought processes, students gain a deeper understanding of operations and improve their strategic problem-solving skills, moving beyond reliance on key words.

MPG2 — Mathematical Communication Teachers can help students recall multiple problem types learned in earlier grades and applying them to more complex problems. They can introduce solution methods with worked-out examples, explaining each step and the decisions made during the problem-solving process. By demonstrating how to solve similar problems, discussing each decision, and engaging students with guiding questions, teachers can effectively connect prior knowledge to new, contextual problem-solving methods.

MPG4 — Mathematical Connections To integrate this goal, teachers can relate the current lesson on dividing fractions and mixed numbers to previous lessons. For example, teachers can remind students of the skills they learned in Lesson 1.04, which involved dividing whole numbers by fractions and fractions by whole numbers, and show how those skills are relevant to the current lesson. Teachers can also show how the concept of division is applicable across different mathematical areas, such as in algebra and geometry, and even in other disciplines, such as physics or economics.

MPG5 — Mathematical Representations Teachers can integrate this goal by encouraging the use of various models to represent division problems involving fractions and mixed numbers. These can include fraction bars, number lines, area models, or students’ own models. Teachers can also connect these models to real-world examples and contextual problems to help students visualize and better understand the problem. This can also facilitate discussion among students about the connections between different models and how they each help to represent and solve division problems.

Content standards

6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1b — Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.*

6.CE.1a — Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.*

6.CE.1c — Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.*

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Prior connections 5.CE.1 — The student will estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition, subtraction, multiplication, and division with whole numbers.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.03 Multiply fractions and mixed numbers Grade 6 — 1.04 Divide fractions and whole numbers

Tools You may ﬁnd these tools helpful: • Fraction bars

Student lesson & teacher guide Divide fractions and mixed numbers Students: Page 41

1.05 Divide fractions and mixed numbers After this lesson, you will be able to... • divide fractions. • divide mixed numbers. • explain the effect of dividing a fraction by a number between 0 and 1.

Divide fractions and mixed numbers We’ve divided fractions and whole numbers, but what does it look like to divide a fraction by another fraction? Let’s look at

. Let’s start by asking, “How many groups of

make up .

First we can draw a diagram of .

1.05 Divide fractions and mixed numbers mathspace.co Then we can split it into groups of .

Divide fractions and mixed numbers We’ve divided fractions and whole numbers, but what does it look like to divide a fraction by another fraction? Let’s look at

. Let’s start by asking, “How many groups of

make up .

First we can draw a diagram of .

Then we can split it into groups of . We can see that there are 4 equal groups of

in . So

We can also apply the method of rewriting division as multiplication by the reciprocal like we did when dividing with whole numbers. Multiply

by the reciprocal of

Multiply the numerators together and denominators together Evaluate the multiplication Divide Both methods give us the same result. We can apply the same process to dividing with mixed numbers, because remember a mixed number is just a different form of a fraction. We just have to convert the mixed number into an improper fraction ﬁrst.

Exploration Students: Page 42 1.05 Divide fractions and mixed numbers mathspace.co

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 1.05 to answer these questions. 1.

Explain how the model is showing the division.

How does the model relate to the final fraction?

What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

What do you notice about dividing a fraction or mixed number by a fraction between 0 and 1?

When a fraction or mixed number is divided by a fraction between 0 and 1, the result is larger than the original fraction or mixed number.

Example 1 Evaluate each expression. a

Create a strategy Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Mathspace Virginia Apply the idea SOL Grade 6 Teacher Edition mathspace.co Multiply by the reciprocal Multiply numerators and denominators

Suggested student grouping: In pairs Students will use a GeoGebra applet to visualize the process of division as it relates to fractions. They will create various division expressions, then observe the corresponding animated model. This will help them understand how division with fractions works. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Explain how the model is showing the division. The model represents the original fractions in the expression using fraction bars. In the first step, both fractions are divided into same size of pieces. This is done by creating equivalent fractions with common denominators. 2. How does the model relate to the final fraction? The final fraction is the result of how many times the second fraction fits into the first fraction. 3. What is the relationship between the numbers in the original division expression and the numbers in the final fraction? The numerator in the final fraction represents the product of the numerator of the first fraction (when it is written as an improper fraction) and the denominator of the second fraction. The denominator in the final fraction represents the product of the denominator of the first fraction and the numerator of the second fraction. 4. What do you notice about dividing a fraction or mixed number by a fraction between 0 and 1? The quotient is always larger than the original fraction or mixed number. Purposeful questions • What happens in the first two steps of the animation? Is this necessary to get to the final result? • How does changing the numbers in the division expression affect the model and the final fraction? • Why do you think it’s helpful to visualize the division of fractions in this way? Possible misunderstandings • Students may think that, because the animation creates equivalent fractions with common denominators, finding common denominators is a necessary step in dividing fractions. Make students aware that this step simply allows us to understand the division better, but it is not required for finding the quotient without models.

Create a step-by-step procedure using algorithmic thinking Targeted instructional strategies Incorporate algorithmic thinking by guiding students to develop a clear, step-by-step procedure for dividing fractions and mixed numbers. Encourage them to articulate each step of the process, from converting mixed numbers to improper fractions to finding reciprocals and performing multiplication. Have students write down their algorithm and test it with various examples to ensure it works consistently. This approach not only helps them understand the mechanics of division but also builds their ability to think systematically. Here’s an exemplar set of steps: 1. Convert any mixed numbers to improper fractions. 2. Keep the first fraction as it is (the dividend). 3. Find the reciprocal of the second fraction (the divisor) by swapping its numerator and denominator. 4. Multiply the first fraction by the reciprocal of the second fraction. 5. Simplify the resulting fraction if possible. 6. Convert back to a mixed number if possible or if necessary.

1.05 Divide fractions and mixed numbers mathspace.co

Explain how the model is showing the division.

How does the model relate to the final fraction?

What is the relationship between the numbers in the original division expression and the numbers in the final fraction?

What do you notice about dividing a fraction or mixed number by a fraction between 0 and 1?

Examples

a fraction Students:When Page 42 or mixed number is divided by a fraction between 0 and 1, the result is larger than the original fraction or mixed number.

Example 1 Evaluate each expression. a

Create a strategy Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea Multiply by the reciprocal Multiply numerators and denominators Evaluate

Reﬂect and check We can use an area model to verify the answer:

Overlap the models and count the number of shaded parts in the row and column. The column has 5 shaded parts and the row has 8 shaded parts.

Mathspace

So, the area model shows .

Virginia SOL Grade 6

mathspace.co Purpose Show students how to divide fractions by turning the division problem into a multiplication problem using the reciprocal of the second fraction.

Expected mistakes Students might find the reciprocal of the first fraction rather than the second fraction. Remind them that the second number is the one that divides the first, so the second fraction is always the one that we find the reciprocal of.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 43 b

Create a strategy Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea Multiply by the reciprocal Multiply numerators and denominators Evaluate Simplify

Example 2 Purpose Evaluate and write your answer in its simplest form. the concept of multiplying by the reciprocal, and further Challenge students to divide fractions by applying a the result when possible. simplifying Create a strategy Concrete-Representational-Abstract (CRA) approach

use with Example 1

Rewrite the mixed numbers as improper fractions, then divide the fractions.

Targeted instructional strategies

Concrete: Begin engaging students with physical fraction tiles or bars. Provide them with pieces representing Apply theby idea tenths and fifths. Ask students to take four pieces of fractions to model . Then, have them use pieces representing Rewrite as improper to create an equivalent fraction of

. Ask students to determine how many groups of

Multiply by the reciprocal of

can fit into

. Guide

them to discover that they only used 8 of the 11 pieces needed to make a full group. Simplify

Representational: Have students draw area models to represent

and

. Instruct them to draw a rectangle

and denominators divided into 5 equal parts and shadeMultiply 4 partsnumerators to represent . Next, draw two rectangles divided into 10 equal

parts and shade 11 parts for

. Then, guide students to divide each part of the first rectangle in two to create

b 10 equal parts. The shaded areas help them visualize that a grid with needed do fit.

does not fit into , but 8 and 11 pieces

Create a strategy

Abstract: Introduce the abstract mathematical method for dividing fractions. Show students how to divide Convert the mixed number into an improper fraction, then rewrite as multiplication.

by multiplying by the reciprocal. Highlight how this matches the result they found using the manipulatives the idea and areaApply models. This demonstrates the consistency between the concrete, representational, and abstract First, convert the mixed number into its improper fraction form. approaches. Rewrite the number Connecting the stages: Encourage students tomixed reflect on how each stage helped them understand dividing fractions. Discuss how the physical manipulatives made the concept tangible, the area models provided a visual Evaluate the multiplication representation, and the abstract calculations confirmed the result mathematically. Emphasize the importance of using different strategies to deepen Evaluate their understanding and to monitor their thinking. Guide them to apply this the addition approach to other fraction division problems, choosing the representation that helps them the most.

1.05 Divide fractions and mixed numbers mathspace.co

b Multiply numerators and denominators

Create a strategy Evaluate Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd. Simplify

the43 idea Students:Apply Page

Multiply by the reciprocal

Example 2

Multiply numerators and denominators

Evaluate and write your answer in its simplest form. Evaluate a

Simplify

Create a strategy Rewrite the mixed numbers as improper fractions, then divide the fractions.

Example 2 Apply the idea Evaluate and write your answer in its simplest form. Rewrite as improper fractions a Multiply by the reciprocal of

Create a strategy

Simplify Rewrite the mixed numbers as improper fractions, then divide the fractions. Multiply numerators and denominators

Apply the idea

Rewrite as improper fractions b

Multiply by the reciprocal of

Purpose Create ahow strategy Show students to convert mixedSimplify numbers to improper fractions and divide the fractions. Convert the mixed number into an improper fraction, then rewrite as multiplication. Multiply numerators and denominators

Students: Pages 43–44 Apply the idea

First, convert the mixed number b

into its improper fraction form. Rewrite the mixed number

Create a strategy

Evaluate the multiplication Convert the mixed number into an improper fraction, then rewrite as multiplication. Evaluate the addition

Apply the idea First, convert the mixed number

into its improper fraction form. Rewrite the mixed number

1.05 Divide fractions and mixed numbers mathspace.co

Evaluate the multiplication Evaluate the addition Then rewrite the division as multiplication. Divide improper fractions Multiply by the reciprocal Multiply the numerators and denominators Evaluate

Example 3 Purpose A meter long roll of fabric to be cutainto sections of equal length . How many piecesand of fabric will there be? as Demonstrate to students how toisconvert mixed number into an improper fraction rewrite division multiplication. Create a strategy

Rewrite the mixed numbers as improper fractions then perform division.

Apply the idea 86

Reﬂect and check

Mathspace Virginia SOL Grade 6 Teacher Edition Rewrite as improper fractions mathspace.co Multiply by the reciprocal of

To check the reasonableness of our answer, let’s use estimation. We can round the length of the fabric roll to 8 meters, and the sections we’re cutting to 3 meters long each. Dividing these, we get , which is just less

Advanced learners: Use open questions to explore division of mixed numbers use with Example 2

Targeted instructional strategies

Encourage students to deepen their understanding by engaging them in open-ended questions related to dividing mixed numbers. For example, ask them, “If the quotient of two mixed numbers is two mixed numbers be?”

, what could the

Allow students to create different pairs of mixed numbers that, when divided, result in this quotient. This strategy promotes critical thinking as they experiment with various combinations and discover patterns in the relationships between the dividends and divisors. Additionally, invite them to analyze how changing the numerator or denominator of the fractions affects the quotient. By allowing students to choose their own numbers and encouraging them to explore and discuss their findings, you support them in developing a deep

Stronger and clearer each time

use with Example 2

English language learner support Ask students to individually write a response to the question, “How does the quotient compare to the dividend when dividing a proper fraction, improper fraction, or mixed number by a fraction between 0 and 1?” Students can refer to example 1 part (a), example 2 part (b), and an example like

to formulate their answers.

Encourage them to use mathematical vocabulary they’ve learned, such as “quotient,” “reciprocal,” “dividend,” and “divisor.” Then, pair students up and have them share their explanations with a partner, listening carefully and asking clarifying questions. Prompt them to consider if dividing by a fraction between 0 and 1 results in a number larger or smaller than the original number and why that happens. After discussing with their partner, have students rotate to a new partner to share and refine their ideas again. Then rewrite the division as multiplication.

Finally, have students return to their original written explanation and revise it, making their reasoning stronger Divide improper fractions and language clearer based on the feedback and new ideas they have gathered from their peers. This iterative process will help students deepen their understanding of fraction division while developing their mathematical Multiply by the reciprocal language proficiency. Multiply the numerators and denominators Evaluate

Students: Page 44 Example 3 A

meter long roll of fabric is to be cut into sections of equal length

. How many pieces of fabric will there be?

Create a strategy Rewrite the mixed numbers as improper fractions then perform division.

Apply the idea

Reﬂect and check Rewrite as improper fractions

To check the reasonableness of our answer, let’s use estimation. We can round the length of the fabric roll

Multiply by the reciprocal of

to 8 meters, and the sections we’re cutting to 3 meters long each. Dividing these, we get , which is just less

Write as a single fraction Divide out the common factor

than 3 because

. This estimation shows that 3 is a

reasonable answer.

Evaluate the division There will be 3 pieces of fabric.

Idea summary

1.05 Divide fractions and mixed numbers To divide one fraction by another, multiply the ﬁrst fraction by the reciprocal of the second. mathspace.co

Purpose Show students how to divide mixed numbers by converting them into improper fractions. Reflecting with students Encourage students to verify their final solution for reasonableness by using multiplication. After finding that there are 3 pieces, they should multiply

to confirm it matches the original roll length. A precise

response includes units at every step and verification through multiplication, while an imprecise response might omit units and skip the checking process. By modeling precise language and thorough verification, you help students develop a habit of mathematical precision.

Visual fraction models to support understanding of dividing mixed numbers use with Example 3

Student with disabilities support

For students who struggle with conceptual processing and visual-spatial reasoning, use visual fraction models Then rewrite the division as multiplication. to enhance their understanding of dividing mixed numbers. Provide students with length models, such as strips of paper or ribbon, representing

Divide improper fractions

meters of fabric. Then, have them use smaller strips representing

by thefits reciprocal meters to measure how many times Multiply this length into the larger piece. Multiply numerators and denominators Remind students that, for division, we want the to find how many groups of

fit into

. Encourage students

to physically place the smaller strips Evaluate along the larger one to see that it fits exactly three times. This hands-on activity helps students visualize the division process and makes the abstract calculation more concrete.

Example 3

Individually dividing common parts of mixed numbers Address student A meter long rollmisconceptions of fabric is to be cut into sections of equal length

use with Example 3

. How many pieces of fabric will there be?

When dividing mixed numbers, one common mistake is to divide the whole numbers separately from the Create a strategy fraction parts, rather than properly converting the mixed number to an improper fraction. Rewrite the mixed numbers as improper fractions then perform division.

For example,

Apply the idea

checkmust be divided separately Misconception: Reﬂect Whole and numbers To check the reasonableness of our answer, let’s use Rewrite as improper fractions from fraction parts estimation. We can round the length of the fabric roll

Multiply by the reciprocal Wrongofquotient to 8 meters, and the sections we’re cutting to 3 meters long each. Dividing these, we get , which is just less

Ensure that students haveWrite reviewed the skill of converting mixed numbers to improper fractions so they do not as a single fraction than 3 because . This estimation shows that 3 is a make this mistake. Divide out the common factor

reasonable answer.

Evaluate the division

Students:There Page will 44 be 3 pieces of fabric.

Idea summary To divide one fraction by another, multiply the ﬁrst fraction by the reciprocal of the second.

Practice What do you remember? 1

Is each statement true or false? a

The reciprocal of a whole number is the whole number over 1.

The reciprocal of a fraction can be found by swapping the numerator and denominator

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

To divide one fraction by another, multiply the second fraction by the reciprocal of the ﬁrst.

Rewrite each expression using multiplication:

b a Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 44

Mathspace Virginia SOL Grade 6 mathspace.co

Practice Students: Pages 44–47

What do you remember? 1

Is each statement true or false? a

The reciprocal of a whole number is the whole number over 1.

The reciprocal of a fraction can be found by swapping the numerator and denominator

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

To divide one fraction by another, multiply the second fraction by the reciprocal of the ﬁrst.

Rewrite each expression using multiplication: b

This picture represents 5 chocolate bars. Exactly how many

Use the fraction bars to evaluate How many

are in

d are in 5?

The area models show

and . Each model has been divided further to create equal sized pieces.

Use the model to divide

. Fill in the blanks:

In , there are ⬚ groups of

and

of another group. Therefore,

1.05 Divide fractions and mixed numbers mathspace.co

Let’s practice 6

Consider the problem

Use the circle to represent

Split the model you made in part (a) into groups of

Use the model to divide

Is the quotient greater than, less than, or equivalent to the ﬁrst fraction?

Consider the problem

Use the fraction bars to represent .

Use the model to divide

by .

Use the number lines to divide each expression. Express your answer in simplest form. a 0

b 0

2 5

1 5

3 5

4 5

7 5

6 5

8 5

9 5

11 5

c 0

1 3

2 3

1 3

2 3

Use the area models to divide each expression. Express your answer in simplest form. a 5 3 1 6

b 7 6 1

1 3

1 4

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

c 1

5 9

7 5

d 1

1 7

2 3

Evaluate each of the expressions: i

iii

Use your answers to select the true statement: A When dividing any number by a proper fraction, the quotient is always larger than the original number. B When dividing any number by a proper fraction, the quotient is sometimes larger than the original number. C When dividing any number by a proper fraction, the quotient is never larger than the original number.

Divide each expression. Express your answer in simplest form. a

Divide each expression, giving your answer as a proper fraction or an improper fraction: a

Divide each expression, giving your answer as a mixed number: a

1.05 Divide fractions and mixed numbers mathspace.co

Let’s extend our thinking 16

When

Athina thinks that the quotient of

is divided by a number, the quotient is . Explain how to ﬁnd the number. is less than the quotient of

Mikki thinks the quotient of both expressions will be equivalent. Who is correct? Explain your thinking. 18

Consider the expression

Without evaluating, will the quotient be greater than, less than, or equivalent to 19

Evaluate

Benjamin wants to engrave his name on a metal plate. There is

There are three sizes of letters to choose from: a

cm wide,

cm space on the plate.

cm wide and

cm wide.

Find the maximum number of letters that Benjamin can engrave on the plate using the following sizes: i

? Explain your thinking.

cm wide

Which letter sizes could he use for his name?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

iii

cm wide

Answers

3 groups

1.05 Divide fractions and mixed numbers

What do you remember? 1 a False

b True

c True

d False

2 a

4 groups 0

3 4 4

5 There are 2 groups of Therefore,

and

of another group. 9 a

6 a

3 pieces

c We can see that there are 3 equal groups of So,

30 pieces

d The quotient is greater than the ﬁrst fraction. Let’s practice 7 a

28 pieces

30 pieces

b c

c 2

70 pieces

63 pieces

14 pieces

8 a

10 a i

iii

b A

2 groups 0

24 pieces

11 a

Answers mathspace.co

12 a

17 Both Athina and Mikki are incorrect.

e 9

13 a

14 a

15 a 1

, which is greater than

18 By looking at the first fraction and the reciprocal of the second fraction, we can see that the resulting numerator and denominator would be greater than 100. This would mean that the quotient will be greater than 1, which is greater than the first fraction. 19 We first solve

Now, we divide the quotient by the third fraction.

Let’s extend our thinking

Thus,

16 Dividing by a number is the same as multiplying by the reciprocal of the number. Let the number be n, Then,

20 a i 12 letters

which can be simpliﬁed to

Comparing the numerators, we see that 15 ÷ 5 = 3. Similarly for the denominator, 4n ÷ 5 = 4. We know that 20 ÷ 5 = 4, which means n = 5. Therefore,

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b cm and

. ii 9 letters cm wide

iii 6 letters

1.06 Solve problems with fraction operations Subtopic overview Lesson narrative In this lesson, students will learn to solve problems involving all four operations with fractions. The lesson includes real-world scenarios such as recipes, fabric measurements, and ribbon usage. Students will identify keywords to determine the correct operation, use estimation for reasonableness, and apply strategies to add, subtract, multiply, and divide fractions. They will practice with mixed numbers, improper fractions, and visual models, reinforcing their problem-solving skills. By the end, students should confidently solve practical problems involving fractions.

1.06 Solve problems with fraction Learning objectives operations Students: Page 48

After this lesson, you will be able to... • estimate solutions to real-world problems involving fraction operations. • solve real-world problems involving fraction operations. • justify solutions to real-world problems involving fraction operations with clear explanations.

Solve problems with fraction operations

Key vocabulary We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in 

fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition. estimate We can use keywords to help us work out which operation we need to use to solve the problem. Here are the four operations and some common keywords that relate to them:

Essential understanding

Addition Subtraction Multiplication Division Determining the correct operation is essential to writing expressions to solve contextual problems with real numbers. more less product equally shared add subtract by in each all together how many left times per total difference groups of divided by This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Standards

Estimation can be a useful strategy for solving real-world problems, especially if the context of the problem doesn’t

Mathematical process goals require us to be exact.

MPG1 — Mathematical Problem Solving Teachers can integrate this goal into their instruction by presenting students with single and multistep word problems Example 1 involving fraction and mixed number operations. Encourage students to apply problem-solving strategies such as drawing diagrams, using number lines, or creating tables to work through the problems. Allowing students to work At a party, Bill makes a drink by combining of water with juice concentrate. through these problems can help develop their problem-solving skills. Teachers can also encourage students to create solve ownof problems based on real-world scenarios. Teachers should model effective practices, Whatand is the totaltheir amount the drink? such as multiplying by the reciprical of the fraction when performing fraction division, converting mixed numbers to improper Create a fractions strategyand validating solutions. By guiding students to understand, plan,execute, and review their problem-solving processes, teachers can develop students’ abilities to estimate, justify, and arrive at reasonable Identify the keyword in the story. The word “total” tells us we need to add the amounts for each part of the drink. solutions in real-world contexts. 1.06 Solve problems with fraction operations mathspace.co

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can integrate this goal by having students share their problem-solving strategies and justifications with their peers. Encourage students to use the language of mathematics, including specialized vocabulary and symbolic notation, to explain their reasoning and the steps they took to solve the problems.

Teachers can integrate this goal by highlighting realworld examples and applications of fraction and mixed number operations, such as in cooking, shopping, and measuring. This helps students see the relevance of their mathematical skills to everyday life and allows them to make connections between the math they are learning and the world around them.

MPG3 — Mathematical Reasoning Teachers can foster mathematical reasoning by guiding students through contextual problems involving fraction operations. This lesson helps students make and test predictions, justify their problem-solving steps, and evaluate their conclusions. Teachers should encourage students to think about the operations involved, such as how the meaning of the operation applies to the problem. By explaining their reasoning and discussing their thought processes, students gain a deeper understanding of operations and improve their strategic problem-solving skills, moving beyond reliance on key words.

MPG5 — Mathematical Representations Teachers can integrate this goal by encouraging the use of various models to represent conextual problems involving operations with fractions and mixed numbers. These can include fraction bars, number lines, area models, or students’ own models. Teachers can also connect these models to real-world examples and contextual problems to help students visualize and better understand the problem. This can also facilitate discussion among students about the connections between different models and how they each help to represent and solve each contextual problem.

Content standards 6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context. 6.CE.1d — Estimate, determine, and justify the solution to single-step and multistep problems in context that involve addition and subtraction with fractions (proper or improper) and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Steps in solving problems Targeted instructional strategies Here are some general steps to guide student in solving math problems: • Read the problem carefully and understand what it is asking for. • Identify the known and unknown quantities in the problem. • Choose the appropriate mathematical operations to use to solve the problem. • Write out the problem in mathematical notation, and work through the problem step-by-step. • Check your answer to make sure it makes sense and is in the correct units. • If the answer does not seem reasonable, go back and check your work or try a different method to solve the problem. Some students may benefit from having a problem-solving checklist displayed on an anchor chart in the classroom. Or other students might prefer to make their own checklists next to each problem, to ensure they are not missing any steps.

Collect and display English language learner support Ask students to work in pairs and listen for and collect words and phrases as keywords that indicate the mathematical procedures required to solve problems involving operations on fractions. Consider grouping language and display these on an anchor chart for students to recognize the different terms they may encounter. Addition Subtraction Multiplication Division

plus, add, sum, total, combined, altogether minus, subtract, difference, take away, less, fewer times, multiplied by, product of, twice, thrice divided by, shared equally, split into, in each

It’s important to note that this list is not all-inclusive and different students may use or find different terms. As they identify new keywords, allow them to continue to update the chart.

1.06 Solve problems with fraction operations mathspace.co

Student lesson & teacher guide Problem solving with fractions In the previous subtopics, students have solved problems involving fractions. This includes being able to add, subtract, multiply, and divide fractions. This lesson focuses on the application of the operations of fractions in real life. Keywords are revisited to help students work out which operation they need to use to solve the problem.

Students: Page 48

1.06 Solve problems with fraction operations After this lesson, you will be able to... • estimate solutions to real-world problems involving fraction operations. • solve real-world problems involving fraction operations. • justify solutions to real-world problems involving fraction operations with clear explanations.

Solve problems with fraction operations We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition. We can use keywords to help us work out which operation we need to use to solve the problem. Here are the four operations and some common keywords that relate to them: Addition

Subtraction

Multiplication

Division

less

product

equally shared

add

subtract

in each

all together

how many left

times

per

total

difference

groups of

divided by

Estimation can be a useful strategy for solving real-world problems, especially if the context of the problem doesn’t require us to be exact.

Example 1

Use the STEAM cycle with fraction operations At a party, Bill makes a drink by combining

Targeted instructional strategies

of water with

juice concentrate.

What is the total amount of the drink?

Present students with the scenario of modifying a recipe to serve a different number of guests, involving fractional amounts of ingredients. Create a strategy Identify the keyword in the story. word “total” tells us wequantities need to addof the amounts for each part of the drink.need to Ask: Begin by asking students how The they might adjust the ingredients in a recipe if they serve more or fewer people than the original recipe specifies. Encourage them to define the problem, consider possible reasons for adjustments, and explore different perspectives on scaling recipes.

Imagine: Facilitate a brainstorming session where students generate ideas on how to adjust the ingredient amounts using fractions. Encourage them to use visual models or manipulatives, such as fraction tiles or diagrams of measuring cups, to predict how changing the quantities will affect the recipe.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 48

Mathspace Virginia SOL Grade 6 mathspace.co

1.06 Solve problems with fraction operations Plan: Guide students to create a step-by-step plan for adjusting the recipe. Help them establish criteria (e.g., maintaining the recipe’s proportions) identify After this lesson, you willand be able to... constraints (e.g., the total amount of an ingredient available). Assist them in determining the data needed,problems such asinvolving the original • estimate solutions to real-world fractioningredient operations. amounts and the desired number of • solve real-world problems involving fraction operations. servings. • justify solutions to real-world problems involving fraction operations with clear explanations.

Create and test: Support students as they calculate the new quantities for each ingredient, using fraction operations to adjust the amounts. Have them create a visual representation, like a scaled-up or scaled-down ingredient list, and, if possible, test their calculations by preparing the recipe or simulating the measurements Solve problems with fraction operations with manipulatives. We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in

Improve:fractions Encourage students to reflect onthetheir bythe asking questions about the accuracy of a cup. If we wanted to know totalresults volume of ingredients, we could use fraction addition.and practicality of their adjusted recipe. Facilitate collaboration to refine their calculations and justify any changes. Prompt them We can use keywords to help us work out which operation we need to use to solve the problem. Here are the four to communicate revised plans clearly,that perhaps creating a final, adjusted recipe card. operationstheir and some common keywords relate toby them: By guiding students through this STEAM cycle, you help them apply fraction operations to a real-world context, Addition Subtraction Multiplication Division enhancing their understanding and problem-solving skills inequally adjusting recipes. more less product shared add

subtract

in each

all together

how many left

times

per

difference

groups of

divided by

total Examples

Estimation can be a useful strategy for solving real-world problems, especially if the context of the problem doesn’t

Students:require Pages 48–49 us to be exact. Example 1

At a party, Bill makes a drink by combining

of water with

juice concentrate.

What is the total amount of the drink?

Create a strategy Identify the keyword in the story. The word “total” tells us we need to add the amounts for each part of the drink.

Apply the idea Add the values Split the mixed numbers into whole and fraction parts Add the whole parts Multiply for a common denominator Evaluate the multiplication Add the numerators over the common denominator 48

Mathspace Virginia SOL Grade 6 mathspace.co

Evaluate the addition Rewrite as a mixed number

Reﬂect and check We could use estimation to see if our exact answer seems reasonable. For example,

is close to

, and if we add

we get 7. So our answer should be a little less than 7, which it is.

Example 2 Jack is making bags for his friends. He has If each bag requires

of fabric.

of fabric, how many bags can he make?

Express your answer as an improper fraction.

1.06 Solve problems with fraction operations mathspace.co

Create a strategy Identify the keyword in the story. The word “each” tells us we need to divide the length of fabric into equal sized

Multiply for a common denominator Evaluate the multiplication Add the numerators over the common denominator

Purpose Evaluate operations, the addition specifically addition, to solve real-world problems. Demonstrate to students how to apply fraction Rewrite as a mixed number Reflecting with students Encourage students to include units of measure at each step of their calculations to maintain mathematical Reﬂect and check precision. Additionally, ask them to evaluate the reasonableness of their answer. One way to do this is by We the couldtotal use estimation seehelps if our exact seemsanswer reasonable. For example, is close to , and if we add estimating amount. to This verifyanswer that their is sensible and precise. we get 7. So our answer should be a little less than 7, which it is.

Students: Page 49 Example 2 Jack is making bags for his friends. He has If each bag requires

of fabric.

of fabric, how many bags can he make?

Express your answer as an improper fraction.

Create a strategy Identify the keyword in the story. The word “each” tells us we need to divide the length of fabric into equal sized pieces for each bag.

Apply the idea Divide the values Rewrite as multiplication using the reciprocal Multiply the numerators and denominators separately Simplify

Purpose Demonstrate to students how to apply fraction operations, specifically division, to solve real-world problems. Expected mistakes Students may see the word “each” and think that indicates multiplication. Remind multiplication 1.06 Solve problemsstudents with fractionthat operations 49 mathspace.co represents repeated addition. In this scenario, we have a total of

of fabric, and we use

m to make a single bag. Ask them what

operation this represents and what operation represents making several bags (repeated subtraction is represented by division).

100

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Advanced learners: Challenge students to write a multiplication problem use with Example 2

Targeted instructional strategies

Encourage advanced students to deepen their understanding of fraction operations by having them rewrite the original word problem so that it requires multiplication to solve. For instance, they can restate the problem by specifying the number of bags Jack wants to make and asking how much fabric he will need in total.3. Interpret the result. A result within the healthy range indicates a normal body temperature, while a result outside this range suggests a potential health issue. One example is: “Jack only has enough fabric to make

bags for his friends. If each bag requires

fabric, how much fabric does he have?” Encourage students to solve their new problem and verify that it aligns with the original scenario (by showing that Jack has

of fabric).

This exercise challenges students to think critically about how division and multiplication are related and how word problems can be structured differently yet involve the same numerical relationships. By engaging in this activity, students will enhance their ability to translate between mathematical operations and deepen their conceptual understanding of fractions and their applications.

Students: Page 50 Example 3 Jamal has

of ribbon. After using some for a project, he has

left. How much ribbon did he use?

Create a strategy To ﬁnd out how much ribbon Jamal used, we need to subtract the amount of ribbon he has left from the total amount he started with. This means we will subtract two fractions:

Apply the idea First, we ﬁnd a common denominator for the two fractions. We can do this by multiplying the denominators together which gives us 6 ⋅ 5 = 30. Multiply for common denominator Evaluate the multiplication Subtract the numerators Jamal used

of ribbon for his project.

Idea summary

Purpose Use keywords to help you identify which operation to use: Demonstrate to students how to apply fraction operations, specifically subtraction, to solve real-world problems. Addition more add all together total

Subtraction less subtract how many left difference

Multiplication product by times groups of

Division equally shared in each per divided by

Practice What do you remember? 1

Evaluate and simplify: a

Determine the operation required to solve each contextual problem.

1.06 Solve problems with fraction operations d mathspace.co h

101

Support visual-spatial processing using a number line

use with Example 3

Student with disabilities support To support students with visual-spatial processing difficulties, use a number line to represent the fractions in the problem visually. Provide fraction bars divided into sixths and fifths that correspond to the denominators in the problem. Begin by helping students convert the fractions to have a common denominator. This can be done by dividing the fifths into 6 equal parts and dividing the sixths into 5 equal parts. This will create 30 total parts in one whole. Then, ask them to determine the equivalent fraction for , which is Example 3

left. How much ribbon did he1 use?

0 After using some for a project, he has of ribbon.

Jamal has

Create a strategy

Have students shade in ribbon on theJamal sixths barwetoneed represent thethe total ribbon Jamal Then, To ﬁnd out how much used, to subtract amount of ribbon hehas. has left from on the the totalsame amountbar converted to a common have them represent . he started with. This denominator, means we will subtract two fractions:

to show the ribbon Jamal has left. Guide

students to determine how many thirtieths Jamal used by counting the number of parts between Apply the idea

and .

First, we ﬁnd a common denominator for the two fractions. We can do this by multiplying the denominators together 1 which gives us 6 ⋅ 5 0 = 30. Multiply for common denominator

By manipulating the fraction models, students can more easily understand the scenario represented in the word Evaluate the multiplication problem. Subtract the numerators used Students:Jamal Page 50

of ribbon for his project.

Idea summary Use keywords to help you identify which operation to use: Addition more add all together total

Subtraction less subtract how many left difference

Multiplication product by times groups of

Division equally shared in each per divided by

Practice What do you remember? 1

Evaluate and simplify: a

Determine the operation required to solve each contextual problem. of a mile and runs

of a mile to get to school. How many miles has Oana traveled?

Oana walks

Webster felt dehydrated. He drank one third of a cup. What fraction of a cup does Webster have left?

Shea has

pounds of beads. She shares the beads evenly between 3 friends. How many pounds of beads

will each friend receive? d 50

102

In a class with 40 students,

Mathspace Virginia SOL Grade 6 mathspace.co

of the class are boys. How many boys are in the class?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice Students: Pages 50–53

What do you remember? 1

Evaluate and simplify: a

Determine the operation required to solve each contextual problem. of a mile and runs

of a mile to get to school. How many miles has Oana traveled?

Oana walks

Webster felt dehydrated. He drank one third of a cup. What fraction of a cup does Webster have left?

Shea has

pounds of beads. She shares the beads evenly between 3 friends. How many pounds of beads

will each friend receive? d 3

In a class with 40 students,

of the class are boys. How many boys are in the class?

A chemistry experiment requires

cup of distilled water, but Trace only has

cup. Write an expression to

represent how much more distilled water Trace needs for the experiment. 4

A food market is open for

hours in the morning and

hours in the evening. Write an expression to

represent the number of hours the food market is open altogether. 5

Selena has

gallons of paint. She plans to use

of the paint for an art project and the rest for a sciene project.

Write an expression to represent the amount of paint Selena plans to use for the science project. 6

A recipe requires cookies? A

of a cup of ﬂour for one batch of cookies. How much ﬂour is needed to make 8 batches of

cups of ﬂour

A glass holds

cups of ﬂour

of a cup of lemonade. If you have 3 cups of lemonade, about how many glasses can be ﬁlled

with lemonade? A 8

3 glasses

Melinda has

5 glasses

of a quart of milk in her fridge. After drinking

2 glasses

6 glasses

of a quart of milk, how much milk is left in

Melinda’s fridge? Express your answer in simplest form.

1.06 Solve problems with fraction operations mathspace.co

103

Let’s practice 9

For each scenario: i

Estimate the solution.

Eboni has

Find the exact solution in simplest form.

m of fabric. After she cuts off some fabric for a dress, she has

m left.

About how much fabric did she use for the dress? A bag of oranges weighing

the oranges given to the

lb was divided equally among

a group of children. What was the mass of

group?

For a school bake sale, Farouk prepares a chocolate mix by mixing

pounds of chocolate powder with

pounds of sugar. Find the total amount of pounds in the mix. Helena takes

minutes to drive from her home to the local shopping center. She spends

of this time

waiting at traﬃc lights. Find the number of minutes she spends waiting. 10

A casual cyclist travels at a speed of

miles per hour uphill. The uphill route is

miles long. Find the total

hours the cyclist took to cycle the uphill route. 11

Mayumi works as an accountant and is at the oﬃce for

hours each day. She spends

hours eating lunch

with her coworkers. How much time does she spend actually working in one day? hours

A 12

hours

Paolo is starting his jewelry business. He bought

hours

bags of beads. If a single bag weighs

8 hours ounces, what is

the total combined weight of the beads Paolo bought? 13

In a school debate, each speaker was allowed for

minutes. How many speakers participated if the debate went

minutes?

Xia is making bags for her friends. She has bags can she make?

yards of fabric. If each bag requires

If a hiker has

quarts during a break, how much water is left in the hiker’s bottle?

of a cake was left after a party. Bella ate

16 17

quarts of water and drinks

A recipe calls for

yards of fabric, how many

of what was left. What fraction of the whole cake did Bella eat?

cup of sugar, but Wei only has

cup of sugar. How much more sugar does Wei need to

complete the recipe? Is your answer reasonable? Justify your solution. 18

At the beginning of the day, Qing’s 32 oz water bottle was full. She drank some of the water before lunch, which left the bottle only

104

full. After lunch, she drank

of what was left, then reﬁlled the bottle.

How many ounces of water did Qing drink before lunch?

How many ounces of water did she need to add to the bottle to reﬁll it after lunch?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

During a school science fair, Quentin needs to complete two experiments. The ﬁrst experiment requires 24 minutes to set up, and the second experiment takes time for each experiment by experiments.

A baker has 960 grams of ﬂour.

minutes. If Quentin manages to reduce the setup

minutes to save time for analysis, calculate the new total setup time for both of the ﬂour is needed to make cookie dough. If

times the amount of ﬂour

needed for the dough is needed for frosting and decorations, does the baker have enough ﬂour for both the cookie dough and the frosting and decorations? Explain your answer. 21

Ursula tutors students in mathematics for

hours each weekday. On Wednesdays, due to additional

commitments, she tutors for only a third of her usual weekday time.

Calculate the total tutoring hours for Ursula from Monday to Friday, considering her reduced hours on Wednesday.

If Ursula were to cut her hours in half on Mondays, how many hours would she work in one week?

Ayako is organizing a 50-minute workout session. The session includes two types of exercises: one lasts minutes and the other lasts

minutes.

Explain how to ﬁnd the total duration for a routine consisting of three sets of the ﬁrst exercise and four sets of the second exercise.

If the remainder of the time is spent on rest breaks or water breaks, how much time is spent resting or drinking water?

Let’s extend our thinking 23

Xander is planning a fundraising event for his school. He sold additional

of the original number of tickets. If the initial number of tickets was 120:

Find the total number of tickets sold.

Find the number of tickets he bought.

After selling and buying tickets, he realized he needed more, so he bought How many tickets does he have now?

of the current total tickets.

For a project, Akram needs to cut a metal rod so that the longer piece is three times the length of the shorter piece. If the rod is

of the tickets he had and then bought an

yards long, explain how to ﬁnd the length of the shorter piece.

In Fady’s Bakery, one recipe of blueberry muﬃns serves 4 people and requires

cups of sugar. If Fady is

preparing to serve 22 people for a dinner party, how much sugar will he need? 26

Myoung is planning a study session for her ﬁnals. The session is divided between two subjects: the ﬁrst requires

minutes per review cycle, and the second needs

Myoung does three review cycles for the ﬁrst subject, takes a second subject, then takes a practice test for including the break and the test.

minutes per review cycle. hour break, does two review cycles for the

hours. Calculate the overall duration of her study session

1.06 Solve problems with fraction operations mathspace.co

105

Zainab explains that if each juice box contains that amount of juice. She multiplies

cup by

cup of juice, then a pack of 6 juice boxes will contain

times

to ﬁnd the total amount of juice in the pack.

Dwight disagrees and says that 6 juice boxes will contain 6 times that amount of juice. He multiplies to ﬁnd the total amount of juice.

cup by 6

Who is correct? Justify your answer. 28

Create a contextual problem with both multiplication and division, or both addition and subtraction. Use fractions for the quantities.

106

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

14 8 bags

1.06 Solve problems with fraction operations

quarts of water

What do you remember? 1 a

d 6

17 Answers may look like: To ﬁnd out how much more sugar Wei needs, we need to

e 3

2 a Addition

b Subtraction

c Division

d Multiplication

subtract

from :

Therefore, Wei needs recipe.

cup more sugar to complete the

4 Total hours the market is open

This answer is reasonable because

cup is less than half

a cup, which is the amount of sugar Wei already has. It makes sense that she would need less than what she already has in order to complete the recipe.

5 cups of ﬂour

6 C

18 a

7 B 5 glasses

19 51 minutes

quarts of milk

20 No, the baker does not have enough ﬂour. Let’s practice 9 a i is approximately 1 and

is around a half, so we

can estimate the solution to be

of 960 grams

is 640 grams of ﬂour. times 640 grams is 960 grams. This means they would need a total of 1600 grams, so they do not have enough ﬂour. 21 a

ii b i is approximately 1, so we can estimate the solution to be ii

. or

c i

is approximately 3 and

is approximately 3.

We can estimate the solution to be 3 + 3 = 6. ii

pounds or

pounds d i

is approximately 6, so we can estimate the

solution to be ii 10

hours.

minutes or

minutes

22 a T o determine the total duration of the routine, calculate the individual duration for each type. For the ﬁrst exercise, the total duration is 3 ×

. For the second

exercise, the total duration is

. Combining the

two products provides the total duration.

11 A 12

minutes

13 13 speakers

Answers mathspace.co

107

Let’s extend our thinking

27 Answers may look like:

23 a 90 tickets

Dwight is correct because if each juice box contains cup of juice, then a pack of 6 juice boxes will contain 6 times that amount of juice.

b 48 tickets

c A fter selling and buying tickets, Xander had a total of 120 − 90 + 48 = 78 tickets. He then bought × 78 = 29.25 more tickets. However, since he cannot

We can multiply

buy a fraction of a ticket, he might have bought either 29 or 30 tickets. Hence, he now has either 78 + 29 = 107 or 78 + 30 = 108 tickets.

juice in the pack.

24 We know the longer piece is equal to three groups of the shorter piece. So if we have one short piece, and then three groups of the short piece to equal the longer piece, we need to ﬁnd the total length of the rod, divided into four equal groups. Since the rod is piece is: 25

yards long, the length of the shorter yards.

cups

26 Overall duration of the study session

108

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

cup by 6 to ﬁnd the total amount of

Amount of juice in the pack =

cup × 6 boxes = 4 cups

So, there will be 4 cups of juice in the pack. 28 Answers may look like: Erina is planning to place ﬂower pots on her balcony. She estimates that about

of the balcony will be occupied.

She adds decorations around each ﬂower pot, estimating that this would contribute an additional

to the occupied

part of the balcony. What is the total portion occupied? If she later decides to remove

of the ﬂower pots to

create more space, what will be the remaining occupied portion?

Topic 1 Assessment: Operations with Fractions 1

The rectangle of blocks represents what happens when ⋅ .

Use the blocks to evaluate

The following rectangle is broken into two halves: Which model could be used to calculate

⋅ 12

What is the product of

⋅

and

Carl has

A 5

÷ 2?

Find the value and express in simplest form: a

is multiplied by .

m of ribbon. After he uses some ribbon for a present, he has

m left. How much ribbon did he use

on the present?

Carl used

m of ribbon.

Carl used

m of ribbon.

Carl used

m of ribbon.

Carl used

m of ribbon.

Danielle is studying for three tests. She studies Science for for

hours, English for

of an hour and Visual Arts

of an hour. How many hours does she spend studying?

Lea is working on projects that require

yards of fabric per project. Lea has 32 yards of fabric. What is the

greatest number of projects that Lea can complete with this fabric? A

138

A bag of pears weighing child?

Sarah baked a batch of cookies, and the recipe calls for

kg was divided among 5 children. What was the mass of the pears given to each cup of sugar. She wants to make a smaller batch

and uses only cup of sugar. Explain how multiplying the original recipe’s sugar amount by making a smaller batch of cookies.

is related to

Topic 1 Assessment: Operations with Fractions mathspace.co

109

Performance Task 10

Consider the following diagram. n

Explain how the diagram below represents both of the equations below. ?÷

110

= n

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

?⋅

Performance Task

Answers Topic 1 Assessment: Operations with Fractions

so ? is

10 The shaded parts (which represents ?) is 4 of 7 equal parts of the entire diagram (which represents the unknown n), of n. To ﬁnd n the value of the whole rectangle,

we can “undo” the

6.CE.1a

using division. This equation can be

re-written as ? ÷ 7 = n.

2 B

We can also see the entire model as

6.CE.1a 3 a 9

6.CE.1b

of the shaded part,

since the shaded part is divided into 4 equal pieces and the entire rectangle is 7 of those pieces. So the diagram also represents ? ⋅

= n.

6.CE.1a, 6.CE.1b, MP2, MP3, MP5

4 C 6.CE.1b 5 B 6.CE.1d hours

6.CE.1d 7 D 6.CE.1e 8

kg 6.CE.1e

9 Answers will vary. Sample answer: When Sarah multiplies the original recipe’s sugar amount by , she is using only half the amount of sugar, which is related to making a smaller batch of cookies. 6.CE.1c

Answers mathspace.co

111

2 Fractions, Decimals, & Percents Big ideas • Percents are useful for comparing a quantity to a whole amount. • Real numbers are either rational or irrational.

Chapter outline 2.01 2.02 2.03 2.04

Percents as fractions (6.NS.1) Percents as decimals (6.NS.1) Convert between fractions, decimals, and percents (6.NS.1) Compare and order fractions, decimals, and percents (6.NS.1)

116 133 142 153

Only 1 % of the world’s water is fresh and accessible. That’s a tiny fraction!

2. Fractions, Decimals, & Percents Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 2.NS.3 — The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into equal-sized parts (halves, fourths, eighths, thirds, and sixths).

5.NS.1 — The student will use reasoning and justification to identify and represent equivalency between fractions (with denominators that are thirds, eighths, and factors of 100) and decimals; and compare and order sets of fractions (proper, improper, and/or mixed numbers having denominators of 12 or less) and decimals (through thousandths).

6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

Big ideas and essential understanding Percents are useful for comparing a quantity to a whole amount. 2.01 — Percent means “per 100” and can be represented as a fraction with a denominator of 100. 2.02 — 100% is equivalent to 1 whole. This relationship can be used to convert between decimals and percents. 2.03 — 100% is equivalent to 1 whole or

2.04 — 100% is equivalent to 1 whole or . This relationship can be used to convert between fractions, decimals, and percents. Real numbers are either rational or irrational. 2.04 — Comparing rational numbers in different forms can easily be done by converting them first to the same form.

. This relationship can be used to convert between fractions, decimals, and percents.

Standards 6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents. 6.NS.1a — Estimate and determine the percent represented by a given model (e.g., number line, picture, verbal description), including percents greater than 100% and less than 1%.* 2.01 Percents as fractions 114

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.NS.1b — Represent and determine equivalencies among decimals (through the thousandths place) and percents incorporating the use of number lines, and concrete and pictorial models.* 2.02 Percents as decimals

6.NS.1e — Use multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100), with and without models. Justify solutions orally, in writing 6.NS.1d — Represent and determine equivalencies or with a model. Ordering may be in ascending or among decimals, percents, fractions (proper or improper), descending order.∗ and mixed numbers that have denominators that are 12 or 2.04 Compare and order fractions, decimals, and less or factors of 100 incorporating the use of number percents lines, and concrete and pictorial models. 2.03 Convert between fractions, decimals, and percents 6.NS.1c — Represent and determine equivalencies among fractions (proper or improper) and mixed numbers that have denominators that are 12 or less or factors of 100 and percents incorporating the use of number lines, and concrete and pictorial models.* 2.01 Percents as fractions

Future connections 6.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs.

7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

2. Fractions, Decimals, & Percents mathspace.co

115

2.01 Percents as fractions Subtopic overview Lesson narrative In this lesson, students will learn to convert percents to fractions and vice versa. They begin by understanding that percent means “per 100” and how to represent percents using fractions with 100 as the denominator. Students will practice converting between percents and fractions, including simplifying fractions and working with improper fractions. The lesson includes interactive models, such as grids and number lines, to visualize these concepts. By the end, students should be able to confidently convert and understand the relationship between percents and fractions.

Learning objectives

2.01 Percents as fractions

Students: Page 56

After this lesson, you will be able to... • estimate and write percents from models. • convert percents to fractions. • convert fractions to percents.

Percents as fractions Percent means “one part out of every 100, “per 100” or how many “out of 100”. In other words, 1% is equal to one Key vocabulary 

hundredth or approximation



greatest common divisor (GCD)



benchmark percent

Percents canpercent be represented with a 10 × 10 grid which has 100 total squares. Each square represents 1% or 1 square out of 100 total squares. 

Essential understanding

percent of squares = number of shaded squares%

represents 6% the grid. Percent means “per 100” and can be representedSo, as 6a shaded fractionsquares with a denominator ofof100. fraction of squares =

Standards

So, the six shaded blue squares represent

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. Percents can represent amounts that are less than a whole or greater than a whole, with 100% representing one whole. • PercentsMPG3 less than a whole will beReasoning smaller than 100% and — Mathematical MPG1 — Mathematical Problem Solving represent part of a whole. They can also be represented by Teachers can encourage students to justify their Teachers can integrate this goal by presenting students fractions less than 1. 25% in writing, a larger model,than or orally with real-world problems that require the conversion of 100% • Percentsreasoning greater than a wholeusing will be 100% when and estimating and converting fractions to percents and percents to decimals and vice versa. For example, they represent a quantity greater than the original. They can also be percents to fractions. Teachers should also encourage can present a problem where students have to calculate represented by fractions greater than 1. 1 100%

Mathematical process goals

students to use appropriate key vocabulary terms, such a discount price or determine a tax rate. Teachers can 150% also create problems that involve estimating, determining, as greatest common divisor or benchmark percents, when applicable. and justifying solutions, encouraging students to apply mathematical concepts to solve these problems. Estimating percents can be helpful for quick calculations or when precise measurements are not necessary. When estimating a percent, it is helpful to be comfortable with calculating the benchmark percents of 0%, 25%, 50%, and 100%. 116

Mathspace Virginia SOL Grade 6 25% Teacher Edition mathspace.co

50%

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can highlight the connections between the current lesson and previous lessons about fractions, decimals, and place value. They can also emphasize the connection between mathematical concepts and real-world contexts, such as the use of percents in financial calculations. Representing fractions and percents on a numberline can help students connect their understanding of percents, fractions, and their equivalencies. By utilizing visual models and number lines, teachers can guide students in representing and converting percents and fractions. This approach also helps reinforce the understanding of benchmark fractions. Additionally, hands-on activities, such as using sticky notes on large number lines or creating human number lines, can support students in visualizing and comprehending the relationships between different representations, enhancing their ability to make mathematical connections.

Teachers can introduce various models (like number lines, hundred grids, fraction circles, base 10 blocks, colored counters, and bar models) to represent equivalent relationships among fractions and percents. They can demonstrate the conversion process using these models and provide practice problems requiring the use of these models. Teachers can also encourage students to use models when dealing with real-world problems, such as calculating discounts or tax rates.

Content standards 6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents. 6.NS.1a — Estimate and determine the percent represented by a given model (e.g., number line, picture, verbal description), including percents greater than 100% and less than 1%.*

6.NS.1c — Represent and determine equivalencies among fractions (proper or improper) and mixed numbers that have denominators that are 12 or less or factors of 100 and percents incorporating the use of number lines, and concrete and pictorial models.*

Prior connections 2.NS.3 — The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into equal-sized parts (halves, fourths, eighths, thirds, and sixths).

Future connections 7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

Lesson Preparation Tools You may find these tools helpful: • 10 × 10 grids • Fraction circles • Number lines • Tape diagrams

2.01 Percents as fractions mathspace.co

117

Student lesson & teacher guide Percents as fractions Students will explore the concept of percentages, focusing on understanding percents as parts of 100, and how to visually represent them using area models and grids. Students will use practical methods for converting percentages to and from fractions using visual aids and double number lines.

Students: Pages 56–57

2.01 Percents as fractions After this lesson, you will be able to... • estimate and write percents from models. • convert percents to fractions. • convert fractions to percents.

Percents as fractions Percent means “one part out of every 100, “per 100” or how many “out of 100”. In other words, 1% is equal to one hundredth or

. Percents can be represented with a 10 × 10 grid which has 100 total squares. Each square represents 1% or 1 square out of 100 total squares. percent of squares = number of shaded squares% So, 6 shaded squares represents 6% of the grid. fraction of squares = So, the six shaded blue squares represent

Percents can represent amounts that are less than a whole or greater than a whole, with 100% representing one whole. • Percents less than a whole will be smaller than 100% and represent part of a whole. They can also be represented by fractions less than 1. • Percents greater than a whole will be larger than 100% and represent a quantity greater than the original. They can also be represented by fractions greater than 1.

100%

25% 100% 1 150%

Estimating percents can be helpful for quick calculations or when precise measurements are not necessary. When estimating a percent, it is helpful to be comfortable with calculating the benchmark percents of 0%, 25%, 50%, and 100%. 25% 50% 75% 1

118

100%

Double number lines can be used with benchmark percents to represent both percents less than 100% and greater than 100%. On this number line, 100% of the quantity is 24 minutes. And we can easily see other percents by looking at the values that line up. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 56

Mathspace Virginia SOL Grade 6 mathspace.co

50% 75% 1

100%

Mathspace Virginia SOL Grade 6 0 mathspace.co time (min)

25%

50%

75%

100%

125%

150%

100%

Benchmark percents and their fraction equivalents will be helpful in converting between percents and fractions as well as estimating percents. To convert any percent to a fraction, remember percent means “per 100”. Create a fraction with the percent quantity as the numerator and 100 as the denominator. This fraction may be reduced.

To convert any fraction to a percent, there are two methods. For the ﬁrst method, create an equivalent fraction with 100 as the denominator. The numerator is the percent.

For the second method, divide the numerator by the denominator, and then multiply by 100%.

Example 1

Use a Concrete-Representational-Abstract (CRA) approach Consider the grid shown. Targeted instructional strategies Concrete: Begin by engaging your students with hands-on activities to explore percents and fractions. Provide each student with a 10 × 10 grid (a hundreds chart) that represents 100%. Give them colored tiles, counters, or markers to physically cover or shade a certain number of squares on the grid. For example, ask students to shade 25 squares to represent 25%. This concrete manipulation helps them see that percent means “per 100” and connects directly to fractions, as they observe that 25 out of 100 squares is the fraction

. Encourage

them to experiment with different percents and discuss how the shaded portions relate to fractions. a How many squares are shaded?

Representational: Transition to the representational stage by having students draw models of the grids. Provide Create strategy graph paper soa they can create accurate 10 × 10 grids and shade in squares to represent various percents. Count the shaded squares in the row and the number of rows. Ask them to label each drawing with both the percent and the corresponding fraction

Introduce number lines marked from 0 to 100 and have students plot points to represent different percents and Apply the idea their equivalent fractions. can also draw pie or bar models to visualize parts of a whole. These visual There are 5 rows of 10They squares shaded so there arecharts 50 squares shaded. representations help students make connections between the physical models and abstract concepts. 5 ⋅ 10 = 50 Abstract: Finally, guide your students to convert between percents and fractions using mathematical symbols and operations. Teach them that to convert a percent to a fraction, they can write the percent number over 100 and simplify the fraction if possible

. To convert a fraction to a percent, show them how

to divide the numerator by the denominator and multiply by 100

. Provide a variety

of practice problems for them to solve, and encourage them to explain their reasoning. This abstract stage solidifies their understanding and allows them to apply the concepts without relying on physical models. 2.01 Percents as fractions mathspace.co

2.01 Percents as fractions mathspace.co

119

Help your students make connections between all three stages to deepen their understanding. When solving abstract problems, encourage them to visualize or refer back to their drawings or the grids they shaded earlier. Ask questions like, “How does the fraction

relate to the 50 squares you shaded on the grid?” or “Can you

draw a quick sketch to help you convert this fraction to a percent?” By linking each stage, you support students in monitoring their thinking and choosing the most effective representation for each problem. This approach helps them build a strong foundation and boosts their confidence in converting between percents and fractions.

Vocabulary exercise: collect and display English language learner support As students work on converting percents to fractions and vice versa, listen for the words and phrases they use to describe the process. Collect these terms—such as “percent,” “per 100,” “numerator,” “denominator,” “simplify,” “improper fraction,” and “equivalent fractions”—and display them prominently in the classroom. Create a visual word wall or chart that pairs each term with a definition and an image or example. For instance, use a 10 × 10 grid with 25 squares shaded to represent 25%, alongside the fraction number lines showing the placement of fractions and their equivalent percents.

simplified to . Include

Encourage students to refer to this display during activities and discussions. Update the display with any new terms or student-generated explanations that arise. This visual reference will support students in connecting mathematical concepts with the vocabulary, aiding both their understanding and language development.

Assuming the numerator is always the percent value Address student misconceptions Additional InformatioStudents may mistakenly believe that to convert a fraction to a percent, they can simply take the numerator as the percent value, ignoring the denominator. For example, when converting

to a

percent, they might incorrectly think it is 3%, assuming the numerator directly represents the percent. This misconception arises from not fully grasping that “percent” means “per 100” and that the fraction must be equivalent to a fraction with a denominator of 100 to find the correct percent value. To address this misconception, encourage students to convert fractions to equivalent fractions with a denominator of 100. Show them how to find a number that, when multiplied by the denominator, results in l100, and then multiply both the numerator and denominator by that number. For instance, multiply both the numerator and denominator of

by 20 to get

, which is 60%. Use visual representations like a 10 × 10

grid where students shade in the appropriate number of squares to match the fraction. For , students would shade 60 out of 100 squares, visually reinforcing that

is equivalent to 60%. By using these visual models,

students can better understand the relationship between fractions and percents and avoid relying solely on the numerator.

120

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

To convert any fraction to a percent, there are two methods. For the ﬁrst method, create an equivalent fraction with 100 as the denominator. The numerator is the percent.

Examples For the second method, divide the numerator by the denominator, and then multiply by 100%. Students: Page 57

Example 1 Consider the grid shown.

a How many squares are shaded?

Create a strategy Count the shaded squares in the row and the number of rows.

Apply the idea There are 5 rows of 10 squares shaded so there are 50 squares shaded. 5 ⋅ 10 = 50

Purpose Show students how to use multiplication as a strategy for counting objects arranged in a rectangular pattern.

Students: Page 58 2.01 Percents as fractions mathspace.co

Purpose Show students how to convert a fraction to a percentage using a real-world scenario of finding the percentage of a grid that is shaded.

2.01 Percents as fractions mathspace.co

121

Students: Page 58

Purpose Show students that they can convert percentages to fractions and use the concept of equivalent fractions to simplify the fraction, in order to understand the relationship between a fraction and its representation on a grid model.

Modeling problem-solving through think-alouds

use with Example 1

Student with disabilities support Begin the lesson by demonstrating your thought process aloud as you solve each part of the problem. As you examine the grid, verbally explain how you count the shaded squares by identifying the number of rows and columns and using multiplication to find the total. When converting the fraction of shaded squares to a percentage, articulate each step, such as simplifying the fraction and relating it to the concept of “percent” meaning per hundred. This think-aloud approach helps students who struggle with processing multiple steps or organizing their thoughts by providing a clear, step-by-step model to follow. Encourage students to try this strategy themselves or with a partner, speaking their reasoning out loud as they work through similar problems. This technique not only supports their understanding but also builds their confidence in problem-solving.

Students: Page 58

122

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Show students how to translate visual representation of fractions into mathematical expressions, and help them understand the concept of whole numbers and fractions combined.

Students: Page 59 b Estimate the percent represented.

Create a strategy Convert the fraction to an approximate percent by considering the benchmark percent that is closest to our fraction.

Apply the idea Given that our fraction that best represented the model was

, we know the 4 whole can be written as 400%.

Visually looking at the last circle representing , this is larger than 25% = and smaller than 50% = . Since the b Estimate the percent represented. shaded part is near the middle of the two benchmarks, we can estimate this as about 38%. Therefore, the shaded portion is approximately 438%. Create a strategy Convert the fraction to an approximate percent by considering the benchmark percent that is closest to our fraction.

Reﬂect and check

When estimating, Apply the idea some variation is expected. Any estimates between 35% and 45% would be appropriate answers. Given that our fraction that best represented the model was

, we know the 4 whole can be written as 400%.

cVisually Whatlooking exact percent of one whole circle is shaded? , this is larger than 25% = at the last circle representing

and smaller than 50% = . Since the

Purposeshaded part is near the middle of the two benchmarks, we can estimate this as about 38%. Create a strategy Therefore, the shaded portion is approximately Show students how to approximate fractions as438%. percentages using benchmark values, fostering their Calculate the exact percent by dividing the number of shaded parts by the total parts in one whole circle and understanding of by the relationship between fractions and percents. multiplying 100%. Reﬂect and check

estimating, Students:When Page Apply the59 idea some variation is expected. Any estimates between 35% and 45% would be appropriate answers. Number of shaded parts: 22 parts: 25 percent of one whole circle is shaded? cTotal What exact 22 ÷ 5 = 4.4 Divide the number of shaded parts by the total parts in one circle

Create a strategy = 4.4 ⋅ 100%

Multiply 4.4 by 100%

Calculate the exact percent by dividing the number of shaded parts by the total parts in one whole circle and = 440% Evaluate multiplying by 100%.

Reﬂect and check Apply thepercent idea is very close to our approximation. Consider ways to make an even better estimate. Our exact Number of shaded parts: 22 Total parts: 25 22 ÷ 5 = 4.4

Example 3

= 4.4 ⋅ 100%

= 440% Write 24% as a fraction.

Divide the number of shaded parts by the total parts in one circle Multiply 4.4 by 100% Evaluate

Reﬂect check Create aand strategy Our exact percent is very close to our approximation. Consider ways to make an even better estimate. To convert the percent into fraction, rewrite the percent as a fraction out of 100.

2.01 Percents as fractions mathspace.co

Apply the idea

Example 3 Write 24% as a fraction.

Write the percent as a numerator over the denominator 100

123

cNumber What of exact percent of22 one whole circle is shaded? shaded parts: Total parts: 25

Create a strategy

22 ÷ 5 = 4.4 Divide the number of shaded parts by the total parts in one circle Calculate the exact percent by dividing the number of shaded parts by the total parts in one whole circle and = 4.4 ⋅ 100% Multiply 4.4 by 100% multiplying by 100%. = 440% Evaluate

Apply the idea Reﬂect and check

Number of shaded parts: 22 Our exact percent is very close to our approximation. Consider ways to make an even better estimate. Total parts: 25 22 ÷ 5 = 4.4

Divide the number of shaded parts by the total parts in one circle

= 4.4 ⋅ 100%

Multiply 4.4 by 100%

= 440% Evaluate PurposeExample 3 Show students how to calculate the exact percentage of a whole that is represented by a specific quantity. Write 24% as a fraction. Reﬂect and check Our exact percent is very close to our approximation. Consider ways to make an even better estimate.

Students:Create Pages 59–60 a strategy

To convert the percent into fraction, rewrite the percent as a fraction out of 100.

Apply the idea Example 3 Write 24% as a fraction.

Write the percent as a numerator over the denominator 100

Create a strategy To convert the percent into fraction, rewrite the percent as a fraction out of 100.

Apply the idea Write the percent as a numerator over the denominator 100 2.01 Percents as fractions mathspace.co

Reﬂect and check To simplify the fraction

, we ﬁnd the greatest common divisor (GCD) of 24 and 100, which is 4.

Dividing both the numerator and the denominator by 4, we can simplify our fraction:

2.01 Percents as fractions is 59 We can model these fractions on a grid where the denominator mathspace.co the total number of squares.

On a 10 by 10 grid representing 100%, shading 24 squares shows the original percentage.

We can create a 5 by 5 grid to represent the simpliﬁed fraction

Each square in this grid represents 4 squares from the 10 × 10 grid. Therefore, shading 6 squares corresponds to the same proportion of the whole.

This visualization helps conﬁrm the equivalence of 24% to same value or percent of the whole.

. Fractions can be simpliﬁed while representing the

Example 4 Purpose Show students to convert a percentage to a fraction and simplify it, reinforcing the concept that Write ashow a percent. percentages and fractions are different ways to represent the same value. Create a strategy We need to ﬁnd an equivalent fraction with the denominator equal to 100. What number can we multiply 5 by to get 100? Then be sure to multiply the numerator and denominator by the same factor.

Apply the idea 124

Mathspace Virginia SOL Grade 6 Teacher WhatEdition can we multiply 5 by to get a denominator of 100? mathspace.co Multiply both the numerator and the denominator by 20 Evaluate

the original percentage.

Reflecting with students Emphasize the importance of simplifying the result to its lowest terms to promote mathematical precision. After rewriting 24% as

We can create a 5 by 5 grid to represent the simpliﬁed fraction

, encourage students to identify the greatest common divisor (GCD) of the numerator and Each square in this grid represents 4 squares from the 10 × 10 grid. Therefore, shading 6 squares corresponds to the same proportion of the whole.

denominator, which in this case is 4. By dividing both the numerator and denominator by 4, they simplify the fraction to

Highlight that presenting the fraction in its simplest form not only shows attention to detail but can also make it easier to compare with other fractions. Explain that simplifying fractions is a critical step in ensuring their answers are precise and universally understood. Reinforce that expressing results in the most reduced form is a key aspect of precise mathematical communication. This visualization helps conﬁrm the equivalence of 24% to same value or percent of the whole.

. Fractions can be simpliﬁed while representing the

Students: Pages 60–61 Example 4 Write

as a percent.

Create a strategy We need to ﬁnd an equivalent fraction with the denominator equal to 100. What number can we multiply 5 by to get 100? Then be sure to multiply the numerator and denominator by the same factor.

Apply the idea What can we multiply 5 by to get a denominator of 100? Multiply both the numerator and the denominator by 20 Evaluate 40 for every 100 is 40%

Reﬂect and check When we are unsure how to get a denominator of 100 directly, we can divide 100 by the current denominator to ﬁnd the multiplying factor. 100 ÷ 5 = 20 60 Mathspace Virginia Grade 6 should be multiplied by 20 to create an equivalent fraction with a denominator So, both numerator and SOL denominator of 100.mathspace.co This method is especially helpful for converting fractions where the denominator is not a simple factor of 100.

Reﬂect and check When we are unsure how to get a denominator of 100 directly, we can divide 100 by the current denominator to ﬁnd

Example 5 Purposethe multiplying factor. 20 Show students how convert fractions to percentages by= creating an equivalent fraction with the denominator Write 250% as atomixed number in its simplest form. 100 ÷ 5 of 100. This process helps students understand the concept of percent as a ratio out of 100. So, both numerator and denominator should be multiplied by 20 to create an equivalent fraction with a denominator Create a strategy of 100. This method is especially helpful for converting fractions where the denominator is not a simple factor of 100.

Students:ToPage write a 61 percent as a fraction divide by 100. Then convert the improper fraction into a mixed number. Apply the idea

Example 5

Divide by 100

Write 250% as a mixed number in its simplest form. Simplify the fraction

Create a strategy

Convert to a mixed number To write a percent as a fraction divide by 100. Then convert the improper fraction into a mixed number.

Reﬂect and check Apply the idea

A double number line is a powerful tool for understanding the conversion between percents and other amounts. Let’s draw a double number line to represent Dividethis by relationship: 100 1 Simplify the fraction

Convert to a mixed number 100% 150% 200% 250% 300%

Reﬂect and check

2.01 Percents as fractions mathspace.co

The top number line represents the mixed number amounts. The bottom line of the double number line represents A double number line is a powerful tool for understanding the conversion between percents and other amounts. Let’s those amounts as percents. This model clearly shows that 250% aligns with , which matches our calculation. draw a double number line to represent this relationship:

125

Write 250% as a mixed number in its simplest form.

Create a strategy To write a percent as a fraction divide by 100. Then convert the improper fraction into a mixed number.

Apply the idea Divide by 100 Simplify the fraction Convert to a mixed number

Reﬂect and check A double linehow is a to powerful tool for understanding the conversion between anddenominator other amounts. Let’s When we number are unsure get a denominator of 100 directly, we can divide 100 bypercents the current to ﬁnd drawmultiplying a double number the factor. line to represent this relationship: 1

100 ÷ 5 = 20 2

So, both numerator and denominator should be multiplied by 20 to create an equivalent fraction with a denominator of 100. This method is especially helpful for converting fractions where the denominator is not a simple factor of 100. 100% 150% 200% 250% 300%

The top number line represents the mixed number amounts. The bottom line of the double number line represents those amounts Example 5 as percents. This model clearly shows that 250% aligns with

, which matches our calculation.

Write 250% as a mixed number in its simplest form.

CreateIdea a strategy summary

PurposeTo writePercent a percent as a “per fraction bymany 100. Then convert theisimproper fraction into a mixed number. means 100”divide or how “out of 100”. 1% equal to one hundredth. Show students how to convert a percentage to a mixed number in its simplest form, highlighting the importance Apply the idea of understanding the relationship between percentages and fractions. We can convert any percentDivide into aby fraction 100 by writing the percent value as the numerator and 100 as the

denominator. Advanced learners: Linking percentages to fractions and We can convert any fractionSimplify into a percent by ﬁnding its equivalent fraction that has a denominatoruse of 100. the fraction with Example 5 ratios to deepen understanding After this, we can write the value in the numerator followed by the % symbol to represent the percent. Targeted instructional strategies Convert to a mixed number

Encourage students to explore the deeper connections between percentages, fractions, and ratios. After Reﬂect andtocheck converting 250% the mixed number , prompt students to consider how this conversion represents a numberhow line is a powerful toolover for understanding percents and other amounts.and Let’sinvite ratio of 5A :double 2. Discuss percentages 100% relatethe to conversion improperbetween fractions and mixed numbers, draw a double number line to represent this relationship: 2.01 Percents fractions 61 students to represent these relationships on a number line to visualize the progression fromas0% upwards. mathspace.co

3 as a 250% increase in production output, Explore real-world contexts where percentages1 exceed 2100%, such and ask students to create their own examples. This will help advanced learners see how percentages, fractions, and ratios are interconnected, enhancing their conceptual 100% 150% 200% 250% 300% understanding of proportional relationships.

The top number line represents the mixed number amounts. The bottom line of the double number line represents amounts Students:those Page 61 as percents. This model clearly shows that 250% aligns with

, which matches our calculation.

Idea summary Percent means “per 100” or how many “out of 100”. 1% is equal to one hundredth.

We can convert any percent into a fraction by writing the percent value as the numerator and 100 as the denominator. We can convert any fraction into a percent by ﬁnding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the % symbol to represent the percent.

2.01 Percents as fractions mathspace.co

126

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice Students: Pages 62–65

What do you remember? 1

What is the meaning of the % symbol when placed after a number?

Fill in the blanks to make each statement true: a c e

29% is equivalent to 29 out of ⬚.

⬚% is equivalent to 73 out of 100.

465% is equivalent to 465 out of ⬚.

b d f

53% is equivalent to ⬚ out of 100.

150% is equivalent to ⬚ out of 100.

0.75% is equivalent to ⬚ out of 100.

Each grid is equal to one whole. For each of the grids shown: i

How many squares are shaded?

What percentage of the grid or grids is shaded?

iii

What fraction of the grid or grids does this percentage represent?

15 out of the 25 class members were sick at home. Determine the percentage of students who were sick at home.

Out of 400 students, 397 lived locally. Determine the percentage of students who did not live locally.

Estimate the percentage represented by the shaded region on the number line.

Figure out the percentage represented by the shaded part on the number line.

2.01 Percents as fractions mathspace.co

127

Let’s practice 8

Look at the picture to help you answer the question. A drought affects 8 out of 12 midwestern states. What percentage of the midwestern states are affected by the drought?

Mario completed of his workday. Use the tape diagram to represent what percentage of the workday Mario has worked.

%

%

How does your model prove that your answer is correct? 0

SOL

For each model: i

Estimate the percent represented.

Determine what exact percentage is shaded.

For each model: i

Estimate the percent represented.

What exact percentage of one whole circle is shaded?

For each: i

Represent the percentage on the grid.

Which fraction in its simpliest form is equivalent to the percent?

65%

128

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Michelle eats 40% of a cookie.

SOL

For each: i

Represent the fraction on the grid.

What percent is equivalent to the fraction?

Miguel watches

of a movie.

Write each percentage as a fraction in its simplest form: a

68%

13%

55%

37%

94%

36%

2.5%

Write each percentage as a mixed number in its simplest form: a

275%

430%

280%

123%

109%

450%

360%

285%

Write each percentage as an improper fraction in its simplest form: a

125%

175%

225%

123%

120%

265%

515%

330%

For each fraction: i

Write an equivalent fraction with a denominator of 100.

Write it as a percentage.

Write the mixed numbers as percentages: a

Write the improper fractions as percentages: a

Represent 40% on the number line and ﬁnd its equivalent fraction. 0

2.01 Percents as fractions mathspace.co

129

Show 55% on the number line and determine the equivalent fraction for this percentage. 0

Let’s extend our thinking 22

Consider the fraction . Explain if

Find the closest whole percent to .

A class has 32 students, and 12 of them are left-handed. a

How many of the students in the class are not left-handed?

What percentage of the class is left-handed?

What percentage of the class is not left-handed?

Jasper won a match in a table tennis competition. The winner of a match was the ﬁrst player to win 4 games and the winner of each game was the ﬁrst player to win 11 points. The scores were 11 − 7, 8 − 11, 11 − 5, 11 − 4, 2 − 11, 11 − 8. a

What fraction of the games did Jasper win?

How many points were played in total?

What fraction of the points did Jasper win?

What percentage of the points did Jasper lose?

In a game, a player scored 24 out of the 30 points scored by her team. Express the points scored by the rest of her team as: a

130

has an equivalent fraction with denominator of 100, and a whole number as its numerator.

A percentage

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A simpliﬁed fraction

Answers

12 a i

b i

2.01 Percents as fractions What do you remember? 1 “out of 100” 2 a 100

b 53

e 100

c 73

d 150

0.75

3 a i 71

ii 71%

iii

b i 50

ii 50%

iii

c i 67

ii 67%

iii

d i 175

ii 175%

iii

4 60% 5 0.75% 6 80%

13 a i

ii 28%

b i

ii 25%

7 25% Let’s practice 8 66.67% 9 0%

20%

40%

60%

100%

The model shows that

80%

= 60%.

10 a i About 50%

ii 40%

b i About 40%

ii 37.5%

c i About 33.33%

ii 35%

11 a i About 200%

ii 220%

b i About 300%

ii 350%

c i About 400%

ii 375%

14 a

15 a

16 a

Answers mathspace.co

131

17 a i

ii 50%

b i

ii 50%

c i

ii 12.5%

d i

ii 25%

e i

ii 80%

f i

ii 75%

g i

ii 40%

h i

ii 70%

20 0

Let’s extend our thinking 22 a I t is not possible since 100 is not a multiple of the denominator 3. b 67% 18 a 325% e 360% 19 a 260% e 120%

132

b 180%

c 750%

d 640%

275%

g 370%

h 218%

b 266.67%

c 375%

d 240%

137.5%

g 150%

h 436%

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

23 a 20 students

b 37.5%

24 a

b 100

25 a 20%

c 62.5% c

d 46%

2.02 Percents as decimals Subtopic overview Lesson narrative In this lesson, students will learn to convert between percents and decimals. The lesson begins with visual representations, such as 10x10 grids and number lines, to illustrate the relationship between percents and decimals. Students will practice converting percents to decimals by dividing by 100 and converting decimals to percents by multiplying by 100. The lesson includes various examples using models such as grids and numberlines to reinforce these concepts. By the end, students should confidently convert and understand the equivalence between percents and decimals.

Learning objectives

2.02 Percents as decimals

Students: Page 66

After this lesson, you will be able to... • convert percents to decimals. • convert decimals to percents.

Percents as decimals Let’svocabulary use the same visual representations we used for comparing fractions and percents to explore the relationship Key between decimals and percents.



decimal



 thousandths place percent In a 10 × 10 grid, each box represents 0.01.

Since 14 boxes are shaded, this model represents 14% or 0.14.

Essential understanding

We could also write as a fraction:

100% is equivalent to 1 whole. This relationship can be used to convert between decimals and percents.

Standards Double number lines can helpful toVirginia model equivalencies between percentsofand decimals. Bechmark percents and This subtopic addresses thebefollowing 2023 Mathematics Standards Learning standards. decimals can make problem solving more eﬃcient.

Mathematical process goals

0 Solving 0.25 0.5 MPG1 — Mathematical Problem

0.75

1 whole 1.25 — 1.5 1.75 2 Reasoning MPG3 Mathematical

Teachers can encourage students to justify their Teachers can integrate this goal by presenting students in writing, with real-world problems that 0% require the conversion of 25% 50% 75% 100% reasoning 125% 150% 175% using 200% a model, or orally when estimating and converting percents to decimals or percents to decimals and vice versa. For example, they 100% decimals to percents. Teachers should also encourage can present a problem where students have to calculate students to use appropriate key vocabulary terms when a discount price or determine a tax rate. Teachers We can convert between decimals and percentages by taking advantage of the hundredths place value. We know applicable. can also create problems that involve estimating, that 1% represents , or 1 hundredth, which we can write in decimal form as 0.01. determining, and justifying solutions, encouraging students apply mathematical concepts to solve We cantoconvert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to these problems. decreasing the place value of each digit by two places, and removing the % symbol. For example, 83% =

which can be described as 83 hundredths. This is also 0.83 when written as a decimal.

Percents as decimals To convert from a decimal into a percentage, we can just reverse the above steps. We can 2.02 convert any decimal into a 133 mathspace.co percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol.

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can highlight the connections between the current lesson and previous lessons about fractions, decimals, and place value. They can also emphasize the connection between mathematical concepts and realworld contexts, such as the use of percents in financial calculations. Representing fractions and percents on a numberline can help students connect their understanding of percents, decmials, and their equivalencies. By utilizing visual models and number lines, teachers can guide students in representing and converting percents and decimals. This approach also helps reinforce the understanding of benchmark fractions. Additionally, hands-on activities, such as using sticky notes on large number lines or creating human number lines, can support students in visualizing and comprehending the relationships between different representations, enhancing their ability to make mathematical connections.

Content standards 6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

6.NS.1b — Represent and determine equivalencies among decimals (through the thousandths place) and percents incorporating the use of number lines, and concrete and pictorial models.*

Prior connections 5.NS.1 — The student will use reasoning and justification to identify and represent equivalency between fractions (with denominators that are thirds, eighths, and factors of 100) and decimals; and compare and order sets of fractions (proper, improper, and/or mixed numbers having denominators of 12 or less) and decimals (through thousandths).

Future connections 7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 2.01 Percents as fractions

Tools You may find these tools helpful: • 10 × 10 Grid • Double number lines 134

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide Percents as decimals Students will explore the relationship between decimals and percents using visual representations and learn the method of converting between percents and decimals.

Students: Page 66

2.02 Percents as decimals After this lesson, you will be able to... • convert percents to decimals. • convert decimals to percents.

Percents as decimals Let’s use the same visual representations we used for comparing fractions and percents to explore the relationship between decimals and percents. In a 10 × 10 grid, each box represents 0.01. Since 14 boxes are shaded, this model represents 14% or 0.14. We could also write as a fraction:

Double number lines can be helpful to model equivalencies between percents and decimals. Bechmark percents and decimals can make problem solving more eﬃcient. 0

0.25

0.5

0.75

25%

50%

75%

100%

1 whole 1.25 1.5

1.75

125%

175%

200%

150%

100%

We can convert between decimals and percentages by taking advantage of the hundredths place value. We know that 1% represents

, or 1 hundredth, which we can write in decimal form as 0.01.

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the % symbol. For example, 83% =

which can be described as 83 hundredths. This is also 0.83 when written as a decimal.

To convert from a decimal into a percentage, we can just reverse the above steps. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol. For example, 0.08 is 8 hundredths or

= 8%.

A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages such as 0.0035 which is 0.35%. Remember to attach the % symbol to decimal at the same time as increasing the place values.

2.02 Percents as decimals mathspace.co 66

Mathspace Virginia SOL Grade 6 mathspace.co

135

Visualizing decimals and percentages Targeted instructional strategies 1. Use of Visual Aids: Visual aids are incredibly valuable in teaching mathematical concepts. For this lesson, visual aids such as a 10 × 10 grid can be used to illustrate the conversion of decimals to percentages. Each box in the grid can represent 0.01, and the shaded boxes can represent a decimal. When the decimal is converted to a percentage, students can visually see that the shaded boxes also represent that percentage out of 100. 2. Using Double Number Lines: Double number lines are another great visual tool to model the concept of equivalence between decimals and percents. On one number line, the students can mark decimals from 0 to 1 in increments of 0.1. On the parallel number line, they can mark percentages from 0% to 100% in increments of 10%. The students can then draw lines connecting equivalent values on the two number lines, e.g., 0.1 and 10%, 0.2 and 20%, and so on. This exercise visually demonstrates that decimals and percentages are two different representations of the same value.

Guided diagram annotation Student with disabilities support Provide students with partially completed 10 × 10 grids and number lines to help them visualize and annotate the relationship between percents and decimals. For example, give them a grid where 30% of the squares are shaded, and have students write the corresponding decimal 0.30 next to it. Similarly, offer a number line marked with key percentages like 0%, 25%, 50%, 75%, and 100%, but only some of the decimal equivalents filled in. Encourage students to fill in the missing decimals and add their own notes or highlight the conversions directly on the diagrams. This hands-on activity allows students to actively engage with the visual models and supports deeper understanding. Display an example on the board showing a shaded grid with annotations, so students can see how to connect the percent with its decimal form.

Align Decimals and Percentages Address student misconceptions Students may incorrectly believe that a decimal value directly translates to the same numerical value in percentages. For example, they might think that 0.5 in decimal form translates to 5% in percentage form. To correct this misconception, teachers can emphasize the understanding that 1 whole (or 1.0 in decimal form) is equivalent to 100% in percentage form. From there, it should be clearer that 0.5 in decimal form, which is half of 1, aligns with 50% in percentage form, which is half of 100%. Teachers can use visual tools such as double number lines to illustrate this alignment. It’s also important to remind students that the conversion from decimals to percentages involves multiplying the decimal by 100, which effectively shifts the decimal point two places to the right. This is another way to help them see why 0.5 in decimal form aligns with 50% in percentage form.

136

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examples Students: Page 67

Example 1 Write 54% as a decimal.

Create a strategy To convert a percentage as a decimal, we can ﬁrst think of 54% as 54 hundredths.

Apply the idea

Reﬂect and check

54% is

Let’s use a 10 × 10 grid to illustrate converting 54% to decimal.

or 54 hundredths.

54 hundredths can be written as a decimal as 0.54

The grid represents a whole. Since there are 100 squares, each square represents 1% or 0.01. Adding the 54 shaded squares that each represent 0.01 gives us 0.54, which conﬁrms our original conversion of 54% to a fraction.

Example 2 Purpose Write 0.314 as ato percentage. Show students how convert a percentage into a decimal. Create strategy Reflecting witha students To write a decimal a percentage multiply by and addpoint the % symbol. Encourage students to as explore why moving the100 decimal two places to the left converts a percentage to a decimal. Explain that the percent symbol (%) means “per hundred,” so dividing by 100 is essential in the Apply the idea

conversion process. Ask students to consider how 0.314 ⋅ 100 = 31.4

Multiply by 100

, and relate this to shifting the decimal

point. You might illustrate =this by writing the%number 31.4% Add the symbol 54.0 and showing how moving the decimal point transforms it into 0.54. Have students try this method with other percentages like 7% or 125% to observe the pattern. This reflection will help them understand the underlying concept and reinforce the relationship between percentages and decimals.

Idea summary

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to

Compare and the connect decreasing place value of each digit by two places, and removing the % symbol.

use with Example 1

EnglishWe language learner support can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol. Encourage students to solve the problem of writing 54% as a decimal using their own methods. Some students might divide 54 by 100, others might move the decimal point two places to the left, and some might think of 54% as 54 hundredths. Write these different approaches on the board for everyone to see. Facilitate a class discussion where students compare and connect these methods, highlighting how they are similar and different. Prompt students to explain their reasoning and how each method leads to the same result of 0.54. This process helps students understand the concept more deeply and connects the mathematical procedures to the 2.02 Percents as decimals 67 language used to describe them. mathspace.co

2.02 Percents as decimals mathspace.co

137

Students:Example Page 67 1 Write 54% as a decimal.

Example 2 Create a strategy

Write 0.314 as a percentage. To convert a percentage as a decimal, we can ﬁrst think of 54% as 54 hundredths.

Create a strategy Apply the idea

Reﬂect and check

To write a decimal as a percentage multiply by 100 and add Let’s the %use symbol. a 10 × 10 grid to illustrate converting 54% to 54% is or 54 hundredths. decimal. Apply the ideacan be written as a decimal as 0.54 54 hundredths 0.314 ⋅ 100 = 31.4 = 31.4%

Multiply by 100 Add the % symbol

Idea summary

Purpose We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to Show studentsdecreasing how to convert percentage into decimal. the placeavalue of each digit by atwo places, and removing the % symbol. We can convert any decimal into a percentage by multiplying the decimalaby 100, which is equivalent The grid represents whole. Since there are 100 to squares,

Advanced learners: understanding through tasks increasing the place Deepening value of each digit by two places, andsquare attaching a %multilayered symbol. each represents 1% or 0.01.

use with Example 2

Targeted instructional strategies

Adding the 54 shaded squares that each represent 0.01 gives us 0.54, which conﬁrmsdeeper our original conversion of Encourage students to extend the initial problem by adding layers that promote thinking. After 54% to a fraction.

converting 0.314 to 31.4%, have students explore how changing the decimal affects the percentage. For instance, ask them to convert decimals like 0.0314, 3.14, and 31.4 into percentages and observe the patterns that emerge. Encourage them to explain why multiplying by 100 converts a decimal to a percentage and how this Example 2 2.02 Percents as decimals 67 relates to place value and powers of ten. mathspace.co

Write 0.314 a percentage. Invite students to as create their own decimals to convert, predict the percentage equivalents, and then verify their predictions. You might use a place value chart or a number line to visually demonstrate how moving the decimal Create a strategy point impacts the value and its percentage. By layering the task in this way, you help advanced learners uncover To write a decimal as a percentage multiply by 100 and add the % symbol. deeper mathematical relationships and enhance their conceptual understanding of decimals and percentages.

Apply the idea 0.314 ⋅ 100 = 31.4

Students: Page 67

= 31.4%

Multiply by 100 Add the % symbol

Idea summary We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the % symbol. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a % symbol.

Practice

2.02 Percents as decimals mathspace.co

Students: Pages 68–70

What do you remember? 1

138

Draw a picture that represents 0.75. What percentage does your picture represent?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

What is 10% as a decimal?

Plot each number on the number line. 0

0.5

35%

1.5

0.6

120%

Let’s practice 4

Use the models to convert each percent to an equivalent decimal. a

20%

99%

100 90 80 70 60 50 40 30 20 10

67%

40%

0 1 2 3 4 5 6 7 8 9 10

Use the diagrams to help you represent an equivalent percent for each decimal. a

0.07

0.3

0.54

0.19

2.02 Percents as decimals mathspace.co

139

Write each percentage as decimal. a

22%

54%

50%

270%

500%

0.4%

0.1%

6897%

Write each decimal as percentage. a

0.25

1.1

6.43

0.006

13.04

0.083

60.3

0.063

387%

0.0387%

Which decimal is equivalent to 63%? A

0.63

3.87%

37%

Write the equivalent decimals for the percentages. Explain how you solved using pictures, numbers, and words. a

Which percent is equivalent to 3.87? A

6.3

136% = ⬚

46% = ⬚

10% = ⬚

Describe and correct the error in converting 0.87 to a percent.

5684% = ⬚

0.87 = 00.87 = 0.0087%

Let’s extend our thinking 12

Plot each number on the number line. Determine any equivalancies. 0

0.5

1.5

0.1

100%

0.10

110%

0.01

1.1

Using this diagram, or otherwise, explain why 38.5% = 0.385

Explain why is it necessary to multiply by 100% when converting a decimal to a percentage.

Explain why is it necessary to divide by 100% when converting from percentage to a decimal.

For each, ﬁll in the boxes with the missing numbers: a

140

Express each as a decimal, rounding your answer to three decimal places where necessary: a

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

c 0.10

2.02 Percents as decimals

We convert 10% into a decimal by dividing it by 100, which decreases the place value of each digit by two places, and removing the % symbol.

What do you remember?

0.0

0.1

0.2 0.3 0.4 0.5 0.6

0.7

0.8 0.9

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Pictures vary; 75% 2 0.1 3 0

0.5

1.5

4 a 0.20

b 0.99

c 0.67

d 0.40

5 a 7%

b 30%

c 54%

d 19%

6 a 0.22

b 0.54

7 a 25% e 0.6%

We convert 5684% into a decimal by dividing it by 100, which decreases the place value of each digit by two places, and removing the % symbol.

56.84.% = 56.84

Let’s practice

e 5

Pictures will vary.

d 56.84

c 0.5

d 2.7

0.004

g 0.001

h 68.97

b 110%

c 643%

d 800%

g 1304%

h 8.3%

2400%

Pictures will vary.

11 The error is that 0.87 was divided by 100 instead of multiplied by 100. The decimal point was moved two spaces to the left instead of two spaced to the right. The correct answer is 87% Let’s extend our thinking 12 0

0.5

1.5

0.1 and 0.10 are equivalent. 0.01 and 1% are also equivalent. 1 and 100% are equivalent. 1.1 and 110% are equivalent.

8 B 9 C 10 a 1.36 We convert 136% into a decimal by dividing it by 100, which decreases the place value of each digit by two places, and removing the % symbol.

13 To convert any percentage into a decimal, divide the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the % symbol. So, 38.5% represents 38.5 hundredths. This is 0.385 when written as a decimal. 14 Multiplying by 100% introduces the % symbol without changing the number’s value. 15 Dividing by 100% removes the % symbol without changing the number’s value. 16 a 0.2% = 0.002 = b 803.5% = 8.035 = 8

Pictures will vary.

17 a 0.485

b 0.46

e 0.043

= =8

b 0.082

c 0.068

d 0.026

0.135

g 0.222

h 0.618

We convert 46% into a decimal by dividing it by 100, which decreases the place value of each digit by two places, and removing the % symbol.

0.46

Pictures will vary.

Answers mathspace.co

141

2.03 Convert between fractions, decimals, and percents Subtopic overview Lesson narrative In this lesson, students will learn to convert between fractions, decimals, and percents. They start by understanding the basic relationships: 1% = 1001 = 0.01. The lesson includes interactive explorations using a 10x10 grid to visualize conversions and various examples to practice these skills. Students will solve problems such as writing fractions, decimals, and percents for given values and completing conversion tables. By the end, students should confidently convert between these forms, using strategies like multiplying or dividing by 100.

2.03 Convert between fractions, Learning objectives decimals, and percents Students: Page 71

After this lesson, you will be able to... • convert between decimals, percents, fractions, and mixed numbers.

Convert between fractions, decimals, and percentages We now know that decimals, fractions and percentages are just different ways of showing the same value: Key vocabulary 

1% can be written as decimal

or 1 hundredth or 0.01 

equivalent



fraction

 percent

Interactive exploration Explore online to answer the questions Essential understanding

100% is equivalent to 1 whole or 100/100. This relationship can be used to convert between fractions, decimals, and mathspace.co percents. Use the interactive exploration in 2.03 to answer these questions. 1.

What is the decimal and percentage equivalent of

Standards

2. A single column of small squares is what fraction of the larger square? This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. 3. What do you notice when the ‘Show equivalent fraction’ checkbox appears?

Mathematical process goals There are some common conversions that we can remember to help convert between percentages, fractions and MPG3 —us Mathematical Reasoning MPG1 — Mathematical Problem Solving decimals. Teachers can encourage students to justify their Teachers can have students apply their understanding

in writing, using a model, or orally when of fractions, decimals, and percentsFractions to solve problems Decimals reasoning Percentages estimating and converting between decimals, percents, in real-world contexts. For instance, students might 100% fractions, and mixed numbers. Teachers should also be asked to calculate a store discount by converting a1.0 encourage students to use appropriate key vocabulary percentage to a decimal. 0.5 50% terms when applicable.

142

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

0.25

25%

0.2

20%

0.125

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can demonstrate how the principles of fractions, decimals, and percents relate to other areas of mathematics. For example, students could be shown how understanding fractions can aid in the comprehension of ratios or probability. By utilizing visual models and number lines, teachers can guide students in representing and converting percents, decimals, and fractions. This approach also helps reinforce the understanding of benchmark conversions. Additionally, hands-on activities, such as using sticky notes on large number lines or creating human number lines, can support students in visualizing and comprehending the relationships between different representations, enhancing their ability to make mathematical connections.

Teachers can ask students to represent their understanding of fractions, decimals, and percents in various ways, such as drawing number lines or creating pictorial models. They might also have students convert word problems into mathematical equations using these forms.

Content standards 6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

6.NS.1d — Represent and determine equivalencies among decimals, percents, fractions (proper or improper), and mixed numbers that have denominators that are 12 or less or factors of 100 incorporating the use of number lines, and concrete and pictorial models. ∗

6.CE.1 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 2.01 Percents as fractions Grade 6 — 2.02 Percents as decimals

2.03 Convert between fractions, decimals, and percents mathspace.co

143

Student lesson & teacher guide Convert between fractions, decimals and percentages Understanding 1% is a useful concept for students in working with percents as it provides a foundation for understanding other percent values and for performing operations. In this idea, a grid model is used to aid students learn that a percentage is representative of a number per one hundred.

Students: Page 71

2.03 Convert between fractions, decimals, and percents After this lesson, you will be able to...

2.03 Convert between fractions, decimals, and decimals, percents Convert between fractions, and percentages • convert between decimals, percents, fractions, and mixed numbers.

We now know that decimals, fractions and percentages are just different ways of showing the same value: or 1 hundredth or 0.01 1% can be written as After this lesson, you will be able to... • convert between decimals, percents, fractions, and mixed numbers.

Interactive exploration

Explore online to answer the questions

Exploration Convert between fractions, decimals, and percentages mathspace.co now know that decimals, fractions and percentages are just different ways of showing the same value: Students:WePage 71 Use the interactive exploration in 2.03 to answer these questions. or 1 hundredth or 0.01 1% can be written as 1. What is the decimal and percentage equivalent of

Interactive exploration A single column of small squares is what fraction of the larger square?

Explore to answer What doonline you notice whenthe thequestions ‘Show equivalent fraction’ checkbox appears?

mathspace.co There are some common conversions that we can remember to help us convert between percentages, fractions and Use the interactive exploration in 2.03 to answer these questions. decimals. ?

What is the decimal and Fractions percentage equivalent of Decimals

A single column of small squares is what fraction 1.0 of the larger square? 100%

What do you notice when the ‘Show equivalent fraction’ checkbox appears? 0.5 50%

Percentages

There are some common conversions that we can remember to help us convert between percentages, fractions and decimals. 0.25 25% Suggested student grouping: In pairs Percentages In this exploration, students will use Fractions a grid-based Decimals applet 0.2 to understand 20%the conversion between fractions, 1.0 decimals, and percentages. By manipulating the grid and observing100% changes, they will make connections between these different numerical representations.0.125 0.5 50%

144

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

0.1

10%

0.05 0.25 0.04 0.2

5% 25% 4% 20%

0.125 0.1 0.05

10%

2.03 Convert between fractions, decimals, and percents mathspace.co

is 0.4 and the percentage equivalent is 40%.

2. A single column of small squares is what fraction of the larger square? A single column of small squares is of the total squares.

of the larger square because the grid is 10 × 10 and one column is

3. What do you notice when the ‘Show equivalent fraction’ checkbox appears? The ‘Show equivalent fraction’ checkbox appears when the fraction can be simplified. The grid can be divided into a smaller number of equal parts, rather than 100 small squares. Purposeful questions • How can you divide the grid into five equivalent parts? If we wanted two of those five parts, how many small squares would be shaded? • How does the percentage relate to the fraction and the decimal? • Can you always see the ‘Show equivalent fraction’ checkbox? Why or why not? Possible misunderstandings • Students might not know how to find the decimal and percentage equivalent of them that

using the applet. Remind

represents 2 of 5 equal parts shaded, so they need to divide the grid into 5 equal parts.

Use algorithmic thinking to write procedures for converting between forms Targeted instructional strategies Students should begin by learning what a percentage is and how it can be expressed as both a fraction and decimal. Show students a grid with one hundred squares and color in a few, for example, 3. Ask students how the shaded value could be represented as a fraction out of the 100 squares. Remind students that 1% means

, so

would be equivalent to 3%. Students should also be prompted to

express the fraction as a decimal. Encourage students to develop and articulate step-by-step procedures—or algorithms—for converting between the different forms. Have them test their algorithms with various examples to ensure accuracy and refine the steps as needed. • Percentage to Fraction - write the percentage value as the numerator and 100 as the denominator (simplify if possible) • Fraction to percentage - Write an equivalent fraction with a denominator of 100. Then, write the value in the numerator with a percentage symbol, %. • Decimal to percentage - Multiply the decimal by 100. Then, write the percentage symbol at the end. Encourage students to practice these algorithms with different numbers to reinforce their understanding. Ask them to explain each step and justify why it is necessary, fostering deeper comprehension and the ability to debug and improve their algorithms. This focus on developing, testing, and refining procedures will enhance their algorithmic thinking skills.

2.03 Convert between fractions, decimals, and percents mathspace.co

145

2.03 Convert between fractions, decimals, and percents

Graphic organizer

Student with disabilities support

Students can create a graphic organizer to help them remember the steps for each type of conversion. Have students create 4 columns on a blank sheet of paper. They should label each column with one of the following: fraction to percentage, percentage to fraction, decimal to percentage, and percentage to decimal. Students After lesson,in you will be able to... should then work an this example each column, highlighting the steps needed for each conversion. It may be • convert between decimals, percents, fractions, and mixed numbers. helpful to provide students with the paper and examples already in the correct columns.

Convertwith between fractions, decimals, and percentages Decimals percentages We now know that decimals, fractions and percentages are just different ways of showing the same value: Address student misconceptions 1% can be written as

or 1 hundredth or 0.01

Students may see a decimal percentage such as 0.25% and think this is the same as 25% because they incorrectly believe percentages need to have at least some whole number part. They may also use the incorrect Interactive exploration conversion procedure. Explore online to answer the questions

Give examples of percentages with decimals such as 1.35% and 0.25% in exercises to make students understand that mathspace.co a decimal percentage is a fraction of 1%. Let students understand that 0.25% is a very small percentage equivalent to 0.0025 as a decimal number. Use the interactive exploration in 2.03 to answer these questions. ?

What is the decimal and percentage equivalent of

A single column of small squares is what fraction of the larger square?

Students: Page 71 do you notice when the ‘Show equivalent fraction’ checkbox appears? 3. What There are some common conversions that we can remember to help us convert between percentages, fractions and decimals. Fractions

Decimals

Percentages

1.0

100%

0.5

50%

0.25

25%

0.2

20%

0.125 0.1

10%

0.05

0.04

2.03 Convert between fractions, decimals, and percents mathspace.co

146

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examples Students: Page 72 Example 1 Write the fraction, decimal, and percent that represents this illustration.

Create a strategy First, count the number of parts in one whole circle. Then, determine how many parts are shaded. This will give the fraction. Next convert the fraction into a decimal and percent.

Apply the idea To ﬁnd the fraction: Number of parts in a whole = 8; Number of shaded parts = 27 Fraction = To ﬁnd the decimal: Decimal = 27 ÷ 8 = 3.375

Divide 27 by 8 Evaluate

To ﬁnd the percent: Percent = 3.375 ⋅ 100% = 337.5%

Multiply the decimal by 100% Evaluate

Example 2 Purpose Convert between fractions and decimalsas to acomplete theatable shown.and a percent. Show students how to percentages, write a visual representation fraction, decimal, Write the answers as mixed number percentages and simpliﬁed mixed numbers where necessary.

Reflecting with students Fraction Decimal Percentage Encourage advanced learners, or any students who are ready, to create their own visual models representing 11% fractions, decimals, and percents greater than one. Invite 0.83 them to choose different shapes, divide them into equal parts, and shade more parts than the total in a single whole to represent improper fractions. After they’ve designed their illustrations, have them calculate and record the corresponding fraction, decimal, and percent for their creation. Create a strategy This activity allows students to explore how visual representations can depict quantities exceeding a whole To convert fractions and decimals to percentages multiply by 100%. To convert percentages to fractions or decimals, and understand the relationship between the parts and the whole. By designing their own problems, students divide by 100. engage more deeply with the concept, discover patterns, and make connections between different numerical forms.

Mathspace Virginia SOL Grade 6 mathspace.co

2.03 Convert between fractions, decimals, and percents mathspace.co

147

Decimal = 27 ÷ 8

Divide 27 by 8

= 3.375

Evaluate

To ﬁnd the percent: Percent = 3.375 ⋅ 100%

Multiply the decimal by 100%

= 337.5% Students: Pages 72–73

Evaluate

Example 2 Convert between percentages, fractions and decimals to complete the table shown. Write the answers as mixed number percentages and simpliﬁed mixed numbers where necessary. Fraction

Decimal

Percentage 11%

0.83

Create a strategy To convert fractions and decimals to percentages multiply by 100%. To convert percentages to fractions or decimals, divide by 100.

Apply the idea Write as a part out of 100 (Fraction) Convert to a decimal Multiply by 100 Evaluate Attach percent symbol Convert to a fraction 72

Mathspace Virginia SOL Grade 6 mathspace.co Multiply by 12.5 to make the denominator equal to 100

Evaluate Convert to percent Convert to a decimal Fraction

Decimal

Percentage

0.11

11%

0.83

83%

0.625

62.5%

Idea summary

Purpose Decimals, fractions and percentages can be converted between each other using the relationship: Challenge students to convert between fractions, decimals, and percentages. 1% =

= One Hundredth = 0.01

Expected mistakes Students may struggle to change eigths into hundredths.

Practice What do you remember?

148

Write

Write 0.65 as both a fraction and percent.

Is 4.4 equivalent to

Which four answer choices represent the illustration?

as both a decimal and percent.

? How do you know?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co A 3.75% B E

375%

0.375

C G

D 475%

3.75

Apply the idea Write as a part out of 100 (Fraction) Convert to a decimal

Matching activity: information gap for conversions

use with Example 2

Multiply by 100

English language learner support

Evaluate

Prepare a set of cards where each card displays Attach percent either symbol a fraction, a decimal, or a percentage from the table Convert to a fraction . Distribute the cards so that each student receives one

card. Explain to students that their task is to find classmates who have the equivalent forms of their number, Multiply by 12.5 to make the denominator equal to 100 forming groups where each member has a different representation (fraction, decimal, or percentage) of the same value. Encourage students to walk around the room and communicate by asking questions and sharing Evaluate information to identify their matches Convert (e.g., “Itohave , does anyone have the equivalent decimal or percentage?”). percent Provide and model sentence stems to support their conversations, such as “Do you have a decimal that Convert to a decimal equals...?” or “My percentage is..., what fraction matches it?” This information gap activity requires students Fraction Decimal Percentage to use mathematical vocabulary and collaborate to complete the task, enhancing their understanding of 0.11 11% conversions while practicing English language skills.

Students: Page 73

0.83

83%

0.625

62.5%

Idea summary Decimals, fractions and percentages can be converted between each other using the relationship: 1% =

= One Hundredth = 0.01

Practice What do you remember? Practice 1

Write

as both a decimal and percent.

Students: Pages 73–75 2

Write 0.65 as both a fraction and percent.

Is 4.4 equivalent to

Which four answer choices represent the illustration?

What do you remember?

? How do you know?

Write

Write 0.65 as both a fraction and percent.

Is 4.4 equivalent E 375%to

Which four answer choices represent the illustration?

as both a decimal and percent.

3.75%

? How doF you know? 0.375

3.75

475%

2.03 Convert between fractions, decimals, and percents mathspace.co

3.75%

375%

C 0.375

3.75

475%

2.03 Convert between fractions, decimals, and percents mathspace.co

149

Let’s practice SOL

Look at each fraction: i

Represent the fraction on the grid.

What decimal is equivalent to the fraction?

iii

What percent is equivalent to the fraction?

SOL

Henry eats

of a pizza.

Write a decimal that represents the amount of pizza that Henry eats.

Write a percent that represents the amount of pizza that Henry eats.

Represent your thinking using a number line. 0

Complete the table with equivalent fractions, decimals, and percentages to represent how much of each picture is colored. Make sure decimals are rounded to the nearest hundredth and fractions are written in their simplest form. Picture

150

Fraction

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Decimal

Percent

Select all of the numbers represented by the tape diagram below.

25%

20%

1.4

2.5

=⬚ D

0.04

9.6 =

0.25

Convert between percentages, fractions and decimals to complete the table: Fraction

Decimal

Percentage 1% 99%

Convert between percentages, fractions and decimals to complete the table: Fraction

Decimal

Percentage

0.83

SOL

Select each number that can be placed in the blank to make this statement true. A

35%

40%

0.4

400%

2.50

0.25%

60% =

0.7% = 0.07 and

Which statement is true? A

= 6.67% and 0.4

and 0.60

and 960%

In a charity walk, Alice completes 7 out of the 10 miles of the total event. Express the distance completed by the rest of the participants as: a

A simpliﬁed fraction

A percentage

A decimal

Let’s extend our thinking 14

What is a real life scenario where we may have to convert between fractions, percents, or decimals?

Express these as a fraction and either a percent or a decimal:

70%

62.5%

47.5%

0.73

85%

0.834

0.543

12%

A teacher has 9 pencils. Five of the pencils are red. a

What fraction represents the ratio of red pencils to the total number of pencils?

What decimal repersents the ratio of red pencils to the total number of pencils?

What percent represents the ratio of red pencils to the total number of pencils?

A group of friends decides to split the cost of a meal, equally. The ﬁrst friend pays

of the total bill, while the

second friend pays 0.25 of the bill. The third friend pays 37.5% of the bill. Is this a fair split? Explain your answer using pictures, numbers and/or words.

2.03 Convert between fractions, decimals, and percents mathspace.co

151

Answers 2.03 Convert between fractions, decimals, and percents

6 a 0.8

b 80%

c 0

Picture

Fraction

Decimal

Percent

0.75

75%

0.8

80%

0.67

67%

What do you remember? 1 Decimal: 0.625, Percent: 62.5% 2 Fraction: 3 Yes.

, Percent: 65% written as a decimal is 4.4.

4 B, C, D, E Let’s practice

8 A, B, G

5 a i

Fraction

Decimal

Percentage

0.01

0.99

99%

0.125 10 ii 0.27

Fraction

Decimal

Percentage

0.28

28%

0.83

83%

iii 27%

b i

0.875 11 B, C 12 D ii 0.005

iii 0.5%

b 0.3

13 a

c 30%

c i Let’s extend our thinking 14 Answers vary. You may think about budgeting a percent of your money or comparing amounts/cost of different things represented by fractions, decimals, or percents. Converting can help us compare across representations. 15 a

ii 1.333

iii 133.3%

d i

and 0.7

b and 0.625

and 0.475

and 73%

and 0.85

and 83.4%

g 16 a

and 54.3%

b 0.56

and 0.12

c 55.56%

17 This is not a fair split because the ﬁrst and third friend are each paying 37.5% of the total bill while the second friend is only paying 25%. ii 0.56

152

iii 56%

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

converts to 0.375 which is 37.5% as

a percent while 0.25 converts to 25% as a percent.

2.04 Compare and order fractions, decimals, and percents Subtopic overview Lesson narrative In this lesson, students will learn to compare and order fractions, decimals, and percents. They start by reviewing symbols and concepts like greater than, less than, ascending, and descending order. Using number lines, students will visualize and compare different forms. They will convert fractions, decimals, and percents to the same form for easier comparison. Problems include placing values in order and comparing numbers using benchmarks and conversions. By the end, students should be able to confidently compare and order these forms.

Learning objectives Students: Page 76

Key vocabulary 

ascending order



descending order

Essential understanding Comparing rational numbers in different forms can easily be done by converting them first to the same form.100% is equivalent to 1 whole or 100/100. This relationship can be used to convert between fractions, decimals, and percents.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal into their instruction by presenting students with real-world problems that require them to apply their understanding of comparing and ordering fractions, decimals, and percents. For example, teachers can pose a problem about a sale at a store, where students must determine the best deal by comparing and ordering various discounts expressed as fractions, decimals, and percents.

2.04 Compare and order fractions, decimals, and percents mathspace.co

153

MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

To integrate this goal, teachers can encourage students to explain their reasoning orally and in writing when solving problems. This can include justifying why they chose a particular strategy for comparing and ordering numbers, explaining their process for converting numbers to a common form, or discussing how they used a model to aid in their understanding.

To integrate this goal, teachers can emphasize the connection between comparing and ordering rational numbers and the strategies outlined in the standards, such as using benchmarks, number lines, and equivalency to compare and order rational numbers. For example, teachers can remind students of their prior work with these strategies in previous lessons, and show how these strategies can be applied to new contexts.

MPG5 — Mathematical Representations Teachers can integrate this goal into instruction by teaching students how to use various models, such as number lines, hundred grids, and pie charts, to represent and compare fractions, decimals, and percents. Teachers can provide practice problems that require students to use these models, and encourage students to draw their own models to assist in their understanding.

Content standards 6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents. 6.NS.1e — Use multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100), with and without models. Justify solutions orally, in writing or with a model. Ordering may be in ascending or descending order.∗

Future connections 7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 2.03 Convert between fractions, decimals, and percents

154

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide Compare and order fractions, decimals, percents Students recall the terminologies related in comparing quantities such as greater than, less than, ascending order and descending order. Students are encouraged to make sense with the quantities, plot rational numbers of different forms on a number line and convert to a common form and make comparison more efficient.

Students: Pages 76–77

2.04 Compare and order fractions, decimals, and percents mathspace.co

155

To avoid this, we can compare fractions, decimals and percents algebraically if we convert them to be in the same form. Begin by looking at the given list and deciding which form would be most helpful to make the comparisons.

You may think it makes the most sense to make comparisons as decimals, changing them to be 0.1, 0.6, and 0.5. While someone else might think it makes the most sense to convert them all to percentages, such as 10%, 60%, and 50%. Which form you choose doesn’t matter as long as it is the same for each number in the list.

Example 1 Fill in the ⬚to with or > to makeform a correct comparison between the numbers. Convert a <common a 3.5 ⬚ 35% Targeted instructional strategies

Having students practice conversions regularly can help them become more fluent. Start with simple Create a strategy conversions, such as converting fractions and decimals or percents and fractions, then gradually Write the numbers in the samebetween form to easily see which is larger. introduce more complex conversions. Apply the idea

Provide students with plenty of examples and practice problems. Encourage them to explain their thought To convert decimal to percentage, we just multiply by 100% process and steps while solving the problems. This way, they can reinforce their understanding and also learn 3.5 ⋅ 100% = 350% Multiply by 100 from their mistakes. 350% > 35%

Compare

Also, remind students This means that: that when converting to a common form for comparison, they should choose the form that they find the most comfortable and intuitive. Whether3.5that is fractions, decimals or percents, what matters most > 35% is that they understand the concept and can apply it correctly. Reﬂect and check We could have converted the percentage to a decimal and compared the values. To convert a percentage to a decimal, divide by 100. Use benchmarks

Student with disabilities support Divide by 100 Convert to a decimal A useful strategy for students who might struggle with direct conversion between fractions, decimals, and decimal values, we still ﬁnd that is greater than 0.35, percentsComparing is to usethe benchmarks. Benchmarks are3.5known values that conﬁrming students our areprevious familiarsolution. with and can easily 3.5 > 0.35 compare to.

Start by identifying a few common benchmarks such as 50%

, 25%

, and 75%

b 35% ⬚

. Have students practice comparing other fractions, decimals, or percents to these benchmarks.

Create a strategy

For example, if a student is asked to compare 70% and , they can recognize that Write the numbers in the same form to easily see which is larger.

75% is greater than 70%, they can conclude that Apply the idea

is equivalent to 75%. Since

is greater than 70%.

This strategy reduces the cognitive load to ofconvert havingboth to perform precise conversions between different To compare the numbers, we ﬁrst need values to the same form. We can convert both values to forms and fractions with a denominator of 100. allows students to make accurate comparisons based on familiar, easily understood benchmarks. It also helps to reinforce their understanding of the Convert relationship fractions, decimals, and percents. 35% tobetween a fraction with a denominator of 100 Convert

156

to a fraction with a denominator of 100

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

2.04 Compare and order fractions, decimals, and percents mathspace.co

Examples You may think it makes the most sense to make comparisons as decimals, changing them to be 0.1, 0.6, and 0.5. someone Students:While Page 77 else might think it makes the most sense to convert them all to percentages, such as 10%, 60%, and 50%. Which form you choose doesn’t matter as long as it is the same for each number in the list.

Example 1

To avoid this, we can compare fractions, decimals and percents algebraically if we convert them to be in the same form. looking given list and deciding which formthe would be most helpful to make the comparisons. Fill in Begin the ⬚ by with < or > at to the make a correct comparison between numbers. a 3.5 ⬚ 35%

You may think it makes the most sense to make comparisons as decimals, changing them to be 0.1, 0.6, and 0.5. Create a strategy While someone else might think it makes the most sense to convert them all to percentages, such as 10%, 60%, and Write the numbers in the samedoesn’t form tomatter easily as see which 50%. Which form you choose long as itisislarger. the same for each number in the list.

Apply the idea

Example 1 To convert decimal to percentage, we just multiply by 100% 3.5with ⋅ 100% Multiplycomparison by 100 Fill in the ⬚ < or=>350% to make a correct between the numbers. 350% > 35% Compare a 3.5 ⬚ 35% This means that: 3.5 > 35%

Create a strategy

Write the numbers in the same form to easily see which is larger.

Reﬂect and check Apply the idea We could have converted the percentage to a decimal and compared the values. To convert a percentage to a To convert decimal to percentage, we just multiply by 100% decimal, divide by 100. 3.5 ⋅ 100% = 350% Multiply 100 Divide byby100 350% > 35%

Compare Convert to a decimal

This means that: Comparing the decimal values, we still ﬁnd that 3.5 is greater than 0.35, conﬁrming our previous solution. 3.5 > 35% 3.5 > 0.35

Reﬂect and check b 35% We could⬚have converted the percentage to a decimal and compared the values. To convert a percentage to a decimal, divide by 100.

Purpose Create a strategy Divide in by different 100 Show students how to compare numbers forms (decimals and percentages) by converting them to Write the numbers in the same form to easily see which is larger. Convert to a decimal the same form. Comparing the decimal values, we still ﬁnd that 3.5 is greater than 0.35, conﬁrming our previous solution. Apply the idea

Students: Pages 77–78

3.5 > 0.35 To compare the numbers, we ﬁrst need to convert both values to the same form. We can convert both values to fractions with a denominator of 100. b 35% ⬚

Create a strategy

Convert 35% to a fraction with a denominator of 100 Convert

to a fraction with a denominator of 100

Write the numbers in the same form to easily see which is larger.

Apply the idea To compare the numbers, we ﬁrst need to convert both values to the same form. We can convert both values to 2.04 Compare and order fractions, decimals, and percents fractions with a denominator of 100. Convert 35% to a fraction with a denominator of 100 Convert

mathspace.co

to a fraction with a denominator of 100

2.04 Compare and order fractions, decimals, and percents mathspace.co

157

Now that both values are in the same form, we can easily compare them:

Therefore, 35% < .

Reﬂect and check We can also compare these numbers using benchmark values. Let’s compare each value to . We 35% is less than . form, we can easily compare them: Nowknow that both values are in50% the =same We also know

is larger than

Since 35% 35% is less Therefore, < than .

and

since 3 is more than half of 5. is more than , we know 35% < .

Reﬂect and check ⬚ also 0.8 compare these numbers using benchmark values. Let’s compare each value to . cWe can

PurposeWe know 35% is less than 50% = . Create a strategy Check students’ understanding of how to compare fractions and percentages by converting them to the same Write the numbers in the same form to easily see which is larger. form. We also know is larger than since 3 is more than half of 5. Apply35% theisidea Since less than

Students: Page 78 c

and

is more than , we know 35% < . Multiply by

to get a denominator of 100

Convert 0.8 to fraction

⬚ 0.8

Convert

Create a strategy

to a fraction with a denominator of 100

Compare fractions Write the numbers in the same form to easilythe seetwo which is larger. Therefore, < 0.8 Apply the idea

Reﬂect and check

Multiply by

to get a denominator of 100

We can also compare these valuesConvert using a 0.8 benchmark by seeing how far each is from 1 whole. to fraction We know

less than 1, and 0.8 is 0.2 less than 1. to a fraction with a denominator of 100 Convert

= 0.4 > 0.2, we know that

Since

Therefore,

< 0.8. Compare the two fractions

< 0.8

Example Reﬂect and2check We can also compare these values using a benchmark by seeing how far each is from 1 whole. Arrange , 40% and 0.5 in descending order using percentages. We know is less than 1, and 0.8 is 0.2 less than 1. to a percentage. a First, convert Since = 0.4 > 0.2, we know that < 0.8.

Create a strategy Convert

to be a fraction out of 100.

Example 2 Purpose Challenge toand compare a fraction andusing a decimal by converting them to a common form. Mathspace Virginia Grade 6 78 students Arrange , 40% 0.5SOL in descending order percentages. mathspace.co

a First, convert

to a percentage.

Create a strategy Convert

158

to be a fraction out of 100.

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Reﬂect and check We can also compare these numbers using benchmark values. Let’s compare each value to . We know 35% is less than 50% = . We also know

is larger than

since 3 is more than half of 5.

Advanced learners: Design their own problems Since 35% is less than

and

is more than , we know 35% < .

use with Example 1

Targeted instructional strategies

Empower advanced learners by inviting them to create their own problems involving comparisons of numbers in different fractions, and percentages. Encourage students to choose numbers that interest ⬚ 0.8 c forms—decimals, them and construct comparison statements for their peers to solve. For example, they might pose a problem like Createthan a strategy “Is greater 85%?” This activity allows students to delve deeper into the relationships between various Write the numbers in theAs same form to easily seeproblems, which is larger. numerical representations. they design their prompt them to think about what makes comparisons tricky and how different forms can be converted or interpreted. By creating and solving these custom problems, Apply the idea students not only reinforce their understanding but also engage in higher-order thinking and creativity. Multiply by

to get a denominator of 100

Convert 0.8 to fraction

Critique, correct, and clarify Convert English language learner support

use with Example 1 to a fraction with a denominator of 100

Present students with statements and work that contain errors in comparing numbers in different forms. Compare the two fractions Ask students to work individually or in pairs to identify and correct the mistakes in these statements. Encourage Therefore, < 0.8 them to explain their reasoning and the steps they took to arrive at the correct comparison. By correcting the errors, students notcheck only deepen their understanding of how to compare decimals, fractions, and percentages Reﬂect and but alsoWe practice using precise mathematical language. This exercise helps English language learners articulate can also compare these values using a benchmark by seeing how far each is from 1 whole. their thought processes and strengthens their mathematical vocabulary. We know

less than 1, and 0.8 is 0.2 less than 1.

= 0.4 > 0.2, we know that

Since

< 0.8.

Students: Pages 78–79 Example 2 Arrange

, 40% and 0.5 in descending order using percentages.

a First, convert

to a percentage.

Create a strategy Convert

to be a fraction out of 100.

Mathspace Virginia SOL Grade 6 mathspace.co

Purpose Check students can convert a fraction to a percentage.

2.04 Compare and order fractions, decimals, and percents mathspace.co

159

Students: Page 79

Purpose Check students can convert a decimal to a percentage. Expected mistakes Students may mistakenly interpret 0.5 as 5% instead of 50% because they might think that converting a decimal to a percentage only involves attaching a percent sign or moving the decimal point one place to the right. This misconception arises from a lack of understanding of the relationship between decimals and percentages, specifically that converting a decimal to a percentage requires multiplying by 100. To address this misconception, emphasize to students that converting a decimal to a percentage involves multiplying the decimal by 100, which effectively shifts the decimal point two places to the right. Demonstrate this process step by step using 0.5 × 100 = 50%. Encourage students to practice this conversion with various decimals to reinforce the concept. Additionally, relate 0.5 to the familiar fraction , and remind students that

. By connecting decimals to

fractions and percentages, students can develop a more robust understanding of the relationships between these representations.

Students: Page 79

160

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 79

Purpose Students demonstrate that they can order rational numbers of different forms in descending order. Reflecting with students Ask students what form will they find easier for making comparisons. Have students share their different preferences and give justification.

Verify answers with code

use with Example 2

Targeted instructional strategies Show students how to write a simple Python program that compares fractions, decimals, and percents and arranges them in descending order. They can use this code to verify their solutions to the parts in this example. An example code is shown: 1

# Step 1: Define the numbers

fraction = 9 / 10

percentage = 0.40

decimal = 0.5

# Step 2: Create a list of the numbers

numbers = [fraction, percentage, decimal]

# Step 3: Sort the list in descending order

sorted_numbers = sorted(numbers, reverse=True)

# Step 4: Convert each number to a percentage and print

10 print(“Numbers in descending order (as percentages):”) 11 for num in sorted_numbers: 12 percentage_form = num * 100 13 print(f”{percentage_form}%”) Note that they will need to input the percentage as a fraction or a decimal since the % symbol in Python is used as the modulus operator (for finding the remainder when dividing two numbers).

2.04 Compare and order fractions, decimals, and percents mathspace.co

161

Students: Page 80

Idea summary When we need to compare a list of fractions, decimals and percents, converting them to be in the same form can make it easier to compare. Look at the given list and decide which form would be most helpful to make the comparisons in.

Practice What do you remember? Practice 1

Complete the following using the words ‘divide’ or ‘multiply’.

Students: Pages a To80–82 convert a decimal into a percentage you ⬚ by 100%. b

To convert a percentage into a decimal you ⬚ by 100%.

Match the following items with their equivalent benchmark fraction, decimal, or percent:

What do you remember?

Percentage

4 Express each percentage as a: Decimal

75% 5 Consider the statement: 69% is less than 0.63

75%

68%

Convert 0.63 to a percentage.

Now, state whether the statement is true or false.

0.2

Fraction

20%

Fraction

50%

8.5%

0.8%

iii iv

0.25

Fraction

0.075

Express decimals as a: 1.63 is greater than 16.3% 6 each Consider the statement: a Convert 1.63 to a percentage. Percentage b

0.98

Now, state whether the statement is true or false.

0.43

0.185

0.8%

Express each percentages as a: i

Decimal

Fraction

75%

8.5%

68%

Consider the statement: 69% is less than 0.63 a

Convert 0.63 to a percentage.

Now, state whether the statement is true or false.

Consider the statement: 1.63 is greater than 16.3% a

Convert 1.63 to a percentage.

Now, state whether the statement is true or false. 80

162

0.25

Express each decimal as a:

To convert a percentage into a decimal you ⬚ by 100%.

0.2

To convert a decimal into a percentage you ⬚ by 100%.

Match theafollowing items with their benchmark decimal, or dpercent: 0.98 b equivalent 0.43 c fraction, 0.075 0.185 a

75%

Completei the20% following using theiiwords 50% ‘divide’ or ‘multiply’. iii a

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice SOL

Consider the shown models:

SOL

Name the percent in each model.

Write which model represents a greater percent and justify your reasoning.

Place the fractions, decimals, and percentages on the number line in the correct location. Explain your thinking. , 1.25, 0.625, 1.2% 0

0.91 and 82%

8.9 and 879%

0.82 and 83%

0.925 and 88%

SOL

0.31 and 45%

Fill in the ⬚ with < or > to make a correct comparison between the numbers.

SOL

For each pair of numbers state which number is greater.

4.5 ⬚ 45%

188.3% ⬚ 1.881

0.897 ⬚ 80%

0.39 and 26%

6.7% ⬚ 0.061

48.1% ⬚ 4.71

97.4% ⬚ 1.01

Determine whether each statements is true or false. a

> 132%

> 68%

> 64%

> 131%

> 90%

> 154%

> 11%

< 80%

, 25%, 0.5

0.893, , 53%

, 100%, 0.5

, 2.8, 240%

Write each list of numbers in ascending order. a

, 20%, 1

, 100%, 0.275

, 100%, 0.2 , 20%, 0.8

, 25%, 1

Write each list of numbers in descending order. a

, 70%, 0.8

, 60%, 0.3

, 30%, 0.6 , 100%

Which of these numbers is between

, 0.847

and 0.85 on a number line?

, 75%, 0.5

45%

0.80

2.04 Compare and order fractions, decimals, and percents mathspace.co

163

SOL

List the numbers in order from least to greatest. Numbers

Least ⬚

0.875% 0.888 SOL

Arrange the numbers in order from greatest to least. 25%

⟶ ⬚

Greatest ⬚

Greatest

0.40

Least

On her ﬁrst Mathematics test, Juliet scored 16 out of 25. On the second Mathematics test, she scored 27 out of 40. On her ﬁnal Mathematics test, Juliet scored 49 out of 50. Arrange her scores in ascending order.

Using the models provided, order the following numbers from largest to smallest: , 0.60, 45%. For each number, justify your answer with reference to the models.

greater than or less than 90%? Justify your reasoning using benchmarks.

Model 1

Model 3

Model 2

Let’s extend our thinking 20

For each list of numbers, ﬁnd: i

The largest value

iii

The value closest to 0.5

92%,

63%,

164

The smallest value

, 0.1, 0.365, 60.1%

71%,

, 0.7, 0.99, 50.8%

, 0.6, 0.689, 65.5%

88%,

, 0.9, 0.83, 81.4%

Arrange the set in ascending order.

For each sets of numbers: i

Plot the numbers on a number line.

, 8%

, 0.225, −20%, −0.025

−0.4, 0.175, −4.95%, 1.25%,

c 22

17.5%,

For each sets of numbers: i

Plot the numbers on a number line.

Arrange the set in descending order.

0.351, −0.5, , −25%

0.15, −0.555, 5%,

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Joshua makes 75% of his shots, his sister makes made the most shots?

Finn asked the class presidents of classes A and B how many students of their section will join the school fair.

of her shots, and his friend makes 0.65 of his shots. Who

The president of class A gave the number 0.4. The president of class B gave the number

Explain how to ﬁnd the number of students that will join in class A given that both sections have total number of students of 20.

State which section has the higher number of students that will join the school fair.

2.04 Compare and order fractions, decimals, and percents mathspace.co

165

Answers

13 a , 0.8, 70%

2.04 Compare and order fractions, decimals, and percents What do you remember? 1 a Multiply 2 a iv

b Divide b ii

c iii

3 a i 98%

c 100%, 0.5,

d 75%, 0.5,

e 60%, 0.3,

g 2.8, , 240%

100%, ,

d i 14 C

, 0.6, 30%

15 b i 43%

c i 7.5%

d i 18.5%

4 a i 0.75

b i 0.68

c i 0.085

d i 0.008

⟶

Least

Greatest 0.888

0.875% 16

Greatest 0.40 Least

25%

17 Test 1, 64%. Test 2, 67.5%. Juliet did better on Test 2.

5 a 63%

b False

18 Less than 90%. Benchmark reasoning:

6 a 163%

b True

(or 100% ), while 90% is only 10% closer to 100%.

Let’s practice 19

7 a T he percent in the bar model is 70% and in the pie chart is 37.5%. b T he bar model represents a greater percent because 70% > 37.5%.

away from 1

away and is

, 0.60, 45%

The hundred’s grid shows that 60 squares (60%) are shaded more than 45 squares (45%). The pie chart shows is greater than 60% (since

is close to 83.33%, which is

more than 60%). Therefore, the order from largest to

8 0

9 a 0.91 e 83% 10 a > e > 11 a True e False

0.5

b 8.9 f

c 45%

d 0.39

c >

d <

b True

c True

d True

False

g True

h False

b 0.2, 100%,

c 25%, 0.5,

d 25%, , 1

e 0.275,

0.925

b > f

1.5

12 a 20%, , 1

, 100%

g 53%, , 0.893

166

20%, , 0.8

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

smallest is , 0.60, 45%. Let’s extend our thinking 20 a i 92%

ii 0.1

iii

b i 0.99

iii 50.8%

c i 0.689

iii 0.6

d i 0.9

ii 81.4%

iii 81.4%

21 a i

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ii 0.03, 8%, b i

0.1

ii 0.76, 0.8, , 89%, 1

c i

−0.5 −0.4 −0.3 −0.2 −0.1

ii −20%, d i ii 22 a i

0.1

0.2

0.3

0.4 0.5

24 a Notice that 0.4 =

, −0.025, 17.5%, 0.225

−0.5 −0.4 −0.3 −0.2 −0.1

0.1

23 Joshua made the most shots with a success rate of 75%.

0.2

0.3

0.4

0.5

and it is equivalent to

×2=

This means that 8 out of the total of 20 students will join the school fair. b Section B

, −0.4, −4.95%, 1.25%, 0.175,

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

ii 0.35, , −25%, −0.5 b i ii

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

, 0.15,

0.1 0.2 0.3 0.4 0.5 0.6 0.7

, −0.55

Answers mathspace.co

167

Topic 2 Assessment: Fractions, Decimals, & Percents 1

Determine the percent represented by each given model. a

0 1 2 3 4 5 6 7 8 9 10

Determine the best estimate for the percentage represented by each model. a

A 50%

C 0.5%

0.05%

A 100%

120%

$6,000

C 150%

6000%

$5,000 $4,000

Total Raised: $6,000 $2,000 Our Goal: $5,000 $1,000 $3,000

Write the following as decimals: a

168

0.02%

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

139.5%

For each of the following pairs of numbers: i

Convert the decimal to a percentage.

Which number is greater?

0.865 and 68%

0.007 and 1%

Which of these numbers is between 0.05% and 0.3 on a number line? A

SOL

Using this diagram, or otherwise, explain why

Which percentage is equivalent to ?

SOL

A 8

68%

0.004

80%

54%

2.5%

= 20%.

125%

13%

Numbers

Least ⬚

0.325% 0.3

SOL

⟶ ⬚

Greatest ⬚

Which list of numbers is arranged from least to greatest? A

List the number in order from least to greatest.

SOL

355%

Write the following as percentages: a

0.4

Write the following as proper fractions or mixed numbers in their simplest form: A

145%

, 120%, 1.111

120%, , 1.111

1.111, 120%,

Write the numbers in descending order.

Justify your solution.

, 330%, 0.332

1.111, , 120%

For each list of numbers: i a

, 20%, 1

Topic 2 Assessment: Fractions, Decimals, & Percents mathspace.co

169

Complete the blanks on each numberline. 0.2

1.6

100%

200%

−

For each of the following, ﬁll in the boxes with the missing numbers: a

0.4% = 0.⬚ =

303.5% = ⬚ =

Performance Task 15

Objects in space can be incredibly far apart from each other. Our normal measurements on Earth aren’t practical to use when measuring many distances in space. Scientists use a unit called a light year, which is the distance light travels in one year. However, many planets are far closer to Earth than distant galaxies. Scientists will often measure planet distances from the sun in light minutes or light hours as opposed to light years, since those numbers are smaller and easier to comprehend. This table shows the distance of each object from the Sun at a given point in time.

Uranus

Distance from the Sun (light minutes) 162

Earth Jupiter Venus Mercury Mars

43 6.01 3.3 12.75

Saturn Neptune Earth’s moon Asteroid

170

Order each object from closest to the Sun to furthest away.

The moon orbits the Earth, which means it spends some time closer to the Sun than the Earth, and sometimes farther away. When these measurements were taken, was the Earth’s moon between the Sun and the Earth or farther away from the Sun than the Earth? Justify your thinking.

The asteroid is headed toward the Sun. What is the next planet it will pass? Justify your thinking.

NASA is designing a new rocket that can carry enough fuel to travel 140 light minutes. If the rocket must get to the planet and back to Earth, could it reach Saturn?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

b i 330%,

Topic 2 Assessment: Fractions, Decimals, & Percents 1 a 13%

b 40%

c 70%

d 60%

6.NS.1a 2 a C

6.NS.1a b 0.0002

ii We can rewrite them all in the same form. For example, if we converted them all to percentages, the original list would be: 33.2%, 165%, 330% This is in ascending order, so we need to go in the reverse order. 6.NS.1d, 6.NS.1e

b B

3 a 0.08

, 0.332

c 1.395

0.2

0.7

1.6

6.NS.1b 4 a i 86.5%

ii 0.865

b i 0.7%

100% 20%

200% 160%

70%

ii 1% 0

6.NS.1b, 6.NS.1e 5 D

1 5

7 10

3 5

4 5

6.NS.1d, 6.NS.1e

6.NS.1b, 6.NS.1e . We can group 100 into ﬁve groups of 20.

6 20% means

When we shade 20 blocks, we are shading one of the ﬁve groups. So 20% =

= .

14 a 0.4% = 0.004 =

b 303.5% = 3.035 =

6.NS.1d

6.NS.1c Performance Task

7 B

15 a M ercury, Venus, Earth’s moon, Earth, Mars, Jupiter, Asteroid, Saturn, Uranus, Neptune

6.NS.1c 8 a

b The moon is between the Sun and the Earth because is less than

6.NS.1c 9 a 83%

b 467%

c 65%

d 262%

6.NS.1c 10

Least 0.325%

⟶

Greatest

0.3

6.NS.1d, 6.NS.1e

c T he asteroid (12.7917 light minutes away) will pass Mars (12.75 light minutes away) next since it is the planet that is next closest to the Sun. d The distance from Earth to Saturn is

−

To travek there and back, the distance is 2⋅

. The rocket carrys enough fuel for

the trip. 6.CE.1b, 6.NS.1b, 6.NS.1c, 6.NS.1e, MP1, MP4

11 D 6.NS.1d, 6.NS.1e 12 a i 1,

, 20%

ii We can rewrite them all in the same form. For example, if we converted them all to percentages, the original list would be: 60%, 20%, 100%. From there we can order the percentages as 100%, 60%, 20% which gives the correct order.

Answers mathspace.co

171

3 Integers & Exponents Big ideas • Integers are a subset of the set of real numbers. • Real numbers are either rational or irrational. • Changing the form of an expression or equation can reveal information that was previously unknown. • Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations.

Chapter outline 3.01 3.02 3.03 3.04 3.05

Identify and represent integers (6.NS.2) Compare and order integers (6.NS.2) Introduction to exponents (6.NS.2) Patterns with perfect squares (6.NS.3) Powers of 10 and place value (6.NS.2)

176 193 205 217 229

The power of 10 helps scientists measure tiny molecules and vast galaxies!

3. Integers & Exponents Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 4.NS.2 — The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to seven digits. 4.CE.2 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using multiplication with whole numbers, and single-step problems, including those in context, using division with whole numbers; and recall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.

5.PFA.1 — The student will identify, describe, extend, and create increasing and decreasing patterns with whole numbers, fractions, and decimals, including those in context, using various representations.

Big ideas and essential understanding Integers are a subset of the set of real numbers. 3.01 — The set of integers is made up of all of the whole numbers and their opposites (including 0).

3.05 — The power of ten relates to the number of zeros before or after the decimal point because we use a base 10 place value system.

Real numbers are either rational or irrational. 3.02 — A number line is a tool that can be easily used for comparing the values of integers.

Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 3.03 — Repeated multiplication can be written using a base (number that is being multiplied by itself repeatedly) and an exponent (the number of times the base is written and then multiplied against itself).

Changing the form of an expression or equation can reveal information that was previously unknown. 3.04 — The special relationship between square roots and perfect squares and cube roots and perfect cubes allows us to reveal the simplest form of a radical expression.

174

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Standards 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers. 6.NS.2a — Represent integers (e.g., number lines, concrete materials, pictorial models), including models derived from contextual situations, and identify an integer represented by a point on a number line. 3.01 Identify and represent integers 3.03 Introduction to exponents 6.NS.2b — Compare and order integers using a number line. 3.02 Compare and order integers 6.NS.2c — Compare integers, using mathematical symbols (>, <, =). 3.02 Compare and order integers

6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares. 6.NS.3a — Recognize and represent patterns with bases and exponents that are whole numbers. 3.03 Introduction to exponents 6.NS.3b — Recognize and represent patterns of perfect squares not to exceed 202, by using concrete and pictorial models. 3.04 Patterns with perfect squares 6.NS.3c — Justify if a number between 0 and 400 is a perfect square through modeling or mathematical reasoning. 3.04 Patterns with perfect squares

6.NS.3d — Recognize and represent powers of 10 with whole number exponents by examining patterns in place value. 3.05 Powers of 10 and place value

Future connections 7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

8.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.

7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

7.NS.3 — The student will recognize and describe the relationship between square roots and perfect squares.

A.EO.3 — The student will derive and apply the laws of exponents.

A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

3. Integers & Exponents mathspace.co

175

3.01 Identify and represent integers Subtopic overview Lesson narrative In this lesson, students will learn to identify and represent integers using various methods. They will explore horizontal and vertical number lines as well as integer chips to visualize and model integers. The lesson includes interactive explorations where students plot integers on a number line and use integer chips to model integer expressions. Real-world examples, such as temperature changes and financial transactions, help students understand positive and negative values. By the end, students should confidently represent integers and understand their applications in real-world contexts.

Learning objectives Students: Page 86

Key vocabulary 

integer



negative



positive

 whole number

Essential understanding The set of integers is made up of all of the whole numbers and their opposites (including 0).

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can integrate this goal into their instruction by presenting students with real-world problems involving integers, such as calculating temperature changes or comparing elevations. Encourage students to apply their understanding of integers and number lines to solve these problems.

Teachers can enhance students' understanding of integers by encouraging them to justify their reasoning and use relevant vocabulary such as positive, negative, and integer. Students can explain the meaning of integers, provide real-world examples, and discuss the necessity of negative numbers. This approach helps students articulate their understanding and connect mathematical concepts to practical situations.

176

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can integrate this goal by helping students make connections between their prior knowledge of whole numbers and the new concept of integers. Additionally, teachers can use horizontal and vertical numberlines to represent integers to help students visualize whole numbers and their opposites. Also, facilitate discussions on what whole numbers are and what they represent or how integers relate to other mathematical concepts and disciplines, such as science, by discussing real-world examples with temperature and elevation.

Teachers can integrate this goal by providing opportunities for students to represent integers using a variety of models, such as horizontal and vertical number lines, concrete materials, integer chips, algebra tiles, and pictorial models. Encourage students to choose the most appropriate representation for a given situation and to make connections among different representations.

Content standards 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

6.NS.2a — Represent integers (e.g., number lines, concrete materials, pictorial models), including models derived from contextual situations, and identify an integer represented by a point on a number line.

Prior connections 4.NS.2 — The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to seven digits.

Future connections 8.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.

Lesson Preparation Tools You may ﬁnd these tools helpful: • Laminated number lines • Integer chips

3.01 Identify and represent integers mathspace.co

177

Student lesson & teacher guide Identify integers Students are introduced to the concept and definition of integers.

Students: Pages 86–87

178

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

There are other ways we can represent integers. Including vertical number lines, integer chips, and pictures. 5 4 3 2 1 0 −1 −2 −3 −4 −5

In this vertical number line, the numbers above zero are positive integers while the numbers below zero are negative integers. The point plotted on the number line represents positive 1.

Integer chips can help us model integer expressions. Each chip represents a single integer unit. Integer chips come in all kinds of colors, in this case the green chips represent +1 and the red chips represent −1. +1 +1 +1

This grouping has 5 green chips representing +1 each. The total grouping represents +5.

−1

−1 −1

−1 −1 −1

This grouping has 7 red chips representing −1 each. The total grouping represents −7.

−1

Example 1

Exploration Is each number an integer? Students: Page 86 Create a strategy For each number, we can ask ourselves: is it a positive or negative whole number, or 0?

Apply the idea • −2 is a negative whole number. So, it is an integer. •

is a positive fraction and is not a whole number. So, it is not an integer.

• 0.4 is a positive decimal and is not a whole number. So, it is not an integer. • 6 is a positive whole number. So, it is an integer. •

is a positive mixed number. So, it is not an integer.

Therefore, −2 and 6 are the only integers from the list of numbers.

3.01 Identify and represent integers mathspace.co

179

Suggested student grouping: Small groups In this exploration, students will use an applet to gain a practical understanding of how numbers are represented on a number line and the significance of the distance between the integers Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What is the significance of the distance between each labeled integer on the number line? Each labeled tick on the number line is 5 units away from the next. This distance represents the difference between two consecutive integers on the scale. 2. How do you plot a given number on the number line? To plot a number on the number line, identify the tick marks that the number would fall between. For this number line, if the given number is 7, it’s between 5 and 10. So, you would plot it between the ticks for 5 and 10 on the number line. Purposeful questions • If you want to plot the number 12, where would it be placed on this number line, and how do you know? • What do you notice about the spacing between the tick marks, and how does that help you understand the number line? • Can you describe how to use the number line to compare the sizes of two numbers, such as 7 and 14? Possible misunderstandings • Assuming each tick mark represents 1 unit instead of 5 units. Guide students to observe the labels and calculate the differences to understand the correct scale. • Believing that only the labeled integers exist on the number line. Emphasize that all real numbers, including those between the labels, are represented on the number line.

Number lines of different scales Targeted instructional strategies Have students look at number lines with different scales and divisions. Show some number lines where only the even integers are numbered and some with every half numbered. Ensure different divisions so students don’t get used to number lines with every other tick representing only an integer. For example: −10

−8

−3

−2.5

−6

−2

−4

−1.5

−1

−2

−0.5

0.5

1.5

2.5

Challenging scales Address student misconceptions When students get in a routine of plotting points on a number line, they often immediately draw a line and number it from −10 to 10 without first looking at the numbers they need to plot. Discuss first looking at the numbers they need to plot, how to determine the scale, and to identify the integer represented by points on ticks with no labels. Provide exercises that use number lines with challenging scales.

180

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−1

Examples Students: Page 87

−1 −1

−1

This grouping has 7 red chips representing −1 each. The total grouping represents −7.

−1 −1

−1

Example 1 Is each number an integer?

Create a strategy For each number, we can ask ourselves: is it a positive or negative whole number, or 0?

Apply the idea • −2 is a negative whole number. So, it is an integer. •

is a positive fraction and is not a whole number. So, it is not an integer.

• 0.4 is a positive decimal and is not a whole number. So, it is not an integer. • 6 is a positive whole number. So, it is an integer. •

is a positive mixed number. So, it is not an integer.

Therefore, −2 and 6 are the only integers from the list of numbers.

Purpose 3.01 Identify and represent 87 Students to demonstrate that they understand the definition of integers and can identify them integers among various mathspace.co types of numbers. Reflecting with students Encourage students to communicate precisely by using clear and accurate definitions of mathematical terms. Start the lesson by having them articulate the definition of an integer—for example, “An integer is a whole number that can be positive, negative, or zero, without fractions or decimals.” Prompt them to apply this definition to each number in the list, explaining their reasoning explicitly. For instance, they might say, “−2 is an integer because it is a negative whole number without fractions or decimals.” By emphasizing precise language and definitions, you help students deepen their understanding and ensure they can accurately identify integers in various forms.

Students: Page 88 Example 2 Consider the number −10. a Represent −10 on a number line.

Create a strategy We can ask ourselves: is it a positive or negative whole number, or 0? This will help determine where the number belongs on the number lines.

Apply the idea Since −10 is negative, place it 10 units left of 0 on the number line. −12 −10 −8

−6

−4

−2

b Represent −10 with integer chips.

Purpose Create a strategy Show students how to represent negative integers in different ways.

Decide on colors for your positive and negative integer chips. Determine whether you need positive or negative chips to represent −10. 3.01 Identify and represent integers

mathspace.co

Apply the idea We need 10 negative integer chips to represent the number −10.

181

belongs on the number lines.

Apply the idea Since −10 is negative, place it 10 units left of 0 on the number line. −12 −10 −8

Students: Page 88

−6

−4

−2

b Represent −10 with integer chips.

Create a strategy Decide on colors for your positive and negative integer chips. Determine whether you need positive or negative chips to represent −10.

Apply the idea Example 2 We need 10 negative integer chips to represent the number −10. Consider the number −10. a Represent −10 on a number line.

−1

−1 −1

Create a strategy

−1

−1 −1

−1

We can ask ourselves: is it a positive or negative whole −1 number,−1or 0? This will help determine where the number belongs on the number lines. −1

Apply the idea Since −10 is negative, place it 10 units left of 0 on the number line. −12 −10 −8 −6 −4 −2 0 2 4 6 PurposeExample 3 Show students to line, represent negative integers in different ways. For each how number determine where the point plotted is.

b a Represent −10 with integer chips.

Think-aloud for0 understanding negative numbers −12 −8modeling −4 4 8 12

use with Example 2

Student disabilities support Create awith strategy Create a strategy

Decide on colors for your step-by-step positive and negative integer chips. Determine you need positive or negative Model your thinking process as you represent −10 on awhether number line and with integer chips. Speak Every second tick is−10. labeled with a multiple of 4. Since there are 4 units between every second tick, each tick is chips to represent aloud your reasoning: 2 units apart. • “Since Applynegative the idea numbers are to the left of zero, I will start at 0 and count 10 units to the left to place −10 on Apply the idea the number We need 10 line.” negative integer chips to represent the number −10. The point 2 units to thefor left−1, of −4. is at −6. • “Each red is chip stands so so I’llitneed 10 red chips to represent −10.” −1

−1 By verbalizing each step, you help students follow the logical sequence and understand why each action is −1 −1 themselves, either individually or with partners, taken. Encourage students to practice this think-aloud strategy −1 −1 −1 to articulate their own understanding and identify any areas of confusion. −1 88

Mathspace

−1 −1

Virginia SOL Grade 6

mathspace.co Students: Page 88

Example 3 For each number line, determine where the point plotted is. a

−12

−8

−4

Create a strategy Every second tick is labeled with a multiple of 4. Since there are 4 units between every second tick, each tick is 2 units apart.

Apply the idea The point is 2 units to the left of −4. so it is at −6.

Purpose Check students ability to accurately read the scale on a horizontal number line and to identify the integer not Mathspace Virginia SOL Grade 6 88 explicitly labeled. mathspace.co 182

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Expected mistakes Students may struggle to interpret the number that isn’t labeled and think that the number is a fraction greater than the whole number before it or 1 greater than the number before it. For example, if a point is between 4 and 8, students may think that the number is 4.5 or 5.

Students: Page 89 b 5

−5

Create a strategy Remember that numbers above zero are positive numbers and numbers below zero are negative numbers on a vertical number line. The marked numbers are 0 and 5. The segment between 0 and 5 is divided into 5 parts so each part is equivalent to one unit.

Apply the idea The point is located above zero on the vertical number line so it is positive. The point is on the third mark. The integer is positive 3.

Idea summary

Purpose An integer is a positive or negative whole number, or 0. Check students ability to accurately read the scale on a vertical number line and to identify the integer not On a number line positive integers are to the right of 0 and negative integers are to the left of 0. explicitly labeled. Reflecting with students Ask students what words can be associated Integers in the real world when moving up or moving down on a vertical number line. Integers are extremely useful numbers because they can be used to describe many things in the real-world.

Steps to identify on(−)unnumbered ticks about a real-world situation. The positive sign (+) andintegers negative sign tell us important information

use with Example 3

Targeted instructional strategies 50

Sometimes the point on a number line representing an integer is on an unnumbered tick such as the following: 40 20

−8 30

−4

The following steps can be followed to identify an a number line. represents a temperature Forinteger example,on a positive temperature 0

above zero, while a negative temperature represents a temperature

1. Identify the scale or distance marked numbers on the number line. −10 between two consecutive below zero.

−20 From 4, the next number labeled to the right is 8 so the distance between them is 8 − 4 = 4. Check that the −30 two consecutive marks on the number line such as between 0 and 4. distance is true between any

2. Count the number of parts−40 the line is divided between two consecutive marked numbers. −50

The length between 0 to −4 is divided in two parts. 3. Divide the scale by the number of parts. With a scale of 4 with 2 parts, 4 ÷ 2 = 2. That means every tick represents 2 units. Check by counting by 2 units to the right or 2 units to the left. The point on the number line above is between 0 and −4. It is 2 units to the3.01 left of 0,and thus the point Identify represent integersis −2. 89 mathspace.co

3.01 Identify and represent integers mathspace.co

183

0 The marked numbers are 0 and 5. The segment between 0 and 5 is divided into 5 parts so each part is equivalent to one unit.

Apply the idea −5

The point is located above zero on the vertical number line so it is positive.

Students:The Page point 89 is on the third mark. The integer is positive 3. Create a strategy

Remember that numbers above zero are positive numbers and numbers below zero are negative numbers on a vertical number line.

Idea summary

The marked numbers are 0 and 5. The segment between 0 and 5 is divided into 5 parts so each part is equivalent to An integer is a positive or negative whole number, or 0. one unit. On a number line positive integers are to the right of 0 and negative integers are to the left of 0.

Apply the idea The point is located above zero on the vertical number line so it is positive.

Integers realThe world The point is onin thethe third mark. integer is positive 3.

Integers inarethe real world Integers extremely useful numbers because they can be used to describe many things in the real-world. Thelearn positive (+) and negative sign (−) tell importantfocusing information about a real-world Students will thesign real-world applications ofus integers, particularly on thesituation. significance of positive and Idea summary negative signs. 50 An integer is a positive or negative whole number, or 0. 40

On a89–90 number line positive integers are to the right of 0 and negative integers are to the left of 0. Students: Pages 20 30 10

For example, a positive temperature represents a temperature above zero, while a negative temperature represents a temperature Integers are extremely useful can be used to describe many things in the real-world. −10 numbers because they below zero. The positive sign (+) and negative sign (−) tell us important information about a real-world situation. −20

Integers in the real world 0 −30 50 −40 40 −50 20 30 10

For example, a positive temperature represents a temperature above zero, while a negative temperature represents a temperature below zero.

0 −10 −20

3.01 Identify and represent integers mathspace.co

−30

−40 −50

In some situations, the positive sign indicates an increase, while the negative sign indicates a decrease.

Coffee Shop Menu

Coffee Shop Menu Espresso

Americano

Latte

Choco Latte

$10

Caramel Latte

$12

Espresso +1 −1

3.01 Identify and represent integers $5 mathspace.co

Americano

Latte

Choco Latte

$10

Caramel Latte

$11

For example, let’s say that a local coffee shop is increasing the prices of some of their menu items by $1. We can quickly describe this situation by writing +1. The positive sign indicates an increase, and the 1 indicates going up by 1. Similarly, a decrease in the price of a menu item by $1 can be represented by the number −1.

Example 4 Write an integer to represent the statement: A loss of $52.

Create a strategy 184

We will start by identifying whether our integer should be positive or negative based on the situation. Since we are representing a loss of money, we will have a negative integer. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Apply the idea

We know that our integer will be negative since we are representing a loss. Since it is a loss of $52, the integer that represents the statement is −52.

Use the STEAM cycle with integers Targeted instructional strategies Introduce a scenario where students track daily temperature changes over a week, using integers to represent fluctuations above and below zero degrees. Ask: Begin by asking students how they can represent temperature changes that are above and below zero, encouraging them to share ideas about using positive and negative numbers in real-world contexts. Imagine: Encourage students to brainstorm ways to visualize these temperature changes, such as plotting points on horizontal or vertical number lines or using integer chips to model gains and losses in temperature. In some situations, positive an increase, the negative sign helping indicates a decrease. Plan: Guide students to the plan how sign theyindicates will represent thewhile temperature data, them set criteria for accuracy and choose appropriate tools like number lines, counters, or drawings to illustrate integers.

Create and test: Have students create their representations of the temperature fluctuations using their chosen Coffee Shop Menu Coffee Shop Menu methods, and then test their models by explaining them to classmates and checking for correct integer Espresso Espresso $5 $5 representation. Americano

Americano

Improve: Facilitate a discussionLatte where students$8reflect on their models, receive $9 feedback, and collaborate to Latte Choco Latte Choco Latte $10 $10 refine their representations, justifying any changes to enhance clarity and accuracy. −1 Caramel Latte

$12

Caramel Latte

$11

Examples For example, let’s say that a local coffee shop is increasing the prices of some of their menu items by $1. We can Students:quickly Pagedescribe 90 this situation by writing +1. The positive sign indicates an increase, and the 1 indicates going up by 1. Similarly, a decrease in the price of a menu item by $1 can be represented by the number −1.

Example 4 Write an integer to represent the statement: A loss of $52.

Create a strategy We will start by identifying whether our integer should be positive or negative based on the situation. Since we are representing a loss of money, we will have a negative integer.

Apply the idea We know that our integer will be negative since we are representing a loss. Since it is a loss of $52, the integer that represents the statement is −52.

Example 5 Purpose Show students how integers can be usedbytothe represent situations. Let the location of a city be represented integer 0, real-world and let a point 7 km to the east of the city be represented by the integer 7. What integer represents the point 4 km to the west of the city?

Reflecting with students Create a strategy Encourage advanced learners, or any students who are ready, to deepen their understanding of integers Use a number represent the“−52” information. by presenting them line withtothe integer and asking them to create a variety of real-world situations that could be represented by this number. Invite them to think beyond financial losses, exploring contexts such as Apply the idea temperatures (e.g., a drop of 52 degrees), elevations (e.g., 52 feet below sea level), or points in a game (e.g., a penalty of 52 points). Facilitate a discussion where students share their scenarios and explore the similarities 4 km west 7 km east and differences between them, highlighting how integers function in diverse situations. −8 −7 −6 −5 −4 −3 −2 −1 0

1 2 3 4 5 6 7 8

−4 represents the point 4 km to the west of the city.

Mathspace Virginia SOL Grade 6 mathspace.co

3.01 Identify and represent integers mathspace.co

185

representing a loss of money, we will have a negative integer.

Apply the idea We know that our integer will be negative since we are representing a loss. Since it is a loss of $52, the integer that represents the statement is −52.

Students: Page 90 Example 5

Let the location of a city be represented by the integer 0, and let a point 7 km to the east of the city be represented by the integer 7. What integer represents the point 4 km to the west of the city?

Create a strategy Use a number line to represent the information.

Apply the idea 4 km west

7 km east

−8 −7 −6 −5 −4 −3 −2 −1 0

1 2 3 4 5 6 7 8

−4 represents the point 4 km to the west of the city.

Purpose90 Mathspace Virginia SOL Grade 6 mathspace.co Challenge students to apply their understanding of integers to represent and solve real-world problems.

Vocabulary exercise: collect and display

use with Example 5

English language learner support As students work through the problem, listen for how they describe the concepts of “east,” “west,” “positive,” “negative,” and “represent.” Collect the different terms and phrases students use and display them in a visible area of the classroom. If students struggle with these terms, provide visual aids such as a compass rose showing “east” and “west,” and a number line with the city at zero, positive integers to the east, and negative integers to the west. Connect the directional words to the mathematical representations by showing how moving east corresponds to positive numbers and moving west corresponds to negative numbers. Encourage students to refer to these visuals when discussing the problem to help them build their mathematical vocabulary. This display will serve as a reference that students can use during this lesson and future discussions about integers and direction.

Students: Page 91

Idea summary A rise or increase in value will be represented by a positive integer. A loss or decrease in value will be represented by a negative integer.

Practice What do you remember? 1

What is an integer?

For the point on the number line −10

186

−5

a What number is represented by the point on the number line? Mathspace b Virginia SOL Grade 6 Teacher Edition Is the number positive, negative or neither? mathspace.co c Is the number an integer? SOL

Which of these is an integer?

Practice Students: Pages 91–95

What do you remember? 1

What is an integer?

For the point on the number line −10

What number is represented by the point on the number line?

Is the number positive, negative or neither?

Is the number an integer?

Which of these is an integer? B

A 4

−5

7.2

−13

Identify each value that represents an integer. Select all that apply. B

−42

E 5

What integer does the point furthest to the left on the number line lie on? −10

−5

−7

−4

Which arrow shows the location of −7 on the number line? 0 O P

A 8

Which is the closest location of point B on the number line? −3

Which value represents receiving $33: +33 or −33?

3.01 Identify and represent integers mathspace.co

187

Let’s practice 9

Determine the integer that the point is plotted on for each horizontal number lines: a

−4 −3 −2 −1 0

−10

−5

−10

−5

−8

−4

10 8 6 4 2 0 −2 −4 −6 −8 −10

5 0

−5 −5

−10

−4

−10

−5

−6

−8

−20

−10

−12

Plot each integer on a horizontal number line: a

−1

−5

−12

−3

Plot each integer on a vertical number line: a

−2

−4

−12

−8

−23

−20

Does each number line show the number −19? a

−16

−8

−16

−32

−24

188

Determine the integer that the point is plotted on for each vertical number lines:

5 4 3 2 1 0 −1 −2 −3 −4 −5

f 5

−3

d 5

−4 −3 −2 −1 0

−6

−5

−29

−26

d −8

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−24

−18

−12

−6

Write an integer to represent each statement: a

A price rise of $17.

A loss of $176.

A temperature drop of 5 °F.

58 °F above 0 °F.

An elevation of 980 ft.

Grew by 9 cm last year.

14 more candies today than yesterday

Deposited $100 to his account.

Descending 9 ﬂoors.

A weight loss of 14 kg.

Depositing $75 in to a bank account.

m A weight gain of 8 kg. o 15

Withdrawing $145 from a bank account.

A distance of 320 ft below sea level.

Ascending 10 ﬂoors.

A proﬁt of $650.

Using the key below, identify the integer represented in the picture and explain your thinking. Key = +1 = −1

When Frankie woke up, the temperature was −4 °C. During the day it rose by 11 °C. Which number line correctly shows this temperature change? A

−5

What integer represents 40° below zero? 50°F 40° 30° 20° 10° 0° −10° −30° −20° −40° −50°

40 units

3.01 Identify and represent integers mathspace.co

189

Use the image below to answer the following: a

What integer represents the water level?

What integer represents the vertical height of the boat?

What integer represents the depth of the hook?

8 7 6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8

A mountain peak is measured to have a height of 5000 m above sea level. An ocean trench is found that is the same distance below sea level. a

If the height above sea level of the mountain peak is represented by the number 5000, state the integer that represents the location of the trench.

In this scenario, state what the integer 0 represents.

Let’s extend our thinking 20

Consider the given vertical number line: a

If point C is +3, then which point is −3?

Is point F positive, negative or neither?

Which points represent positive integers?

A B +3 D E F G

Sarah plots the number 7 on a horizontal number line. Eric tells her to move 3 units to the right and then 6 units to the left. a

Draw a number line showing the initial position of the point.

Draw a number line showing the ﬁnal position of the point.

Identify the number Sarah ends up on.

If zero lies between a and d, explain how to ﬁnd one set of possible values for a, b, c, and d.

b c

190

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The image shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line. 5 m above 4 m above 3 m above 2 m above 1 m above Surface 1 m below 2 m below 3 m below 4 m below 5 m below 6 m below 7 m below 8 m below

5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8

What integer represents 5 m below the surface?

Bill is at a location represented by the integer −3, and he goes up 5 m. Describe Bill’s new location.

Using the key below, identify the integer represented in the picture and explain your thinking. Key = +1 = −1

3.01 Identify and represent integers mathspace.co

191

Answers

3.01 Identify and represent integers What do you remember?

g h 10

−10

−5

−20

−10

1 Any positive or negative whole number or zero. 2 a 0

b Neither

c Yes

3 C 4 B, C and E 5 −8

13 a No

b No

c Yes

d No

14 a 17

b −176

c −5

d 58

6 A

e 980

g 14

h 100

7 B

−9

−14

k 75

n 10

o −145

p 650

b +10 or 10

c −6

d +11 or 11

m 8

8 −33

15 a −9 Let’s practice

16 C

9 a 2 e −5

b −1

c 2

d 8

−10

g 3

h −6

b −6

c 3

d −7

−8

g 3

h −10

10 a 1 e −5

11 a

17 −40° 18 a 0

b 2 or +2

19 a −5000

b Being at sea level.

20 a E

b −5

c A, B, C

0 1 2 3 4 5 6 7 8 9 10 −10

−5

b 0 1 2 3 4 5 6 7 8 9 10

d −5

c 4

e −10

−5

22 Possible answer: a = 5, b = 1, c = −1, d = −4

23 a −5

f −12 −8

−4

b Bill is at 2 m above the surface.

g −5 −4 −3 −2 −1 0 1 2 3 4 5

h −5 −4 −3 −2 −1 0 1 2 3 4 5

192

b Negative

21 a

12 a

c −8

Let’s extend our thinking

−5 −10

−320

5 4 3 2 1 0 −1 −2 −3 −4 −5

c d 5 4 3 2 1 0 −1 −2 −3 −4 −5

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

10 5 0 −5 −10

24 The integer represented in the picture is − 3. This is because each black dot represents + 1 and each white dot represents − 1, according to the key provided. In the picture, there are 6 black dots and 9 white dots. If we pair each black dot with a white dot, they cancel each other out because + 1 (black dot) and −1 (white dot) together equal 0. After canceling out 6 pairs of black and white dots, we are left with 3 unpaired white dots. Since each white dot represents −1, the three unpaired white dots together represent −3. Therefore, the integer being depicted is −3, illustrating that the net effect of the dots is negative.

3.02 Compare and order integers Subtopic overview Lesson narrative In this lesson, students will learn to compare and order integers. They begin by using number lines to understand the relative positions of integers. Students will practice using inequality symbols (<, >, =) to compare integers and order them from least to greatest or vice versa. The lesson includes explorations with interactive applets to compare integers dynamically using a horizontal number line. Problems include identifying the largest or smallest integer, arranging numbers in ascending or descending order, and solving real-world scenarios to compare and order integers involving temperature and financial data. By the end, students should confidently compare and order integers.

Learning objectives

3.02 Compare and order integers

Students: Page 96

After this lesson, you will be able to... • compare integers using number lines and inequality symbols. • use a number line to order integers in ascending or descending order.

Compare and order integers We can use the following symbols to compare integers: Key vocabulary 

> means “is greater than”, such as 7 > 2  descending order ascending order < means “is less than”, such as −2 < 4



inequality symbol

= means “is equal to”, such as −6 = −6 The number that is furthest right on the number line is always greater than the numbers on its left. Essential understanding

A number line is a tool that can be easily used for comparing the values of integers. −8 −7 −6 −5 −4 −3 −2 −1 0

−4 is the furthest to the left, therefore it is the smallest integer plotted on this number line.

Standards 8 is the furthest to the right, therefore it is the largest integer plotted on this number line.

This subtopic addresses thesmallest following Virginiais2023 Standards of 8. Learning standards. Writing the numbers from to largest calledMathematics ascending order: −4, 0, 3, Writing the numbers from largest Mathematical process goalsto smallest is called descending order: 8, 3, 0, −4.

MPG3 — Mathematical Reasoning Interactive exploration Teachers canExplore incorporate reasoning into the lesson by asking students to justify their stepsl in the onlinemathematical to answer the questions process of comparing and ordering integers orally, in writing, or using a mode. Have students explain why certain mathspace.co rules apply when comparing integers using number lines or mathematical symbols. Additionally, teachers can faciliate discussions around identifying opposite integers and explaining different situations where an integer and its opposite interactive exploration in 3.02 to answer this question. wouldUse be the used. 1.

What happens to the inequality symbol when the order of the integers is swapped?

Example 1 Which is the largest number marked on the number line?

3.02 Compare and order integers mathspace.co

193

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can make connections between this lesson and previous lessons on number systems, as well as real-world applications, such as comparing temperatures or financial gains and losses. Encourage students to recognize the relationship between integers and other mathematical concepts they have learned. Additionally, teachers can use horizontal and vertical numberlines to represent, compare, and order integers to help students visualize whole numbers and their opposites. Also, facilitate discussions on what whole numbers are and what they represent or how integers relate to other mathematical concepts and disciplines, such as science, by discussing real-world examples with temperature and elevation.

Teachers can promote the use of mathematical representations by having students use horizontal and vertical number lines, inequality symbols, and contextual examples to represent, compare, and order integers. Encourage students to make connections between different representations and understand the importance of using various methods

Content standards 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

6.NS.2c — Compare integers, using mathematical symbols (>, <, =).

6.NS.2b — Compare and order integers using a number line.

Prior connections 4.NS.2 — The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to seven digits.

Future connections 7.NS.2 — The student will reason and use multiple strategies to compare and order rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.01 Identify and represent integers

Tools You may ﬁnd these tools helpful: • Index cards • Number line • Integer chips

194

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide Compare and order integers Students are introduced to the symbols used to compare integers and the concept of ordering integers in ascending and descending order. They learn that the number furthest right on the number line is always greater than the numbers on its left. In the exploration, students will manipulate an applet to compare two integers on a number line and observe the changes to the inequality symbol when the order of the integers is swapped.

Students: Page 96

3.02 Compare and order integers After this lesson, you will be able to... • compare integers using number lines and inequality symbols. • use a number line to order integers in ascending or descending order.

3.02 Compare and order integers Compare and order integers

We can use the following symbols to compare integers: After this lesson, you will be able to... > means “is greater than”, such as 7 > 2 • compare integers using number lines and inequality symbols. < means “is lessathan”, such < 4 integers in ascending or descending order. • use number lineasto−2 order = means “is equal to”, such as −6 = −6 The number that is furthest right on the number line is always greater than the numbers on its left.

Compare and order integers

−8 −7 −6 −5 −4 −3 −2 −1 0

We can use the following symbols to compare integers:

> greater than”, as 7 > it2 is the smallest integer plotted on this number line. −4means is the “is furthest to the left,such therefore < “is lessto than”, such therefore as −2 < 4 it is the largest integer plotted on this number line. 8 means is the furthest the right, = meansthe “isnumbers equal to”,from suchsmallest as −6 =to −6largest is called ascending order: −4, 0, 3, 8. Writing The number that is furthest right on number line isdescending always greater than on its left. Writing the numbers from largest to the smallest is called order: 8,the 3, 0,numbers −4. −8 −7 −6 −5 −4 −3 −2 −1 0 Interactive exploration

Explore online to answer the questions

−4 is the furthest to the left, therefore it is the smallest integer plotted on this number line. Exploration

mathspace.co 8 is the furthest to the right, therefore it is the largest integer plotted on this number line. Students:Writing Pagethe96 numbers from smallest to largest is called ascending order: −4, 0, 3, 8.

Use the interactive exploration in 3.02 to answer this question. Writing the numbers from largest to smallest is called descending order: 8, 3, 0, −4. 1. What happens to the inequality symbol when the order of the integers is swapped?

Interactive exploration Explore online to answer the questions

Example 1

mathspace.co

Which is the largest number marked on the number line? Use the interactive exploration in 3.02 to answer this question. 1.

What happens to the symbol when is swapped? 0 the order5 of the integers 10 15 −10inequality−5

Create a strategy Recall that the further an integer is to the right on a number line, the larger the integer is.

Example 1

Apply Which isthe theidea largest number marked on the number line? The integer farthest to the right on the number line is 13. So, the largest number is 13. −10

Create a strategy

−5

3.02 Compare and order integers mathspace.co

195

Suggested student grouping: In pairs Students will be using an interactive applet to compare integers on a number line. They will be moving the points of the integers and observing the changes in the inequality symbol when the order of the integers is swapped. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What happens to the inequality symbol when the order of the integers is swapped? When the order of the integers is swapped, the direction of the inequality symbol also switches. So, if the inequality was 3 > −1, when swapped, it becomes −1 < 3. Purposeful questions • What do you notice about the inequality symbol when you move one integer past another on the number line? • How does changing the positions of the integers on the number line affect the inequality statement between them? • Can you explain how the direction of the inequality symbol relates to the positions of the integers on the number line? Possible misunderstandings • Confusion about negative integers being greater than positive integers due to their absolute values (e.g., believing that −5 > 3 because 5 is greater than 3). Revisit the number line to see that negative integers are to the left of positive integers, reinforcing that numbers further to the right are greater regardless of absolute value.

Larger or smaller? Targeted instructional strategies When comparing and ordering integers, using a number line is a great visual strategy to help students see the value of the number compared to 0. Using a number line will not always be efficient so encouraging students to learn to mentally compare the values of the integers is important. An example activity to reinforce this is a mix, pair, share. Have index cards each labeled with one integer from −15 up to 15. Give one card to each student, and have students roam around the room, while playing some fun music. When the music pauses, students will pair up with the person closest to them. Have them compare the integers on their index cards, using mathematical language like, “My integer is 5 ticks above 0” or “My integer is 7 wholes below 0. The student with the larger integer, remains standing ‘big and tall’ while the student with the smaller integer crouches down, ‘low and small’.

Physically accessible items Student with disabilities support Ensure that students have a physical laminated number line in front of them, so they can easily compare or order integers. Provide large dry erase markers that are easier for students with limited motor skills to use. You can even have students place objects like small blocks or game chips as points on the number line if drawing poses too much of a challenge. Another option for more easily drawing points is to use dot markers (or bingo daubbers) on an oversized copy of a number line.

196

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The number that is furthest right on the number line is always greater than the numbers on its left. −8 −7 −6 −5 −4 −3 −2 −1 0

−4 is the furthest to the left, therefore it is the smallest integer plotted on this number line. 8 is the furthest to the right, therefore it is the largest integer plotted on this number line. Direction matters Writing the numbers from smallest to largest is called ascending order: −4, 0, 3, 8. Address student misconceptions

Writing the numbers from largest to smallest is called descending order: 8, 3, 0, −4.

Students may generalize that the more digits a number has, the larger the number is. They may struggle to realize that with Interactive negative numbers the opposite is true. The more negative a number is, the farther it is to the exploration Explore onlinethus to answer the questions left of the number line and the smaller the number.

mathspace.co Use the interactive exploration in 3.02 to answer this question. Examples 1.

What happens to the inequality symbol when the order of the integers is swapped?

Students: Page 96 Example 1

Which is the largest number marked on the number line? −10

−5

Create a strategy Recall that the further an integer is to the right on a number line, the larger the integer is.

Apply the idea The integer farthest to the right on the number line is 13. So, the largest number is 13.

Purpose96 Mathspace Virginia SOL Grade 6 mathspace.co Ensure students are able to identify the integer and compare positive and negative numbers plotted on the number line.

Greater than and less than

use with Example 1

Student with disabilities support A fun way to help remember greater than and less than symbols is by associating them with the alligator’s mouth. Imagine an alligator that loves eating larger amounts of food. The symbol > for greater than is similar to an alligator’s mouth that opens to the left. For example, we know that 4 is greater than 3 and so in symbols, it is 4 > 3 where the alligator’s mouth opens to 4. The symbol < for less than that is written between two numbers means that the number on the left is less than the number on the right. So, −8 < −2 translated into words is “−8 is less than −2” where the alligator’s mouth opens to the right, the greater number.

Students: Page 97

Example 2 Consider the numbers −3 and −9. a Graph −3 and −9 on the number line.

Create a strategy We can see that −3 and −9 are both negative and so will be to the left of 0.

Apply the idea To plot the point −3, start at 0 and count left 3 places. To plot the point −9, we can start at −5 and jump left a further 4 places. 3.02 Compare and order integers mathspace.co −10

−5

197

Consider the numbers −3 and −9. a Graph −3 and −9 on the number line.

Create a strategy We can see that −3 and −9 are both negative and so will be to the left of 0.

Example 2

Apply the idea To plot thethe point −3, start 0 and Consider numbers −3at and −9. count left 3 places. To plot the point −9, we can start at −5 and jump left a further 4 places. a Graph −3 and −9 on the number line.

Create a strategy

−10

−5

We can see that −3 and −9 are both negative and so will be to the left of 0. b Insert either < or > to make a true statement.

Apply the idea

−3 ⬚ − 9

PurposeTo plot the point −3, start at 0 and count left 3 places. To plot the point −9, we can start at −5 and jump left a further 4 places. ability to plot two negative numbers correctly on a number line. Check students Create a strategy

−10 numbers,−5 0 If Page we are comparing two negative the number closer to zero will5 be the larger10number. 2 Students:Example 97

Consider the numbers −3 and −9. Apply the idea b Graph Insert either or >on tothe make a trueline. statement. We can see previous number line that −3 is closer to 0 which means it is larger than −9. So, −3 > −9. a −3from and< the −9 number −3 ⬚ − 9

Create a strategy

We can see that −3 and −9 are both negative and so will be to the left of 0. Create a strategy

Example 3

If we are comparing two negative numbers, the number closer to zero will be the larger number. Arrange the following numbers in ascending order: To plot the the point To plot the point −9, we can start at −5 and jump left a further Apply idea −3, start at 0 and count left 3 places. 11, −25, 19, −15, 28 4 places. We can see from the previous number line that −3 is closer to 0 which means it is larger than −9. So, −3 > −9.

Apply the idea

Create a strategy

−10

−5

Ascending means ordering from smallest to largest. We can do this by going from left to right on the number line.

Example 3 PurposebApply Insert either the idea< or > to make a true statement. Ensure students are able compare two negative Arrange the following numbers in ascending order: numbers −3 ⬚ − 9 on a number line and use symbols to show Plot the points on the to number line: relationship between them. 11, −25, 19, −15, 28 Create a strategy

−30 −25 −20 −15 −10

−5

Reflecting with students If we areacomparing Create strategy two negative numbers, the number closer to zero will be the larger number. Encourage students read statement as a sentence, and consider if it reads true. −3 is greater than −9. Arrange the list to from leastthe to greatest Ascending means ordering from smallest to largest. We can do this by going from left to right on the number line. Does that sound Is −3 closer to 0? −25, −15, 11, 19, 28 Apply the reasonable? idea We canthe seeidea from the previous number line that −3 is closer to 0 which means it is larger than −9. So, −3 > −9. Apply

Students: Page 97

Plot the points on the number line:

Example 3

−30 −25 −20 −15 −10

−5

Arrange the list from least to greatest Arrange the following numbers in ascending order: −25, −15, 11, 19, 28 11, −25, 19, −15, 28

3.02 Compare and order integers mathspace.co

Create a strategy Ascending means ordering from smallest to largest. We can do this by going from left to right on the number line.

Apply the idea Plot the points on the number line: −30 −25 −20 −15 −10

−5

3.02 Compare and order integers mathspace.co

Arrange the list from least to greatest −25, −15, 11, 19, 28

198

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Encourage students to use their knowledge of integers on a number line to put the numbers into ascending order. Expected mistakes Students may not consider the negative sign in plotting points on the number line and so mistakenly order the numbers. They may also arrange the numbers in ascending order instead of descending order. Challenge this misconception by asking students which temperature is higher, −40°F or −60°F?

Advanced learners: Deepen understanding through multi-layered tasks use with Example 3 Targeted instructional strategies After students have arranged the given numbers in ascending order, add layers of complexity to deepen their understanding. Encourage them to consider how adding decimals or fractions, such as −15.5 or 19.75, would affect the ordering of the numbers. Invite students to explore the concept of absolute value and discuss how it relates to the distance from zero on the number line. Challenge them to create their own sets of numbers, including both rational and irrational numbers, and arrange these in ascending order. This approach helps students connect the ordering of integers to the broader real number system, fostering a deeper understanding of numerical relationships.

Students: Page 98

Example 4 The melting point of krypton is −157 °C. The melting point of radon is −71 °C. Write an inequality comparing the two melting points.

Create a strategy

Apply the idea

A negative integer with a smaller number has a greater value than a negative integer with a larger number. It is further to the left of 0 on a number line.

−71 is a negative integer with a smaller number which means it has a greater value than −157. −157 °C < −71 °C

Idea summary Purpose The sizes of integers can be compared using inequality symbols. Check students’ of how to compare negative numbers using inequalities and apply this Theunderstanding symbol < represents the phrase is less than. understandingThe in symbol real-world context. > represents the phrase is greater than. The symbol = represents the phrase is equal to.

Discussion supports: comparing negative temperatures

use with Example 4

English language learner support Facilitate a class discussion to help students articulate their understanding of comparing negative temperatures Practice using inequalities. Provide sentence stems to support their use of mathematical language and vocabulary related to temperatures and inequalities. For example: What do you remember? • “When comparing −157°C and −71°C, the temperature that is warmer is ⬚ because...” • “Since1 −157°C is less than/greater than −71°C, the correct inequality is...” Which point on the number line represents the greatest integer? • “In negative numbers, a number with a smaller/larger absolute value is actually ⬚ on the number line because...” L J M K • “The melting point of krypton is ⬚ than the melting point of radon, so we write the inequality as...” A

Encourage students to use these stems to discuss their reasoning with partners or small groups. This will help 2 State the greatest number plotted on the following number line: them practice using comparative language and deepen their understanding of how negative numbers are ordered, especially in real-world contexts like temperature. −10

State the smallest number plotted on the following number line: −20

−5

−15

−10

−5

Determine if the following statements are true or false.

3.02 Compare and order integers mathspace.co

199

Write an inequality comparing the two melting points.

Create a strategy

Apply the idea −71 is a negative integer with a smaller number which means it has a greater value than −157.

A negative integer with a smaller number has a greater value than a negative integer with a larger number. It is Students:further Page 98left of 0 on a number line. to the

−157 °C < −71 °C

Idea summary The sizes of integers can be compared using inequality symbols. The symbol < represents the phrase is less than. The symbol > represents the phrase is greater than. The symbol = represents the phrase is equal to.

Practice What do you remember? Practice 1

Which point on the number line represents the greatest integer?

Students: Pages 98–100 A

What do you remember? 2

−5

200

State the smallest number plotted on the following number line:

−20

C −15

−10

−5

State the greatest number plotted on the following number line: 4

State the greatest number plotted on the following number line: −10

Which point on the number line represents the greatest integer? 3

Determine if the following statements are true or false. a

−10 is always −5to the left of0a positive integer 5 on a number 10 line. A negative integer

Zero is considered a positive integer.

State the csmallest number plotted onsmaller the following A negative integer is always than zero. number line: d

On a number line, every integer to the right of another integer is always greater.

A positive integer is always −20 −15 larger −10than a−5negative 0 integer. 5

The number zero is always located between the positive and negative integers on a number line.

integer is either positive,are negative, or zero. Determineg if Every the following statements true or false. h

An integer’s value decreases as it moves to the right on a number line.

A negative integer is always to the left of a positive integer on a number line.

Zero is considered a positive integer.

Mathspace VirginiaisSOL Grade 6smaller than zero. 98negative A integer always

On a number line, every integer to the right of another integer is always greater.

A positive integer is always larger than a negative integer.

The number zero is always located between the positive and negative integers on a number line.

Every integer is either positive, negative, or zero.

An integer’s value decreases as it moves to the right on a number line.

mathspace.co

For each of the following pairs of numbers: i

Plot the numbers on a number line.

State which number is larger.

4 and 8

2 and −7

17 and 0

−4 and −9

For each of the following pairs of numbers: i

Plot the numbers on a number line.

State which number is smaller.

0 and 6

−8 and 0

7 and 16

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−1 and −3

Let’s practice 7

Write a statement using < or >, that compares the two integers plotted on the following number lines: a

−8 −6 −4 −2 0

−8

−2

−5

−8

−4

−8 −6 −4 −2 0

d 0

f 5

−2

h −14 −12 −10 −8 −6 −4 −2

Plot the pair of numbers on a number line.

Complete each number sentence with < or >, that should appear between each pair to make the statement true.

3⬚7

15 ⬚ 4

4 ⬚ −7

−3 ⬚ −9

0 > −1

23 < −47

For each of the following statements: i

Write each number sentence in words.

State whether the number sentence is true or false.

−8 < −4

7 > −5

Identify whether <, >, or = should appear between each pair of numbers to make each statement true: 9 ⬚ −2

−29 ⬚ −74

0 ⬚ −6

−45 ⬚ 22

−7 ⬚ −8

−2 ≥ 14

14 > −2

−2 > 14

11 > −9

−9 ≥ 11

−9 = 9

32 ⬚ −40

14 ≤ −2

−9 > 11

−4

For each of the following pairs of numbers:

−6

−12

−10

−6 ⬚ 6

State whether the following statements are true or false:

−30 ⬚ −30

State the greatest number in each set: a

−19, 13, 6

−16, 0, −12

−20, −3, −15

13, −14, 2

−7, −17, −12

0, −8, 7

35, −36, −40

−99, −87, −71

State the smallest number in each set: a

−14, −4, −21

−7, −24, 10

0, −4, 5

34, −36, −2

−12, 0, −2

−23, −2, 13

−9, 21, −2

16, 25, −27

4, 59, −15

59, 4, −15

−7, 6, 7, −5, 3

17, −22, 13, 0, −3

22, −6, 5, 18, −2

−27, 12, 0, −8, 16

Which of these lists the numbers from greatest to least? A

−15, 4, 59

4, −15, 59

Let’s extend our thinking 15

Arrange the following numbers in ascending order: a

11, −25, 19, −15, 29

13, −7, 0, −4, 8

Arrange the following numbers in descending order: a

2, −11, 7, −4, 18

−15, 6, 0, −7, 9

3.02 Compare and order integers mathspace.co

201

Explain how to complete the following patterns: a c e g

5, 3, 1, ⬚, ⬚, ⬚

−11, −6, −1, ⬚, ⬚, ⬚

25, 22, 19, ⬚, ⬚, ⬚

31, 25, 19, ⬚, ⬚, ⬚

b d f h

−12, −10, −8, ⬚, ⬚, ⬚

7, 9, 11, ⬚, ⬚, ⬚

−6, −10, −14, ⬚, ⬚, ⬚

−15, −8, −1, ⬚, ⬚, ⬚

Calgary has a temperature of 5° C while Montreal has a temperature of −11° C. Identify the state which has a higher temperature. Explain your reasoning.

For each of the following scenarios, write a sentence to compare the two quantities. Example: Neville has $8 in savings. Iain has $14 saved. Two possible answers: • Neville has less savings than Iain. • Iain has greater savings than Neville.

202

Ray’s account balance is −$10. Mohamad’s account balance is −$13.

Jenny has made $74 selling ice creams. Irene has made $119 selling lemonade.

Uther has read 33 books this year. His sister Patricia has read 12 books.

Tobias has 778 songs on his cell phone. Marge has 525 songs on her tablet.

Honolulu has a temperature of 98 °F and Ottawa has a temperature of 38 °F.

The melting point of helium is −458 °F. The melting point of xenon is −169 °F.

Charlie has made $38.92 this week from his part time job. Quiana has made $80.37 from her online craft store.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

9 a i −8 is less than −4

3.02 Compare and order integers What do you remember?

b i 7 is greater than −5

ii True

c i 0 is greater than −1

ii True

d i 23 is less than −47

ii False

10 a >

1 K

e >

2 13

11 a False

3 −16

e False

4 a True e True

b False

c True

d True

12 a 13

True

g True

h False

e −7

5 a i

ii 8

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ii 17 −5

ii 2 10

ii −4

d i −10

b <

c >

d >

g <

h =

b False

c True

d False

True

g False

h False

b 0

c −3

d 13

g 35

h −71

b −24

c −4

d −36

−23

g −9

h −27

14 D Let’s extend our thinking

c i −10

13 a −21 e −21

b i

−5

15 a −25, −15, 11, 19, 29

b −7, −4, 0, 8, 13

c −7, −5, 3, 6, 7

d −22, −3, 0, 13, 17

16 a 18, 7, 2, −4, −11

c 22, 18, 5, −2, −6 6 a i

ii 0

0 1 2 3 4 5 6 7 8 9 10

b i

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ii 7 ii −8

c i −10

−5

ii −3

d i −5

Let’s practice 7 a −6 < −2 or −2 > −6

b 4 > −2 or −2 < 4

c −8 < −5 or −5 > −8

d 7 > −2 or −2 < 7

e −10 < 0 or 0 > −10

g −10 < 4 or 4 > −10

h −6 > −14 or −14 < −6

7 > 4 or 4 < 7 ii 3 < 7

8 a i 0 1 2 3 4 5 6 7 8 9 10

b i

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ii 15 > 4 ii 4 > −7

c i −10

−5

−10

−5

d i ii −3 > −9

ii True

b 9, 6, 0, −7, −15 d 16, 12, 0, −8, −27

17 a 5 − 2 = 3 and 3 − 2 = 1, so the pattern can be completed by subtracting 2 from each result. Repeating the operation will complete pattern: 5, 3, 1, −1, −3, −5. b − 12 + 2 = −10 and −10 + 2 = −8, so the pattern can be completed by adding 2 to each result. Repeating the operation will complete pattern: −12, −10, −8, −6, −4, −2. c − 11 + 5 = −6 and −6 + 5 = −1, so the pattern can be completed by adding 5 to each result. Repeating the operation will complete pattern: −11, −6, −1, 4, 9, 14. d 7 + 2 = 9 and 9 + 2 = 11, so the pattern can be completed by adding 2 to each result. Repeating the operation will complete pattern: 7, 9, 11, 13, 15, 17. e 25 − 3 = 22 and 22 − 3 = 19, so the pattern can be completed by subtracting 3 from each result. Repeating the operation will complete pattern: 25, 22, 19, 16, 13, 10. f

6 − 4 = −10 and −10 − 4 = −14, so the pattern can be − completed by subtracting 4 from each result. Repeating the operation will complete pattern: −6, −10, −14, −18, −22, −26.

g 31 − 6 = 25 and 25 − 6 = 19, so the pattern can be completed by subtracting 6 from each result. Repeating the operation will complete pattern: 31, 25, 19, 13, 7, 1. h − 15 + 7 = −8 and −8 + 7 = −1, so the pattern can be completed by adding 7 to each result. Repeating the operation will complete pattern: −15, −8, −1, 6, 13, 30.

Answers mathspace.co

203

18 5 is greater than −11. Calgary has a higher temperature than Montreal. 19 a Ray has less debt than Mohamad.

Mohamad has greater debt than Ray.

b Jenny has greater sales than Irene.

Irene has less sales than Jenny.

c Uther has read more books than Patricia.

Patricia has read lesse books than Uther.

d Tobias has more songs on his cellphone than Marge. Marge has less songs on her tablet than Tobias on his cell phone. e Honolulu has a higher temperature than Ottawa.

Ottawa has lower temperature than Honolulu.

Helium has a lower melting point than xenon.

Xenon has higher melting poingt than helium.

g Charlie has less earning than Quiana.

Quiana has greater earning than Charlie.

204

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

3.03 Introduction to exponents Subtopic overview Lesson narrative In this lesson, students will be introduced to exponents. They will learn that an exponent indicates how many times a base number is multiplied by itself. The lesson covers various examples and includes explorations where students identify patterns in exponentiation, such as recognizing that any number to the power of 1 is the number itself, and any number to the power of 0 is 1. Students will also practice converting between exponential and expanded forms. By the end, students should recognize and represent patterns with bases and exponents that are whole numbers.

Learning objectives

3.03 Introduction to exponents

Students: Page 101

After this lesson, you will be able to... • identify and explain the meaning of the base and exponent of a power. • represent repeated multiplication using exponents. • find the value of a number raised to an exponent. • recognize and represent patterns with bases and exponents that are whole numbers.

Introduction to exponents An exponent (or power) is a small number placed in the upper right hand corner of another number to note how Key vocabulary many times a base is being multiplied by itself.

 expanded  exponent base form 10 is the base exponential For example, in the expression 103 the number term and the number 3 is the exponent (or form index or 3 power). The expression 10 is the same as 10 ⋅ 10 ⋅ 10, or the number 10 multiplied 3 times.  power 

Exponent/power

Essential understanding

10 ⋅ 10 ⋅ 10

Repeated multiplication can be written using a base (number that is being multiplied by itself repeatedly) and an Multiplied 3 timesagainst itself). exponent (the number of times the base is written and then multiplied Base

In the above expression, we call 103 the exponential form and 10 ⋅ 10 ⋅ 10 the expanded form of the expression.

Standards We often encounter a power of 2 when measuring area. Consider the area of a square, for example, which is given by This subtopic the following Virginia Mathematics standards. side length addresses times side length. A number, e.g. 52023 with an exponent (orStandards power) of of 2, Learning can be expressed as 52, and can be read as “5 to the power of 2” or “ﬁve squared”.

Mathematical process goals

A number, e.g. 10 to the power of 3, can be expressed as 103, and can be read as “ten cubed”. A power of 3 is involved in calculationsReasoning like measuring the volume of a cube.MPG4 — Mathematical Connections MPG3 — Mathematical

Teachers can integrate this goal by encouraging students x squared to reason and make predictions about the values of expressions with whole number bases and exponents. 2 x They can also guide students in making logical x x conclusions about the use of exponents in representing large numbers and their real-world applications.x ⋅ x = x2

This goal can be integrated into instruction by connecting x cubed x understanding of multiplication as repeated students' x addition to the concept of exponents as repeated x x3 multiplication. Teachers can also relate the concept of exponents to real-world applications like calculating area, x ⋅ x ⋅ x = and x3 population growth. volume,

A base to the power of any other number, e.g. 34, can be read as “three to the power of four”, and means that the 3.03 Introduction to exponents base number is multiplied by itself the number of times shown in the exponent. 34 = 3 ⋅ 3 ⋅ 3 ⋅ 3

mathspace.co

To evaluate or simplify an exponential expression, the only step we need to take is completing the multiplication.

205

MPG5 — Mathematical Representations Teachers can integrate this goal by having students create and interpret different representations of exponential expressions, such as numerical, pictorial, or symbolic representations. For example, they can illustrate 23 as three repeated multiplications of 2 or as a cube with a side length of 2. Teachers can also encourage students to connect different representations, such as linking the numerical pattern in a multiplication table to the symbolic representation of an exponent.

Content standards 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

6.NS.3a — Recognize and represent patterns with bases and exponents that are whole numbers.

Prior connections 4.CE.2 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using multiplication with whole numbers, and single-step problems, including those in context, using division with whole numbers; and recall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.

Future connections 7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation. A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.3 — The student will derive and apply the laws of exponents. A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.01 Identify and represent integers

Student lesson & teacher guide Exponent notation Students are introduced to exponent notation and its parts. Students are to use new terminology and practice moving between exponential form to expanded form. They are then to take part in an exploration to demonstrate the meaning of exponents and effect of changing the value of exponents.

206

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Pages 101–102

3.03 Introduction to exponents After this lesson, you will be able to... • identify and explain the meaning of the base and exponent of a power. • represent repeated multiplication using exponents. • find the value of a number raised to an exponent. • recognize and represent patterns with bases and exponents that are whole numbers.

Introduction to exponents An exponent (or power) is a small number placed in the upper right hand corner of another number to note how many times a base is being multiplied by itself. For example, in the expression 103 the number 10 is the base term and the number 3 is the exponent (or index or power). The expression 103 is the same as 10 ⋅ 10 ⋅ 10, or the number 10 multiplied 3 times. Exponent/power

10 ⋅ 10 ⋅ 10 Multiplied 3 times

Base

In the above expression, we call 103 the exponential form and 10 ⋅ 10 ⋅ 10 the expanded form of the expression. We often encounter a power of 2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number, e.g. 5 with an exponent (or power) of 2, can be expressed as 52, and can be read as “5 to the power of 2” or “ﬁve squared”. A number, e.g. 10 to the power of 3, can be expressed as 103, and can be read as “ten cubed”. A power of 3 is involved in calculations like measuring the volume of a cube. x squared

Exploration x2

Complete the following table of values using a pattern: 20

22 4

x cubed x

23 8

x ⋅ x = x2

x ⋅ x ⋅ x = x3

A base the powerthe of pattern any other 34, canthe be table. read as “three to the power of four”, and means that the 1. to Describe younumber, used toe.g. complete base number is multiplied by itself the1 number of times shown in the exponent. 2. What do you notice about 2 ? 34 = 3 ⋅ 3 ⋅ 3 ⋅ 3 3. What do you notice about 20 ? To evaluate or this simplify an exponential expression, the only we need to take is completing the multiplication. 4. Test observation by ﬁlling in a new table with step a different base. Do you notice the same thing? 5.

3 ⋅ to 3 ⋅complete 3⋅3 34 = try Now the entire table if the base is 1. What do you notice? = 81 Simplify the multiplication

Example 1

3.03 Introduction to exponents mathspace.co

101

Identify the base of 32.

Create a strategy Use the base and exponent deﬁnition: baseexponent

Apply the idea baseexponent = 32 The base of the expression is 3.3.03 Introduction to exponents mathspace.co

Example 2

207

Anatomy and meaning of exponent notation Targeted instructional strategies Break down exponential notation into basic components so that students can understand them on a fundamental level. For any given exponential notation, show that it consists of: • Base • Exponent or power and that exponential notation can be expanded so that the base number is multiplied by itself the number of times shown in the power. Exponent/power

10 10 10 Multiplied 3 times

Base

Misunderstanding Expanded Form Address student misconceptions Students may think that the base and exponent should be multiplied together so that 23 = 2 × 3 rather than correctly multiplying 2 × 2 × 2.

Exploration Students: Page 102

Exploration Complete the following table of values using a pattern: 20

22 4

23 8

Describe the pattern you used to complete the table.

What do you notice about 21 ?

What do you notice about 20 ?

Test this observation by ﬁlling in a new table with a different base. Do you notice the same thing?

Now try to complete the entire table if the base is 1. What do you notice?

Any number raised to the power of 1 is equal to the original number. And 1 raised to any power is still 1 because 1 times itself any number of times will always be 1. Any number raised to the power of groups 0 is 1. Though there is debate among mathematicians about whether 00 = 1 or is Suggested student grouping: Small Studentsundeﬁned. are exploring the power of numbers, particularly focusing on how the base and the exponent affect the outcome. The students complete a table that requires them to use a pattern to fill in missing values of powers Example of 2. They1then investigate what happens when the base is changed and when the base is 1.

20 1

Identify the 2 2 base of232.

Create a strategy

24 16

Use the base and exponent deﬁnition: baseexponent

208

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Example 2

Apply the idea baseexponent = 32 The base of the expression is 3.

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Describe the pattern you used to complete the table. Each time the exponent increased by 1, the result doubled. So, the next number in the sequence can be found by simply doubling the previous number. 2. What do you notice about 21? 2 to the power of 1 is just 2 itself. So it can be observed that any number to the power of 1 is the number itself. 3. What do you notice about 20? 2 to the power of 0 is 1. This seems to be true for any number to the power of 0, except for 0 itself. 4. Test this observation by filling in a new table with a different base. Do you notice the same thing? Yes, the same observation holds. For example, if we try it with 3 and 5 as the base, we find that for each, the number to the power of 1 is the number itself and the number to the power of 0 is 1. 5. Now try to complete the entire table if the base is 1. What do you notice? If the base is 1, no matter what the exponent is, the result is always 1. Purposeful questions • What patterns do you observe in the results as the exponents increase in your table? • How does each result compare to the previous one when you increase the exponent by 1? • If you change the base from 2 to another number, like 3 or 5, how does that affect the pattern you’ve observed? • What do you think would be the value of 20? Do you think that would be the same or different for other bases?Exploration • If you change the base to 1 what happens? Complete the following table of values using a pattern:

20 21 22 23 Possible misunderstandings

4 8raised to the power of zero equals zero. Explain that any non-zero number raised • Believing that any number 1. Describe the pattern you and useduse to complete theintable. to the power of zero equals one, patterns the table to illustrate this concept. 1 2. that What do you notice 2? • Assuming increasing theabout exponent always multiplies the result by 2, regardless of the base. Help ? which the result increases is actually the base itself, highlighting the What do you notice students3.understand that the about factor20by 4. between Test this observation ﬁlling in rate a newof table with a different base. Do you notice the same thing? relationship the basebyand the growth. 5.

Now try to complete the entire table if the base is 1. What do you notice?

Any number raised to the power of 1 is equal to the original number. And 1 raised to any power is still 1 because Examples 1 times itself any number of times will always be 1.

Students:Any Page 102 number raised to the power of 0 is 1. Though there is debate among mathematicians about whether 00 = 1 or is undeﬁned.

Example 1 Identify the base of 32.

Create a strategy

Apply the idea

Use the base and exponent deﬁnition: baseexponent

baseexponent = 32 The base of the expression is 3.

Example 2 Purpose Identify the exponent of 46. Check that students can identify the base of an exponent. Create a strategy Use the base and exponent deﬁnition: baseexponent

Apply the idea baseexponent = 46

3.03 Introduction to exponents The exponent of the expression is 6. mathspace.co

Example 3

209

Identify the base of 32.

Exploration Create a strategy

Apply the idea

Use the base and exponent deﬁnition: baseexponent Complete the following table of values using a pattern:

Students: Page 102 0 1 2

22 4

23 8

baseexponent = 32 The base of the expression is 3.

Describe the pattern you used to complete the table.

What do you notice about 21 ?

Example 2

0 3. the What do youofnotice Identify exponent 46. about 2 ?

Test this observation by ﬁlling in a new table with a different base. Do you notice the same thing?

Create strategy ideanotice? 5. a Now try to complete the entire table if the base is Apply 1. Whatthe do you Use the base and exponent deﬁnition: baseexponent

baseexponent = 46

exponent the expression is 6. Any number raised to the power of 1 is equal to the original The number. And 1 of raised to any power is still 1 because 1 times itself any number of times will always be 1. Any number raised to the power of 0 is 1. Though there is debate among mathematicians about whether 00 = 1 or is Example undeﬁned. 3

Purpose Write 75 ⋅ 64 in expanded form. Example 1 can identify the exponent of an expression. Check that students Create a strategy Identify the base of 32.

Clarify, critique andhow correct meaning exponent notation Use the exponent to know many times the baseof should be multiplied by itself. English Create alanguage strategy learner support

use with Example 2

Apply the idea

base andincorrect exponent statement: deﬁnition: base theexponent fifth power Display Use the the following “I exponent know that 105 is read as ten tobase = 32 and means adding 75 ⋅ 64 = 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 Multiply each of the bases by themselves the number of times 5 groups of 10.” indicated byThe thebase exponent of the expression is 3.

Invite one or two students to share their critiques and corrected explanations with the class. Listen for amplify the language students use to describe what should happen when 10 is added 5 times or 102and Mathspace Virginia SOL Grade 6 Example 2 mathspace.co when 10 is multiplied by 5 rather than multiplying 10, 5 times. Take time to clarify this important distinction. 6

the exponent of 4 . This willIdentify help students understand transforming a number in exponential form to expanded form and use correct language in reading and writing numbers in exponential form.

Create a strategy

Apply the idea

Use the base and exponent deﬁnition: baseexponent

Students: Page 102

baseexponent = 46 The exponent of the expression is 6.

Example 3 Write 75 ⋅ 64 in expanded form.

Create a strategy Use the exponent to know how many times the base should be multiplied by itself.

Apply the idea 75 ⋅ 64 = 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6

102

Multiply each of the bases by themselves the number of times indicated by the exponent

Mathspace Virginia SOL Grade 6 mathspace.co

Purpose Ensure that students can convert numbers written in exponential form into expanded form.

210

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Guided diagram annotation to expand exponents

use with Example 3

Student with disabilities support Provide students with a visual organizer that guides them through expanding exponential expressions. Start by displaying 75 ⋅ 64 at the top of the diagram. Draw arrows from 75 pointing to five blank boxes, each representing a factor of 7, and from 64 pointing to four blank boxes for the factors of 6. Encourage students to fill in the boxes by writing the base number repeatedly according to its exponent. As students fill in the diagram, prompt them to read aloud each part, reinforcing the connection between the exponent and the number of times the base is multiplied.

Students: Page 103

Example 4 Write 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 in exponential form.

Create a strategy

Apply the idea

To write repeated multiplication of the same number in exponential form, count how many times the number is multiplied by itself. This will be the exponent. The base will be the number that is multiplied repeatedly. Example 4

The number 8 is multiplied by itself 5 times, so in exponential form, this is written as 85.

Write 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 in exponential form.

Example 5

Create a strategy

Apply the idea

Purpose Given the table of multiplication values: To write repeated of the same number in The number 8 is multiplied by itself 5 times, so in Show students how to use exponential notation to represent repeated multiplication of the same number. exponential form, count how many times the number is exponential form, this is written as 85.

Exponential form Expanded form Evaluate multiplied by itself. This will be the exponent. The base 41 Students:willPage be the103 number that is multiplied repeatedly. 42 43 44 Example 5 45 46 Given the table of values: a Complete the table of values. Exponential form Expanded form Evaluate 41 Create a strategy Apply the idea 42 3 The 4 expanded form shows the base being multiplied by Exponential form itself 44 repeatedly, the number of times equivalent to the 41 exponent. To evaluate, we calculate the result of this 45 42 multiplication. 6 4 43 44 a Complete the table of values. 45 46

Create a strategy

Apply the idea

Expanded form 4 4⋅4 4⋅4⋅4 4⋅4⋅4⋅4 4⋅4⋅4⋅4⋅4 4⋅4⋅4⋅4⋅4⋅4

Evaluate 4 16 64 256 1024 4096

The expanded form shows the base being multiplied by Exponential form Expanded form Evaluate itself repeatedly, the number of times equivalent to the 41 4 4 b What do you notice about the numbers in the “Evaluate” column? exponent. To evaluate, we calculate the result of this 42 4⋅4 16 multiplication. 43 4⋅4⋅4 64 Create a strategy 4 ⋅ 4 ⋅relationships 4⋅4 256 Observe the pattern formed by the numbers in the “Evaluate”4 column to identify4any or sequences. 45 4⋅4⋅4⋅4⋅4 1024 6 4 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 4096 Apply the idea Reﬂect and check Each number in the “Evaluate” column is four times the We will continue to explore this concept of exponential number before it. This pattern reﬂects the fact that as the growth throughout our mathematics courses. It b What do you notice about in the column? exponent grows larger by 1 wethe arenumbers multiplying by “Evaluate” 4 an demonstrates how quickly values can grow as the additional time.

Create a strategy

exponent increases, which has many real world applications.

Observe the pattern formed by the numbers in the “Evaluate” column to identify any relationships or sequences. 3.03 Introduction to exponents mathspace.co

Apply the idea

Reﬂect and check

Each number in the “Evaluate” column is four times the

We will continue to explore this concept of exponential

211

a Complete the table of values.

Create a strategy

Apply the idea

The expanded form shows the base being multiplied by itself repeatedly, the number of times equivalent to the Purposeexponent. To evaluate, we calculate the result of this multiplication.

Exponential form

Expanded form

Evaluate

41 4 4 42 4⋅4 16 3 Show students how to convert between exponential form 4and expanded form, expressions 4 ⋅ 4 ⋅ 4and how to evaluate 64 44 4⋅4⋅4⋅4 256 in exponential form. 45 4⋅4⋅4⋅4⋅4 1024 46 4⋅4⋅4⋅4⋅4⋅4 4096 Students: Page 103

b What do you notice about the numbers in the “Evaluate” column?

Create a strategy Observe the pattern formed by the numbers in the “Evaluate” column to identify any relationships or sequences.

Apply the idea

Reﬂect and check

Each number in the “Evaluate” column is four times the number before it. This pattern reﬂects the fact that as the exponent grows larger by 1 we are multiplying by 4 an additional time.

We will continue to explore this concept of exponential growth throughout our mathematics courses. It demonstrates how quickly values can grow as the exponent increases, which has many real world applications.

Purpose Challenge students to recognize and understand the pattern of exponential growth. 3.03 Introduction to exponents

103

mathspace.co

Reflecting with students Invite advanced learners, or any students who are ready, to extend the table by exploring patterns with negative exponents. Encourage them to add rows for 40, 4−1, 4−2, and so on, completing the “Expanded form” and “Evaluate” columns for these new entries. They should notice that as the exponent decreases by −1, the evaluated value is divided by 4 each time, leading to fractions like

and

Students: Page 104

Idea summary An exponent (or power) notes how many times a base is being multiplied by itself. A base to the power of any other number means that the base number is multiplied by itself the number of times shown in the exponent.

Practice What do you remember? 1

Identify the base of 29.

Identify the exponent of 106.

Write each exponential form with base 9: a

9⋅9⋅9

9⋅9⋅9⋅9

9⋅9⋅9⋅9⋅9

5⋅5⋅5⋅5

5⋅5⋅5

311

113

Write each expression in expanded form: a

9⋅9 35

Show why 20 = 1.

Let’s practice SOL

212

Which of the following is equivalent to 53?

A 3⋅3⋅3⋅3⋅3 B 5⋅3 Mathspace Virginia SOL Grade 6 Teacher Edition SOL 7 Which of the following is equivalent to 11 ⋅ 11 ⋅ 11? mathspace.co A 3 ⋅ 33 B 11 ⋅ 33 8

Write two squared in exponential form.

Practice Students: Pages 104–106

What do you remember? 1

Identify the base of 29.

Identify the exponent of 106.

Write each exponential form with base 9: a

9⋅9⋅9

9⋅9⋅9⋅9

9⋅9⋅9⋅9⋅9

5⋅5⋅5⋅5

5⋅5⋅5

311

113

102

3⋅3

Write each expression in expanded form: a

9⋅9 35

Show why 20 = 1.

Let’s practice SOL

Which of the following is equivalent to 53? A

SOL

3⋅3⋅3⋅3⋅3

5⋅3

Which of the following is equivalent to 11 ⋅ 11 ⋅ 11? A

3 ⋅ 33

11 ⋅ 33

Write two squared in exponential form.

Write each expression in expanded form:

Write each expanded form in exponential form: a

2⋅2⋅2⋅2

7⋅7⋅7⋅7⋅7⋅7

Complete the table shown: Words Six to the power of four Eight cubed Eleven to the power of three Twenty three to the power of ﬁve Seven squared Fifteen to the power of four

Explain how to evaluate 45.

Evaluate:

Expanded form

Exponential form

122

3.03 Introduction to exponents mathspace.co

213

SOL

Claudia wrote the values of the powers of 4 that she knew 41 = 4 42 = 16 43 = 64 44 = 256 45 = ? What is the value of 45? A

260

1024

625

For each table of values: i

Complete the table of values.

What do you notice about the numbers in the “Evaluate” column?

Exponential form 21 22 23 24 25 26

Expanded form

Evaluate

Exponential form 51 52 53 54 55 56

Expanded form

Evaluate

Which is larger 123 or 127 and explain why without calculating.

Ethan and Soﬁa want to write 182 in expanded form. Ethan wrote 18 ⋅ 2, and Soﬁa wrote 18 ⋅ 18. a

Who wrote it correctly?

If they solved the expanded form, would they get the same answer? Explain.

Let’s extend our thinking 18 19

Using the concept of exponents, explain how to ﬁnd the missing base in the equation ⬚4 = 16.

Using the concept of exponents, explain how to ﬁnd the missing exponent in the equation 5⬚ = 125.

If a population of 3 rabbits triples every hour, represent the population after 4 hours.

Write each expression in expanded form: a e

214

54 ⋅ 96 6

4 ⋅9

43 ⋅ 82

c 2

77 ⋅ 13

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

663 ⋅ 924 1

47 ⋅ 62

75 ⋅ 64

912 ⋅ 822

Write each expression in exponential form: a

5⋅5⋅5

4⋅4⋅4⋅4⋅4

9⋅9⋅2⋅2

12 ⋅ 12 ⋅ 9 ⋅ 9 ⋅ 9

13 ⋅ 13 ⋅ 13 ⋅ 13 ⋅ 5 ⋅ 5

11 ⋅ 7 ⋅ 11 ⋅ 7 ⋅ 7

3⋅4⋅3⋅3⋅4

6⋅2⋅6⋅6⋅2

Aaliyah calculated the ﬁrst 6 powers of 3 and put the results in the table: 31 3

32 9

33 27

34 81

35 243

36 729

What would Aaliyah get if she continued and tried to evalute 37?

What is the last digit Aaliyah would get after evaluating 39?

Aaliyah used a calculator to evaluate 311. After seeing 19 683 as answer from the calculator, she realized that she made an error in putting the input to the calculator. Is Aaliyah’s statement correct? Explain your answer.

Sam is wanting to solve a Tower of Hanoi puzzle with 8 disks. He can calculate the minimum number of moves to ﬁnish the puzzle by solving the expression 28 − 1. a

Evaluate 28.

Now, ﬁnd the minimum number of moves to ﬁnish Tower of Hanoi with 8 disks.

3.03 Introduction to exponents mathspace.co

215

Answers

ii Each number in the “Evaluate” column is double the number before it.

3.03 Introduction to exponents

b i

Exponential form

Expanded form

Evaluate

5⋅5

What do you remember?

1 2

5⋅5⋅5

125

2 6

5⋅5⋅5⋅5

625

3 a 9 2

b 93

c 94

4 a 3⋅3⋅3⋅3⋅3

b 3⋅3⋅3⋅3

c 3⋅3 5

d 95

23 = 2 ⋅ 2 ⋅ 2 = 8 2 =2=2

÷2

20 = 1

3125

5⋅5⋅5⋅5⋅5⋅5

15 625

16 127 is larger than 123 because the more times you multiply a number by itself, the bigger it gets. 127 means multiplying 12 by itself seven times while 123 means multiplying 12 by itself three times, so it ends up being a much bigger number.

÷2 ÷2

22 = 2 ⋅ 2 = 4

5⋅5⋅5⋅5⋅5

ii Each number in the “Evaluate” column is 5 times the number before it.

d 3⋅3⋅3⋅3⋅3⋅3⋅3

24 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16

17 a Sofia

Let’s practice

b No. Ethan would get 36 while Soﬁa would get 324.

6 D Let’s extend our thinking

7 D

18 Since 16 is not a very large number, we can use trial and error to ﬁnd a number that when multiplied by itself 4 times, the result is 16. We may ﬁnd that 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16.

8 22 9 a 5 ⋅ 5 d 7⋅7⋅7 4

c 10 ⋅ 10

e 6

b 7

10 a 2 11

b 5⋅5⋅5⋅5⋅5 c 3

8⋅8⋅8⋅8

19 125 is the result of multiplying 5 by itself for a number of times.

Expanded form

Exponential form

Six to the power of four

6⋅6⋅6⋅6

Eight cubed

8⋅8⋅8

20 3 ⋅ 34 or 35

Eleven to the power of three

11 ⋅ 11 ⋅ 11

113

21 a 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9

Twenty three to the power of five

23 ⋅ 23 ⋅ 23 ⋅ 23 ⋅ 23

235

7⋅7

15 ⋅ 15 ⋅ 15 ⋅ 15

154

Words

Seven squared Fifteen to the power of four

12 As the base is 4 and the power is 5, you multiply 4 by itself 5 times. So, 45 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4. 13 a 3 e 512

b 1

c 64

d 216

g 64

h 144

So multiply 5 by itself until reaching 125: 5 ⋅ 5 ⋅ 5 = 125. The missing exponent is 3.

b 4⋅4⋅4⋅8⋅8 c 66 ⋅ 66 ⋅ 66 ⋅ 92 ⋅ 92 ⋅ 92 ⋅ 92 d 7⋅7⋅7⋅7⋅7⋅6⋅6⋅6⋅6 e 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 9 ⋅ 9 f

77 ⋅ 77 ⋅ 77 ⋅ 77 ⋅ 77 ⋅ 13 ⋅ 13

g 47 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 h 91 ⋅ 91 ⋅ 82 ⋅ 82 22 a 5 3 4

b 45 2

e 13 ⋅ 5

11 ⋅ 7

c 92 ⋅ 22

d 122 ⋅ 93

h 63 ⋅ 22

g 3 ⋅4

23 a 2187

14 C 15 a i

216

The missing base is 2.

b 3 Exponential form

Expanded form

Evaluate

2⋅2

2⋅2⋅2

2⋅2⋅2⋅2

2⋅2⋅2⋅2⋅2

2⋅2⋅2⋅2⋅2⋅2

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

c A aliyah’s statement is correct. We can notice that the last digits of any whole number powers of 3 are 3, 9, 7, and 1, after this the pattern repeats. This means that the last digit of the 5th to 11th powers of 3 are 3, 9, 7, 1, 3, 9, 7. Because the last digit she got is 3 instead of 7, she made an error in putting the input to the calculator. 24 a 256

b 255

3.04 Patterns with perfect squares Subtopic overview Lesson narrative In this lesson, students will explore patterns with perfect squares. They begin by using interactive sliders to investigate the first 12 perfect squares and understand the relationship between side length and the total number of squares. The lesson defines perfect squares as numbers that can be written as an integer squared. Students will practice evaluating squares and determining if given numbers are perfect squares using grids. By the end, students should recognize and represent patterns of perfect squares up to 202 using models and justify if numbers between 0 and 400 are perfect squares.

Learning objectives Students: Page 107

Key vocabulary 

perfect square

Essential understanding The special relationship between square roots and perfect squares and cube roots and perfect cubes allows us to reveal the simplest form of a radical expression.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can integrate this goal by encouraging students to use tiles, geoboards, virtual manipulatives, or create their own drawings of perfect squares on grid paper to justify whether a number is a perfect square. Additionally, teachers can faciliate discussions on whether zero to the zero power is a perfect square and provide real-world examples in which students must identify a correct response and justification of whether or not a number is a perfect square.

Teachers can make connections to prior lessons and knowledge by reminding students about their understanding of patterns with whole numbers, fractions, and decimals, and how recognizing perfect squares is an extension of that knowledge.

3.04 Patterns with perfect squares mathspace.co

217

MPG5 — Mathematical Representations Teachers can help students understand perfect squares by using concrete models like square arrays, area models, and diagrams. Encouraging students to connect these representations and construct squares using manipulatives allows them to observe patterns and derive meaning. This hands-on approach helps students recognize perfect squares, understand their properties, and differentiate between numbers by comparing areas. Constructing perfect squares enables students to identify similarities and differences, reinforcing their understanding of mathematical structures and patterns. This method enhances their ability to visually and conceptually grasp the concept of perfect squares.

Content standards 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

6.NS.3c — Justify if a number between 0 and 400 is a perfect square through modeling or mathematical reasoning.

6.NS.3b — Recognize and represent patterns of perfect squares not to exceed 202, by using concrete and pictorial models.

Prior connections 5.PFA.1 — The student will identify, describe, extend, and create increasing and decreasing patterns with whole numbers, fractions, and decimals, including those in context, using various representations.

Future connections 7.NS.3 — The student will recognize and describe the relationship between square roots and perfect squares.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons:

Tools You may ﬁnd these tools helpful: • Printed 10 x 10 grid sheets • Colored pencils or markers

218

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide Patterns with perfect squares Students: Page 107

3.04 Patterns with perfect squares mathspace.co

219

Exploration Students: Page 107

Suggested student grouping: In pairs Students will be using sliders to explore the first 12 perfect squares. They will observe the relationship between the side length of the square and the total number of smaller squares it contains, which will lead them to the definition of a perfect square. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What is the relationship between the side length and the total number of smaller squares? The total number of smaller squares equals the side length times itself, which is what we mean when we say a number is “squared”. 2. What do you think a perfect square number is? A perfect square number is the product of a number multiplied by itself. Purposeful questions • As you adjust the slider, what happens to the total number of smaller squares when the side length increases by one unit? • Can you describe any patterns you observe between different side lengths and their corresponding total number of smaller squares? • How would you determine the total number of smaller squares for a square with a side length of 10 without counting each square individually? Possible misunderstandings • Students may think that the total number of smaller squares increases by the same amount each time the side length increases (assuming a linear relationship). Guide students to calculate the differences between total numbers for successive side lengths to reveal that the increase is not constant.

220

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Use a Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Engage students with physical manipulatives to explore perfect squares. Provide each student with square tiles or blocks to build squares with side lengths from 1 to 20. Encourage them to construct squares by arranging the tiles into equal rows and columns, forming perfect squares. For instance, they can build a square with a side length of 5 tiles, resulting in a total of 25 tiles used. As they build each square, have them count the total number of tiles to discover the pattern of perfect squares. This hands-on activity allows students to physically see and understand the relationship between the side length and the area of the square. Representational: Transition from the concrete manipulatives to representational drawings. Have students draw the squares they constructed on grid paper, shading in the areas to represent each perfect square up to 202. Encourage them to label the side lengths along the edges and write the total number of squares inside each drawn square. Use visual aids such as charts or diagrams that display the sequence of perfect squares. For example, create a table with side lengths from 1 to 20 in one column and the corresponding perfect squares in the next column. This visual representation helps students see the patterns and relationships in a different form. Abstract: Guide students to express perfect squares using mathematical symbols and notation. Teach them that a perfect square is a number that can be written as n2, where n is an integer. Have them practice calculating squares of numbers, such as 72 = 49, and write these expressions. Introduce methods to determine if a number between 0 and 400 is a perfect square, such as finding the square root and checking if it is a whole number. Encourage them to justify their reasoning using mathematical language and symbols. Help students make connections between the concrete, representational, and abstract stages to deepen their understanding. For example, when they compute 92 = 81, remind them of the square they built with 9 tiles on each side and the drawing they made on grid paper.

Stronger and clearer each time English language learner support Begin by asking students to individually write an explanation of what a perfect square is and how to determine if a number between 0 and 400 is a perfect square. Encourage them to include examples and visual representations, such as grids or models, to illustrate their explanation. Then, have students pair up and share their explanations with a partner. Instruct them to listen carefully and provide constructive feedback or ask clarifying questions. After the discussion, have students revise their original explanations, making them stronger and clearer by incorporating new ideas or mathematical language they learned from their partner. Repeat this process by having students share with a new partner or in small groups, further refining their explanations. This routine supports English language learners by providing multiple opportunities to practice and improve both their understanding of perfect squares and their ability to express mathematical concepts in English. Students will learn about the concept of perfect squares, understanding that they are numbers that can be represented as an integer raised to the power of 2. They will see that perfect squares can be visualized as actual squares.

3.04 Patterns with perfect squares mathspace.co

221

Examples Students: Page 108

Purpose Demonstrate to students that an exponent of 2 indicates a number is being multiplied by itself. Expected mistakes Students might mistakenly believe that 92 means 9 multiplied by 2, resulting in an answer of 18. This confusion arises when students interpret the exponent as a multiplier rather than as an indicator of how many times the base is used as a factor. To address this misconception, emphasize that an exponent tells us how many times to multiply the base by itself, not by the exponent. Encourage students to rewrite expressions with exponents as repeated multiplication, such as 92 = 9 × 9. Use visual representations like square arrays or area models to illustrate this concept. For example, show a square with sides of length 9 units to represent 92, and explain that the area of the square (81 square units) demonstrates the result of multiplying 9 by itself.

Students: Pages 108–109

222

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

If we build out a 14 × 14 grid, we get 196 squares so we need to add 4 more squares to get to 200. This is not enough units to create a 15 × 15 grid so 200 is not a perfect square. 14

Reﬂect and check Building out grids can be helpful ways to determine and justify whether a number is a perfect square. It is important to consider whether we have enough leftover units to build a larger grid, so it’s always a good idea to try building a grid with a 1 unit increase in length and width.

Idea summary Purpose Perfect squares are numbers raised to the power of two or can be obtained by multiplying an integer by itself. Challenge students to apply their understanding of perfect squares and use their spatial awareness to visually determine if a number is a perfect square.

Practice

Reflecting with students Encourage students to design their own problems related to perfect squares to deepen their understanding Whatpatterns. do you Invite remember? and explore them to choose different numbers and determine whether they are perfect squares, justifying 1theirComplete reasoning. Have students analyze what characteristics these numbers share and how they relate the table, one column has been done for you. to their square roots. Prompt students to consider how adding or subtracting from a perfect square affects its Number 1 2 3 4 5 6 7 8 9 10 status and whether patterns emerge when they manipulate these numbers. By crafting their own problems, Square of number 49 students engage more deeply with the concept, discover new patterns, and take ownership of their learning. 2

Complete the table below: Exponential form 112

Expanded form

Evaluate

13 ⋅ 13 142 15 ⋅ 15 172 20 ⋅ 20

3.04 Patterns with perfect squares mathspace.co

223

If we build out a 14 × 14 grid, we get 196 squares so we need to add 4 more squares to get to 200. This is not enough units to create a 15 × 15 grid so 200 is not a Facilitate mathematical communication with sentence frames perfect square.

use with Example 2

Student with disabilities support

Encourage students to articulate their reasoning when determining if a number14is a perfect square by using math talk sentence frames. Provide frames such as “I believe ⬚ is a perfect square because ⬚ or “To check if ⬚ is a perfect square, I ⬚.” Display these on a classroom poster or hand out personal reference cards so students can easily access them during discussions and written work. Begin by modeling how to use these frames, 14 square because thinking aloud as you solve an example problem. For instance, say “I know that 169 is a perfect 13 times 13 equals 169.” Reﬂect and check Building out grids can be helpful ways to determine and justify whether a number is a perfect square. It is important to

Students:consider Pagewhether 109 we have enough leftover units to build a larger grid, so it’s always a good idea to try building a grid with a 1 unit increase in length and width.

Idea summary Perfect squares are numbers raised to the power of two or can be obtained by multiplying an integer by itself.

Practice What do you remember?

Practice 1

Complete the table, one column has been done for you. Number

Students: Pages Square 109–111 of number 2

7 49

Complete the table below:

What do you remember? Exponential form

Expanded form

Evaluate

112

⋅ 13 done for you. Complete the table, one column has13been 142

Number Square of number 172

15 ⋅ 15

7 49

20 ⋅ 20

Complete the table below: Exponential form 112

Expanded form

Evaluate

13 ⋅ 13 142 15 ⋅ 15 172 20 ⋅ 20 3

Where do all of the perfect squares lie on this multiplication chart? Why?

3.04 Patterns with perfect squares mathspace.co

Multiplication table 1 2 3 4 5

224

1 1 2 3 4 5

2 2 4 6 8 10

3 3 6 9 12 15

4 4 8 12 16 20

5 5 10 15 20 25

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6 6 12 18 24 30

7 7 14 21 28 35

8 8 16 24 32 40

9 9 18 27 36 45

10 10 20 30 40 50

11 11 22 33 44 55

12 12 24 36 48 60

109

Multiplication table 6 7 8 9 10 11 12

1 6 7 8 9 10 11 12

2 12 14 16 18 20 22 24

3 18 21 24 27 30 33 36

4 24 28 32 36 40 44 48

5 30 35 40 45 50 55 60

6 36 42 48 54 60 66 72

7 42 49 56 63 70 77 84

8 48 56 64 72 80 88 96

9 54 63 72 81 90 99 108

Explain how you know whether a number is a perfect square.

Write two squared in exponential form.

10 60 70 80 90 100 110 120

11 66 77 88 99 110 121 132

12 72 84 96 108 120 132 144

Let’s practice 6

Determine whether the numbers are perfect squares: a

225

Tania was playing around with pennies and made 4 different arrangements. Which arrangement of pennies makes a perfect square? Explain. A

6 pennies

9 pennies

14 pennies

20 pennies

Explain why 0 is a perfect square.

For the values below, determine if the number is a perfect square. Justify your answer using a diagram. a

121

For the values below, determine if the number is a perfect square. Justify your answer mathematically or with words. a

144

What is the area of the square traﬃc sign in square inches? 22 in

22 in

A square painting measures 4 feet on each side. What is the area of the painting?

Write three perfect squares that have values greater than 120 and less than 200.

3.04 Patterns with perfect squares mathspace.co

225

A gardener is planning to arrange tiles in a square pattern in a section of the garden. If she uses 100 tiles for the pattern, how many tiles will be in each row?

Which is a perfect square between 81 and 144? A

121

A square patio has an area of 225 m2. Find the dimensions of the patio.

Fill in the missing numbers in the pattern: a c

4, 9, 16, ⬚, 36, 49, ⬚

25, 36, ⬚, 64, 81, ⬚, 121, 144, ⬚, 196

114

81, ⬚, 121, 144, ⬚, 196, 225, ⬚, 289, 324

How many blocks do you need to add to: a

Square 6 to get Square 7?

Square 9 to get Square 10?

Square 15 to get Square 16?

Square 3

Square 4

Square 5

Square 6

Let’s extend our thinking 19

I am a two-digit perfect square number. The sum of my digits is 13. What perfect square number am I?

Consider the equation 72 = ⬚2. The missing number is between what two whole numbers?

Barney was considering the pattern of square numbers and thought he spotted a pattern existed when calculating the differences between each pair of square numbers. a

Calculate the differences between consecutive square numbers. 12 – 02

22 – 12

iii 32 – 22

42 − 32

What pattern did Barney identify when he looked at the answers from part (a)?

Barney isn’t sure if the pattern will continue. Explain whether you think it will or will not continue forever.

Consider the ﬁgure: Number

Square Number

100

First Difference

Second Difference

226

Complete the ﬁgure by ﬁlling in the boxes with the differences between each pair of numbers above.

Describe the pattern that exists in the ﬁrst difference row.

Describe the pattern that exists in the second difference row.

Neil was working out 22 ⋅ 52 and thought that he could simplify the expression using the fact that 2 ⋅ 5 = 10 a

Evaluate 22 ⋅ 52 by ﬁrst evaluating each square.

Now, using the fact 2 ⋅ 5 = 10 evaluate 102.

Is 22 ⋅ 52 = (2 ⋅ 5)2 a true statement?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

9 a Yes

3.04 Patterns with perfect squares What do you remember? 1

Number

Square of number

25 36

81 100

Exponential form

Expanded form

Evaluate

112

11 ⋅ 11

122

13 ⋅ 13

169

142

14 ⋅ 14

196

152

15 ⋅ 15

225

172

17 ⋅ 17

289

202

20 ⋅ 20

400

3 All of the perfect squares on a multiplication chart lie along the diagonal that runs from the top left corner (starting with 1 ⋅ 1) to the bottom right corner (ending with 12 ⋅ 2, or as far as the chart goes). This diagonal represents the squares of the numbers from 1 up to the maximum number on one side of the chart, in this case, 12. A perfect square is the product of a number multiplied by itself. In the multiplication chart, each entry is the product of the number from the corresponding row and the number from the corresponding column. The diagonal line where the row number and column number are the same represents cases where a number is multiplied by itself, thus yielding a perfect square. 4 Example answer: If there is a whole number than can be squared to get the original number, then you know it is a perfect square.

72 or 7 ⋅ 7 = 49

b No

6 ⋅ 4 = 24

c Yes

112 or 11 ⋅ 11 = 122

d No

5 22 Let’s practice 6 a No e No

b Yes

c Yes

d No

Yes

g Yes

h No

7 B It makes a perfect square because there are the same number of pennies on each side and there are no empty spaces in the middle. 8 A perfect square is defined as the product of an integer with itself. In the case of 0, it can be represented as 0 ⋅ 0, which equals 0. This satisfies the definition of a perfect square since 0 is an integer, and when multiplied by itself, the result is 0.

10 a Y es, The square root of 144 is 12 (12 ⋅ 12 = 144), making it a perfect square. It meets the definition of a perfect square, which is a number that is the square of an integer. b N o, the square root of 72 is not an integer (it’s approximately 8.49). Since there is no whole number that, when squared, equals 72, it is not a perfect square. c N o, the square root of 50 is also not an integer (about 7.07). There is no integer that, when squared, equals 50, so it cannot be considered a perfect square. d Y es, he square root of 81 is 9 (9 ⋅ 9 = 81, making it a perfect square. It perfectly fits the definition of a perfect square as a number that is the square of an integer.

Answers mathspace.co

227

11 484 in2

This layer can be viewed as adding two lots of the previous side length plus one corner square.

12 16 ft

13 144, 169, 196

Previous side length

14 10 tiles 15 B 16 15 m

Previous side length

17 a 4, 9, 16, 25, 36, 49, 64 b 81, 100, 121, 144, 169, 169, 225, 256, 289, 324 c 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 18 a 13 blocks

b 19 blocks

c 31 blocks

This pattern increases by two at each stage starting with one. This makes a sequence of odd numbers. 22 a Number

Let’s extend our thinking

Square Number

100

19 49 First Difference

20 8 and 9 21 a i 1

ii 3

iii 5

Second Difference

iv 7

b We obtain the sequence of odd numbers: 1, 3, 5, 7, 9,...

b The set consists of odd numbers from 3 to 19.

c E xample answer: To grow the square at each stage we need to add on a new L-shaped layer.

c Consistently, the difference is 2.

9 7 5 3 1 1

4 +3

228

9 +5

16 +7

25 +9

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

23 a 100

b 100

c Yes

3.05 Powers of 10 and place value Subtopic overview Lesson narrative In this lesson, students will explore the powers of 10 and their impact on place value. They will start by moving sliders in an interactive applet to observe patterns in numbers when multiplied by powers of 10. The lesson covers how exponents indicate the number of times 10 is multiplied by itself and how this affects place value. Students will practice writing large numbers as powers of 10 and identify the place value of digits in large numbers. By the end, students will confidently recognize and represent powers of 10.

Learning objectives

3.05 Powers of 10 and place value

Students: Page 112

After this lesson, you will be able to... • identify powers of 10. • use patterns in place value to write, represent and evaluate powers of 10.

Powers of 10 and place value Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. We just looked Key vocabulary 

at special properties of the exponent 2, now let’s look at some special properties of 10.  exponent  place value decimal

 power

Interactive exploration Explore online to answer the questions

Essential understanding mathspace.co

The power of ten relates to the number of zeros before or after the decimal point because we use a base 10 place value Use system. the interactive exploration in 3.05 to answer these questions. 1.

What patterns do you notice in the numbers? 0

2. Why do you think 10 = 1? Standards

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. For any power of ten, the expanded form will have the same number of tens as the power. The number that it Mathematical goals evaluates to willprocess have the same number of zeros as the exponent. The following table demonstrates another way to think of some of the powers of ten. MPG3 — Mathematical Reasoning MPG1 — Mathematical Problem Solving Power Ten numeral) In Words Teachers can promote mathematical reasoning by asking Teachers canof integrate this goalMeaning into their instructionValue by (basic 5 10 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 100,000 One hundred thousand presenting students with real-world problems that involve students to make predictions about patterns in powers 104 of place value and 10 ⋅ powers 10 ⋅ 10 ⋅ 10 10,000 Ten thousand of 10 and then verify or refute their hypotheses. For the use of 10. For example, 3 10 ⋅ 10a problem where1,000 One thousand example, they could ask students to predict the number they10could ask students10 to⋅ solve they 2 10different ⋅ 10 100 One hundred of zeros in 104 based have10to convert between metric units using on the patterns they have observed 1 1 2 3 10 10 10 Ten powers of 10. in 10 , 10 , and 10 . 100 1 1 One We can see that the exponent relates to the place value of the 1. The larger the exponent, the larger the place value.

Example 1 3.05 Powers of 10 and place value mathspace.co

If you have a 1 in the hundred thousands place, what power of 10 does this represent?

Create a strategy

229

MPG4 — Mathematical Connections Teachers can support students in making connections between different mathematical concepts by linking the concept of powers of 10 to place value, and showing how these concepts apply to both whole numbers and decimals. Teachers can help students connect their understanding of place value and powers of 10 by using concrete examples, such as counting zeros in large numbers and using place value tables. This approach reinforces the concept of exponents and their application in real-world contexts, enhancing students' comprehension and problem-solving skills.

Content standards 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

6.NS.3d — Recognize and represent powers of 10 with whole number exponents by examining patterns in place value.

Future connections 7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.03 Introduction to exponents

Tools You may ﬁnd these tools helpful: • Place value mats

230

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide 3.05 Powers of 10 and place value Powers of 10 and place value

Students are reminded of the concept ofbe exponents, After this lesson, you will able to... specifically focusing on the base 10. In the interactive exploration, they will identify patterns and understand why 10 raised to the power of 0 equals 1. Additionally, they will • identify powers of 10. • use patterns in place value to write, represent and evaluate powers of 10. learn how the exponent in powers of ten relates to place value.

Students: Page 112

Powers of 10 and place value

3.05 Powers of 10 and place value

Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. We just looked at special properties of the exponent 2, now let’s look at some special properties of 10.

Interactive exploration Explore online to answer After this lesson, you willthe bequestions able to...

• identify powers of 10. •mathspace.co use patterns in place value to write, represent and evaluate powers of 10. Use the interactive exploration in 3.05 to answer these questions. 1. What patterns do you notice in the numbers? Powers of 10 and place value 0

2. Why do you think 10 = 1? Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. We just looked at special properties of the exponent 2, now let’s look at some special properties of 10. For any power of ten, the expanded form will have the same number of tens as the power. The number that it evaluates Interactive to will have the same number of zeros as the exponent. exploration Explore to answeranother the questions The following tableonline demonstrates way to think of some of the powers of ten.

Power of Ten Value (basic numeral) mathspace.co Meaning 105 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 100,000 4 10Use ⋅ 10 ⋅ 10 ⋅ 10 10,000 the interactive10 exploration in 3.05 to answer these questions. 103 10 ⋅ 10 ⋅ 10 1,000 1. What patterns do you notice in the numbers? 102 10 ⋅ 10 100 1 2. Why do you10 think 100 = 1? 10 10 100 1 1 After this lesson, you will be able to...

In Words One hundred thousand Ten thousand One thousand One hundred Ten One

3.05 Powers of 10 and place value For can any see power ten, the expanded willplace have value the same number tensthe as the power. the Thelarger number it value. • identify powers of 10. We thatofthe exponent relatesform to the of the 1. The of larger exponent, thethat place evaluates •touse willpatterns have theinsame zerosrepresent as the exponent. placenumber value toofwrite, and evaluate powers of 10. The following Example 1 table demonstrates another way to think of some of the powers of ten.

Exploration Power of Ten

Meaning

Value (basic numeral)

In Words

5 have a 1 in the hundred thousands place, what power of 10 does this represent? If10 you ⋅ 10 ⋅ place 10 ⋅ 10 ⋅ 10value 100,000 One hundred thousand Powers of 10 10and

4 Students:Recall Page 112 10 10 ⋅(or 10 power) ⋅ 10 ⋅ 10tells us the number 10,000of times to multiply a Ten thousand that an exponent certain number by itself. We just looked 3 Create a strategy 10 ⋅ 10 ⋅ 10 One thousand at10special properties of the exponent 2, now let’s1,000 look at some special properties of 10. 2 To the power we can use a place value table. 10determine 10 ⋅of1010 that a 1 in the hundred 100 thousands place represents, One hundred Each 101 place to the right10of 1 increases the power of1010 by 1. Ten

100

Interactive exploration

1 to answer the questions 1 Explore online

One

We can see that the exponent relates to the place value of the 1. The larger the exponent, the larger the place value.

mathspace.co

Example 1 Use the interactive exploration in 3.05 to answer these questions. do you notice in place, the numbers? If you1.Mathspace haveWhat a 1 inpatterns the hundred thousands what power of 10 does this represent? 112 Virginia SOL Grade 6 2. Why do you think 100 = 1? mathspace.co

Create a strategy To determine the power of 10 that a 1 in the hundred thousands place represents, we can use a place value table. For any power of ten, the expanded form will have the same number of tens as the power. The number that it Each place to the right of 1 increases the power of 10 by 1. evaluates to will have the same number of zeros as the exponent. The following table demonstrates another way to think of some of the powers of ten. Power of Ten Meaning 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 105 104 10 ⋅ 10 ⋅ 10 ⋅ 10 103 10 ⋅ 10 ⋅ 10 102 Mathspace Virginia 10 ⋅SOL 10 Grade 6 112 101 mathspace.co 10 100 1

Value (basic numeral) 100,000 10,000 1,000 100 10 1

In Words One hundred thousand Ten thousand One thousand 3.05 Powers of 10 and place value One hundred mathspace.co Ten One

231

Suggested student grouping: Small groups Students will be using the slider to change the exponent of 10 and observe the expanded form and the equivalent standard form of the number. They are expected to identify patterns in the numbers and explain the mathematical concept behind the equality 100 = 1. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What patterns do you notice in the numbers? As the index increases, the value of the power of 10 increases by a multiple of 10. The number of zeros in the standard form of the number is equal to the index in the expanded form. 2. Why do you think 100 = 1? Consider the sequence of powers of 10 : 101 = 10, 102 = 100, and so forth. As we increase the exponent by 1, we multiply the previous result by 10. Conversely, decreasing the exponent by 1 means dividing the previous result by 10. When we follow this pattern backwards from 101, dividing by 10 gives

= 100 = 1. Thus, 100 = 1.

Purposeful questions • What pattern do you observe in the value of 10n as you increase the exponent n? • How does decreasing the exponent by 1 affect the standard form of 10n? • In the expanded form, how is the exponent related to the number of times 10 is multiplied? Possible misunderstandings • Believing that 100 = 0 because zero represents nothing. Explain that any nonzero number raised to the zero power equals 1, emphasizing the pattern of division by 10 leading to 100 = 1. • Thinking that the exponent indicates the total number of digits in the standard form minus one. Encourage students to compare several examples to see that the number of zeros actually equals the exponent when the base is 10.

Analyze patterns and draw conclusions about place value Targeted instructional strategies Use pattern analysis to help students recognize how multiplying by powers of 10 affects place value. Guide them to observe that each time they multiply a number by 10, all digits shift one place to the left, and a zero is added at the end. Have them record these observations in a table to organize the data and make the patterns more visible. Ten Thousands 100 101 102 103 101

Thousands

Hundreds

Tens

1 0

1 0 0

1 0 0 0

Ones 1 0 0 0 0

Encourage them to predict the outcome of multiplying by higher powers of 10 based on the patterns they’ve identified. This will help them make predictions and draw conclusions about the relationship between exponents and place value.

232

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Powers of 10 and place value Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. We just looked at special properties of the exponent 2, now let’s look at some special properties of 10.

Interactive exploration Explore online to answer the questions

Stronger and clearer each time mathspace.co English language learner support

Begin by asking write a brief of questions. what happens to a number when it is multiplied by a Use the students interactive to exploration in 3.05explanation to answer these power of 10. Then, have students pair up and share their explanations with a partner. Encourage them to listen 1. What patterns do you notice in the numbers? carefully, ask clarifying questions,0 and offer specific feedback to help improve each other’s explanations. 2.

Why do you think 10 = 1?

After discussing with their first partner, have students find a new partner and repeat the process, sharing their now-refined explanations. Finally, invite students to revise their original written explanations, incorporating any For any power of ten, the expanded form will have the same number of tens as the power. The number that it new insights or vocabulary they have gained from their peers. evaluates to will have the same number of zeros as the exponent. Emphasize use table of precise mathematical terms such as “exponent,” The the following demonstrates another way to think of some of the powers“place of ten. value,” “digit movement,” and “powers of Power 10.” This routine will help students articulate theirnumeral) understanding more clearly and strengthen both of Ten Meaning Value (basic In Words their mathematical reasoning language. 10 ⋅ 10and ⋅ 10 use ⋅ 10 ⋅ of 10 academic 100,000 One hundred thousand 105 104 103 102 101 100

10 ⋅ 10 ⋅ 10 ⋅ 10 10 ⋅ 10 ⋅ 10 10 ⋅ 10 10 1

Examples

Students: Pages 112–113

10,000 1,000 100 10 1

Ten thousand One thousand One hundred Ten One

We can see that the exponent relates to the place value of the 1. The larger the exponent, the larger the place value.

Example 1 If you have a 1 in the hundred thousands place, what power of 10 does this represent?

Create a strategy To determine the power of 10 that a 1 in the hundred thousands place represents, we can use a place value table. Each place to the right of 1 increases the power of 10 by 1.

Apply the idea Hundred Thousands 1

Ten Thousands 0

Thousands 0

Hundreds 0

Tens 0

Ones 0

There are ﬁve places to the right of 1, from ten thousands to ones. So, 1 in the hundred thousands place is 105. 112

Mathspace

Virginia SOL Grade 6

Reﬂect and check mathspace.co

The power of 10 that a 1 in the hundred thousands place represents is 105 because 100, 000 = 105. This is because moving each place to the left in the place value table multiplies the value by 10, starting from 100 = 1 for the ones place.

Example 2 Purpose Show students howexactly to determine power 10number that a of digit in as a specific place A library has 1, 000, 000the books. Writeof this books a power of 10. value represents using a place value table. Create a strategy

Reflecting with1, students To write 000, 000 as a power of 10, we will count the number of zeros to the left of the decimal point. This number To promote attention to mathematical precision in this example, emphasize to your students the importance of will be the exponent for 10. accurately identifying the correct place value of the digit. Encourage them to use a place value chart to carefully Apply ReﬂectIllustrate and check determine thatthe theidea 1 is indeed in the hundred thousands place. how being off by even one place can The number 1, 000, 000 has six zeros to the left of the Writing large as aplace power for of 10the makes significantly affect the answer—for example, mistaking the hundred numbers thousands tenthem thousands place 6 decimal point, indicating that the library’s book 5 4 collection easier to read and understand. In this scenario, 10 easily would change the power of 10 from 10 to 10 , which represents a tenfold difference. can be represented as 106 books. shows that the library has a very large collection of books. Show them how to count the exact number of places from the ones place to the hundred thousands place, ensuring they understand that the exponent in 105 corresponds to these five places. By highlighting how a small error in identifying place value can lead to an incorrect and imprecise answer, you help students appreciate the Example 3 necessity of precision. Encourage them to double-check their work and verify that each digit is in the correct place, fostering carefulexponent. and accurate mathematical practices. Find the missing 10⬚ = 10, 000, 000

3.05 Powers of 10 and place value mathspace.co

Create a strategy To ﬁnd the missing exponent, we can count the number of zeros in the number 10, 000, 000. This number is the value

233

Color-coding place value to support visual-spatial processing

use with Example 1

Student with disabilities support Use color-coding to help students visually distinguish each place value in the place value table. Assign a specific color to each column—for example, red for ‘Hundred Thousands’, orange for ‘Ten Thousands’, yellow for ‘Thousands’, green for ‘Hundreds’, blue for ‘Tens’, and purple for ‘Ones’. Apply the idea

When you Hundred place the number 1 in the hundred thousands column and zeros in theTens others, Ones highlight the 1 and Thousands Ten Thousands Thousands Hundreds the ‘Hundred Thousands’ column with processing 1 0 in red. This visual0differentiation 0 aids students0 who struggle 0 complex numerical information by making the structure of place value more accessible. 5 There are ﬁve places to the right of 1, from ten thousands to ones. So, 1 in the hundred thousands place is 10 .

Encourage students to use the same color-coding in their own notes and exercises, reinforcing their Reﬂect and check understanding of how each place relates to a specific power of 10.

Students: Page 113

Example 2 A library has exactly 1, 000, 000 books. Write this number of books as a power of 10.

Create a strategy To write 1, 000, 000 as a power of 10, we will count the number of zeros to the left of the decimal point. This number will be the exponent for 10.

Apply the idea

Reﬂect and check

The number 1, 000, 000 has six zeros to the left of the decimal point, indicating that the library’s book collection can be represented as 106 books.

Writing large numbers as a power of 10 makes them easier to read and understand. In this scenario, 106 easily shows that the library has a very large collection of books.

Example 3 Purpose Make students aware exponent. that that they can represent large numbers or the position of a digit in a large number as a Find the missing power of 10. 10⬚ = 10, 000, 000 Create a strategy Confusing the number of digits with the exponent

use with Example 2

To ﬁnd thestudent missing exponent, we can count the number of zeros in the number 10, 000, 000. This number is the value Address misconceptions of the missing exponent for 10.

Students may think that the exponent in 10n represents the total number of digits in the number, including the the idea leading Apply digit. For example, when asked to write 1, 000, 000 as a power of 10, they might observe that there are 7 The number 10, 000, 000conclude has seven that zeros.1,Therefore, missing seven digits and incorrectly 000, 000the = 10 . exponent that makes 10⬚ = 10, 000, 000 true is 7. 7

= 10, 000, 000 10 exponent To address this misconception, emphasize that the corresponds to the number of zeros following the leading 1 in powers of 10. Encourage students to count the zeros after the 1, not the total number of digits. Use a place value chart to visually represent the number 1, 000, 000. Show that each zero increases the place value summary by a power ofIdea 10: ones 100, tens 101, hundreds 102, thousands 103, up to millions 106. You might draw a chart 6 number of zeros after the 1 is the same For a power ten, as zeros the exponent. with columns labeled 100ofto 10the and place the digit 1 under 106 with in the other columns. This visual aids 6 The power of ten changes the place value of the 1. students in seeing that 1, 000, 000 has six zeros and is equal to 10 .

3.05 Powers of 10 and place value mathspace.co

234

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

113

Apply the idea

Reﬂect and check

The number 1, 000, 000 has six zeros to the left of the decimal point, indicating that the library’s book collection can be the represented Apply idea as 106 books.

Writing large numbers as a power of 10 makes them easier to read and understand. In this scenario, 106 easily shows that the library has a very large collection of books.

Students: Page 113

Hundred Thousands 1

Ten Thousands 0

Thousands 0

Hundreds 0

Tens 0

Ones 0

Example 3 places to the right of 1, from ten thousands to ones. So, 1 in the hundred thousands place is 105. There are ﬁve Find the and missing exponent. Reﬂect check ⬚

= 10, 000, 000is 105 because 100, 000 = 105. This is because The power of 10 that a 1 in the hundred thousands10place represents moving each place to the left in the place value table multiplies the value by 10, starting from 100 = 1 for the ones place.

Create a strategy

To ﬁnd the missing exponent, we can count the number of zeros in the number 10, 000, 000. This number is the value of the missing exponent for 10.

Example 2

Apply the idea The number 000,1,000 has seven zeros. Therefore, the of missing that 10⬚ = 10, 000, 000 true is 7. A library has 10, exactly 000, 000 books. Write this number booksexponent as a power ofmakes 10. 107 = 10, 000, 000

Create a strategy To write 1, 000, 000 as a power of 10, we will count the number of zeros to the left of the decimal point. This number will be the exponent for 10.

Idea summary

Purpose power of ten, the number of zeros after the 1 isReﬂect the same as the exponent. Apply For the aidea and check Students demonstrate that they can interpret the place value of digits in large numbers using powers of 10.

The power ten has changes the place the 1. Writing large numbers as a power of 10 makes them The number 1, 000,of000 six zeros to thevalue left ofofthe decimal point, indicating that the library’s book collection easier to read and understand. In this scenario, 106 easily Advanced learners: Extend to scientific notation useofwith Example 3 can be represented as 106 books. shows that the library has a very large collection books.

Targeted instructional strategies Expand students’ grasp of exponents by introducing them to scientific notation. After they determine that 3.05 Powers of 10 and place value 113 7 107 = 10,Example 000, 000,3encourage students to express this number in scientific notation as 1 × 10 . Provide a variety mathspace.co of large and small numbers, such as 0.00001 or 5, 300, 000, and have students engage in productive struggle to Find thethem missing try to convert to exponent. scientific notation (e.g., 1 × 10−5 and 5.3 ×106, respectively). 10⬚ = 10, 000, 000

Discuss how the exponent reflects the number of places the decimal point moves and the significance of the coefficient being a number between 1 and 10. Connect this concept to real-world applications, like expressing Create a strategy the speed of light or the size of microscopic organisms, to illustrate the practicality of using scientific notation in To ﬁnd the exponent, we can count thestudents number ofdeepen zeros in the number 10, 000, 000. of This number is the valuevalue, various fields. By missing exploring scientific notation, their understanding exponents, place of the missing exponent for 10. and the efficiency of representing extremely large or small numbers in mathematics and science. Apply the idea The number 10, 000, 000 has seven zeros. Therefore, the missing exponent that makes 10⬚ = 10, 000, 000 true is 7.

Students: Page 113

107 = 10, 000, 000

Idea summary For a power of ten, the number of zeros after the 1 is the same as the exponent. The power of ten changes the place value of the 1.

3.05 Powers of 10 and place value mathspace.co

113

3.05 Powers of 10 and place value mathspace.co

235

Practice Students: Pages 114–115

What do you remember? 1

SOL

Match each power of 10 to its correct numerical value. a

106

100

1, 000, 000

iii

1, 000

103

100, 000

102

10, 000

How should 105 written in a place value chart? A

Hundreds 1

Tens 0

Thousands

Hundreds 0

1 C

Ten Thousands 1

Thousands

Hundred Thousands

Ones 0 Hundreds 0

Ten Thousands 0

Tens 0

Ones 0

Thousands 0

Hundreds 0

Tens 0

Ones 0

If you have a 1 in the millions place, what power of 10 does this represent? A

Tens 0

1 3

Ones

104

105

106

107

10, 000

250

Select all that are powers of 10. A

100

200

1, 000

100, 000

Which is smaller: 105 or 107? Explain why.

Let’s practice

SOL

How many zeros should we use when writing 109 as a whole number? Explain your answer.

If a city has a population of 1, 000, 000 express this population count as a power of 10.

A certain bacteria culture grows to 100, 000 cells in one day. Express this number as a power of 10.

Xian placed the numeral 1, 000 in the place value chart. Thousands 1

Hundreds 0

Tens 0

What is 1, 000 written in powers of 10?

236

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Ones 0

SOL

What is the value of 107? 101 = 10 A

102 = 100

1, 000

103 = 1, 000

104 = 10, 000

100, 000

1, 000, 000

10, 000, 000

Find the missing exponent. 10⬚ = 10, 000

Find the missing exponent. 10⬚ = 100, 000, 000, 000

Find the missing exponent. 10⬚ = 100, 000, 000

Express 6 × 107 as a whole number.

Let’s extend our thinking 4, 000 = ⬚ × 103

Complete the statement:

The observable universe is estimated to be about 93, 000, 000, 000 light-years in diameter. Write this number as a power of 10.

Write 45, 000 as a power of 10.

Write 1, 090, 000 as two different powers of 10.

Write as whole numbers:

2 × 105

3 × 102

7 × 103

8 × 101

6 × 106

5 × 107

1 × 104

The distance between two stars is approximately 9 × 107 meters. Express the distance as a whole number.

3.05 Powers of 10 and place value mathspace.co

237

Answers 3.05 Powers of 10 and place value What do you remember?

10 D 11 104 = 10, 000 12 1011 = 10, 000, 000, 000 13 108 = 10, 000, 000

1 a ii. 1, 000, 000

b iv. 100, 000

c v. 10, 000

d iii. 1, 000

e i. 100

14 60, 000, 000 Let’s extend our thinking

2 D

15 4

3 C

16 9.3 × 1010

4 A, C, E, F

17 4.5 × 104

5 105

18 1, 090 × 103 and 1.09 × 106

Let’s practice 6 9 zeros, because the power of 10 is 9.

19 a 200, 000

d 10, 000

e 80

7 106

g 50, 000, 000

8 105

20 90, 000, 000 meters

9 103

238

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b 300

c 7, 000

6, 000, 000

Topic 3 Assessment: Integers & Exponents 1

Use a number line to determine the smallest number in each set: a

−7, −24, 10

−12°C or 4°C

−16°C or −7°C

34, −36, 0

3°C or −14°C

9°C and 19°C

b −6 −4 −2

−8 −6 −4 −2 0 2 4 6 8

−10

−8

−6

−4

−2

d −8 −6 −4 −2 0 2 4 6 8

Use the symbols <, >, or = to compare the integers below. a e

Write an expression using an inequality sign, < or >, that compares the two integers plotted on each number lines: a

1, −1, 5

For each pair of temperatures, write an expression using an inequality sign, < or >, to compare which temperature is greater: a

−14, 0, −21

58 ⬚ 63

−71 ⬚ 31

9⬚−2

−6 ⬚ 6

c g

What is the integer represented on each number line? a

−4 −3 −2 −1 0

2 3

0⬚−6

150 ⬚ − 150 −7 ⬚ − 8

b −5

−5

0⬚5

d −6

−3

5 0

−5 −5

SOL

Select all the correct answers. Identify each value that represents an integer. A E

B −2

1.2

−4

Plot each integer on a vertical number line: a

SOL

−10

−2

Use a number line to place the following integers in ascending order. 5, 17, −4, 0, 1, −20

Topic 3 Assessment: Integers & Exponents mathspace.co

239

Consider the expression 45.

In this expression, the base is ⬚, and the exponent is ⬚.

Write each expression in expanded form: a

0.133

In the pattern shown, what is the value of 25?

Is 75 a perfect square? Justify your answer.

Adeline and Delia want to write 152 in expanded form. Adeline wrote 15 ⋅ 2, and Delia wrote 15 ⋅ 15.

Who wrote it correctly?

If they evaluated the expanded form, would they get the same answer? Explain.

101 ⬚

102 ⬚

103 ⬚

104 ⬚

Millions

Hundred Thousands

Ten Thousands

Thousands

Using the pattern in the table, draw a model for the 8th perfect square.

Hundreds

Perfect Square

240

23 8

24 16

Tens

Ones

105 ⬚

22 4

Use this place value chart to represent 106. Ten Millions

21 2

Complete the table: 100 ⬚

20 1

Express each of statements as an integer: a

A price increase of $17

25 ft below sea level

A temperature drop of 5°C

Getting paid $50

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Model

Performance Task 18

A common claim says that a sheet of paper cannot be folded in half more than seven times. But is this true? How many times can you fold a piece of paper?

Every time you fold a piece of paper in half, you get a rectangle with half the area. a

Complete the table. Total Rectangle Subdivisions

Fraction of the Original Area

Decimal of the Original Area

Percent of the Original Area

0 folds

20 = 1

1.0

100%

1 folds

21 = 2

0.5

50%

2 folds 3 folds 4 folds 5 folds 6 folds 7 folds b

About what percentage of the original area is the paper after 6 folds?

If you know a piece of paper is about

of its original area, can you determine how many times it has been

folded without unfolding it? If so, how many times has it been folded? d

In 2002, Britney Gallivan, then a junior in high school, folded a single piece of paper in half 12 times. She currently holds the Guinness World Record for the most times to fold a sheet of paper in half. What perctange of the original area did this paper have after 12 folds?

Topic 3 Assessment: Integers & Exponents mathspace.co

241

Answers

9 In this expression, the base is 4, and the exponent is 5. 6.NS.3a

Topic 3 Assessment: Integers & Exponents 1 a −21

b −24

c −1

10 a 5 ⋅ 5

d −36

c 0.13 ⋅ 0.13 ⋅ 0.13

6.NS.2a, 6.NS.2b 2 a −12°C < 4°C

b −16°C < −7°C

c 3°C > −14°C

d 9°C < 19°C

c −8 < −5 or −5 > −8

d 7 > −2 or −2 < 7

6.NS.3c 13 a Delia

4 a 58 < 63

b −71 < 31

c 0<5

d 150 > −150

e 9 > −2

g 0 > −6

h −7 > −8

b No. Adeline would get 30 while Delia would get 225. 6.NS.3b

−6 < 6 14

6.NS.2c c −4

b 2 f

d 0

100

101

102

103

104

105

100

1000

10 000

100 000

6.NS.3d

−7

6.NS.2a

6 B, D, E, and G 6.NS.2a

Ten Millions Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

5 4 3 2 1 0 −1 −2 −3 −4 −5

6.NS.3d 16

10 5 0 −5

6.NS.3b

−10

17 a 17 6.NS.2a

6.NS.2a 8

−20

−15

−10

−5

From least to greatest, we have −20, −4, 0, 1, 5, 17. 6.NS.2a, 6.NS.2b

242

⋅

12 No, 82 = 64 and 92 = 81. Since 75 is between 64 and 81, it cannot be a perfect square.

6.NS.2b, 6.NS.2c

⋅

6.NS.3a b 4 > −2 or −2 < 4

7 a

⋅

11 25 = 32

3 a −6 < −2 or −2 > −6

e 3

6.NS.3a

6.NS.2c

5 a 2

b 8⋅8⋅8⋅8

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b −25

c −5

d 50

Performance Task 18 a

Percent Fraction Total Decimal of of the Rectangle of the the Original Original Original SubArea Area Area divisions 0 folds

20 = 1

1.0

100%

1 folds

21 = 2

0.5

50%

2 folds

22 = 4

0.25

25%

3 folds

23 = 8

0.125

12.5%

4 folds

24 = 16

0.0625

6.25%

5 folds

25 = 32

0.03125

3.125%

6 folds

26 = 64

0.015625

1.5625%

7 folds

27 = 128

0.0078125 0.78125%

b 1.5625% c Yes. A paper has

of the original are when it has

been folded 5 times. d 0.02441406% 6.NS.1b, 6.NS.1d, 6.NS.3a, MP1, MP4, MP5

Answers mathspace.co

243

4 Operations with Integers Big ideas • The properties of real numbers can be applied to many types of expressions. • Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. • Integers are a subset of the set of real numbers.

Chapter outline 4.01 4.02 4.03 4.04 4.05

Add and subtract integers (6.CE.2) Multiply and divide integers (6.CE.2) Absolute value of integers (6.CE.2) Real-world problems with integers (6.CE.2) Integers in the coordinate plane (6.MG.3)

248 266 280 292 304

Mount Everest, the highest point on Earth, is 29,032 feet above sea level, while the Mariana Trench, the deepest part of the ocean, is 36,070 feet below sea level. That’s a huge difference in integers!

4. Operations with Integers Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 4.CE.1 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers.

5.CE.1 — The student will estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition, subtraction, multiplication, and division with whole numbers.

Big ideas and essential understanding The properties of real numbers can be applied to many types of expressions. 4.01 — Every real number has an additive inverse or opposite. This property can be used to rewrite subtraction using addition. 4.02 — Every real number has a multiplicative inverse or reciprocal. This property can be used to rewrite division using multiplication. Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 4.04 — Determining the correct operation is essential to writing expressions to solve contextual problems with real numbers.

Integers are a subset of the set of real numbers. 4.03 — The absolute value of a number is the distance of that number from 0 on the number line. Every number and its opposite have the same absolute value. 4.05 — Integers can be used to describe points in 2-dimensional space using ordered pairs or coordinates (x, y) where the x-coordinate is the horizontal distance from a central point called the origin and the y-coordinate is the vertical distance from the origin.

Standards 6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context. 6.CE.2a — Demonstrate/model addition, subtraction, multiplication, and division of integers using pictorial representations or concrete manipulatives.∗ 4.01 Add and subtract integers 4.02 Multiply and divide integers

6.CE.2b — Add, subtract, multiply, and divide two integers.∗ 4.01 Add and subtract integers 4.02 Multiply and divide integers 6.CE.2c — Simplify an expression that contains absolute value bars ∣∣ and an operation with two integers and represent the result on a number line. 4.03 Absolute value of integers

246

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.CE.2d — Estimate, determine, and justify the solution to one and two-step contextual problems, involving addition, subtraction, multiplication, and division with integers. 4.01 Add and subtract integers 4.02 Multiply and divide integers 4.04 Real-world problems with integers 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers. 6.NS.2d — Identify and describe the absolute value of an integer as the distance from zero on the number line. 4.03 Absolute value of integers 6.MG.3 — The student will describe the characteristics of the coordinate plane and graph ordered pairs. 6.MG.3a — Identify and label the axes, origin, and quadrants of a coordinate plane. 4.05 Integers in the coordinate plane

4.05 Integers in the coordinate plane 6.MG.3c — Graph ordered pairs in the four quadrants and on the axes of a coordinate plane. Ordered pairs will be limited to coordinates expressed as integers. 4.05 Integers in the coordinate plane 6.MG.3d — Identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane. Ordered pairs will be limited to coordinates expressed as integers. 4.05 Integers in the coordinate plane 6.MG.3e — Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers. 4.05 Integers in the coordinate plane

6.MG.3b — Identify and describe the location (quadrant or the axis) of a point given as an ordered pair. Ordered pairs will be limited to coordinates expressed as integers.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers. 7.MG.4 — The student will apply dilations of polygons in the coordinate plane. 7.PFA.2 — The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables.

8.MG.3 — The student will apply translations and reflections to polygons in the coordinate plane. 8.PS.3 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on scatterplots. A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

4. Operations with Integers mathspace.co

247

4.01 Add and subtract integers Subtopic overview Lesson narrative In this lesson, students will learn to add and subtract integers using number lines and integer chips. They will explore these concepts through interactive activities and visual models. The explorations include plotting integers on a number line, using arrows to represent addition and subtraction, and employing integer chips to model and solve problems. Examples and strategies will guide students in adding and subtracting positive and negative integers, understanding opposites, and applying these skills to real-world scenarios. By the end, students should confidently perform addition and subtraction with integers.

Learning objectives

4.01 Add and subtract integers

Students: Page 118

After this lesson, you will be able to... • represent addition and subtraction of integers using models and pictures. • add and subtract two integers. • find and justify the solultion to real-world problems involving addition and subtraction of integers.

Add integers We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated Key vocabulary

by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with − are negative.  negative  opposite difference The sign of an integer givesit integer a direction. We can imagine that for every integer on the number line there is an arrow  positive going from 0 to that integer. sum 

For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

Essential understanding

Interactive exploration

Every real number has an additive inverse or opposite. This property can be used to rewrite subtraction using Explore online to answer the questions addition.

mathspace.co

Standards Use the interactive exploration in 4.01 to answer these questions. This subtopic addresses the followingadding Virginia 2023 Mathematics Standards of Learning standards. 1. What direction represents a positive integer? 2.

What direction represents adding a negative integer?

Mathematical process goals

3. What are the ways you can create a sum that is positive? MPG1 —4.Mathematical What are theProblem ways youSolving can create a sum that is negative?

Teachers helpare students develop mathematical skills in this lesson by providing real-life 5.canWhat the ways you can create a sumproblem-solving that is 0? situations where integer operations are used, such as temperature changes, elevation differences, or financial 6. What happens when 0 is one of the numbers being added? transactions. Teachers can encourage students to use multiple strategies, including graphic organizers, step-by-step processes, and concrete or pictorial models, before transitioning to standard algorithms for adding and subtracting integers. Additionally, teachers can ask guiding to arrows foster discussion on problem-solving strategies, The addition of integers can be represented byquestions adding their on the number line. When we combine the the lengths and directions of two arrows, we get a third arrow whose length andideas. direction corresponds to the sum. This is reasonableness of solutions, and effective communication of mathematical This approach ensures students adding two (orand more) integers results in another integer. canbecause, estimate, determine, justify their always solutions effectively. The image shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the Mathspace Virginia SOL Grade 6 Teacher Edition order of addition does not affect the result?

248

mathspace.co

2 8

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can help students develop their reasoning skills by discussing the rules for adding and subtracting integers with the same and different signs, as well as the concept of subtracting an integer as adding its opposite. Teachers can ask students to explain the “why” behind these rules and encourage them to use logical reasoning to analyze their solutions.

Teachers can integrate this goal into their instruction by using various representations, such as number lines, integer chips, counters, or other manipulatives to help students visualize and understand integer operations. Teachers can make a larger numberline out of tape on the floor that students can physically move on to help demonstrate integer addition of subtraction. When using integer chips, teachers should help students understand zero pairs (a positive and negative form of the same number). They can also encourage students to represent and describe mathematical ideas and relationships using different methods, such as visual models, symbolic notation, and verbal explanations.

MPG4 — Mathematical Connections Teachers can help students make connections between the rules for adding and subtracting integers and their prior knowledge of adding and subtracting whole numbers. They can also encourage students to relate the concepts of integer addition and subtraction to real-world situations, making connections to other disciplines, such as science and social studies.

Content standards 6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context. 6.CE.2a — Demonstrate/model addition, subtraction, multiplication, and division of integers using pictorial representations or concrete manipulatives.∗

6.CE.2b — Add, subtract, multiply, and divide two integers.∗ 6.CE.2d — Estimate, determine, and justify the solution to one and two-step contextual problems, involving addition, subtraction, multiplication, and division with integers.

Prior connections 4.CE.1 — The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.01 Identify and represent integers

4.01 Add and subtract integers mathspace.co

249

Tools You may find these tools helpful: • Number line

• Integer chips

• Chess pawn or game chip

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Model addition and subtraction of integers using the number line Targeted instructional strategies Emphasize that for a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left. To show how addition of integers is represented on the number line, one can start modeling addition of a negative number and a positive number. For example, to model 4 + (−9) on the number line: • Place the tail of arrow on 0 and draw an arrow 4 units to the right of 0 stopping on 4. • Start the next arrow from the head of the first arrow and draw the second arrow 9 units to the left. • Stop at −5 since you are counting 9 units down from 4. • Mark the distance between the last arrow head from 0 to find the equivalent number of the expression. −9

−5 −7

−6

−5

−4

−3

−2

−1

Information gap English language learner support Pair up students and provide each partner with a different piece of an integer addition or subtraction problem. For example, give one student a number line with arrows showing movement but without numerical labels, and give the other student a set of integer chips arranged to represent the same problem without an accompanying equation. Students must communicate with each other to share their information and collaboratively determine the integers involved and the operation being performed. Encourage them to use specific mathematical vocabulary such as “positive,” “negative,” “add,” “subtract,” “increase,” “decrease,” “opposite,” and “zero pairs.” Model for students how to ask effective questions and explain their reasoning clearly. This activity creates a need for precise communication, helps students practice using mathematical language, and deepens their understanding of integer operations through collaborative problem-solving.

Represent the problem physically Student with disabilities support To assist students that may struggle to draw number lines and arrows, it can be useful to physically model the problem on a given number line. Give students a physical copy of a number line and an item to represent the starting point, for example a sentence strip with a number line drawn and a chess pawn. Begin with a problem with two positives, such as 4 + 5. Have students place the pawn at 4 and move the pawn 5 spaces to the right. Next, have students model a positive plus a negative, such as 3 + (−6), where students place the pawn at 3 and move 6 spaces to the left. Have students model a negative plus a positve, such as −1 + 2, and lastly a negative plus a negative, such as −3 + (−3)..

250

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

4.01 Add and subtract integers After this lesson, you will be able to... • represent addition and subtraction of integers using models and pictures. • add and subtract two integers. • find and justify the solultion to real-world problems involving addition and subtraction of integers.

Student lesson & teacher guide Add integers integers Students:Add Pages 118–119

We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with − are negative. The sign of an integer gives it a direction. We can imagine that for every integer on the number line there is an arrow going from 0 to that integer.

4.01 Add and subtract integers

For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

Interactive After this lesson,exploration you will be able to... online to answer the questions •Explore represent addition and subtraction of integers using models and pictures. • add and subtract two integers. •mathspace.co find and justify the solultion to real-world problems involving addition and subtraction of integers.

Use the interactive exploration in 4.01 to answer these questions. What direction represents adding a positive integer? Add1. integers

2. What direction represents adding a negative integer? We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated 3. What are the ways you can create a sum that is positive? by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with − are negative. 4. What are the ways you can create a sum that is negative? The sign of an integer gives it a direction. We can imagine that for every integer on the number line there is an arrow ways you can create a sum that is 0? going5.fromWhat 0 to are thatthe integer. 6. What happens when 0 is one of the numbers being added? For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left. The addition of integers can be represented by adding their arrows on the number line. When we combine the Interactive exploration lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to the sum. This is Exploretwo online to answer the questions because, adding (or more) integers always results in another integer. The image shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the mathspace.co order of addition does not affect the result? Use the interactive exploration in 4.01 to answer these6 questions.

What direction represents adding a positive integer?

What direction represents adding a negative integer?

−3 −2

−1

the ways you can createWe a sum is positive? What3. if weWhat wantare to add a negative integer? use that the same approach, the only difference being that the arrows are pointing different directions. The image shows that 4 is + (−9) = −5, which is the same result that we get from (−9) + 4. 4. inWhat are the ways you can create a sum that negative? 5.

What are the ways you can create a sum that −9is 0?

What happens when 0 is one of the numbers −5 being added? −7

−6

−5

−4

−3

−2

−1

4 0

The addition of integers can be represented by adding their arrows on the number line. When we combine the To think of this process more simply, we can plot the ﬁrst integer on the number line and move the direction and lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to the sum. This is number of spacesVirginia indicated by the6 second integer. Mathspace SOL Grade 118 because, adding two (or more) integers always results in another integer. mathspace.co When we add a positive number we move to the right. The image shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the −1 + 4 = 3 order of addition does not affect the result? 6

2 8 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

When we add a negative number we move to the left. What if we want to add a negative integer? We use the same approach, the only difference being that the arrows are pointing in different directions. The image shows that 44 ++ (−3) (−9) == 1−5, which is the same result that we get from (−9) + 4. −9

−10 −9 −8 −7 −6 −5 −4 −3−5 −2 −1 0 1 2 3 4 5 6 7 8 9 10 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

Example 1 118

Mathspace Virginia SOL Grade 6 mathspace.co

Find the value of −7 + 13.

Create a strategy Draw a model with arrows using a number line.

4.01 Add and subtract integers mathspace.co

251

Add integers We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with − are negative.

Exploration The sign of an integer gives it a direction. We can imagine that for every integer on the number line there is an arrow going from 0 to that integer.

Students: Page 118

For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.01 to answer these questions. 1.

What direction represents adding a positive integer?

What direction represents adding a negative integer?

What are the ways you can create a sum that is positive?

What are the ways you can create a sum that is negative?

What are the ways you can create a sum that is 0?

What happens when 0 is one of the numbers being added?

The addition of integers can be represented by adding their arrows on the number line. When we combine the lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to the sum. This is because, adding two (or more) integers always results in another integer. Suggested student grouping: Small groups The image shows how 6 are + 2 =given 8 is represented usingmovable the addition of arrows the numberdifferent line. Can you see how the can In this exploration, students a graph with points thatonrepresent integers. They order of addition does not affect the result? drag these points and observe the sum as they move the points. They will observe how the sum changes when 2 sum. This will help them understand the adding positive or negative integers and when zero is6included in the 8 concept of addition in integer number line. −3 −2

−1

Ideal student responses What if we want to add a negative integer? We use the same approach, the only difference being that the arrows are These ideal responses may differ from other correct student responses. Less formal responses can be pointing in different directions. The image shows that 4 + (−9) = −5, which is the same result that we get from (−9) + 4. connected with the more precise mathematical language presented here. −9

1. What direction represents adding a positive integer? −5 Moving to the right along the number line represents adding1 a positive integer. −7 −6 −5 −4 −3 −2 −1 0 2 3 4 5 2. What direction represents adding a negative integer? Moving to the left along the number line represents adding a negative integer. 118 Mathspace Virginia SOL Grade 6 3. What aremathspace.co the ways you can create a sum that is positive? A sum is positive if we add two positive integers together or if the positive integer is greater than the negative integer. 4. What are the ways you can create a sum that is negative? A sum is negative if we add two negative integers together or if the negative integer is greater than the positive integer. 5. What are the ways you can create a sum that is 0? A sum is zero if we add a positive integer and a negative integer of the same value. 6. What happens when 0 is one of the numbers being added? When zero is added to any number, the sum remains the same as the original number. This is because zero is the identity element for addition. Purposeful questions • What happens when you add a positive and a negative integer of the same value? • How does the order of the integers affect the sum?

252

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

When we add a negative number we move to the left. 4 + (−3) = 1

Examples Students: Pages 119–120

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Example 1 Find the value of −7 + 13.

Create a strategy Draw a model with arrows using a number line.

Apply the idea We start by drawing an arrow for −7 in the number line. −7 −8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9

Adding a positive integer means we move to the right. So, to draw the arrow that represents adding 13, we need to count 13 units to the right starting at −7. −7

−8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9

Finally, we draw a third arrow which starts at 0 and ends at the tip of the second arrow. This third arrow represents the sum of −7 and 13 which is 6. −7

13 6

−8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9

−7 + 13 = 6

Evaluate

Reﬂect and check What if we start by drawing ﬁrst the arrow that represents 13 then the arrow that represents adding −7? −7

13 −3 −2 −1 0

2 3 4 5 6 7 8 9 10 11 12 13 14

Notice that when we draw the third arrow that represents the sum, we get the same answer. −7

4.01 Add and subtract integers mathspace.co

6 −3 −2 −1 0

119

2 3 4 5 6 7 8 9 10 11 12 13 14

This shows that −7 + 13 is the same as 13 + (−7). The order of drawing the arrows does not affect the sum.

Example 2 Purpose Find the value + (−6). Show students how of to11use a number line to visualize addition of positive and negative integers. Create a strategy Adding a negative integer means we will move to the left on the number line.

Apply the idea Plot 11 on the number line: −2 −1

From 11 we want to move 6 units to the left. −2 −1

11 + (−6) = 5

Evaluate

4.01 Add and subtract integers mathspace.co

253

Advanced learners: Extend integer addition to algebraic concepts

use with Example 1

Targeted instructional strategies Encourage students to connect the visual representation of integer addition on the number line to broader algebraic ideas.and After demonstrating how to find the value of −7 + 13 using the number line, prompt students to Reﬂect check explore What how ifthis operation reflects the properties of real13numbers, such the commutative we start by drawing ﬁrst the arrow that represents then the arrow thatasrepresents adding −7? and associative properties. Discuss how adding a positive number moves13to the right on the number line, while adding a −7 negative number moves to the left, and relate this to the concept of additive inverses. Introduce simple algebraic equations like x + (−7) = −3 6 and −2 −1ask 0 students 1 2 3 4 to 5 solve 6 7 for 8 9x,10connecting 11 12 13 14 the solution back to the number line model. Encourage students to consider how these integer operations apply in real-world contexts, such Notice that when we drawor thechanges third arrow represents the we get the same answer. to algebraic reasoning and as financial gains and losses inthat temperature. Bysum, linking integer addition real-life applications, you help advanced learners deepen13their understanding and see the interconnectedness −7 6 of mathematical concepts. −3 −2 −1 0

2 3 4 5 6 7 8 9 10 11 12 13 14

shows that −7 + 13 is the same as 13 + (−7). The order of drawing the arrows does not affect the sum. Students:This Page 120

Example 2 Find the value of 11 + (−6).

Create a strategy Adding a negative integer means we will move to the left on the number line.

Apply the idea Plot 11 on the number line: −2 −1

From 11 we want to move 6 units to the left. −2 −1

11 + (−6) = 5

Evaluate

Reﬂect and check We can combine the adjacent signs by writing: 11 + (−6) = 11 − 6 =5

Adding a negative 6 is the same as subtracting 6 Subtract

Example 3 Purpose Find the value −12 + (−8).number plus a negative number can easily be solved by subtracting the first Show students that aofpositive number by the second number. Create a strategy

Adding a negative integer means we move to the left on the number line.

120

254

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Reﬂect and check We can combine the adjacent signs by writing: 11 + (−6) = 11 − 6

Adding a negative 6 is the same as subtracting 6

=5 Students: Pages 120–121

Subtract

Example 3 Find the value of −12 + (−8).

Create a strategy Adding a negative integer means we move to the left on the number line.

Apply the idea Plot −12 on a number line: 120

Mathspace Virginia SOL Grade 6 mathspace.co

−25

−20

−15

−10

−5

−20

−15

−10

−5

From −12, move 8 units to the left. −25

−12 + (−8) = −20

Reﬂect and check −12 + (−8) = −12 − 8

Adding a negative is the same as subtracting a positive

Idea summary

Purpose When ﬁnding the sum of two integers, we can use a number line. Show studentsThe how sumtoof:add negative integers using a number line. •

two positive integers is another positive integer

Analyzing patterns adding negative • a positive integerin and a negative integer mayintegers be positive or negative

use with Example 3

• two negative integers is another negative integer Targeted instructional strategies

Apply the idea • an integer and its opposite is 0

To deepen students’ understanding of adding negative integers, use pattern analysis to deepen students’ Plot −12 on a number line: understanding of adding negative integers. Begin by presenting several addition problems involving negative numbers, such as −4 + (−6), −9 +−25 (−3), and−20 −7 + (−5).−15Have students solve −10 −5 these0problems individually and Subtract integers then pair up to compare their answers. Encourage them to discuss any patterns they notice, such as the sums We have how to the model addition with between number lines. we will explore chips. Integer chips are From −12,looked move 8atunits to becoming more negative and theleft. relationship theNow, addends and theinteger result. another way to model integers and use the fact that the sum of two opposite integers is 0.

Guide them to see that adding two negative numbers results in a negative sum whose absolute value is the −25 −20 −15 −10 −5 0 sum of the absolute values of exploration the addends. Facilitate a class discussion to generalize this pattern into a rule for Interactive adding negativeExplore integers, helping them make predictions online to answer the questions −12 + (−8) and = −20draw conclusions based on their observations.

mathspace.co Students: Page 121 −12 + (−8) = −12 − 8 Reﬂect and check

Adding negative is the same as subtracting a positive Use the interactive exploration in 4.01 toaanswer these questions. 1.

Which problems were the simplest?

Why did we have to add zero pairs for some problems and not others?

3. Idea If wesummary are subtracting two negative numbers, how do the integer chips work? When ﬁnding the sum ofone twonegative integers,and we one can positive use a number line. 4. If we are subtracting integer, how do the integer chips work? The sum of: • two positive integers is another positive integer Now that explored subtraction with integer chips, exploreorsubtraction • we’ve a positive integer and a negative integer may let’s be positive negative on a number line. When we added integers• ontwo a number line we went in the direction indicated by the sign of the number. But the subtraction operation negative integers is another negative integer tells us •to reverse theand direction of the integer that follows. an integer its opposite is 0

Subtract integers We have looked at how to model addition with number lines. Now, we will explore integer chips. are 121 4.01 Add andInteger subtractchips integers another way to model integers and use the fact that the sum of two opposite integers is 0. 4.01 Addmathspace.co and subtract integers

mathspace.co

Interactive exploration Explore online to answer the questions

255

Idea summary

Interactive exploration When ﬁnding the sum of two integers, we can use a number line. Explore online to answer the questions

The sum of:

• two positive integers is another positive integer mathspace.co • a positive integer and a negative integer may be positive or negative • two negative integers is another negative integer Use the interactive exploration in 4.01 to answer these questions. • an integer and its opposite is 0 Students: Pages 121–122 1. Which problems were the simplest?

Subtract integers 2.

Why did we have to add zero pairs for some problems and not others?

If we are subtracting two negative numbers, how do the integer chips work? Subtract integers 3.

4. If we are subtracting one negative and one positive integer, how do the integer chips work? We have looked at how to model addition with number lines. Now, we will explore integer chips. Integer chips are another way to model integers and use the fact that the sum of two opposite integers is 0. Now that we’ve explored subtraction with integer chips, let’s explore subtraction on a number line. When we added integers on a number line we went in the direction indicated by the sign of the number. But the subtraction operation Interactive exploration tells us to Explore reverse online the direction of the that follows. to answer theinteger questions When we subtract a positive number we move to the left, because we’re reversing the direction indicated by positive 4.mathspace.co −2 − 4 = −6 Use the interactive exploration in 4.01 to answer these questions. 1.

Which problems were the simplest?

−10zero −9 −8pairs −7 −6 −5 −3 −2problems −1 0 1 2 and 3 4not 5 others? 6 7 8 9 10 Why did we have to add for−4 some

4.01 Add and subtract integers mathspace.co

121

3. If we are subtracting two negative numbers, how do the integer chips work? When4.we If subtract negative number we move thepositive right, because reversing the direction indicated by −3. we are asubtracting one negative andtoone integer,we’re how do the integer chips work? 4 − (−3) = 7 Now that we’ve explored subtraction with integer chips, let’s explore subtraction on a number line. When we added integers on a number line we went in −8 the−7direction indicated −10 −9 −6 −5 −4 −3 −2 −1 0 by 1 2the 3 sign 4 5 of6 the 7 8number. 9 10 But the subtraction operation tells us to reverse the direction of the integer that follows. Let’s look at 3 − 5. We can see from the number line that this is actually the same as 3 + (−5). In other words, subtracting 5 is the same as adding the opposite of 5. −5

5 3

4.01 Add and subtract integers mathspace.co

−2 −6

−5

−4

−3

−1

−2

Let’s look at 7 − (−2). Starting at 7, we move to the right 2 because we’re reversing the direction of −2. From the number line we can see this gives us the same result as adding 7 + 2. −2

7 −3 −2 −1

9 4

Example 4 Find the value of 8 − 7.

Create a strategy Subtracting a negative integer means we will move to the left on the number line.

Apply the idea Plot 8 on a number line: 0

From 8, move 7 units to the left. 0

8−7=1

256

Mathspace Virginia SOL Grade 6 Teacher Edition 122 Mathspace Virginia SOL Grade 6 mathspace.co mathspace.co

121

• • • •

two positive integers is another positive integer a positive integer and a negative integer may be positive or negative two negative integers is another negative integer an integer and its opposite is 0

Exploration Subtract integers

Students:WePage 121 have looked at how to model addition with number lines. Now, we will explore integer chips. Integer chips are another way to model integers and use the fact that the sum of two opposite integers is 0.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.01 to answer these questions. 1.

Which problems were the simplest?

Why did we have to add zero pairs for some problems and not others?

If we are subtracting two negative numbers, how do the integer chips work?

If we are subtracting one negative and one positive integer, how do the integer chips work?

Now that we’ve explored subtraction with integer chips, let’s explore subtraction on a number line. When we added integers on a number line we went in the direction indicated by the sign of the number. But the subtraction operation tells us to reverse the direction of the integer that follows. Suggested student grouping: In pairs

Students will be using integer chips to visually understand the process of adding and subtracting integers. Through the use of integer chips, students will set up expressions and see the animations to understand how the operations work. The expressions students will be dealing with are: 4 + 2, 5 + (–3), 5 – 3, 2 – 4, –1 – 3, and −3 − (−2). Ideal student responses

4.01 Add and subtract integers mathspace.co

121

These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Which problems were the simplest? The problems 4 + 2 and 5 − 3 were the simplest because they involved addition and subtraction of positive numbers, which is more straightforward. 2. Why did we have to add zero pairs for some problems and not others? We had to add zero pairs for some problems when we were dealing with subtraction of negative numbers or when the number we were subtracting was larger than the number we were subtracting from. This is because when subtracting, we are removing chips, and if there are not enough chips to remove, we need to add zero pairs. 3. If we are subtracting two negative numbers, how do the integer chips work? If we are subtracting two negative numbers, we add the opposite of the second number (which is a positive number) to the first number. This is because subtracting a negative is the same as adding a positive. 4. If we are subtracting one negative and one positive integer, how do the integer chips work? If we are subtracting a positive number from a negative number, we add more negative chips. This is because subtracting a positive number is the same as adding a negative number. Purposeful questions • How does the position of the negative sign change the operation in an expression? • How is adding zero pairs helpful in understanding integer operations? Possible misunderstandings • Subtracting a negative number makes the value more negative. Show with integer chips that subtracting a negative is the same as adding a positive, which increases the total value. • Zero pairs have a numerical value and change the total sum. Emphasize that zero pairs add to zero and do not affect the total, reinforcing that they represent zero.

4.01 Add and subtract integers mathspace.co

257

Examples Students: Page 122

7 −3 −2 −1

9 4

Example 4 Find the value of 8 − 7.

Create a strategy Subtracting a negative integer means we will move to the left on the number line.

Apply the idea Plot 8 on a number line: 0

From 8, move 7 units to the left. 0

8−7=1

Virginia SOL Grade 6 Purpose122 Mathspace mathspace.co Demonstrate to students how to use a number line to perform simple subtraction operations.

Students: Page 123

Purpose Show students that subtracting a negative integer is the same as adding its opposite.

258

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Confusing subtraction of negatives with subtraction of positives

use with Example 5

Address student misconceptions Students might mistakenly treat subtracting a negative number the same as subtracting a positive number. For instance, they may calculate 3 − (−9) as 3 − 9, resulting in −6, because they overlook the effect of the double negative. To help resolve this misconception, reinforce the concept that subtracting a negative is equivalent to adding a positive. Encourage students to rewrite expressions like 3 − (−9) as 3 + 9 to simplify the operation. Use a number line to visually demonstrate this principle: starting at 3, show that subtracting −9 means moving 9 units to the right, arriving at 12. Incorporate an image of a number line with arrows illustrating the movement from 3 to 12 to make the concept more tangible.

Students: Page 123

Purpose Show students how to translate a real-world scenario involving ascending and descending into a problem of addition and subtraction of integers on a number line.

Students: Page 124

4.01 Add and subtract integers mathspace.co

259

Purpose Challenge students to apply their understanding of integers on a number line to solve a real-world problem involving ascending from a negative position. Reflecting with students Discuss the significance of negative numbers in this context and how they represent positions below the surface. Invite students to explore alternative scenarios, such as what Nadia’s position would be if she descended further or ascended to the surface.

Students: Page 124

Practice Students: Pages 124–127

What do you remember? 1

State whether each of the following numbers is an integer: a e

−4

9.0

d 0

0.333 …

State which values on the number line below represent integers. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

260

Do we move to the left or to the right of a number line when: a

Adding a positive integer

Adding a negative integer

Subtracting a positive integer

Subtracting a negative integer

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Is the result of each operation positive, negative or either positive or negative? a

Adding two positive integers

Subtracting two positive integers

Adding two negative integers

Subtracting two negative integers

Adding two different signed integers

Subtracting two different signed integers

Use the plotted point shown to determine each of the following values. −10

−5

6 units left of the plotted point

5 units right of the plotted point

2 units right of the plotted point

4 units left of the plotted point

Evaluate: a

38 − 7

22 − 8

9 + 19

32 − 17

46 + 54

520 − 18

72 + 74

129 + 85

13 − (−2)

Subtraction can be rewritten as addition. Example: 3 − 4 = 3 + (−4) Rewrite each subtraction problem as addition. a

−3 − 7

4 − 11

−11 − (−6)

Write an integer to represent each of the following statements: a

A price rise of $17

A loss of $176

A temperature drop of 5 °F

58 °F above 0 °F

An elevation of 980 ft

Grew by 3 in last year

Descending 9 ﬂoors

Depositing $75 in to a bank account

A distance of 320 ft below sea level

Ascending 10 ﬂoors

Withdrawing $145 from a bank account

A proﬁt of $650

Let’s practice 9

Write and solve the addition problem represented by the model. a

−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Use the plotted point on the number line to add each pair of integers. −15

−5 + 4

−10

−5 + (−8)

−5

−5 + 11

−5 + 5

4.01 Add and subtract integers mathspace.co

261

Use the plotted point on the number line to subtract each pair of integers. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

a 12

1−4

1 − (−5)

1 − 10

1−1

Evaluate: a

14 + 7

−97 + 39

−40 + (−105)

109 + (−191)

−825 + 1174

12 345 − 5432

48 392 − 30 000

−24 150 + 18 230

Evaluate: a

−12 − 3

36 − 53

5 − (−198)

288 − 677

3000 − 1495

−2027 − (−973)

51 000 − 48 400

−32 000 − 41 000

Evaluate: a

7 + (−5)

−51 + 23

120 − (−52)

95 − (−95)

(−17) + (−17) + (−18)

(−18) − (−11) − (−17)

−76 − 29 + 112

175 + (−345) + 28

Felix is trying to determine the smaller result between −27 + 38 and 41 + (−32). He decides to use a number line to ﬁnd each sum and then plot the resulting points on a number line to determine the smaller sum. a

Describe how Felix will ﬁnd each sum using the number line, including the directions he will move and how many units he will move in each sum.

Plot each sum on a number line and state which sum is smaller.

Use a number line or integer chips to describe how both −6 − 7 and 5 − 11 both result in negative answers.

Determine whether the given situation would represent addition or subtraction.

The total length of 3 pencils with lengths 6 in, 7 in, and 9 in.

Total volume of water if one can holds 8 oz and the other contains 12 oz.

The change for $10 bill if you buy a notebook worth $2.

The change in a plant’s height if it measured 14 inches last month and measures 17 inches this month.

The difference in temperatures from a high of 45 °F and a low of −12 °F.

The balance in a bank account after a deposit of $108 and a second deposit of $62.

Amy solved a subtraction problem and wrote her work as: 53 − (−26) = −53 + (−26) = −79 Describe and correct the error in her steps.

Valentina dives off a platform that is 8 m above the water. She descends 14 m before returning to the surface. a

Plot the integer that represents Valentina’s initial position on the diving platform on a number line.

Write the integer that represents her lowest position throughout the dive.

Write the greatest depth that Valentina reaches during the dive.

One evening in Moscow the temperature fell to −3° C. By midday the next day, the temperature was forecast to be 15° C. a

Plot the temperature during the evening on a number line.

Plot the temperature the next day on the same number line.

Find the expected rise in temperature.

Liquid nitrogen can be used to make ice cream. The ice cream mixture starts at 26° C, then rapidly cools to −185° C after liquid nitrogen is added to it. State the integer that represents this change in temperature.

262

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s extend our thinking 22

Each letter has been assigned to an integer. A word must be formed using these letters and each letter can only be used once. The table shows the assignment of the integers: Letter Integer

−3

−2

−1

The score of the word is the sum of the integers assigned to its letters. Find the score of each word: a 23

NET

EAR

CARES

Each bottle of orange juice is supposed to contain 250 mL. Nine bottles are inspected by the quality assurance department. The results of the inspection are shown in the table: Bottle Amount below or above the required volume (mL) a

−7

−11

−1

−5

Find the actual volume of: i

RAIN

Bottle 1

Bottle 2

iii

Bottle 6

Bottle 10

Calculate the total volume contained in the 10 bottles.

The table shows the proﬁt of Eiichiro’s Ramen from August to December. The goal of Eiichiro’s Ramen was to have a total proﬁt of $10 from August to December. August $13

September −$10

October −$11

November $3

December $14

Did Eiichiro’s Ramen reach their goal? Explain your answer using number lines. 25

Write in order from greatest to least: −5 − (−8), 7 + (−12), 10 − 18, −16 + 21, 31 + (−31)

A dolphin is 12 feet below the surface of the water. It swims up and jumps out of the water to a height of 6 feet above the surface. Find the vertical distance the dolphin travels. Explain your answer using number lines.

Consider the integer −18:

Write two integers with different signs that have a sum of −18.

Write two integers with the same sign that have a sum of −18.

Consider the diagram at the right: Place the integers −3, −2, −1, 0, 2, 3, 4 and 5 in the circles so that any three circles in a straight line add up to 3. 1

Consider the diagram at the right: Place the integers −5, −4, −3, −2, −1, 0, 1, 2 and 3 in the circles so that any three circles in a straight line add up to −3.

4.01 Add and subtract integers mathspace.co

263

Answers

16 For both representations, ﬁrst change both subtraction problems to addition, creating the problems −6 + (−7) and 5 + (−11). Both representations are shown below. Both representations show that when the operations are performed, the answers are negative.

4.01 Add and subtract integers What do you remember? 1 a Yes e No

b Yes

c No

d Yes

Yes

g Yes

h No

2 −6, 0, 10 3 a To the right

b To the left

c To the left

d To the right −6 + (−7)

4 a Positive

5 + (−11)

b Either positive or negative c Negative d Either positive or negative

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

−15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

e Either positive or negative f

b 5

c 0

d −6

6 a 31

b 14

c 28

d 15

g 146

h 214

e 100

502

7 a −3 + (−7)

b 4 + (−11)

c −11 + (6)

d 13 + (2)

8 a 17

b −176

c −5

d 58

e 980

g −9

h 75

−320

k −145

9 a −6 + 9 = 3

17 a Addition

19 a

c 6

d 0

11 a −3

b 6

c −9

d 0

12 a 21

b −58

c −145

d −82

g 18 392

h −5920

c 203

d −389

g 2600

h −73 000

b −28

c 172

d 190

g 7

h −142

6913

b −17

e 1505

14 a 2 e −52

−1054

The smaller sum is 41 + (−32), which is 9.

264

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−5

Addition

c 6 m

20 a

−18 −15 −12 −9 −6 −3

c 18 °C 21 −211 °C

For the sum 41 + (−32), Felix would start at 41 on the number line, then move 32 units to the left to ﬁnd the sum. 6

−10

b −6

15 a F or the sum −27 + 38, Felix would start at −27 on the number line, then move 38 units to the right to ﬁnd the sum.

53 − (−26) = 53 + 26 = 79

d −4 + (−5) = −9

b −13

13 a −15

d Subtraction

e Subtraction

18 To write subtraction as addition, the subtraction sign changes to addition and the sign of the second number changes. Amy changed the sign of the ﬁrst number instead of the second. The correct work would be

650

10 a −1

b Addition

c Subtraction

b 7 + (−2) = 5

c −8 + (−4) = −12

e 349

−6 + (−7)

Let’s practice

Either positive or negative

5 a −8

5 + (−11)

Let’s extend our thinking 22 a 1

b −2

23 a i 254 mL ii 243 mL

c 1

d 2

iii 250 mL

iv 251 mL

b 2499 mL 24 No. In order to determine if the total profit reach $10, we need to find the total profit from August to December using the number line. Remember when adding a positive number we move the point to the right of the number line and when adding a negative number we move the point to the left of the number line.

First, ﬁnd the total proﬁt from August to September. Locate 13 on the number line.

• −26 and 8

• −9 and −9

• −25 and 7

• −10 and −8

• −24 and 6

• −11 and −7

• −23 and 5

• −12 and −6

• −22 and 4

• −13 and −5

−15

−10

−5

Since we are adding −10, we need to move the point, 10 units going to the left which resulted to 3.

−15

−10

−5

Then, we need to add −11 for the month of October. Since we are adding −11, we need to move the point, 11 units going to the left which resulted to −8.

−15

−10

−5

27 a Possible answers:

28 Example answer:

−1 4 −3

Now, we need to add 3 for the month of November. Since we are adding 3, we need to move the point, 3 units going to the right.

−15

−10

−5

−15

−5

5 −2

2 3

29 Example answer: −3 −2

2 −1

−5 −10

0 1

Lastly, we need to add 14 for the month of December. Since we are adding 14, we need to move the point, 14 units going to the right which resulted to 9.

b Possible answers:

Since the total proﬁt is only $9, Eiichiro’s Ramen did not reach their goal.

−4

0 1

25 −16 + 21, −5 − (−8), 31 + (−31), 7 + (−12), 10 − 18 26 18 feet. In order to determine the vertical distance, we need to ﬁnd the total distance the dolphin travels from below the surface up to above the surface. Let 0 be the level with the surface. The green dot represents −12 which means the dolphin is 12 feet below the surface. The blue dot represents 6 which means the dolphin is 6 feet above the surface. We can count from the number line from −12 up to 6 units to ﬁnd the vertical distance which is equal to 18 feet. 20

−10

−20

Answers mathspace.co

265

4.02 Multiply and divide integers Subtopic overview Lesson narrative In this lesson, students will learn to multiply and divide integers. They start with interactive explorations to discover patterns in determining the signs of products and quotients when multiplying and dividing integers with different signs. The lesson includes visual models such as arrays to illustrate these operations. Practice problems involve first determining the sign, then performing the operation, and finally checking the result using a model. By the end, students should be proficient in multiplying and dividing integers and understanding the associated sign rules.

Learning objectives

4.02 Multiply and divide integers

Students: Page 128

After this lesson, you will be able to... • represent multiplication and division of integers using models. • multiply and divide two integers. • find and justify the solution to real-world problems involving multiplication and division of integers.

Multiply integers

Key vocabulary Interactive exploration 

array



factor

Explore online to answer the questions

mathspace.co



dividend



divisor



product



quotient

Use the interactive exploration in 4.02 to answer these questions.

Essential 1. Whatunderstanding is the sign of the product of two positive integers? 2. number If you are at the product 3 ⋅ (−4), how manyThis tilesproperty are in 3 groups −4 tiles? Every real haslooking a multiplicative inverse or reciprocal. can beofused to rewrite division using multiplication. 3. What is the sign of the product of one positive and one negative integer? 4.

Is the product of 3 ⋅ (−4) the same as −4 ⋅ 3? Check this for other products.

What is the sign of the product of two negative integers?

Standards

This subtopic addresses the following Virginia Mathematics Standards Learning standards. comes down to Unlike adding and subtracting integers, where2023 we can use the number line or of counters, multiplication looking at the sign of factors.

Mathematical process goals

MPG1 — Mathematical Problem Solving

We have seen that the product of two positive integers is a positive integer.

Teachers can integrate this goal into their instruction by providing students with contextual problems that involve The product of a positive integer and a negative integer is a multiplication and division of integers. Teachers can encourage students to apply learned rules about operations negative integer. with integers and to use multiple strategies, including graphic organizers, step-by-step processes, and concrete or pictorial models, before transitioning to standard algorithms for adding and subtracting integers. Additionally, teachers can ask guiding questions to foster discussion on problem-solving strategies, the reasonableness of The product two approach negative integers a positivecan integer. solutions, and effective communication of mathematical ideas.ofThis ensuresisstudents estimate, determine, and justify their solutions effectively.

Example 1 266

Mathspace

Virginia SOL Grade 6 Teacher Edition

Find the value of: mathspace.co

−4 ⋅ 5

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can encourage students to share their strategies for solving integer multiplication and division problems, both verbally and in written form. This can be done through class discussions, small group work, or student presentations.

Teachers can integrate this goal into their instruction by connecting the concepts of multiplication and division of integers to students’ prior knowledge of multiplication and division with whole numbers. Additionally, they can help students make connections between the rules for integer operations and real-life situations where these operations are applied.

MPG3 — Mathematical Reasoning Teachers can help students develop their reasoning skills by discussing the rules for multiplying and dividing integers with the same and different signs. They can help students compare the rules for adding and subtracting integers to the rules for multiplying and dividing integers to see how they differ. Teachers can ask students to explain the “why” behind these rules and encourage them to use logical reasoning to analyze their solutions.

MPG5 — Mathematical Representations Teachers can integrate this goal into their instruction by providing students with opportunities to represent integer multiplication and division in four different ways; using a concrete representations first, a pictorial representation second, representation on a numberline third, and lastly using the algorithim. Teachers can also encourage students to make connections among different representations and to use the most appropriate representation for each problem they encounter.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.01 Identify and represent integers Grade 6 — 3.02 Add and subtract integers

4.02 Multiply and divide integers mathspace.co

267

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Hands-on manipulatives for integer operations Student with disabilities support Provide students with two-color counters or integer chips to physically represent positive and negative integers. Designate one color (e.g., yellow) for positive integers and another color (e.g., red) for negative integers. Start by guiding students through routines where they model multiplication and division problems using these counters. For example, to solve −3 × 2, have students create two groups of three red counters each, demonstrating the total negative value. For division, show how to separate counters into equal groups, such as dividing eight red counters into groups of four to find −8 ÷ 4 = −2. Encourage students to manipulate the counters and explain each step aloud, reinforcing their understanding of how the signs affect the operations. As students gain confidence, present more complex problems and have them use the counters independently to find solutions.

4.02 Multiply and divide integers Student lesson & teacher guide After this lesson, you will be able to...

Multiply integers • represent multiplication and division of integers using models. multiply and divide two integers. Students begin by •reviewing how to multiply whole numbers and exploring what happens when we include negative • find and justify the solution to real-world problems involving multiplication and division of integers. integers in multiplying numbers.

Students:Multiply Page 128integers

Interactive exploration Explore online to answer the questions

mathspace.co 4.02 Multiply and divide integers Use the interactive exploration in 4.02 to answer these questions. 1.

What this is thelesson, sign ofyou thewill product of two After be able to...positive integers?

2. 4.

you are looking at the product 3 ⋅ (−4), many tiles models. are in 3 groups of −4 tiles? •If represent multiplication and division of how integers using •What multiply and divide two integers. is the sign of the product of one positive and one negative integer? • find and justify the solution to real-world problems involving multiplication and division of integers. Is the product of 3 ⋅ (−4) the same as −4 ⋅ 3? Check this for other products.

What is the sign of the product of two negative integers?

Multiply integers Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to exploration looking at Interactive the sign of factors. Explore online to answer the questions

mathspace.co

We have seen that the product of two positive integers is a positive integer.

The product of a positive integer and a negative integer is a negative Use the interactive exploration in 4.02 to answer these integer. questions. 1.

What is the sign of the product of two positive integers?

2. 3.

If you are looking at the product 3 ⋅ (−4), how many tiles are in 3 groups of −4 tiles? The product of two negative integers is a positive integer. What is the sign of the product of one positive and one negative integer?

Is the product of 3 ⋅ (−4) the same as −4 ⋅ 3? Check this for other products.

5. What Example 1 is the sign of the product of two negative integers? Find the value of: Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to −4 ⋅ 5 looking at the sign of factors.

268

We have seen that the product of two positive integers is a positive Mathspace SOL Grade 6 Teacher Edition Create Virginia a strategy integer. mathspace.co Determine the sign of the product, then multiply 4 and 5 and apply the sign. The product of a positive integer and a negative integer is a negative integer.

4.02 Multiply and divide integers After this lesson, you will be able to... • represent multiplication and division of integers using models. • multiply and divide two integers. • find and justify the solution to real-world problems involving multiplication and division of integers.

Exploration Students: Page 128

Multiply integers Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.02 to answer these questions. 1.

What is the sign of the product of two positive integers?

If you are looking at the product 3 ⋅ (−4), how many tiles are in 3 groups of −4 tiles?

What is the sign of the product of one positive and one negative integer?

Is the product of 3 ⋅ (−4) the same as −4 ⋅ 3? Check this for other products.

What is the sign of the product of two negative integers?

Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to looking at the sign of factors.

Suggested student grouping: In pairs We have seen that the product of two positive integers is a positive Students will manipulate an applet to explore theinteger. multiplication of integers. Sliders are used to select the two integers to be multiplied, while the checkboxes are used to change the sign of the integers. Blue tiles represent The product of a positive integer and a negative integer is a positive integers, while orange tiles represent negative integers. The product of the integers is shown as negative integer. an array. Ideal student responses

The product of two negative integers is a positive integer.

These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. Example 1 1. What is the sign of the product of two positive integers? The Find signthe of value the product of two positive integers is positive. of: −4 ⋅tiles 5 2. If you are looking at the product 3 ⋅ (− −4), how many are in 3 groups of −4 tiles? There are a total of 12 orange tiles, which represents −12.

strategy 3. WhatCreate is theasign of the product of one positive and one negative integer? theproduct sign of the then multiply 4 andnegative 5 and apply the sign. The Determine sign of the ofproduct, one positive and one integer is negative.

4. Is the product of 3 ⋅ (− −4) the same as −4 ⋅ 3? Check this for other products. Yes, both products are the same. This also applies to other products. 5. What is the sign of the product of two negative integers? The sign of the product of two negative integers is positive. 128

Mathspace

Virginia SOL Grade 6

Purposeful questions mathspace.co

• Does changing the order of the factors affect the sign of the product? Possible misunderstandings • Students may mistake the relationship of the sign and the color of the tiles.

4.02 Multiply and divide integers mathspace.co

269

Error analysis: critique, correct, and clarify

Multiply integers

English language learner support

Interactive exploration

Utilize the “Critique, Correct, and Clarify” routine to strengthen your students’ understanding of multiplying and Explore online to answer the questions dividing integers with different signs. Provide them with an example of incorrect mathematical work to analyze. For instance, present the following incorrect solution: mathspace.co Problem: (−4) × (−3)

Use the interactive exploration in 4.02 to answer these questions. 1.

Student’s incorrect work: (−4) × (−3) = −12

What is the sign of the product of two positive integers?

you are looking at theconcluded product 3 ⋅ (−4), many tiles times are in 3agroups of −4equals tiles? a negative. Ask your Explain that 2. the Ifstudent incorrectly thathow a negative negative 3. What is the sign of the product of one positive and one negative integer? students to work individually or in small groups to identify the mistake, correct the calculation, and clarify the 4. Is the of result. 3 ⋅ (−4) the same as −4them ⋅ 3? Check thisprecise for othermathematical products. reasoning behind theproduct correct Encourage to use vocabulary such as “product,” 5. What isand the sign of the product of two negative “positive,” “negative,” “same signs.” This activity willintegers? help students deepen their understanding of the sign rules for multiplying and dividing integers while enhancing their ability to communicate mathematical reasoning clearly and accurately. Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to looking at the sign of factors. Wegeneral have seen that of themultiplying product of two positive integers a positive After the exploration, students are introduced to the rules integers with theis same signs and integer. different signs. The product of a positive integer and a negative integer is a negative integer.

Examples Students: Pages 128–129

The product of two negative integers is a positive integer.

Example 1 Find the value of: −4 ⋅ 5

Create a strategy Determine the sign of the product, then multiply 4 and 5 and apply the sign.

Apply the idea The sign of the product is negative because we are multiplying a negative and a positive integer. 4 ⋅ 5 = 20

Evaluate

We determined that the sign of the product should be negative, so the answer is −20. 128

Mathspace

Virginia SOL Grade 6

mathspace.co Reﬂect and check

Let’s visualize the multiplication of a negative and a positive integer using an array. In this case, we are multiplying −4 and 5. In the array, we have 4 rows and 5 columns, which represent the multiplication of −4 and 5. Each row has 5 orange tiles, representing negative integers. When we count the total number of orange tiles, we have 20 orange tiles, which represent the product of −4 and 5. Since orange tiles represent negative integers, our product is −20 which matches our previous answer.

−4 ⋅ 5 = −20

−4

Example 2 Purpose the value of: that they can multiply a positive integer by a negative integer to find a negative product. StudentsFind demonstrate −7 ⋅ (−5)

Create a strategy 270

We haveVirginia the product two negative integers, Mathspace SOL of Grade 6 Teacher Edition so the product will be positive. mathspace.co

Apply the idea −7 ⋅ (−5) = 35

Evaluate

Apply the idea Using patterns to understand multiplication with negative numbers

use with Example 1

The sign ofinstructional the product is negative because we are multiplying a negative and a positive integer. Targeted strategies 4 ⋅ 5 = 20

Evaluate

Encourage students to discover the rules for multiplying negative and positive integers by exploring patterns in We determined that the sign of the product should be negative, so the answer is −20. multiplication. Begin by having students calculate a series of products, such as 1 × 5, 0 × 5, −1 × 5, −2 × 5, and so on. Ask students to record their results and observe how the products change as the multiplier decreases. Reﬂect and check Guide them to notice that each time the multiplier decreases by 1, the product decreases by 5. This pattern Let’s visualize the multiplication of a negative and a positive integer using an array. 5 illustrates thatcase, multiplying by a negative number reverses the direction on the number line, resulting in a In this we are multiplying −4 and 5. −4 ⋅ 5 = −20 negative product when multiplied by a positive number. By identifying and discussing these patterns, students can develop deeper understanding of whywhich a negative times a positive equals a negative, helping them apply In the a array, we have 4 rows and 5 columns, represent the multiplication −4 this ruleofwith confidence future problems. −4 and 5. Each rowinhas 5 orange tiles, representing negative integers. When we count the total number of orange tiles, we have 20 orange tiles, which represent the product of −4 and 5. Since orange tiles represent negative integers, our product is −20 which matches our previous answer.

Students: Page 129

Example 2 Find the value of: −7 ⋅ (−5)

Create a strategy We have the product of two negative integers, so the product will be positive.

Apply the idea −7 ⋅ (−5) = 35

Evaluate

Reﬂect and check Let’s visualize the multiplication of the two negative integers by using an array representation with blue tiles for positive integers and orange tiles for negative integers. −5 −7 ⋅ (−5) = 35

−7

In the array, there are 35 blue tiles, which represent positive integers. This shows that when we multiply two negative integers, like −7 and −5, the product is a positive integer, in this case, 35.

Purpose Students demonstrate that they can multiply two negative integers to find a positive product. 4.02 Multiply and divide integers

129

mathspace.co

4.02 Multiply and divide integers mathspace.co

271

Believing that multiplying two negative numbers gives a negative result use with Example 2

Address student misconceptions

Students may think that when multiplying two negative integers, the product should be negative. They might incorrectly apply the rule that “a negative times a positive is negative” to this scenario, leading them to conclude that −7 × (−5) = −35 instead of 35. This misconception arises from a misunderstanding of the rules governing the multiplication of integers and confusion about how negative numbers interact. To address this misconception, encourage students to explore patterns in multiplication involving negative numbers. For example, create a sequence where one factor remains constant while the other decreases: 5 × 3 = 15 5 × 2 = 10 5×1=5 5×0=0 5 × (−1) = −5 5 × (−2) = −10 Observe how the product decreases by 5 each time the second factor decreases by 1. Then, repeat the pattern with −5: −5 × 3 = −15 −5 × 2 = −10 −5 × 1 = −5 −5 × 0 = 0 −5 × (−1) = 5 −5 × (−2) = 10 Point out that as the second factor decreases, the product increases, turning from negative to positive when multiplying two negatives. Discuss with students how this pattern demonstrates that multiplying two negative numbers results in a positive product. Visual aids like number lines can also help students visualize how the products change. By guiding students to discover the pattern themselves, you can help them build a stronger understanding of the rules for multiplying integers.

Students: Page 130 Example 3 A submarine dives 22 m each minute for 16 minutes. What integer represents the total depth of the dive after 16 minutes?

Create a strategy Diving 22 meters is represented by the integer −22. We can think of this as repeated addition because an extra 22 m is added for each minute the diver is under. That means we can also think of this as multiplication.

Apply the idea Depth of the dive = −22 ⋅ 16 = −352 m

Set up the equation Evaluate

Reﬂect and check We can check the reasonableness of our answer by thinking about the signs. We knew diving 22 m was a negative integer because the diver is going under water. And time was positive because time is always moving forward. Multiplying a positive and negative integer results in a negative integer, which is what we got. This makes sense because the diver ends up deeper underwater.

Idea summary

272

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

We have seen that the product of two positive integers is a positive integer. The product of a positive integer and a negative integer is a

Example 3 PurposeA submarine dives 22 m each minute for 16 minutes. What integer represents the total depth of the dive after Show students that understanding the relationship between positive and negative integers is essential in real16 minutes? life scenarios such as calculating the total depth of a dive. Create a strategy

Reflecting with students Diving 22 meters is represented by the integer −22. We can think of this as repeated addition because an extra 22 m Encourage advanced anyis students are tothink create their own submarine diving problems is added for eachlearners, minute theor diver under. Thatwho means weready, can also of this as multiplication. by changing variables like the dive rate, duration, or introducing ascent phases. Invite them to choose different Apply the idea depths per minute, vary the number of minutes diving or ascending, and even incorporate pauses at certain Example 3 Depth of the own dive =scenarios, −22 ⋅ 16 Set upwill thenaturally equation delve deeper into integer operations and the depths. By designing their students = −352 m Evaluate real-world application positive and negative numbers. Have represents them calculate total depth A submarine divesof22 m each minute for 16 minutes. What integer the totalthe depth of the dive after after each phase and explain how each action (dive or ascent) affects the overall depth using integers. Encourage students to 16 minutes? Reﬂect and check share their problems and solutions with the class, fostering a collaborative learning environment where they can We can check the reasonableness of our answer by thinking about the signs. We knew diving 22 m was a negative a strategy discuss Create and compare different mathematical approaches.

integer because the diver is going under water. And time was positive because time is always moving forward. Diving 22 meters is represented by the integer −22. We can think of this as repeated addition because an extra 22 m Multiplying aeach positive andthe negative integer in a negative which is what we got. This makes sense added for minute diver is under.results That means we can integer, also think of this as multiplication. Students:isbecause Page 130 the diver ends up deeper underwater.

Apply the idea Depth of the dive = −22 ⋅ 16

Idea summary = −352 m

Set up the equation Evaluate

We have seen that the product of two positive integers is a positive integer. We can check the reasonableness of our answer by thinking about the signs. We knew diving 22 m was a negative integer because the diver is going under water. And was positive because timeand is always moving forward. Thetime product of a positive integer a negative integer is a negative integer. Multiplying a positive and negative integer results in a negative integer, which is what we got. This makes sense

Reﬂect and check

because the diver ends up deeper underwater. The product of two negative integers is a positive integer.

Idea summary We have seen that the product of two positive integers is a positive integer.

Divide integers

The same principles that help us to multiply integers also apply to divide. The product of a positive integer and a negative integer is a negative integer.

Divide integers

Interactive exploration

Students begin by Explore reviewing the multiplying online to rules answerof the questions integers, then exploring how it relates to the division of integers. The product of two negative integers is a positive integer.

Students: Pagesmathspace.co 130–131

Use the interactive exploration in 4.02 to answer these questions. 1. What is the sign of the quotient of two positive integers? Divide integers 2. What is thethat signhelp of the of integers two negative integers? The same principles us quotient to multiply also apply to divide.

Interactive exploration Explore online to answer the questions

mathspace.co

We have seen that the quotient of two positive integers is a positive number. The quotient of a positive integer and a negative integer is a negative number.

Mathspace Virginia SOL Grade 6 Usemathspace.co the interactive exploration in 4.02 to answer these questions.

130

The quotient of two negative integers is a positive number. What is the sign of the quotient of two positive integers?

2. What is the sign apply of theto quotient of two negative The same sign properties both multiplication and integers? division. However, unlike multiplication, division of two integers does not always result in another integer.

Example 4 Find the value of: 130

Mathspace Virginia SOL Grade 6 mathspace.co

48 ÷ (−6)

Create a strategy

We have the quotient of one positive and one negative number, so the quotient will be negative. 4.02 Multiply and divide integers mathspace.co Apply the idea Reﬂect and check 48 ÷ (−6) = −8

Evaluate

We can check if we have determined the correct sign, by using multiplication. Would −8 times −6 equal 48?

273

negative integer.

The product of two negative integers is a positive integer.

Exploration Students:Divide Page 130 integers The same principles that help us to multiply integers also apply to divide.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.02 to answer these questions. 1.

What is the sign of the quotient of two positive integers?

What is the sign of the quotient of two negative integers?

Suggested student grouping: In pairs Students will manipulate an applet to explore the division of integers. The horizontal slider selects the divisor, 130 Mathspace Virginia SOL Grade 6 the vertical slider selects the quotient, and the checkboxes change the sign of the integers. Blue tiles represent mathspace.co positive integers, while orange tiles represent negative integers. The dividend is shown as an array. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What is the sign of the quotient of two positive integers? The sign of the quotient of two positive integers is positive. 2. What is the sign of the quotient of two negative integers? The sign of the quotient of two negative integers is also positive. Purposeful questions • How are the rules used in multiplying integers related to the ones used in dividing integers? Possible misunderstandings • Since the applet is very similar to the one used theseen previous may need clarification We in have that theexploration, quotient of twostudents positive integers is a positive number. about what the blue and orange tiles represent. The quotient of a positive integer and a negative integer is a negative number.with the same signs and different signs. Students are introduced to the general rules of dividing integers

Examples

The quotient of two negative integers is a positive number.

Students:The Page same 131 sign properties apply to both multiplication and division. However, unlike multiplication, division of two integers does not always result in another integer.

Example 4 Find the value of: 48 ÷ (−6)

Create a strategy We have the quotient of one positive and one negative number, so the quotient will be negative.

Apply the idea 48 ÷ (−6) = −8

Reﬂect and check Evaluate

Example 5 274

Mathspace Virginia SOL Grade 6 Teacher Edition Evaluate: mathspace.co

We can check if we have determined the correct sign, by using multiplication. Would −8 times −6 equal 48?

We have seen that the quotient of two positive integers is a positive number. The quotient of a positive integer and a negative integer is a negative number. Purpose Students demonstrate that they can divide a positive integer by a negative integer horizontally, finding a negative quotient. The quotient of two negative integers is a positive number.

Expected mistakes same sign properties apply to both and division. However, unlike of two StudentsThe may overlook the negative signmultiplication when dividing a positive number bymultiplication, a negativedivision number, leading them integers does not always result in another integer. to mistakenly conclude that the quotient is positive. For example, they might compute 48 ÷ (−6) = 8 instead of the correct result, −8. This misunderstanding arises from not applying the rule that a positive divided by a Example 4 negative yields a negative result. To address misconception, emphasize the rules for dividing integers with different signs. Remind students Find this the value of: 48 ÷ (−6)is negative. that when dividing numbers with unlike signs, the quotient Additionally, use a number line to visually demonstrate how dividing by a negative number changes the Create a strategy direction. For instance, show that starting at zero and moving 48 units in the positive direction, then dividing into We have the quotient of one positive and one negative number, so the quotient will be negative. groups of −6, results in −8 groups. By combining visual aids with clear explanations of the sign rules, you can help students internalize the correct procedures for dividing withand negative Apply the idea Reﬂect check numbers. 48 ÷ (−6) = −8

Students: Page 131

Evaluate

We can check if we have determined the correct sign, by using multiplication. Would −8 times −6 equal 48?

Example 5 Evaluate:

Create a strategy We have the quotient of two negative integers, so the quotient will be positive.

Apply the idea

Reﬂect and check Evaluate

We can check if we have determined the correct sign, by using multiplication. Would 6 times −10 equal −60?

Idea summary Purpose We have seen that the quotient of two positive integers is a positive Students demonstrate that they can divide a negative integer by another negative integer vertically, finding a number. positive quotient. The quotient of a positive integer and a negative integer is a negative number. Exploring patterns in division with negative numbers

use with Example 5

Targeted instructional strategies Encourage students to discover the rules for dividing negative numbers by examining The quotient of two negative integers is a positivepatterns number. in division problems. Begin by creating a table with examples such as

= 6,

= –6,

= –6, and

= 6. Have

students calculate each quotient and note the signs of the dividend, divisor, and result. Facilitate a class discussion where students share their observations about how the signs affect outcome ofintegers each division 4.02the Multiply and divide 131 mathspace.co problem. Guide them to recognize that when both the dividend and divisor have the same sign, the quotient is positive, and when they have different signs, the quotient is negative. By identifying this pattern themselves, students will understand why

= 6 without merely memorizing rules. This approach promotes critical

thinking and helps students internalize the concept through exploration and pattern recognition.

4.02 Multiply and divide integers mathspace.co

275

Create a strategy We have the quotient of two negative integers, so the quotient will be positive.

Apply the idea

Reﬂect and check

Students: Page 131

We can check if we have determined the correct sign, by using multiplication. Would 6 times −10 equal −60?

Evaluate

Idea summary We have seen that the quotient of two positive integers is a positive number. The quotient of a positive integer and a negative integer is a negative number.

The quotient of two negative integers is a positive number.

4.02 Multiply and divide integers mathspace.co

131

Practice Students: Pages 132–134

What do you remember? 1

Determine whether the following is true or false: a

The number which divides the other number is known as the divisor.

The number to be divided into is known as the quotient.

The product of the divisor and quotient is the dividend.

If the quotient has no remainder, then the quotient and divisor are factors of the dividend.

Detemine whether multiplication or division should be used to ﬁnd each of the following. Do not solve. a

The total length of 5 ribbons, each measuring 7 inches long

Amount to pay if you buy 14 packs of notebook paper and each pack costs $2 each.

The length of a strip of paper 36 inches long that will be cut into strips of 6 inches in length

The number of number of glasses needed if a pitcher which holds a half gallon, or 64 fl oz, if each glass holds 8 fl oz.

Evaluate: a

4⋅5

8 ⋅ 94

12 ⋅ 67

52 ⋅ 173

546 ÷ 6

828 ÷ 3

208 ÷ 16

285 ÷ 19

State whether the following expressions will be positive or negative: a

The product of two negative numbers

The quotient of a positive number and negative number

(−147) ⋅ 183

(−11) ⋅ (−85)

157 ⋅ (−14)

35 ⋅ 8

85 ÷ (−5)

−29 ⋅ 61

(−2432) ÷ 32

1587 ÷ (−23)

Write an equation representing the multiplication on the number line below: 0

276

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

9 10 11 12 13 14 15 16

Consider the equation 6 ⋅ 4 = 24, to ﬁnd the value of the following: a

(−6) ⋅ 4

6 ⋅ (−4)

(−6) ⋅ (−4)

Consider the equation 42 ÷ 7 = 6, to ﬁnd the value of the following: a

(−42) ÷ 7

42 ÷ (−7)

(−42) ÷ (−7)

Let’s practice 8

The counters below represent the integer −20. a

Use the counters to demonstrate the operation −20 ÷ 10, and include the solution in your answer.

Use your demonstration from part a to explain why a negative divided by a positive is negative.

Identify the expressions that have equivalent solutions: A

4 ⋅ −8

−2 ⋅ −16

−8 ⋅ 4

−1 ⋅ −32

−3 ⋅ 13

5 ⋅ (−9)

−9 ⋅ 0

−3 ⋅ 1

(−4) ⋅ (−4)

6 ⋅ (−6)

12 ⋅ 3

6 ⋅ 12

11 ⋅ (−1)

−4 ⋅ (−11)

8 ⋅ (−12)

25 ⋅ (−10)

−12 ⋅ (−12)

m −23 ⋅ 13

7 ⋅ (−320)

−22 ⋅ (−131)

Evaluate: a

8÷4

0 ÷ (−36)

d (−56) ÷ (−4)

(−50) ÷ (−25)

Complete the following equations: e

−32 ⋅ 1

Evaluate:

i 12

⬚ ⋅ (−5) = 55

⬚ ÷ (−6) = −9

b f

−6 ⋅ ⬚ = −66 ⬚ ÷ 5 = −7

c g

⬚ ⋅ 8 = −56 ⬚ ÷ 5 = −11

d h

Renee made 6 withdrawals of $95 each from her bank account.

7 ⋅ ⬚ = −84

(−72) ÷ ⬚ = 9

Renee wants to estimate the total withdrawn her bank account without checking her balance. How could she estimate this total, and what value would it be?

Write an integer expression that represents how to determine the actual total withdrawn from Renee’s bank account.

Determine the total withdrawn in Renee’s bank account as both an integer, then describing this change in context.

Zach’s score in a video game was changed by −105 points because he missed some goals. He got −15 points for each missed goal. a

Write an expression that Zach could use to determine the number of missed goals.

Determine the number of missed goals.

Find the integer in the following statements: a

An integer when multiplied by −12, gives −108.

An integer when divided by −7, gives 13. 4.02 Multiply and divide integers mathspace.co

277

Let’s extend our thinking 16

Evaluate: a

4 ⋅ (−7) ⋅ 5

15 ÷ (40 ÷ (−8))

(8 ⋅ (−15)) ÷ 4

−8 ⋅ (−3)

−5 ⋅ (−3) ⋅ (−7)

(−36) ÷ ((−6) ÷ (−3))

(18 ⋅ (−3)) ÷ (−2)

(−126) ÷ ((−6) ⋅ (−3))

Evaluate: a

12 ⋅ 3 ⋅ 4

−8 ⋅ (−5) ⋅ (−3) ⋅ 6

Cooper is trying out a new city design program that will build towers of blocks in yellow and red. The program uses multiplication to duplicate patterns, but the sign of the original values indicates color. He tries multiplying his original pattern by both 2 and −2 and the patterns are shown below:

Pattern 1

Pattern 2

Which pattern uses 2 as a multiplier? Which pattern uses −2 as a multiplier? Explain your thinking.

Cooper loads a city that a friend has designed. This city shows at town square with the design shown. Write a series of expressions using → between each new operation that would create this sequence of blocks. Choose where you would like to begin.

Josie answers questions in two online trivia games about animals. In each quiz, she loses points when she gives a wrong answer. The table shows the number of points lost in each quiz for incorrect answers and the total number of points Josie lost per game. a

In which game did she have more incorrect answers?

Josie is competing against Lily to see who loses less points at the end of both trivia games. Lily ﬁnished both games and lost a total of 80 points. Josie has ﬁnished the farm animals game and is still playing the zoo animal game. How many more questions can Josie miss and still win?

Farm animals Zoo animals

Points lost per wrong answer −4 −6

Explain why there is no integer n so that n ⋅ n = −1.

The product of three distinct integers is −21. Determine all of the possible integers.

278

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Total points lost −28 −36

Answers

11 a 2

4.02 Multiply and divide integers

b False

2 a Multiplication

e 91 4 a Positive

d Division

b −9

c −11

d −10

−6

g 14

h 2

k 9

−503

12 a −11 ⋅ (−5) = 55

b −6 ⋅ 11 = −66

c −7 ⋅ 8 = −56

d 7 ⋅ (−12) = −84

e 54 ÷ (−6) = −9

g −55 ÷ 5 = −11

h (−72) ÷ (−8) = 9

−35 ÷ 5 = −7

13 a T he value would be close to $600 since $100 is a close amount to $95 and 100 ⋅ 6 = 600.

b 752

c 804

d 8996

276

g 13

h 15

b −95 ⋅ 6

b Negative

c Negative

d Positive

g Negative

h Negative

c − 95 ⋅ 6 = −570, which means that after 6 withdrawals of $95, Renee’s bank account has decreased by $570.

e Negative

Positive

Negative

d True

b Multiplication

c Division 3 a 20

c True

−3

What do you remember? 1 a True

e 0

14 a

b 7 goals

15 a 9

b −91

5 5 ⋅ 3 or 3 ⋅ 5 6 a −24

b −24

c 24

7 a −6

b −6

c 6

Let’s extend our thinking

Let’s practice 8 a −20 ÷ 10 = −2

16 a 144

b −140

17 a −18

b −3

18 a −720

b −30

c 24

d −105

c 27

d −7

19 a P attern 1 was multiplied by −2 since multiplying by a negative value changes the sign, and the color of the blocks changed. Since the color of the blocks stayed the same in Pattern 2, the multiplier would be 2. b Sample answer: 2 ⋅ 3 → 6 ⋅ −1 → −6 ÷ 6 → −1 ⋅ 2 b W e are dividing 20 negative counters into 10 groups. The problem wants to know the value in an individual group. There are 2 negative counters inside of each group. If we divide any amount of negative values into groups, the values inside of each group will still be negative. 9 A, C, E 10 a −45 e −36 44

m −299

b 0

c −3

d 16

g 72

h −11

−96

k −250

n −2240

o 2882

20 a Farm animals

b 2 questions

21 When an integer is multiplied to itself, the result is always positive. As −1 is negative there is no integer which we can multiply to itself and get −1. 22 We know that 21 has factors 1, 3 and 7. So the three distinct integers with the product of −21 are: −1, 1 and 21; − 1, 3 and 7; 1, −3 and 7; 1, 3 and −7.

144

Answers mathspace.co

279

4.03 Absolute value of integers Subtopic overview Lesson narrative In this lesson, students will learn about the absolute value of integers. They will understand that absolute value represents the distance from zero on a number line, irrespective of direction. The lesson includes interactive explorations using number lines and practical examples such as temperature and elevation. Students will practice finding absolute values, simplifying expressions involving absolute values, and solving real-world problems. Activities involve evaluating absolute values, comparing them, and applying order of operations. By the end, students should confidently interpret and use absolute values in various contexts.

Learning objectives Students: Page 135

Key vocabulary 

absolute value



distance

Essential understanding The absolute value of a number is the distance of that number from 0 on the number line. Every number and its opposite have the same absolute value.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG2 — Mathematical Communication

Teachers can integrate problem-solving into the lesson by providing contextual problems that require the use of absolute value, such as comparing distances or calculating differences in temperature. Students will need to apply their understanding of absolute value and its properties to determine the correct solutions to these problems.

Teachers can encourage mathematical communication by having students explain their thought process when solving absolute value problems, both verbally and in writing. Students can also work in pairs or small groups to discuss and justify their solutions, using the correct mathematical language and notation.

280

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can promote mathematical reasoning by asking students to analyze and justify the properties of absolute value (e.g., why is ∣x∣ always greater than or equal to 0?). This will require students to use logical reasoning and make connections between their understanding of absolute value and the number line.

Teachers can integrate this goal into the lesson by having students represent absolute value problems using a variety of methods, such as number lines, visual models, and symbolic notation. Students can also practice translating between different representations (e.g., from a number line to an equation) to strengthen their understanding of the concept and improve their ability to communicate mathematically.

Content standards 6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context. 6.CE.2c — Simplify an expression that contains absolute value bars ∣∣ and an operation with two integers

6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers. 6.NS.2d — Identify and describe the absolute value of an integer as the distance from zero on the number line.

and represent the

result on a number line.

Future connections 7.PFA.2 — The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables.

Tools You may find these tools helpful: • Number line

4.03 Absolute value of integers mathspace.co

281

Student lesson & teacher guide Absolute value of integers Students: Page 135

282

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Exploration Students: Page 135

Suggested student grouping: Individual Students are to use an applet which shows the absolute value, or distance from zero for different integers on the number line. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the absolute value of a positive number? The absolute value of a positive number is the distance between the number and zero on the number line. 2. What do you notice about the absolute value of a negative number? The absolute value of a negative number is the distance between the number and zero on the number line. 3. Can the absolute value of a number ever be a negative number? Why or why not? No. Absolute value is the distance a number is from zero. If you count the number of units from zero to the number, the number of units is its absolute value. You could be on the right or left side of zero, but the number of units you count represents the distance or absolute value, and that will always be a positive number. Purposeful questions • Given a number, let’s say 4, what number will give the same absolute value? • If numbers have the same absolute value, what can you say about the numbers? • What is the absolute value of zero? Possible misunderstandings • Believing that absolute value changes negative numbers into positive numbers by altering their sign. Explain that absolute value represents distance from zero without regard to direction, so it doesn’t change the original number but describes how far it is from zero. • Thinking that the absolute value of zero is undefined or cannot be determined. Clarify that zero is zero units away from zero, so its absolute value is zero. • Assuming that absolute value can be negative if dealing with negative numbers or large magnitudes. Emphasize that distance cannot be negative, so absolute values are always non-negative regardless of the original number.

4.03 Absolute value of integers mathspace.co

283

Warm-up exercise Targeted instructional strategies Math warm ups are the opening routines to get kids participating. These can be used to practice skills you are trying to teach as well as review previously taught skills and concepts that will help in the current lesson. As an opening exercise, make use of the number line again as students work individually to list two rational numbers that are the same distance from zero. Students find as many pairs as possible and reach a conclusion about what must be true for every pair of numbers that lie that same distance from zero. The following questions can be asked to scaffold learning and let students understand absolute value. • What are some examples you found (pairs of numbers that are the same distance from zero)? • What is the relationship between each pair of numbers? • How does each pair of numbers relate to zero? Define absolute value, then ask for the absolute value of a positive number and its opposite.

Stronger and clearer each time English language learner support To support students to recognize the similarities and differences between “absolute value” and “opposite,” ask students to respond to the prompt “How is a number’s absolute value the same and different than its opposite?” Including diagrams with explanation are encouraged as they meet with 2–3 other partners for feedback. Anticipate student with prompts such as “Can you show on your diagram . . .” or “Why are the values the same for this number?”. To strengthen the group’s final response, encourage students to borrow ideas and language from each partner.

Absolute value and opposite numbers Address student misconceptions Students may confuse absolute value with opposites, thinking that absolute value changes the sign of the number. Remind students that absolute value represents the distance of the number from 0 without worrying about the sign or direction. Use some concrete examples: in a board game, 2 jumps are needed to make 2 moves to the forward and 2 moves backward; a car’s odometer would put read 4 but ends in two different places when a car travels 4 miles north or 4 miles south; $8 is involved when saving $8 and spending $8 but in one case you gain it and in the other you lose it. Absolute value is the amount or value involved, not including the sign. Following the exploration, the symbol for absolute value is discussed. Let the students realize that the absolute value of either a positive or negative number is positive.

Examples Students: Page 136

284

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Students demonstrate that they can identify the absolute value of a negative number. Reflecting with students Ask the students to explain whether ∣−155∣ = − ∣155∣.

Students: Page 136

Purpose Challenge students to compare the absolute of a given number to other integers and to absolute value of other integers. Expected mistakes Students may compare the number inside the absolute value sign thinking for example that ∣−20∣ is bigger than ∣−30∣. Remind students to evaluate the absolute value first to ensure they are working with the value of the full expression.

Students: Page 136

4.03 Absolute value of integers mathspace.co

285

Purpose Challenge students to order absolute values of integers including 0 and negative numbers.

Think-aloud modeling to deepen understanding

use with Example 3

Student with disabilities support Model your thought process step-by-step as you evaluate each absolute value and order the results. Begin by explaining that the absolute value of a number represents its distance from zero on the number line, which is always a non-negative value. For example, say aloud, “The absolute value of 19 is 19 because it is 19 units from zero. The absolute value of −31 is 31 because it is 31 units from zero.” By verbalizing each step, you help students understand the reasoning behind the calculations. Encourage students to practice this strategy by working in pairs, taking turns to “think aloud” as they solve each part of similar problems. This approach supports students who may struggle with conceptual understanding or maintaining attention by providing a clear, auditory model of the problem-solving process.

Students: Page 137

Example 4 Evaluate ∣7 − 11∣ and represent the solution on a number line.

Create a strategy We need to follow Order of Operations, so we will need to do the subtraction inside the absolute value bars ﬁrst.

Apply the idea ∣7 − 11∣ = ∣−4∣

Subtract the inside of absolute value bars

The distance between −4 and 0 is 4 on our number line −8 −7 −6 −5 −4 −3 −2 −1 0

Reﬂect and check We can also think about this as the distance between 7 and 11 on a number line which is 4. 0

Example 5 Evaluate

286

and represent the absolute value step on a number line.

Mathspace Virginia SOL Grade 6 Teacher Edition Create a strategy mathspace.co We need to follow Order of Operations, so we will take the absolute value ﬁrst.

−8 −7 −6 −5 −4 −3 −2 −1 0

Reﬂect and check

PurposeWe can also think about this as the distance between 7 and 11 on a number line which is 4. Show students that absolute value can be interpreted as the distance between two numbers on a number line 0

Students: Page 137

Example 5 Evaluate

and represent the absolute value step on a number line.

Create a strategy We need to follow Order of Operations, so we will take the absolute value ﬁrst.

Apply the idea −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

We can see that 12 is 12 units away from 0 on our number line. So, ∣12∣ = 12. Find the absolute of 12 using number line above Divide −12 by 2

Reﬂect and check Would the answer be different if we changed the problem to

Purpose Show students how to apply the order of operations to problems involving negative numbers and absolute values, and how to represent these steps on a number line.

Students: Page 138

4.03 Absolute value of integers mathspace.co

137

4.03 Absolute value of integers mathspace.co

287

Purpose Demonstrate to students how understanding of absolute values can be applied to real-life situations. Reflecting with students Encourage advanced learners, or any students who are ready, to not only select the correct statements but also to justify their reasoning for each one. Ask them to explain why Laura has a greater elevation than Fred by discussing the meanings of positive and negative elevations in real-world contexts. Prompt students to prove that Laura is further from sea level than Fred by calculating the absolute values of their elevations and comparing them. Challenge them to articulate why Fred, being below sea level, does not have a greater elevation despite the numerical value of his depth. Encourage them to use illustrations to support their reasoning. By requiring students to provide detailed justifications, you help them develop stronger logical reasoning skills and a deeper understanding of absolute value and negative numbers. This approach also supports their ability to construct mathematical arguments and communicate their thinking clearly.

Students: Page 138

Practice Students: Pages 138–140

What do you remember? 1

Represent the following integers on the number line shown: −7, 8, 0, 3, −3 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Consider this number line: −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

288

How far is the number 3 from 0?

Now, state the value of ∣3∣.

How far is the number −7 from 0?

Now, state the value of ∣−7∣.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Which value does not belong? A

∣15∣

−15

For each of the following pairs of numbers: i

Plot the pair of numbers on a number line.

Complete each number sentence with < or >, that should appear between each pair to make the statement true.

8 ⬚ 17

20 ⬚ 19

1⬚−6

−9 ⬚ − 5

Identify whether <, >, or = should appear between each pair of numbers to make each statement true: −56 ⬚ − 28

−4 ⬚ − 6

−35 ⬚ 35

92 ⬚ − 104

1001 ⬚ 650

−7 ⬚ − 5

4 + (−13)

−37 (−1)

5 − 11

−8 ⋅ 25

−24 + (−30)

17 + (−8)

−82 (9)

0 − (−74)

−18 + 35

108 ÷ (−4)

−28 − 28

0 (745)

m −92 + 0

−333 ÷ (−3)

−40 ⋅ (−40)

a e 6

∣−15∣

−2 ⬚ 0

Evaluate each expression.

−50 ⬚ − 50

Let’s practice 7

Which two points on the number line shown have the same absolute value? Justify your reasoning using the deﬁnition of absolute value. A

−10

∣20∣

∣−37∣

∣65∣

∣−155∣

∣x∣ = 9

∣x∣ = 4

∣x∣ = 2

∣x∣ = 7

∣x∣ = 11

∣x∣ − 8 = 0

∣x∣ − 10 = 0

∣x∣ − 3 = 1

Identify the sign, <, > or =, that should appear between each pair of numbers to make each statement true: ∣9∣ ⬚ ∣−2∣

−∣29∣ ⬚ − ∣74∣

∣−9∣ ⬚ ∣0∣

∣−7∣ ⬚ ∣7∣

35 ⬚ ∣−50∣

∣−45∣ ⬚ ∣22∣

∣−7∣ ⬚ ∣−10∣

∣−43 − 8∣

∣−7∣ − 7

∣24∣ − ∣−38∣

−13 + ∣−25∣

−3 ⋅ ∣62∣

3 ⋅ ∣−6∣

Evaluate each value. a e

Find the possible values of x for each value:

−5

Evaluate and state what additional integer has an equivalent absolute value. a

∣−1∣ ⬚ − ∣30∣

Evaluate each of these numbers and order the results, from smallest to largest. a

∣30 − 2∣, ∣15∣, ∣23∣

∣−33∣, ∣−11∣, ∣−22∣

∣21∣, ∣−7∣, ∣−49∣, ∣10 − 50∣

∣8∣, ∣0∣, ∣7∣, ∣−9∣

Identify each value smaller than 14 and plot on the number line provided. A

−14

∣−14∣ − (−14)

∣−17∣ − 2

∣−23 + 17∣

−15

−10

−5

20 4.03 Absolute value of integers mathspace.co

289

Today, Santiago has a peak temperature of 37 °C, while Minneapolis has a peak temperature of −11 °C. Using the concept of number lines, explain which city had the least peak temperature.

Amelia has a bank account balance of −$18. Marge has a bank account balance of −$55. Using the concepts of absolute values, explain who has a bigger debt.

Aurelia has a bank account balance of $13. Then she spent $54. Using the concepts of absolute values, calculate how much debt she has. Represent your answer on a number line.

Let’s extend our thinking 17

Michael was solving the problem ∣−8 − 12∣. His work and reasoning is shown below: ∣−8 − 12∣ = ∣−8∣ − ∣12∣ = 8 − 12

Find the absolute value of each term Subtract

= −4 He used a number line to evaluate −8 − 12 and realized he made a mistake. Explain what the number line would show, and correct his error in thinking. 18

If ∣a + 8∣ = 20, what values could a be equal to? Explain your answer.

Evaluate. ∣32 − 95 + 6∣ + ∣−15∣ − 4

Serena is setting up automatic deposits and withdrawals for her bank account that will occur on a regular basis. She wants to set up deposits of $150 each month and withdrawals of $50 each week. She will only do this if the absolute value of the deposits is greater than the absolute value of the withdrawals. Reminder: there are 12 months and 52 weeks in a year.

Calculate how much money Serena will deposit and withdraw in a year, writing each as an integer.

Compare the absolute values found in the previous part. Does this plan make sense for Serena’s budget?

A palindrome is a word or sentence that reads the same forwards and backwards. Using the number line shown, follow the instructions to graph and label the points to discover the mystery palindrome. What is the mystery term? • E: ∣−4∣ − 6 • R: −2 ⋅ ∣−4∣ and ∣−2 ⋅ −4∣ • C: Any integer with an absolute value of 4 • A:

and an integer the same distance away from 0 −10

290

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

14 Minneapolis had the least peak temperature. On the number line below, we can see both temperatures and −11 is lower than 37.

4.03 Absolute value of integers

−15 −10

What do you remember? 1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

2 a 3

b 3

c 7

d 7

−5

15 Each account balance are represented with negative signs, so taking the absolute value of each number will show who has the bigger debt.

3 C

The absolute value of −18 is 18, while the absolute value of −55 is 55. This shows that 55 is bigger than 18.

4 a i

So, Marge has the bigger debt of $55. 0

ii 8 < 17 b i

0 11

ii 20 > 19 c i

−10

−5

ii 1 > −6 d i

−10

−5

5 a < e >

b <

c >

d <

g >

h =

b 37

6 a −9

c −6

d −200

e −54

g −738

h 74

−27

k −56

o 0

p 1600

m −92

n 111

c 65, −65

b 37, 37

b x = −4 and 4

c x = −2 and 2

d x = −7 and 7

e x = −11 and 11

g x = −10 and 10

h x = −4 and 4

e −42

Michael should have subtracted −8 − 12 before ﬁnding the absolute value. His work would look like Evaluate subtraction

= 20

Find absolute value

20 a S erena will deposit $1800, which is the integer 1800. She with withdraw $2600, which is the integer −2600. b ∣ 1800∣ < ∣−2600∣. Since the absolute value of the withdrawals is greater than the absolute value of deposits, this plan does not make sense for Serena’s budget. 21 Race car R

C 0

A 5

R 10

x = −8 and 8

c >

d <

g >

h >

b 0

c −14

d 12

17 Michael would begin at −8 on a number line and move to the left 12 units, where he would end at −20, which has an absolute value of 20.

−10

b = f

11 a 51

19 68

d 155, 155

9 a x = −9 and 9

e <

18 a = 12 or −28 because we know that both ∣−20∣ = 20 and ∣20∣ = 20. We need to choose values of a that evaluate to both 20 and −20.

7 A and E have the same absolute value. Absolute value is the distance away from 0 on a number line, and A and E are both 8 units away from 0.

10 a >

Let’s extend our thinking

Let’s practice

8 a 20, −20

∣− 8 − 12∣ = ∣−20∣

ii −9 < −5

16 Aurelia owes $41.

−186

12 a 15, 23, 28

b 11, 22, 33

c 7, 21, 40, 49

d 0, 7, 8, 9

13 A, C, F −15

−10

−5

Answers mathspace.co

291

4.04 Real-world problems with integers Subtopic overview Lesson narrative In this lesson, students will learn to solve real-world problems involving integers. They will explore various scenarios such as temperature changes, financial transactions, and time delays. Students will practice identifying key terms related to integer operations, estimating solutions, and applying integer addition, subtraction, multiplication, and division. Problems include determining the sign, performing the operation, and verifying the results using models like number lines. By the end, students should confidently apply integer operations to real-world contexts.

4.04 Real-world problems with Learning objectives integers Students: Page 141

After this lesson, you will be able to... • estimate solutions to real-world problems involving operations with integers. • solve one and two-step real-world problems involving operations with integers. • justify solutions to problems involving operations with integers.

Real-world problems with integers We can use our knowledge of addition and subtraction on the number line to describe how the real world quantities Key vocabulary 

change. estimate

1 unit Positive direction

Essential understanding

−10

−5

Determining the correct essential writing expressionsusing to solve contextual problems real numbers. We can also talk aboutoperation changes inisthe quantitytowe are representing integer operations. Given with a starting temperature and some change in a certain direction, what is the ﬁnal temperature? Given a starting balance and an ending balance of money in an account, what has been the amount and sign of the change?

Standards When using integers to represent real-world situations, it is important to correctly identify any positive and negative integers as addresses well as the correct operations. This subtopic the following Virginia 2023 Mathematics Standards of Learning standards. Table of common key words to look for:

Mathematical process goals

Term Key Words: Positive proﬁt, above zero temperature MPG1 — Mathematical excess, Problem Solving Negative owes, below level Teachers can integrate debt, this goal into theirsea instruction by presenting students with real-life integer situations Addition plus, more than, total, sum, combined and guiding them through the process of identifying the appropriate operations and strategies for solving the Subtraction difference, less than, decreased problems. Teachers canminus, encourage students to apply learnedby rules about operations with integers and to use Multiplication times, graphic product organizers, of, double/triple multiple strategies, including step-by-step processes, and concrete or pictorial models, before Division to standardquotient, per, for split, ratio, into transitioning algorithms adding and subtracting integers. Additionally, teachers can ask guiding

questions to foster discussion on problem-solving strategies, the reasonableness of solutions, and effective Estimation can be a good strategy for solving real-world problems if the context doesn’t require an exact solution. communication of mathematical ideas. This approach ensures students can estimate, determine, and justify their solutions effectively.

Example 1

292

Mathspace

Virginia SOL Grade 6 Teacher Edition

mathspace.co Tara is waiting for the next ﬂight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34-minute delay. She takes a nap and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?

MPG2 — Mathematical Communication

MPG5 — Mathematical Representations

Teachers can integrate this goal into their instruction by encouraging students to share their reasoning, strategies, and solutions with their peers through class discussions, small group work, or presentations. Additionally, teachers can provide opportunities for students to practice using mathematical language and notation when explaining their thought processes and justifying their solutions.

Teachers can integrate this goal into their instruction by using various representations, such as number lines, integer chips, counters, or other manipulatives to help students visualize and understand integer operations. Teachers can make a larger numberline out of tape on the floor that students can physically move on to help demonstrate integer operations. They can also encourage students to represent and describe mathematical ideas and relationships using different methods, such as visual models, symbolic notation, and verbal explanations.

MPG3 — Mathematical Reasoning Teachers can integrate this goal into their instruction by emphasizing the importance of estimating the result of integer operations before performing the actual calculation. They can also guide students through the process of checking the reasonableness of their solutions and provide opportunities for students to practice using logical reasoning to analyze and evaluate their work.

Content standards 6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

6.CE.2d — Estimate, determine, and justify the solution to one and two-step contextual problems, involving addition, subtraction, multiplication, and division with integers.

Future connections 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 4.01 Add and subtract integers Grade 6 — 4.02 Multiply and divide integers

Tools You may find these tools helpful: • Number line • Step-by-step graphic organizer

4.04 Real-world problems with integers mathspace.co

293

Student lesson & teacher guide Real-world problems with integers Students will extend their understanding of operations on integers, representing real-world changes in quantities. They will learn to correctly identify both positive and negative integers in these operations and use estimation as a strategic approach to solving real-life problems.

Students: Page 141

4.04 Real-world problems with integers After this lesson, you will be able to... • estimate solutions to real-world problems involving operations with integers. • solve one and two-step real-world problems involving operations with integers. • justify solutions to problems involving operations with integers.

Real-world problems with integers We can use our knowledge of addition and subtraction on the number line to describe how the real world quantities change. 1 unit Positive direction −10

−5

We can also talk about changes in the quantity we are representing using integer operations. Given a starting temperature and some change in a certain direction, what is the ﬁnal temperature? Given a starting balance and an ending balance of money in an account, what has been the amount and sign of the change? When using integers to represent real-world situations, it is important to correctly identify any positive and negative integers as well as the correct operations. Table of common key words to look for: Term Positive Negative Addition Subtraction Multiplication Division

Key Words: excess, proﬁt, above zero temperature debt, owes, below sea level plus, more than, total, sum, combined minus, difference, less than, decreased by times, product of, double/triple quotient, per, split, ratio, into

Estimation can be a good strategy for solving real-world problems if the context doesn’t require an exact solution.

Example 1 Tara is waiting for the next ﬂight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34-minute delay. She takes a nap and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?

Create a strategy Add the delay and subtract her sleep time from her wait time.

294

Apply the idea Waiting time = 64 + 34 − 23

Set up the equation

= 98 − 23

Perform 64 + 34

= 75 minutes

Evaluate

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 4.04 Real-world problems with integers mathspace.co

141

Comprehension exercise: three reads English language learner support Provide small groups of students with a variety of problems with different vocabulary terms. 1. On the first read, have students identify the numbers in the problem with any units. 2. On the second read, students should go back to those numbers in the problem and read further out, looking for key math words that can be given or displayed to the students in an organizer. Students should label these keywords with the corresponding sign or operation.

4.04 Real-world problems with integers

3. On the third read, students should write down the values from the problem with appropriate signs and operations and read through the problem again to find any missing terms or values that are needed to make a solution.

Vocabulary meets the number line After this lesson, you will be able to...

Student with disabilities • estimate solutions support to real-world problems involving operations with integers. • solve one and two-step real-world problems involving operations with integers. Set up a template for students to solve problems which may include the following: • justify solutions to problems involving operations with integers. • A space for keywords in the problem (a word bank at the top of keywords collected across the lesson may be useful) • A space for the symbol or operation the word represents Real-world problems withbyintegers • A space for the number represented the keyword attached We canline use our knowledge of addition andifsubtraction on the number line to describe how the real world quantities • A number to help with the solution the problem uses an operation change.

A number line may not be needed for every problem since some problems only need the integer represented 1 unit Positive direction by the problem. −10

−5

Mixing uptalk definitions and order We can also about changes in the quantityof wevalues are representing using integer operations. Given a starting temperature and some change in a certain direction, what is the ﬁnal temperature? Given a starting balance and an Address student misconceptions ending balance of money in an account, what has been the amount and sign of the change?

Similar-sounding words such as ascend and can oftenidentify have their definitions switched. When usingvocabulary integers to represent real-world situations, it isdescend important to correctly any positive and negative Studentsintegers also may need clarification on the order of writing the values of the numbers if there are multiple as well as the correct operations. numbersTable in the same problem confuse negative signs for subtractions if operations need to be done in the of common key wordsand to look for: problem. Term

Key Words:

excess, above zero terms temperature It can bePositive helpful to provide a listproﬁt, of vocabulary the students can use as a guide. Negative Addition Subtraction Multiplication Division

Examples

Students: Page 141

debt, owes, below sea level plus, more than, total, sum, combined minus, difference, less than, decreased by times, product of, double/triple quotient, per, split, ratio, into

Estimation can be a good strategy for solving real-world problems if the context doesn’t require an exact solution.

Create a strategy Add the delay and subtract her sleep time from her wait time.

Apply the idea Waiting time = 64 + 34 − 23

Set up the equation

= 98 − 23

Perform 64 + 34

= 75 minutes

Evaluate

4.04 Real-world problems with integers 141 mathspace.co 4.04 Real-world problems with integers

mathspace.co

295

Purpose Students demonstrate that they can use keywords in a problem to complete multiple operations on a non-zero starting number with no number line provided. Expected mistakes Students may not know that a delay will add time to the total, and sleeping would take away the amount of time. Take time to discuss the context of the problem with students to make sure everyone understands it. Encourage students to share their own experiences with being delayed.

Decompose the problem into smaller parts

use with Example 1

Targeted instructional strategies Students may not be sure how to approach the problem as there are multiple numbers in the instructions. Encourage students to break down the problems into smaller components and solve them one component at a time. For example, this is a possible way to break down this problem. 1. Make sense of quantities by identifying what each number represents: • 64 represents the initial time she had to wait • 34 represents the additional time she had to wait • 23 represents the time that has passed 2. Write an expression for the time left to wait: Waiting time = Initial waiting time + Additional waiting time − Time passed = 64 + 34 − 23 3. Evaluate 64 + 34 to find the total waiting time Waiting time = Initial waiting time + Additional waiting time − Time passed = 64 + 34 − 23 = 9 8 − 23 4. Evaluate 98 − 23 to find the remaining time to wait Waiting time = Initial waiting time + Additional waiting time − Time passed = 64 + 34 − 23 = 98 − 23 = 75 5. Write a conclusion: Tara has 75 minutes left to wait for her flight.

Students: Page 142 Example 2 A science club has a budget of $600 for a new project. The club members decide to spend $450 on laboratory equipment and the rest on protective gear. If each set of protective gear costs $25, how many sets of protective gear can the club buy with the remaining budget?

Create a strategy Our goal here is to ﬁgure out how many sets of protective gear the science club can buy after spending part of their budget on laboratory equipment. We will subtract the amount of money spent on lab equipment from the total budget. This will let us know how much they have left to spend on protective gear. We will divide this amount by the cost of protective gear to ﬁnd how many they can purchase.

Apply the idea 296

The total budget for the project is $600. The club decided to spend $450 of that budget on laboratory equipment. We can ﬁnd how much money is left by subtracting the amount spent on equipment from the total budget. Mathspace Virginia SOL Grade 6 Teacher Edition Set up the equation mathspace.co Money leftover = 600 − 450 = 150 dollars

Subtract

We have now ﬁgured out that we have $150 left in the club’s budget. We know that each set of protective gear costs

Our goal here is to ﬁgure out how many sets of protective gear the science club can buy after spending part of their budget on laboratory equipment. We will subtract the amount of money spent on lab equipment from the total budget. This will let us know how much they have left to spend on protective gear. We will divide this amount by the cost of protective gear to ﬁnd how many they can purchase.

Apply the idea The total budget for the project is $600. The club decided to spend $450 of that budget on laboratory equipment. We can ﬁnd how much money is left by subtracting the amount spent on equipment from the total budget. Money leftover = 600 − 450 = 150 dollars

Set up the equation Subtract

We have now ﬁgured out that we have $150 left in the club’s budget. We know that each set of protective gear costs $25, so to ﬁnd how many sets of protective gear we can purchase we will divide our remaining money, $150, by the cost of each set of protective gear, $25 Number of sets of gear = 150 ÷ 25 = 6 sets of gear

Set up the equation Divide

The science club can purchase 6 sets of protective gear with the remaining budget.

Example 2

Idea summary

PurposeA science club has budget of $600 a new project. The club members to spend $450 Answers thata are integers can for be positive or negative. When solvingdecide a problem, the sign of on thelaboratory integer Demonstrate how basic arithmetic canset be applied tohave solve real-world problems budget equipment and the the rest on protective gear. each protective gear costs $25, how or many sets or ofinvolving protective gear determines location ofoperations a thing orIfperson, orof whether we a proﬁt or loss, savings debt. can the club buy with the remaining budget? management. Term Key Words: Positive

excess, proﬁt, above zero temperature

Create a strategy Negative debt, owes, below problems sea level Advanced learners: Generalizing with algebra

use with Example 2

Our goal here is to ﬁgure out how many sets of protective gear the science club can buy after spending part of their budget on laboratory equipment. Subtraction minus, difference, less than, decreased by Encourage advanced learners generalize the problem byfrom introducing variables to represent the much quantities We will subtract the amount to of money spentof, on lab equipment the total budget. This will let us know how Multiplication times, product double/triple have left to spend on protective gear. We will divide this amount by the cost of protective gear to ﬁnd how many involved.they Instead of working with specific numbers, have students let “B” represent the total budget, “E” the Division quotient, per, split, ratio, into they can purchase. amount spent on equipment, “C” the cost per set of protective gear, and “N” the number of sets that can be Addition plus, more than, total, sum, combined Targeted instructional strategies

purchased. Guide them to formulate the general equation N = Apply the idea

to represent the situation.

Thestudents total budget for the project is $600. Theunderstand club decided the to spend $450 of that budget on laboratory equipment. This helps recognize patterns and underlying mathematical structure of budget Practice Weproblems. can find howInvite much them moneyto is left by subtracting the amount spent on equipment from theoutcome total budget. allocation explore how changing each variable affects the by substituting Money = 600 − 450 Set upnot the only equation different values for “B”,leftover “E”, and “C”. This approach deepens their understanding of the original What do you remember? = 150 dollars Subtract problem but also equips them to solve more complex budgeting and resource allocation problems in the future. We have now a figured that we have $150such left inas thea club’s budget.or Wepie know that each set of protective gear costs is If possible, include visualout representation, bar model chart, to illustrate how the budget 1 so Write thehow integer that represents eachgear situation. $25, to find many sets of protective we can purchase we will divide our remaining money, $150, by the divided and how the variables a 15° degrees belowrelate zero to each other. b A deposit of $74 cost of each set of protective gear, $25 c 11 feet below seaoflevel Number of sets gear = 150 ÷ 25 e

A withdrawal of $50

Students: Page 142

= 6 sets of gear

d up A gain of 39 yards Set the equation f A loss of 3 feet Divide

The club Virginia can purchase 6 sets of protective gear with the remaining budget. 142 science Mathspace SOL Grade 6 mathspace.co

Idea summary Answers that are integers can be positive or negative. When solving a problem, the sign of the integer determines the location of a thing or person, or whether we have a proﬁt or loss, or savings or debt. Term Positive Negative Addition Subtraction Multiplication Division

Practice What do you remember? 1

142

Write the integer that represents each situation. a

15° degrees below zero

11 feet below sea level

A deposit of $74 4.04 Real-world problems with integers A gain of 39 yards mathspace.co

A withdrawal of $50

A loss of 3 feet

Mathspace

Virginia SOL Grade 6

297

Practice Students: Pages 142–146

What do you remember? 1

298

Write the integer that represents each situation. a

15° degrees below zero

A deposit of $74

11 feet below sea level

A gain of 39 yards

A withdrawal of $50

A loss of 3 feet

Match each verbal situation to its integer operation. a

A diver swimming at 8 feet below sea level descends 15 more feet

−8 + 15

A loss of 8 yards followed by a gain of 15 yards

−15 ⋅ 8

A hiker at an elevation of 8 feet ascends an additional 15 feet

iii

8 + 15

The total change in a bank account balance for a $15 withdrawal each day for 8 days

−8 − 15

18− (−6)

−33− (−7)

28 ⋅ (−7)

−18 + (−13)

Evaluate each integer operation. a

−9 + 13

−9 − 18

−9 ⋅ (−15)

−19 + 11

6− (−15)

∣−119 + (−87) ∣

(−7) (−5)

Simplify each expression. a

−∣−25∣

∣−43∣ − (−3)

∣−36∣ − ∣−19∣

∣−23∣ ⋅ ∣12∣

Determine whether the ﬁnal answer will be positive or negative. Do not solve. a

The local movie theater reported losses of $475 each day for three days. What was the loss for the three days?

An elevator is on the twelfth floor. It goes down 8 floors and then up 2 floors. What floor is the elevator on now?

On a test, Ruby scores 25 points for the questions she answered correctly and gets −2 points for each incorrect answer. Ruby answered 4 questions incorrectly. How many points did Ruby score?

The elevation of Mt. Everest is 29 028 feet. The elevation of the Dead Sea is −485 feet. What is the difference in the elevation between Mt. Everest and the Dead Sea?

Jessie has $58 left on her checking account. If she writes a check for $65, what will Jessie’s balance be?

Joseph reported that the coldest day on record for his town was ﬁve times colder than yesterday’s temperature, −4 °C. What was the temperature of the coldest day on record in Joseph’s town?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice 6

Valentina and Maria want to meet at the cinema to watch a movie. Valentina lives 3 km west of the cinema, and Maria lives 5 km east of the cinema. a

In the number lines below, the distance between each tick is 1 km. Which of the following number lines show the location of Valentina and Maria in relation to the cinema? A West Cinema East Maria’s House

Valentina’s House

West

Valentina’s House

C West Cinema East

Cinema

Maria’s House

West

Maria’s House

East

Valentina’s House Cinema

Valentina’s House

East Maria’s House

Draw a new number line that shows the location of Valentina and Maria using integers, where 0 is the cinema, east is the positive direction and west is the negative direction of the coordinate plane.

Calculate how far Maria would have to travel to get to Valentina’s house.

At the start of the month, Laura owes $60 to the bank. A week later, her employer deposits $130 into her account. a

Draw a number line that shows the change in Laura’s account balance.

Suppose that next month Laura now has $130 in her account, and she spends $60 on rent. Draw a number line that shows the change in Laura’s account balance for this month.

The following diagram shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line: a

State the integer that represents 3 m above the surface.

State the integer that represents 4 m below the surface.

If Nadia is initially 2 m above the surface, and descends 6 m in the elevator, state the integer that represents her new location.

If Nadia is at a location represented by the integer −4, and ascends 3 m, describe his new location.

5 m above 4 m above 3 m above 2 m above 1 m above surface 1 m below 2 m below 3 m below 4 m below 5 m below 6 m below 7 m below 8 m below

5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8

Mike has $100 in his wallet. He has 3 nieces and 2 nephews. a

Explain how to calculate the total amount of money he could give to his nieces and nephews if he gave $15 to each.

Explain how to calculate the amount of money each person gets if he shares his money equally with each of his nieces and nephews.

The planet Anarres experiences drastic changes in temperature over the course of the day. At sunrise, the temperature is measured to be −250 °F, and it warms to a temperature of −110 °F by midday. Calculate the difference of temperature from sunrise to midday.

A submarine is initially 45 ft below sea level. It then descends 30 ft straight down to its ﬁnal position. a

If sea level is represented by the integer 0, and upwards is the positive direction, ﬁnd the integer that represents the submarine’s ﬁnal position.

Describe the ﬁnal position of the submarine.

A bird soaring at a height of 25 ft above sea level sees a school of ﬁsh and dives down 32 ft. Vivian watches the bird dive and estimates that the bird will end up 55 feet below sea level since 25 + 30 = 55. Is this a reasonable estimate? Explain your thinking. 4.04 Real-world problems with integers mathspace.co

299

Mair’s cable company automatically deducts $48 from her bank account each month. Her current bank balance is $160. Write two different ways to use integer operations to write the problem. Calculate her account balance after 4 cable bill payments, if she doesn’t make any deposits. Include the account balance in your answer.

Jack observes the elevation of a hot-air balloon at 2000 feet is descending at a rate of −120 feet per minute. a

What is the elevation change of the hot-air balloon after 12 minutes?

Jack extends his observation and predicts that after 30 minutes, the hot air balloon will have descended 3600 feet. Is this a reasonable estimate? Explain your thinking.

Theo tracked the supply spending of a company and noticed the budget for paper in a company changed by −$96 in 6 days. Theo wanted to determine the amount spent on each of those 6 days if spending was the same each day. He wrote: −$96 ⋅ 6 = −$576 Theo realized he made a mistake and wants to ﬁx it. Describe and ﬁx his mistake.

Overnight, the temperature decreased 4 °F each hour for 6 hours then rose 6 degrees each hour for 3 hours as the sun rose. Which operation and answer show the correct integer operation that represents the overall change in temperature? A

−24 + 18 = 6

−4 + 6 = 2

24 + 18 = 42

−24 + 18 = −6

A female brown bear weights 600 pounds. After hibernating for 6 months, she weighs only 420 pounds. Determine her monthly change in weight if she lost the same amount of weight each month.

Ms. Jones has an account balance of −$75. She earns $104 for babysitting and tutoring. She works 4 hours as a babysitter and 3 hours as a tutor. a

If she deposits her money into her account, what would her new account balance be?

If she earned $56 for babysitting, how much does she earn per hour? How much does she earn for tutoring per hour?

If she babysits for 2 more hours and tutors for 1 more hour, what will her new bank account balance be after depositing the money?

Let’s extend our thinking 19

A clothing store was selling jackets for $68 each. The store changed the price by these amounts over 3 months: +$27, −$12, −$11

What is a reasonable estimate for the price of the jacket? Explain your reasoning.

What is the actual current price for a jacket?

Three contestants are heading into the ﬁnal round of a game show. Eileen has 39 points and Roxanne has −3 points. If Oprah has 14 more points than Eileen, calculate how many points separate Roxanne and Oprah.

300

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Victoria’s bank account has a balance of −$350 at the beginning of the year. She earns $1100 a month, and deposits it into her bank account. She also withdraws $200 a month to pay for rent. a

Write an expression to represent how to calculate the change to Victoria’s bank account over the course of a 12 month year, explaining your choice in what operations to include.

Use your expression from the previous part to calculate the total change in Victoria’s bank account, writing your answer as both an integer and describing the integer in context of the context.

Find the account balance at the end of the year.

In golf, your score for each hole is determined by comparing it to the par (or expected) number of shots for that hole. Taking one less shot than expected will give a score of −1, one more shot will give +1, and so on. Ivan played 18 holes of golf, and the frequency of each score he obtained is shown in the adjacent table:

Find Ivan’s total score for his 6 scores of +2.

Find Ivan’s total score for his 5 scores of −1.

Find Ivan’s total score for the total 18 holes.

Score −1 Par +1 +2

Frequency 5 3 4 6

The motion of a robot to the left or right along a straight line is determined by whether the robot is facing left or right, and whether it is stepping forwards or backwards. Use a number line where 1 unit represents 1 step and the positive direction is to the right to answer the following questions: a

For each of the following, ﬁnd the integer that represents the ﬁnal position: i

The robot starts at 0, facing to the right, and takes 5 steps forward.

The robot starts at 0, facing to the left, and takes 13 steps forward.

Facing Right Right Left Left

Stepping Forward Backward Forward Backward

Moving Right Left Left Right

iii The robot starts at 0, facing to the left, and takes 11 steps backwards. b

Create a set of instructions for the robot to have a ﬁnal position of: i

−8

iii

−23

Jenny is going to buy 8 notebooks, 4 folders, 1 pack of pencils, and 1 lunch box for school. She will use a $10 gift card, then pay the rest of the total. a

As a way to estimate, Jenny rounded the price of pencils, folders, and notebooks to $5 and the lunch box to $10. Jenny calculates her total as 8 ⋅ $5 + 4 ⋅ $5 + $5 + $10 = $75.

Back-to-School Savings 6 pk. Pencils

Folder

After subtracting $10 from her estimate for the gift card amount, she would pay $65. Is this a reasonable estimate? Explain your thinking. b

Calculate the actual amount Jenny will need to pay for this purchase after using the $10 gift card.

How different were the estimate and the actual cost? Did Jenny choose an accurate method to estimate?

Spiral Notebook

Lunch Box

4.04 Real-world problems with integers mathspace.co

301

Answers

14 a −1440 ft b T his is not a reasonable estimate. The balloon is only 2000 feet in the air and cannot travel down 3600 feet. Jack can only predict values less than 2000 feet.

4.04 Real-world problems with integers What do you remember? 1 a −15

15 Since $96 needs to be split over $6 days, Theo needed to divide instead of multiply. His work should be

c −11

d 39

2 a iv. −8 − 15 b i. −8 + 15

c iii. 8 + 15

d ii. −15 ⋅ 8

3 a 4

b 35

c 24

d −26

This means the company spent $16 a day on paper. 16 D

b 74

e −50

−3

e −27

g −196

h −31

135

−8

k 21

b 46

c 206

d 8

c Positive

d Positive

4 a −25 e 17

5 a Negative e Negative

17 30 pounds 18 a $29

276

b She earns $14 for babysitting and $16 for tutoring.

b Positive f

Negative

Let’s extend our thinking

Let’s practice

19 a S ample answer: A reasonable estimate would be $75. By ﬁnding the closest value of 5 for each, we ﬁnd the estimate by solving 70 + 25 − 10 − 10. This gives an estimate of a cost of $75.

6 a D b

Valentina’s House −6

−4

−2

Maria’s House 0

c $73

b $72

20 56 points

c 8 km 7 a

−100 −80 −60 −40 −20

100

21 a T he expression 12 ⋅ 1100 + 12 ∗ (−200) ﬁnds the total amount deposited and withdrawn over a year by multiplying the values given by the number of months, and combines the two amounts to ﬁnd the total change. b T he change is 10 800, which represents an additional $10 800 to Victoria’s bank account. c $10 450

8 a 3 c −4

100 120 140 160 180 200

22 a 12

b −5

c 11

b −4

23 a i 5

ii −13

iii 11

d 1 m below the surface

b i Example answers: • The robot starts at 0, facing to the right, and takes 8 steps backwards. • The robot starts at −3, facing to the left, and takes 5 steps forward.

9 a M ultiply the number of his nieces and nephews which is 5 to $15. b D ivide $100 by the number of his nieces and nephews which is 5. 10 140 °F 11 a −75 b T he ﬁnal position of the submarine is 75 ft below sea level. 12 This is not a reasonable estimate. Vivian should subtract her values since the bird begins above sea level then falls. This would give an estimate of 25 − 32 = −7, or 7 feet below sea level. 13 $160 − $48 − $48 − $48 − $48 = −$32 and $160 − 4 ⋅ $48 = −$32

302

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

ii Example answers: • The robot starts at 0, facing to the right, and takes 15 steps forward. • The robot starts at 20, facing to the left, and takes 5 steps forward. iii Example answers: • The robot starts at 0, facing to the right, and takes 23 steps backwards. • The robot starts at −15, facing to the left, and takes 8 steps forward.

24 a J enny’s estimate is not entirely reasonable. She overestimated the cost of pencils, folders, and notebooks. leading to an overestimation of the total cost by the same amount. b T he cost of the items will be $31, so after the $10 gift card, Jenny will owe $21. c T here is a $44 difference in Jenny’s estimate and the actual amount owed. Jenny did not choose an accurate way to estimate.

Answers mathspace.co

303

4.05 Integers in the coordinate plane Subtopic overview Lesson narrative In this lesson, students will explore the coordinate plane, understanding its structure and how to plot points using ordered pairs (x, y). They will learn about the x-axis, y-axis, origin, and the four quadrants. The lesson includes interactive explorations to describe locations and identify patterns in each quadrant. Students will practice plotting points, identifying coordinates, and determining distances between points. By the end, students will confidently navigate and use the coordinate plane for various applications.

Learning objectives Students: Page 147

Key vocabulary 

coordinate plane



coordinates



horizontal

 ordered pair



origin



quadrant



vertical

 x-axis



y-axis

Essential understanding Integers can be used to describe points in 2 dimensional space using ordered pairs or coordinates (x,y) where the x-coordinate is the horizontal distance from a central point called the origin and the y-coordinate is the vertical distance from the origin.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

304

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Mathematical process goals MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can encourage students to share their strategies for identifying and labeling the axes, origin, and quadrants. By discussing their thought processes and using appropriate mathematical vocabulary, students can deepen their understanding of the coordinate plane.

Teachers can help students make connections between their knowledge of integers and the coordinate plane by emphasizing the role of integers in determining the positions on the axes. Students can also explore connections between the coordinate plane and other mathematical concepts, such as area and perimeter.

MPG5 — Mathematical Representations Teachers can incorporate this goal into instruction by creating a large model of the coordinate plane in the classroom and having students label the axes, four quadrants, and the origin. Additionally, teachers can provide students with ordered pair cards and ask them to move to the quadrant or axis to show where that ordered pair would be plotted. Teachers should consistently review coordinate plane vocabulary (coordinates, ordered pair, axies, quadrants, horizontal, vertical, etc.) and encourage students to use this vocabulary when describing the location of a point in the coordinate plane. Students can also create their own models of the coordinate plan with geoboards and/or grid paper with labels. Teachers can also refer to objects in the class and ask students to describe features of the objects as horizontal or vertical to help provide real-world application (e.g., Is the bottom edge of a door running horizontal or vertical?).

Content standards 6.MG.3 — The student will describe the characteristics of the coordinate plane and graph ordered pairs. 6.MG.3a — Identify and label the axes, origin, and quadrants of a coordinate plane. 6.MG.3b — Identify and describe the location (quadrant or the axis) of a point given as an ordered pair. Ordered pairs will be limited to coordinates expressed as integers.

6.MG.3d — Identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane. Ordered pairs will be limited to coordinates expressed as integers. 6.MG.3e — Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers.

6.MG.3c — Graph ordered pairs in the four quadrants and on the axes of a coordinate plane. Ordered pairs will be limited to coordinates expressed as integers.

Prior connections 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

Future connections 8.PS.3 — The student will apply the data cycle (formulate 8.MG.3 — The student will apply translations and questions; collect or acquire data; organize and represent reflections to polygons in the coordinate plane. data; and analyze data and communicate results) with a A.F.1 — The student will investigate, analyze, and focus on scatterplots. compare linear functions algebraically and graphically, and model linear relationships. 7.MG.4 — The student will apply dilations of polygons in the coordinate plane.

4.05 Integers in the coordinate plane mathspace.co

305

Tools You may find these tools helpful: • Ruler • Dry-erase coordinate planes • Graph paper • Coloured markers

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Begin by engaging students with a large, physical coordinate grid laid out on the classroom floor or playground. Use tape or chalk to create the x-axis and y-axis, clearly labeling the origin where they intersect. Assign positive and negative directions along each axis and label the quadrants I through IV. Provide students with numbered floor tiles or markers that they can place at specific points on the grid. Have students physically stand on or place objects at various coordinates, such as (2, 3) or (−4, −1), to model plotting points. This handson approach helps students understand the axes, origin, quadrants, and how ordered pairs indicate positions on the plane. Representational: Provide students with graph paper and have them draw their own coordinate planes, including the x-axis, y-axis, origin, and quadrants. Ask them to label each part appropriately. Using the points they plotted on the physical grid, have students draw and label these points on their graph paper. Encourage them to use dots or small circles to represent points and to draw lines or arrows to indicate movement from the origin to each point. As they plot points, have them note the corresponding ordered pairs. This visual representation reinforces their understanding of how the coordinate plane works. Abstract: Introduce the concept of ordered pairs (x, y) as a way to symbolize the points they’ve been plotting. Show how each point on their graph corresponds to an ordered pair of numbers. Discuss how the x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance. Have students write out the ordered pairs next to each plotted point on their graphs. Help students see how the visual points and the numerical ordered pairs are connected.

306

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Critique, clarify and correct English language learner support Display examples of ordered pairs graphed on the coordinate plane along with incorrect ordered pairs around the room. Common errors when graphing or identifying ordered pairs include switching the x and y coordinates as well as miscounting. Have students describe the errors made with graphing, paying close attention to the vocabulary used when describing the errors. Note the usage of the words x-axis, y-axis, x-coordinate, and y-coordinate when students are describing the errors and their corrections.

Provide tools to help with lining up coordinates Student with disabilities support Students may struggle with counting due to visual-spatial processing challenges. Give students rulers or even small post-it notes to place as a vertical and horizontal marker if they are struggling to keep their eyes tracking along a grid line.

Transposing Coordinates Address student misconceptions Students tend to forget which order the x and y-coordinates should be listed in. They also list the coordinates without parentheses. One way to help students remember to look first at the x-value before the y-value when plotting is to use the acronym “x-y” as a reminder. When introducing the concept of plotting, make sure to emphasize that “x” comes before “y” in the alphabet and in the coordinates.

Student lesson & teacher guide Introduction to the coordinate plane Students are introduced with the definition of terms related to graphing within the coordinate plane. These terms include: coordinate plane, coordinates, ordered pair, x-axis, y-axis and origin. Students will then take part in an exploration to investigate how the coordinate plane can be used to describe the location of different objects using an applet.

4.05 Integers in the coordinate plane mathspace.co

307

Students: Pages 147–148

The coordinates of a point are given in relation to the origin. In the image, we can see that the cat is 6 units to the right of the origin, and 2 units above the origin. So we can say the cat has the coordinates (6, 2). The x-coordinate is 6 and the y-coordinate is 2. y 4 (6, 2)

3 2 1 0

9 10

Notice that the order of the numbers is important. It would be incorrect to say the cat has the coordinates (2, 6). These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin. Coordinates are always written with parentheses in the form (x, y) where the ﬁrst number, x, is the x-coordinate and the second number, y is the y-coordinate. Notice that the x-coordinate also tells us how far a point is from the y-axis and the y-coordinate tells us how far a point is from the x-axis. In the image of the cat, the coordinates (6, 2) tell us the cat is 6 units from the y-axis and 2 units from the x-axis.

Example 1 308

Mathspace SOL Grade 6 Teacher ConsiderVirginia the coordinate plane shown: Edition mathspace.co

y 10 9 8

Exploration Students: Page 147

Suggested student grouping: Small groups In this exploration, students will use a provided applet to draw a coordinate plane over a given area. They will then observe and describe the location of various objects both with and without the coordinate plane. This will help students understand the utility of a coordinate plane in providing a clear and unified system for describing locations. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. How would you describe the location of each object without the coordinate plane drawn? Without the coordinate plane, the location of an object can be described based on its position relative to other objects or the boundaries of the area. For example, one might say an object is in the top-right corner or in the middle. 2. How would you describe the location of each object with the coordinate plane drawn? With the coordinate plane drawn, the location of an object can be described using coordinates. For example, the apple is at the position (5, 5) on the coordinate plane. 3. Which way of describing locations is more clear? Describing locations using a coordinate plane is clearer because it provides a standard, precise way to specify any location within the area, rather than relying on relative and potentially vague descriptions. Purposeful questions • Why do you think coordinate planes are used widely in mathematics and other fields? • Can you think of any real-world applications where coordinate planes are useful? Possible misunderstandings • Students may confuse the x and y coordinates on the plane, or may not understand that the first number in an ordered pair refers to the x-coordinate (horizontal direction) and the second number refers to the y-coordinate (vertical direction). After the exploration, describing the coordinates to locate an object is introduced and students are guided on how the coordinates should be written.

4.05 Integers in the coordinate plane mathspace.co

309

and the y-coordinate is 2. Notice that the order of the numbers isy important. It would be incorrect to say the cat has the coordinates (2, 6). These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin. Coordinates are always written with parentheses in the form (x, y) where the ﬁrst number, x, is the x-coordinate and 4 the second number, y is the y-coordinate. (6, 2) 3

Examples Notice that the x-coordinate also tells2 us how far a point is from the y-axis and the y-coordinate tells us how far a is from the x-axis. Students:point Page 148

x 0 (6, 2) tell us the cat is 6 units from the y-axis In the image of the cat, the coordinates and 2 units from the x-axis. 0 1 2 3 4 5 6 7 8 9 10

Notice that the Example 1 order of the numbers is important. It would be incorrect to say the cat has the coordinates (2, 6). These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin. Consider the coordinate plane shown: Coordinates are always written with parentheses in the form (x, y) where the ﬁrst number,y x, is the x-coordinate and 10 the second number, y is the y-coordinate. 9

Notice that the x-coordinate also tells us how far a point is from the y-axis and the y-coordinate tells us how far a 8 point is from the x-axis. 7 6 2 units from the x-axis. In the image of the cat, the coordinates (6, 2) tell us the cat is 6 units from the y-axis and 5 4 3 2 1 0

Example 1 Consider the coordinate plane shown:

a What object has coordinates (1, 4)?

Create a strategy Use the numbers on the axes in locating the coordinates.

b What object has coordinates (10, 1)?

y 0 1 2 3 4 5 6 7 8 9 10 x 10 9 8 7 6 Apply the idea 5 Start at (0, 0). Move 1 space 4 to the right, then 4 spaces up. 3 (1, 4) is a star. The object with coordinates 2 1 0 0 1 2 3 4 5 6 7 8 9 10 x

PurposeCreate a Whataobject has coordinates (1, 4)? strategy Apply the idea StudentsUse willthedemonstrate that they can use to identify location within the firstthen quadrant of the numbers on the axes in locating the coordinates coordinates. Start at (0, 0).aMove 10 space to the right, 1 space up. Create a strategy Apply the idea coordinate plane. The object with coordinates (10, 1) is beach ball. Use the numbers on the axes in locating the coordinates.

Students: Page 148

Start at (0, 0). Move 1 space to the right, then 4 spaces up. The object with coordinates (1, 4) is a star.

b What object has coordinates (10, 1)?

148

Mathspace Virginia SOL Grade 6 mathspace.co

Create a strategy

Apply the idea

Use the numbers on the axes in locating the coordinates.

Start at (0, 0). Move 10 space to the right, then 1 space up. The object with coordinates (10, 1) is beach ball.

Purpose 148 Mathspace Virginia SOL Grade 6 Students willmathspace.co demonstrate that they can use coordinates to identify a location within the first quadrant of the coordinate plane.

Students: Page 149 c What are the coordinates of the bicycle?

Create a strategy

Apply the idea

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical axis to identify the y-coordinate.

The number on the horizontal axis directly below the bicycle is 3 and across the vertical axis is 6. So, the coordinates are (3, 6).

Example 2 Plot the point (6, 3) onto the coordinate plane.

310

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co Create a strategy Use the numbers on the axes to move on the coordinate plane.

Apply the idea Start at (0, 0). Plot the point 6 space to the right, then 3 spaces up.

c What are the coordinates of the bicycle?

Purpose Show students the position of a point on a coordinate plane be determined by the location of the point Create athat strategy Apply thecan idea relative Follow to thethe x-axis. and x-axis. grid line up to the horizontal axis to identify The number on the horizontal axis directly below the the x-coordinate. Then, follow the grid line across the

Students:vertical Pageaxis 149 to identify the y-coordinate.

bicycle is 3 and across the vertical axis is 6. So, the coordinates are (3, 6).

Example 2 Plot the point (6, 3) onto the coordinate plane. cCreate Whataare the coordinates of the bicycle? strategy

Apply the idea

Use the numbers on the axes to move on the coordinate Create plane. a strategy

Start at (0, 0). Plot the point 6 space to the right, then 3 Apply spaces the up. idea

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical axis to identify the y-coordinate.

The on point the horizontal axis described directly below This number will be the on the plane by (6,the 3). bicycle is 3 and across the vertical axis is 6. So, the y coordinates 9are (3, 6). 8 7

Example 2

6 5

Plot the point (6, 3) onto the coordinate plane.

4 3

Create a strategy

2 Apply the idea

Use the numbers on the axes to move on the coordinate plane.

x Start at (0, 0). Plot the point 6 space to the right, then 3 spaces up. 1 2 3 4 5 6 7 8 9

This will be the point on the plane described by (6, 3). y

9 Example 3 8 Purpose 7 Consider the point with coordinates (9, 4). Challenge students to use their knowledge of coordinates to plot a6point in the first quadrant of the coordinate a How far is the point from the x-axis? plane. 5 4

Create strategy Apply the idea Reflecting witha students 3 To ﬁnd the distance of a point from the x-axis, consider For the point 4), the y-coordinate is 4. Thus, the 2(9, not Ask students if there is a situation where the order of coordinates will matter.

the y-coordinate of the point. The distance to the x-axis is measured vertically, making it equal to the absolute Students:value Page 149 of the y-coordinate.

distance from1 the x-axis is ∣4∣ = 4 units. 1

2 3 4 5 6 7 8 9

b How far is Example 3 the point from the y-axis?

Create athe strategy Consider point with coordinates (9, 4).

Apply the idea

To determine ofthe a point from the y-axis, a How far is the the distance point from x-axis? examine the x-coordinate. This distance is measured horizontally, making it equal to the absolute value of the Create a strategy x-coordinate. To ﬁnd the distance of a point from the x-axis, consider the y-coordinate of the point. The distance to the x-axis is measured vertically, making it equal to the absolute value of the y-coordinate.

Given the point (9, 4), the x-coordinate is 9. Therefore, the distance from the y-axis is ∣9∣ = 9 units.

Apply the idea For the point (9, 4), the y-coordinate is 4. Thus, the distance from the x-axis is ∣4∣ = 4 units. 4.05 Integers in the coordinate plane mathspace.co

149

b How far is the point from the y-axis?

PurposeCreate a strategy Apply the idea Show students howthe todistance determine the distance of a point from thex-coordinate coordinates the point. To determine of a point from the y-axis, Given an the axis point using (9, 4), the is 9.of Therefore, the examine the x-coordinate. This distance is measured horizontally, making it equal to the absolute value of the x-coordinate.

distance from the y-axis is ∣9∣ = 9 units.

4.05 Integers in the coordinate plane

4.05 Integers in the coordinate plane 149 mathspace.co mathspace.co

311

Create a strategy

Apply the idea

To ﬁnd the distance of a point from the x-axis, consider the y-coordinate of the point. The distance to the x-axis is measured vertically, making it equal to the absolute Students:value Page 149 of the y-coordinate.

For the point (9, 4), the y-coordinate is 4. Thus, the distance from the x-axis is ∣4∣ = 4 units.

b How far is the point from the y-axis?

Create a strategy

Apply the idea

To determine the distance of a point from the y-axis, examine the x-coordinate. This distance is measured horizontally, making it equal to the absolute value of the x-coordinate.

Given the point (9, 4), the x-coordinate is 9. Therefore, the distance from the y-axis is ∣9∣ = 9 units.

4.05 Integers in the coordinate plane

149

mathspace.co Purpose Show students how to determine the distance of a point from an axis using the coordinates of the point.

Reflecting with students Challenge advanced learners to extend this problem by calculating the distance from the point (9, 4) to other lines, such as y = x or x = 5, using the perpendicular distance formula. This will help them see how the coordinates of a point relate to its position relative to various lines. To support this exploration, provide a coordinate grid showing the point (9, 4), the axes, and additional lines, with perpendicular distances indicated; this visual aid will help them internalize these geometric relationships.

Students: Page 150

Example 4 Write the coordinates of the point that is 5 units to the right of (9, 6).

Create a strategy Plot the given coordinates, then move horizontally by the required number of units.

Apply the idea 10 9 8 7 6 5 4 3 2 1

Plot (9, 6) on the coordinate plane and move 5 units to the right.

The new coordinates are (14, 6). (9, 6)

(14, 6) 5 units

x 1 2 3 4 5 6 7 8 9 10 11 12 1314

Reﬂect and check Another way to ﬁnd the coordinates of the new point is by realizing that moving right will increase the x-coordinate so we need to add 5 to the x-coordinate. New coordinates = (9 + 5, 6) = (14, 6)

Add 5 to 9 Evaluate

Example 5 Purpose Point A has the coordinates (3, 6), and point B has the coordinates (8, 6). What is the distance between A and B? Show students how to find the coordinates of a point that has been moved a certain distance in a specific directionCreate on a coordinate a strategy plane. Plot the points on the coordinate plane and then count the horizontal units from point A to point B.

312

Apply the idea SOL Grade 6 Teacher Edition Mathspace Virginia mathspace.co Plot the point A and point B. The distance between them is 5 units. y

Reﬂect and check Another way to ﬁnd the distance between the two points is by realizing that point A and point B have the same y-coordinate. Therefore, the distance between them is the

Add 5 to 9

= (14, 6)

Students: Page 150

Evaluate

Example 5 Point A has the coordinates (3, 6), and point B has the coordinates (8, 6). What is the distance between A and B?

Create a strategy Plot the points on the coordinate plane and then count the horizontal units from point A to point B.

Apply the idea

Reﬂect and check

Plot the point A and point B. The distance between them is 5 units.

Another way to ﬁnd the distance between the two points is by realizing that point A and point B have the same y-coordinate. Therefore, the distance between them is the difference in their x-coordinate.

Distance = (8 − 3)

8 7

Subtract 3 from 8 Evaluate

5 units

5 4 3 2 1

x 1

150

2 3 4 5 6 7 8 9

Mathspace Virginia SOL Grade 6 mathspace.co

Purpose Show students that they can determine the distance between two points on a coordinate plane by plotting the points and counting the units between them.

Assuming zero distance when coordinates are the same

use with Example 5

Address student misconceptions Students might mistakenly believe that if two points share the same x-coordinate or y-coordinate, the distance between them is zero. For example, since points A(3,6) and B(8,6) both have a y-coordinate of 6, they might conclude that there is no distance between them along the y-axis and therefore the total distance is zero. Emphasize to students that sharing one identical coordinate means the points are aligned along that axis, but there can still be a distance between them along the other axis. Encourage students to look at which coordinates differ—in this case, the x-coordinates. Have them plot the points on a coordinate plane to visually see the horizontal distance between them. Explain that the distance is found by calculating the difference between the x-coordinates: 8 − 3 = 5 units. Reinforce the idea that even when one set of coordinates is the same, there can still be a measurable distance along the axis with differing coordinates.

Students: Page 151

Idea summary The coordinate plane is used to describe the location of actual points called coordinates in a two-dimensional space. The coordinates are pair of numbers that are in the form of

(x, y) x y

is the ﬁrst number which is found in the x-axis is the second number which is found in the y-axis

Quadrants in the coordinate plane

4.05 Integers in the coordinate plane Now that we know how to graph points with positive coordinates, let’s see what happens if we extend the axes of a mathspace.co coordinate plane in both directions.

Interactive exploration

313

Interactive exploration Explore online to answer the questions

Idea summary mathspace.co The coordinate plane is used to describe the location of actual points called coordinates in a two-dimensional space.

Use thein interactive in 4.05 to answer these questions. Quadrants theexploration coordinate The coordinates are pair of numbers thatplane are in the form of 1.

Drag P into the section labeled 1st quadrant. What do you notice about the coordinates? Is that true for

Repeat for the other 3 quadrants. What do you notice about the points in each quadrant? Are your is the second number which is found in the y-axis observations true for everyy point in that quadrant?

Students are introduced to the definition of a quadrant before (x, they y) take part in an exploration to identify the every point in the 1st quadrant? characteristics of the coordinates in eachx quadrant. is the ﬁrst number which is found in the x-axis

Students: Page 151 3. Drag P along the y-axis. What do you notice about the coordinates? Is that true for every point on the y-axis?

4. Drag P along the x-axis. What do youplane notice about the coordinates? Is that true for every point on Quadrants in the coordinate the y-axis? Now that we know how to graph points with positive coordinates, let’s see what happens if we extend the axes of a coordinate plane in both directions. The coordinate plane is divided into four distinct regions, called quadrants.

Interactive The 1st quadrant is on the exploration top right. The x-coordinate and y-coordinate of a point Explore online to answer in the 1st quadrant are both positive.the questions

y-axis Quadrant 2

Quadrant 1 x-axis

The quadrants are numbered in an counterclockwise direction: mathspace.co summary • 2ndIdea quadrant: x-coordinates are negative, y-coordinates are positive • 3rd The quadrant: both plane coordinates coordinate is usedare to negative describe the location of actual points called coordinates in a two-dimensional Origin the interactive explorationare in positive, 4.05 to answer these questions. • Use 4th space. quadrant: x-coordinates y-coordinates are negative P an intoaxis, theare section quadrant. do you true for4 Quadrant 3 Is that Quadrant Points1.that lie on like (−5,oflabeled 0)numbers or (0, 1st 4),that are are not in What any form quadrant. TheDrag coordinates pair the of notice about the coordinates? every point in the 1st quadrant? Points on the x-axis have a y-coordinate of 0. (x, y) 2. Repeat for the other 3 quadrants. What do you notice about the points in each quadrant? Are your Points on the y-axis have an x-coordinate of 0. observations true for everyx point in that is the ﬁrst quadrant? number which is found in the x-axis 3.

y is which is found inIsthe Drag P along the y-axis. What dothe yousecond notice number about the coordinates? thaty-axis true for every point on the y-axis?

4. Drag P along the x-axis. What do you notice about the coordinates? Is that true for every point on Exploration the y-axis?

Quadrants in the coordinate plane

Students:Now Page 151 that we know how to graph points with positive coordinates, let’s see what happens if we extend the axes of a The coordinate is divided into four distinct regions, called quadrants. coordinate planeplane in both directions.

y-axis

4.05 Integers in the coordinate plane 151 The 1st quadrant is on the top right. The x-coordinate and y-coordinate of a point mathspace.co Quadrant 2 Quadrant 1 in the 1st quadrant are both positive. Interactive exploration

x-axis

Explore to answer questions The quadrants areonline numbered in an the counterclockwise direction: • 2nd quadrant: x-coordinates are negative, y-coordinates are positive mathspace.co • 3rd quadrant: both coordinates are negative • 4th quadrant: x-coordinates are positive, y-coordinates are negative Usethat the interactive exploration in or 4.05 Points lie on an axis, like (−5, 0) (0, to 4),answer are notthese in anyquestions. quadrant.

Origin Quadrant 3

Quadrant 4

Dragx-axis P into the asection labeledof1st0.quadrant. What do you notice about the coordinates? Is that true for Points1.on the have y-coordinate every point in the 1st quadrant? Points on the y-axis have an x-coordinate of 0. 2. Repeat for the other 3 quadrants. What do you notice about the points in each quadrant? Are your observations true for every point in that quadrant? 3.

Drag P along the y-axis. What do you notice about the coordinates? Is that true for every point on the y-axis?

Drag P along the x-axis. What do you notice about the coordinates? Is that true for every point on the y-axis?

The coordinate plane is divided into four distinct regions, called quadrants.

4.05 Integers in the coordinate plane mathspace.co y-axis

151

The 1st quadrant is on the top right. The x-coordinate and y-coordinate of a point

Quadrant 2 Quadrant 1 Suggested grouping: In pairs in thestudent 1st quadrant are both positive. x-axisin the In this exploration, will move point P around in the Geogebra applet to observe the changes The quadrantsstudents are numbered in an counterclockwise direction: • 2nd quadrant: x-coordinates are P negative, are positive coordinates. They will start with point at the y-coordinates origin and then move it into different quadrants and along the • 3rdThe quadrant: both arehow negative x and y axis. goal is to coordinates understand the coordinate values change based on the quadrantOrigin and axis in • 4th quadrant: x-coordinates are positive, y-coordinates are negative which point P is located.

Points that lie on an axis, like (−5, 0) or (0, 4), are not in any quadrant.

Quadrant 3

Quadrant 4

Points on the x-axis have a y-coordinate of 0. Points on the y-axis have an x-coordinate of 0.

314

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

4.05 Integers in the coordinate plane

151

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Drag P into the section labeled 1st quadrant. What do you notice about the coordinates? Is that true for every point in the 1st quadrant? In the 1st quadrant, both the x and y coordinates are positive. This is true for all points in the 1st quadrant. 2. Repeat for the other 3 quadrants. What do you notice about the points in each quadrant? Are your observations true for every point in that quadrant? In the 2nd quadrant, x is negative and y is positive. In the 3rd quadrant, both x and y are negative. In the 4th quadrant, x is positive and y is negative. These observations are true for all points in their respective quadrants. 3. Drag P along the y-axis. What do you notice about the coordinates? Is that true for every point on the y-axis? When P is on the y-axis, the x-coordinate is always 0 . This is true for every point on the y-axis. 4. Drag P along the x-axis. What do you notice about the coordinates? What do you notice about the coordinates? Is that true for every point on the y-axis? When P is on the x-axis, the u-coordinate is always 0. This is true for every point on the x-axis. Purposeful questions • How do the signs of the coordinates relate to the quadrant the point is in? • Why do you think the x-coordinate is always 0on the y-axis and the y-coordinate on the x-axis? Possible misunderstandings • Students may think that the coordinate plane may only have positive x-values and y-values. • Students may believe that the coordinate plane may have limited x-values and y-values.

Use the STEAM cycle with integer coordinates Targeted instructional strategies Stir students’ interest with a treasure map activity where they use the coordinate plane to locate hidden treasures. Ask: Begin by asking students how they might use a coordinate grid to find specific locations on a map. Encourage them to define the problem by exploring how points are represented with ordered pairs and discuss different perspectives on navigating the coordinate plane. Imagine: Invite students to brainstorm strategies for locating treasures using coordinates. Have them analyze patterns in plotting points in different quadrants and predict how changing the x or y values affects a point’s position. Encourage them to share their ideas through sketches or by modeling with graph paper, fostering creative thinking. Plan: Guide students in creating a plan to locate the treasures. Assist them in setting criteria such as correctly identifying the axes, origin, and quadrants, and acknowledge constraints like limited ability to travel or the number of treasures to find. Determine how they will collect and record data—perhaps by creating a table of coordinates or using a digital graphing tool. Create and test: Support students as they plot the coordinates of the treasures on the coordinate plane. Provide tools like graph paper, rulers, or interactive software. Have them test their accuracy by checking if the plotted points lead to the correct treasure locations. Improve: Facilitate a review session where students reflect on their plotting methods and results. Encourage them to collaborate on improving their techniques, such as paying closer attention to the signs of the coordinates or scaling the axes differently. Have them justify their design changes and communicate how these adjustments lead to more accurate plotting. By guiding students through this STEAM cycle, you help them actively engage with the coordinate plane, enhancing their problem-solving skills and understanding of graphing concepts.

4.05 Integers in the coordinate plane mathspace.co

315

Use color-coded quadrants and graph paper Student with disabilities support Use color-coded quadrants to help students with visual-spatial processing difficulties distinguish between the different regions on the coordinate plane. Provide graph paper to help students visualize and track movements on the coordinate plane.

Examples Students: Page 152

Example 6 What are the coordinates of the point shown in the coordinate plane?

y 8 6 4 2

−8 −6 −4 −2 −2

Example 6

−4 −6

What are the coordinates of the point shown in the coordinate plane?

−8 y 8 6 4

Create a strategy

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the verticalx −8 −6 −4 −2 2 4 6 8 axis to identify the y-coordinate. −2 −4

Apply the idea

−6

The number on the horizontal axis directly above the point is 4 and across the y-axis is −6. So, the−8 coordinates are (4, −6).

Create a strategy

Example 7 Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical Purposeaxis to identify the y-coordinate. What are how the coordinates of the point shownof in the coordinate y Show students to read the coordinates a point on a plane? coordinate plane by tracking along the horizontal Give the coordinates in the form (x, y). and vertical axes. Apply the idea 5 The number on the horizontal axis directly above the point is 4 and across the y-axis is −6. So, the coordinates are

Students:(4,Page −6). 152

x −5

Example 7

5 −5

What are the coordinates of the point shown in the coordinate plane?

Give the coordinates in the form (x, y). 5

Create a strategy Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the −5 5 positive or negative direction.

Apply the idea

−5

The point is located 2 spaces to the left, then 1 space down. So, the coordinates are (−2, −1).

Create a strategy 316

Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the Mathspace Virginia SOL Grade 6 Teacher Edition positive or negative direction. mathspace.co

Apply the idea Virginia SOL Grade 6 152 Mathspace mathspace.co

−5

5 −5

Create a strategy Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the positive or negative direction.

Apply the idea The point is located 2 spaces to the left, then 1 space down. So, the coordinates are (−2, −1).

Purpose Show students how to correctly identify the coordinates of a point in a 2-dimensional space using a coordinate plane. 152

Mathspace

Virginia SOL Grade 6

mathspace.co Expected mistakes Students might mistakenly write the coordinates of the point as (−1, −2) instead of (−2, −1). This happens when they confuse the horizontal (x-axis) and vertical (y-axis) positions, assigning the x-coordinate to the vertical movement and the y-coordinate to the horizontal movement. They may think that the first number corresponds to moving up or down, and the second number corresponds to moving left or right. To address this misconception, remind students that the x-coordinate always represents how far they move left or right from the origin, and the y-coordinate represents how far they move up or down. Encourage students to start at the origin, move along the x-axis first to reach the correct horizontal position, and then move parallel to the y-axis to reach the point.

Students: Page 153

Example 8 Plot the point (−9, 3) on the coordinate plane.

Create a strategy

Apply the idea

The ﬁrst coordinate tells us how far to the right (positive) or left (negative) the point is from the origin.

Starting from the origin, go 9 units in the left direction and then 3 units in the upward direction.

The second coordinate tells us how far above (positive) or below (negative) the point is from the origin.

5 x −5

5 −5

Example 9 Purpose In which quadrant does the point (3, −2) lie? Show students how to correctly identify the coordinates of a point in a 2-dimensional space using a coordinate plane. Create a strategy Recall the characteristic of each quadrants: • 1st quadrant: positive x and positive y. • 2nd quadrant: negative x and positive y. • 3rd quadrant: negative x and negative y. • 4th quadrant: positive x and negative y.

Apply the idea Since the coordinates have positive x and negative y, the point (3, −2) lies in 4th quadrant.

Reﬂect and check

4.05 Integers in the coordinate plane We can plot the point (3, −2) to see which quadrant it is in: mathspace.co y 4

317

x −5

5 −5

Students: Page 153

Example 9 In which quadrant does the point (3, −2) lie?

Create a strategy Recall the characteristic of each quadrants: • 1st quadrant: positive x and positive y. • 2nd quadrant: negative x and positive y. • 3rd quadrant: negative x and negative y. • 4th quadrant: positive x and negative y.

Apply the idea

Reﬂect and check

Since the coordinates have positive x and negative y, the point (3, −2) lies in 4th quadrant.

We can plot the point (3, −2) to see which quadrant it is in: y 4 3 2 1 −4 −3 −2 −1 −1 −2

x 1

4 (3, −2)

−3 −4

Purpose Show students that they can identify the quadrant in which a point lies based on its coordinates in a Cartesian coordinate system. 4.05 Integers in the coordinate plane 153 mathspace.co

Using visual aids to reinforce quadrant identification on the coordinate plane Targeted instructional strategies

use with Example 9

Students often struggle with identifying the correct quadrant for a point due to confusion about the signs of the coordinates and the numbering of quadrants. Provide students with a clear, labeled diagram of the coordinate plane, highlighting each quadrant and noting the sign combinations for (x, y) in each one. Encourage students to create their own version of this diagram in their notes, including arrows to indicate the positive and negative directions of the axes. Explain that the quadrants are numbered in a counterclockwise direction starting from the upper right quadrant (Quadrant I). Reinforce this by guiding students through plotting points with different sign combinations and identifying their quadrants. For example, show that the point (3, −2) lies in Quadrant IV because it has a positive x-coordinate and a negative y-coordinate. To cater to visual learners, consider using a color-coding system where each quadrant is shaded a different color to help students visually differentiate them. Incorporate interactive activities like matching coordinate pairs to their corresponding quadrants or completing a table that lists the signs of x and y for each quadrant. Encourage students to refer back to their diagrams when determining the quadrant of a point, reinforcing the relationship between the signs of the coordinates and their quadrant locations.

318

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 154 Example 10 What is the distance between A (6, 8) and B (6, −4)?

Create a strategy Since the x-coordinates are the same, ﬁnd the difference of the y-coordinates.

Apply the idea Distance = 8 − (−4)

Example 10 =8+4

Reﬂect and check Subtract −4 from 8 Combine the adjacent signs

We can check the distance between the two points by graphing:

= 12 units Evaluate What is the distance between A (6, 8) and B (6, −4)?

9 y 8 7 6 Create a strategy 5 4 Since the x-coordinates are the same, ﬁnd the difference of the y-coordinates. 3 2 1

Apply the idea

Distance = 8 − (−4)

A (6, 8)

12 units x

Reﬂect and check

Subtract −4 from 8

=8+4

Combine the adjacent signs

= 12 units

Evaluate

1 2 3 4 5 6 7 8 9 −1−1 We can check the distance between the two points by −2 graphing: −3 −4 B (6, −4) −5 y 9 −6 A (6, 8) 8 7 6 5 4 3 12 units 2 1 x

Example 11 Purpose On which axis does point (0, −4) lie? 1 2 3 4 plane, 5 6 7 especially 8 9 −1 coordinate Show students how to calculate the distance between two points on−1a when the −2 x-coordinates the same. −3 Create are a strategy −4

−5 Plot the point (0, −4) on a coordinate plane to determine which axis it lies on. −6

B (6, −4)

Students: Page 154

Apply the idea

Example 11

Reﬂect and check If one of the coordinates is 0, then the point lies on the axis of the coordinate that is not 0.

y 4 3

On which axis does point (0, −4) lie? 2

Create a strategy

For point (0, −4), the x-coordinate is 0 and the y-coordinate is −4. Therefore, the point must lie on the y-axis.

−4 −3 −2 −1 1 2 3 4 Plot the point (0, −4) on a coordinate plane to determine which axis it lies on. −1

Apply the idea

−2

Reﬂect and check

−3

(0, −4) −4 y 4

If one of the coordinates is 0, then the point lies on the axis of the coordinate that is not 0.

For point (0, −4), the x-coordinate is 0 and the y-coordinate is −4. Therefore, the point must lie on the y-axis.

We can observe that the point 2 (0, −4) lies on the y-axis. 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

154

(0, −4)

Mathspace Virginia SOL Grade 6 mathspace.co

We can observe that the point (0, −4) lies on the y-axis.

154

Mathspace Virginia SOL Grade 6 mathspace.co

4.05 Integers in the coordinate plane mathspace.co

319

Purpose Show students that they can determine the axis on which a point lies based on its coordinates. Reflecting with students Ask students to explore additional points that lie on the y-axis by selecting different values for y while keeping x = 0, such as (0, 2), (0, −3), and (0, 5). Encourage them to plot these points on a coordinate plane to visualize their positions. Guide students to observe that all these points align vertically along the y-axis. Then, prompt them to identify what all these points have in common regarding their coordinates. Help them conclude that points on the y-axis always have an x-coordinate of 0. Similarly, have students consider points on the x-axis by keeping y = 0 and varying x, and recognize that these points lie horizontally along the x-axis. This reflection will help students generalize that a point lies on an axis if one of its coordinates is 0, deepening their understanding of the coordinate plane.

Students: Page 155

Idea summary The coordinate plane is divided into 4 quadrants. y-axis Quadrant 2

Quadrant 1 x-axis

Origin Quadrant 3

Quadrant 4

Points that lie on an axis, like (−5, 0) or (0, 4), are not in any quadrant. Points on the x-axis have a y-coordinate of 0. Points on the y-axis have an x-coordinate of 0. The point (0, 0) is the origin.

Practice What do you remember? Practice 1

Tell whether the integers below would be placed to the left or to the right of 0 on a number line.

Students: Pages 155–159 a −4 2

−11

−2

State whether each of the following statements is true or false.

The coordinate plane is used to describe the location of actual points, not regions, in a two-dimensional What do youa remember? space.

a 2

320

The number line laying down horizontally on the plane is the y-axis.

The coordinates b 7of the point at whichcthe x-axis 16 and the y-axis intercept d −11 are (0, 0).

The x-coordinate and y-coordinate of any point in the ﬁrst quadrant are both positive.

Tell whether integers belowalong would be placed to the left or to the right of 0 on a number line. c the The vertical distance the y-axis is the y-coordinate. −4

−2

State whether each of the following statements is true or false. a

The coordinate plane is used to describe the location of actual points, not regions, in a two-dimensional space.

The number line laying down horizontally on the plane is the y-axis.

The vertical distance along the y-axis is the y-coordinate.

The coordinates of the point at which the x-axis and the y-axis intercept are (0, 0).

The x-coordinate and y-coordinate of any point in the ﬁrst quadrant are both positive.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Select the coordinate plane that correctly labels the origin, axes, and 4 quadrants. A

y-axis Quadrant IV Origin (0, 0)

5 4 3 2 1

−5 −4 −3 −2 −1 −1

Quadrant II Origin (0, 0) −5 −4 −3 −2 −1 −1 −2 Quadrant III −3 −4 −5

y-axis Quadrant I

Quadrant I

Origin (1, 1)

x-axis

−5 −4 −3 −2 −1 −1

1 2 3 4 5

−2 Quadrant IV −3 −4 −5

Quadrant IV

Quadrant I

x-axis 1 2 3 4 5

−2 Quadrant III −3 −4 −5

Points with a negative y-coordinate.

Points with a negative x-coordinate.

Points where the x-coordinate and y-coordinate have the same sign.

Points with a negative x-coordinate and a positive y-coordinate.

Points with a negative x-coordinate and a negative y-coordinate.

Quadrant IV

5 4 3 2 1

Quadrant II

x-axis 1 2 3 4 5 Quadrant III

Match each coordinate point to the quadrant it belongs to: (7, 8)

Quadrant III

(−3, 4)

Quadrant IV

(2, −1)

iii

Quadrant I

(−5, −6)

Quadrant II

(0, 6)

On which axis do the following points lie? a

SOL

−5 −4 −3 −2 −1 −1

Quadrant II

5 4 3 2 1

State the quadrant(s) that have:

Origin (0, 0)

x-axis

y-axis

y-axis Quadrant II

Quadrant I

1 2 3 4 5

−2 Quadrant III −3 −4 −5

(0, −9)

(10, 0)

(−1, 0)

Which ordered pair best represents point A on the grid?

(−5, 4)

(3, 7)

(−6, 2)

(4, 5)

4 2

−8 −6 −4 −2 −2

x 2

−4 −6 −8

4.05 Integers in the coordinate plane mathspace.co

321

Consider the following coordinate plane: 4

Fill in the coordinates of the following points: a

2 1 B −4 −3 −2 −1 −1 C −2

B(⬚, ⬚) x, y

C(⬚, ⬚) x, y

A(⬚, ⬚) x, y

x 1

−3

D(⬚, ⬚)

−4

x, y

Let’s practice 9

SOL

The point (−8, 0) is located on the x-axis in the coordinate plane. a

Tell whether this statement is true or false.

Explain your thinking.

Graph and label the points on the coordinate plane below using the ordered pairs. 10 y 9 8 7 6 5 4 3 2 1 −10 −9 −8 −7 −6 −5 −4 −3 −2

SOL

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

2 3 4 5 6 7 8 9 10

A (4, 0)

B (0, 8)

C (4, 6)

D (7, 3)

E (7, 7)

F (0, 0)

G (−9, 10)

H (10, −2)

I (−6, −5)

J (−1, 0)

K (8, 2)

L (−8, 3)

Identify the ordered pair for each point graphed below in the coordinate plane.

D I

y 9 8 C 7 6 5 4 3 2 1

L −9 −8 −7 −6 −5 −4 −3 −2 −1−1 J −2 −3 −4 −5 −6 F E −7 −8 −9

322

x 1

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

x 1

SOL

Use the numbers −4, −1, 0, 2, 6 to create an ordered pair representing a point located on the y-axis.

Consider the given points on the number plane:

(⬚, ⬚)

Which graphed point is best represented by:

SOL

(9, −3)

(−4, −7)

(2, 4)

(−8, 8)

10 8 6 4 2

P x

−10−8 −6 −4 −2 −2 −4 −6 S −8 −10

2 4 6 8 10 Q

Rubiel placed the point of his marker on the origin of a regular coordinate plane. He marked a point after moving his marker 3 units to the right and 5 units down. Which ordered pair identiﬁes where Rubiel marked his point? A

SOL

(3, 5)

(3, −5)

(−3, −5)

Using the coordinate plane below, answer the following questions: a

Describe point P in terms of distance from the x- and y-axis.

Describe point M in terms of distance from the x- and y-axis.

Describe the distance from point Q to point S on their shared horizontal line.

Describe the distance from point N to point L on their shared vertical line.

N L R −12−10−8−6−4

(−3, 5) y 12 T 10 8 6 4 2

−2 −4 −6 Q −8 −10 −12

2 4 6 8 10 12

For each description below: i

Find the coordinate of the point.

Graph the point in a coordinate plane.

The point 9 units below the origin.

The point 3 units to the left of the origin.

The point 4 units to the left of (−3, 6).

The point 7 units to the right of (−1, −2).

The point 2 units to the right and 2 units below the point (2, 5).

The point 6 units to the left and 5 units above the point (4, −4).

4.05 Integers in the coordinate plane mathspace.co

323

Let’s extend our thinking 17

10 y 9 8 7 6 5 4 3 2 1

Graph the ordered pairs on the coordinate plane shown. a

(−3.5, 6.5)

(0, −4.7)

(1, 8.2)

(−5.9, −9)

−10 −9 −8 −7 −6 −5 −4 −3 −2

For each of the following:

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

x 1

2 3 4 5 6 7 8 9 10

Determine the coordinate of the ordered pair based on the description provided.

Plot the point on the coordinate plane.

The ordered pair is 8 units away from the y-axis.

The ordered pair is located in the third quadrant.

The y-value of the ordered pair is equal the x-value plus 5.

To get from the ordered pair to the coordinates (10, −3), you would move 18 units to the right.

The ordered pair is 3 units away from the x-axis.

Describe in words, how to get to the point (−9, 4) from the point (10, 6) in the coordinate plane.

The locations of different places around Linda’s town are represented in the coordinate plane.

y 800 m

Which place is the farthest vertically from the origin? Explain how you know.

600 m Park

400 m School

200 m

−400 m −200 m −800 m

x 200 m

400 m

−600 m Home

A student graphed the following ordered pairs in the coordinate plane: a

(5, −7)

(−6, −4)

(8, 1)

(−2, 0)

Find, correct, and explain the mistake the student made when graphing the ordered pairs.

324

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−400 m Mall

−200 m

10 8 A 6 4 2

2 4 6 8 10 −10−8 −6 −4 −2 −2 C D −4 −6 −8 −10

ensure the point is on the y-axis, and the second number can be any of the given numbers except 0.

Answers 4.05 Integers in the coordinate plane

13 a Q

What do you remember?

14 B

1 a To the left of 0.

b To the right of 0.

c To the right of 0.

d To the left of 0.

b False

c True

d True

3 B 4 a Third quadrant, Fourth quadrant b Second quadrant, Third quadrant c First quadrant, Third quadrant d Second quadrant

c Point Q is located at (−7, −8) and Point S is at (3, −8). Since they share the same y-coordinate, (−8), they are on the same horizontal line. The distance between them is the absolute difference between their x-coordinate: ∣3 − (−7)∣ = 10 units.

e Third quadrant b iv Quadrant II

c ii Quadrant IV

d i Quadrant III

6 a y-axis

b x-axis

c y-axis

d x-axis

d Point N is at (−12, 11) and point L is at (−12, 4). They share the same x-coordinate, (−12), meaning they are on the same vertical line. The distance between them is the absolute difference of their x-coordinate: ∣11 − 4∣ = 7 units.

7 C 8 a (2, 3)

b (−4, 0)

c (−2, −2)

d (3, −4)

Let’s practice 9 a The statement is true.

16 a i (0, −9)

b P oints on the x-axis have a y-coordinate of 0 because they are not up or down from the origin, but to the left or right. Since the point (−8, 0) has a y-coordinate of 0, it lies on the x-axis. The x-coordinate of −8 shows it is 8 units to the left of the origin on the x-axis. 10

L −10 −9 −8 −7 −6 −5 −4 −3 −2 I

11 a (6, 4)

b (3, 9)

10 y 9 B 8 7 6 5 4 3 2 J 1 F −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

ii −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11

x 1

K x

A 1

2 3 4 5 6 7 8 9 10 H

b i (−3, 0) ii

3 2 1

−4 −3 −2 −1

c (0, 7)

d (−9, 6)

−1

x 1

−2

e (−2, −8)

(−6, −7)

g (3, −2)

h (4, −7)

−3

(−5, 4)

(−7, −2)

k (8, −2)

(−2, 0)

−4

d R

b Point M is located at (10, −7) on the coordinate plane. This means that point M is 10 units away from the y-axis in the positive direction, placing it to the right of the y-axis. It is also 7 units away from the x-axis in the negative direction, placing it below the x-axis. Therefore, point M is in the fourth quadrant, where the x-coordinate is positive, and the y-coordinate is negative.

e True

5 a iii Quadrant I

c P

15 a Point P is located at (9, 9) on the coordinate plabe. This means that point P is 9 units away from the x-axis and 9 units away from the y-axis, placing it in the ﬁrst quadrant where both x- and y coordinates are positive.

e To the left of 0. 2 a True

b S

12 The ordered pair that represents a point on the y-axisusing the given numbers is (0, −4), (0, −1), (0, 2), (0, 6). Note: The ﬁrst number in the pair must be 0 to

Answers mathspace.co

325

Let’s extend our thinking

c i (−7, 6) ii

7 6 5 4 3 2 1

10 y 9 C 8 7 6 5 4 3 2 1

−8 −7 −6 −5 −4 −3 −2 −1−1

−10 −9 −8 −7 −6 −5 −4 −3 −2

−2 −3 −4 D

d i (6, −2) ii

y 2 1 −2 −1 −1

x 1 2 3 4 5 6 7

−3 −4

e i (4, 3)

18 a i T here are two possibilities: (8, y) or (−8, y), where y can be any value since the description does not specify the distance from x-axis.

3 2 1

4 3

A′ −8 −6 −4 −2 −1

2 1

−1

x 2 3 4 5 6 7 8 9 10

ii Choose any y-value for your point since the distance from the y-axis is speciﬁed but not from the x-axis. For example, (8, 0), start at the origin, move 8 units to the right, and plot your point there. If choosing (−8, 0), start at the origin, move 8 units to the left, and plot your point.

−2

1 −1 −2 −3 −4 −5 B −6 −7 −8 −9 −10

A x 2 4 6 8

−2 −3 −4

−2

b i I n this quadrant, both x- and y-values are negative. So, any ordered pair where x < 0 and y < 0 would satisfy and example of a coordinate pair (−x, −y) where x and y are positive numbers.

f i (−2, 1) ii

y 4 3 2 1 −4

−3

−2

−1

−1 −2

ii Since points in the third quadrant have both negative x- and y-coordinates, start at the origin, move left (negative x direction) and down (negative y direction) to plot a point like (−5, −5). 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 A −5 −6 −7

c i T his implies a direct relationship between y = x + 5. The speciﬁc coordinates aren’t given, so this can represent many possible points, e.g., (0, 5), (1, 6).

326

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

ii This description represents a linear relationship between x- and y-coordinates. To plot this, choose several x-values and calculate their corresponding y-values using y = x + 5. For example, if x = 0, y = 5; plot this point. If x = −5, y = 0 plot this as well. 7 y 6 5 A 4 3 2 1 x

B −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

d i M oving 18 units to the right means subtracting 18 from the x-coordinate of (10, −3). Therefore, the starting point would be (10 − 18, −3) = (−8, −3).

19 To get from the point (10, 6) to the point (−9, 4) in the coordinate plane, move 19 units to the left to change the x-coordinate from 10 to −9, and then move 2 units down to adjust the y-coordinate from 6 to 4. 20 The place farthest vertically from the origin is the Mall. This is because the Mall’s y-coordinate is −800 meters, which has the largest absolute value compared to the y-coordinates of the other locations, indicating it is the farthest in vertical distance from the origin. 21 The mistakes made when graphing the ordered pairs include reversing the signs and positions of the x- and y-coordinates. Correctly, Point A should be at (5, −7), Point B at (−6, −4), Point C at (8, 1), and Point D at (−2, 0). These corrections ensure each point accurately reﬂects its given ordered pair in the coordinate plane.

ii Based on the answer from part i, (−8, −3) is the point to plot. Start at the origin, move 8 units to the left and 3 units down to reach the starting point (−8, −3). Plot this point to represent the starting location before moving. 7 y 6 5 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

e i B eing 3 units away from the x-axis implies a y-coordinate of 3 or −3, while the x-coordinate can be any value. Thus, the pair could be (x, 3) or (x, −3), where x can be any value. ii Since the points are deﬁned by their distance from the x-axis, choose any x-value. If plotting a point like (0, 3), start at the origin and move 3 units upwards. To plot (0, −3), start at the origin and move 3 units downwards. The x-coordinate can vary, and you could also plot points like (5, 3) or (−4, −3), depending on the x-value you choose. y 4 3

2 1 −4 −3 −2 −1 −1 D

−2 −3

x 1 2 3 4 5 6 7 B

−4

Answers mathspace.co

327

Topic 4 Assessment: Operations with Integers 1

When Frankie woke up, the temperature was −4°C. During the day it rose by 11°C. Which number line correctly shows this temperature change? A

−5

Simplify the following expressions and represent the answer on a number line: a

−∣5 − 8∣

∣−100 − 55∣

∣−6 + 6∣

Marc claims both 6 and −6 have the same absolute value. Explain how he could be correct using a number line.

Find the value of each expression:

−3 + 4

−5 – 5

14 + (−14)

−5 ⋅ 5

175 + 50

−3 ⋅ −2

−4 (0)

12 ⋅ −2

−50 ÷ 5

−13 + 0

5 − 225

Select all of the numbers below that would give the following expression a negative answer:

−7

−14

−9

−7 − ⬚ = C

Fill in the blanks to demonstrate how to use counters to solve problems. Start with 4 positive counters.

Add __ negative counters.

4 + (−2)

So 4 + (−2) = __ .

328

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

What is the total number of counters?

________

2 + (2 − 2)

In a game of tug-of-war, a team wins by pulling the flag over its goal line. The flag begins at 0. During a game, the flag moves 8 ft to the right, 12 ft to the left, and 13 ft back to the right. Write an integer expression for the distance of the flag from 0.

Did a team win? Justify your answer.

How far could each team move the rope and in what direction in order for the game to be won in 2 tugs of the rope.

−25

−15

−10

−5

Use the signs <, > or = to determine which of each pair of numbers is a greater distance from 0: a

−20

Goal line

∣9∣ ⬚ ∣−2∣

∣−9∣ ⬚ ∣0∣

∣−7∣ ⬚ ∣−10∣

∣−1∣ ⬚ −∣30∣

Considering absolute value as the distance from zero, ﬁnd the possible values of x for each of the following: a

∣x∣ = 9

∣x∣ = 7

Martie wants to buy enough jelly beans that she can divide them evenly between 5 people and give each person 14 jelly beans. How many jelly beans should she buy in total?

Consider the coordinate plane shown: a

(9, 7)

(2, −2)

iii (−7, −4)

(0, 5)

y 9 8 7 J 6 E 5 4 3 C L 2 1 x H B I −9−8−7−6−5−4−3−2 −1 −1 1 2 3 4 5 6 7 8 9 G

State the letter that has the following coordinates:

Write the coordinates of each point: B

iii F

−2 −3 −4 −5 −6 −7 F −8 −9

Consider the three points: A (−2, 8), B (6, 6), and C (−5, 0). a

On the same coordinate plane: i

Highlight the x-axis in blue.

iii Plot A, B, and C on a coordinate plane. b

Which point is closest to the origin?

Which point is closest to the y-axis?

Highlight the y-axis in purple.

Identify the origin and label it with O.

Topic 4 Assessment: Operations with Integers mathspace.co

329

Which of the following is the best description of the point (0, 7) in the coordinate plane? A

Quadrant II

x-axis

y-axis

(−5, 5)

(6, −8)

In which quadrant do the following points lie? a

Quadrant I

(5, 7)

(−8, −2)

A vertical line is drawn through the point (−3, 4). Which of the following points lies on that line? A

(2, 4)

(2, −3)

(−3, −4)

(−2, −4)

Performance Task 16

The states of different substances change at different temperatures. Most commonly, we use the states of solid, liquid, and gas. The temperature at which a solid becomes a liquid is called its melting point. The melting point of ice into water is 0°C. Let’s explore how we can use absolute value to explore melting points.

330

Ice Lead Mercury Leather Plastic Carbon Dioxide Butter Glycerin

Melting Point (°C) 0 621 −39 550 130 −78 35 −16

Which substance has the highest melting point? Which has the lowest? Create a model to justify your reasoning.

Order each substance’s melting point from closest to farthest away from ice’s melting point.

The coldest temperature ever recorded in Virginia was –34°C on January 22, 1985. Would a mercury thermometer work in that region? Explain.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

moves 10 ft to the left, then moves 11 ft to the right, the ﬂag lands on the right goal line.

Topic 4 Assessment: Operations with Integers 1 C

6.CE.2a, 6.CE.2b, 6.CE.2d 8 a >

6.CE.2a

b >

9 a x = ±9

b x = ±7

6.NS.2d

−5 −4 −3 −2 −1 0 1 2 3 4 5

10 70 jellybeans

b 155

6.CE.2b, 6.CE.2d

25 50 75 100 125 150 175

11 a i J b i (5, 0)

c 0

ii D

iii K

iv E

ii (−2, 2)

iii (−3, −7)

iv (6, −6)

6.MG.3d

−5 −4 −3 −2 −1 0 1 2 3 4 5

12 a

9 8 A 7 6 5 4 3 2 1

d 3 −5 −4 −3 −2 −1 0 1 2 3 4 5

6.CE.2c 3 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

9 10

−5−4−3−2 −1 −1

6.NS.2d 4 a 1 e 225 −10

x O 1 2 3 4 5 6 7 8 9

b Point C b −10

c 0

d −25

g 0

h −24

−13

k −220

c Point A

6.MG.3a, 6.MG.3c, 6.MG.3e 13 D 6.MG.3b

6.CE.2b 5 A, C, and D 6.CE.2b Start with 4 positive counters.

−2 −3 −4 −5

Both 6 and −6 are the same distance from 0. Since absolute value is the distance from zero, both these integers have an absolute value of 6.

d >

6.NS.2d

2 a −3

c <

14 a 1st quadrant

b 3rd quadrant

c 2nd quadrant

d 4th quadrant

6.MG.3b What is the total number of counters?

Create 2 zero pairs

Add 2 negative counters.

15 C 6.MG.3e Performance Task 16 a Lead has the highest melting point.

4 So 4 + (−2) = 2.

4 + (−2)

2 + (2 − 2)

6.CE.2a, 6.CE.2b

Carbon Dioxide has the lowest melting point. This number line conﬁrms this because 621 is farthest to the right and −78 is farthest to the left. Carbon Glycerin dioxide Butter

7 a 0 + 8 − 12 + 13 b N o. The expression never reaches 10 or −10 which is the required distance from 0 for a team to win. c A nswers vary. Since the ﬂag is already 9 ft to the right, the team on the right should pull atleast 1 ft greater than the team on the left. For example, if the ﬂag

−100

Mercury

100

Plastic

200

300

400

Leather

Lead

500

600

Ice

b G lycerin, Butter, Mercury, Carbon Dioxide, Plastic, Leather, Lead c Yes. Mercury would be a liquid at this temperature. 6.NS.2a, 6.NS.2d, MP4, MP5

Answers mathspace.co

331

Ratios & Proportional 5 Relationships Big ideas If two quantities are proportional, the relationship can be represented in a variety of ways.

Chapter outline 5.01 5.02 5.03 5.04

Introduction to ratios (6.PFA.1) Equivalent ratios and ratio tables (6.PFA.1) Unit rates (6.PFA.2) Proportional relationships (6.PFA.2)

336 356 373 391

Equivalent ratios help in shopping! If 3 apples cost $6, then 6 apples will cost $12. The ratio of apples to dollars stays the same!

5. Ratios & Proportional Relationships Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 5.PFA.1 — The student will identify, describe, extend, and create increasing and decreasing patterns with whole numbers, fractions, and decimals, including those in context, using various representations.

6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

Big ideas and essential understanding If two quantities are proportional, the relationship can be represented in a variety of ways. 5.01 — A ratio is a comparison of two numbers or measurements.

5.03 — Unit rates provide a way to compare different quantities by expressing the ratio of two measurements with a denominator of one, making it easier to understand and compare different ratios.

5.02 — Equivalent ratios are useful in understanding real-world situations. Two ratios are equivalent if there is a nonzero number that can be multiplied by both quantities in one ratio to equal the corresponding quantities in the second ratio.

5.04 — A proportional relationship is a collection of equivalent ratios.

Standards 6.PFA.1 — The student will use ratios to represent relationships between quantities, including those in context. 6.PFA.1a — Represent a relationship between two quantities using ratios. 5.01 Introduction to ratios 6.PFA.1b — Represent a relationship in context that makes a comparison by using the notations a/b, a:b, and a to b. 5.01 Introduction to ratios

334

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.PFA.1c — Represent different comparisons within the same quantity or between different quantities (e.g., part to part, part to whole, whole to whole). 5.01 Introduction to ratios 6.PFA.1d — Create a relationship in words for a given ratio expressed symbolically. 5.01 Introduction to ratios 6.PFA.1e — Create a table of equivalent ratios to represent a proportional relationship between two quantities, when given a ratio. 5.02 Equivalent ratios and ratio tables

6.PFA.1f — Create a table of equivalent ratios to represent a proportional relationship between two quantities, when given a contextual situation. 5.02 Equivalent ratios and ratio tables

6.PFA.2c — Determine whether a proportional relationship exists between two quantities, when given a table of values, context, or graph. 5.04 Proportional relationships

6.PFA.2 — The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).

6.PFA.2d — When given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph. 5.03 Unit rates 5.04 Proportional relationships

6.PFA.2a — Identify the unit rate of a proportional relationship represented by a table of values, a contextual situation, or a graph. 5.03 Unit rates 6.PFA.2b — Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate. 5.03 Unit rates 5.04 Proportional relationships

6.PFA.2e — Make connections between and among multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs. 5.04 Proportional relationships

Future connections 7.PFA.1 — The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic functions.

5. Ratios & Proportional Relationships mathspace.co

335

5.01 Introduction to ratios Subtopic overview Lesson narrative In this lesson, students will be introduced to ratios and their various representations. They will explore how ratios compare two quantities, using different forms such as “a to b,” “a : b,” and as fractions. The lesson includes practical examples, visual models, and an exploration of part-to-part, part-to-whole, and whole-to-whole ratios. Students will practice writing ratios, interpreting them in real-world contexts, and using tape diagrams to solve problems. By the end, students should understand how to express and compare quantities using ratios.

Learning objectives

5.01 Introduction to ratios

Students: Page 162

After this lesson, you will be able to... • represent relationships as ratios using a/b, a:b and a to b notations. • make part to part, part to whole, and whole to whole comparisons. • write real-world situations to represent a ratio expressed symbolically.

The language of ratios

Key vocabulary Exploration

 part-to-whole  quantity part-to-part Consider the following image that shows ingredients for a batch of smoothies:  tape diagram  whole-to-whole 

 ratio

Essential understanding A ratio is a comparison of two numbers or measurements.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. 1. What kind of comparisons can you make between the number of strawberries and the number of bananas in the image? Mathematical process goals MPG1 — Mathematical Problem-Solving A ratio can compares the relationship two values. Itproblem-solving tells us how much there of one thing compared to with Teachers help students developbetween their mathematical skills byispresenting word problems another. In other words, a ratio is an association between two or more quantities. varying structures and information presentations. This approach includes problems with different unknown quantities, The ratiosteps, of strawberries bananas data, is 6 toand 4. The of bananas to strawberries is 4 to additional irrelevant to numerical dataratio presented in charts or diagrams. By6.doing so, students learn to identify relevant information, mathematical and there applyare appropriate We can also associate the twounderstand quantities bythe saying for every 6structure, strawberries, 4 bananas.problem-solving For every strategies. Forthere example, problems can be framed with given ratios, diagrams, irrelevant information, or multiple 4 bananas, are 6 ratio strawberries. steps to enhance students’ comprehension and flexibility in solving real-world mathematical problems. We can also write ratios in the form a : b which is read as “a to b”. If we want to describe the relationship between the number strawberries to the number of bananas, we could write the ratio as 6 : 4. Finally, ratios can be represented as a fraction. 336

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

We can describe the ratio of bananas to strawberries as the fraction .

The order that the words are written correspond to the order of the values in the ratio, so it is important that pay attention to the order.

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can encourage students to express their mathematical ideas and reasoning verbally and in writing. For example, when explaining the different notations for ratios, teachers can ask students to verbalize their understanding in their own words or write it down.

To integrate this goal, teachers can emphasize the connections between fractions and ratios, showing students how both are used to represent relationships between quantities. For example, they could demonstrate how a ratio of 3 to 5 can also be expressed as a fraction, 3/5.

MPG3 — Mathematical Reasoning Teachers can integrate this goal by creating activities that require logical reasoning. For example, they could ask students to justify why the ratio of apples to oranges in a basket can be represented in different ways but still means the same.

MPG5 — Mathematical Representations To help students understand ratios, teachers can incorporate various mathematical representations into their instruction. They should emphasize that ratios compare two quantities and can be written in different forms such as using a colon (a : b), the word “to” (a to b), or fraction notation

. By using counters, part-part-

whole mats, and real-world examples, teachers can illustrate part-to-part and part-to-whole relationships. Concrete activities, such as arranging items to match ratios, reinforce students’ understanding and correct common misconceptions.

Content standards 6.PFA.1 — The student will use ratios to represent relationships between quantities, including those in context.

6.PFA.1c — Represent different comparisons within the same quantity or between different quantities (e.g., part to part, part to whole, whole to whole).

6.PFA.1a — Represent a relationship between two quantities using ratios.

6.PFA.1d — Create a relationship in words for a given ratio expressed symbolically.

6.PFA.1b — Represent a relationship in context that makes a comparison by using the notations a/b, a:b, and a to b.

6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

6.PFA.2 — The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).

5.01 Introduction to ratios mathspace.co

337

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.01 Review: Simplify fractions and mixed numbers

Tools You may ﬁnd these tools helpful: • Manipulatives (beads, counters, cubes) • Highlighters

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Use a Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Engage students with hands-on activities using physical objects to explore ratios. Provide items like colored counters, blocks, or beads in two different colors. For example, give each student 4 red beads and 6 blue beads. Ask them to group and compare the quantities of each color. Encourage them to arrange the beads side by side to visually represent the ratio of red to blue beads. Representational: Guide students to represent the ratios through drawings and diagrams. Have them draw pictures of the beads they used, using colored pencils or markers to match the bead colors. For instance, they can draw 4 red circles and 6 blue circles. Introduce tape diagrams or bar models to visually represent the ratios. Assist them in labeling each part of the diagram with the corresponding quantity. Abstract: Teach students to express ratios using numbers and symbols. Show them the different notations: “a to b,” “a : b,” and “ .” Using the bead example, demonstrate how to write the ratio of red beads to blue beads as “4 to 6,” “4 : 6,” and “ .” Provide practice with real-world situations where they write ratios symbolically.

Three reads English language learner support Provide a variety of writing ratios problems for students to try, with different students receiving different problems. • On the first read of the problem, students should identify the values being directly compared in the problem • On the second read, students should find the order and units to compare the values. • On the third read, students should find any information about how to represent the answer and write their ratio.

338

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Switching the values Address student misconceptions It is common for students to make mistakes when writing ratios, including writing the values in the wrong order. If a ratio is written as “whole : part” instead of “part : whole” or as “part B : part A,” it may cause confusion or inaccuracies when using the ratio in calculations or comparisons. To help students remember the correct order of writing ratios, it may be helpful to provide examples and practice problems that emphasize the correct order of the values. For instance, you can give students a few simple ratio problems and ask them to identify the order of the values. You can also create word problems that require students to write ratios in the correct order to find a solution. Another strategy is to teach students to use the colon symbol (:) to separate the values in a ratio and to remember that the first value always represents the part being compared to the whole. You can also use visual aids, such as diagrams or bar models, to help students understand the concept of ratios and how they represent the relationship between different values. Finally, it is important to provide students with plenty of opportunities to practice writing and using ratios in various contexts. This can help build their confidence and fluency in using ratios correctly.

5.01 Introduction to ratios mathspace.co

339

Student lesson & teacher guide The language of ratios Students will be introduced to the concept of ratios as a comparison between two values. Starting with a simple exploration activity, they will exame an image of strawberries and bananas and make comparisons between the two. Students will learn that ratios can be represented in different ways including in the form a : b and as fractions.

Exploration Students: Page 162

5.01 Introduction to ratios After this lesson, you will be able to... • represent relationships as ratios using a/b, a:b and a to b notations. • make part to part, part to whole, and whole to whole comparisons. • write real-world situations to represent a ratio expressed symbolically.

The language of ratios Exploration Consider the following image that shows ingredients for a batch of smoothies:

What kind of comparisons can you make between the number of strawberries and the number of bananas in the image?

A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another. In other words, a ratio is an association between two or more quantities. The ratio of strawberries to bananas is 6 to 4. The ratio of bananas to strawberries is 4 to 6. Suggested student grouping: Individual In this activity, students arethe asked to observe an image and make comparisons theevery number of We can also associate two quantities by saying for every 6 strawberries, there are 4between bananas. For 4 bananas, are 6 present strawberries. strawberries and there bananas in the image. We can also write ratios in the form a : b which is read as “a to b”.

Ideal student responses If we want to describe the relationship between the number strawberries to the number of bananas, we could write These ideal responses the ratio as 6 : 4. may differ from other correct student responses. Less formal responses can be connected with thecan more precise mathematical Finally, ratios be represented as a fraction. language presented here.

1. WhatWekind of comparisons youtomake between the number can describe the ratio of can bananas strawberries as the fraction . of strawberries and the number of bananas in the image? The order that the words are written correspond to the order of the values in the ratio, so it is important that pay The attention numbertoofthe strawberries is greater than the number of bananas. There are 2 more strawberries order. than bananas. Additionally, the ratio of strawberries to bananas is 6 : 4 or 3 : 2, meaning that for every 6 strawberries, we need 4 bananas or for every 3 strawberries, we need 2 bananas.

340

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

162

Mathspace

Virginia SOL Grade 6

The language of ratios Exploration Consider the following image that shows ingredients for a batch of smoothies:

Purposeful questions • How many different ways can you represent the comparison of strawberries and bananas? Which way is the most efficient? Possible misunderstandings • Students may confuse the different forms of representing ratios. They may also simplify 6 : 4 to 3 : 2. Remind students that we are looking for the simplest way to compare the two quantities. 1.

What kind of comparisons can you make between the number of strawberries and the number of

Students: Page bananas 162 in the image? A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another. In other words, a ratio is an association between two or more quantities. The ratio of strawberries to bananas is 6 to 4. The ratio of bananas to strawberries is 4 to 6. We can also associate the two quantities by saying for every 6 strawberries, there are 4 bananas. For every 4 bananas, there are 6 strawberries. We can also write ratios in the form a : b which is read as “a to b”. If we want to describe the relationship between the number strawberries to the number of bananas, we could write the ratio as 6 : 4. Finally, ratios can be represented as a fraction. We can describe the ratio of bananas to strawberries as the fraction . The order that the words are written correspond to the order of the values in the ratio, so it is important that pay attention to the order.

Examples Students: Page 163 Mathspace Virginia SOL Grade 6 Example 1 mathspace.co 162

Write the ratio of butterﬂies to ladybugs in three ways.

Create a strategy Consider all the ways we talked about how to compare quantities using both words and symbols.

Apply the idea

Reﬂect and check

The ratio of butterﬂies to ladybugs is 3 to 8.

Consider whether 8 : 3 would describe the desired ratio.

Using ratio notation, we can also express the ratio as

This would mean that there are 8 butterﬂies for every 3 ladybugs. This does not match the image since there should be more ladybugs than butterﬂies. The order of the numbers in a ratio is important.

either 3 : 8 or .

Example 2 5.01 Introduction Lily describes the ratio between two quantities at the beach as . Use words to describe the relationship that the to ratios mathspace.co ratio may represent and draw a picture to represent the description of your ratio.

Create a strategy

341

Purpose Students demonstrate that they can express a ratio in different ways using both words and symbols, and understand the relationship between the quantities in a ratio. Create a strategy

Consider the ways we talked about how to compare quantities using both words and symbols. Reflecting with all students To support students in developing mathematical precision, emphasize the crucial role of the order of values in Apply the idea a ratio. Encourage students to always pay close attentionReflect to howand thecheck ratio is phrased—in this case, “butterflies The ratio of butterflies to ladybugs is 3 to 8. Consider whether 8 : that 3 would describe desired to ladybugs”—and ensure they list the number of butterflies first. Point out writing thethe ratio as 8ratio. to 3 would Using ratio notation, we can also express the ratio as This would mean that there are 8 butterflies for every represent “ladybugs to butterflies,” which conveys a different relationship. Use specific examples to illustrate 3 ladybugs. This does not match the image since there either 3 : 8 or . how switching the order changes the meaning of the ratio, potentially leading to misunderstandings. should be more ladybugs than butterflies. The order of

Students: Page 163

the numbers in a ratio is important.

Example 2 Lily describes the ratio between two quantities at the beach as . Use words to describe the relationship that the ratio may represent and draw a picture to represent the description of your ratio.

Create a strategy Remember that ratios describe the relationships between two quantities. In a fraction, the numerator describes one quantity and the denominator describes the other.

Apply the idea We have 5 of one quantity and 4 of another. At the beach, she could be noticing the number of sea shells and the number of sand dollars.

We could say there are 5 seashells for every 4 sand dollars; or the ratio of seashells to sand dollars is

or 5 : 4.

5.01 Introduction to ratios

163

mathspace.co Purpose Show students that they can interpret ratios in real-world contexts and visually represent these ratios.

Reflecting with students Students may think ratios always represent parts of a whole. They might think that the ratio

means one

quantity is a fraction of another or expect the quantities to sum up to a whole. This misunderstanding can make it challenging for them to interpret ratios where the numerator and denominator do not sum to a familiar total or when the fraction is greater than one. Emphasize the importance of context and explain to students that

in this context means there are 5 units of

one quantity (e.g., seashells) for every 4 units of another quantity (e.g., sand dollars). Use visual aids, such as drawing groups of 5 seashells alongside groups of 4 sand dollars, to illustrate the comparison between the two quantities. Encourage students to express ratios in different forms, like “5 to 4” or “5 : 4,” to help them distinguish ratios from fractions that represent parts of a whole.

342

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Advanced learners: Explore equivalent ratios with double number lines use with Example 2 Targeted instructional strategies Introduce double number lines as a visual tool to help students understand and represent the ratio . Draw two parallel number lines, one for each quantity in the ratio—for example, one for seashells and one for sand dollars. Mark increments on the seashells number line every 5 units and on the sand dollars number line every 4 units to reflect the given ratio. Challenge students to use the double number lines to find equivalent ratios by extending the lines and adding more increments. This will help them see how the quantities increase proportionally—when there are 10 seashells, there are 8 sand dollars; when there are 15 seashells, there are 12 sand dollars, and so on. This visual representation reinforces the concept of equivalent ratios and how the ratio describes a consistent relationship between the two quantities. Present real-world problems for students to solve using the double number lines, such as “If Lily found 20 seashells, how many sand dollars did she find?” They should locate 20 on the seashells number line and see the corresponding value on the sand dollars number line. This method supports students who may struggle with abstract concepts by providing a concrete visual that makes ratios more accessible and easier to interpret.

Students: Page 164

Idea summary A ratio compares the relationship between two quantities. It tells us how much there is of one thing compared to another. We can also write ratios in the form a : b which is read as “a to b” or write them as the fraction

Ratios as comparisons Idea summary

also use compare quantities by using part to part ratios, part to whole ratios, and whole to whole ratios. RatiosWeascanAcomparisons ratio compares the relationship between two quantities. It tells us how much there is of one thing compared These ratios can describe different relationships in the same scenario.

to another. Students will learn how to compare quantities using part to part ratios, part to whole ratios, and whole to whole ratios. Recall this recipe for smoothies: We can also write ratios in the form a : b which is read as “a to b” or write them as the fraction

Students: Pages 164–165

Ratios as comparisons We can also use compare quantities by using part to part ratios, part to whole ratios, and whole to whole ratios. These ratios can describe different relationships in the same scenario. Recall this recipe for smoothies: A part to part ratio describes the ratio between two parts of a whole. In this recipe, the number of strawberries and the number of bananas are both parts of the whole group of fruit. The ratio of strawberries to bananas is 6 : 4. A part to whole ratio describes the relationship between one quantity and the total group of quantities. For example, if we wanted to describe the ratio of bananas to the total amount of fruit in this recipe, we could say “for every 4 bananas, we need 10 total fruit for this batch of smoothies”. We could write it as 4 : 10. We now want to decide between two recipes for a batch of smoothies. Consider these two recipes: Recipe 1

Recipe 2

A part to part ratio describes the ratio between two parts of a whole. In this recipe, the number of strawberries and the number of bananas are both parts of the whole group of fruit. The ratio of strawberries to bananas is 6 : 4. A part to whole ratio describes the relationship between one quantity and the total group of quantities. For example, if we wanted to describe the ratio of bananas to the total amount of fruit in this recipe, we could say “for every 4 bananas, we need 10 total fruit for this batch of smoothies”. We could write it as 4 : 10. We now want to decide between two recipes for a batch of smoothies. Consider these two recipes: Recipe 1 Recipe 2 5.01 Introduction to ratios A whole to whole ratio is a ratio that compares the total of one quantity to the total of another. In this example, mathspace.co Recipe A has 10 total fruit, and Recipe B has 6 total fruit. So, the ratio of the total fruit in Recipe A to the total fruit in Recipe B is 10 : 6.

343

Recipe 2

A whole to whole ratio is a ratio that compares the total of one quantity to the total of another. In this example, Recipe A has 10 total fruit, and Recipe B has 6 total fruit. So, the ratio of the total fruit in Recipe A to the total fruit in Recipe B is 10 : 6. A part to part ratio can also compare between parts of one group to parts of another group. For example, for every 6 strawberries needed in Recipe A, we need 4 strawberries for Recipe B. This ratio can be written as 6 : 4 or .

164

344

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examples Students: Page 165

Purpose Students demonstrate that they can write a ratio comparing a part to a whole. Expected mistakes Students may write the ratio comparing the number of shaded trees to the number of unshaded trees rather than to the total number of trees. For example, they might count 2 shaded trees and 13 unshaded trees and write the ratio as 2 : 13 instead of 2 : 15. To address this misconception, emphasize the difference between part-to-part and part-to-whole ratios. Encourage students to carefully read the question and highlight phrases like “to the total number of trees.” Use the diagram to label the counts: 2 shaded trees, 13 unshaded trees, and 15 total trees. Additionally, it may be helpful for students to first write the ratio in words: Shaded trees : Total trees and then substitute the numerical values. Reflecting with students The complementary ratio would represent the ratio of the unshaded trees to the larger whole. Encourage advanced learners to develop a formula to find the complementary ratio without directly counting the unshaded trees. Challenge them to express the relationship mathematically. They may start by writing it verbally as: Number of Unshaded Trees = Total Trees − Shaded Trees and then as a formula: U = T − S. Then, have them formulate the complementary ratio as “Unshaded Trees to Total Trees”, which becomes U : T or (T − S) : T in formula form. By writing these relationships algebraically, students can generalize the strategy to any similar situation without needing to count each time. This exercise deepens their understanding of part-whole relationships and enhances their ability to abstract and generalize mathematical concepts. Encourage students to apply this formula to different problems, reinforcing their skills in creating and using mathematical models. By transitioning from concrete numbers to abstract formulas, they strengthen their problem-solving abilities and appreciate the utility of mathematical generalization.

5.01 Introduction to ratios mathspace.co

345

Students: Page 165

Purpose Students demonstrate that they can write a part to part ratio.

Students: Page 166

Purpose Students demonstrate that they can accurately count and compare quantities to form a whole to whole ratio.

346

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 166

Purpose Show students how to interpret and use a tape diagram to understand ratio relationships in real-world scenarios.

Students: Pages 166–167

Apply the idea In the tape diagram, Emma had 5 parts. We established that each part was worth 3 miles. Emma

In total, Emma’s weekly mileage is 5 ⋅ 3 = 15 miles.

Example 5 Purpose Thestudents ratio 3 : 4 represents a comparison between itemsmeaningful at a grocery store. Challenge to breakdown a given ratio into parts and use it to calculate an unknown value. a Create a context using this ratio in a part to part comparison.

Create a strategy A part to part ratio describes the ratio between two parts of a whole. Consider any two quantities that are “part” of the grocery store.

Apply the idea The ratio of oranges to apples in the grocery store is 3 : 4. This means for every 3 oranges, there are 4 apples in the grocery store.

Reﬂect and check

5.01 Introduction to ratios Remember that the ratio 3 : 4 does not mean there are only 3 oranges and 4 apples in the grocery store. mathspace.co

347

Use color-coding to enhance understanding of tape diagrams

use with Example 4

Student with disabilities support To support students who struggle with visual-spatial processing, use color-coding on tape diagrams and calculations to highlight the relationship between Emma’s and Riley’s mileages. Color each of Emma’s five parts one color (e.g., pink) and each of Riley’s three parts another color (e.g., yellow). This visual distinction helps students easily see the proportion of miles each runner completes. When working through the calculations, encourage students to use the same colors to highlight or underline the numbers associated with each runner. For example, calculating how many miles each part represents, students can write or highlight Emma’s Apply when the idea numbersInin pink and Riley’s numbers in yellow. Provide colored pencils or markers the tape diagram, Emma had 5 parts. We established that each part was worth 3 miles. so students can actively participate in shading the diagrams and annotating their calculations consistently. Emma

total, Emma’s Students:InPage 167 weekly mileage is 5 ⋅ 3 = 15 miles.

Apply the idea In the tape diagram, Emma had 5 parts. We established that each part was worth 3 miles. Example 5 Emma between 3 3 at a grocery 3 3 The ratio 3 : 4 represents a comparison items store.

a Create a context using this ratio in a part to part comparison. In total, Emma’s weekly mileage is 5 ⋅ 3 = 15 miles.

Create a strategy A part to part ratio describes the ratio between two parts of a whole. Consider any two quantities that are “part” of the grocery 5 store. Example

Apply the The ratio 3 :idea 4 represents a comparison between items at a grocery store. The ratio ofa oranges to apples theingrocery store 3 : 4. This means for every 3 oranges, there are 4 apples in the a Create context using this in ratio a part to partiscomparison. grocery store.

Create a strategy Reﬂect and check

A part to part ratio describes the ratio between two parts of a whole. Consider any two quantities that are “part” of Remember the ratio 3 : 4 does not mean there are only 3 oranges and 4 apples in the grocery store. the grocerythat store.

Apply the idea The ratio of oranges to apples in the grocery store is 3 : 4. This means for every 3 oranges, there are 4 apples in the grocery store.

Reﬂect and check

This group would also represent the ratio 3 : 4 because for every 3 oranges, there are 4 apples. Remember that the ratio 3 : 4 does not mean there are only 3 oranges and 4 apples in the grocery store. b Create a context using this ratio in a part to whole comparison.

Purpose Create a strategy Apply the idea Students demonstrate that they can interpret and apply the concept of ratios in real-world contexts.

A part to whole ratio describes the relationship between We could compare the amount of white bread to the total one quantity and the total group of quantities. amount of bread sold at the grocery store. group167 would also represent the ratio 3 : 4 because for every 3 oranges, there are 4 apples. Students:This Page The ratio 3 : 4 would mean that for every 3 white breads sold, there are 4 total breads sold. b Create a context using this ratio in a part to whole comparison. c Create a context using this ratio in a whole to whole comparison.

Create a strategy

Apply the idea

A part to whole ratio describes the relationship between Create a strategy one quantity and the total group of quantities. A whole to whole ratio is a ratio that compares the total of one quantity to the total of another.

We could compare the amount of white bread to the total Apply the idea amount of bread sold at the grocery store. We could compare the total number of frozen vegetables The ratio 3 : 4 would mean that for every 3 white breads to the total number of fresh vegetables. sold, there are 4 total breads sold. The ratio of 3 : 4 would mean that for every 3 frozen vegetables, there are 4 fresh vegetables.

c Create a context using this ratio in a whole to whole comparison.

Create a strategy 348

Apply the idea

5.01 Introduction to ratios mathspace.co

167

A whole to whole ratio is a ratio that compares the total of We could compare the total number of frozen vegetables Mathspace Virginia SOL Grade 6 Teacher Edition one quantity to the total of another. to the total number of fresh vegetables. mathspace.co The ratio of 3 : 4 would mean that for every 3 frozen vegetables, there are 4 fresh vegetables.

Apply the idea The ratio of oranges to apples in the grocery store is 3 : 4. This means for every 3 oranges, there are 4 apples in the grocery store.

PurposeReﬂect and check that thethat ratiothey 3 : 4 can doescreate not mean there are only part-to-whole 3 oranges and 4 ratios apples in grocery store. StudentsRemember demonstrate and interpret in the a real-world context. Reflecting with students Encourage your students to delve deeper into the ratio by calculating the percentage of white bread sold compared to the total bread sales. Ask them, “If the ratio is 3 : 4, what fraction or percentage of the total bread sold is white bread?” Guide them to see that of the bread sold is white bread, which equates to 75%. Then, group would also represent the ratio 3 : 4 because for every 3 oranges, areit’s 4 apples. initiate aThis discussion about the reasonableness of this scenario: “Do youthere think realistic that 75% of all bread sold in a grocery store is white bread? Why or why not?” This willbprompt them to consider real-world sales patterns and the diversity of consumer preferences. Create a context using this ratio in a part to whole comparison. Additionally, encourage them to adjust the ratio to reflect a more plausible scenario, perhaps 3 : 7 or 3 : 10, and discuss how changing the numbers affects the proportion Create a strategy Apply of thewhite idea bread sold. This exercise will help them understand howratio ratios represent real-world situations importance of of considering context A part to whole describes the relationship between We and couldthe compare the amount white bread to the totalwhen one quantity and the total group of quantities. amount of bread sold at the grocery store. interpreting mathematical relationships.

Students: Page 167

The ratio 3 : 4 would mean that for every 3 white breads sold, there are 4 total breads sold.

c Create a context using this ratio in a whole to whole comparison.

Create a strategy

Apply the idea

A whole to whole ratio is a ratio that compares the total of We could compare the total number of frozen vegetables one quantity to the total of another. to the total number of fresh vegetables. The ratio of 3 : 4 would mean that for every 3 frozen vegetables, there are 4 fresh vegetables.

5.01 Introduction to ratios mathspace.co

167

Purpose Students demonstrate that they can create a real-life context using a given ratio in a whole-to-whole comparison. Expected mistakes Students might think that the ratio 3 : 4 means there are exactly 3 frozen vegetables and exactly 4 fresh vegetables in the store, rather than understanding that the ratio represents a proportional relationship between the two quantities. They may not realize that the ratio can apply to any equivalent quantities that maintain the same relative proportions. To address this misconception, emphasize that ratios describe how two quantities relate to each other proportionally, and these quantities can be scaled up or down while maintaining the same ratio. Encourage students to create a table of equivalent ratios, such as 6 : 8, 9 : 12, and 12 : 16 to see how the relationship between frozen and fresh vegetables remains consistent.

Students: Page 168

Idea summary A ratio can represent different comparisons within the same quantity or between different quantities. • • •

part to whole - compares part of a whole to the entire whole part to part - compares part of a whole to another part of the same whole whole to whole - compares all of one whole to all of another whole

Practice What do you remember? 1

The ratio of vowels to consonants in a word is 4 to 9. Are there more vowels or consonants5.01 in the word? Introduction to ratios Explain. mathspace.co

Given a ratio of 3 cats to 5 dogs. a

What does the ﬁrst number in the ratio represent?

349

Practice Students: Pages 168–171

What do you remember? 1

The ratio of vowels to consonants in a word is 4 to 9. Are there more vowels or consonants in the word? Explain.

Given a ratio of 3 cats to 5 dogs.

What does the ﬁrst number in the ratio represent?

What does the second number in the ratio represent?

What is the ratio between 3 cats and 5 dogs?

You are comparing apples to oranges in a fruit bowl. Three-fourths of the fruit are apples. Is the fraction same as

Which of the following represent the ratio of cats to dogs? Select all that apply.

A E 5

the

? Explain.

9 cats to 5 dogs

5:9

5 cats to 9 dogs

D 9:5

5 : 14

Which ratio does not belong with the other three? Explain your reasoning. 3 parts to 7 parts

3 out of every 7

3 for each 7

3 for every 7

Let’s practice 6

Express the following ratios as word statements: a

4:3

3:4

5:7

6:9

10 : 2

7:8

1 : 12

11 : 15

Create a set of items with squares and stars. The set of items should represent a 2 : 3 ratio for the number of squares to the number of stars.

For each of the following diagrams:

350

Write the ratio of circles to triangles.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Write the ratio of triangles to circles.

For each of the following diagrams: i

Write the ratio of shaded squares to unshaded squares.

Write the ratio of shaded squares to total squares.

For each of the following diagrams: i

Write the ratio of shaded sections to unshaded sections.

Write the ratio of shaded sections to total sections.

Consider each ratio written as a fraction. In each fraction, the numerator represents one quantitiy and the denominator represents another quantity. Write a scenario that could match each ratio. a

Consider each scenario. i

Express each pair of quantities as a ratio written in the form a : b.

Express each pair of quantities as a ratio written as a fraction .

There are 11 cats to 6 dogs.

In a classroom there are 12 girls to 5 boys.

When baking a cake, you need 4 cups of sugar for every 5 cups of ﬂour.

There are 3 red marbles to 2 blue marbles.

The Great Dane weighed twice as much as the Labrador.

In a working day, a bricklayer lays 50 bricks every hour.

A student has two bags of sour candies. They separate them by color. Bag 1: Bag 1 has 24 total candies. There are 4 green candies, 6 blue candies, 12 red candies and 2 yellow candies. Bag 2: Bag 2 has 20 total candies. There are 4 green candies, 3 blue candies, 9 red candies and 4 yellow candies. Find each of the ratios: a

Green to red in Bag 1

Blue to red in Bag 2

Total in Bag 1 to total in Bag 2

Total yellow to the total in Bag 1 & Bag 2

5.01 Introduction to ratios mathspace.co

351

For each of the following diagrams: i

Find the fraction of the shaded region of chickens to the unshaded region of chickens.

Find the fraction of the shaded region of chickens to the total number of chickens.

When making pasta, you need 5 cups of plain ﬂour to 8 eggs. a

Write down the ratio of eggs to ﬂour.

Write down the ratio of ﬂour to eggs.

Use the table to write the ratio. Explain what the ratio means. a

dramas to movies

comedies to movies

movies : action

movies : dramas

Movie Drama Comedy Action

Number 5 6 4

A student describes the relationship between two groups of objects at school with the ratio 2 : 11. Use words to describe what the student may be describing.

In a bag, there are 20 marbles. If there are 5 green marbles, 6 blue marbles, 2 red marbles, and 7 yellow marbles, ﬁnd the ratios of the colored marbles below: a

352

Green to red.

Blue to red.

Green to blue.

Red to blue.

Blue to green.

Green to the total number of marbles.

Yellow to the total number of marbles.

Red to the total number of marbles.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s extend our thinking 19

The ratio of your monthly allowance to your friend’s monthly allowance is 6 : 4. The monthly allowances total $80. The ratio 6 : 4 is diplayed using a tape diagram. You

Your friend

How much money does each part represent? Justify your reasoning.

What is your allowance? Justify your reasoning.

iii

What is your friend’s allowance? Justify your reasoning.

Marisol is comparing 2 different cookie recipes. The first recipe call for 3 cups of flour. The second recipe calls 2 cups of flour. She has 15 cups of flour and wants to make the same number of batches of each recipe. a

Draw a tape diagram that could represent this scenario.

How many batches of each recipe can she make? Justify your reasoning.

A fruit salad contains 4 apples, 6 oranges, and 8 strawberries. Explain what the ratio 6 : 18 would represent.

Identify all of the ratios that could be used to represent the number of lawns that could be mowed in each number of hours. Eddie can mow 4 lawns in 7 hours 4:7

4 to 7

8 to 14

2:3

1:2

12 : 21

A rectangular pool has a length of 12 meters and a width of 6 meters. Lucidia states that the ratio of the length to the width of the pool is 12 : 6. Annalee states that the ratio of the width to the length of the pool is 6 : 12. Determine who is correct and explain your reasoning.

A bridge is made up of 3 lengths,

, and

Find the total length of the bridge.

Write the lengths of

as a ratio.

Write the lengths of

as a ratio.

Write the lengths of

, as shown in the ﬁgure: 1m P

4m Q

7m R

as a ratio.

James and Emma scored goals in their basketball game in the ratio 4 : 3. a

Find the fraction of the goals scored by James.

Find the fraction of the goals scored by Emma.

Harris and Bella divided chocolates between them in the ratio 5 : 4. a

Find the fraction of chocolates that Bella gets.

Find the fraction of chocolates that Harris gets.

If there are 27 chocolates, describe how you might ﬁnd how many chocolates Harris will get?

5.01 Introduction to ratios mathspace.co

353

Answers

12 a i 11 : 6

5.01 Introduction to ratios

b i 12 : 5

What do you remember?

c i 4 : 5

d i 3 : 2

e i 2 : 1

f i 50 : 1

1 There are more consonants than vowels in the word. The ratio of vowels to consonants is 4 : 9, which means that for every 4 vowels, there are 9 consonants. Since 9 is greater than 4, the number of consonants is greater than the number of vowels. 2 a The number of cats being compared. b The number of dogs being compared.

13 a The ratio of green to red marbles is

c 3 : 5

b The ratio of blue to red marbles is .

3 No, the fraction is not the same as . The fraction represents three parts out of four, which is less than one whole part. On the other hand, represents four parts out of three, which is more than one whole part and thus represents a larger quantity than . 4 A, B, F 5

3 parts to 7 parts

3 out of every 7

marbles is

14 a i

3 for every 7

b i

✓

c i

d i

Let’s practice b Three to four

c Five to seven

d Six to nine

e Ten to two

g One to twelve

h Eleven to ﬁfteen

Seven to eight

d The ratio of yellow marbles to the total number of

3 for each 7

✓

6 a Four to three

c The ratio of green to blue marbles is

15 a 8 : 5

b 5:8

16 a The ratio of dramas to movies is

. This means that

out of every 15 movies, 5 is a drama. b The ratio of comedies to movies is

. This means that

more than half of the movies, speciﬁcally 6 out of 15, are comedies. 8 a i 3 : 15

ii 15 : 3

b i 6 : 10

ii 10 : 6

9 a i 13 : 7

ii 13 : 20

b i 9 : 11

ii 9 : 20

10 a i 11 : 4

ii 11 : 15

b i 14 : 2

ii 14 : 16

c i 15 : 1

ii 15 : 16

11 a Answers vary. 5 candies for $7 b Answers vary. 3 miles every 20 minutes c Answers vary. 13 miles per 1 hour d 12 donuts for $8 seconds

354

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

c T he ratio of movies to action movies is 15 : 4. This indicates that for every 15 movies in the collection, 4 are action movies. d T he ratio of movies to dramas is 15 : 5, which means for every 15 movies, 5 are dramas. 17 This could refer to various scenarios at school, such as having 2 pencils for every 11 pens, 2 books for every 11 notebooks, or any other comparison between two different groups of objects where the ﬁrst group has 2 items for every 11 items in the second group. 18 a 5 : 2 e 6 : 5

b 6:2

c 5:6

d 2:6

5 : 20

g 7 : 20

h 2 : 20

Let’s extend our thinking 19 i

ecause there are 10 total parts, you know that 1 part B represents $80 ÷ 10 = $8.

ii 6 parts represent $8 ⋅ 6 = $48.

So, your monthly allowance is $48.

iii 4 parts represent $8 ⋅ 4 = $32.

4:7

4 to 7

8 to 14

12 : 21

23 Both Lucidia and Annalee are correct. The order in which they state the length : width and the width : length are stated correctly. 24 a 12 m

b 1:7

25 a

26 a

c 1 : 12

d 1:4:7

So, your friend’s monthly allowance is $32.

20 a

Recipe 1 15 total cups of flour Recipe 2

b T here are 5 total parts in the diagram and 15 cups of ﬂour.

c T here are 9 total parts in the ratio. There are three times that many chocolates. If we multiply 3 by 5, which is Harris’ part, we get “15 chocolates for Harris.

= 3, so she could make 3 batches of each recipe.

21 The ratio of the number of oranges to the total number of fruits in the salad can be written as “oranges : total fruits”. Since the total number of fruits is 4 + 6 + 8, the ratio can be written as 6 : 18.

Answers mathspace.co

355

5.02 Equivalent ratios and ratio tables Subtopic overview Lesson narrative In this lesson, students will learn about equivalent ratios and ratio tables. They will explore how to create equivalent ratios by multiplying or dividing both terms of a ratio by the same number. The lesson includes practical examples and a demonstration of simplifying ratios. Students will also learn to use ratio tables to solve problems and find unknown values by maintaining proportional relationships. By the end, students should be able to create a table of equivalent ratios to represent a proportional relationship when given a ratio or a contextual situation.

Learning objectives Students: Page 172

Key vocabulary 

equivalent fractions



equivalent ratios



greatest common factor (GCF)



proportion



ratio table



simplified ratio

Essential understanding Equivalent ratios are useful in understanding real-world situations. Two ratios are equivalent if there is a nonzero number that can be multiplied by both quantities in one ratio to equal the corresponding quantities in the second ratio.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal into the instruction of equivalent ratios by presenting students with complex problems that require the understanding and application of equivalent ratios to solve. For example, teachers can provide word problems that require students to find missing values in ratio tables or to determine if two ratios are equivalent. 356

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can help students develop mathematical connections by guiding them through the process of creating tables of equivalent ratios to represent proportional relationships. This involves identifying the given ratio, creating equivalent ratios by multiplying or dividing both sides of the ratio by the same number, and repeating this process until the table is filled. MPG3 — Mathematical Reasoning Emphasizing multiplicative thinking over additive thinking Teachers can facilitate mathematical reasoning in their helps students understand proportional relationships. instruction by prompting students to justify their answers. Additionally, discussing real-world applications of ratios, For example, when students complete a ratio table, comparing fractions and ratios, and translating ratios teachers can ask them to explain how they knew which between symbolic and word forms can deepen students’ number to multiply or divide by to find the missing values. understanding and connections. Teachers can foster mathematical communication by encouraging students to explain their thought processes when simplifying ratios or creating equivalent ratios. Teachers can ask questions like “Why did you choose to divide by that number?” or “Can you explain how you knew these two ratios were equivalent?”

MPG5 — Mathematical Representations Teachers can meet this goal by using a variety of representations to teach equivalent ratios. For example, using ratio tables to visually show how equivalent ratios are created, or using real-world examples to contextualize the concept of ratios. Teachers can also ask students to represent their solutions in different ways, such as through diagrams, equations, or written explanations.

Content standards 6.PFA.1 — The student will use ratios to represent relationships between quantities, including those in context.

6.PFA.1e — Create a table of equivalent ratios to represent a proportional relationship between two quantities, when given a ratio.

6.PFA.1f — Create a table of equivalent ratios to represent a proportional relationship between two quantities, when given a contextual situation.

6.NS.1 — The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

Future connections 6.PFA.2 — The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 1.01 Review: Simplify fractions and mixed numbers Grade 6 — 5.01 Introduction to ratios

5.02 Equivalent ratios and ratio tables mathspace.co

357

Tools You may ﬁnd these tools helpful: • Highlighters • Laminated double number lines

Student lesson & teacher guide Equivalent ratios Students will learn about equivalent ratios and how they are useful in real-world situations They will also explore the concept of simplified ratios and how to find them by identifying the greatest common factor.

Students: Pages 172–173

358

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A special type of equivalent ratio is a simplified ratio. Simplified ratio A ratio that has no equivalent ratio with smaller integer values. The two integers in the ratio have a greatest common factor of 1. Therefore, all ratios equivalent to the simplified ratio have to be multiples of it. To simplify a ratio, we can identify the greatest common factor (or the largest integer that can evenly divide both numbers in the ratio) and divide both parts of the ratio by it. Let’s say we have a recipe for 5 cakes using 25 cups of flour. The ratio of the number of cakes to the number of cups of flour is 5 : 25. To simplify the ratio we can see that both 5 and 25 can be evenly divided by 5.

The simpliﬁed ratio tells us that one cake requires 5 cups of ﬂour. This is very useful information for planning ingredients for lots of cakes.

Example 1

Aligning The ratio of and tablesMatching to chairs is 1 : 2. If there are 14 chairs, how many tables are there? Targeted instructional strategies Create a strategy

Two core skills of equivalent ratios are finding equivalent fractions and writing ratios from word problems. Multiply the both sides of ratio by a number to get the equivalent ratio.

Begin by having students write ratios using words, then vertically beneath fill in the ratio and the given ApplyThis the could idea include examples like “1 apple to 3 oranges” and “2 blue balls to 5 red balls.” information. The ratio 1 : 2 says that each table two chairs. Ask students to use multiplication tohas change one part of the ratio, such as changing 2 blue balls to 6 blue balls. 1 : 2 = ⬚ : 14

Rewrite the equivalent ratio

Have students make the same change to the other part of the ratio. Use this idea to have students practice 1:2=1⋅⬚:2⋅⬚ What number should be multiplied to 2 to become 14 changing the ratios to different values. 1:2=1⋅7:2⋅7 = 7 : 14

Multiply by 7

Evaluate

So, if there are 14 chairs, then we will need 7 tables.

Hands-on manipulatives to explore equivalent ratios Student with disabilities support Introduce students2 to physical manipulatives like colored counters or linking cubes to explore equivalent Example ratios. Provide two different colors to represent each term of the ratio—for instance, red for the first term and The ratio of players to teams is 60 : 10. If there are only 12 students present, how many teams can be made? blue for the second. Have students build a model of a given ratio, such as 2 red counters to 3 blue counters. Then, guide them to create new sets by multiplying or dividing both quantities by the same number, showing Create a strategy ratios like 4 red to 6 blue or 1 red to 1.5 blue. This concrete activity helps students visualize how scaling both Divide both sides of the ratio by a number to get the equivalent ratio. terms maintains the ratio’s equivalency. Encourage students to verbalize their process using language like, “I multiplied parts by 2 to get 4 and 6.” Applyboth the idea 60 : 10 = 60 ÷ 5 : 10 ÷ 5

Divide by 5

= 12 : 2

Evaluate

So, if there are 12 students, then there will be 2 teams.

5.02 Equivalent ratios and ratio tables 5.02 Equivalent ratios and ratio tables 173 mathspace.co mathspace.co

359

numbers in the ratio) and divide both parts of the ratio by it. Let’s say we have a recipe for 5 cakes using 25 cups of flour. The ratio of the number of cakes to the number of cups of flour is 5 : 25. Tospecial simplifytype the of ratio we can see 5 and 25 can be evenly divided by 5. A equivalent ratiothat is aboth simplified ratio.

Examples Simpliﬁed ratio

simpliﬁed ratio tells us that one cake requires 5 cups of ﬂour. This is very useful information for planning Students:The Page A ratio173 that has no equivalent ratio with smaller integer values. ingredients for lots of cakes.

The two integers in the ratio have a greatest common factor of 1. Therefore, all ratios equivalent to the simplified ratio Example 1 have to be multiples of it. To simplify a ratio, we can identify the greatest common factor (or the largest integer that can evenly divide both The ratio of tables to chairs is 1 : 2. If there are 14 chairs, how many tables are there? numbers in the ratio) and divide both parts of the ratio by it. Let’s sayawe have a recipe for 5 cakes using 25 cups of flour. The ratio of the number of cakes to the number of cups Create strategy of flour is 5 : 25. Multiply the both sides of ratio by a number to get the equivalent ratio. To simplify the ratio we can see that both 5 and 25 can be evenly divided by 5.

Apply the idea The ratio 1 : 2 says that each table has two chairs. The simpliﬁed ratio tells us that one cake requires 5 cups of ﬂour. This is very useful information for planning 1 : 2 = ⬚ : 14 Rewrite the equivalent ratio ingredients for lots of cakes. 1:2=1⋅⬚:2⋅⬚ What number should be multiplied to 2 to become 14 1:2=1⋅7:2⋅7

Example 1

= 7 : 14

Multiply by 7

Evaluate

So, there 14 chairs, then needare 7 tables. Theifratio ofare tables to chairs is 1we : 2.will If there 14 chairs, how many tables are there?

Create a strategy Multiply the both sides of ratio by a number to get the equivalent ratio.

PurposeExample 2 the idea Write anApply equivalent ratiotogiven initial The ratio of players teamsan is 60 : 10. Ifcomparison there are onlyof 12values. students present, how many teams can be made? The ratio 1 : 2 says that each table has two chairs.

Reflecting with students 1 : 2 = ⬚ : 14 Rewrite the equivalent ratio Create a strategy Finding Divide equivalent ratios is only possible astolong as the of tables and chairs can keep the same 2 = 1 of ⋅ ⬚the : 2ratio ⋅ ⬚ by a number What number benumber multiplied both1 :sides get should the equivalent ratio. to 2 to become 14 relationship. Encourage advanced learners, or any students who are ready, to find values of tables or chairs that 1:2=1⋅7:2⋅7 Multiply by 7 would not allow for the relationship to stay equivalent. Apply the idea = 7 : 14 Evaluate 60 ÷ 5then : 10 we ÷ 5will need Divide by 5 So, if there60 are: 10 14 =chairs, 7 tables. = 12 : 2 Evaluate

Students: Page 173

So, if there are 12 students, then there will be 2 teams.

Example 2 The ratio of players to teams is 60 : 10. If there are only 12 students present, how many teams can be made? 5.02 Equivalent ratios and ratio tables mathspace.co

Create a strategy

173

Divide both sides of the ratio by a number to get the equivalent ratio.

Apply the idea 60 : 10 = 60 ÷ 5 : 10 ÷ 5

Divide by 5

= 12 : 2

Evaluate

So, if there are 12 students, then there will be 2 teams.

Purpose 5.02 Equivalent ratios and ratio tables 173 mathspace.co Challenge students to use a given ratio to find an equivalent relationship using division instead of multiplication.

360

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 174 Example 3 Simplify the ratio 10 : 24.

Create a strategy Simplify the ratio by dividing each part by a common factor to ﬁnd the equivalent ratio.

Apply the idea Divide by a common factor Evaluate

Example 4 Purpose StudentsWrite demonstrate that they cansimpliﬁed simplifyratio. a given ratio using common factors. 54 cents to $3.00 as a fully Reflecting witha students Create strategy Example 3 Encourage students to explore real-life scenarios where simplifying a ratio is beneficial. For instance, discuss Convert the dollar value into cents, then divide by a common factor to simplify. how in cooking, reducing a recipe’s ingredient ratios simplifies scaling it up or down for different serving sizes. Simplify the ratio 10 : 24. Ask them to consider Apply the idea situations like mixing solutions or creating models, where maintaining the same proportion Create a simplified strategy is crucial, and= a100 ratio makes calculations more straightforward. $1.00 cents Simplify the byabout dividingexamples each part by adollar common factor to ﬁnd the equivalent ratio. ratios (like scale 1 : 100) help in Guide students to ratio think such as value map reading, where simplified Convert into cents understanding distances easily. By identifying these situations, students will appreciate how simplifying ratios Divide by 6 Apply the idea not only makes mathematical computations easier but also enhances their ability to visualize and interpret Evaluate Divide bycontexts. a common factor relationships between quantities in various

Students: Page 174 Idea summary

Evaluate

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other Example 4 ratio. A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values. Write 54 cents to $3.00 as a fully simpliﬁed ratio.

Create a strategy

Ratio tables Convert the dollar value into cents, then divide by a common factor to simplify. We can use a ratio table to represent a series of equivalent ratios.

Apply the idea For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of ﬂour, we could write this as a $1.00 ratio: 2= :100 1. cents

Convert dollar value into cents

In a ratio table we have: Sugar Flour

2 1

4 2

6 3

8Divide by 6 4 Evaluate

We can also use a ratio table to help us determine unknown values. For example, if we wanted to ﬁnd out how much ﬂour is needed when we use 12 tablespoons of brown sugar, we have the following: SugarIdea2summary 4 6

Purpose Flour ⬚the ratios can be increased or decreased by some multiple to be equal to 1 2 equivalent 3 Two ratios are if4one of Challenge students toratio. perform extra step of converting cents to dollars before simplifying the ratio. the other 174

Mathspace

Virginia SOL Grade 6

A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values.

mathspace.co Expected mistakes Students may simplify without first converting to common units resulting in the incorrect ratio 18 : 1. Emphasize to students the importance of converting all quantities to the same unit before simplifying a Ratio tables ratio. Encourage students to identify the units involved and decide which unit would be most practical for We can use a ratio table to represent a series equivalent ratios. the calculation—typically converting dollars to of cents or vice versa. Walk them through the example by first

For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of ﬂour, we could write this as a ratio: 2 : 1. In a ratio table we have: Sugar Flour

2 1

4 2

6 3

8 4

5.02 Equivalent ratios and ratio tables mathspace.co

361

Example 4 Create a strategy Simplify ratio dividing eachsimpliﬁed part by a ratio. common factor to ﬁnd the equivalent ratio. Write 54the cents to by $3.00 as a fully

Apply idea Createthe a strategy

converting $3.00 to 300 cents, so the ratio becomes 54 cents to 300 cents. This makes it clear that both by a common factor factor to simplify. Convert the dollar value into cents,Divide then divide by a common quantities are now comparable and can be simplified accurately. Evaluate

Apply the idea Reflecting with students $1.00 = 100 cents Lead students in a discussion of the advantages and disadvantages of converting to cents to dollars versus Convert dollar value into cents dollars to cents. Example 4 Divide by 6

Students:Write Page 174 54 cents to $3.00 as a fully simpliﬁed Evaluate ratio. Create a strategy Convert the dollar value into cents, then divide by a common factor to simplify.

Idea summary

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio. $1.00 = 100 cents A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values. Convert dollar value into cents

Apply the idea

Divide by 6

Ratio tables

Evaluate

We can use a ratio table to represent a series of equivalent ratios.

Ratio tables For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of ﬂour, we could write this as a ratio: 2 Idea : 1. how summary Students will learn to use a ratio table to represent a series of equivalent ratios and to determine unknown Two ratios equivalent if one of the ratios can be increased or decreased by some multiple to be equal to In a ratio table we are have: values. the other ratio. Sugar 2 4 6 8 A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values. Students: Pages 174–175 Flour 1 2 3 4

We can also use a ratio table to help us determine unknown values. For example, if we wanted to ﬁnd out how much ﬂour is needed when we use 12 tablespoons of brown sugar, we have the following:

Ratio tables

Sugar 2 4 6 8 12 We can use a ratio table to represent a series of equivalent ratios. ⬚ Flour 1 2 3 4 For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of ﬂour, we could write this as a ratio: 2Mathspace : 1. 174 Virginia SOL Grade 6 mathspace.co

In a ratio table we have: Sugar Flour

2 1

4 2

6 3

8 4

2 1

4 2

6 3

8 4

12 ⬚

We can determine the corresponding amount of ﬂour to 12 tablespoons of brown sugar by ﬁnding equivalent ratios. 174

Mathspace Virginia SOL Grade 6 mathspace.co 6 : 3 = 12 : ⬚ Equivalent ratios

6:3=6⋅2:3⋅2

Multiply both parts of the ratio by 2

= 12 : 6

Therefore, for every 12 tablespoons of brown sugar, we can use 6 cups of ﬂour. You may have noticed that there was another way to ﬁnd this using the table. Sugar Flour

2 1

4 2

6 3

8 4

12 ⬚

We can see in the table that for 4 cups of ﬂour we need 2 tablespoons of brown sugar, and for 8 cups of ﬂour we need 4 tablespoons of brown sugar. We know that 4 + 8 = 12 so we could have added 2 + 4 to get the 6 tablespoons of brown sugar.

Example 5 The table shows the ratio of dogs to cats:

362

Dogs to Cats 9 : 5 18 : 10 Mathspace Virginia SOL Grade 6 Teacher Edition 27 : mathspace.co 45 : : 50

Information gap English language learner support Create an information gap activity to engage students in using ratio tables and mathematical language. Prepare pairs of cards where each student has only partial information about a ratio table related to a real-world situation. For example, one student has the first column filled out, and the other has the second column filled We can determine the corresponding amount of ﬂour to 12 tablespoons of brown sugar by ﬁnding equivalent ratios. out. Pair up the students and explain that they need to complete their ratio tables by asking their partner for the 6 : 3 = 12 : ⬚ Equivalent ratios missing information. Encourage them to use key vocabulary such as “ratio,” “equivalent ratios,” “proportional 6:3=6⋅2:3⋅2 Multiply both parts of the ratio by 2 relationships,” and “scaling up or down” in their conversations. = 12 : 6

Model questions they might ask, like “What value have for the second quantity when the first is 4?” or Therefore, for every 12 tablespoons of brown sugar,do weyou can use 6 cups of flour. “How did you find that equivalent ratio?” This activity creates a purpose You may have noticed that there was another way to find this using the table. for communication and helps students practice both the mathematical concepts and the related English vocabulary. By requiring them to exchange Sugar 2 4 6 8 12 information verbally, you2support their language development in the context of learning about equivalent ratios. ⬚ Flour 1 3 4 We can see in the table that for 4 cups of flour we need 2 tablespoons of brown sugar, and for 8 cups of flour we need 4 tablespoons of brown sugar.

Students: Page 175

We know that 4 + 8 = 12 so we could have added 2 + 4 to get the 6 tablespoons of brown sugar.

Example 5 The table shows the ratio of dogs to cats: Dogs 9 18 27 45

to : : : : :

Cats 5 10

a Complete the table of equivalent ratios.

Create a strategy We can ﬁnd the equivalent ratios by multiplying or dividing both sides of a ratio by the same value.

Apply the idea 9:5=9⋅3:5⋅3

Dogs 9 18 27 45 90

to : : : : :

Multiply by 3

= 27 : 15

Evaluate

=9⋅5:5⋅5

Multiply by 5

= 45 : 25

Evaluate

= 9 ⋅ 10 : 5 ⋅ 10

Multiply by 10

= 90 : 50

Evaluate

Cats 5 10 15 25 50

Purpose Show students how to complete ratio tables by finding equivalent ratios. 5.02 Equivalent ratios and ratio tables mathspace.co

175

5.02 Equivalent ratios and ratio tables mathspace.co

363

Reflecting with students Encourage students to explore and articulate the constant ratio relationship between dogs and cats in the table. Ask them to express this relationship as an equation, such as Cats = ⋅ Dogs, and discuss how this equation models the proportional relationship represented in the table. Prompt students to use this equation to calculate the number of cats for any given number of dogs, including the 270 dogs scenario from the problem. Additionally, have students consider how they could reverse the equation to find the number of dogs if the number of cats is known, reinforcing the concept of inverse operations in proportional relationships. This reflection will help students generalize the pattern observed and apply their understanding of ratios and proportions to a variety of situations.

Students: Page 176

Purpose Challenge students apply their understanding of ratios to predict and calculate values in a proportional relationship.

Recognizing repetition: Generalize equivalent ratios

use with Example 5

Targeted instructional strategies Students might find it challenging to see how multiplying both quantities in a ratio by the same number generates equivalent ratios. Begin by posing the initial problem and have students complete the ratio table individually, encouraging them to observe any patterns in the numbers. Ask them to note how each entry in the “Dogs” column relates to the previous one, and similarly for the “Cats” column. Facilitate a class discussion where students share the patterns they’ve noticed, such as each number in the “Dogs” column being a multiple of 9 and each in the “Cats” column being a multiple of 5. Guide them to articulate that the ratios are formed by multiplying both quantities by the same factor. This repetition highlights the proportional relationship between the two quantities. Encourage students to generalize this pattern by predicting other equivalent ratios beyond the table. For instance, ask what the numbers would be if the number of dogs is multiplied by 4. This helps them understand that any ratio of the form 9 ⋅ k : 5 ⋅ k will be equivalent, where k is any positive integer. By recognizing and generalizing the repeated multiplication, students deepen their understanding of how equivalent ratios are formed.

364

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 176

Purpose Ensure students are capable of applying their understanding of ratios to real-world situations Expected mistakes Students might not understand that in a proportional relationship, when one quantity increases by a certain factor, the other quantity should increase by the same factor. They might calculate the earnings for 18 cakes by not recognizing that the earnings should be triple the amount earned for 6 cakes. Emphasize the importance of proportional reasoning by showing students how to use multiplication to scale quantities. Encourage students to compare the ratios of cakes sold to earnings and observe that the ratio remains constant. Use visual aids like ratio tables or double number lines to illustrate how both quantities increase proportionally. Have students practice multiplying the number of cakes by factors like 2 and 3, and calculating the corresponding earnings to reinforce the concept. This helps students see that if the number of cakes is multiplied by a factor, the earnings should also be multiplied by the same factor.

Students: Page 176

Purpose Ensure students are capable of applying their understanding of ratios to real-world situations

5.02 Equivalent ratios and ratio tables mathspace.co

365

Students: Page 176

Purpose Show students how they can use proportional reasoning to compare quantities and make decisions.

Students: Page 177

Practice Students: Pages 177–180

What do you remember? 1

Deﬁne: a

Equivalent ratios

Simpliﬁed ratios

How can you ﬁnd two ratios that describe the same relationship? Give examples to support your explanation.

For the following equivalent ratios, determine the multiplier to go from the ﬁrst ratio to the second.

1 : 2 and 3 : 6

and

4 : 10 and 2 : 5

8 : 4 and 12 : 6

7 to 1 and 21 to 3

and

1 : 4 and 5 : 20

15 to 5 and 12 to 4

Determine whether each pair of ratios is equivalent. a

2 : 7 and 4 : 14

9 to 12 and 6 to 8

3 to 10 and 6 to 30

4 to 16 and 32 to 8

6 : 4 and 18 : 12

and

10 : 100

Write down three ratios that are equivalent to the following ratios: a

366

and

3:1

6 to 10

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Write at least 2 ratios to describe each collection. a

Consider the ratio 5 : 7. Can you create an equivalent ratio by adding the same number to each quantity in the ratio? Explain.

Which ratio does not belong with the other three? Explain your reasoning. 2:3

4:6

6:8

12 : 18

Let’s practice 9

Complete the pairs of equivalent ratios:

256 to ⬚ = 16 to 3

20 to 5 = ⬚ to 10 396 : ⬚ = 18 : 12

30 : 115 = 6 : ⬚

⬚ : 225 = 460 : 900

37 : 49 = ⬚ : 1323

4 : 10

8:4

20 : 40

100 : 2000

10 : 20

15 : 40

18 : 22

36 : 48

16 : 24

45 : 35

35 : 5

450 : 350

a e 10

Simplify each ratio:

Simplify each ratio: a

540c to $3.00

5 km to 420 m

120 cm to 5 m

11 years to 28 months

Write as simpliﬁed ratios:

20 days to 5 weeks

10 minutes to 120 seconds

15 000 mL to 5 L

3 L to 39 000 mL

Complete the equivalent ratios: a

1 3 6

5 15

2 6 8

3 6 12

6 12 48 60

d 4 3

20 16 12 8

5.02 Equivalent ratios and ratio tables mathspace.co

367

18 10 6

i 12

SOL

27 21 15

2 15 25 35

14 1 2 4 8

16 8

24 12

10 8 6

8 l

1 45 135

20 16 12 8

324 108

81 27

The ratio of students to teachers competing in a charity race is 9 : 4. If 54 students take part in the race, how many teachers took part?

Tricia and Luigi invest money into a business in the ratio . If Tricia has invested $2400, how much has Luigi invested?

At a sporting event, the ratio of security guards to spectators is 1 to 120. If 8400 spectators attend the event, how many security guards will be required?

There are 6 fiction books and 18 nonfiction books on a shelf. Which three ratios represent the relationship of the number of fiction books to the total number of books on the shelf? A E

2 : 10

6 : 18

1 to 4

1:3

If each ratios are equivalent to 5 : 28, ﬁnd the value of x: a

50 : x

60 : x

80 : x

x : 84

110 : x

x : 168

x : 252

x : 504

Complete the ratio table for the ratio 4 : 6.

Complete the ratio table for the ratio

6 4

7 11

The table shows the ratio of dogs to cats: a

Complete the table of equivalent ratios.

If there are 270 dogs, how many cats would there be?

Simplify the ratio of the number of dogs to cats from part (b).

18 16

28 33

Dogs 9 18 27 45

to : : : : :

35 55 Cats 5 10

Sarah created the following ratio tables but made a mistake in each. Describe and correct her error in creating the ratio tables. a

368

3 to 9

A B 3 7 6 12 13 20

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A 5 25 125

B 3 9 27

Every 10 pens cost $4. No. of pens Cost ($)

Complete the ratio table.

How much would you expect to pay for 5 pens?

Find the cost of buying 90 pens.

To convert US dollars, USD ($), to Philippine Peso, PHP, we can use the ratio table below. Every USD($) is worth 56 PHP. USD ($) PHP

Complete the ratio table.

Hector has just returned from holiday with 840 pesos. How many US dollars can he exchange this for?

How many pesos will you be able to buy with $11?

Let’s extend our thinking 25

Two kinds of pine trees, Bristlecone and Aleppo, are planted in rows. In each row, the ratio of Bristlecone to Aleppo is 10 : 9. Altogether, 600 Bristlecone pine trees are planted. Explain how to ﬁnd the number of planted Aleppo pines.

A painter wants to create a certain color by mixing two different colors of paint, Vespa and Nitro, in the ratio 4 : 1. He uses 4 liters of the Nitro color. a

How many liters of the Vespa color must he use?

How many liters of paint will he have altogether once the two colors are combined?

The ratio of y : x is 5 : 3. Find two more points that could fall on this line. x y

0 0

3 5

y 22 20 18 16 14 12 10 8 6 4 2

9 15

Ryan and Valerie are preparing for a party. Ryan blows up 12 balloons in 15 minutes. Valerie blows up 24 balloons in 28 minutes. Assume that both keeps blowing up balloons at a constant rate. a

15 12

Complete the table for the number of balloons Valerie blows up for each time period: Time (minutes) No. of balloons

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Complete the table for the number of balloons Ryan blows up for each time period: Time (minutes) No. of balloons

84 96

Who is blowing up balloons faster?

5.02 Equivalent ratios and ratio tables mathspace.co

369

David and Justin both travel to school by riding scooters. David travels 8 km in 4 minutes. Justin travels 35 km in 16 minutes. Assume that both travel at a constant speed. a

Complete the table for the distance traveled by David for each time period: Time (minutes) Distance (kilometers)

16 32

48 70

140

80 175

Who travels faster? Explain your answer.

To convert US dollars, USD ($), to Japanese yen, JPY (¥), we can use the ratio table below: USD ($) JPY (¥)

370

Complete the table for the distance traveled by Justin for each time period: Time (minutes) Distance (kilometers)

4 8

3 305.40

Complete the ratio table.

Oriana has $40.00, and wants to buy a dress that costs ¥4080. Explain whether she can afford the dress.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

Let’s practice

c 30 : 115 = 6 : 23

What do you remember? 1 a A nswers may vary. Ratios can be equivalent if one of them is the increase or decrease of the other ratio by multiplying or dividing both terms of the other ratio by the same number. b Answers may vary.

2 To find two ratios that describe the same relationship, you can multiply or divide both terms of the first ratio by the same non-zero number. This process is known as finding equivalent ratios. For example, if you have the ratio 2 : 3, multiplying both terms by 2 gives you 4 : 6, which describes the same relationship. Similarly, if you start with the ratio 5 : 10, dividing both terms by 5 results in 1 : 2, another ratio that describes the same relationship. Both pairs, 2 : 3 and 4 : 6, as well as 5 : 10 and 1 : 2, show the proportionality between two quantities, maintaining the same relationship despite being represented by different numbers. b

e 3

g 5

e 256 to 48 = 16 to 3

g 115 : 225 = 460 : 900

h 37 : 49 = 999 : 1323

10 a 2 : 5 e 1 : 2 2 : 3

Simpliﬁed ratios are equivalent ratios where one of the ratios is the result of simplifying the other ratio.

3 a 2

b 20 to 5 = 40 to 10

9 a

5.02 Equivalent ratios and ratio tables

396 : 264 = 18 : 12

b 2:1

c 1:2

d 1 : 20

3:8

g 9 : 11

h 3:4

9:7

k 7:1

9:7

11 a

b 250 : 21

c 6 : 25

d 33 : 7

5:1

g 3:1

h 1 : 13

12 a 9 : 5 e 4 : 7 13 a

b Equivalent

c Not Equivalent

d Equivalent

e Not Equivalent

g Equivalent

h Not Equivalent

6 a Ratio of bees to lady bugs is 6 : 9 or 2 : 3.

Ratio of lady bugs to total bugs is 9 : 15 or 3 : 5.

b Ratio of pennies to quarters is 10 : 5 or 2 : 1.

324

243

4 a Equivalent

Not Equivalent g

5 a Example answer: 6 : 2, 9 : 3, 30 : 10 b Example answer: 3 to 5, 12 : 20, 60 to 100 c Example answer:

, 6 : 15,

d Example answer: 1 : 10, 5 : 50, 100 : 1000

Ratio of quarters to total coins is 5 : 15 or 1 : 3. 7 No, you cannot create an equivalent ratio by simply adding the same number to each quantity in the ratio 5 : 7. Equivalent ratios are created by multiplying or dividing both terms of the ratio by the same non-zero number. Adding the same number to each term changes the relationship between the two quantities, resulting in a different ratio that does not maintain the same proportion as the original 5 : 7 ratio. 8 6:8

108

135

14 24 teachers 15 $5400 16 70 security guards 17 A, D, G

Answers mathspace.co

371

18 a x = 280 e x = 616 19

21 a

b x = 336

c x = 448

d x = 15

Let’s extend our thinking

x = 30

g x = 45

h x = 90

25 Notice that 600 is 60 times of 10, which is the value of Bristlecone pines in the ratio. This means the number of Aleppo pines is 9 ⋅ 60 = 540.

26 a 16 liters

Dogs

Cats

27 Answers vary. Some points include (6, 10), (12, 20), and (15, 25). 28 a

c 9 : 5

29 a

22 a T he ratios between A and B are not consistent. To correct this, ensure that the increments in A are proportional to the increments in B.

b T he ratios between A and B are not consistent. The values in row B should be the result of multiplying the value in row A by a constant factor. No. of pens

Cost ($)

b $36 24 a

372

Time (minutes)

112

No. of balloons

Time (minutes)

Distance (kilometers)

Time (minutes)

Distance (kilometers)

105

140

175

c J ustin, because he travels 35 km in 16 minutes, while David only travels 32 km in 16 minutes. 30 a USD ($) JPY (¥)

c $2

USD ($)

PHP

112

168

224

280

b 616 pesos

Time (minutes) No. of balloons

c Valerie

b 150 cats

23 a

b 20 liters

c $15

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

1 101.80

203.60 305.40 407.20 509.00

b O riana cannot afford the dress, because ¥4080 ≈ $40.08 which is more than the $40 that she has.

5.03 Unit rates Subtopic overview Lesson narrative In this lesson, students will learn to solve problems using unit rates and their applications. They begin by exploring the concept of a unit rate as a comparison of two different quantities where one of the quantities is one unit. The lesson includes practical examples, such as speed (miles per hour) and price (cost per item). Students will learn to identify the unit rate of a proportional relationship represented by a table of values, a contextual situation, or a graph. An exploration is included where students analyze different scenarios to identify and calculate unit rates. By the end, students should be able to calculate and use unit rates to solve contextual problems effectively.

Learning objectives Students: Page 181

Key vocabulary 

compound unit



unit rate



constant rate



rate

 ratio

Essential understanding Unit rates provide a way to compare different quantities by expressing the ratio of two measurements with a denominator of one, making it easier to understand and compare different ratios.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate problem-solving goals into their instruction by creating real-world scenarios that require students to apply their understanding of ratios and proportional relationships. For example, teachers can present a problem where students must use the concept of unit rate to determine the cost per item when given a total cost and quantity.

5.03 Unit rates mathspace.co

373

MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can provide students with opportunities to solve real-world and mathematical problems using ratio and rate reasoning by making tables of equivalent ratios relating quantities with whole number measurements and finding a missing value in a ratio table. Teachers can help students understand the concept of a unit rate, which is represented as a/b for a ratio with b ≠ 0. They can use rate language in context, for example, explaining that a recipe with a ratio of 3 cups of flour to 4 cups of sugar means there are 3/4 cups of flour per cup of sugar. Similarly, they can illustrate that paying $75 for 15 hamburgers translates to a rate of $5 per hamburger.

Teachers can enhance students’ understanding of unit rates by presenting multiple representations simultaneously, allowing students to connect symbols with physical models or diagrams. Encourage students to identify patterns and use them to make conjectures. For example, ask students to identify a pattern in a given table and generalize their findings. Additionally, teachers can create activities that connect the concept of ratios and unit rates to other mathematical topics or real-world contexts. For example, they can relate the concept of unit rates to speed or price comparisons, helping students see the relevance of what they are learning. Teachers should provide feedback on their responses and use strategies such as reversibility and flexibility to help students deeply engage with the concept of unit rates. This approach helps students see how quantities vary together and apply learned strategies to solve problems effectively.

MPG5 — Mathematical Representations Teachers can integrate this goal by encouraging students to represent ratios and unit rates in various ways, such as through tables, graphs, or word problems. They can also ask students to interpret and create their own representations. For example, after finding the unit rate from a given ratio, students can be asked to represent the same information in a table or a graph.

Content standards 6.PFA.2 — The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values). 6.PFA.2a — Identify the unit rate of a proportional relationship represented by a table of values, a contextual situation, or a graph.

6.PFA.2b — Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate. 6.PFA.2d — When given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph.

Prior connections 6.PFA.1 — The student will use ratios to represent relationships between quantities, including those in context.

374

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 5.01 Introduction to ratios Grade 6 — 5.02 Equivalent and simplified ratios

Tools You may ﬁnd these tools helpful: • Scientific calculator • Cubes or beads

Student lesson & teacher guide Unit rates Students may be using compound unit rates but are unaware of how they can be useful in solving real-world problems involving multiple quantities and rates of change. This section explores finding compound unit rates in various forms, and uses these unit rates to answer questions about other possible values connected to the rate.

Students: Page 181

5.03 Unit rates mathspace.co

375

Exploration Students: Page 181

Suggested student grouping: Small groups Students will explore different rates input the names of the items and their amounts and use the sliders to adjust the number lines. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. If a cyclist rode 5 miles in 20 minutes, at a steady rate. How long would it take to ride 10 miles? Given that the cyclist travels 5 miles in 20 minutes, the rate of speed is 1 mile every 4 minutes. Therefore, to travel 10 miles at the same rate, it would take 40 minutes. Purposeful questions • How does adjusting the sliders on the number lines help you to understand the relationships between the quantities involved in calculating the unit rate? • When comparing the unit rates, what factors should be considered to determine which is more efficient? Possible misunderstandings • Students may think that they must compare the given amounts rather than finding a unit rate. This could lead to incorrect comparisons.

Use the STEAM cycle for unit rates Targeted instructional strategies Spark student interest by asking them if they have ever been down the toilet paper aisle at a grocery or department store. Ask them how many different options they have seen and how they would decide which one to choose. Show them photos or physical packaging or pricing signs from the supermarket that use emotive language like “Great value!” or “Now with more sheets per roll!” Use the STEAM cycle to explore this problem using unit rates. Ask: Have students create a specific question. For example, “What package has the best value for toilet paper roll when comsidering the price per regular roll equivalent?” Imagine: Encourage students to brainstorm different methods for comparing costs, such as calculating unit rates, creating tables, or graphing the data, and share their ideas with the class. Plan: Guide students to develop a plan to collect necessary data on various toilet paper brands, including criteria like price, number of rolls, and number of equivalent regular rolls, determining how they will calculate and compare unit rates.

376

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Create and Test: Support students as they gather data, calculate the unit rates, and represent their findings using tables or graphs, testing their methods for clarity and accuracy. Improve: Facilitate a discussion for students to reflect on their results, identify any issues, and collaborate on refining their calculations or representations to make a more informed comparison. For example, have them consider a different comparison of unit price per square foot or sheet. Additionally, students could explore other products and how they might approach finding a unit rate, such as price per pound for meat or price per unit for brooms.

Three reads English language learner support Provide students with many word problems involving unit rates and use the three reads strategy: • On the first read, students should mark the numbers in the problem with their units, noting the order that the numbers are written in • On the second read, students should highlight or mark the words in the problem that indicate whether the problem is asking for a unit rate in math or words • On the third read, students should put these components together and read to make sure that there are no other items that need to be considered. At this point, it is time to start solving the problem.

Physical items Student with disabilities support Have beads or other physical items that represent the original value and then divide them into groups to represent the unit rate. Students can use these groups to complete a table. It may also benefit to have a stepby-step guide of determining where values go in a unit rate to make it easier for students to set up their rates.

Switching vocabulary Address student misconceptions One common misconception when finding the unit rate is to only look at the denominator of the given fraction or ratio, and assume that it represents the unit. However, this is not always the case. The unit rate is the rate per one unit of the given quantity, so it is important to determine what unit the denominator represents. For example, if the given fraction is , the denominator of 1 could represent 2 units of time, 2 units of distance, 2 units of weight, or any other unit depending on the context of the problem. To find the unit rate, you need to divide the numerator by the denominator and include the appropriate unit of measurement. For instance, if 2 represents 2 units of time, then the unit rate would be per 1 unit of time or 3 units/time.

= 3 units of the quantity

5.03 Unit rates mathspace.co

377

Students: Pages 181–182

Since he can run 10 meters in 1 second, we say the unit rate is 10 meters per second or 10 m/s. We can calculate how far the sprinter runs in 1 second by dividing the 100 meters evenly between the 10 seconds. Let’s represent that as a rate: Sprinter’s speed = Sprinter’s speed = 10 m/s This calculation tells us that the sprinter runs 10 meters in one second. A unit rate can also be found from a ratio. If a dance game at the arcade cost $1.25 to play for 5 minutes, the ratio of cost to time played is 1.25 : 5. To ﬁnd the unit rate, divide both quantities of the ratio by 5 to ﬁnd the ratio for a single minute: ÷5 Time (min.) Cost ($)

0.25 1.25 ÷5

The unit rate is $0.25 per minute played. To determine how much it would cost to play the dance game for 20 minutes, we can multiply our unit rate by 20: ×20 Time (min.) Cost ($)

0.25 1.25 5.00 ×20

It would cost $5.00 to play the dance game for 20 minutes.

Example 1 A tap ﬁlls up a 240-liter tub in 4 hours. a Which is the compound unit for the rate of water ﬂow?

Create a strategy

Apply the idea

The rate of water ﬂow represents the number of liters that ﬂows from the tap each hour.

The unit is liters per hour, L/hr.

b What is the ﬂow of the water as a unit rate?

Create a strategy 378

We can ﬁnd the rate of water ﬂow in liters per hour by dividing the capacity of the tub in liters by the number of Mathspace Virginia SOL Grade 6 Teacher Edition hours passed. mathspace.co

Apply the idea

The unit rate is $0.25 per minute played. To determine how much it would cost to play the dance game for Sprinter’s speed = 10 m/s 20 minutes, we can multiply our unit rate by 20: This calculation tells us that the sprinter runs 10 meters in×20 one second. (min.) 20 A unit rate can also be found from a ratio. If Time a dance game1 at the cost $1.25 to play for 5 minutes, the ratio of 5 arcade cost to time played is 1.25 : 5. To ﬁnd the unit rate, both1.25 quantities 5.00 of the ratio by 5 to ﬁnd the ratio for a single Costdivide ($) 0.25 minute:

Examples

×20

Students: Page 182

It would cost $5.00 to play the dance game for 20(min.) minutes. Time 1 Cost ($)

÷5 5

0.25 1.25

Example 1

÷5

A tapunit ﬁllsrate up ais240-liter tubminute in 4 hours. The $0.25 per played. To determine how much it would cost to play the dance game for 20 multiplyunit ourfor unitthe rate byof 20: a minutes, Which is we the can compound rate water ﬂow? ×20

Create a strategy

Apply the idea

Time (min.)

The rate of water ﬂow represents the number of liters The unit is liters per hour, L/hr. Cost ($) 0.25 1.25 5.00 that ﬂows from the tap each hour. ×20

b would Whatcost is the ﬂow to of play the water as a unit rate? It $5.00 the dance game for 20 minutes.

Purpose Create a strategy Make students aware Example 1 that the compound unit can be generated from the given quantities and measurements in We can find the rate of water flow in liters per hour by dividing the capacity of the tub in liters by the number of the problem. hours passed. A tap fills up a 240-liter tub in 4 hours.

Reflecting with students a Which the compound unit for the rate of water ﬂow? Apply theisidea Ask students why we use liters per hour rather than hours per liter. Discuss how the unit rate would change if strategy Apply the idea we usedCreate hoursaper liter insted. The rate of water ﬂow represents the number liters Evaluate theofdivision

The unit is liters per hour, L/hr.

ﬂows 182 from the tap each hour. Students:that Page

b What is the ﬂow of the water as a unit rate? 182

Mathspace Virginia SOL Grade 6 mathspace.co

Create a strategy

We can ﬁnd the rate of water ﬂow in liters per hour by dividing the capacity of the tub in liters by the number of hours passed.

Apply the idea

Evaluate the division

Purpose182 Mathspace Virginia SOL Grade 6 mathspace.co Check that students know how to use a compound unit to create the unit rate.

Students: Page 183

Example 2 A car travels 320 km in 4 hours. a Complete the table of values. Time taken (hours) Distance traveled (kilometers)

4 320

Create a strategy

Apply the idea

Since the time is being divided by 2 each time, divide the distance traveled by 2 for each new distance in the table.

Divide 320 by 2 Evaluate Divide 160 by 2 Evaluate 5.03 Unit rates This means that the tables of values is given by: mathspace.co Time taken (hours) Distance traveled (kilometers)

4 320

2 160

1 80

379

A car travels 320 km in 4 hours. a Complete the table of values. Time taken (hours) Distance traveled (kilometers)

4 320

Create a strategy

Apply the idea

Since the time is being divided by 2 each time, divide the distance traveled by 2 for each new distance in the table.

Divide 320 by 2 Evaluate Divide 160 by 2 Evaluate

Example 2

This means that the tables of values is given by:

A car travels 320 km in 4 hours.

Time taken (hours) Distance traveled (kilometers)

a Complete the table of values. Time taken (hours) 4 Distance traveled (kilometers) 320 b What is the speed of the car as a unit rate?

4 320

2 160

1 80

Apply PurposeCreate Create a a strategy strategy Apply the the idea idea Since the time is being divided by 2 each time, divide the Demonstrate to students how to use the relationship between quantities to find the equivalent ratio Divide by 12and Use the table from part (a) to ﬁnd what the distance was Divide 80320 km by hr distance traveled by 2 for each new distance in the table. after 1 hour. complete a table of values. Evaluate Evaluate

Reflecting with students Divide 160 by 2 When creating a unit rate, the values being divided also have their units divided, which Evaluate creates the wording of Example 3 the unit rate of “ ⬚ per ⬚.” Ask students how knowing both the unit rate numerically and in units help us solve This means that the tables of values is given by: other problems? Henry bikes 45 miles in 3 hours. a What is the speed of the bike in miles per hour?

Students: Page 183

Time taken (hours) Distance traveled (kilometers)

4 320

2 160

1 80

Create a strategy b What the is speed of the a unitWe rate? Miles perishour the unit ratecar foras speed. need to ﬁnd the distance traveled in 1 hour.

Create a strategy Apply the idea

Apply the idea

Use the table from part (a) to ﬁnd what the distance was after 1 hour.

Divide 80 km by 1 hr

Evaluate Substitute the values of the distance and time in the expression Consider what factor to divide 3 by to get 1

Example 3 Purpose Divide by to get the unit rate Henry bikes 45 miles 3 hours. Make students aware thatinthey can use the table of values to find the unit rate. a What is the speed of the bike inEvaluate miles per hour?

Understanding compound units Speed as unit rate Create a strategy Targeted instructional strategies

use with Example 2

Miles per hour is the unit rate for speed. We need to ﬁnd the distance traveled in 1 hour.

A compound unit takes two unconnected units and writes them as a rate of “⬚ per ⬚.” Students have 5.03 Unit rates 183 experienced Applycompound the idea units in their everyday experience. mathspace.co

Provide students with several examples of real life ratios they may experience, such as two cars making a trip, with one driving 100 miles in 2 hours and the second driving 130 miles in 3 hours. Ask students how they could Substitute the values of the distance and time in the expression tell which car was driving faster at the time. Show students how to find the compound unit (miles per hour) of each car, giving further problems to try, such as 220 miles in 4 hours, 210 miles in 3.25 hours, and 105 miles in Consider what factor to divide 3 by to get 1 1.5 hours. Divide by

to get the unit rate

Evaluate Speed as unit rate

5.03 Unit rates mathspace.co

380

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

183

b What is the speed of the car as a unit rate?

Create a strategy

Apply the idea

Use the table from part (a) to ﬁnd what the distance was after 1 hour.

Divide 80 km by 1 hr Evaluate

Students: Page 183 Example 3 Henry bikes 45 miles in 3 hours. a What is the speed of the bike in miles per hour?

Create a strategy Miles per hour is the unit rate for speed. We need to ﬁnd the distance traveled in 1 hour.

Apply the idea

Substitute the values of the distance and time in the expression Consider what factor to divide 3 by to get 1 Divide by

to get the unit rate

Evaluate Speed as unit rate

5.03 Unit rates mathspace.co

183

Purpose Show students how to calculate the unit rate of speed from given distance and time, reinforcing the concept of unit rates.

Students: Page 184 b If Henry travels at this constant rate, what distance will Henry travel in 2 hours?

Create a strategy Set up a table from the ratio original and create equivalent ratios in the table to solve the problem.

Apply the idea We can set up a table from the ratio 45 : 3. miles hours

45 3

Divide both values by 3 to ﬁnd the distance for 1 hour. miles hours

15 1

45 3

Use that unit rate to solve for the remaining missing values. miles hours

15 1

30 2

45 3

60 4

From the table, we can see that in 2 hours, Henry can travel 30 miles.

Example 4 Purpose Iain feels how like buying some ice-cream his friends. Show students to use a ratio tablefor tohimself solve and problems involving constant rates. • A box of 6 Cornettos costs $7.20 • A box of 4 Paddle pops costs $6.40

a How much does each Cornetto cost?

Create a strategy 5.03 Unit rates mathspace.co

Find the unit price for each Cornetto.

Apply the idea Divide the total cost by the number of Cornettos

381

hoursthe idea 1 Apply

We a table fromfor thethe ratio 45 : 3. missing values. Usecan thatset unituprate to solve remaining miles miles hours hours

15 1

30 2

45 45 3 3

60 4

Dividethe both values to ﬁnd for 1 hour. table, we by can3see thatthe in 2distance hours, Henry can travel 30 miles. Students:From Page 184 miles hours

15 1

45 3

Example Use that unit4rate to solve for the remaining missing values. 30some45 60 for himself and his friends. Iainmiles feels like15 buying ice-cream hours 1 2 3 4 • A box of 6 Cornettos costs $7.20 • A box of 4 Paddle pops costs $6.40 From the table, we can see that in 2 hours, Henry can travel 30 miles. a How much does each Cornetto cost?

Create a strategy Example 4

Find the unit price for each Cornetto. Iain feels like buying some ice-cream for himself and his friends. Apply theofidea • A box 6 Cornettos costs $7.20 • A box of 4 Paddle pops costs $6.40 Divide the total cost by the number of Cornettos a How much does each Cornetto cost? Divide the numerator and denominator by 6 to get the unit price

Create a strategy Find the unit price for each Cornetto. Evaluate Cornettos cost $1.20 per piece.

Apply the idea

Divide the total cost by the number of Cornettos b How much does each Paddle pop cost? Divide the numerator and denominator by 6 to get the unit price

PurposeCreate a strategy Show students how to calculate the unitEvaluate price of an item given the total cost and quantity. Find the unit price of Paddle pop. $1.20 per piece. Students:Cornettos Pagescost 184–185 b How much does each Paddle pop cost?

Create a strategy Mathspace SOL Grade 184 the Find unit priceVirginia of Paddle pop. 6 mathspace.co

Apply the idea Divide the total cost by the number of Paddle pops Divide the numerator and denominator by 4 to get the unit price 184

Mathspace Virginia SOL Grade 6 mathspace.co

Evaluate

Paddle pops cost $1.60 per piece.

Apply the idea

Divide the total cost by the number of Paddle pops c Which type of ice-cream is the better buy? Divide the numerator and denominator by 4 to get the unit price Purpose Create a strategy Show students how to calculate the unit price of an item given the total cost and the number of items. Evaluate The better buy is the ice-cream is the one with a lower unit price.

pops cost $1.60 per piece. Students:Paddle Page 185

Apply the idea

Compare the unit prices of Cornettos and Paddle pops and choose the lower unit price. c Which type of ice-cream is the better buy?

$1.40 < $1.60

Cornettos are a better buy than Paddle Pops. Create a strategy The better buy is the ice-cream is the one with a lower unit price.

Apply Idea the idea summary Compare the unit of Cornettos and Paddle pops and choose thewith lower unit price. A rate is a prices measure of how quickly one measurement changes respect to another.

382

< $1.60 of 1, such as 2 feet per second or 5 miles per When rates are expressed as a quantity with $1.40 a denominator hour, they are called unit rates. Cornettos are a better buy than Paddle Pops. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice Idea summary

c Which type of ice-cream is the better buy?

Create a strategy The better buy is the ice-cream is the one with a lower unit price.

Apply the idea Compare the unit prices of Cornettos and Paddle pops and choose the lower unit price. $1.40 < $1.60 Cornettos are a better buy than Paddle Pops.

Idea summary

Purpose A rate is a measure of how quickly one measurement changes with respect to another. Show students thatrates theyare can apply their understanding of unit prices to as make decisions value expressed as a quantity with a denominator of 1, such 2 feetinformed per second or 5 miles about per Apply When the idea for money in real-world situations. hour, they are called unit rates. Divide the total cost by the number of Paddle pops

Advanced learners: Deepening price analysis withbymultilayered tasks Divideunit the numerator and denominator 4 to get the unit price

use with Example 4

Targeted instructional strategies

Practice

Evaluate

Encourage students to extend the problem by adding layers of complexity that require deeper analysis. For pops cost $1.60 per piece. instance,Paddle ask them to consider how buying larger quantities might influence the unit price—what if there’s a What do you remember? discount for purchasing multiple boxes, or if a larger pack size is available? Challenge them to determine the price perc 1ice cream ifofrate there’s a 10% discount on one of the products. Invite students to compare not only the unit Select the in each Which type ice-cream isset: the better buy? prices but also factors like quality, brand or nutritional content, analyzingD how these factors might i A 3:4 B 15reputation, cm C 6 cm/min affect the “better buy.” Encourage them to create graphs or tables to visually represent how unit price changes Create strategy ii a A 22% B $15/hr C 25 : 35 D 125 cm with quantity or discounts; for example, a line unit price versus number of units purchased. 2 graph The better is theper ice-cream is the with a lowershowing unit price. C 17 : 3 D 12 hrs iii Abuy$1.50 liter B one 91 cm This approach allows advanced learners to apply their understanding in more complex, situations, iv A $22.50 B 112.5 cm C 65 km per hr D 10real-world days thethinking idea fosteringApply critical and decision-making skills. 2 The speed of a car is usually measured using the units mi/hr. Write suitable units for each rate: Compare the unit prices of Cornettos and Paddle pops and choose the lower unit price. a The price of gasoline b Goals scored in a soccer game $1.40 < $1.60 c Heart rate d Pay rate at a part-time job

Students:Cornettos Page 185 are a better buy than Paddle Pops. 3

Which rates are examples of unit rates? A

$2 per min

5 men/3 days

95 km per hour

14 days in 2 weeks

Idea summary A rate is a measure of how quickly one measurement changes with respect to another. When rates are expressed as a quantity with a denominator of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. 5.03 Unit rates 185 mathspace.co

Practice What do you remember? Practice 1

Select the rate in each set:

Students: Pages i A185–190 3:4

15 cm

6 cm/min

A 22%

$15/hr

25 : 35

125 cm

iii

A $1.50 per liter

91 cm2

17 : 3

12 hrs

A $22.50

112.5 cm

65 km per hr

10 days

What do you remember? 1

speed of a car is usually measured using the units mi/hr. Write suitable units for each rate: Select2 theThe rate in each set: a

The price of gasoline

Goals scored in a soccer game

A 3 :c 4 Heart rate

A3 22% B of$15/hr Which rates are examples unit rates?

iii

A $2 perliter min A $1.50 per

2 B cm 5 men/3 days 91

CC 9517km : 3per hour

A $22.50

112.5 cm

15 cm

C Pay rate at a part-time job

25 : 35 65 km per hr

6 cm/min

125 cm

14 days weeks D in 122hrs

10 days

5.03 Unit rates 185 rates 5.03 Unit mathspace.co

mathspace.co

383

The speed of a car is usually measured using the units mi/hr. Write suitable units for each rate: a

The price of gasoline

Goals scored in a soccer game

Heart rate

Pay rate at a part-time job

95 km per hour

Which rates are examples of unit rates? A

$2 per min

5 men/3 days

14 days in 2 weeks

8:4=⬚:1

Consider each scenario. i

Describe each situation in your own words.

Write the rate indicated by each situation. What are the units?

45 mi

Population in City

2100

Population

1800 1500 1200 900 600 300 0

2012

2013

2014

2015 2016 Year

2017

2018

February 2019 Length: 108 in

Complete the following pairs of equivalent ratios: a e

February 2020 Length: 120 in

1:⬚=3:6

⬚ : 1 = 3 : 12

b f

2 : 5 = ⬚ : 2.5 1 : ⬚ = 18 : 9

c g

3:9=1:⬚

⬚ : 20 = 1 : 4

A sprinter runs 100 m in 10 seconds. Determine whether the statements are true:

⬚ : 1 = 32 : 8

The quantities compared are distance and time.

The unit of speed can be expressed as s/m

The unit rate is 10 m/s

In 10 seconds, the runner runs 1 m.

Let’s practice 7

Justin earns $420 in 6 hours. a

What two quantities are being compared?

What is the ratio for the two quantities of Justin’s earnings?

What is the unit rate?

A three episode TV series had a total run time of 72 minutes. Find the rate of run time per episode.

A carton of 10 eggs cost $6.50. Calculate the cost of 1 egg in cents.

A 0.75 L bottle of milk costs $1.05. Calculate the cost of milk per liter.

384

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Lynette can read 35 pages of books in 20 minutes. a

How many pages of books can she read in 1 minute?

Create a table of values for the situation.

Write the rate as a unit rate: a

1040 books per 13 libraries

85 people per 5 buses

234 kilometers per 4.5 hours

9 cakes per 72 guests

For each table, ﬁnd the unit rate. a

x 1 2 3 y 2 4 6

x 2 3 4 y 3 4.5 6

4 8

5 10

x y

1 0.4

2 0.8

8 12

6 9

x 100 200 300 y 2 4 6

x 2 4 6 8 y 10 20 30 40

400 8

y 500 10

10 50

3 1.2 3

5 2 5

x y

1 20

2 40

3 60

4 80

5 100

x y

2 14

3 21

4 28

5 35

7 49

For each scenario: i a

Find the unit rate.

Complete the table

Tricia reads 20 pages of her book in 2 hours.

A car travels 200 mi in 4 hours.

Time (hours) 2 3 4 Time taken (hours) Number of pages read 20 Distance traveled (miles) c

A toy manufacturer makes 720 toys in 9 hours.

Time (hours) 9 10 11 Number of toys 720 15

4 1.6

4 200

Tina burns 640 calories by jogging 8 km. Distance (km) Calories burned

8 640

Create a scenario that matches each set of double number lines. a

Miles 0

0 5 Hour

20 25

Jumps 0

30 60 90 120 150

30 45 60 75

0 10 20 30 40 50 Second

Books reads 0 9 18 27 36 45

0 4 Week

0 3 6 Dog walked

Hour 0 1

5.03 Unit rates mathspace.co

385

Myrtle drives the same number of miles to and from work each day, as shown on the graph below. Miles to Work 130 miles Miles 120 miles 110 miles 100 miles 90 miles 80 miles 70 miles 60 miles 50 miles 40 miles 30 miles 20 miles 10 miles

Days

1 days 2 days 3 days 4 days 5 days

Based on the graph, what is the unit rate of miles driven per day? Explain your thinking. 17

The seventh grade choir sold pizzas as a fundraiser. The choir teacher created the graph below for the students. Pizza Sale Proﬁts y 44$ 40$ 36$ 32$ 28$ 24$ 20$ 16$ 12$ 8$ 4$

x 4 Pizzas 8 Pizzas

Based on the graph, what is the unit rate of proﬁt for the pizzas? A

$0.56 per pizza

$4.00 per pizza

18 pizzas per $100

$1.80 per pizza

Tina counts 26 heart beats over a 20-second interval. Find Tina’s heart beat in one minute.

A cyclist travels 70 yards per 10 seconds. a

Find the distance the cyclist travels in 1 minute.

Create a table of values for the situation.

A 1.35 kg pack of hamburger costs $15. If 1 kg = 1000 g how many grams of hamburger does each dollar buy?

Luke writes 2400 words in 20 minutes.

386

Determine Luke’s writing speed in words per second.

Create a table of values for the situation.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examine the table. Is the rate between time spent on a cell phone and the cost of Wi-Fi constant? Explain your reasoning.

Time (minutes) 4 5 12 15

Cost (in cents) 12 24 48 96

Jackson and Carlos were running around a track. They started running at the same time. When Jackson had run 5 laps, Carlos had run 2 laps. The table shows the laps Jackson and Carlos completed. Jackson 5 15 25

Carlos 2 6 10

Is the ratio between Jackson’s and Carlos’ number of laps consistent? Explain your reasoning. 24

A supermarket sells two different brands of eggs, Happy Hens and Sunny Side Up. • Happy Hens eggs cost $6.00 for 12 eggs. • Sunny Side Up eggs cost $1.50 for 5 eggs.

Calculate the cost in cents per egg from Happy Hens.

Calculate the cost in cents per egg from Sunny Side Up.

Which brand sells its eggs at a cheaper price?

Jade and Blake create videos at a constant rate each week. The proportional relationship between the number of videos created each week by Jade and Blake is represented in the tables below. a

Complete the missing value in each ratio table.

Determine who is making videos at a faster rate.

Jade Blake Videos Week(s) ? 1 48 3 80 5 120 11 26

Videos ? 30 60 120

Week(s) 1 2 4 8

A store has two different brands of trash bags. Which brand of trash bags is the better deal? Explain your thinking. Brand A Trash Bags Cost (in Dollars) Number of Trash Bags ? 1 3.60 20 9.00 50

Brand B Trash Bags Cost (in Dollars) ? 3.06 15.30

Number of Trash Bags 1 18 90

5.03 Unit rates mathspace.co

387

Let’s extend our thinking 27

Explain how to convert 120 km/hr into meters per minute.

Neville is deciding between four cars to buy. He wants to buy the most fuel eﬃcient one. a

Complete the table: Car Magnum Falcador Canyonero Civil

b 29

liters used 13.405 20.71 6.336 2.754

Distance travelled (km) 38.3 54.5 19.2 8.1

L/km

Which car is the most fuel eﬃcient?

Iain feels like buying some ice-cream for himself and his friends. • A box of 8 Cornettos costs $18.00 • A box of 5 Magnums costs $7.25 Which type of ice-cream is cheaper individually? By how much?

Isabelle is buying juice for her nephew’s birthday party. • A 3.2 L bottle of apple juice costs $13.76 • A 2.1 L bottle of orange juice costs $6.30 Which juice is the best buy? Justify your reasoning.

A tap ﬁlls up a 240 L tub in 4 hours. a

Find the rate of water ﬂow of the tap.

Complete the table of values: Time (hours) Liters of water

c 32

4 240

How many liters of water can a tap ﬁll up in

hours? Explain your reasoning.

Avril is running from John in a game of tag. Avril runs 30 yd every 5 seconds and John runs 21 yd every 3 seconds. Who is faster? Explain your answer.

Adam really likes apples and eats 4 per day. a

Find the number of apples that Adam eats in one week.

If Adam buys 44 apples, how many days will that last him?

A worm takes 15 seconds to travel 20 cm. a

Find the unit rate of the worm’s distance traveled in meters per second.

Find the rate of the worm’s time spent in seconds per meter distance traveled.

Explain why these two numbers are different.

Consider the scenarios: • Han paid $124.62 for 67 L of petrol in Humbleton. • Amelia paid $82.08 for 57 L of petrol in Dunkilderry.

388

Calculate the cost per liter of petrol in Humbleton.

Calculate the cost per liter of petrol in Dunkilderry.

In which suburb is petrol cheaper?

Han buys 30 L in Statesota, and this petrol station matched the best price between Humbleton and Dunkilderry. How much did he pay?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

Let’s practice 7 a T he amount in dollars Justin earns and the time in hours.

5.03 Unit rates

b 420 : 6

What do you remember? 1 i

ii B

c $70 per hour or $70/hr iii A

iv C

2 a dollars per gallon

b goals per game

c beats per minute

d dollars per hour

8 24 min/episode 9 65 cents/egg 10 $1.4/L

3 A, C 4 a i T ravel Time Scenario: This situation involves a car journey where the time taken to travel between two points is illustrated. The clock and car indicate a change in time and possibly distance covered in that period. ii Travel Time Scenario:

11 a 1.75 pages b

Time (minutes)

Number of 1.75 3.5 pages read

• Units: Miles per hour (mph) or kilometers per hour (km/h), depending on the distance unit. b i P opulation Growth Scenario: This scenario depicts the growth in population of a city over a certain time period. It involves numerical data that likely shows the starting population, the increased number, and the time frame over which this change occurs. ii Population Growth Scenario: • Rate: The rate here is the population growth rate of the city. • Units: People per year, indicating how many people are added to the city’s population annually. c i G rowth of a Crocodile: This illustrates the growth of a crocodile in terms of its length over a year. It shows a comparison between the initial length of the crocodile and its length after a speciﬁed time period, indicating growth.

c 52 km/hr 13 a 2 e

b 17 people/bus cakes/guest

d b 0.4

c 1.5

g 5

h 7

18 20

7 10.5 14 17.5 21 24.5 28 31.5 35

12 a 80 books/library

• Rate: The rate could be the speed of the car, which would be calculated based on the distance covered over the time change indicated by the clocks.

14 a i 10 pages/hr ii

Time (hours)

Distance traveled (miles)

b i 50 mi/hr ii

Time taken (hours)

Distance traveled (miles)

200

100

c i 80 toys/hr ii

Time (hours)

Number of toys

720

800

880

d i 80 cal/km ii

Distance (km)

Calories burned

640

560

480

ii Growth of a Crocodile: • Rate: This rate is the growth rate of the crocodile in terms of length. • Units: Inches per year, showing how much the crocodile’s length increases each year. 5 a 1 : 2 = 3 : 6

b 2 : 5 = 1 : 2.5

c 3 : 9 = 1 : 3

d 8:4=2:1

e 0.25 : 1 = 3 : 12

g 5 : 20 = 1 : 4

h 4 : 1 = 32 : 8

6 a True

b Not true

1 : 0.5 = 18 : 9

c True

d Not true

15 a Answers vary. Doug can run 30 miles in 5 hours. b Answers vary. Marie can read 9 books every 4 weeks. c A nswers vary. Paulina can jump 15 times every 10 seconds. d Answers vary. Matt can walk 3 dogs every 1 hour. 16 The graph shows a relationship with a constant rate. Each day, the number of miles increases by 25, so the unit rate is 25 miles per day. 17 B The unit rate of proﬁt for the pizzas is $4 per pizza. 18 78

Answers mathspace.co

389

19 a 420 yd b

Let’s extend our thinking

Time (seconds)

Number of distance traveled (yards)

140

210

280

350

420

20 90 g

27 First, convert km to m by multiplying the value by 1000 (since there are 1000 meters in 1 kilometer). So, 120 km/hr is equal to 120 000 m/hr. Next, convert hr to min by dividing the value by 60 (since there are 60 minutes in 1 hour). So, 120 000 m/hr is equal to 2000 m/min. This means that 120 km/hr is equivalent to 2000 m/min. 28 a

21 a 2 words/second b

Time (seconds)

120

300

600

900 1200

Number of 120 words written

240

600 1200 1800 2400

Distance travelled (km)

L/km

13.405

38.3

0.35

20.71

54.5

0.38

Canyonero

6.336

19.2

0.33

Civil

2.754

8.1

0.34

Car

liters used

Magnum Falcador

22 No, the relationship between the time and the cost is not consistent because it does not have a unit rate. After 4 minutes, the cost is 12 cents which suggests a unit rate of 3 cents per minute but after 12 minutes, the cost is 48 which suggests a unit rate of 4 cents per minute.

29 Magnum is cheaper by 80 cents

23 Yes, Jackson and Carlos ran at a consistent rate. The unit rate at which Jackson runs is 2.5 laps for every lap Carlos runs. Using the unit rate, we can conﬁrm this by multiplying the number of laps Carlos runs by 2.5.

The cost per liter for orange juice is calculated by dividing

2 ⋅ 2.5 = 5

6 ⋅ 2.5 = 15

10 ⋅ 2.5 = 25

24 a 50 cents/egg

b 30 cents/egg

c Sunny Side Up 25 a Jade Blake Videos Week(s) Videos Week(s) 16

120

b J ade makes videos at a faster rate since their unit rate is greater. 26 To ﬁnd the cost per bag for Brand A, we can divide the cost by the number of bags. For 20 bags at $3.60, the cost per bag is $3.60/20 = $0.18 per bag. To ﬁnd the cost per bag for Brand B, we can divide the cost by the number of bags. For 18 bags at $3.06, the cost per bag is $3.06/18 = $0.17 per bag. Brand B offers a better deal at $0.17 per bag compared to Brand B’s $0.18 per bag.

b Canyonero

30 The cost per liter for apple juice is calculated by dividing

the cost by the volume: $3.00 per L.

31 a 6 0 L/hr b Time taken (hours) Liters of water

240

180

120

hours, the tap can ﬁll up 90 L of water. This is c I n found by using the rate of water ﬂow (60 L/hr) and multiplying by 1.5 hours. 32 John, because his speed is 7 yd/s while Avril’s speed is 6 yd/s. 33 a 28 apples 34 a

meters per second

b 11 days b 75 s/m

c T hese two numbers are different because they describe opposite relationships. The ﬁrst measures how much distance is covered per unit of time, indicating speed. The second measures how much time is required to cover a unit of distance, indicating duration. c Dunkilderry

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

, which simpliﬁes to

Orange juice is the best buy because it has a lower cost per liter ($3.00 per L) compared to apple juice ($4.30 per L). This means Isabelle gets more juice for her money with the orange juice.

35 a $1.86/ L

390

, which simpliﬁes to

the cost by the volume: $4.30 per L.

b $1.44/ L d $43.20

5.04 Proportional relationships Subtopic overview Lesson narrative In this lesson, students will explore proportional relationships and how to represent them using ratio tables and graphs. They will learn to identify proportional relationships by examining whether the ratio between quantities remains constant. The lesson includes examples where students create ratio tables to find unknown values and use graphs to visually represent proportional relationships. Students will explore continuous versus discrete contexts and understand how to interpret proportional graphs. By the end, students should be able to recognize and represent proportional relationships using various methods.

Learning objectives

5.04 Proportional relationships

Students: Page 191

After this lesson, you will be able to... • identify a proportional relationship when given a table of values, graph or context. • find the unit rate and create (or complete) a table of values or graph for a proportional relationship representing a real-world context. • compare multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.

Proportional relationships We say that two quantities have a proportional relationship if the values are always represented by the same ratio. Key vocabulary When two quantities are proportional, we can use a ratio table to show equivalent ratios and ﬁnd unknown values.  origin linear If a cookie recipe calls for 2 cups of sugar for every 4 cups of ﬂour, we could write this as the ratio 4 : 2.  proportional relationship  ratio table Putting this in a ratio table, we have: 

Cups of sugar (x)

Essential understanding

Cups of flour ( y)

2 ×2 2 4

×2

This relationship is proportional becauseof the ratio, , ofratios. ﬂour to sugar is constant: A proportional relationship is a collection equivalent

Standards Let’s consider another scenario. The cost to rent a scooter and time rented are shown in the table below: This subtopic addresses the following Virginia 2023 Mathematics Time (min) 3 6 Standards 9 15 of Learning standards. Cost ($) 5 7 9 13

Mathematical process goals

This relationship is not proportional because the ratios between cost and time are not constant: MPG2 — Mathematical Communication

This can be incorporated during the ‘Analyzing tables for proportional relationships’ stage. Teachers can encourage students explain reasoning verbally in writing - for instance, why they table or does noty). We seetowhat eachtheir relationship looks like inor a graph by turning each column from believe the tablea into andoes ordered pair (x, represent a proportional relationship. This promotes the use of mathematical language and the expression of 14 Cost ($) 14 Flour (cups) mathematical ideas. 13 12 11 10 9 8 7 6 5 4 3 2

13 12 11 10 9 8 7 6 5 4 3 2

5.04 Proportional relationships mathspace.co

391

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers incorporate this goal into their instruction by presenting multiple representations simultaneously, allowing students to connect symbols with physical models or diagrams. Encourage students to identify patterns and use them to make conjectures. For example, ask students to identify a pattern in a given table and generalize their findings.

Teachers can foster mathematical representations by encouraging students to use multiple representations and share their solution processes. Activities like completing tables and graphing proportional relationships help students connect verbal descriptions, tables, and graphs. Teachers should address common misconceptions, such as omitting the origin in graphs and selecting appropriate axes. By using relevant contexts, such as global warming data, students can engage more meaningfully. Additionally, tools like geoboards can provide concrete experiences before students draw graphs, enhancing their understanding of proportional relationships.

Content standards 6.PFA.2 — The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values). 6.PFA.2b — Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate.

6.PFA.2c — Determine whether a proportional relationship exists between two quantities, when given a table of values, context, or graph. 6.PFA.2d — When given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph.

6.PFA.2e — Make connections between and among multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.

6.PFA.1 — The student will use ratios to represent relationships between quantities, including those in context.

Future connections A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

7.PFA.1 — The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

392

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Predict the graph of proportional relationships using pattern analysis Targeted instructional strategies Encourage students to collect data from various proportional relationships and organize it into ratio tables. Guide them to perform pattern analysis by examining how the ratios between quantities remain constant. Have them identify patterns in how one quantity scales with another and use these patterns to predict missing values or extend the tables. Facilitate activities where students use the patterns they’ve discovered to anticipate the shape and characteristics of the graphs representing these relationships. Provide graph paper or graphing software so they can plot the data and visually confirm their predictions. By linking the numerical patterns in the tables to the visual patterns on the graphs, you’ll help students deepen their understanding of proportional relationships and enhance their ability to make informed predictions. Guide students to draw conclusions from their pattern analysis by discussing how the constant ratio influences the proportional relationship. Encourage them to explain how this consistent pattern results in a straight line that passes through the origin when graphed. By articulating their observations, students will better understand the underlying principles of proportional relationships and how to apply this knowledge to solve problems in different contexts.

Student lesson & teacher guide Proportional relationships Students are introduced to proportional relationships and how they are represented by ratios and ratio tables. They will also understand how to translate these relationships into graphs, distinguishing between proportional and non-proportional scenarios. Furthermore, students will differentiate between continuous and discrete relationships.

Students: Pages 191–192

5.04 Proportional relationships After this lesson, you will be able to... • identify a proportional relationship when given a table of values, graph or context. • find the unit rate and create (or complete) a table of values or graph for a proportional relationship representing a real-world context. • compare multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.

Proportional relationships We say that two quantities have a proportional relationship if the values are always represented by the same ratio. When two quantities are proportional, we can use a ratio table to show equivalent ratios and ﬁnd unknown values. If a cookie recipe calls for 2 cups of sugar for every 4 cups of ﬂour, we could write this as the ratio 4 : 2. Putting this in a ratio table, we have: Cups of sugar (x) Cups of flour ( y)

2 ×2 2 4

×2

This relationship is proportional because the ratio, , of ﬂour to sugar is constant:

5.04 Proportional relationships mathspace.co Let’s consider another scenario. The cost to rent a scooter and time rented are shown in the table below: Time (min)

393

When two quantities are proportional, we can use a ratio table to show equivalent ratios and ﬁnd unknown values. If a cookie recipe calls for 2 cups of sugar for every 4 cups of ﬂour, we could write this as the ratio 4 : 2. Putting this in a ratio table, we have: Cups of sugar (x) Cups of flour ( y)

2 ×2 2 4

×2

This relationship is proportional because the ratio, , of ﬂour to sugar is constant:

Let’s consider another scenario. The cost to rent a scooter and time rented are shown in the table below: Time (min) Cost ($)

3 5

6 7

9 9

15 13

This relationship is not proportional because the ratios between cost and time are not constant:

We see what each relationship looks like in a graph by turning each column from the table into an ordered pair (x, y). 14 Flour (cups) 13 12 11 10 9 8 7 6 5 4 3 2 1 1

14 Cost ($) 13 12 11 10 9 8 7 6 5 4 3 2 1

Sugar (cups)

2 3 4 5 6 7 8 9

Graph shows the proportional relationship between cups of ﬂour and cups of sugar.

Time (min)

2 3 4 5 6 7 8 9

Graph shows the non-proportional relationship between cost of a scooter and time.

In both graphs, we connected the points with a line because the values in between the points make sense for the context. Fore example, we could make a recipe with 3.5 cups of sugar or (depending on the renting rules) we could probably rent a scooter for 7.25 minutes. If the contexts were changed so the values in between the points did not make sense, we would not connect them. For example, if we were comparing the ratio of flour : eggs in a recipe we would probably not use a fraction of an 5.04 Proportional relationships 191 egg. Or if the scooter rental only allowed us to rent for specific amounts of time, then it would not make sense to mathspace.co calculate cost times outside of that. We call the relationship continuous if it would make sense to include the values between points and discrete if only the points make sense in the context. Two different graphs can be represented by the data depending on the order we choose. The above graph shows the relationship between cups of flour and cups of sugar where y represents the cups of flour and x represents the cups of sugar. This graph represents the ratio of y to x as 4 : 2. Sugar (cups) 6 5

We can also create a graph for the ratio of y to x as 2 : 4 where y represents the cups of sugar and x represents the cups of ﬂour. Notice the similarities and differences between the two graphs. They both pass through (0, 0) but the ratio 4 : 2 is much steeper than the ratio 2 : 4.

4 3 2 1

Flour (cups) 1 2 3 4 5 6 7 8 9 10 11

It is important to state which quantity is represented by x and which quantity is represented by y. A graph is proportional if the graph is linear, meaning it looks like a straight line, and it passes through the origin, (0, 0). The graph of a proportional relationship always includes (0, 0) because we can create the equivalent ratio 0 : 0 by multiplying both parts of the ratio by 0. For the sugar and ﬂour example, this would mean that a recipe that calls for 0 cups of ﬂour would need 0 cups of sugar.

Example 1 If the ratio of y to x is represented by 3 : 1, plot the ratio as a single point on the coordinate plane.

y 4

394

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

3 2

Use a Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives to model proportional relationships. Provide students with two types of objects, such as red and blue counters, beads, or blocks. Instruct them to create groups where the ratio of red to blue objects remains constant. For example, have them start by grouping 1 red counter with 2 blue counters. Then, ask them to create additional groups that maintain this ratio: 2 reds with 4 blues, 3 reds with 6 blues, and so on. Encourage students to physically build multiple sets, each time increasing the quantities while keeping the ratio of red to blue objects the same. As students manipulate the objects, ask guiding questions like, “How many blue counters do you have when you have 2 red counters?” and “What you notice about the relationship between the number of red andweblue counters in each If thedo contexts were changed so the values in between the points did not make sense, would not connect them. group?” For example, if we were comparing the ratio of flour : eggs in a recipe we would probably not use a fraction of an ratio This hands-on activity helps students see that even though the total number of objects changes, the egg. Or if the scooter rental only allowed us to rent for speciﬁc amounts of time, then it would not make sense to between the two types of objects remains constant. By physically arranging and counting the objects, students calculate cost times outside of that. concretely experience how equivalent ratios form a proportional relationship. We call the relationship continuous if it would make sense to include the values between points and discrete if only

Representational: Transition manipulatives to drawings and visual representations. Have students create the points make sense in from the context. ratio tables the can groups they made thedepending counters.onEncourage toThe draw pictures of the counters Two based differenton graphs be represented by with the data the order wethem choose. above graph shows relationship between cupsthe of ﬂour and cups the to cups of ﬂour and x represents the from for eachthe group or simply record numbers in of a sugar table.where Then,y represents guide them plot the pairs of numbers cupson of sugar. graphto represents ratio ofShow y to x them as 4 : 2.how each point represents a set of quantities that their tables graphThis paper create athe graph. have the same ratio. Help them draw a straight line through the origin that connects the points, illustrating the Sugar (cups) 6 proportional relationship visually. 5

We can Teach also create a graphhow for the of y to x proportional as 2 : 4 where y Abstract: Move on to abstract symbols and equations. students toratio represent 4 represents the cups of sugar and x represents the cups of ﬂour. relationships using equations like y = kx, where k is the constant of proportionality. Show them how to find the Notice the similarities and differences between the two graphs. 3 unit rate (the constant ratio) from their ratio tables and use it as k in the equation. Practice solving problems They both pass through (0, 0) but the ratio 4 : 2 is much steeper 2 using these equations without relying on manipulatives or drawings. Encourage students to compare different than the ratio 2 : 4. 1 proportional relationships by examining their equations, unit rates, and how they appear in graphs and tables. Flour (cups) This helps students recognize and represent proportional relationships using abstract mathematical notation. 1 2 3 4 5 6 7 8 9 10 11 It is important to state which quantity is represented by x and which quantity is represented by y. A graph is proportional if the graph is linear, meaning it looks like a straight line, and it passes through the origin, (0, 0).

Examples

The graph of a proportional relationship always includes (0, 0) because we can create the equivalent ratio 0 : 0 by

Students:multiplying Pages both 192–193 parts of the ratio by 0. For the sugar and ﬂour example, this would mean that a recipe that calls for 0 cups of ﬂour would need 0 cups of sugar.

Example 1 If the ratio of y to x is represented by 3 : 1, plot the ratio as a single point on the coordinate plane.

y 4 3 2 1 x 1

Create a strategy A ratio of the form y : x, means that the x represents the horizontal position and y the vertical position of the point.

192

Mathspace Virginia SOL Grade 6 mathspace.co

5.04 Proportional relationships mathspace.co

395

Apply the idea The horizontal position is 1, and the vertical position is 3. y 4 (1, 3)

3 2 1

x 1

Example 2 Purpose Consider the given graph: red Show students that they can translate a ratio into a point on a coordinate plane. 15

Model a think-aloud strategy Student with disabilities support

use with Example 1

Apply the idea

Model your thought position processis step-by-step asposition you translate the ratio 3 : 1 into a point on the coordinate plane. The horizontal 1, and the vertical is 3. Begin by saying aloud, “The ratio y : x is 3 : 1, so for every 1 unit of x, y is 3 units.” Explain how this means x = 1 5 y and y = 3. Next, demonstrate plotting the point (1, 3) on the graph, describing each action as you do it: “I move green 4 1 unit along the x-axis, then 3 units up the y-axis, and place my point here.” 5

(1, 3) Encourage students this think-aloud strategy in pairs, taking turns explaining their reasoning as they a What ratio ofto y :practice x has been plotted? 3 plot different ratios. This approach helps students who struggle with conceptual understanding and maintaining 2 of the problem-solving process. Over time, students will internalize attentionCreate by providing a clear, verbal model a strategy this method and apply it independently. Use the graph to ﬁnd the ratio, y : x. Often, we choose the point where x = 1 if y is a whole number as well. This will

give us the most simpliﬁed ratio.

the idea Students:Apply Page 193

x 1

The point (1, 2) lies on the line. At this point x = 1 and y = 2. So, the ratio is 2 : 1.

Reﬂect and check

Example 2

We could have chosen any point on the line in our graph. They all represent equivalent ratios. For example, if we chose thethe point (5, 10), we know: Consider given graph: red

So, the ratio 10 : 5 is an equivalent ratio to 2 : 1.

5 green 5

a What ratio of y : x has been plotted?

5.04 Proportional relationships mathspace.co

193

Create a strategy Use the graph to ﬁnd the ratio, y : x. Often, we choose the point where x = 1 if y is a whole number as well. This will give us the most simpliﬁed ratio.

Apply the idea The point (1, 2) lies on the line. At this point x = 1 and y = 2. So, the ratio is 2 : 1.

396

Reﬂect and check

Mathspace Virginia SOL Grade 6 Teacher Edition We could have chosen any point on the line in our graph. They all represent equivalent ratios. For example, if we mathspace.co chose the point (5, 10), we know:

a What ratio of y : x has been plotted?

Create a strategy Use the graph to ﬁnd the ratio, y : x. Often, we choose the point where x = 1 if y is a whole number as well. This will give us the most simpliﬁed ratio.

Apply the idea The point (1, 2) lies on the line. At this point x = 1 and y = 2. So, the ratio is 2 : 1.

Reﬂect and check We could have chosen any point on the line in our graph. They all represent equivalent ratios. For example, if we chose the point (5, 10), we know:

So, the ratio 10 : 5 is an equivalent ratio to 2 : 1.

Purpose Show students how to determine the ratio represented by a graph and understand that any point on the line represents an equivalent ratio. 5.04 Proportional relationships

193

mathspace.co Reflecting with students Encourage students to use multiple points on the line to determine the ratio. Discuss how all of the points on the line have a common ratio when simplified.

Students: Page 194

Purpose Challenge students to interpret ratios graphically and understand the relationship between the ratio and the real-world context it represents. Reflecting with students Encourage advanced learners to connect the ratio represented on the graph to the broader mathematical concepts of linear functions and slope. Highlight that the ratio y : x = 2 : 1 corresponds to the equation y = 2x, which is a linear function with a slope of 2. Invite students to explore how this constant ratio indicates a proportional relationship and how it’s graphically represented by a straight line passing through the origin. Ask them to consider how changing the ratio would affect the slope and the equation of the line—for instance, what happens to the graph if the ratio is 3 : 1 or 1 : 2? Encourage them to graph these new ratios to observe the differences.

Students: Page 194

5.04 Proportional relationships mathspace.co

397

Purpose Challenge students to apply their understanding of ratios and proportions to identify whether a relationship is proportional by examining a table of values.

Students: Pages 194–195

Apply the idea Fractions are equivalent if we can multiply the numerator and denominator by the same number. Let’s determine if the ratio of each coordinate pair is equivalent to the unit rate:

All of the ratios are equal because we can create each fraction by multiplying

by a constant.

Since the ratio of y : x is constant, this table represents a proportional relationship. c

y 20

Purpose Challenge students to apply their understanding of ratios and proportions to identify whether a relationship is 15 proportional by examining a table of values. 10 5 x

398

15 6 Teacher 20 Virginia5 SOL10Grade Edition

Mathspace mathspace.co

Create a strategy

Apply the idea

the ratio of each coordinate pair is equivalent to the unit rate:

Expected mistakes StudentsApply may the neglect idea to simplify each fraction and may not recognize that they are equivalent. Remind students that fractions must be fully siplified determine if relationships are proportional. Fractions are equivalent if we canto multiply the numerator and denominator by the same number. Let’s determine if All theof ratios equal because can create each theof ratio eachare coordinate pair is we equivalent to the unitfraction rate: by multiplying

by a constant.

Students:Since Page 195of y : x is constant, this table represents a proportional relationship. the ratio c

y 20 15

10 ratios are equal because we can create each fraction by multiplying All of the

by a constant.

Since the 5 ratio of y : x is constant, this table represents a proportional relationship. x

Create15a strategy

Apply the idea

In a proportional relationship, the coordinates will fall on 10 a straight line that goes through the origin.

The graph shows a straight line that goes through the origin. Therefore, this graph represents a proportional relationship.

5 x

10y 8

6 Purpose show students that they4can determine whether a relationship is proportional by examining a graph on a Create a strategy2 Apply the idea coordinate plane. x

In a proportional relationship, the coordinates will fall on −8 −6 −2goes through 2 4 6 the 8 origin. a straight line−4 that −2

Students: Page 195

−4

The graph shows a straight line that goes through the origin. Therefore, this graph represents a proportional relationship.

−6

−8 y 8 6 4

Create a strategy2

x In a proportional relationship, the coordinates will fall on −8 −6 −2 goes 2 through 4 6 8 a straight line−4 which the origin. −2 −4

Apply the idea The graph shows a straight line that does not go through the origin. Therefore, this graph does not represent a proportional relationship.

−6 −8

5.04 Proportional relationships mathspace.co

195

Create a strategy

Apply the idea

In a proportional relationship, the coordinates will fall on a straight line which goes through the origin.

The graph shows a straight line that does not go through the origin. Therefore, this graph does not represent a proportional relationship.

5.04 Proportional relationships mathspace.co

195

Purpose Show students that they can determine whether a relationship is proportional by examining a graph on a coordinate plane.

5.04 Proportional relationships mathspace.co

399

Students: Page 196 e Mark has $500 in savings and the amount in his savings account doubles every year.

Create a strategy In a proportional relationship, a situation with increase or decrease at the same rate. It will also make sense to have 0 of both quantities.

Apply the idea At the start (time 0), Mark had saved $500. If the amount of money doubles every year, this is not a constant rate. We Mark can create a table of values to the conﬁrm this:in his savings account doubles every year. e has $500 in savings and amount Years passed

0 1 2 3 4 500 1000 2000 4000 8000 In a proportional relationship, a situation with increase or decrease at the same rate. It will also make sense to have 0 The ﬁrst year, the account grows by $500. The next year, the account grows by $1000. Since the rate of increase is of both quantities. not constant and there are $500 in his account at the beginning, this situation is not proportional.

Create strategy Totalasavings

Apply the idea Reﬂect and check

At the start (time 0), Mark had saved $500. If the amount of money doubles every year, this is not a constant rate. We could have determined this relationship was not proportional from the start. The point (0, 0) could not fit this We can create a table of values to the confirm this:in his savings account doubles every year. e Mark has because $500 in savings and amount relationship, Mark has $500 at the start which means this relationship begins at (0, 500). Years passed 0 1 2 3 4 Create strategy 500 Totalasavings 1000 2000 4000 8000 f aItproportional just started raining, and itarains half an every or hour all day. at the same rate. It will also make sense to have 0 In relationship, situation withinch increase decrease The first year, the account grows by $500. The next year, the account grows by $1000. Since the rate of increase is both quantities. Purposeof not constant and there are $500 in his account at the beginning, this situation is not proportional.

Create ahow strategy Apply the idea Show students to determine whether a real-world situation involving a constant rate of change represents Apply the idea In a proportional relationship, a situation with increase At time 0, there were 0 inches of rain. Then, the rain fell Reﬂect and check a proportional relationship.

or at the will also fit the pattern to of the same amount eachyear, hour.this is not a constant rate. At decrease the start (time 0),same Markrate. had Itsaved $500. If the amount money doubles every We could have determined this relationship was not proportional from the start. The point (0, 0) could not fit this havecan 0 of both aquantities at some We create table of values to point. confirm this: Time passed begins 0 at (0, 1 500).2 3 4 because Mark has $500 at the start which means this relationship Students:relationship, Page 196 Years passed 0 1 2 3 4 1 2 Total rain 0 Total savings 500 1000 2000 4000 8000 f It just started raining, and it rains half an inch every hour all day. Since the rate of increase constant and there are is The first year, the account grows by $500. The next year, the account grows by $1000.isSince the rate of increase 0 inches of rain at the beginning, this situation is not constant and there are $500 in his account at the beginning, this situation is not proportional. Create a strategy Apply the idea proportional. In a proportional relationship, a situation with increase At time 0, there were 0 inches of rain. Then, the rain fell Reflect and check or decrease at the same rate. It will also fit the pattern to the same amount each hour. We determined relationship was not proportional from the start. The point (0, 0) could not fit this havecould 0 of have both quantities at this some point. Time passed begins 0 at (0, 1 500).2 3 4 Example relationship,4because Mark has $500 at the start which means this relationship Total rain 0 1 The ratio table represents the proportional relationship between number of pens purchased and cost.

fa It just started all day.the rate of increase is constant and there are Complete theraining, table: and it rains half an inch every hour Since 0 inches of rain at the beginning, this situation is Pens 10 20 30 40 50 Apply the idea proportional. Cost (dollars) 11.60 29.00 58.00 In a proportional relationship, a situation with increase At time 0, there were 0 inches of rain. Then, the rain fell or decrease at the same rate. It will also ﬁt the pattern to the same amount each hour. Create strategy have 0 ofa both Example 4 quantities at some point. Time passed 0 1 2 3 4 PurposeIn a ratio table, every column must represent an equivalent ratio. We can multiply or divide the number of pens by any 1cost. 2 Total rain 0 overand valueratio asthat long we do therate same the costrelationship in column. how students aasrepresents constant oftochange inthat a real-world context, like rainfall time, is indicative of a The table the proportional between number of pens purchased

Create a strategy

proportional relationship. a Complete the table: Pens

Students: Pages 196–197 Cost (dollars)

20 11.60

Since the rate of increase is constant and there are 0 inches of rain at the beginning, this situation is 50 proportional. 29.00 58.00

Create a strategy

Example 4

In a ratio table, every column must represent an equivalent ratio. We can multiply or divide the number of pens by any 196 Mathspace Virginia SOL Grade 6 value mathspace.co as long as we do the same to the cost in that column. The ratio table represents the proportional relationship between number of pens purchased and cost. a Complete the table: Pens Cost (dollars)

20 11.60

50 29.00

58.00

Create a strategy 400

In table, every column must6represent an equivalent ratio. We can multiply or divide the number of pens by any Mathspace Virginia SOL Grade 196a ratio Mathspace Virginia SOL Grade 6 Teacher Edition value mathspace.co as long as we do the same to the cost in that column. mathspace.co

The ratio table represents the proportional relationship between number of pens purchased and cost. a Complete the table: Pens Cost (dollars)

20 11.60

50 29.00

58.00

Create a strategy In a ratio table, every column must represent an equivalent ratio. We can multiply or divide the number of pens by any value as long as we do the same to the cost in that column.

Apply the idea Since 20 pens cost $11.60, 10 pens (half of 20) cost $11.60 ÷ 2 = $5.80. 30 pens cost 3 times more than 10 pens. 40 pens cost 4 times more than 10 pens.

Apply the idea 58 = 29 ⋅ 2, so the number of pens is 100.

Since 20 pens cost $11.60, pens6 (half of 20) cost $11.60 ÷ 2 = $5.80. 196 Mathspace Virginia SOL10Grade Pens 10 20 30 40 50 mathspace.co 30 pens cost 3 times more than 10 pens. Cost (dollars) 5.80 11.60 17.40 23.20 29.00 40 pens cost 4 times more than 10 pens.

100 58.00

58 = 29 ⋅ 2, so the number of pens is 100.

Reﬂect and check

Pens 20 by dividing 30 the cost 40by the number 50 of pens. 100 We could have also found10the unit rate Cost (dollars) 5.80 11.60 17.40 23.20 29.00 58.00

Apply the idea We couldand thencheck multiply $0.58 by any number of pens to ﬁnd the total cost. Reﬂect

20 pens cost $11.60, 10 pens (half of 20) cost $11.60 ÷ 2 = $5.80. PurposeSince We could have also found the unit rate by dividing the cost by the number of pens. 30 pens cost 3 times more than 10 pens. Demonstrate to students how to use proportional reasoning to complete a ratio table in real-world context. b pens Calculate costmore of buying pens. 40 cost the 4 times than 90 10 pens.

=could 29 ⋅ 2, so the number of pens is number 100. Students:58 Page 197 We then multiply $0.58 by any of pens to ﬁnd the total cost.

Create a strategy

Apply the idea

Reﬂect check Create aand strategy

Apply the idea

Pens 10 20 30 40 50 100 We know the cost of of 10 pens is $5.80. Consider how Cost = 9 ⋅ 5.80 Find the cost of 9 lots of 10 pens Cost (dollars) 5.80 11.60 17.40 23.20 29.00 58.00 many groups the of 10 pens have. b Calculate cost of we buying 90 pens. = $52.20 Evaluate

cWe How you expect payby fordividing 5 pens?the cost by the number of pens. couldmuch havewould also found the unittorate We know the cost of of 10 pens is $5.80. Consider how Cost = 9 ⋅ 5.80 Find the cost of 9 lots of 10 pens many groups of 10 pens we have. = $52.20 Evaluate

Create a strategy

We could thenthe multiply $0.58 any so number of pens to ﬁnd the total cost. 5 pens is half amount of 10bypens they will cost half as much. c How much would you expect to pay for 5 pens?

Apply the idea

Reﬂect and check

Purposeb Calculate the cost of buying 90 pens. Create a strategy We could also use the unit rate we found in part (a). The Cost = 5.80 ÷ 2 Halve the cost of 10 pens Show students how to apply the proportional relationshipunit derived from theForratio table, to calculate the cost of a rate was $0.58. 5 pens: 5 pens is half the amount of 10 pens so they will cost half as much. = $2.90 Evaluate Create a strategy Apply the idea non-table value. We know the cost of of 10 pens is $5.80. Consider how Apply the idea groups have. Students:many Page 197of 10 pens weHalve Cost = 5.80 ÷ 2 the cost of 10 pens = $2.90

Evaluate

Cost = 9 ⋅ 5.80

= $52.20 Evaluate We could also use the unit rate we found in part (a). The unit rate was $0.58. For 5 pens: $0.58 ⋅ 5 = $2.90

c How much would you expect to pay for 5 pens?

Example 5

$0.58 ⋅ 5 = $2.90 Find the cost of 9 lots of 10 pens

Reﬂect and check This is the same result as our other strategy.

This is the same result as our other strategy.

Create a strategy

Zoe eats 6 sour candies every minute. 5 pens is half the amount of 10 pens so they will cost half as much. a Does this situation represent a proportional relationship?

Example 5 Apply the idea

Create a strategy

Reﬂect and check Apply the idea

Create a strategy

Apply the idea

In a proportional relationship, a situation will increase or decrease at the same rate. It will also ﬁt the pattern to have 0 of both quantities at some point.

If 0 minutes has passed, Zoe will have eaten 0 candies. The ratio of candy eaten to time is always 6 : 1 so this situation represents a proportional relationship.

Create a strategy

Apply the idea

In a proportional relationship, a situation will increase or decrease at the same rate. It will also ﬁt the pattern to have 0 of both quantities at some point.

If 0 minutes has passed, Zoe will have eaten 0 candies. 5.04 Proportional relationships The ratio of candy eaten to time is always 6 : 1 so mathspace.co this situation represents a proportional relationship.

We could also use the unit rate we found in part (a). The Cost = 5.80 ÷ 2 Halve the cost of 10 pens Zoe eats 6 sour candies everyaminute. In a proportional relationship, situation will increase or If 0 minutes passed, will have eaten 0 candies. unit rate washas $0.58. For 5Zoe pens: = $2.90 Evaluate decrease at the same rate. It will aalso ﬁt the pattern to The ratio of candy eaten to time is always 6 : 1 so this a Does this situation represent proportional relationship? ⋅ 5 = $2.90 have 0 of both quantities at some point. situation represents $0.58 a proportional relationship. This is the same result as our other strategy.

Example 5 Purpose 5.04 Proportional relationships 197 mathspace.co Show students applyevery the proportional relationship derived from the ratio table, to calculate the cost of a Zoe eats how 6 sourto candies minute. non-table value. a Does this situation represent a proportional relationship? 5.04 Proportional relationships mathspace.co

197

401

Reﬂect and check We could have also found the unit rate by dividing the cost by the number of pens.

We could then multiply $0.58 by any number of pens to ﬁnd the total cost.

Discussion supports for proportional reasoning

use with Example 4

English language learner support b Calculate the cost of buying 90 pens.

Provide students with discussion prompts and sentence stems to facilitate meaningful conversations about completing the ratio table and calculating the costs of pens. For example, offer sentence stems like “To find the Create a strategy Apply the idea cost of ⬚ pens, I ⬚,” or “The relationship between the number of pens and the cost is ⬚.” Encourage students We know the cost of of 10 pens is $5.80. Consider how Cost = 9 ⋅ 5.80 Find the cost of 9 lots of 10 pens to use key vocabulary andEvaluate “divide.” During group discussions, many groups of 10such pens as we “ratio,” have. “proportion,” “unit rate,” “multiply,” = $52.20 prompt students to explain their reasoning by asking questions like “How did you determine the cost for 90 pens?” or “What pattern do you notice in the ratio table?” These supports help students articulate their c How much would you expect to pay for 5 pens? mathematical thinking while practicing English language skills. Additionally, consider creating a visual word bank on the board with relevant mathematical terms and phrases Create a strategy that students can reference discussions. Encourage students to build on each other’s ideas by using 5 pens is half the amountduring of 10 pens so they will cost half as much. phrases like “I agree with ⬚ because ⬚,” or “I have a different approach; I think ⬚.” This strategy not only Apply the idea Reﬂect and check reinforces their understanding of proportional relationships but also promotes a collaborative and language-rich We could use the unitthoughts. rate we found in part (a). The Cost = 5.80 ÷ 2 Halve the cost of 10 pens classroom environment where English learners feel supported in also sharing their = $2.90

Evaluate

unit rate was $0.58. For 5 pens:

$0.58 ⋅ 5 = $2.90

Students: Page 197

This is the same result as our other strategy.

Example 5 Zoe eats 6 sour candies every minute. a Does this situation represent a proportional relationship?

Create a strategy

Apply the idea

In a proportional relationship, a situation will increase or decrease at the same rate. It will also ﬁt the pattern to have 0 of both quantities at some point.

If 0 minutes has passed, Zoe will have eaten 0 candies. The ratio of candy eaten to time is always 6 : 1 so this situation represents a proportional relationship.

Purpose 5.04 Proportional relationships Show students that constant rates of change represent proportional relationships. mathspace.co

197

Reflecting with students Encourage students to attend to precision. When defining variables in this context, have students clearly state that x represents time in minutes and y represents the number of sour candies eaten. When creating the table, ensure students label the columns as “Time (minutes)” and “Candies Eaten” to include both the variable and its unit. On the graph, prompt students to accurately label the axes with both the variable and the unit—such as “Time (minutes)” on the horizontal axis and “Candies Eaten” on the vertical axis. When writing the equation y = 6x, have them explain that for every minute (x), Zoe eats 6 candies (y), reinforcing the relationship between the variables and their units. By attending to these details, students will communicate their mathematical reasoning more precisely and develop a deeper understanding of the proportional relationship.

402

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 198 b Create a table and a graph that represent this situation.

Create a strategy Let y be the number of candies Zoe eats and x be the time in minutes. We can use the ratio of y : x, which is 6 : 1, to create both a table and a graph.

Apply the idea The ratio ofa candies to atime is always 6 : 1 in this b Create table and graph that represent thistable: situation. x (Time)

2 3 4 5 10 12 18 24 30 60 Let y be the number of candies Zoe eats and x be the time in minutes. We can use the ratio of y : x, which is 6 : 1, to To graph, we will assume that Zoe is continuously eating the candy. We will use our ratio to graph and connect our create both a table and a graph. points with a line:

Create a strategy 6 y (Candies)

y 30 The ratio of candies to time is always 6 : 1 in this table: 27 x (Time) 1 2 3 424 5 10 21 y (Candies) 6 12 18 24 30 60 18 15 To graph, we will assume that Zoe is continuously eating the candy. We will use our ratio to graph and connect our 12 points with a line: 9 6 y (1, 6) 30 3 x 27 1 2 3 4 5 6 7 8 9 24 21 18 Reﬂect and check 15 12 y In this problem, we assumed Zoe was eating the candy 9 30 continuously, so we drew the line on the graph to connect 6 27 (1, 6) points. If we instead knew that she was eating each piece the 3 x 24 instantaneously at each minute we would not connect the points 21 1 2 3 4 5 6 7 8 9 because the values in between would have no meaning since she 18 does not eat additional candy between each minute. 15 12 Reﬂect and check 9 In this problem, we assumed Zoe was eating the candy 6 y (1, 6) 30 continuously, so we drew the line on the graph to connect 3 x 27 the points. If we instead knew that she was eating each piece 1 2 3 4 5 6 7 8 9 24

Apply the idea

instantaneously at each minute we would not connect the points

21 because the values in between would have no meaning since she 18 does not eat additional candy between each minute. 15 c Compare the characteristics that are easier or harder to see from the written context, the table, and the graph. 12 Purpose 9 Create6a strategy Challenge students(1,to 6) represent a proportional relationship using a table and a graph Consider relationship that are important. Use the representation from parts (a) 3 the different parts of a proportional x

and (b) to answer the question.

1 2 3 4 5 6 7 8 9 Students: Pages 198–199

Apply the idea context makes it easy to seethat theare connection We can easily see and the unit rate of cTheCompare the characteristics easier orbetween harder tothe seeratio fromand thereal-life. written context, the table, the graph. 6 candies per minute. It is harder to see that the situation includes the point (0, 0).

Create a strategy Consider the different parts of a proportional relationship that are important. Use the representation from parts (a) SOL Grade 6 198 and (b)Mathspace to answerVirginia the question. mathspace.co

198

Mathspace Virginia SOL Grade 6 mathspace.co

5.04 Proportional relationships mathspace.co

403

c Compare the characteristics that are easier or harder to see from the written context, the table, and the graph.

Create a strategy Consider the different parts of a proportional relationship that are important. Use the representation from parts (a) and (b) to answer the question.

Apply the idea The context makes it easy to see the connection between the ratio and real-life. We can easily see the unit rate of 6 candies per minute. It is harder to see that the situation includes the point (0, 0). The table highlights that every ratio of y : x is equivalent and we can see how many candies were eaten for each of the different minutes. However, the table won’t have all of the possible values. 198 Mathspace Grade 6 the relationship goes through the origin. It is easy to see that the relationship The graph makesVirginia it easySOL to see that mathspace.co

increases at a constant rate because it is linear, but it takes a little more work to see what the rate actually is.

Idea summary Purpose We can use ratio tables to determine unknown values by multiplying or dividing. All of the values in the table will be equivalent ratios. Challenge students to compare and contrast the information that is easier or harder to see from the written The table highlights that every ratio of y : x is equivalent and we can see how many candies were eaten for each of The graph of a proportional extended far enough) is a straight line that passess through the context,the thedifferent table, and the graph in arelationship real-world(ifscenario. minutes. However, the table won’t have all of the possible values.

origin (0, 0). The graph makes it easy to see that the relationship goes through the origin. It is easy to see that the relationship Students:increases Page 199 at a constant rate because it is linear, but it takes a little more work to see what the rate actually is.

Practice Idea summary

What We do can youuse remember? ratio tables to determine unknown values by multiplying or dividing. All of the values in the table 1

will be equivalent ratios. For each of the following representations, how can you tell if a relationship is proportional? The graph of a proportional relationship (if extended far enough) is a straight line that passess through the i Graph ii Table iii Context origin (0, 0). Find the missing value in each case, given that the two quantities are in proportion: a

Practice 3

If the following ratios are equivalent to 5 : 8, ﬁnd the missing value: 10 : ⬚

What do you remember?

⬚ : 96

Practice

2.5 : ⬚

⬚:1

Given that the relationship is proportional, explain how to use unit rate to ﬁnd the value missing in the table.

Forxeach of how can you tell if a relationship is proportional? ⬚ 5 the following 7 10 representations, 15 i yGraph15 ii Table iii Context 21 30 36 45

Students: Pages 199–205 2 5

Find the missing value in each case, given that the two quantities are in proportion: Which of the following represents a proportional relationship?

The number of pages in ba book and the color of the book cover. c What do youaA remember? 3

For each of representations, how can you tell if a relationship is proportional? a the 10 : following ⬚ b ⬚ : 96 d ⬚:1 D The amount of gas in a car’s tank and the color of theccar.2.5 : ⬚

Graph ii isTable Context 4 Given that the relationship proportional, explain how to iii use unit rate to ﬁnd the value missing in the table.

i 2

7 5

404

b Consider the given table:

10 : ⬚

⬚ : 96

2.5 : ⬚

⬚:1

C The height of a building and the number of doors in the building. a Graph the relationship shown by the table. The amount of gas a car’s tank and the color of the car. Given thatbDthe relationship is inproportional, Does the relationship include (0, 0)? explain how to use unit rate to ﬁnd the value missing in the table. 6 cSarah drove 180between miles in the 3 hours to of reach her destination. What was average speed for the trip?the ratio of Is the ratio values y and corresponding values ofher x consistent? If so, determine

x y

Which of the following represents a proportional relationship? 0 1of pages 2 in a 3book and the color of the book cover. A xThe number If the following ratios are equivalent to 5 : 8, ﬁnd the missing value: 0 3 a bag6of apples 9 and the number of apples in the bag. B yThe weight of

a 4

Sarah drove 180 miles in 3 hours to reach her destination. What was her average speed for the trip?

x ⬚ 5 7 each 10 case, 15 Find the missing value in are in proportion: A B given that the two quantities C D

a 3

B The weight of a bag of apples and the number of apples in the bag. If theThe following are equivalent 5 : 8, ﬁnd the missing value: C height ratios of a building and the to number of doors in the building.

⬚

y : x. A 15 21 30 36 B 45 d State whether the relationship is proportional.

Consider the given table: Which7 of the following represents a proportional relationship?

x 0 1 3 The number of pages in 2a book and the color of the book cover

The weight of a bag of apples and the number of apples in the bag

a Graph relationship by the table. The height of athe building andshown the number of doors in the building

b Does the relationship include (0, 0)? The amount of gas in a car’s tank and the color of the car

5.04 Proportional relationships mathspace.co

199

Is the ratio between the values of y and corresponding values of x consistent? If so, determine the ratio of y : x. Mathspace Virginia SOL Grade 6 Teacher Edition d State whether the relationship is proportional. mathspace.co c

5.04 Proportional relationships mathspace.co

199

Sarah drove 180 miles in 3 hours to reach her destination. What was her average speed for the trip? A

Consider the following table: x y

0 0

1 3

2 6

3 9

Graph the relationship shown by the table.

Does the relationship include (0, 0)?

Is the ratio between the values of y and corresponding values of x consistent? If so, determine the ratio of y : x.

State whether the relationship is proportional.

The number of cupcakes eaten by guests at a party is shown on the graph: a

How many cupcakes are eaten by 3 guests?

How many cupcakes are eaten by 2 guests?

Is the ratio between the number of guests and the corresponding values of cupcakes eaten consistent? If so, determine the ratio of x : y that has been plotted.

State whether the graph represents a proportional relationship.

No. of cupcakes

10 9 8 7 6 5 4 3 2 1

No. of guests 1

Match each table with its missing value. a

x 1 2 3 y 2 4

4 8

x y

9 3

24 8

39 13

x 9 15 21 y 12 16 20 28

x y

1 6

2 12

5 30

Let’s practice 10

Consider the following tables of proportional relationships. i

Find the unit rate.

Complete the missing values to show a proportional relationship between x and y.

x 1

y 5 15

6 8 50

x 4

40 50

y 6 12 48 60

x 3 12 24 48

y 1 2 4 16

x 324 108 36 12 4

y 81

5.04 Proportional relationships mathspace.co

405

A trail mix recipe states that 2 cups of peanuts should be mixed with a of raisins.

of a cup

Peanuts

Raisins

Complete the given table.

4 8 10 12 12

It requires approximately 8 mL of polyethylene to create 20 plastic bags. a

Find the unit rate of polyethylene used per plastic bag.

Complete the following table to demonstrate this relationship. mL bags

c 13

SOL

How much polyethylene would be required to create 5 plastic bags.

Ivan can wash 75 cars in 5 days. a

Create a graph that matches this scenario.

At this rate, how many cars can Ivan wash in 11 days?

A bus travels at a speed of 80 km/hr. How far does the bus travel in 5 hours?

State whether each situation is proportional. Explain your reasoning.

Lisa is trying to save money. She starts with no money in her account. After 3 weeks, she had saved $234.21. After 8 weeks, she had saved $624.56.

The amount of money a restaurant makes doubles every hour.

Two games cost $15. Five games cost $37.50.

Two chickens laid 7 eggs. Five chickens 13.

For each of the following table of values, state whether or not they represent a proportional relationship. a

x 1 2 3 y 2 4 6

4 8

5 10

x y

0 0

1 6

2 11

3 4

x 1 2 3 y 3 5 7

4 9

5 11

x y

0 0

1 2.5

2 5

3 7.5

x 1 2 3 y 2 3 5

4 7

6 9

x y

16 5.5

12 3.5

8 1.5

4 30

4 0.5

4 10 0 2.5

Examine the table. Time (minutes) 5 8 12 24

Cost (in cents) 25 40 60 120

Does a proportional relationship exist between the time spent on a cell phone and the cost of Wi-Fi? Explain your reasoning.

406

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The cost of eggs is shown in the table below. Number of Eggs Cost (dollars) a

48 11.04

36 eggs

Is the cost proportional to the number of eggs?

A tap ﬁlls up a 240 L tub in 4 hours. a

Find the rate of water ﬂow of the tap.

Complete the table of values:

4 240

How many liters of water can a tap ﬁll up in

hours?

Harry and Carl love reading. They both read at a constant rate. Harry reads 16 books every 12 weeks. Carl has kept a table of his reading habits which is shown below: Number of weeks Number of books read a

12 20

24 40

36 60

48 80

Complete the following table for Harry: Number of weeks Number of books read

36 8.64

12 eggs

Time (hours) Liters of water

24 6.00

Find the cost per egg for each quantity: i

12 3.12

12 16

48 32

60 80

Determine who reads more quickly.

The tables represent a proportional relationship with a constant unit rate of change of y with respect to x. Table 1:

Table 2:

x x 1 2 3 4 y y 3 6 9 12

2 8

3 12

4 16

5 20

Which table has a greater unit rate of change of y with respect to x? 22

State whether each of the following graphs represents a proportional relationship. a

y 20

4 3

1 −4 −3 −2 −1 −1 −2

x 1

5 −20−15 −10 −5 −5

x 5 10 15 20

−10

−3

−15

−4

−20

5.04 Proportional relationships mathspace.co

407

y 25

6 4

−8 −6 −4 −2 −2

2 4 6 8

−4 −6

−8

x 5 10 15 20 25 30 35 40 45

5 10 15 20 25 30 35 40 45 −15 −10 −5

For each of the following graphs: i

Determine the ratio y : x.

Describe the ratio as a unit rate for the given values.

The graph shows the amount of time it takes Kate to make beaded bracelets. No. of bracelets made

The graph shows the number of liters of gas used by a ﬁghter jet per second. 8

Liters of gas

4 3

Time (hours) 1

408

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Flight time (seconds) 2

The graph shows the number of liters of ice cream per tub.

Liters of ice cream

Distance (miles) 6

6 5

Time (minutes)

No. of tubs 4

The graph shows the distance Natalia swam per minute.

16 20 24 28

15 20 25 30 35

For each graph: i

Is the relationship shown proportional?

Explain your reasoning?

Spoons of chocolate 11 10 9 8 7 6 5 4 3 2 1

No. of fish caught 14 12 10 8 6 4

Cups of milk 1

No. of hours 1

The ratio of y : x is 6 : 2. By ﬁnding two points that represent equivalent ratios to 6 : 2, graph the ratio on a coordinate plane.

The table and the graph represent a proportional relationship with a constant unit rate of change of y with respect to x. x y

2 13

4 26

6 27

16.5

7 45.5

11 5.5 −3 −2

−1 −5.5

x 1

−11 −16.5 −22

Which describes a greater unit rate of change of y with respect to x, the table or the graph?

5.04 Proportional relationships mathspace.co

409

Let’s extend our thinking 27

The coordinate plane shows the parking fee for various numbers of hours parked: a

Does the graph represents a proportional relationship? Explain your reasoning.

What should have been the hourly cost of parking if the ﬁrst three ordered pairs are only considered in the graph?

If the relationship is proportional, the parking fee of $18 should be for how many hours?

20 18 16 14 12 10 8 6 4 2

Parking fee ( y)

No. of hours (x) 1 2 3 4 5 6 7 8 9 10

The graph shows the cost, in dollars, of printing x digital photos: a

Is the cost of printing digital photos proportional to the number of photos printed?

Determine the ratio of the number of photos to the cost of printing digital photos.

At this rate, how much would it cost to print 70 digital photos?

Cost (y) 3

1 No. of photos (x) 1 2 3 4 5 6 7 8 9 10

The table and the graph represent a proportional relationship with a constant unit rate of change of y with respect to x. x y

16 35.2

48 105.6

y 11 8.8 6.6 4.4 2.2

144 316.8

Murphy says that the table has a greater unit rate than the graph because the unit rate of the table is 3 while the graph is only 2.2.

−10

Raven said that the table and the graph has the same unit rate which is 2.2. Which student is incorrect? Explain.

The cost y of buying x pounds of fruit is shown on the graph. One graph shows the cost of buying x pounds of apples, and the other shows the cost of buying x pounds of bananas. a

Which graph has the greater unit rate?

Which fruit costs more per pound?

11 10 9 8 7 6 5 4 3 2 1

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Cost

Apples

Bananas

410

−5 −2.2 −4.4 −6.6 −8.8 −11

No. of pounds 5 6 7

Consider the scenarios: • Han paid $124.62 for 67 L of petrol in Humbleton. • Amelia paid $82.08 for 57 L of petrol in Dunkilderry. a

Calculate the cost per liter of petrol in Humbleton.

Calculate the cost per liter of petrol in Dunkilderry.

In which suburb is petrol cheaper?

Han buys 30 L in Statesota, and this petrol station matched the best price between Humbleton and Dunkilderry. How much did he pay?

5.04 Proportional relationships mathspace.co

411

Answers

b i 1.5

5.04 Proportional relationships What do you remember? 1 i

or a graph to be proportional, it must go through the F origin and increase at a constant rate which makes it linear.

c i

2 a

3 a 16

b 60

c 4

d 0.625

d i

Peanuts

4 Since the relationship is proportional, we know the ratio between y and x is constant. First, ﬁnd the unit rate, which is 3. For the ratio of x : y to be 1 : 3, the number in the blank must be 12.

12 48

324

243

108

Raisins

4 8

6 B

7 a y 9 8 7 6 5 4 3 2 1

12 a mL/plastic bag or 0.4 mL/plastic bag

b Yes.

c Yes. 3 : 1

d Yes.

b 4 cupcakes are eaten by 3 guests. c Yes. 1 : 2 d Yes. b A

c B

d D

Let’s practice ii

0.4

0.8

1.6

bags

13 a

8 a 6 cupcakes are eaten by 3 guests.

9 a C

b mL

c 2 mL

x 1 2 3 4 5 6 7 8 9

412

5 B

10 a i 5

ii F or a graph to be proportional, (0, 0) must ﬁt the pattern in the table and the ratio between the 2 quantities must be constant. iii For a context to be proportional, it must make sense that 0 of one quantitiy means you have 0 of the other quantiity. Quantities must also be changing at a constant rate.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

190 days 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10

(11, 165)

cars

1 2 3 4 5 6 7 8 9 10 11 12 13 14

b 165 cars 14 400 km

24 a i Yes

15 a Y es, for both numbers of weeks, she had saved $78.07 per week. Lisa also starts with $0.00 in her account. A constant rate and starting with $0.00 at 0 weeks means this relationship is proportional.

ii Explain your reasoning? It increases at a constant rate and passes through the origin.

b N o, the rate they earn money keeps increasing as time passes. c Y es, for both numbers of games, the rate per game is $7.50 per week. Zero games would also cost $0.00. A constant rate and starting with $0.00 at 0 weeks means this relationship is proportional.

b i No ii Explain your reasoning? The line drawn does not pass the origin (0, 0). There is also no common ratio. 25 Example answer:

d N ot, the rate is not constant between the number of chickens and the number of eggs laid. 16 a Proportional

18 16 14 12 10 8 6 4 2

b Not proportional

c Not proportional

d Proportional

e Not proportional

Not proportional

17 Yes, a proportional relationship exists because the cost per minute remains constant. Each minute costs 5 cents which is conﬁrmed by dividing the cost by the time for each pair. 18 a i $0.26

Let’s extend our thinking

19 a 60 L/hr

27 a No. The points do not form a straight line.

Time taken (hours)

b $2

Liters of water

240

180

120

28 a Yes

c 90L 20 a

x 2 4 6 8 10 12 14 16 18

26 The table

ii $0.24

b No

Number of weeks

Number of books read

30 a Apples

21 Table 2

31 a $1.86/L

e No 23 a i 5 : 1

b Yes f

c No

d Yes

b 4:1

c $17.50

29 Murphy is incorrect. Murphy computed the unit rate by dividing x values to its preceding x values or y values to its preceding y values. He should have divided y value to its corresponding x values.

b Carl

22 a Yes

c 9 hours

c Dunkilderry

b Apples b $1.44/L d $43.20

No ii 5 bracelets per hour

b i 1 : 2

liters per second

c i 1 : 4

liters per tub

d i 1 : 5

miles per minute

Answers mathspace.co

413

Topic 5 Assessment: Ratios & Proportional Relationships 1

Express each pair of quantities as a ratio: a

24 pounds to 42 pounds

90 hours to 150 hours

18 apples to 54 apples

Create a situation, in words, that each ratio could represent. a

2:3

5:3

Write both a ratio and a fraction to represent each grouping. Explain what it means. a

Frogs to turtles

Basketballs to soccer balls

414

A class had adopted tadpoles as class pets. Currently, thirty-seven are sill tadpoles and thirteen are frogs. a

Write the part-to-part ratio of tadpoles to frogs.

Write the part-to-whole ratio of frogs to total class pets.

How many more tadpoles must turn into frogs to have a frog to tadpole ratio of 1 : 1?

A train travels 120 kilometers in 4 hours and continues for another 180 kilometers in 6 hours. a

Create a ratio table to represent the ratio of time to distance for the ﬁrst 6 hours.

Use the ratio table to ﬁnd the distance travelled in the ﬁrst 2 hours.

Find the unit rate.

For each of the following ratios: i

Create a ratio table with at least four equivalent ratios.

Find the unit rate.

iii

Create a story that the ratio could represent.

2 : 3

3:6

The ratio of players to coaches participating in a community sports event is 7 : 3. If 49 players take part in the event, create a ratio table to determine how many coaches there must be.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Choose all of the following that represent a ratio of 6 : 5. A

1 5 6 30 7 35 12 60

or every 6 minutes James runs, he walks for F 1 minute.

y 22 20 18 16 14 12 10 8 6 4 2

x 2 4 6 8 10 12 14 16 18 2022

Find the missing value in the following ratio tables. a

1 2 6 3 9 4 12 5 15

5 10 15 20 25

4 8 16 20

Determine if the following represent proportional relationships. a

Jen walks 2 dogs per hour.

4 3

2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

SOL

1 2 4 5 6

2 4 16 32 64

A student describes the relationship of two groups of objects at school with the ratio of 5 : 8. Use words to describe what groups of objects the student may be comparing.

Complete the ratio table using the proportional relationship. Cups of sugar Cookies

3 10

5.04 Proportional relationships mathspace.co

415

Performance Task 13

416

Four runners are training for a 15 mile race. They are all planning to run at a consistent pace. Here are their recent speeds: Dante: 4 miles in 25 minutes

Lisa: 3 miles in 20 minutes

Mariella: 2 miles in 12 minutes

Pablo: 5 miles in 28 minutes

How far can Lisa run in one hour?

How fast can Dante run one mile?

To determine who will win a race, would it be better to ﬁnd how fast each runner can run a mile or how many miles they can each run in one hour? Justify your answer.

Who would win a 15 mile race?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

b i

Topic 1 Assessment: Equations & Inequalities Topic 5 Assessment: Ratios & Proportional Relationships 1 a 24 : 42

b 18 : 54

ii 2

iii For every 1 hour that passed, we walk 2 miles.

c 90 : 150

6.PFA.1e, 6.PFA.2a, 6.PFA.1d

6.PFA.1a, 6.PFA.1b 2 a A nswers vary. Martin needs 2 cups of sugar for every 3 cups of ﬂour in his recipe.

7 They need 21 coaches. Players

b A nswers vary. Gabriella needs 5 tablespoons of blue paint for every 3 tablespoons of red paint to make the perfect shade of purple. c A map is drawn to scale where every map represents 1 mile in real life.

inch on the

Coaches

6.PFA.1e, 6.PFA.1f, 6.PFA.2b

6.PFA.1d

8 A, D

3 a Ratio: 2 : 5

6.PFA.2e

Fraction:

For every 2 frogs, there are 5 turtles.

9 a 3 6.PFA.2b

b Ratio: 6 : 4, can also be written 3 : 2 Fraction: , can also be written For every 6 basketballs, there are 4 soccer balls or For every 3 basketballs, there are 2 soccer balls. 6.PFA.1a, 6.PFA.1b

10 a No

11 Answer vary. The student could be comparing the number of students who prefer pens to those who prefer pencils.

c Twelve more tadpoles must turn into frogs.

Distance (km.)

120

180

b 60 km

6.PFA.2b

13 a Lisa can run 9 miles in one hour. b D ante can run one mile in approximately minutes.

6.PFA.1f, 6.PFA.2a, 6.PFA.2d 1.5

Performance Task

c 30 km/hr

Cups of sugar Cookies

6.PFA.1c

6 a i

c No

6.PFA.1d

b 13 : 50

Time (hours)

b Yes

6.PFA.2c

4 a 37 : 13

5 a

b 12

ii 1.5

iii For every 2 eggs in the recipe, we need 3 tablespoons of milk.

= 6.25

c A nswers may vary. If you decide to calculate how many miles they can each run in one hour, this directly compares their speeds over a ﬁxed time, making it clear who is faster. If you decide to calculate how fast they can each run 1 mile, you can multiply this by 15 to ﬁnd each of their race times. d P ablo would win a 15 mile race because he has the highest speed of 10.71 miles per hour and it would take him 84 minutes.

Answers mathspace.co

417

6 Equations & Inequalities Big ideas • Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. • The properties of real numbers can be applied to many types of expressions. • An equals sign indicates an equivalent relationship between two expressions. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 6.01 6.02 6.03 6.04 6.05 6.06

Algebraic expressions (6.PFA.3) Properties of real numbers One-step equations with addition and subtraction (6.PFA.3) One-step equations with multiplication and division (6.PFA.3) Write inequality statements (6.PFA.3) Solutions to inequalities (6.PFA.4)

422 436 453 477 494 506

An algebraic expression with ‘x’ can represent anything from the number of candies in a jar to the cost of your favorite game!

6. Equations & Inequalities Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 5.PFA.2 — The student will investigate and use variables in contextual problems.

6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

Big ideas and essential understanding Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 6.01 — The structure of an expression can reveal important details about the situation it represents.

An equals sign indicates an equivalent relationship between two expressions. 6.03, 6.04 — The properties of equality allow an equation to be manipulated without changing the equivalence of the expressions on either side.

The properties of real numbers can be applied to many types of expressions. 6.02 — The properties of real numbers can be applied to simplify and evaluate algebraic expressions more easily.

A solution set is the collection of all values that make an equation or inequality true. 6.05 — An inequality is a mathematical sentence that compares two expressions and uses the symbols >, <, ≤, or ≥. 6.06 — Inequalities have an infinite number of solutions so their solution sets are often represented on a number line.

Standards 6.PFA.3 — The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable. 6.PFA.3a — Identify and develop examples of the following algebraic vocabulary: equation, variable, expression, term, and coefficient. 6.01 Algebraic expressions

420

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.PFA.3b — Represent and solve one-step linear equations in one variable, using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale). 6.03 One-step equations with addition and subtraction 6.04 One-step equations with multiplication and division 6.05 Write inequality statements

6.PFA.3c — Apply properties of real numbers and properties of equality to solve a one-step equation in one variable. Coefficients are limited to integers and unit fractions. Numeric terms are limited to integers. 6.03 One-step equations with addition and subtraction 6.04 One-step equations with multiplication and division 6.05 Write inequality statements 6.PFA.3d — Confirm solutions to one-step linear equations in one variable using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale). 6.03 One-step equations with addition and subtraction 6.04 One-step equations with multiplication and division 6.PFA.3e — Write a one-step linear equation in one variable to represent a verbal situation, including those in context. 6.03 One-step equations with addition and subtraction 6.04 One-step equations with multiplication and division 6.05 Write inequality statements 6.PFA.3f — Create a verbal situation in context given a one-step linear equation in one variable. 6.03 One-step equations with addition and subtraction 6.04 One-step equations with multiplication and division

6.PFA.4 — The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line. 6.PFA.4a — Given the graph of a linear inequality in one variable on a number line, represent the inequality in two equivalent ways (e.g., x < −5 or 5 > x using symbols. Symbols include <, >, ≤, ≥. 6.06 Solutions to inequalities 6.PFA.4b — Write a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line. 6.06 Solutions to inequalities 6.PFA.4c — Given a linear inequality in one variable, create a corresponding contextual situation or create a number line graph. 6.06 Solutions to inequalities 6.PFA.4d — Use substitution or a number line graph to justify whether a given number in a specified set makes a linear inequality in one variable true. 6.06 Solutions to inequalities 6.PFA.4e — Identify a numerical value(s) that is part of the solution set of a given inequality in one variable. 6.05 Write inequality statements 6.06 Solutions to inequalities

7.PFA.3 — The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

7.PFA.4 — The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable.

6. Equations & Inequalities mathspace.co

421

6.01 Algebraic expressions Subtopic overview Lesson narrative In this lesson, students will learn about algebraic expressions, focusing on translating verbal phrases into algebraic expressions and representing these expressions using algebra tiles. They will explore terms, coefficients, and constants within expressions. The lesson includes an exploration where students define variables to model real-world scenarios, such as the cost of a home renovation. Students will practice creating and simplifying algebraic expressions, understanding how to use visual models like algebra tiles to represent and manipulate these expressions. By the end, students should be able to confidently translate and represent algebraic expressions.

Learning objectives

6.01 Algebraic expressions

Students: Page 208

After this lesson, you will be able to... • identify examples of an equation, variable, expression, term, and coefficient. • write examples of an equation, variable, expression, term, and coefficient. • represent algebraic expressions with visual models.

Translate algebraic expressions We use algebraic expressions when we want to write a number sentence but we don’t know one of the Key vocabulary numbers involved.   equation coefficient For example: What is the total constant weight of a cat and a 10 lb weight?  number  term  variable In this case, let’s use c for the weight of the cat. 

Essential understanding

 expression

c + 10

The structure of an expression can reveal important details about the situation it represents.

10 lb

Standards

weight = cat plus 10Standards = c + 10 This subtopic addresses the following VirginiaTotal 2023 Mathematics of Learning standards.

Mathematical process c + 10 is called an algebraic goals expression which is an expression that contains at least one variable. c is called a variable. This is a symbol used to represent an unknown MPG4 — quantity. Mathematical Connections MPG3 — Mathematical Reasoning Coefficients are the mathematical numerical factor in a termby and are usedTeachers to show how we have. The variable canmany makevariables mathematical connections by ulinking Teachers can support reasoning guiding with a coefficient of 3reasoning is written as which means students to use logical to3u understand the 3 ⋅ u. the concept of variables and algebraic expressions to structure of algebraic expressions. For instance, teachers students’ prior knowledge from 5.PFA.2a. They can 3u = 3also ⋅ urelate these concepts to real-world situations. can ask students to identify the terms, coefficients, and For instance, teachers can discuss how algebraic constants in given algebraic expressions. Students3canthe coefficient expressions can be used to represent real-world also be encouraged to reason why expressions cannot u the variable calculating theblocks total cost be Terms solved,are while equations can be solved to find the in ansituations, a number, variable, product, and/or quotient expression.such Theyasare the building of anof items given the price per item. value of the variable that makes the statement true. expression. Terms are separated by + or − signs. Consider the expression:

422

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

• This is an expression with 2 terms. • The term

has a coeﬃcient of

. The negative belongs with the coeﬃcient.

MPG5 — Mathematical Representations Teachers can assist students in using mathematical representations by instructing them to represent word problems using algebraic expressions. For example, teachers can provide a problem about the number of apples a student has, and guide students to represent this problem with an algebraic expression. Teachers can also encourage students to visualize the problem, perhaps by drawing pictures or with concrete manipulatives such as algebra tiles, to support their understanding of the algebraic representation.

6.PFA.3a — Identify and develop examples of the following algebraic vocabulary: equation, variable, expression, term, and coefficient.

Prior connections 5.PFA.2 — The student will investigate and use variables in contextual problems.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.01 Identify and represent integers

Tools You may find these tools helpful: • Counters • Blocks or beads of different colors

6.01 Algebraic expressions mathspace.co

423

Student lesson & teacher guide Translate algebraic expressions Students: Pages 208–209

6.01 Algebraic expressions After this lesson, you will be able to... • identify examples of an equation, variable, expression, term, and coefficient. • write examples of an equation, variable, expression, term, and coefficient. • represent algebraic expressions with visual models.

Translate algebraic expressions We use algebraic expressions when we want to write a number sentence but we don’t know one of the numbers involved. For example: What is the total weight of a cat and a 10 lb weight? In this case, let’s use c for the weight of the cat.

c + 10

10 lb Total weight = cat plus 10 = c + 10

c + 10 is called an algebraic expression which is an expression that contains at least one variable. c is called a variable. This is a symbol used to represent an unknown quantity. Coeﬃcients are the numerical factor in a term and are used to show how many variables we have. The variable u with a coeﬃcient of 3 is written as 3u which means 3 ⋅ u.

3u = 3 ⋅ u

Exploration 3

the coeﬃcient

In order to write an expression that can be useduto the model the total cost of a home renovation project, Ms. Chen variable deﬁnes the variables: Terms are a number, variable, product, and/or quotient in an expression. They are the building blocks of an Let w represent the separated cost replacing expression. Terms are by +aorwindow, − signs.and p represent the cost of painting a room. 1. the What could these expressions represent in this context? Consider expression: • w

• p

• 3w

• 5p

• w+p

• This anthis expression terms. 2. is In context, with what2do the coefficients describe? • The has a coefficient of write . The with the coefficient. 3. term What expressions could we thatnegative wouldn’tbelongs make sense in this context? • The term 5 has no variable. It is called a constant term. Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Example 1 For the algebraic expression 4x + 23: 208

Mathspace Virginia SOL Grade 6 mathspace.co

a Determine the number of terms.

424

Mathspace SOL Grade 6 Teacher Edition Create Virginia a strategy mathspace.co Terms are separated by plus or minus signs in the expression.

Apply the idea The algebraic expression 4x + 23 contains two terms: 4x and 23.

Exploration Students: Page 209

Exploration In order to write an expression that can be used to model the total cost of a home renovation project, Ms. Chen deﬁnes the variables: Let w represent the cost replacing a window, and p represent the cost of painting a room. 1.

What could these expressions represent in this context? • w

• p

• 3w

• w+p

• 5p

In this context, what do the coeﬃcients describe?

What expressions could we write that wouldn’t make sense in this context?

Expressions and parts of expressions, like factors and coeﬃcients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can Suggested student grouping: In pairsby the expressions. explain the relationships described

Students will be exploring how variables and expressions can be used to model real world situations. They will be asked to interpret Example 1 what different expressions could represent in this context, discuss the meaning of coefficients, and think about what expressions would not make sense. For the algebraic expression 4x + 23:

Ideal student responses a Determine the number of terms. These ideal responses may differ from other correct student responses. Less formal responses can be Create strategy the idea connected witha the more precise mathematical languageApply presented here. Terms are separated by plus or minus signs in the

The algebraic expression 4x + 23 contains two terms:

1. What could these expressions represent in this context? expression. 4x and 23. The expression w represents the cost of replacing one window. Similarly, p represents the cost of painting one room. 3w represents the cost of replacing three windows. 5p represents the cost of painting five rooms. b Identify the coefficientthe of the firstcost term.of replacing one window and painting one room. Finally, w + p represents total 2. In this context, what do the coefficients describe? Apply the idea Create a strategy The The coefficients describe the quantity of each task. For example, in the expression 5p, the coefficient 5 coefficient of a term is the number that is multiplied The first term is 4x, so the coefficient of the first term is 4. represents the cost of painting five rooms. by the variable in the term. 3. What expressions could we write that wouldn’t make sense in this context? c Identifylike the wp constant Expressions or term. wouldn’t make sense in this context, because it wouldn’t be meaningful to multiply

or divide the costs of these different types of work. Expressions like −p − w also wouldn’t make sense in this Create a strategy Apply the idea context. Since we’re cost ofis athe home project, no negative The constant term incalculating an algebraic the expression term renovation In the expression 4x +there 23, theisconstant term is cost, 23. or money not contain any or variable. back,that fordoes painting a room replacing a window. Additionally, expressions that include variables not defined in this context wouldn’t make sense. For example, w + s would not be meaningful, because we don’t know what s represents. Example 2 Purposeful questions A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of purchasing p pineapples.

• How can you write an expression to represent the total cost of replacing three windows and painting two rooms? Create a strategy Apply the idea • What does the coefficient in the expression 5w tell us about the renovation work being done? The total cost changes based on the number of The total cost is $3 times the number of pineapples • Is thepineapples expression w + p meaningful in this context? What does itThis represent, and whyby does it make sense? purchased. purchased. can be represented the algebraic expression of 3p.

6.01 Algebraic expressions mathspace.co

209

6.01 Algebraic expressions mathspace.co

425

Order of words, terms and operations Address student misconceptions One common misconception in translating word expressions to algebraic expressions is the assumption that the order of the words in the phrase or sentence corresponds directly to the order of the terms in the algebraic Exploration expression. In order to writethe an expression that be used tofrom model total costAofcommon a home renovation Ms. Chenthis as 15 − For example, consider phrase “15 is can subtracted a the number”. mistake project, is to translate deﬁnes the variables: x, assuming that “15” must come before “x” in the expression. However, the correct translation is actually x − 15. Let w represent the cost replacing a window, and p represent the cost of painting a room.

To avoid these and other misconceptions, it is important to carefully read and analyze the word expression, 1. What could these expressions represent in this context? paying attention to the order of operations and the meanings of the words. It can also be helpful to start with • w • p • 3w • 5p • w+p simpler examples and gradually work up to more complex ones. 2. In this context, what do the coefficients describe? Exploration 3. What expressions could we write that wouldn’t make sense in this context? In order to write an expression that can be used to model the total cost of a home renovation project, Ms. Chen defines the variables: Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Let w represent the cost replacing a window, and p represent the cost of painting a room. expressions in parts and as a whole while paying attention to the quantities represented by the variables can Students:Viewing Page 209could these expressions represent in this context? 1. theWhat explain relationships described by the expressions.

Examples

• w Exploration

• p

• 3w

• 5p

• w+p

Example 2. In 1this context, what do the coefficients describe? In order to write an expression can be used to model thesense total cost ofcontext? a home renovation project, Ms. Chen 3. What expressions couldthat we write that wouldn’t make in this For the algebraic expression 4x + 23: defines the variables: a Let Determine the number terms. a window, and p represent the cost of painting a room. w represent the costofreplacing Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. 1. What couldin these represent in this context? Viewing partsexpressions and as a whole while paying attention to the quantities represented by the variables can Create aexpressions strategy Apply the idea explain the relationships described by the expressions. • 3w • w • p • 5p 4x + 23 contains • two w +terms: p Terms are separated by plus or minus signs in the The algebraic expression expression. 4x and 23. 2. In this context, what do the coefficients describe?

Example 1 expressions could we write that wouldn’t make sense in this context? 3. What b Identify the coefficient of the For the algebraic expression 4x +first 23:term. and of expressions, PurposeExpressions a Determine theparts number of terms. like factors and coefficients, all have unique meanings in a given context. Create aexpressions strategy in parts and as a whole while paying attention the idea Viewing to the quantities represented by the variables can are Show students how to identify the number of terms in an Apply algebraic expression by recognizing that terms explain the relationships described by the expressions. The coefficient of a term is the number that is multiplied The first term is 4x, so the coefficient of the first term is 4. Create a strategy Apply the idea separated by plus or minus signs.

by the variable in the term. Terms are separated by plus or minus signs in the expression. 1 Students:Example Page 209

The algebraic expression 4x + 23 contains two terms: 4x and 23.

c Identify the constant term. For the algebraic expression 4x + 23: b Identify the coeﬃcient of the ﬁrst term. a Determine the number of terms. Create a strategy

Apply the idea

The constant term in an algebraic expression is the term Create a strategy Create strategy that doesa not contain any variable.

In the expression Apply the idea 4x + 23, the constant term is 23. Apply the idea

The coeﬃcient of a term is the number that is multiplied Terms are separated by plus or minus signs in the by the variable in the term. expression.

The first term is 4x, so the coefficient of the first term is 4. The algebraic expression 4x + 23 contains two terms: 4x and 23.

Example 2 c Identify the constant term. b Identify the coeﬃcient of the ﬁrst term. A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of purchasing p pineapples.

Purpose Create a strategy Apply the idea Createtoa students strategy Apply theinidea Demonstrate a term an algebraic expression. Create a strategy how to identify the coefficient ofApply the idea

The constant term in an algebraic expression is the term The coeﬃcient of a term is the number that is multiplied that not contain variable. The does total cost changesany based on the number of the variable in the term. Students:by Page 209 pineapples purchased.

In the expression 4x + 23, the constant term is 23. The first term is 4x, so the coefficient of the first term is 4. The total cost is $3 times the number of pineapples purchased. This can be represented by the algebraic expression of 3p.

cExample Identify the 2 constant term. 6.01 expressions 209 A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of Algebraic purchasing p pineapples. Create a strategy Apply the idea mathspace.co

The constant term in an algebraic expression is the term Create strategy that doesa not contain any variable.

In the expression 4x + 23, the constant term is 23.

The total cost changes based on the number of pineapples purchased.

The total cost is $3 times the number of pineapples purchased. This can be represented by the algebraic expression of 3p.

Example 2 426

Apply the idea

Mathspace Virginia SOL Grade 6 Teacher Edition A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of purchasing p pineapples. mathspace.co

Create a strategy

Apply the idea

6.01 Algebraic expressions mathspace.co

209

Expressions and parts of expressions, like factors and coeﬃcients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Example 1 Purpose the algebraic expression + 23: Instruct For students how to identify4x the constant term in an algebraic expression. a Determine the number of terms.

Vocabulary building: collect and display

use with Example 1

Create alanguage strategy learner support English

Apply the idea

Terms are separated by plus or minus signs in the

The algebraic expression 4x + 23 contains two terms:

As you explore the expression 4x + 23, encourage students to share their understanding of the terms “term,” expression. 4x and 23. “coefficient,” and “constant term.” As students discuss these concepts, listen for the language they use and collect their descriptions. Write their words and phrases on a chart or board that all students can see. b Identify the coeﬃcient of the ﬁrst term.

For example, students might say, “a term is a piece of an expression separated by plus or minus signs,” “the coefficient is the number multiplied by the variable,” or “the constant term is the number without a variable.” Create a strategy Apply the idea Display The these student-generated definitions alongside formal mathematical definitions. aids, coefficient of a term is the number that is multiplied The first term is 4x, so the coefficient Use of thevisual first term is 4.such as underlining or color-coding by the variable in the term. the terms in the expression (e.g., underline “4x” in blue and “23” in red) to help students visually separate the terms. By connecting student language c Identify the constant term. to mathematical vocabulary and providing visual representations, you support students in developing a deeper understanding of the concepts. Create a strategy

Apply the idea

The constant term in an algebraic expression is the term does not contain any variable. Students:that Page 209

In the expression 4x + 23, the constant term is 23.

Example 2 A local fruit stand charges $3 per pineapple. Write an algebraic expression for the total cost of purchasing p pineapples.

Create a strategy

Apply the idea

The total cost changes based on the number of pineapples purchased.

The total cost is $3 times the number of pineapples purchased. This can be represented by the algebraic expression of 3p.

6.01 Algebraic expressions mathspace.co

209

Purpose Show students how to develop an algebraic expression that represents a real-world situation.

Students: Page 210 Example 3 Write an algebraic expression for the phrase “seven more than twice x”.

Create a strategy

Apply the idea

Translate the terms into mathematical symbols and operations.

The phrase “seven more than” indicates that we need to add 7. The “twice” means multiply by 2, so “twice x” is 2x. We can combine the whole description into a single expression: 2x + 7

Example 4 Purpose The perimeter square canabe writtenphrase as 4s. Explain what each partexpression. of the expression represents. Show students how of toatranslate verbal into an algebraic Create a strategy First, we need to identify the two parts of the expression. The coeﬃcient is 4 and the variable is s. We know that the perimeter of an object is the distance around the outside edges and a square has 4 sides of equal length.

Apply the idea

Reﬂect and check Perimeter = 4s

We can see from the perimeter formula that there are 4 of an unknown quantity s.

6.01 Algebraic expressions mathspace.co Another way to represent the perimeter of a square is s + s + s + s. This shows that to ﬁnd the perimeter of a square, we just need to add the side length to itself

427

Advanced learners: Design their own expressions

use with Example 3

Targeted instructional strategies Encourage students to create their own verbal phrases and translate them into algebraic expressions. After working through “seven more than twice x”, invite advanced learners to invent different phrases like “three times the sum of x and five” or “half of the difference between x and ten,” and write the corresponding expressions. This 3activity allows them to explore how language maps to mathematical operations, deepening Example their understanding of algebraic structure. Write an algebraic expression for the phrase “seven more than twice x”.

Additionally, prompt them to think about whether different phrases can result in the same algebraic expression, which fosters Createanalytical a strategythinking about equivalence and simplification. Apply the ideaBy designing their own problems, studentsTranslate engagethemore with the symbols contentand and develop a richer grasp of than” how indicates verbal descriptions termsdeeply into mathematical The phrase “seven more that we need totranslate operations. add 7. The “twice” means multiply by 2, so “twice x” is 2x. to algebra. We can combine the whole description into a single expression: 2x + 7

Students: Page 210 Example 4

The perimeter of a square can be written as 4s. Explain what each part of the expression represents.

Create a strategy First, we need to identify the two parts of the expression. The coeﬃcient is 4 and the variable is s. We know that the perimeter of an object is the distance around the outside edges and a square has 4 sides of equal length.

Apply the idea

Reﬂect and check Perimeter = 4s

We can see from the perimeter formula that there are 4 of an unknown quantity s.

Another way to represent the perimeter of a square is s + s + s + s. This shows that to ﬁnd the perimeter of a square, we just need to add the side length to itself 4 times.

The coeﬃcient 4 represents the 4 equal length sides of the square. For 4s to be the perimeter, s must represent the length of one side of the square.

Idea summary

Purpose Expressions can be used to represent mathematical relationships. In an expression, sums often represent Show studentstotals howand to coeﬃcients identify and the components a mathematical a realandinterpret factors represent multiplication.of When interpreting anexpression expression in representing context, we world scenario. can use the units to help understand the meaning.

Using hands-on perimeter models

Represent algebraic expressions Targeted instructional strategies We can use algebra tiles to help us visualize algebraic expressions.

use with Example 4

To help The students understand what each part of the expression P = 4s represents, engage them in a hands-on tile x represents an unknown number. The tile +1 represents adding one unit and −1 represents subtracting one unit. activity by constructing squares with equal-length sides. Provide students with four equal-length sticks or strips Positive Negative of paper, each labeled with the variable s to represent the side length. Ask them to arrange these pieces to form a square and discuss how to calculate the perimeter by adding the lengths of all sides: P = s + s + s + s. or − x or + x +x −x Variable tiles Guide them to simplify this sum to P = 4s, highlighting how the coefficient 4 corresponds to the four sides of the square and s represents the length of one side. To reinforce the concept, display a diagram of a square −1 path of the perimeter being added. This with each side labeled s and arrows the square indicating the Unit tilesaround +1 visual and tactile approach helps students make the connection between the geometric figure and the algebraic expression, theirSOL understanding of how each component relates to the shape. Mathspace Virginia Grade 6 210 deepening mathspace.co

428

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

We can see from the perimeter formula that there are 4 a square, we just need to add the side length to itself of anperimeter unknownof quantity s. can be written as 4s. Explain what4 each The a square times.part of the expression represents. The coeﬃcient 4 represents the 4 equal length sides of the square. Create a strategy For 4swe to need be the s must represent length The coeﬃcient is 4 and the variable is s. First, toperimeter, identify the two parts of thethe expression.

Students:ofPage 210 one side of the square.

We know that the perimeter of an object is the distance around the outside edges and a square has 4 sides of equal length.

Apply Idea the idea summary

Reﬂect and check

Another way to represent the perimeter of a square is Perimeter Expressions can be used=to4srepresent mathematical relationships. In an expression, sums often represent s + s + s + s. This shows that to find the perimeter of and the coefficients and factors represent multiplication. When interpreting an expression in context, we We cantotals see from perimeter formula that there are 4 a square, we just need to add the side length to itself can use quantity the unitss.to help understand the meaning. of an unknown 4 times. The coefficient 4 represents the 4 equal length sides of the square.

Represent algebraic expressions

For 4s to be the perimeter, s must represent the length We can use algebra tiles to help us visualize algebraic expressions. of one side of the square. The tile x represents an unknown number. The tile +1 represents adding one unit and −1 represents subtracting one unit.

Represent algebraic expressions

Negative Students are introduced to the use of algebra tiles toPositive visualize and understand algebraic expressions. They learn summary how to representIdea unknown numbers, addition, and subtraction using different tiles and how to identify the terms of or − x sums often represent or +relationships. +x x Variable Expressions can be used totiles represent mathematical In− xan expression, equivalent algebraic expressions. totals and coeﬃcients and factors represent multiplication. When interpreting an expression in context, we can use the units to help understand the meaning. Students: Pages 210–211 +1 −1 Unit tiles

Represent algebraic 210 Mathspace Virginia SOL Gradeexpressions 6 mathspace.co

We can use algebra tiles to help us visualize algebraic expressions. The tile x represents an unknown number. The tile +1 represents adding one unit and −1 represents subtracting one unit. Positive +x

Variable tiles

Negative +x

−x

Unit tiles

−x

−1

This table demonstrates how expressions can be built using the tiles: 210 Mathspace Virginia SOL Grade 6 mathspace.co

Word Expression

Algebraic Expression

three more than x

x+3

three less than x

x−3

Representative with Algebra Tiles +x +1

+x −1

−1

the product of x and three

x×3

Algebra tiles can also help us identify the terms of the equivalent algebraic expression. Let’s break down the algebra tiles below. +x

Notice that there are two different types of algebra tiles. These represent the two terms in the expression. The first term in blue, are the two tiles with the +x. This represents the term 2x where the coefficient is the 2 and the variable is the x. The second term in green, are the five tiles with the +1. This represents the term 5. When we add them together, we get the algebraic expression 2x + 5.

Example 5 Write an equivalent algebraic expression and identify each term for the following:

Create a strategy

Apply the idea

There are many ways to write expressions that are algebraically equivalent by rearranging the terms and combining like terms, but for simplicity, we’ll directly reﬂect the layout shown by the tiles.

From the image, we have one positive variable tile and four positive unit tiles. To express this algebraically we 6.01 Algebraic expressions can write: mathspace.co x+1+1+1+1 Another way to write the expression is to count up the +1

429

Word Expression

Algebraic Expression

three more than x

x+3

three less than x

x−3

Use graphic organizers

Representative with Algebra Tiles +x +1

+x −1

Student with disabilities support

−1

+x the product x and three x×3 To support students whoofstruggle with conceptual processing and language, introduce graphic organizers that help them map verbal phrases to algebraic expressions. Create a+ xtwo-column chart where one side lists common verbal phrases (e.g., “three more than a number”) and the other side shows the corresponding Algebra tiles can also help the terms of the equivalent algebraic expression. down thestudents algebra to add algebraic expressions (e.g., “nus + identify 3”). Provide examples to get them started, and Let’s thenbreak encourage tiles below. to the chart as they encounter new phrases. +x

Examples variable is the x.

second term in green, are the ﬁve tiles with the +1. This represents the term 5. Students:The Page 211 When we add them together, we get the algebraic expression 2x + 5.

Example 5 Write an equivalent algebraic expression and identify each term for the following:

Create a strategy

Apply the idea

From the image, we have one positive variable tile and four positive unit tiles. To express this algebraically we can write: x+1+1+1+1

Another way to write the expression is to count up the +1 tiles and show that we have 4 in total: x+4 There are two terms in this expression and they are the x and the 4. These terms are seperated by the + sign.

Example 6 Purpose Representhow the expression −2xa−visual 5 usingrepresentation algebra tiles. Show students to translate of algebra tiles into an equivalent algebraic expression and identify each term in the expression. Create a strategy

We can use negative variable tiles and negative unit tiles Use repetition to simplify expressions to represent the expression.

Targeted instructional strategies

Apply the idea

−x

−1

−x

−1

use with Example 5 −1

Encourage students to notice the repetition in the visual representation of the four +1 tiles. Begin by having them write an expression that includes each tile individually: x + 1 + 1 + 1 + 1. Ask students to identify the 6.01 Algebraic expressions 211 repeated addition of +1 and discuss how this is equivalent to multiplication. Guide them to mathspace.co simplify the repeated addition by writing it as x + 4, combining the four +1 tiles into a single term. Emphasize how recognizing repetition helps in combining like terms and simplifying expressions. Have students identify the terms in the simplified expression, noting that x and 4 are the terms separated by the + sign. By focusing on the repetition and simplification, students can better understand how to translate visual models into algebraic expressions efficiently.

430

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 211 Example 6 Represent the expression −2x − 5 using algebra tiles.

Create a strategy

Apply the idea

We can use negative variable tiles and negative unit tiles to represent the expression.

−x

−1

−x

−1

6.01 Algebraic expressions mathspace.co

211

Purpose Show students how to visually represent algebraic expressions using algebra tiles, which facilitates their understanding of negative variables and negative units in an algebraic expression. Expected mistakes Students may be unsure how to represent negative variables using algebra tiles. They might think that algebra tiles only represent positive quantities, leading them to depict −2x − 5 incorrectly as 2x − 5 with positive x-tiles. To help students overcome this misconception, remind them that negative x-tiles are the opposites of positive x-tiles. Use a different color for negative tiles (for example, red for negative and green for positive) to clearly distinguish between them. Show side-by-side examples of positive and negative expressions using the tiles, such as representing both 2x + 3 and −2x − 3.

Students: Page 212

Idea summary We can represent algebraic expressions with visual models to better understand them. We can rearrange models of algebraic expressions to generate equivalent expressions.

Practice What do you remember? Practice 1

What is a variable used for in algebra?

Students: Pages 212–215 2

Farmer Jones has sheep, cows, goats and chickens on his farm. Use variables to write an expression for the total number of animals on the farm.

3 Match the following terms with the correct example from the options: What do you remember? i

Variable

Coeﬃcient

iii

Expression

Constant

2x + 5y

What is a avariable used 12 in the termfor 12xin algebra? b b in the term 5b

4 Jones Consider given diagram: Farmer hasthe sheep, cows, goats and chickens on his farm. Use variables to write an expression for the a An algebraic expression using addition to total number of animals on the farm. p p p p p

An algebraic expression using multiplication to from the options: Match thebfollowing terms with the correct example represent the diagram is 5 ⋅ ⬚.

Variable

Constant

125 in the 12x b a diagram b in thethat term 5b c 10 Use term algebra tiles to draw represents each expression:

2x + 5y

represent the diagram is ⬚ + ⬚ + ⬚ + ⬚ + ⬚.

Coeﬃcient

Positive +x

Variable tiles Unit tiles a

x+2

Negative −x

Expression

iii

Write the answer to part (b) in another way.

−x

6.01 Algebraic expressions mathspace.co

−1

2x + 1

3x − 4

−3x + 2

431

Consider the following diagram: a b c

An algebraic expression using addition to represent the diagram is ⬚ + ⬚ + ⬚ + ⬚ + ⬚.

An algebraic expression using multiplication to represent the diagram is 5 ⋅ ⬚. Write the answer to part (b) in another way.

Use algebra tiles to draw a diagram that represents each expression: Positive +x

Variable tiles

x+2

−x

Unit tiles a

Negative or

−x

−1

2x + 1

3x − 4

−3x + 2

Let’s practice 6

Examine the given diagram. a

Use the diagram to label each part with the correct algebraic term. Variable

Coeﬃcient

Expression

Constant

What algebraic term describes the entire mathematical statement x + 8 = 16?

c 7

Write down the number of terms in the following expressions: a

−2x

7y + 1

1024

−7 + (−8) + x + 2x

8y + 4 − (−12y)

5x − 7y + 8xyz − 12

What is the coeﬃcient in the following terms? a

−2y

−3y

8ab

−101pqr

−i

i 9

b f

e 8

x + 8?

How many terms are in

What is the constant term in the following expressions? a

a − 15

8y − 3

11 − u

8 + 5x

3xy + 5 + 4y − x

14 − 20ab + 4a

8y + x = −11

What is the constant term in the following equations? a

3n = 18

y = 5x + 3

10a + 7c − 1 = 16b

3xy + 7 − x = y

What is the difference between an expression and an equation? Provide an example of each in your explanation.

432

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Identify the following as equation or expression. a

3x + 5 = 14

2x + 3

4z = 12

2 − 5y

2y − 7 = 3y + 1

= −24

m − (−4) − 28

Expressions

Variables

In the equation 5xy − 3y = 45x + 6, list all of the: a

14 + 8z − z + 9

Coeﬃcients

Terms

Write a simpliﬁed algebraic expression for the following diagrams: a

Each single box represents p.

Each single box represents b.

Each single box represents y.

ach purple box represents p and each green box E represents g.

Write a simpliﬁed algebraic expression for the following diagrams: a

+1 +1 +1 +x

+x +x +x

−1

−x

−1

Provide an example of each of the following vocabulary terms.

Example Equation Variable Expression Term Coefficient

6.01 Algebraic expressions mathspace.co

433

Describe and correct the error in identifying the number of terms, coeﬃcients, and constants in the algebraic expression

• 3 Coeﬃcients: , x, and y • 1 Term:

• 0 Constants

Let’s extend our thinking 18

Write an algebraic expression for each of the following statements without using a multiplication or division sign: a

Seven multiplied by x

Eight more than 5x

Three less than 8y

The product of 9 and b

The quotient of 8u and 9

Four times y take away 9

7 less than the quotient of 5 and x

The sum of a and 4, divided by 6

Write a word statement for each of the following expressions: a

2b + 18

3x − 5

4n − 15

A teacher asked his students to write an expression to represent the number of red and yellow marbles in the jar. Delaney used the expression r + y to represent the total number of red and yellow marbles. Dalton used 9r + 6y to represent the the total number of red and yellow marbles. Who is correct? Explain your thinking using algebraic vocabulary.

A recipe for a cake calls for 2 cups of ﬂour for every 3 eggs. Write an algebraic expression for the number of cups of ﬂour needed for n eggs.

The sum of the ages of a mother and her daughter is 42 years. If the mother’s age is q years, write an algebraic expression for the daughter’s age.

Write down an algebraic expression for:

The total value of m coins, where each coin is worth p cents.

The length of string remaining when two pieces of length K cm are cut from a piece of string originally L cm long.

Describe and correct the error in writing the phrase as an expression. a

The quotient of 8 and a number y:

16 decreased by a number: x − 16

434

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

15 a x + 3

b 4x + 2

6.01 Algebraic expressions

1 A variable is used to represent an unknown number.

Example

3 i

ii A

iii D

4 a p + p + p + p + p

+x +x +x

17 The error is in identifying the number of coeﬃcients: , x,

−1

−x −x −x

term with a single coeﬃcient

Let’s extend our thinking

18 a 7x e

Let’s practice Constant

b Equation

c 1

d 4

g 3

h 4

b −2

c 6

d −3

e 5

g 9

−101

−1

b 11

c 0 or no constant term

d 8

e −3

g 5

h 14

e −1

c −11

b 3 f

d −4

An equation has an equals sign but an expression doesn’t. One example of an equation is x + 9 = 30 and an example

e Equation

Expression g Equation

13 a 5, −3, 45

h Expression

b 5xy, −3y, 45x, 6

c 5xy − 3y and 45x + 6 14 a 4p

b Expression c Expression d Equation f

b 5b

Delaney is correct because she let r represent the red marbles in the jar and let y represent the yellow marbles in the jar. The total number of red and yellow marbles in the jar would then be r + y. Dalton is correct because he let r represent 1 red marble. Since there are 9 red marbles, 9r represents the total number of red marbles. He then let y represent 1 yellow marble. Since there are 6 yellow marbles, 6y represents the total number of yellow marbles. The total number of red and yellow marbles in the jar would then be 9r + 6y.

22 42 − q

11 Answer may look like:

12 a Equation

of an expression is

20 Both students are correct but their variables represent different quantities.

b 2

9 a −15

4y − 9

d 9b

d The sum of thrice m and 21 divided by 10

c 2

c 8y − 3

c Fifteen subtracted from the product of 4 and n

Variables

Coefficient

b 5x + 8

and no constants.

b 5 subtracted from the product of x and 3

10 a 18

xy contains only one

19 a The sum of twice b and 18

6 a Expression

2x 2 in 2x

and y. In reality, the expression b

8 a 4

2x + 3

iv C

b 5⋅p

5 a +1 +1 +x

e 2

Expression Coefficient

c 5p

7 a 1

Variable Term

2 Possible answers: s + c + g + h or w + x + y + z

d −2x + 1

2x + 3 = 13

Equation What do you remember?

c 3x − 2

23 a p ⋅ m

b L − 2K

24 a T he given expression suggests dividing the number y by 8. However, the phrase “the quotient of 8 and a number y” actually means 8 is being divided by y, thus and not . b T he given expression x − 16 suggests subtracting 16 from a number x. However, the phrase “16 decreased by a number” means that we are taking a number away from 16, which should be x − 16.

d x and y c 6y

d p + 2g

Answers mathspace.co

435

6.02 Properties of real numbers Subtopic overview Lesson narrative In this lesson, students will explore the properties of real numbers, focusing on commutative, associative, identity, and inverse properties. The lesson includes an interactive exploration where students investigate the commutative properties by changing the order of addition or multiplication to see the effects. Another exploration involves choosing different types of numbers and observing the results of adding and multiplying by 0 and 1, and finding inverse operations. By the end, students should be able to use these properties to manipulate and simplify algebraic expressions effectively.

Learning objectives

6.02 Properties of real numbers

Students: Page 216

After this lesson, you will be able to... • identify and apply the associative, commutative, inverse, and identity properties of addition and multiplication.

Commutative properties

Key vocabulary

Interactive exploration



associativeExplore property online to answer the questions



commutative property



identitity property of multiplication



identity property of addition



mathspace.co inverse property of addition



inverse property of multiplication



 real number multiplicative property of zero Use the interactive exploration in 6.02 to answer these questions.

What do you notice when you change the order of the addition?

Essential 2. Whatunderstanding do you notice when you change the order of the multiplication? The properties of real numbers can be applied to simplify and evaluate algebraic expressions more easily. The commutative properties of real numbers are:

Standards

Property Symbols Example Commutative property of addition a+b=b+a −3 + 6 = 6 + (−3) ThisCommutative subtopic addresses following Virginia 2023 Standards Learning standards. propertythe of multiplication a ⋅ bMathematics =b⋅a 6 ⋅ (−3) =of(−3) ⋅6

Mathematical process goals The commutative property is the reason that we can add numbers in any order or multiply numbers in any order. Keep mind that while additionSolving and multipication are commutative, and division are not. MPG2 —subtraction Mathematical Communication MPG1 —inMathematical Problem Let’s see why the commutative property withby subtraction doescan not work. Teachers integrate this goal into their instruction Teachers can integrate this goal into theirapplied instruction 7 − 4 ≠with 4 − 7varying complexities Apply the of commutative with subtraction by encouraging students to justify their solutions to the providing students problem property situations related3 to real numbers. For example, teachers problems by using the language of mathematics. For ≠ −3 Simplify instance,the when students are solving problems, they canNotice provide that require students to with apply thatword if weproblems applied the commutative property subtraction, left and right side are not equal. We can should be asked to identify the mathematical property theadjust properties of real numbers to solve. this by turning the subtraction into a addition of a negative. they are using and explain why it is applicable. 7 − 4 = 7 + (−4) Rewrite subtraction as addition 7 − 4 = −4 + 7 436

Apply the commutative property of addition

= 3 SOL Grade 6 Simplify Mathspace 3Virginia Teacher Edition mathspace.co

We can see that when we convert the subtraction operation to an addition operation, we can still apply the commutative property of addition.

MPG3 — Mathematical Reasoning Teachers can integrate this goal into their instruction by providing opportunities for students to use inductive and deductive reasoning skills when applying the properties of real numbers. For instance, students can be asked to justify why the Commutative property of addition works for all real numbers.

Future connections 6.PFA.3 — The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable.

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Uncovering the properties Targeted instructional strategies Give students the following two problems: 4 + 5 + 6 and 1 × 2 × 3. Ask students to answer the following: 1. Write each problem again with the numbers in a different order. Do the answers stay the same? 2 Does putting parentheses around different pairs of numbers in each problem change the answer? 3. What number could we add to the first problem so it would not change the answer? 4. What number could we multiply in the second problem, and it would not change the answer? Alternatively, give students a different set of problems using negative numbers.

6.02 Properties of real numbers mathspace.co

437

Student & teacher guide 6.02lesson Properties of real numbers Commutative properties After this lesson, you will be able to...

Students start the lesson engaging in an exploration to build an and understanding of the commutative property. • identifyby and apply the associative, commutative, inverse, identity properties of addition and multiplication.

Students: Pages 216–217

Commutative properties

6.02 Properties of real numbers Interactive exploration Explore online to answer the questions

mathspace.co After this lesson, you will be able to... • identify and apply the associative, commutative, inverse, and identity properties of addition and Use the interactive exploration in 6.02 to answer these questions. multiplication. 1.

What do you notice when you change the order of the addition?

What do you notice when you change the order of the multiplication?

Commutative properties The commutative properties of real numbers are:

Interactive exploration

Property Symbols Explore online to answer the questions Commutative property of addition a+b=b+a mathspace.co Commutative property of multiplication a⋅b=b⋅a

Example −3 + 6 = 6 + (−3) 6 ⋅ (−3) = (−3) ⋅ 6

TheUse commutative property is the reason we canthese add numbers the interactive exploration in 6.02 that to answer questions.in any order or multiply numbers in any order. Keep in mind that while addition and multipication are commutative, subtraction and division are not. 1. What do you notice when you change the order of the addition? Let’s see why the commutative property applied with subtraction does not work. 2. What do you notice when you change the order of the multiplication? 7−4≠4−7 Apply the commutative property with subtraction 3 ≠ −3 Simplify The commutative properties of real numbers are: Notice that if we applied the commutative property with subtraction, the left and right side are not equal. We can adjust this by turningProperty the subtraction into a addition ofSymbols a negative. Example 7 − 4property = 7 + (−4)of additionRewrite subtraction Commutative a + b as = baddition +a −3 + 6 = 6 + (−3) 7 − 4property = −4 + 7of multiplication Apply the commutative of 6addition Commutative a ⋅ b = b ⋅property a ⋅ (−3) = (−3) ⋅ 6

3=3 Simplify The commutative property is the reason that we can add numbers in any order or multiply numbers in any order. We can see that when we convert the subtraction operation to an addition operation, we can still apply the Keep in mind that while addition and multipication are commutative, subtraction and division are not. commutative property of addition. Let’s see why the commutative property applied with subtraction does not work. Let’s see why the commutative property applied with division does not work. 7−4≠4−7 Apply the commutative property with subtraction Apply the commutative property with division 3 ≠ −3 Simplify Simplify Notice that if we applied the commutative property with subtraction, the left and right side are not equal. We can adjust this turning the the subtraction into aproperty additionwith of a division, negative.the left and right side are not equal. We can adjust Notice thatby if we applied commutative 7 − 4the = 7division + (−4) into a multiplication Rewrite subtraction as addition this by turning of it’s reciprocal. 7 − 4 = −4 + 7 3=3

Apply the commutative property of addition Rewrite division as multiplication Simplify

the commutative property of multiplication We can see that when we convert Apply the subtraction operation to an addition operation, we can still apply the commutative property ofSOL addition. Mathspace Virginia Grade 6 216 Simplify mathspace.co Let’s see why the commutative property applied with division does not work. We can see that when we convert the division operation to an multiplication operation, we can still apply the Apply the commutative property with division commutative property of multiplication. The commutative property can be Simplify applied to help us evaluate expressions more easily.

Example 1 Find the value of: 6 + (−5) + 5

438

216 Mathspace Virginia SOL Grade 6 Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co Create a strategy mathspace.co Since we can add numbers in any order, let’s add −5 and 5 ﬁrst to make the calculation easier.

Apply the idea

6.02 Properties of real numbers After this lesson, you will be able to... • identify and apply the associative, commutative, inverse, and identity properties of addition and multiplication.

Exploration Students: Page 216

Commutative properties Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 6.02 to answer these questions. 1.

What do you notice when you change the order of the addition?

What do you notice when you change the order of the multiplication?

The commutative properties of real numbers are: PropertyIn pairs Symbols Example Suggested student grouping: Commutative property of addition a + b = b + a −3 + 6 = 6 +or (−3) Students will be manipulating sliders to change the numbers being added multiplied. They will observe the Commutative property of multiplication a⋅b=b⋅a 6 ⋅ (−3) = (−3) ⋅ 6 changes in the squares and array when the operations are reversed. This exploration provides an interactive way for The students to understand the commutative properties of addition and multiplication. commutative property is the reason that we can add numbers in any order or multiply numbers in any order. Keep in mind that while addition and multipication are commutative, subtraction and division are not.

Ideal student responses

Let’s see why the commutative property applied with subtraction does not work.

These ideal responses may differ from other correct student responses. Less formal responses can be 7−4≠4−7 Apply the commutative property with subtraction connected with the more precise mathematical language presented here. 3 ≠ −3

Simplify

1. WhatNotice do you notice when you change the order of the addition? that if we applied the commutative property with subtraction, the left and right side are not equal. We can The adjust outcome doesn’t wheninto thea addition order ofofaddition this by turning change the subtraction a negative.is changed. This is a demonstration of the commutative 7property of addition. − 4 = 7 + (−4) Rewrite subtraction as addition − 4 = −4when +7 Apply thethe commutative of addition 2. What do you7notice you change order ofproperty the multiplication? 3 = 3the order of multiplication Simplify Similar to addition, doesn’t affect the outcome. This is a demonstration of the commutative We can seeproperty that whenof wemultiplication. convert the subtraction operation to an addition operation, we can still apply the commutative property of addition.

Purposeful Let’s questions see why the commutative property applied with division does not work. • How does the commutative property to other mathematical operations? Applyapply the commutative property with division • What are some real-world examples where the commutative property is applicable? Simplify

Possible misunderstandings • Students may assume that the commutative property applies to all mathematical operations. It’s important to clarify that it is only applicable to addition and multiplication.

216 Mathspace Virginia SOL Gradecorrect, 6 Error analysis: Critique, and clarify mathspace.co

English language learner support Present students with an incorrect or unclear explanation of the commutative property, such as “Multiplication is always commutative because 2 × 3 = 6 and 6 ÷ 3 = 2.” Ask students to work individually or in pairs to identify any mistakes or misunderstandings in the statement. Encourage them to discuss what is incorrect and why it doesn’t accurately describe the commutative property. Then, have students rewrite the explanation correctly, ensuring it clearly conveys the concept that changing the order of numbers in addition or multiplication does not change the result. This activity engages students in analyzing mathematical language and helps them develop a clearer understanding of the commutative properties while practicing their English language skills.

6.02 Properties of real numbers mathspace.co

439

Rewrite division as multiplication Apply the commutative property of multiplication Simplify

Examples We can see that when we convert the division operation to an multiplication operation, we can still apply the commutative property of multiplication.

Students: Page 217

The commutative property can be applied to help us evaluate expressions more easily.

Example 1 Find the value of: 6 + (−5) + 5 Notice that if we applied the commutative property with division, the left and right side are not equal. We can adjust

Create a strategy this by turning the division into a multiplication of it’s reciprocal. Since we can add numbers in any order, let’s add −5 and 5 ﬁrst to make the calculation easier. Rewrite division as multiplication

Apply the idea

Apply the commutative property of multiplication 6 + (−5) + 5 = −5 + 5 + 6 Rewrite using the commutative property of addition Simplify =0+6 Evaluate −5 + 5

We can see that when=we 6 convert the division Evaluateoperation to an multiplication operation, we can still apply the commutative property of multiplication.

Reﬂect and check The commutative property can be applied to help us evaluate expressions more easily. We could apply a different property to get the same result:

Example6 1+ (−5) + 5 = 6 + ((−5) + 5) Find the value of:

Rewrite using the associative property of addition

=6+0

Evaluate −5 + 5

Evaluate 6 + (−5) + 5

Create a strategy

2 PurposeExample Since we can add numbers in any order, let’s add −5 and 5 ﬁrst to make the calculation easier. Show students how to useproperty the commutative property Use the commutative of addition to and ﬁll in inverse the missing number: of addition to help with a multistep addition problem.Apply the idea 19 + (−15) = (−15) + ⬚ 6 + (−5) + 5 = −5 + 5 + 6

Rewrite using the commutative property of addition

Reflecting with students = 0 + 6 Evaluate −5 + 5 Create a strategy the idea Encourage students to explore why rearranging the termsApply in the expression 6 + (−5) + 5 makes the problem =6 Evaluate The commutative property of addition means that when We want to write 19 + (−15) therearrange opposite way easier to solve. Ask them how the commutative property of addition allows us to thearound. terms to group we add two numbers, it does not matter what order we 19 sum + (−15)is=zero. (−15) +Discuss 19 −5 and 5Reﬂect together. Highlight that −5 and 5 are additive inverses, and their how recognizing and check add them. and utilizing additive can simplify We could apply ainverses different property to get calculations the same result:in multistep problems. Invite students to consider if there are other pairs of6 numbers where this strategy could be applied. This will help them make + (−5) + 5 = in 6 +different ((−5) + 5) problems Rewrite using the associative property of addition generalizations about the =patterns observed when properties of operations to simplify expressions. 6+0 Evaluate −5 + using 5

Students: Page 217

Evaluate

Example 2

6.02 Properties of real numbers mathspace.co

217

Use the commutative property of addition to ﬁll in the missing number: 19 + (−15) = (−15) + ⬚

Create a strategy

Apply the idea

The commutative property of addition means that when we add two numbers, it does not matter what order we add them.

We want to write 19 + (−15) the opposite way around. 19 + (−15) = (−15) + 19

Purpose To ensure students can apply the commutative property of addition.

440

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.02 Properties of real numbers mathspace.co

217

Students: Page 218

Purpose Show students how to simplify complex multiplication by using the commutative property of multiplication. Expected mistakes Students may incorrectly assume that the commutative property of multiplication does not apply when negative numbers are involved. They might believe that changing the order of factors with negatives will alter the result, leading to hesitation in rearranging the factors to simplify the calculation. Reinforce to students that the commutative property holds true for all real numbers, including negative numbers. Demonstrate this by showing that −20 ⋅ 7 ⋅ 5 can be rearranged to −20 ⋅ 5 ⋅ 7 without changing the product. Use simple examples, such as comparing −3 ⋅ 4 = −12 and 4 ⋅ −3 = −12, to illustrate that the order of multiplication does not affect the outcome. Encourage students to practice rearranging factors in multiplication problems involving negative numbers to build confidence in applying the commutative property universally.

Students: Page 218

6.02 Properties of real numbers mathspace.co

441

Purpose Show students how to identify the commutative property of addition.

Students: Page 218

Associative properties Students explore the associative properties of real numbers, with a focus on addition and multiplication. They also delve into why these properties don’t hold for subtraction and division. Through hands-on examples, they apply the associative properties to evaluate and simplify expressions.

Students: Page 219

Associative properties The associative properties of real numbers are: Property Associative property of addition Associative property of multiplication

Symbols a + (b + c) = (a + b) + c a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

Example 6 + ((−3) + 2) = (6 + (−3)) + 2 6 ⋅ (−3 ⋅ 2) = (6 ⋅ (−3)) ⋅ 2

The associative property is the reason that we can group sums or products of numbers differently and the result remains the same. While addition and multiplication are associative, subtraction and division are not. Let’s see why the associative property applied with subtraction does not work. (7 − 4) + 3 ≠ 7 − (4 + 3) 3+3≠7−7 6≠0

Apply the associative property with subtraction Simplify inside of the parentheses Simplify

Notice that if we applied the associative property with subtraction, the left and right not equal. We can adjust this by turning the subtraction into addition by the opposite. (7 − 4) + 3 = (7 + (−4)) + 3

Rewrite subtraction as addition

(7 − 4) + 3 = 7 + (−4 + 3)

Apply the associative property of addition

3 + 3 = 7 + (−1) 6=6

Simplify inside the parentheses Simplify

We can see that when we convert the subtraction operation to an addition operation, we can still apply the associative property of addition. Let’s see why the associative property applied with division does not work. Apply the associative property with division Simplify inside of the parentheses Simplify Notice that if we applied the associative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it’s reciprocal Rewrite division in the parentheses as multiplication

442

Mathspace Virginia SOL Grade 6 Teacher Edition Rewrite division outside the parentheses as multiplication mathspace.co Apply the associative property of multiplication

Let’s see why the associative property applied with division does not work. Apply the associative property with division Simplify inside of the parentheses Simplify Notice that if we applied the associative property with division, the left and right side are not equal. We can adjust this by turning the division into a multiplication of it’s reciprocal Rewrite division in the parentheses as multiplication Rewrite division outside the parentheses as multiplication Apply the associative property of multiplication Simplify inside of the parentheses Simplify We can apply the associative properties to evaluate and simplify expressions.

Color-code groupings to support understanding Targeted instructional strategies When teaching the associative properties of addition and multiplication, use color-coding for the numbers to emphasize how the groupings have changed. For example, in the expression (a + b) + c = a + (b + c), assign 6.02 Properties realeach numbers 219 a specific color to each variable so students can see that there is a different pair of colorsof in set of mathspace.co parentheses. This strategy supports students with visual-spatial processing challenges by making abstract concepts more concrete and accessible.

Examples Students: Page 220

Purpose Students demonstrate that they can identify the property of integer operations used to show equivalent expressions with an equals sign.

6.02 Properties of real numbers mathspace.co

443

Expected mistakes Having an extra set of parentheses for the negative term in the expression may confuse some students in determining the change from the first expression to the second. Consider using square brackets instead to better differentiate them. Reflecting with students Encourage advanced learners to create more complex expressions with additional sets of parentheses. Challenge them to discover whether the associative property still applies as the number of parentheses increases.

Students: Page 220

Purpose Show students how to use the associative property to simplify calculations.

Students: Page 220

Inverse and identity properties Students: Page 221

444

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Exploration Students: Page 221

Suggested student grouping: Small groups In this exploration, students will be choosing four types of numbers - a negative number, a decimal, a whole number, and a fraction. They will then perform a series of mathematical operations on these chosen numbers and document their observations. Through this process, they will discover some of the fundamental properties of numbers and operations. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Add 0 to each of your chosen numbers. The answers may vary depending on the numbers chosen. However, adding 0 to any number doesn’t change its value. 2. Multiply each of your chosen numbers by 0. The answers may vary depending on the numbers chosen. However, multiplying any number by 0 gives the result of 0. 3. Multiply each of your chosen numbers by 1. The answers may vary depending on the numbers chosen. However, multiplying any number by 1 doesn’t change its value. 4. Try to find a number that you can add to each number and get 0. The answers may vary depending on the numbers chosen. However, the number that can be added to any number to yield 0 is its additive inverse (the negative of the number). 5. Try to find a number that you can multiply each number by and get 1. The answers may vary depending on the numbers chosen. However, the number that can be multiplied with any number (except 0) to yield 1 is its multiplicative inverse (the reciprocal of the number). 6. What patterns did you observe? Adding 0 to any number doesn’t change the number, and multiplying any number by 0 always gives 0. Multiplying any number by 1 also doesn’t change the number. The additive inverse of a number is what you need to add to get 0, and the multiplicative inverse of a number is what you need to multiply to get 1.

6.02 Properties of real numbers mathspace.co

445

Purposeful questions • Why do you think adding 0 to a number or multiplying a number by 1 doesn’t change its value? • What do you notice about the relationship between a number and its additive inverse? • What do you notice about the relationship between a number and its multiplicative inverse? Possible misunderstandings • Students may confuse the concept of additive inverses and multiplicative inverses. It is important to note that additive inverses are achieved by changing the sign of the number, while multiplicative inverses are achieved by taking the reciprocal of the number.

Examples Students: Page 221

Purpose Show students how to identify the inverse property of multiplication.

Confusing identity and inverse properties

use with Example 7

Address student misconceptions Students might confuse the identity property of multiplication with the inverse property of multiplication. They may believe that since the product is 1 in the equation

⋅ 12 = 1, it demonstrates the identity property, thinking

that any multiplication resulting in 1 relates to the identity property. To address this misconception, clarify the differences between the identity and inverse properties. Explain that the identity property involves multiplying any number by 1 to get the same number back (e.g., 12 ⋅ 1 = 12), whereas the inverse property involves multiplying a number by its multiplicative inverse (reciprocal) to get 1 .

446

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 222

Purpose Show students how to identify the inverse property of addition.

Advanced learners: Justifying the inverse property with multiple representations Targeted instructional strategies

use with Example 8

Encourage students to deepen their understanding of the inverse property of addition by using multiple representations to justify why it holds true for all real numbers. Ask them to represent a + (−a) = 0 using number lines, algebraic expressions, and visual models like integer chips or counters. For example, they can draw a number line where starting at zero, moving a units to the right, and then a units to the left brings them back to zero. With integer chips, they can pair positive and negative chips of equal value to show how they result in zero. By exploring these different representations, students will connect the abstract concept of additive inverses to representations, enhancing their conceptual understanding.

Students: Page 222

6.02 Properties of real numbers mathspace.co

447

Purpose Demonstrate to students how the identity property of multiplication and the inverse property of addition can be used to simplify an expression.

Students: Page 222

Purpose Illustrate to students how the inverse property of addition can be used to simplify an expression, even when it involves both addition and subtraction.

Students: Page 223

Idea summary Property Identity property of addition Identity property of multiplication Inverse property of addition

Symbols a + 0 = a and 0 + a = a a ⋅ 1 = a and 1 ⋅ a = a a + (−a) = 0 and (−a) + a = 0

Inverse property of multiplication

a⋅

Multiplicative property of zero

a ⋅ 0 = 0 and 0 ⋅ a = 0

= 1 and

⋅ a = 1; where a ≠ 0

Practice What do you remember? Practice 1

Match the name of the property to its description:

Students: Pages 223–225property of addition a Commutative b

Commutative property of multiplication

Identity property of addition

What do youd remember? Identity property of multiplication e

448

Inverse property of addition

Match the name of the property to its description: f

Inverse property of multiplication

a⋅b=b⋅a

a+0=a

iii

a+b=b+a

a + (−a) = 0

a⋅

a⋅0=0

Commutative property of addition g Multiplicative property of zero

i aa vii ⋅ 1 ⋅=ba = b ⋅ a ii

a+0=a

Commutative property of multiplication

Identity of identity addition iii 1 isaequal + b to = bitself. +a a property The additive is 1 because any number added to

Identity ofmultiplicative multiplication b property Zero has no inverse.

Inverse property of addition

5 + (−5) = 0 and −5 + 5 = 0

Inverse property of multiplication

Multiplicative e 8 ⋅ −12property of zero f 49 ÷ 7

Determine whether each statement is true or false:

Evaluate: a

13 + 8

−7 − 23

a + (−a) = 0

4⋅

a⋅

= 4 and

⋅4=4

a⋅0=0

12 + (−34)

a ⋅⋅ 1(−41) =a gvii −23

4 Complete each statement by ﬁlling in the blank(s): Mathspace a Virginia SOLcommutative Grade 6 Teacher Edition Using the property of addition: mathspace.co i 1+9=9+⬚ ii 19 + 15 = 15 + ⬚ iv 7 + (−23) = −23 + ⬚ v −8 + (−17) = ⬚ + (−8)

iii vi

−25 − (−29)

98 ÷ (−14)

−3 + 8 = 8 + ⬚

a+4=⬚+⬚

Determine whether each statement is true or false: a

The additive identity is 1 because any number added to 1 is equal to itself.

Zero has no multiplicative inverse.

5 + (−5) = 0 and −5 + 5 = 0

4⋅

= 4 and

⋅4=4

Evaluate: a

13 + 8

−7 − 23

12 + (−34)

−25 − (−29)

8 ⋅ −12

49 ÷ 7

−23 ⋅ (−41)

98 ÷ (−14)

Complete each statement by ﬁlling in the blank(s): a

Using the commutative property of addition: i iv

1+9=9+⬚

7 + (−23) = −23 + ⬚

3⋅5=⬚⋅3

19 + 15 = 15 + ⬚

iii

−8 + (−17) = ⬚ + (−8)

Using the commutative property of multiplication: i iv

6 ⋅ (−3) = ⬚ ⋅ 6

9⋅4=4⋅⬚

iii

0 ⋅ 12 = 12 ⋅ ⬚

−3 + 8 = 8 + ⬚

a+4=⬚+⬚

−2 ⋅ 7 = ⬚ ⋅ −2

b⋅8=⬚⋅⬚

Let’s practice 5

Choose the property that is demonstrated by each of the following statements: • Associative property • Commutative property • Inverse property • Identity property a

3⋅

5 + 14 = 14 + 5

−56 ⋅ 1 = −56

(−2 + 4) + 6 = 6 + (−2 + 4)

4 ⋅ (3 ⋅ 6) = (4 ⋅ 3) ⋅ 6

176 + (−176) = 0

−5 ⋅ 7 = 7 ⋅ −5

State whether the following equations are true or false: a

13 ⋅ 12 = 13 ⋅ 3 ⋅ 4

−4 ⋅ 1005 ⋅ 6 = −24 ⋅ 1005

20 ⋅ 75 = 20 ⋅ 3 ⋅ 25

−2 ⋅ 94 ⋅ −6 = 94 ⋅ (−12)

14 ⋅ 18 = 18 ⋅ 14

−6 + 6 − (−7) = 0 + 7

90 − 20 = 20 − 90

90 ÷ 6 ÷ (−3) = 90 ÷ (6 ÷ (−3))

Find the missing numbers in the following equations: a e g

16 ⋅ 15 ⋅ 13 = 15 ⋅ ⬚ ⋅ 16

−6 ⋅ 250 ⋅ 11 = ⬚ ⋅ 250

−3 ⋅ 88 ⋅ ⬚ = 88 ⋅ 36

17 ⋅ (−4) ⋅ 19 = −68 ⋅ ⬚

1230 ⋅ 3 = 123 ⋅ ⬚ ⋅ 3

3 ⋅ (8 ⋅ 5) = 3 ⋅ ⬚

d h

−9 + 9 = ⬚

15 ⋅ 1 + (−15) ⋅ 1

−1 ⋅ ⬚ = 0

⬚ + (−32) = 0 ⬚⋅7=1

11 + ⬚ = 11

41 − 14 + 0

3 ⋅ (1 ÷ 3)

−53 + (−37) + 53

9 + (−9) + 27 − 27

−22 ÷ 22 ⋅ 11

43 + ⬚ = 0

24 ⋅ 50 = 24 ⋅ ⬚ ⋅ 10

21 ⋅ (−250) = 21 ⋅ ⬚ ⋅ 50

Find the missing number in each equation: a

0 + 34 = 34

Evaluate:

⋅⬚=1

−8 ⋅ 1 = ⬚

6.02 Properties of real numbers mathspace.co

449

Choose the operation that would make these statements true.

A 11

−

⋅

20 − 5

−15

Select all expressions that are equivalent to −20 + 5: A

0 ⬚ 29 = 0 and 29 ⬚ 0 = 0

−10 + (−10 + 5)

5 + (−20)

Write an example equation that demonstrates each property: a

Additive Identity Property

Multiplicative Identity Property

Additive Inverse Property

Multiplicative Inverse Property

Answer the following questions: a

Find (32 ÷ 4) ÷ 2 by dividing 32 by 4 then dividing the result by 2.

Find 32 ÷ (4 ÷ 2) by dividing 4 by 2 then dividing 32 by the result.

Is the associative property applicable to division?

Consider the equation: 20 − 7 = 7 − 20

Determine whether the statement is true or false.

Does the commutative property apply to subtraction? Explain your answer.

A student claims that division is commutative and gives the example shown: 1÷5=5÷1 Is the student correct? Explain your answer.

Let’s extend our thinking 16

Consider the following statement: (a − b) − c = a − (b − c) Explain whether the statement is true for all, true for some, or never true for all integers a, b and c.

Imagine you have 10 apples and you want to share them equally, but you are told to share them with zero people. Using this scenario, explain why dividing by zero does not work and what it means for an operation to be “undeﬁned.”

Evaluate using the properties of real numbers:

0.5 + (−0.5) + 0

−2 + (−3 + 2)

1 ⋅ −1.25

3+0

4⋅

⋅ −2

Explain how the properties of real numbers can help you evaluate the following expression using mental math. a

−84 − 97 + 84

(21 + (−34)) − 21

−3 + (−58 − 7)

14 + 3 + 6 + 17

Rufino is given the expression : (−22) ⋅ 6 + (−22) ⋅ 4 Rufino says that he can use the associative property of multiplication to add the 6 and the 4, and then add (−22) + (−22), as it would be easier to multiply −44 ⋅ 10. Is Rufino correct? Explain your reasoning using the properties of real numbers.

450

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

15 The student is not correct. 1 ÷ 5 is equivalent to which cannot be simpliﬁed further. 5 ÷ 1 is equal to 5 since

6.02 Properties of real numbers

anything divided by 1 is itself. commutative.

What do you remember? 1 a iii

b i

c ii

e iv

g vi

b True

c True

2 a False

d vii

d 4

g 943

h −7

4 a i 1

ii 19

iii −3

iv 7

v −17

vi 4 + a iii 7

iv −3

b i 5

ii 9 vi 8 ⋅ b

v 0

(a − b) − c = (6 − 4) − 2 = 2 − 2 = 0

d False

c −22

e −96

Let’s extend our thinking 16 Suppose a = 6, b = 4 and c = 2. Then we have

b −30

3 a 21

≠ 5 so division is not

and a − (b − c) = 6 − (4 − 2) = 6 − 2 = 4 So the statement is not true for a = 6, b = 4 and c = 2. Suppose a = 6, b = 4 and c = 0. Then we have (a − b) − c = (6 − 4) − 0 = 2 − 0 = 2 and a − (b − c) = 6 − (4 − 0) = 6 − 4 = 2 So the statement is true for a = 6, b = 4 and c = 0.

Let’s practice 5 a Identity property

b Inverse property

c Commutative property

d Identity property

e Commutative property

g Inverse property

h Commutative property

This means that (a − b) − c = a − (b − c) for some integers a, b and c

b True

c True

d False

True

g False

h False

b 5

c −66

d −5

g 19

h 40

b 32

c 0

d 0

e 0

g 12

h −8

17 Dividing by zero is like trying to share 10 apples with nobody; it doesn’t make practical sense because there’s no way to distribute the apples to no recipients. Mathematically, division by zero is “undefined” because it is not consistent or meaningful. Division is the operation of finding how many times the divisor fits into the dividend. However, dividing by zero would imply that you are trying to see how many times “nothing” fits into the dividend, which is impossible. We use the term “undefined” since the operation doesn’t produce a result that fits within our established numerical system.

9 a 27

b 1

c −37

d 0

18 a 0

e 0

6 a True e True 7 a 13 e −12 8 a −43

Associative property

e −1.25

−11

b −3 f

c 4

19 a O bserve that the ﬁrst and the third term are opposites, with this we can use the commutative property of addition to reorder the terms.

10 C 11 A, B, D 12 a 16 + 0 = 16 and 0 + 16 = 16

−84 − 97 + 84 = −84 + 84 – 97 Rewrite using the commutative property of addition

b 34 ⋅ 1 = 34 and 1 ⋅ 34 = 34 c 27 + (−27) = 0 and −27 + 27 = 0 d 9 ⋅ 13 a 4

d 0

= 1; and

⋅9=1

b 16

c No

14 a False, 20 − 7 = 13 but 7 − 20 = −13 b T he commutative property does not apply to subtraction because changing the order of the numbers changes the sign of the result. In the given example, 13 is not the same as −13 because one is positive and one is negative.

= −84 + 84 – 97

Evaluate 0 − 97

= −97

Evaluate

b N otice that the ﬁrst and third terms are opposites, it will be easier to evaluate the sum of 21 and −21 as opposites sum to zero. Rewrite using (21 + (−34)) − 21 = −34 + (21 − 21) associative property of addition

= −34 + 0

Evaluate (21 − 21)

= −34

Evaluate

Answers mathspace.co

451

c U sing associative propety of addition, the sum does not change irrespective of how the numbers are grouped. −3 + (−58 − 7) = −58 – 7 Subtract the values −58 and − 7

= −3 + (−65) Add the values

= −68

− 3 + (−65) Evaluate

Or (−3 − 58) − 7 = −3 – 58

Subtract the values − 3 and − 58

= −61 – 7

Add the values − 61 − 7

= −68

Evaluate

d U sing the commutative proroprty of addition in which a + b = b + a, in this problem we can interchange 3 + 6 to 6 + 3 so both sides can equate to 20 + 20. 14 + 3 + 6 + 17 = 14 + 6 + 3 + 17

Interchange 3 + 6 to 6 + 3

= 20 + 20 Add the values 14 + 6 and

= 40 Evaluate

452

3 + 17

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

20 Ruﬁno’s approach is incorrect because he misinterprets the associative property of multiplication. The associative property allows us to change the grouping of numbers in a multiplication operation without affecting the result, but it does not apply to combining terms from separate multiplication operations before performing addition. His method unintentionally blends concepts that are not covered by the associative property.

6.03 One-step equations with addition and subtraction Subtopic overview Lesson narrative In this lesson, students will learn to model and solve one-step equations using addition and subtraction. They will explore various representations, such as balance scales, algebra tiles, and pictorial models, to understand the concept of keeping equations balanced. The lesson includes an interactive applet where students use a balance scale and algebra tiles to visualize and solve equations. They will apply step-by-step strategies for isolating the variable using the addition and subtraction properties of equality. Students will solve equations, verify solutions using substitution, and interpret solutions in real-world contexts. Additionally, they will write one-step linear equations to represent verbal situations and create verbal situations given one-step linear equations which involve addition and subtraction. By the end, students should confidently model, solve, and verify one-step addition and subtraction equations.

6.03 One-step equations with Learning objectives addition and subtraction Students: Page 226

After this lesson, you will be able to... • represent one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations. • solve one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations. • apply properties of real numbers and properties of equality to solve one-step equations involving addition or subtraction. • write a one-step linear equation involving addition or subtraction to represent a verbal or contextual situation. • use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving addition or subtraction. • write a verbal situation in context when given a one-step linear equation involving addition or subtraction.

Model equations An algebraic equation is a mathematical statement that says two expressions are equal.

Key vocabulary There are many ways to represent an algebraic equation. Some of the ways are:    

 • algebra addition property • balance scalesof equality • algebra tiles pictorialtiles models  balance scales equation Balance scales are beneficial because they show that the left and right sides of an equation are equal and so the

inverse operations equation is balanced. subtraction property of equality x−5



pictorial model

property The equationsubstitution shown on this scale is:of equality 

x−5=2

Essential understanding The properties of equality allow an equation to be manipulated without changing the equivalence of the expressions way to represent an algebraic equation is to use Algebra tiles. These tiles represent the variables and units on Another either side. on each side of the equation. Positive

Variable tiles

Negative 6.03 One-step equations with addition and subtraction mathspace.co +x

−x

453

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can integrate mathematical problem-solving into their instruction on solving one-step linear equations in one variable by linking concrete and numerical methods. Start with hands-on activities like using algebra tiles to represent and solve equations visually. Gradually transition to symbolic methods, emphasizing the connection between the two. Ensure students understand and apply the properties of real numbers and equality. Address common misconceptions by highlighting the differences between additive and multiplicative identities and inverses. Incorporate real-life scenarios to make the problems relatable and engaging, reinforcing the practical applications of the concepts.

Teachers can incorporate mathematical reasoning into their instruction on solving one-step linear equations involving addition and subtraction by guiding students through the reasoning process using properties of equality. Present equations such as −12 = h + 9 and

MPG2 — Mathematical Communication Teachers can foster mathematical communication by encouraging students to explain their thought processes and solutions verbally and in writing. For instance, after solving a one-step addition equation, students could be asked to explain the steps they took and why they chose a particular strategy. Teachers could also incorporate specialized vocabulary and symbolic notation into lessons and discussions to enhance students’ mathematical communication skills.

MPG5 — Mathematical Representations Teachers can incorporate mathematical representations into their instruction on solving one-step linear equations involving addition and subtraction by using concrete and pictorial models before moving to algorithms. Using tools like algebra tiles can help students visualize the process. For example, when solving x + 3 = 7, students can physically remove three units from both sides of the model to see the balance. Encouraging students to verbalize the steps, like “subtract three from both sides,” helps reinforce the concept. Utilizing balance scales and number lines can also aid in illustrating the equality and solution process, ensuring students grasp the conceptual underpinnings before applying procedural techniques.

Content standards 6.PFA.3 — The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable. 6.PFA.3b — Represent and solve one-step linear equations in one variable, using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale). 6.PFA.3c — Apply properties of real numbers and properties of equality to solve a one-step equation in one variable. Coefficients are limited to integers and unit fractions. Numeric terms are limited to integers.

454

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.PFA.3d — Confirm solutions to one-step linear equations in one variable using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale). 6.PFA.3e — Write a one-step linear equation in one variable to represent a verbal situation, including those in context. 6.PFA.3f — Create a verbal situation in context given a one-step linear equation in one variable.

Prior connections 5.PFA.2 — The student will investigate and use variables in contextual problems.

6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

Future connections 7.PFA.3 — The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

6.PFA.4 — The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

Tools You may find these tools helpful: • Scientific calculator • Algebra tiles

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Use a Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Engage students with physical manipulatives to model one-step equations involving addition and subtraction. Use balance scales and weights to represent equations, where each side of the scale represents each side of the equation. For example, to model the equation x + 3 = 7, place a block labeled “x” and three unit weights on one side, and seven unit weights on the other side of the scale. Allow students to physically remove the same number of weights from both sides to keep the scale balanced while isolating the variable. Introduce algebra tiles, using a specific tile to represent the variable and unit tiles for constants, so students can manipulate them to solve the equations. Representational: Transition to drawings by having students sketch the manipulatives they used. Encourage them to draw balance scales with symbols for variables and constants. For instance, they can draw a scale with a box (representing “x”) and three circles on one side, and seven circles on the other side. Guide them to show each step of solving the equation in their drawings, like crossing out circles from both sides. This helps students visualize the process of maintaining balance while isolating the variable. Reinforce that these sketches are representations of the concrete models they worked with.

6.03 One-step equations with addition and subtraction mathspace.co

455

Abstract: Move on to solving equations using symbols and numbers. Write the equation x + 3 = 7 on the board and demonstrate how to solve for “x” by subtracting 3 from both sides: x + 3 − 3 = 7 − 3, resulting in x = 4. Explain that this abstract method reflects the actions they took with the manipulatives and drawings. Have students practice solving similar equations using the addition and subtraction properties of equality. Encourage them to check their solutions by substituting the value back into the original equation. Help students connect the concrete, representational, and abstract stages by discussing how each one relates to the others. Point out that removing weights from the scale is like crossing out circles in their drawings, and both actions correspond to subtracting numbers in the equation. Encourage students to explain their thinking at each stage, highlighting how the physical manipulation, the visual representation, and the symbolic computation all work together. This reinforces their understanding and helps them choose the most helpful representation when solving equations.

Student lesson & teacher guide Model equations Students explore the representation of algebraic equations using balance scales, algebra tiles, and pictorial models. They delve into the properties of equality including addition, subtraction and substitution, and learn how to verify solutions using these properties. The lesson also highlights inverse operations, additive inverse property and the concept of zero pairs.

Students: Pages 226–227

6.03 One-step equations with addition and subtraction After this lesson, you will be able to... • represent one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations. • solve one-step linear equations involving addition or subtraction using a variety of concrete manipulatives and pictorial representations. • apply properties of real numbers and properties of equality to solve one-step equations involving addition or subtraction. • write a one-step linear equation involving addition or subtraction to represent a verbal or contextual situation. • use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving addition or subtraction. • write a verbal situation in context when given a one-step linear equation involving addition or subtraction.

Model equations An algebraic equation is a mathematical statement that says two expressions are equal. There are many ways to represent an algebraic equation. Some of the ways are: • balance scales

• algebra tiles

• pictorial models

Balance scales are beneficial because they show that the left and right sides of an equation are equal and so the equation is balanced. x−5

The equation shown on this scale is: x−5=2

456

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co Another way to represent an algebraic equation is to use Algebra tiles. These tiles represent the variables and units on each side of the equation.

Model equations An algebraic equation is a mathematical statement that says two expressions are equal. There are many ways to represent an algebraic equation. Some of the ways are: • balance scales

• algebra tiles

• pictorial models

Balance scales are beneficial because they show that the left and right sides of an equation are equal and so the equation is balanced. x−5

The equation shown on this scale is:

x−5=2

Another way to represent an algebraic equation is to use Algebra tiles. These tiles represent the variables and units on each side of the equation. Positive

Variable tiles

Negative +x

−x

−1

Unit tiles

We can use the key above, to represent an equation with algebra tiles. The equation x − 5 = 2 can be created with this combination of tiles: −1 +x

−1

= +1

Pictorial models can be really helpful for visualizing equations that represent real-world situations. These models can use image of any object represent the variables and units of an equation. 226 anMathspace Virginia SOLto Grade 6 mathspace.co Here is a pictorial model of x − 5 = 2 where the variable x is represented by a backpack and the units are represented by a pencil. The gray pencils are meant to show they have been removed from the situation.

Balance scales, algebra tiles, and pictorial modles are three different ways that we can better understand and represent algebra equations.

Example 1 Write math the equation the algebra tiles. Use talkrepresented sentencebyframes

Student with disabilities support +x reasoning Encourage students to articulate their they=model solve addition and subtraction +1 +1 +1as +1 +1 and +1 +1 +1 one-step +1 equations. Provide sentence frames such as “To keep the equation balanced, I ⬚ because ⬚ or “When I subtract/add ⬚ from/to both sides, the variable becomes ⬚.

Display Create these frames on a classroom poster or give each student a reference sheet for easy access. As students a strategy work with balance scales, algebra tiles, or pictorial models, prompt them to use these frames to explain each step aloud or in writing. BeginCount by modeling use Then the frames +1 tilesproblem, demonstrating how the numberhow of +xto tiles. count theduring numberaofsample to express the reasoning behind each action taken to isolate the variable. This strategy supports students who may struggle with language processing by providing them with structured ways to Apply communicate the idea their mathematical thinking. There is 1 + x tile and 4 + 1 tiles on the left side. There are 5 + 1 tiles on the right side. So, the equation is: x+4=5

Example 2

6.03 One-step equations with addition and subtraction mathspace.co Represent x − 3 = −5 using algebra tiles. Do not solve the equation.

Create a strategy

Apply the idea

457

Pictorial models can be really helpful for visualizing equations that represent= real-world situations. These models can use an image of any object to represent the variables and units of an equation. Here is a pictorial model of x − 5 = 2 where the variable x is represented by a backpack and the units are represented by a pencil. The gray pencils are meant to show they have been removed from the situation.

Examples

Students:Balance Pagescales, 227 algebra tiles, and pictorial modles are three different ways that we can better understand and represent algebra equations.

Example 1 Write the equation represented by the algebra tiles. Balance scales, algebra tiles, and pictorial modles are three different ways that we can better understand and represent algebra equations. +x = +1 +1 +1 +1 +1 +1 +1 +1 +1

Example 1 Create strategyrepresented by the algebra tiles. Write theaequation Count the number of +x tiles. Then count the number of +1 tiles +x

Apply the idea Createis a1 +strategy There x tile and 4 + 1 tiles on the left side. There are 5 + 1 tiles on the right side. So, the equation is: x+4=5 Count the number of +x tiles. Then count the number of +1 tiles

Example 2 PurposeApply the idea Show students how 3to=and interpret algebra tiles toThere form are an 5equation. equation. Represent −5 using algebra tiles. not solve the There is 1 +xx−tile 4 + 1 tiles on the leftDo side. + 1 tiles on the right side. So, the equation is:

Students:Create Pagea 227 strategy

x+4=5

Apply the idea

Use a positive variable tile to represent x and −1 tiles to represent the negative constants.

On the left side of the equation, there is an x which we can represent with one +x tile and a −3 which we can make by using three −1 tiles. On the right side of the Example 2 equation the −5 can be shown using ﬁve −1 tiles. Represent x − 3 = −5 using algebra tiles. Do not solve the equation. −1

Create a strategy

Apply the idea+x

Use a positive variable tile to represent x and −1 tiles to represent the negative constants.

On the left side of the−1equation, there an x which we −1 is −1 −1 can represent with one +x tile and a −3 which we can make by using three −1 tiles. On the right side of the equation the −5 can be shown using ﬁve −1 tiles. 6.03 One-step equations with addition and subtraction 227

−1

−1 +x

−1 −1

−1

mathspace.co

−1

6.03 One-step equations with addition and subtraction mathspace.co

227

Purpose Show students how to use algebra tiles to represent equations visually without solving them. Expected mistakes Students might mistakenly represent negative numbers like −3 and −5 using positive unit tiles instead of negative unit tiles. This often occurs because they focus on the number’s magnitude and overlook the negative sign, or they might not fully understand how to depict negative values with algebra tiles. To address this misconception, clarify that negative numbers are represented with negative unit tiles, which are typically a different color. Encourage students to pay close attention to the signs of the numbers in the equation

458

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 228 Example 3 Write an equation that represents the given balance scale.

x+4

−2

Example 3 Create a strategy

Write an equation that represents the given balance scale. x+4 −2 A balance scale represents an equation that is equal or balanced. The left side of the scale corresponds to the left side of the equation, and the right side of the scale corresponds to the right side of the equation.

Apply the idea On the left side of the balance there is an x + 4 and on the right side of the balance there is an −2. We set the left and right side of the balance to equal:

Create a strategy

x + 4 = −2 A balance scale represents an equation that is equal or balanced. The left side of the scale corresponds to the left side of the equation, and the right side of the scale corresponds to the right side of the equation.

Apply the idea

4 PurposeExample On the left side of the balance there is an x + 4 and on the right side of the balance there is an −2. We set the left and Show students to 6interpret balance scales as equations. right sidehow ofx the Represent + 1 =balance usingtoa equal: pictorial model. x + 4 = −2

Students:Create Pagea 228 strategy

Choose objects that could represent the variable x and the constants 1 and 6. The objects should be related for it to make sense.

Example 4

Apply the idea

Represent x + 1 = 6 using a pictorial model.

Create a strategy

Choose objects that could represent the variable x and the constants 1 and 6. The objects should be related for it to make sense.

Apply the idearepresents x and the coins represent 1 and 6. So, the total number of coins in the purse, x, plus 1 coin The coin purse is equal to 6 total coins.

Idea summary Algebraic equation: mathematical statement that says two expressions are equal. We can represent algebraic equations pictoriallyxwith: The coin purse represents and the coins represent 1 and 6. So, the total number of coins in the purse, x, plus 1 coin is equal• to balance 6 total coins. scales • algebra tiles • pictorial models

Idea summary

Purpose Algebraic equation: mathematical statement that says two expressions are equal. We can represent algebraic Show studentsequations how to pictorially visually with: represent an algebraic equation using objects. •

balance scales

•

algebra tiles

•

pictorial models

Expected mistakes Ask students to explain howSOL theGrade coin6 purse represents the unknown x and the coins represent the known Virginia 228 Mathspace numbers in xmathspace.co + 1 = 6. Guide them to determine how many coins must be inside the purse to satisfy the equation. Discuss how removing one coin from both sides of the model keeps the equation balanced. Encourage them to consider alternative visual representations using different objects or symbols. This reflection helps students begin to connect the visual model to the algebraic process of solving for x. 228

Mathspace Virginia SOL Grade 6 mathspace.co

6.03 One-step equations with addition and subtraction mathspace.co

459

The coin purse represents x and the coins represent 1 and 6. So, the total number of coins in the purse, x, plus 1 coin

Students:is Page equal to228 6 total coins.

Idea summary Algebraic equation: mathematical statement that says two expressions are equal. We can represent algebraic pictorially with: Solveequations equations •

balance scales

•

algebra tiles

•

pictorial models

Interactive exploration Explore online to answer the questions

mathspace.co

Solve equations with addition or subtraction Use the interactive exploration in 6.03 to answer these questions. Students: Pages 229–231 1. What kinds of things can you do that keep the scale balanced? Mathspace Virginia SOL Grade 228 2. What kinds of things can6you do to make the scale unbalanced? mathspace.co

Test this with different types of tiles. Are these observations always true?

Solve equations When working with equations, we must keep the equation balanced or it will no longer be a true statement.

exploration Adding or Interactive subtracting the same amount to both sides keeps the equations balanced. These are two of the properties Explore online to answer the questions of equality. mathspace.co

+1 +1 +1 +1 What kinds of things +x can you do that keep +x +1 the scale balanced?

What kinds of things can you do to make the scale unbalanced?

Test this with different types of tiles. Are these observations always true?

Use the interactive exploration in 6.03 to answer these questions.

+1 +1 +1 +1 +1

When working with equations, we must keep the equation balanced or it will no longer be a true statement. Addition property of equality: adding the same number to both sides of an equation creates an equivalent equation. Adding or subtracting the same amount to both sides keeps the equations balanced. These are two of the properties Example: of equality. If x−2=7 Then We can also visualize this with a scale. If

x−2+2=7+2

+1 +1 +1

+1 +1 +x

+1 +1 +1 +1 +1

+1 +1 +1 +x

−1 −1

+1 +1 +1 +1

Addition property of equality: adding the same number to both sides of an equation creates an equivalent equation. Then Example: If

x − 2 = 7+1 +1 +1

Then −1 −1 +1 +1x − 2 + 2 = 7+1+ 2+1 +1 +1 +1 +1

We can also visualize this with a scale. If +1 +1 +1 If we add the same amount to both sides of the scale it stays balanced. +x

−1 −1

+1 +1 +1 +1

6.03 One-step equations with addition and subtraction mathspace.co

Then

+1 +1 +1 +x

460

−1 −1 +1 +1

+1 +1 +1 +1 +1 +1

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co If we add the same amount to both sides of the scale it stays balanced.

229

Subtraction property of equality: subtracting the same number from both sides of an equation creates an equivalent equation. Example: x+5=7

If Then

x+5−5=7−5

We can also visualize this with a scale. If +1 +x

+1 +1

+1 +1 +1

+1 +1

+1 +1 +1 +1

Then +1 +x

−1

+1 +1 −1 −1

+1 +1 +1 −1 −1 −1

+1 +1 −1 −1

+1 +1 +1 +1 −1 −1

If we take away the same amount from each side of the equation, it will remain balanced. Notice for these equations the number we chose to add or subtract was the opposite of a number in the original equation. This strategy helps us solve equations by utilizing inverse operations and the additive inverse property. Addition and subtraction are opposite or inverse operations that undo one another. If we choose the numbers we add or subtract carefully we can use this to eliminate extra numbers from an equation. For example, with the equation x − 5 = 17 we want to isolate x which requires getting rid of the constant term −5. To do this we can use the inverse operation by adding 5. x − 5 + 5 = 17 + 5 Now, we can see the additive inverse in action: x + 0 = 17 + 5 Now, applying the additive identity: x = 17 + 5 Then we can simplify the right side of the equation: x = 22 We don’t always write out all of these steps, but it is still important to know what is happening algebraically. Once we have solved an equation, we can verify the solution using the substitution property of equality. Substitution property If a = b, then b can be substituted for a in any expression, equation, or inequality. Consider the solution of x = 22 for the equation x − 5 = 17.

230

Mathspace Virginia SOL Grade 6 mathspace.co

6.03 One-step equations with addition and subtraction mathspace.co

461

Exploration Students: Page 229

Solve equations Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 6.03 to answer these questions. 1.

What kinds of things can you do that keep the scale balanced?

What kinds of things can you do to make the scale unbalanced?

Test this with different types of tiles. Are these observations always true?

When working with equations, we must keep the equation balanced or it will no longer be a true statement. Adding or subtracting the same amount to both sides keeps the equations balanced. These are two of the properties

Suggested student grouping: Small groups of equality. Students will be interacting with an applet that represents the equation x = 3. They will have the opportunity to add and remove algebra tiles on a scale and observe the effects this has on the balance of the scale. This exploration will help students understand how different values +1 of x can affect the +1 balance of an equation. +1 Ideal student responses

+1 +1

+1 +x

+1 +1 +1 +1

These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What kinds of things can you do that keep the scale balanced? Keeping theproperty same number of tiles onthe each of the scale keeps it balanced. If xan=equivalent 3, then adding or Addition of equality: adding sameside number to both sides of an equation creates equation. removing the same number of tiles from both sides does not change the balance. Example: 2. What kinds of things can you do to makeIfthe scale unbalanced? x−2=7 Adding or removing tiles from only one side of the scale unbalances it. This shows that the equation is not Then x−2+2=7+2 true for those values of x. We can also visualize this with a scale.

3. Test Ifthis with different types of tiles. Are these observations always true? Yes, these observations hold true regardless of the types of tiles used. The scale stays balanced as long as +1 +1 the same number of tiles are added or removed from both+1sides. +x

−1 −1

+1 +1 +1 +1

Purposeful questions • What do you observe when you add the same number of tiles to both sides of the scale? Why do you think this happens? Then • Is it possible to balance the scale if you add tiles to one side only? What does this tell you about maintaining +1 +1 +1 equality in an equation? −1 −1an+1unbalanced +1 +1 +1 back +1 +1to +1 +1 • How can you manipulate the tiles+xto return scale balance? What mathematical principles does this demonstrate? Possible misunderstandings If we add same amountdepends to both sides thetotal scalenumber it stays balanced. • Thinking thatthe the balance on of the of tiles rather than the equality of both sides. Emphasize that balance is achieved when both sides have equal value, illustrating that the relationship between the sides is more important than the total number of tiles. 6.03 One-step equations with addition and subtraction mathspace.co

462

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

229

Understanding the process with algorithmic thinking Targeted instructional strategies Incorporate algorithmic thinking by guiding students to develop a step-by-step procedure for solving onestep equations while maintaining balance. Begin by reviewing the inverse of each of the four main operations: addition → subtraction, subtraction → addition, multiplication → division, and division → multiplication. Ask students to first identify the operation used in each of the following problems: x + 3 = 5, y − 2 = 6, 9m = 99, and = 10. For this lesson, focus on mastering addition and subtraction operations through this algorithmic approach. Encourage students to use various combinations of positive and negative numbers on both sides of the equation to enhance their understanding and flexibility in applying the algorithm. Have students write down the inverse operation for each problem as part of their algorithm. Emphasize that equations need to stay balanced to remain equal, and the goal is to use the inverse operation to isolate the variable. Guide students to apply their algorithmic steps to balance the equations and find the solutions. Finally, have them verify their solutions by substituting the values back into the equations to confirm that both sides are balanced.

Critique, correct, and clarify English language learner support Present students with an intentionally incorrect solution to a one-step addition or subtraction equation to engage them in the Critique, Correct, and Clarify routine. For example, write on the board: “Solve x − 4 = 10: x = 10 − 4 x=6 “ Ask students to work in pairs to identify and explain the mistake in the solution. Encourage them to discuss what was done incorrectly, correct the error, and clarify the correct steps using proper mathematical vocabulary. This activity helps students practice using terms like “inverse operations,” “isolating the variable,” and “properties of equality.” By analyzing and correcting the mistake, students deepen their understanding of solving one-step equations while simultaneously developing their mathematical language skills.

Working with equations Address student misconceptions Students may use the same operation instead of the inverse when trying to balance or work through an equation. Though the equation is still true, remind the students that when finding the solution of an equation, the goal is to isolate the variable, leaving the variable on one side of the equation. Students explore the properties of equality, including the addition, subtraction, and substitution properties. They practice maintaining balance in equations by adding or subtracting the same number from both sides. Additionally, they delve into the concept of inverse operations and the additive inverse property.

6.03 One-step equations with addition and subtraction mathspace.co

463

Examples Students: Page 231

Purpose Show students how to balance scales using algebraic reasoning.

Students: Pages 231–232

464

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Reﬂect and check Let’s verify the answer using a balance scale model. We start with twenty one +1 tiles on the left side of the scale, then one +x and thirteen +1 tiles on the right side to represent 21 = x + 13. +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1

+1 +1 +1 +1 +x

+1 +1 +1 +1 +1 +1 +1 +1 +1

Based on our solution, 8 = x, we can replace the +x tile with 8 of the +1 tiles. +1 +1 +1 +1 +1 +1 +1

+1 +1 +1 +1 +1 +1 +1

The balance scale is now balanced with both sides having the same number of positive unit tiles (+1), verifying that our solution is correct.

Example 7 Purpose Show students Solve: x −how 1 = 7 to use inverse operations to isolate variables and solve equations. Expected mistakes Create a strategy StudentsThe might believe that the variablesomust always betoon the leftofside of the equation to solve it. This inverse of subtraction is addition, we need to add both sides the equation. misconception can lead them to rewrite the equation or feel confused about how to proceed when the variable idea is on theApply right the side. They may think that an equation like 21 = x + 13 needs to be rearranged before solving. x −1=7 Original that equation To address this misconception, emphasize the equality sign means both sides are equal, and the variable x − 1 + 1 = 7 + 1 Addition property of equalityto focus on isolating the variable, regardless of its can be on either side of the equation. Encourage students x=7+1 Additive inverse position. x=8

Evaluate the addition

Verify solutions with code

use with Example 6

Reﬂect and check

Targeted instructional strategies

Let’s verify the answer using a balance scale model.

Encourage students to verify their solutions by creating simple Python scripts. Show them how to substitute We start with a +x tile and a −1 tile one the left side of the scale, then 7 of the +1 tiles on the right to represent the their solution back equation x − 1into = 7. the original equation using code. For example, guide them to write: 1

x = 8

+1 +1 +1 +x

−1

if 21 == x + 13:

print(‘Solution is correct’)

else:

print(‘Solution is incorrect’) 232

Mathspace

+1 +1 +1 +1

Virginia SOL Grade 6

mathspace.co This allows students to apply computational thinking by creating with code, reinforcing their understanding of solving and checking equations while introducing basic programming concepts.

6.03 One-step equations with addition and subtraction mathspace.co

465

The balance scale is now balanced with both sides having the same number of positive unit tiles (+1), verifying that our solution is correct.

Students: Pages 232–233

Example 7 Solve: x − 1 = 7

Create a strategy The inverse of subtraction is addition, so we need to add to both sides of the equation.

Apply the idea x−1=7 x−1+1=7+1

Original equation Addition property of equality

x=7+1

Additive inverse

x=8

Evaluate the addition

Reﬂect and check Let’s verify the answer using a balance scale model. We start with a +x tile and a −1 tile one the left side of the scale, then 7 of the +1 tiles on the right to represent the equation x − 1 = 7. +1 +1 +1 +x

−1

+1 +1 +1 +1

Based on our solution, x = 8 we can replace the +x tile with 8 of the +1 tiles. +1 +1 +1 +1 232

+1 +1 +1

Mathspace Virginia SOL Grade 6+1 +1 +1 +1 −1 mathspace.co

+1 +1 +1 +1

Remove zero pairs. The scale stays balanced. +1 +1 +1 +1

Zero pair

+1 +1 +1

+1 +1 +1 +1 −1

+1 +1 +1 +1

+1 +1 +1

+1 +1 +1 +1

The balance scale is now balanced with both sides having the same number of positive unit tiles, verifying that our solution is correct.

Example 8 Purpose −4 is the solution the equation −8 + x = −12. Show students how to tosolve basic algebraic equations by using addition property of equality and understanding a operations. Verify using substitution. of inverse Create a strategy Substitute −4 into the equation. The left side of the equation will be equal to the right side if −4 is a solution.

Apply the idea −8 + (−4) = −12 −12 = −12

466

Substitute x = −4 Evaluate the addition

The left side of the equation is equal to the right side of the equation, verifying that −4 is a solution. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co Reﬂect and check We can also use a pictorial model to verify if −4 is the solution to −8 + x = −12. This pictorial model represents the equation −8 + x = −12, where each ice cube represents −1 and the cooler

+1 +1 +1 +1

The balance scale is now balanced with both sides having the same number of positive unit tiles, verifying that our is correct. Students:solution Pages 233–234

Example 8 −4 is the solution to the equation −8 + x = −12. a Verify using substitution.

Create a strategy Substitute −4 into the equation. The left side of the equation will be equal to the right side if −4 is a solution.

Apply the idea −8 + (−4) = −12 −12 = −12

Substitute x = −4 Evaluate the addition

The left side of the equation is equal to the right side of the equation, verifying that −4 is a solution.

Reﬂect and check We can also use a pictorial model to verify if −4 is the solution to −8 + x = −12. This pictorial model represents the equation −8 + x = −12, where each ice cube represents −1 and the cooler represents x. = −1

= +x

We then replace the cooler with four ice cubes, representing x = −4. =

6.03 One-step equations with addition and subtraction mathspace.co

233

We now have twelve ice cubes on both sides of the equation, verifying that −4 is a solution to the equation.

b Verify using a model.

PurposeCreate a strategy Help students understand verifytiles a solution toMake an equation using Represent the equationhow usingtoalgebra on a scale. the left side of thesubstitution. scale to have the same number of unit tiles as the right side by replacing the variable tile with the appropriate tiles.

Reflecting with students Encourage students Apply the ideato communicate precisely by using clear definitions and accepted mathematical terminology throughout When verifying that −4 is the solution to the equation −8 + x = −12, prompt The equation −8 + xtheir = −12 solution. can be represented by: them to explicitly state each step. For example, they can say, “Substitute x = −4 into the equation to get −8 + (−4) = −12.” Then, guide them −1 −1 −1 −1 −x −1 −1 −1 −1 −1 −1 to simplify the left side: “−8 + (−4) = −12,” leading to “−12 = −12.” Emphasize the importance of concluding −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 with a clear statement such as, “Since both sides of the equation are equal, x = −4 is indeed a solution.” By using precise language and detailed explanations, students develop a habit of clarity that enhances their mathematical understanding and communication skills. The given solution is −4, so we replace the variable tile with four negative unit tiles. −1 −1 −1 −1 −1 −1

−1 −1 −1 −1 −1 −1

6.03 One-step equations with addition and subtraction Both sides of the scale now have twelve negative unit tiles, verifying that −4 is a solution to the equation. mathspace.co

467

We then replace the cooler with four ice cubes, representing x = −4. =

now have twelve ice cubes on both sides of the equation, verifying that −4 is a solution to the equation. Students:WePage 234 b Verify using a model.

Create a strategy Represent the equation using algebra tiles on a scale. Make the left side of the scale to have the same number of unit We have twelve icereplacing cubes on the both sides oftile the equation, verifying tiles. that −4 is a solution to the equation. tilesnow as the right side by variable with the appropriate

Apply the idea

b Verify using a model. The equation −8 + x = −12 can be represented by:

Create a strategy Represent the equation using algebra on a−xscale. Make the left to have the same number of unit −1 −1 tiles −1 −1 −1 side −1 of −1 the −1 scale −1 −1 tiles as the right side by replacing the variable tile with the appropriate tiles. −1 −1 −1 −1 −1 −1 −1 −1 −1 −1

Apply the idea The equation −8 + x = −12 can be represented by: The given solution is −4, so we replace the variable tile with four negative unit tiles. −1 −1 −1 −1

−1 −1 −1 −1

−x −1 −1 −1 −1 −1 −1 −1 −1

−1 −1 −1 −1

The is −4,now so we replace variable tiletiles, withverifying four negative unit Bothgiven sidessolution of the scale have twelvethe negative unit that −4 is tiles. a solution to the equation. −1 −1 −1 −1 −1 −1

−1 −1 −1 −1 −1 −1

PurposeExample 9 Show students how to contains verify a500 solution using model. A box of matches matches. The a match box falls to the ground and you count 78 matches on the ground. The rest of of thethe matches are still the box. Both sides scale now haveintwelve negative unit tiles, verifying that −4 is a solution to the equation.

Students:a Page 234 Write an equation that shows the relationship of the number of matches. Create a strategy

Apply the idea

Example 9

The amount of matches that are still in the box is the Let m represent the number of matches still in the box. unknown value. contains 500 matches. The match box falls to the ground and you A box of matches matches on the ground. m count + 78 =78 500 The rest of the matches are still in the box. a Write an equation that shows the relationship of the number of matches.

Create a strategy Virginia SOL Grade 6 234 Mathspace

Apply the idea

mathspace.co

The amount of matches that are still in the box is the unknown value.

Let m represent the number of matches still in the box. m + 78 = 500

Purpose234 Mathspace Virginia SOL Grade 6 mathspace.co Show students how to create an equation from a real world situation involving subtraction.

Students: Page 235 b Solve the equation and interpret the solution.

Create a strategy The inverse of addition is subtraction, so we need to subtract to both sides of the equation.

Apply the idea m + 78 = 500 m + 78 − 78 = 500 − 78

468

Original equation Subtraction property of equality

= 4226 Teacher Edition Mathspace Virginia SOLmGrade mathspace.co This means that there are 422 matches that did not fell out of the box.

b Solve the equation and interpret the solution.

Create a strategy The inverse of addition is subtraction, so we need to subtract to both sides of the equation.

Apply the idea m + 78 = 500

Original equation

m + 78 − 78 = 500 − 78

Subtraction property of equality

m = 422 This means that there are 422 matches that did not fell out of the box.

Example 10 Purpose Demonstrate students how represent to solve the a simple equation and interpret the result in the context of the problem. Write ato situation that could equation x + 5 = 25. Reflecting witha students Create strategy b Solve the equation and interpret the solution.by introducing variables to represent different quantities. For Encourage students to generalize the Think of a situation where there is anproblem unknown value, x added to 5 which would equal 25. example, let students define t as the total number of matches in the box, f as the number of matches that fell Create a strategy out, andApply m as the theidea number of matches remaining. Guide them to formulate the general equation m + f = t. The inverse of addition is subtraction, so we need to subtract to both sides of the equation. The amount of moneyhow in James’ wallet, x, plusvariable the $5 inaffects his handthe is equal to the $25questions he receivedlike, for his birthday. Then, have them explore changing one others. Ask “What happens to m if f increases?” or “How does the equation change if the total number of matches t is unknown?” This approach Apply the idea allows advanced learners to deepen their understanding by exploring variable relationships and prepares m + 78 = 500 Original equation them for moreIdea complex algebraic concepts. It also connects the problem to broader mathematical ideas about m +summary 78 − 78 = 500 − 78 Subtraction property of equality b Solve the equation and interpret the solution. equations and functions.m = 422 Addition property of equality If a = b then a + c = b + c This means that there are 422 matches that did not of the Subtraction property of equality If afell = bout then a − box. c=b−c Create a strategy

Students: Page 235

property of addition + subtract (−a) = 0 and (−a) sides + a = of 0 the equation. The inverse ofInverse addition is subtraction, so we needato to both Identity property of addition a ⋅ 1 = a and 1 ⋅ a = a a + 0 = a and 0 + a = a Apply the idea Substitution property

Example 10

m + 78 = 500

Original equation

m + 78that − 78 = 500 − 78 Subtraction of equality Write a situation could represent the equation property x + 5 = 25. m = 422

Practice

Create a strategy This means that there are 422 matches that did not fell out of the box. Think of a situation where there is an unknown value, x added to 5 which would equal 25.

What do you remember? Apply the idea

Example 10 1 amount Consider this setinofJames’ tiles: wallet, x, plus the $5 in his hand is equal to the $25 he received for his birthday. The of money a Write an expression to represent the tiles at the left side of the equal symbol. Write a situation that could represent the equation x + 5 = 25. b Write an expression to represent the tiles at the right side of the equal symbol. c

Now, write an equation to represent the set of tiles.

summary CreateIdea a strategy

PurposeThink 2 of Write the algebraic that represents each set of tiles: a situation whereequation there is an unknown value, x added to 5 which would equal 25. Addition property of equality enough If a to = bcreate then a +contexts c = b + c around them. Make sure students understand equations a b −1 +1 +1 +1 +1

Subtraction property of equality If a = b then a − c = b +x −c = = +1 +1 +1 +1 +1 +1 of +1 +1 +1 Inverse property addition a + (−a) = 0 and (−a) + a = 0 −1 Students:The Page 235 amount of money in James’ wallet, x, plus the $5 in his hand is equal to the $25 he received for his birthday. Identity property of addition a ⋅ 1 = a and 1 ⋅ a = a Substitution property a + 0 = a and 0 + a = a

Apply the +1 idea +x

Idea summary

6.03 One-step equations with addition and subtraction mathspace.co

PracticeAddition property of equality

Subtraction property of equality Inverse property of addition What do you remember? Identity property of addition Substitution property 1 Consider this set of tiles:

If a = b then a + c = b + c If a = b then a − c = b − c a + (−a) = 0 and (−a) + a = 0 a ⋅ 1 = a and 1 ⋅ a = a a + 0 = a and 0 + a = a

Write an expression to represent the tiles at the left side of the equal symbol.

Write an expression to represent the tiles at the right side of the equal symbol.

c Now, write an equation to represent the set of tiles. Practice 2

Write the algebraic equation that represents each set of tiles:

+1 +1 Whata do you remember? +1

235

Consider this set of tiles:

b +x

−1 −1

6.03 One-step equations with addition and subtraction mathspace.co

Write an expression to represent the tiles at the left side of the equal symbol.

Write an expression to represent the tiles at the right side of the equal symbol.

Now, write an equation to represent the set of tiles.

6.03 One-step equations with addition and subtraction

235

469

Practice Students: Pages 235–239

What do you remember? 1

Consider this set of tiles: a

Write an expression to represent the tiles at the left side of the equal symbol.

Write an expression to represent the tiles at the right side of the equal symbol.

Now, write an equation to represent the set of tiles.

Write the algebraic equation that represents each set of tiles: a

+1 +1 +1 +1

−1

Consider this scale:

Identify whether the scale would remain balanced in each scenario.

Take away an orange from the left side of the scale and add an orange to the right side of the scale.

Take away an orange from the left side of the scale and take away an orange from the right side of the scale.

Add an orange to the left side of the scale and add an apple to the right side of the scale.

Add an orange to the left side of the scale and take away an orange tile from the right side of the scale.

What can you replace x with to make both sides of the scale balanced? +x

470

−1

−1 −1 −1

+1 +1 +1

State the operation needed to solve the following equations: a

d+6=3

y−2=7

x + 28 = 45

u−1=3

s − 2 = −35

t + 3 = −45

x − (−12) = 39

y + (−15) = −6

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice 6

Use the models to determine whether the given value of x is a solution to the equation: a

−1

b =

−1 −1 −1 −1 +1

x = −8; x + 3 = 11

−1 −1

x = −8; x + 1 = −7 7

Solve the equation represented by the following a

+1 +1 +x

−1 +x

−1

+1 −1

Use the model to solve x + 2 = 5. a

Explain how the model represents the equation x + 2 = 5.

How much does one circle weigh? How do you know?

The solution is x = ⬚.

Describe how you could check your answer in part (a).

Which model represents the solution of x + 3 = 8? Explain your answer. A

+1 x

B x

x +1

For each equation: a

Draw a picture and use it solve.

Use the picture to verify the solution.

14 + x = 64

k − 14 = −7

6.03 One-step equations with addition and subtraction mathspace.co

471

SOL

Use the properties of real numbers to rearrange and solve the following equations: a

Rearrange the equation 3 + x = 7 using the commutative property, then solve for x.

Use the associative property to regroup (x + 7) + 9 = 23, then solve for x.

If x + 5 = 21, and you want to ﬁnd x, would you add or subtract 5 from both sides of the equation? Why?

Which of the following is a solution to this number sentence? y − 18 = 3 A

SOL

y = 21

y = −21

y = −15

y = −54

Solve: a

x + 6 = 15

x + 4 = −9

x − 7 = −7

x − (−4) = 10

10 + x = 30

−8 + x = 20

21 = x − 13

7=x−1

Write an equation to represent each verbal situation. Verbal Sentence The sum of a number and nine is equal to thirty. A number decreased by ten is eighteen. Sixteen less than m is equal to twenty six. 37 is the same as h increased by fourteen.

Algebraic Equation

You step onto a moving walkway at the airport and travel forward 15 feet. You step off the walkway at the gate which is located at the 30 foot mark. Write an equation you can use to ﬁnd the distance traveled on the walkway.

Students and staff organized a bake sale to raise money for new library books. Together they raised $3250. The staff contributed $1250. Write an equation you can use to ﬁnd the amount s raised by the students.

Describe the error in writing the sentence as an equation. Verbal Sentence A number t is 7 less than 20

Consider the equation p + 12 = 35. Select the scenario that this equation could represent. A

A park currently has 35 trees. If 12 more trees are planted, how many trees will the park have?

A baker has 12 loaves of bread and bakes enough to have 35 loaves. How many loaves did the baker bake today?

A student has p pages left to read to complete a 35-page reading assignment. If the student has already read 12 pages, how many pages were assigned?

There are p puppies in a pet store. If 12 puppies are adopted, leaving 35 puppies, how many puppies were there to start with?

There are 15 bananas in a fruit box. There are 10 more bananas than pineapples. This can be represented by the equation 15 = p + 10. a

Use the scales to represent this situation.

How many pineapples are in the fruit box?

472

Algebraic Equation t − 7 = 20

p=⬚

Explain why your solution makes sense.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The temperature dropped seven degrees. Now, the thermometer says −8 degrees. The scales shown model this situation. a

Write the equation that represents this situation.

−1

What was the temperature before it dropped?

−1 −1 −1

x=⬚

Explain why your solution makes sense.

−1

Write a real-world world context that each equation could be used to represent. a

x + 4 = 60

y − 5 = 82

53 = r − 19

−5 = p + 13

k + (−8) = −10

Let’s extend our thinking 23

Solve each equation. Justify your work using a property of equality. a

u − 22 = −4

−24 + p = 59

27 + m = −32

Find the unknown number for each of the following: a

When 10 is added to a number, the result is 12.

When 20 is subtracted to a number, the result is 5.

The sum of a number and 39 is 83.

17 less than a number is 25.

Wendy and Astrid are solving the equation x − 2 = 4. The following are the steps of their solutions: Wendy’s solution:

Astrid’s solution:

Step 1:

x−2=4

Step 1:

x−2=4

Step 2:

x−2+2=4−2

Step 2:

x−2+2=4+2

Step 3:

x=2

Step 3:

x=6

Which soution is incorrect? Explain your answer. 26

John has a debt that he’s paying off. After making a payment of $30, he still owes $ − 150. Let d represent the original debt amount before the payment. a

Find the amount of money John originally owed.

Use substitution to verify your solution.

Alex gives Lisa 20 books from his library. Alex now has only 30 books in his library. How many books (b) did Alex have initially? Justify your solution.

6.03 One-step equations with addition and subtraction mathspace.co

473

Answers 6.03 One-step equations with addition and subtraction

9 Model A represents the solution to the equation x + 3 = 8. If we replace the x rectangle with 5 squares and add 3 more squares, both sides will have 8 squares.

What do you remember? 1 a x

b 4

2 a 1 + x = 8

b x−2=5

3 a No

b Yes

c x=4

c No

+1 +1

d No

4 B 10 i

5 a Subtraction

b Addition

c Subtraction

d Addition

e Add 2

g Add (−12)

h Subtract (−15)

Subtract 3 =

Let’s practice 6 a Yes

b No

7 a x = 3

b x=6

8 a O n the left side of the scale, there is one circle and two squares. This combination represents the expression x + 2, where x stands for the circle and 2 stands for the two squares.

On the right side of the scale, there are ﬁve squares, which directly represent the number 5.

The equality sign in the equation indicates that both sides of the equation are balanced, just like a balanced scale. In this case, it means that the weights (or values) on both sides of the scale are equal. b T he model shows that one circle and two squares are equal to ﬁve squares. If we were to replace the circle with squares, we would need three so that both sides would have ﬁve squares. This means that one circle weighs three squares.

x = 50 ii a −1 −1 −1

−1

−1 −1 −1 −1

−1

The solution is x = 3.

c W e can check our answer in part (a) by substituting the circle with the number of squares it is equal to.

−1

−1 −1 −1 −1

−1

−1 −1 −1

−1

−1 −1 −1

−1

−1 −1 −1 −1

−1

b =

−1

+1 +1 +1 +1

As we can see, both sides of the scale are equal, which veriﬁes our answer.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−1

−1 −1 −1 −1

−1

−1 −1 −1

−1

+1 +1

k=7

474

11 a x+3=7

Rearrange using commutative property

x + 3 − 3 = 7 – 3

Subtract 3 from both sides

x=4

Simplify both sides

x + (7 + 9) = 23

Regroup using associative property

x + 16 = 23

Add 7 and 9

x + 16 − 16 = 23 – 16

Subtract 16 from both sides

x=7

Simplify both sides

12 Answer may look like: I would subtract 5 from both sides of the equation. This is because the goal is to isolate the variable (x) on one side of the equation, and subtracting 5 from both sides will eliminate the constant term 5 from the left side. 13 A 14 a x = 9 e x = 20

c T he solution makes sense because if we substitute p = 5, both sides of the equation would be 15.

b x = −13

c x=0

d x=6

x = 28

g x = 34

h x=8

15 Verbal Sentence The sum of a number and nine is equal to thirty.

Algebraic Equation x + 9 = 30

A number decreased by ten is eighteen.

x − 10 = 18

Sixteen less than m is equal to twenty six.

m − 16 = 26

37 is the same as h increased by fourteen.

37 = h + 14

16 x + 15 = 30 17 s + 1250 = 3250 18 The error in the equation is that it incorrectly sets up the relationship. The correct equation should be t = 20 − 7 to reﬂect that t is 7 less than 20, which simpliﬁes to t = 13. The given equation t − 7 = 20 implies t = 27 when solved, which does not match the original sentence.

Having 5 pineapples in the fruit box matches the situtation where the number of bananas is 10 more than this, which is 15. 21 a x − 7 = −8 b x = −1 c T he solution makes sense because if we substitute x = −1, both sides of the equation would be −8. Having − 1° as the initial temperature matches the situtation, because dropping − 1° by − 7° results to − 8°. 22 Answers may vary a C onsider you have an amount of money, x. If you receive 4 more dollars, you will have a total of 60 dollars. This equation represents the amount of money you currently possess. b I magine you spent 5 dollars, leaving you with 82 dollars. This equation represents the original amount of money you had. c S uppose you had a certain number of candies, r. 53 candies remain after giving away 19 of them. Solving this equation helps you ﬁnd how many candies you initially had. d T he temperature is − 5 warmer, after rising 13 degrees. Solving this equation ﬁnd the original temperature, p, before the increase. Let’s extend our thinking

19 B 20 a

23 a u − 22 = −4

Given equation

u = 18

Addition property of equality

b −24 + p = 59 p = 83 c 27 + m = −32 m = −59

b p = 5

d k + (−8) = −10

Given equation Addition property of equality Given equation Subtraction property of equality Given equation

k = −2

Subtraction property of equality

24 a 2

c 44

b 25

d 42

Answers mathspace.co

475

25 Example answer: Wendy’s solution is incorrect. She made a mistake in Step 2 when she added 2 to the left side of the equation and subtracted 2 from the left side of the equation. The correct step is to add 2 to both sides of the equation.

27 We can represent the situation as b − 20 = 30, where b represents the initial number of books. Solving the equation: b − 20 = 30 b − 20 + 20 = 30 + 20

26 a d + 30 = −150 d = −180 b If d was originally $ − 180, after making a payment of $30, the debt would be $ − 150: −180 + 30 = −150, which veriﬁes the solution.

476

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b = 50 Alex initially has 50 books. This number is correct as deducting 20 from 50 would result in 30, the current number of Alex’s books.

6.04 One-step equations with multiplication and division Subtopic overview Lesson narrative In this lesson, students will learn to model and solve one-step equations using multiplication and division. They will explore various representations, such as balance scales and algebra tiles, to understand the concept of maintaining balance in equations. Students will apply the multiplication and division properties of equality to isolate the variable and verify solutions to these equations. Additionally, they will write one-step linear equations to represent verbal situations and create verbal situations given one-step linear equations which involve multiplication and division. By the end, students should confidently model, solve, and verify one-step multiplication and division equations.

6.04 One-step equations with Learning objectives multiplication and division Students: Page 240

After this lesson, you will be able to... • represent one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations. • solve one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations. • apply properties of real numbers and properties of equality to solve one-step equations involving multiplication or division. • write a one-step linear equation involving multiplication or division to represent a verbal or contextual situation. • use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving multiplication or division. • write a verbal situation in context when given a one-step linear equation involving multiplication or division.

One-step equations with multiplication and division Just vocabulary like we saw with addition and subtraction, multiplication and division are also inverse operations. For example, Key multiplying a number by two is the opposite of dividing it by two.  division property of equality balanced equation Looking at a balance scale model again, we can see how we can multiply or divide by the same number on both  equivalent equations  identity property of multiplication sides of an equation to keep it balanced.  inverse property of multiplication  multiplication property of equality 

+x +x +x

+1 +1 +1

Essential understanding

The properties of equality allow an equation to be manipulated without changing the equivalence of the expressions on either side.

Multiplication property of equality: Multiplying each side of an equation by the same number produces an equivalent equation. Example: If Then We can visualize this with a scale.

x 6.04 = 3 One-step equations with multiplication and division x⋅2=3⋅2

mathspace.co

477

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can incorporate mathematical problem-solving into their instruction on solving one-step linear equations in one variable involving multiplication and division by presenting general algorithms that can be broadly applied to similar equations. Teachers should use real-life situations and concrete manipulatives like algebra tiles to help students understand the connection between symbolic, graphical, and numerical methods. Emphasizing the properties of real numbers and properties of equality will help students transition from concrete to abstract problem-solving methods. Encourage students to identify relevant information, create a plan, and verify their solutions to ensure accuracy and understanding.

Teachers can foster mathematical reasoning by guiding students through the process of solving one-step linear equations involving multiplication and division using properties of equality. Provide students with examples and ask them to justify each step in the process, addressing common misconceptions such as confusing the inverse operation needed. Use models like balance scales to represent equations and verbal descriptions to explain the meaning of each step. Encourage students to explain their thinking, explore multiple solution methods, and write equations in various forms to deepen their understanding and reasoning skills. MPG5 — Mathematical Representations

MPG2 — Mathematical Communication

Teachers can incorporate mathematical representations into their instruction on solving one-step linear Teachers can foster mathematical communication by encouraging students to explain their thought processes equations in one variable involving multiplication and division by using concrete and pictorial and solutions verbally and in writing. For instance, after representations. Utilize algebra tiles and balance solving a one-step multiplication equation, students scales to make mathematics visual and provide could be asked to explain the steps they took and why conceptual understanding. Have students write out they chose a particular strategy. Teachers could also incorporate specialized vocabulary and symbolic notation what is happening as they use manipulatives, making connections between the models and the algebraic into lessons and discussions to enhance students’ equations. Employ true/false sentences and ask students mathematical communication skills. to support their reasoning by explaining their thinking. Plotting the variable on a number line can also help students visualize and confirm the solution’s validity.

478

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Prior connections 5.PFA.2 — The student will investigate and use variables in contextual problems.

6.CE.2 — The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

6.PFA.4 — The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 4.02 Multiply and Divide integers Grade 6 — 6.02 Properties of real numbers Grade 6 — 6.03 One-step equations with addition and subtraction

Student lesson & teacher guide Solve equations with multiplication or division Students will learn about the multiplication and division properties of equality, and how they can be used to keep both sides of an equation balanced. They will visualize these concepts using scale models. Additionally, they will explore the application of multiplicative inverse and identity properties to isolate variables in equations.

6.04 One-step equations with multiplication and division mathspace.co

479

Students: Pages 240–242

6.04 One-step equations with multiplication and division After this lesson, you will be able to... • represent one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations. • solve one-step linear equations involving multiplication or division using a variety of concrete manipulatives and pictorial representations. • apply properties of real numbers and properties of equality to solve one-step equations involving multiplication or division. • write a one-step linear equation involving multiplication or division to represent a verbal or contextual situation. • use a variety of concrete manipulatives and pictorial representations to verify solutions to equations involving multiplication or division. • write a verbal situation in context when given a one-step linear equation involving multiplication or division.

One-step equations with multiplication and division Just like we saw with addition and subtraction, multiplication and division are also inverse operations. For example, multiplying a number by two is the opposite of dividing it by two. Looking at a balance scale model again, we can see how we can multiply or divide by the same number on both sides of an equation to keep it balanced.

+x +x +x

+1 +1 +1

Multiplication property of equality: Multiplying each side of an equation by the same number produces an equivalent equation. Example: If Then

x=3 x⋅2=3⋅2

We can visualize this with a scale. +x

240

480

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The scale tells us that x = 3. We can double the values on both sides and the scale will still be balanced. +x

We multiplied the tiles on both sides of the scale by 2 and it is still balanced. We could have also multiplied both sides of the original scale by 3 to create 3 equal groups on each side of the balance and it would still be balanced. +1

If we treat each side of the scale like a group that aligns with the other side, we can keep applying the multiplication property of equality. Division property of equality: Dividing each side of an equation by the same number produces an equivalent equation. Example: If Then

4x = 8 =

We can visualize this with a scale. +x

Notice that the groups on the left and right align where every 2x aligns with 4 unit tiles. We can divide both sides by 2 (removing half of the tiles) and the scale will still be balanced. +x

We can also divide both sides of the original scale by 4, since we can see there are 4 groupings where every x tile is equal to 2 unit tiles. +x

6.04 One-step equations with multiplication and division mathspace.co

241

6.04 One-step equations with multiplication and division mathspace.co

481

Think algorithmically Targeted instructional strategies To incorporate algorithmic thinking into your lesson, guide students to develop a clear, step-by-step procedure for solving one-step equations involving multiplication or division. Encourage them to think about each action systematically and understand how each step works towards isolating the variable. Have students write down their algorithm and apply it to various problems, refining their steps as they encounter different types of equations. An example set of steps might be: 1. Identify the operation affecting the variable (multiplication or division). 2. Determine the inverse operation needed to isolate the variable. 3. Apply the inverse operation to both sides of the equation to maintain balance. 4 Simplify both sides of the equation to find the value of the variable. 5. Verify the solution by substituting it back into the original equation to ensure it holds true.

Compare and connect English language learner support Have students solve one-step equations using addition or subtraction (e.g., x + 7 = 10, y − 4 = 9) and then equations using multiplication or division

. After solving, ask students to compare and

connect the strategies they used for each type of equation. Encourage them to discuss similarities, such as using inverse operations to isolate the variable, and differences in the operations performed. Facilitate a class discussion where students share their methods and reasoning, highlighting how understanding addition and subtraction equations can help with multiplication and division equations. Provide visual representations, like balance scales or algebra tiles, to illustrate the connections between the operations. By comparing and connecting these methods, you help students deepen their understanding of solving equations and enhance their mathematical language skills.

482

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Think-aloud strategy modeling Student with disabilities support Begin by solving a one-step multiplication or division equation in front of the class, verbalizing each thought process as you go. For example, write the equation 5x = 20 on the board and say, “To solve for x, I need to isolate the variable. Since 5 is multiplied by 5, I’ll divide both sides by 5 to keep the equation balanced.” Use visuals like a balance scale diagram to illustrate how performing the same operation on both sides maintains equality. Explain each step, highlighting why you choose a particular operation and how it helps solve the equation. After modeling, pair students and encourage them to solve a similar equation while thinking aloud to each other, describing each step just as you did. This approach helps students internalize the problem-solving process, enhances their conceptual understanding, and improves their ability to focus and articulate their reasoning. Consider displaying an image of a balance scale with equal weights on both sides, showing how dividing both sides keeps the scale balanced.

Examples Students: Pages 242–243

6.04 One-step equations with multiplication and division mathspace.co

483

Apply the idea We must add three +1 tiles on the right side of the scale for every new positive tile on the left side. So, we need to add a total of six +1 tiles to the right side of scale 1. We need 9 positive 1 tiles in place of the question mark to balance scale 2, option D.

Reﬂect and check Apply the idea

We can write this algebraically as: We must add three +1 tiles on the right side of the scale for every new x=3 Write the equation positive tile on the left side. So, we need to add a total of 3x = 3 ⋅ 3 Multiply both sides by 3 six +1 tiles to the right side of scale 1. 3x = 9 Evaluate the multiplication We need 9 positive 1 tiles in place of the question mark to balance scale 2, option D.

Reﬂect andidea Apply the Example 2check We can write this algebraically as:

We must add three +1 tiles on the right side of the scale for every new Purpose Solve 3x = 18 x=3 Write the equation positive tileusing on thethe left side. So, weofneed to add a total of Show students how to balance equations concept a scale 3x = 3 ⋅ 3

Multiply sides by 3 six both +1 tiles to the right side of scale 1. Apply the idea 3x = 9 Evaluate the multiplication Students:To Page 243 weincan divide both sides ofmark the to balance scale 2, option Write Weundo needmultiplication, 9 positive 1 tiles place of the question D. the original equation equation. Divide both sides by 3

Create a strategy

Reﬂect and check

Example 2

Evaluate the division

We can write this algebraically as: x=3 Solve 3x = 18 3x = 3 ⋅ 3

Example 3= 9 Create a 3x strategy

Write the equation Multiply both sides by 3

Apply the idea Evaluate the multiplication To undo multiplication, we can divide both sides of the Solve: equation.

Example 2

Create a strategy

Apply the idea

To undo we can multiply both sides of the Solve 3x division, = 18 equation.

Create a strategy Example 3

Apply the idea

Write the original equation Divide both sides by 3 Evaluate the division Write the original equation Multiply both sides by 8

Evaluate multiplication PurposeTo undo multiplication, we can divide both sides of the Write the the original equation Solve: equation. Show students how to isolate a variable in a simple algebraic equation using division. Divide both sides by 3

strategy Students:Create Pagea 243 Example 4

Apply the idea

Evaluate the division

To undo division, we can multiply both sides of the 5 is a solution to the equation 8x = 40. equation.

Write the original equation

a Verify using Example 3 substitution.

Multiply both sides by 8 Evaluate the multiplication

Solve: Create a strategy

Apply the idea

Substitute 5 into the equation. The left side of the equationashould Create Examplestrategy 4 be equal to the right side if 5 is a solution. To undo division, we can multiply both sides of the

Apply 40the = 40idea

equation. 5 is a solution to the equation 8x = 40. a Verify using substitution.

8 (5) = 40

Substitute x = 5

Evaluate the multiplication Write the original equation The left side of the equation is equal to the right side of the equation, verifying that 5 is a solution. Multiply both sides by 8 Evaluate the multiplication

Create a strategy

6.04 the One-step Apply ideaequations with multiplication and division

243

mathspace.co

Substitute 5 into the equation. The left side of the 8 (5) = 40 Substitute x = 5 equation should Example 4 be equal to the right side if 5 is a 40 = 40 Evaluate the multiplication Purposesolution. The left side of the equation is equal to the right side of 5 is a solution equation 8x = 40. in an equation by using multiplication to cancel out a division. Show students howtotothe isolate a variable the equation, verifying that 5 is a solution. a Verify using substitution.

Create a strategy 484

Substitute 5 into the equation. The left side of the equationVirginia should SOL be equal to 6 the right side if 5 is a Mathspace Grade Teacher Edition solution. mathspace.co

6.04 the One-step Apply ideaequations with multiplication and division

243

mathspace.co

8 (5) = 40

Substitute x = 5

40 = 40

Evaluate the multiplication

The left side of the equation is equal to the right side of the equation, verifying that 5 is a solution.

To undo multiplication, we can divide both sides of the equation.

Write the original equation Divide both sides by 3 Evaluate the division

Expected mistakes StudentsExample might mistakenly believe that to solve the equation 3

= 6, they should subtract 8 from both sides. This

misconception arises from confusing the inverse operations needed to isolate the variable; they may think that Solve: is the inverse of subtraction, subtraction must be used here, even though the operation involved since addition is division. Create strategy Apply the idea To address thisamisconception, remind students that the inverse operation of division is multiplication, not To undo division, we can multiply both sides of the Writethis the original subtraction. Encourage them to identify the operation affecting the variable—in case, xequation is being divided by equation. 8. Guide them to multiply both sides of the equation by 8 to “undo” the division. Multiply both sides by 8

Evaluate the multiplication

Students: Page 243 Example 4 5 is a solution to the equation 8x = 40. a Verify using substitution.

Create a strategy

Apply the idea

Substitute 5 into the equation. The left side of the equation should be equal to the right side if 5 is a solution.

8 (5) = 40

Substitute x = 5

40 = 40

Evaluate the multiplication

The left side of the equation is equal to the right side of the equation, verifying that 5 is a solution.

6.04 One-step equations with multiplication and division mathspace.co

243

Purpose Show students the process of verifying a solution by substitution.

Students: Page 244 b Verify using a model.

Create a strategy Represent the equation using algebra tiles on a scale. Make the left side of the scale to have the same number of unit tiles as the right side by replacing the variable tile with the appropriate tiles.

Apply the idea The equation 8x = 40 can be represented by: +x

The given solution if 5, so we replace each variable tile with 5 unit tiles. Notice that the variable tile is positive, so the resulting 5 unit tiles are positive. +1

equations with multiplication and division +16.04+1One-step +1 +1 +1

mathspace.co

485

The given solution if 5, so we replace each variable tile with 5 unit tiles. Notice that the variable tile is positive, so the resulting 5 unit tiles are positive. +1

Both sides of the scale now have 40 positive unit tiles, verifying that 5 is a solution to the equation.

Purpose Show students how to verify a solution to an equation using a model.

Students: Page 245 Example 5 244

Mathspace Virginia SOL Grade 6 mathspace.co

At a beach fruit stand, fresh squeezed juice is sold at $6 per quart. You and your friends spent a total of $84 on quarts of juice. Write an equation that shows the relationship between the total cost and the number of quarts of juice purchased.

Create a strategy

Apply the idea

The number of quarts of juice purchased is the unknown value.

Let q represent the number of quarts of juice purchased. 6q = 84

PurposeExample 6 Check students’ ability to translate a real-world situation into a mathematical equation and understand the Write a situation that could represent the equation = 52. concept of linear relationship. Create a strategy

Apply the idea

Reflecting with students Think of a situation where there is an unknown value, x, At a craft store, you buy 4 packs of markers. Each pack Encourage advanced learners, or any students who are ready, to build upon the juice stand scenario by creating divided into 4 equal sized groups and each group is contains the same number of markers. In total, you have their own real-world problems that involve writing linear equations. Inviteanthem to think of different contexts, size 52. 52 markers. Write equation that shows the relationship between the number of markers in a pack and the total such as selling homemade crafts at a fair or earning money from mowing lawns, and assign their own prices number of each markers youwrite bought. or rates. For instance, they might consider selling bracelets at $8 and an equation like 8b =96 to represent total sales. By designing their own problems, students can explore how changing variables affects the relationships in Idea summary their equations. Encourage them to represent these relationships graphically, perhaps by plotting the number of items sold against the totalproperty cost toofvisualize relationship. Multiplication equality the If alinear = b then a ⋅ c = b ⋅ c This activity not only deepens their understanding of translating real-world situationsIf into mathematical equations but also fosters creativity and a = b, and c ≠ 0, then Division property of equality engagement. Inverse property of multiplication Identity property of multiplication

486

Substitution property Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice

and a ⋅ 1 = a and 1 ⋅ a = a If a = b, then b can be substituted for a in any expression, equation, or inequality.

purchased.

Create a strategy

Apply the idea

The number of quarts of juice purchased is the Example 5 unknown value.

Let q represent the number of quarts of juice purchased. 6q = 84

Students: Page 245

Example 6

Create a strategy

Write a situation that could represent the equation The number of quarts of juice purchased is the unknown value.

= 52.

Apply the idea Let q represent the number of quarts of juice purchased. 6q = 84

Create a strategy

Apply the idea

Think of a situation where there is an unknown value, x, divided into 4 equal sized groups and each group is size 52.

At a craft store, you buy 4 packs of markers. Each pack contains the same number of markers. In total, you have 52 markers. Write an equation that shows the relationship between the number of markers in a pack and the total number of markers you bought.

Example 6

Write a situation that could represent the equation

Create a strategy

= 52.

Apply the idea

Idea summary

PurposeThink of a situation where there is an unknown value, x, At a craft store, you buy 4 packs of markers. Each pack divided intoMultiplication 4 equal sizedproperty groups and each groupIfisa = b then contains of equality a ⋅ c = bthe ⋅ csame number of markers. In total, you have Check students’ understanding of representing a real life52 situation using a mathematical equation. size 52. markers. Write an equation that shows the relationship If a = b, and c ≠ 0, then Division property of equality between the number of markers in a pack and the total Students: Page 245 number of markers you bought. and Inverse property of multiplication a ⋅ 1 = a and 1 ⋅ a = a If a = b, then b can be substituted for a in any expression, equation, or inequality.

Identity property of multiplication Substitution property Idea summary Multiplication property of equality

If a = b then a ⋅ c = b ⋅ c

Division property of equality

If a = b, and c ≠ 0, then

Practice

and

Inverse property of multiplication

a ⋅ 1 = a and 1 ⋅ a = a If a = b, then b can be substituted for a in any expression, equation, Substitution property or inequality. Write the algebraic equation that represents each set of tiles:

property of multiplication What do Identity you remember? 1

Practice+x

b =

−1

What do you remember?

Practice 1

Write the algebraic equation that represents each set of tiles: a

+x Students: Pages 245–249 = +x

+1 +1

6.04 One-step equations with multiplication and division = −1 −1 −1 mathspace.co

−1

What do you remember? 1

−1

245

Write the algebraic equation that represents each set of tiles: a

+x +1 +1 +1 +x

+1 +1

6.04 One-step equations with multiplication and division = −1 −1 −1 mathspace.co

−1

−1 245

−1

6.04 One-step equations with multiplication and division mathspace.co

487

Consider the scale shown. Which image represents the scale after each side is divided by 2?

+x +x +1 +x

+x +x

State the operation needed to solve the following equations: a

2x = 5

12y = 36

Let’s practice 4

Solve the equation represented by the following a

+x +x

b +x/4

For each pair of scales, Scale 1 has been changed in some way to get Scale 2 so that both scales are balanced. Draw what could go in place of the question mark on Scale 2. Then, use the properties of equality to justify that both scales are balanced. a

Scale 1: +x

Scale 2: +x

488

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Scale 1: +x

Scale 2: +x

Draw a picture to represent each equation and use it to solve for x. a

3x = 18

Draw a picture to determine whether the given value of x is a solution to the equation: a

−5x = 45

x = 7; 5x = 35

x = 12;

The image represents the equation 6b = 24. Jerome says 1 bag must contain −4 coins. Is Jerome correct? Justify your answer.

1¢

Ashton determines that 16 is the solution to the equation

= 8.

How can Ashton conﬁrm his solution is correct? Use a picture to explain your thinking. 10

How would you solve the equation shown?

Multiply both sides of the equation by

Divide both sides of the equation by 5

Multiply both sides of the equation by 5

Divide both sides of the equation by −10

Select the two methods that can be used to solve the algebraic equation: −3x = 9 A

Add 3 to each side.

Multiply each side by −3

Divide each side by −3

Add

Divide each side by 3

Multiply each side by

to each side

6.04 One-step equations with multiplication and division mathspace.co

489

Solve: a

−9t = 3

7t = 56

88 = −11n

12 = −3x

10t = 60

Write an equation to represent each verbal sentence. Verbal Sentence The product of 5 and x is equal to 25. The quotient of p and ten is equal to seven. Twelve times m is equal to ninety six. h divided by eleven equals thirteen.

Algebraic Equation

There are 20 cupcakes on the table. Soleil can bake 12 cupcakes per hour. Write an equation to ﬁnd the number of cupcakes (C) Soleil will have after baking for h hours.

Tony walks 30 000 steps daily. Write an equation that represents the average number of steps Tony takes each hour.

Gian operates a stall at a sports fair. Gian made $458.50 by selling 14 basketballs, 7 soccer balls, and x tennis rackets. a

Find the amount Gian made from selling 14 basketballs and 7 soccer balls.

Write an equation Gian can use to ﬁnd the number of tennis rackets he sold.

$13 $14.50

$35

There are 12 oranges in a basket. The oranges are arranged into three groups. This situation is shown in the model.

+1 +1 +1 +1 +1 +1 +1 +1 x

490

Write an equation to represent this situation.

How many oranges are in each group?

c 18

Use the model to verify your answer.

+1 +1 +1 +1

x=⬚

There are 12 girls in a class. The girls make up one-fourth of the class. This situation can be represented by

Draw a picture to represent this situation.

How many students are in the class?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Consider the following equation: 3x = 42

Solve the equation for x.

Create a real-world scenario where this equation could be applied.

Write a real-world world context that each equation could be used to represent. a

8x = 32

5y = 60

Let’s extend our thinking 21

The price of a kilogram of apples is p dollars. If John bought 5 kilograms of apples for $30, ﬁnd the cost of 1 kilogram of apples.

Solve each equation. Justify your work using a property of equality. a

−8k = 56

12n = −60

Find the unknown number for each of the following: a

When a number is multiplied by 10, the result is 60.

When a number is divided by 8, the result is 9.

The product of 3 and a number is 18.

The quotient of a number and 8 is 12.

Show how you can solve the equation 4x = 12 using the multiplication property of equality.

Bryon is solving the equation

= 6. The following are his steps: Step 1:

Step 2:

⋅2=6÷2

Step 3:

x=3

In which step did Bryon made a mistake? Explain your answer.

6.04 One-step equations with multiplication and division mathspace.co

491

Answers

c 3x = 18

6.04 One-step equations with multiplication and division

What do you remember?

1 a 2x = 10

= −9

x = 6

2 D

3 a Division

b Multiplication

c Division

d Multiplication

4 a x = 3

−x

= −x

−x

7 a Yes b Yes

x = −9 x

x = −8

6 a −5x = 45

−x

b x = 20

5 a Example answer: b Example answer: +1

Let’s practice

−1

8 Jerome is not correct. If we replace each bag with − 4 coins, that gives us − 24 which is not equal to positive 24 coins on the right.

b x 4

−1 ¢ −1 ¢ −1 ¢ −1 ¢ −1 ¢ −1 ¢ −1 ¢ −1 ¢

x = 20

−1 ¢ −1 ¢ −1 ¢ −1 ¢

1¢

9 Ashton can conﬁrm his solution is correct by visually representing the equation using objects such as circles. When 16 is divided into two equal groups, each group has 8 circles. This conﬁrms that x = 16 is a solution to the

10 C 11 C and F

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

≠

1¢

−1 ¢ −1 ¢ −1 ¢ −1 ¢

equation

492

12 a e −8 = n

b t=8

c t = 60

d x = 48

x = −4

g −48 = x

h t=6

Verbal Sentence

b Answers may vary

Algebraic Equation

The product of 5 and x is equal to 25.

5x = 25

c Answers may vary Example answer: A contractor pays his workers y dollars per hour. If he needs to spend exactly 60 dollars for a job, this equation can be used to calculate how many hours of work he is paying for at the rate of 5 dollars per hour.

The quotient of p and ten is equal to seven. 12 m = 96

Twelve times m is equal to ninety six. h divided by elevenequals by thirteen.

d Answers may vary

14 C = 20 + 12h 15

= 1250 steps per hour

16 a $283.50

b 14 (13) + 7 (14.50) + 35x = 458.50

17 a 3x = 12

b x = 4

Example answer: A school has 72 textbooks that need to be evenly distributed among p classrooms. The equation helps determine the number of classrooms each receiving 8 textbooks. Let’s extend our thinking 21 k = $6

c x

Example answer: A bus has 63 seats. If each row in the bus contains an equal number of seats and there are 9 rows, this equation can be used to determine how many seats there are in each row.

22 a −8k = 56

Given equation

k = −7

Division property of equality

Given equation

r = 20

Multiplication property of equality

This shows there are 4 oranges in each group of x which veriﬁes the solution from part (b).

18 a W e can divide the class into 4 groups where one of the groups represents the girls.

Given equation

q = −42

Multiplication property of equality

d 12n = −60

Given equation

n = −5

23 a 6

Division property of equality

b 72

c 6

d 96

24 To solve the equation 4x = 12 by multiplying each side by

Classroom g

the reciprocal of 4, we multiply both sides by , which gives us

b 48 students 19 a x = 14 b T here are 42 beads that need to be split evenly to make 3 bracelets. How many beads will be on each bracelet?

⋅ 4x =

⋅ 12. This simpliﬁes to x = 3, showing

that multiplying each side by the reciprocal of 4 isolates x on one side of the equation, yielding the solution. 25 In which step did Bryon made a mistake? Explain your answer. Example answer: Bryon made a mistake in Step 2. He should have multiplied both sides of the equation by 2, instead of dividing the right side by 2.

20 a Answers may vary Example answer: A shop sells boxes of chocolates, each box containing 8 chocolates. If a customer buys enough boxes to accumulate a total of 32 chocolates, this equation can be used to ﬁnd out how many boxes they purchased.

Answers mathspace.co

493

6.05 Write inequality statements Subtopic overview Lesson narrative In this lesson, students will learn to write inequality statements to compare values using symbols such as, ≥, <, >, ≤. They will understand the meanings of these symbols and how to write inequalities that represent real-world situations. The lesson includes practical examples and exercises where students practice converting verbal statements into inequalities and vice versa. Additionally, students will identify values that are part of a solution set for given inequalities. By the end, students should be able to confidently write and interpret inequality statements in various contexts.

Learning objectives Students: Page 250

Key vocabulary 

greater than



greater than or equal to



inequality



inequality symbol



less than



less than or equal to

Essential understanding An inequality is a mathematical sentence that compares two expressions and uses the symbols >, <, ≤, or ≥.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving In teaching the concept of linear inequalities, teachers can integrate Mathematical Problem Solving by presenting students with real-world problems that require the use of inequalities. For instance, teachers could provide a scenario where a certain budget must not be exceeded, prompting students to create a linear inequality that represents this constraint. This encourages students to apply their understanding of inequalities to solve complex, real-world problems.

494

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

Teachers can encourage Mathematical Communication by having students explain their thought process when solving problems involving inequalities. For example, when students create linear inequalities from verbal situations, teachers could ask them to explain why they chose a particular inequality symbol or how they translated the verbal description into a mathematical expression. Such discussions can help students to clarify their understanding and express their mathematical thinking.

Teachers can incorporate mathematical reasoning into their instruction on writing inequality statements by addressing common misconceptions and emphasizing the conceptual understanding of phrases like “at least” and “at most.” For example, when discussing “at least,” ask students if a certain value (e.g., 101 texts for “at least 100 texts”) fits the inequality to help them understand it should be greater than or equal to. Similarly, use concrete materials such as algebra tiles and balance scales to connect verbal descriptions to inequalities. Encourage students to label their models and substitute numbers to see which values make the inequality true. Asking questions that prompt critical thinking about how the inequality represents the story problem is essential for building a deep understanding.

MPG5 — Mathematical Representations Teachers can incorporate mathematical representations into their instruction on writing inequality statements by using graphic organizers to help students connect verbal descriptions, symbols, explanations, and graphs. For example, present a scenario where a student has a maximum number of items to share and have them complete a four-part organizer that includes writing the inequality, graphing it, and explaining their reasoning. Encourage class discussions on comparison vocabulary like “less than” or “less than or equal to” and use number lines to visualize solutions. To deepen understanding, have students create examples and non-examples of inequalities, and connect these representations to real-world contexts. This approach helps students make meaningful connections between different mathematical representations.

Content standards 6.PFA.4 — The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

6.PFA.4c — Given a linear inequality in one variable, create a corresponding contextual situation or create a number line graph.

6.PFA.4b — Write a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line.

6.PFA.4e — Identify a numerical value(s) that is part of the solution set of a given inequality in one variable.

Prior connections 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

6.PFA.3 — The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable.

Future connections 7.PFA.4 — The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a oneand two-step linear inequality in one variable.

7.PFA.3 — The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

6.05 Write inequality statements mathspace.co

495

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 2.04 Compare and order fractions, decimals, and percents

Student lesson & teacher guide Inequality symbols While an equation represents two sides being equally balanced, an inequality represents one side being larger than the other. The symbols change meaning depending on some factors like values being compared, context of comparison, and units of measurement. This section explores the use words, symbols, and actual values to represent different inequality statements.

Students: Pages 250–251

496

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

If we switch the order so that there are three cats on the left and ﬁve cats on the right, we can say that three cats is less than ﬁve cats. We can write this as an inequality 3 < 5. Both images mean the same thing but are stated differently. If we switch the order of an inequality, we have to change the inequality sign. This is also true with algebraic inequalities. For example, x > 10 means the same thing as 10 < x. In other words, “x is greater than ten” is the same as “ten is less than x”. For example, the expressions x > 5 and 5 > x represent different sets of numbers, while x > 5 and 5 < x represent the same set of numbers. We can use this understanding of inequality symbols to write inequalities that represent real-world situations. Let’s write an inequality to represent the statement: “a student needs to score at least 75 points to pass an exam.” Let s represent the student’s score. The key phrase “at least 75 points” tells us that the lowest passing score is 75. So the student will pass the exam if they score 75 points or if they score more than 75 points. If we use s to represent the score we can write the inequality s ≥ 75. Here are some common phrases and examples used for the different inequality symbols. Inequality Symbol < > ≤ ≥

Vocabulary/Representations

Example

less than, fewer than, under greater than, exceeds, more than less than or equal to, at most, no more than, up to greater than or equal to, at least, no less than

“The speed limit is less than 60 mph.” translates to S < 60 “The temperature is greater than 30 °C” translates to T > 30 “You can spend up to 50 dollars.” translates to C ≤ 50 “You need at least 8 hours of sleep.” translates to H ≥ 8

Example 1

Use a number Concrete-Representational-Abstract (CRA) approach For the sentence ⬚ 0.3. Targeted strategies a Chooseinstructional the mathematical symbol that makes the number sentence true.

Concrete: Begin by engaging students with physical manipulatives to model inequalities in real-world situations. Create a strategy Use items like counters, blocks, or coins to represent different quantities. For example, present a scenario Convert the fraction to a decimal to compare values easily. where one group has 5 blocks and another has 8 blocks. Ask students to physically compare the two sets by placing Apply them side by side to see which group has more or less. Encourage students to use terms like “greater the idea than” or “less than” as they discuss their observations. This hands-on activity helps students understand Convert to decimal inequalities through direct comparison of tangible objects. Compare the decimals So, we can see that

> 0.3.

Reﬂect and check Another true inequality with the same numbers is 0.3 < .

6.05 Write inequality statements mathspace.co

497

Representational: Transition from manipulatives to drawings that represent the quantities. Have students draw pictures or use symbols to depict the blocks they used earlier. For instance, they can draw 5 circles for the first group and 8 circles for the second group. Ask them to compare the drawings just as they did with the physical objects. Begin the transition to the abstract stage by introducing inequality symbols. Show studdents how they can represent the relationship between the two groups. Have students draw the correct symbols between the groups of images. Abstract:If we Move onthe to order usingsoabstract andon numbers switch that theresymbols are three cats the left to represent inequalities without the aid of drawings and ﬁve catsProvide on the right, we can say that threeand cats is or manipulatives. real-world scenarios have students write inequality statements to represent them. less than cats. We can write as an For example, if aﬁve person must be atthis least 13inequality years old to create an account on a website, they can write “x ≥ 13”. 3 < 5. Practice identifying values that satisfy the inequality by substituting different numbers for “x” and checking if the Both images mean thehelps same thing but areapply statedinequalities to various contexts using mathematical notation. inequality holds true. This students differently. If we switch the order of an inequality, we have to change the inequality sign. This is also true with algebraic inequalities.

Vocabulary and For example, x >exercise: 10 means thecollect same thing as 10display < x. In other words, “x is greater than ten” is the same as “ten is less than x”. language learner support English For example, the expressions x > 5 and 5 > x represent different sets of numbers, while x > 5 and 5 < x represent the

While students are working on writing inequalities to represent real-world situations, listen carefully for the same set of numbers. phrases they use to describe each of the four inequality symbols. Create a visual display divided into four We can use this understanding of inequality symbols to write inequalities that represent real-world situations. sections—one for each inequality symbol—and list the students’ phrases in each section. Let’s write an inequality to represent the statement: “a student needs to score at least 75 points to pass an exam.” For example, under “<”, include score. expressions like “less than”, “fewertells than”, “below”; under “≥”,score list phrases such Let s represent the student’s The key phrase “at least 75 points” us that the lowest passing is 75. So the student will pass exam if they score points or if they score more than 75 points. If we use s to represent as “greater than or equal to”,the“at least”, “no less75than”. the score we can write the inequality s ≥ 75.

Encourage students to contribute additional phrases they discover or use. This organized display will Here are some common phrases and examples used for the different inequality symbols. help English language learners connect verbal expressions with mathematical symbols, reinforcing their Inequality understanding of inequalities. Keep the display accessible throughout the lesson Vocabulary/Representations Exampleso students can reference and Symbol build upon it as they practice writing and interpreting inequality statements. <

less than, fewer than, under greater than, exceeds, more than less than or equal to, at most, no more ≤ than, up to greater than or equal to, at least, no Students:≥Page 251less than >

Examples

“The speed limit is less than 60 mph.” translates to S < 60 “The temperature is greater than 30 °C” translates to T > 30

“You can spend up to 50 dollars.” translates to C ≤ 50 “You need at least 8 hours of sleep.” translates to H ≥ 8

Example 1 For the number sentence

⬚ 0.3.

a Choose the mathematical symbol that makes the number sentence true.

Create a strategy Convert the fraction to a decimal to compare values easily.

Apply the idea Convert to decimal Compare the decimals So, we can see that

> 0.3.

Reﬂect and check Another true inequality with the same numbers is 0.3 < .

Purpose Students demonstrate that they can use an inequality symbol to compare different representations of rational 6.05 Write inequality statements 251 mathspace.co numbers. 498

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Expected mistakes Students may forget how to compare a fraction and decimal. Remind them of the different forms a number can be written in and ask them which form they think would be easiest to use for comparing. Reflecting with students Challenge students to find a value that falls between these two numbers.

Students: Page 252 b Write the statement in words.

Create a strategy

Apply the idea

Replace the inequality symbol > with a vocabulary that has the same meaning.

“ is greater than 0.3” or “ is more than 0.3”

b Write the statement in words.

PurposeExample 2 Create a strategy Apply the idea Show students how to convert numerical comparisons into written statements. Write an inequality to represent each of the following situations. Replace the inequality symbol > with a vocabulary that has same meaning. a nthe is greater than 9

“ is greater than 0.3” or “ is more than 0.3”

Create a strategy

Apply the idea

Students: Page 252

The phrase “greater than” tells us which inequality Example 2 symbol to use.

n>9

Write an inequality to represent each of the following situations. b weight of the9package is under 5 kg. Let w be the weight of the package. a The n is greater than

Create a strategy

Apply the idea

word “under” means The phrase “greater than”less tellsthan. us which inequality symbol to use. b Write the statement in words.

w 5 n >< 9

c You must be at least 18 years old to vote in the United States. Let a be the age of the voters. Create a strategy Applyofthe b The weight of the package is under 5 kg. Let w be the weight theidea package. Replace the inequality symbol > with a vocabulary that

the idea “Apply is greater than 0.3” or “ is more than 0.3” PurposeCreate a strategy Create a strategy Apply the idea has same The the phrase “atmeaning. least 18” means you can vote in the US if a ≥ 18statement. Students demonstrate that they can write the inequality corresponding to a verbal The word “under” less you are exactly 18 means years old orthan. if you are older.

w<5

Expected mistakes StudentscdExample may switch variable in United the Explain that important and show them 2be the The must maximum foryears theand ride is Let h beanswer. the heightLet ofathe You atheight least 18 old constant to 120 votecm. in the States. beriders. the ageorder of theis voters. how 9 > n reads as “9 is greater than n” which has the opposite meaning of the original statement. Write an inequality to represent each of the following situations. Create a strategy Apply the idea

Reflecting students a nwith is greater than 9 word “maximum” means highest. So,vote to ride you The phrase “at least 18” means you can in the UScan if ha≤≥120 18 An inequality with a variable means that be that height or but you cannot be taller. values can be replaced for the variable and give a true you are exactly 18shorter years old or if you aredifferent older. Create strategy to give different examples of values Apply thewould idea make the statement true. statement. Askastudents that The phrase “greater than” tells us which inequality

The to maximum height for the ride is 120 cm. Let h be the height of the riders. use. Students:dsymbol Page 252

n>9

Example 3

Create a strategy

Apply the idea

Write a real-world scenario forhighest. each inequality. b The weight of the package is under 5 kg. Let w becan the weight of the package. The word “maximum” means So, to ride you be height or shorter but you cannot be taller. a that x ≤ 20

Create a strategy

Apply the idea

The word means less than. Create a “under” strategy

Apply the idea

h ≤ 120

w<5

We can think of x as an object that can contain up to 20 A sample scenario would be “A group of students is Example 3 pieces of another object. preparing boxes for toy donation. Each box can hold up to c You must be at least 18 years old to vote in the United States. Let toys”. a be the age of the voters. 20 small Write a real-world scenario for each inequality.

PurposeaCreate a strategy Apply the idea x ≤ 20 StudentsThe demonstrate that they can write the inequality corresponding to a verbal statement. phrase “at least 18” means you can vote in the US if a ≥ 18

Create a strategy you are exactly 18 years old or if you are older. 252 Mathspace Virginia SOL Grade 6

Apply the idea

mathspace.co We can think of x as an object that can contain up to 20 A sample scenario would be “A group of students is 6.05 Write inequality statements pieces of another object. preparing boxes for toy donation. Each box can hold up to mathspace.co d The maximum height for the ride is 120 cm. Let h be the 20 height of the riders. small toys”.

Create a strategy

Apply the idea

499

Write an the inequality to represent the following Replace inequality symbol > each with aofvocabulary thatsituations. “ is greater than 0.3” or “ is more than 0.3” has same meaning. b weight of the9package is under 5 kg. Let w be the weight of the package. a The nthe is greater than

Create Create a a strategy strategy

Apply Apply the the idea idea

The phrase word “under” means “greater than”less tellsthan. us which inequality Students:The Page 252 Example 2

w 5 n >< 9

symbol to use.

Write an inequality to represent each of the following situations. c You must be at least 18 years old to vote in the United States. Let a be the age of the voters. a The n is greater than b weight of the9package is under 5 kg. Let w be the weight of the package.

Create a strategy Create a Create a strategy strategy The phrase “at least 18” means you can vote in the US if

Apply the idea Apply Apply the the idea idea

The phrase “greater than” tells which you exactly 18 means years old orthan. ifusyou are inequality older. The are word “under” less symbol to use.

a ≥ 18 n >< 9 w 5

The must maximum foryears the ride Let United h be the heightLet ofathe cd You be atheight least 18 old is to 120 votecm. in the States. beriders. the age of the voters. b The weight of the package is under 5 kg. Let w be the weight of the package.

Purpose Create a a strategy strategy Apply the the idea idea Apply StudentsCreate demonstrate that they can write the inequality corresponding Create a strategy Apply the idea to a verbal statement. The “maximum” means highest. So,vote to ride you The word phrase “at least 18” means you can in the UScan if The word “under” means less than. be that height or18shorter but you cannot taller. you are exactly years old or if you are be older.

ha≤≥120 18 w<5

Students: Page 252

cd You be atheight least 18 old is to 120 votecm. in the States. beriders. the age of the voters. The must maximum foryears the ride Let United h be the heightLet ofathe

Example 3 Create Create a a strategy strategy

Writephrase a real-world scenario for each inequality. The “at least 18” means you can in the UScan if The word “maximum” means highest. So,vote to ride you you are exactly 18 years old or if you are older. a x ≤ 20 be that height or shorter but you cannot be taller.

Create a strategy

Apply Apply the the idea idea 18 ha≤≥120

Apply the idea

d The maximum height for the ride is 120 cm. Let h be the height of the riders. We can think of x as an object that can contain up to 20 A sample scenario would be “A group of students is 3 pieces of another object. preparing boxes for toy donation. Each box can hold up to PurposeExample Create a strategy Apply the idea 20 small toys”. StudentsThe demonstrate that they can write the inequality corresponding to a verbal statement. Writeword a real-world scenario forhighest. each inequality. “maximum” means So, to ride you can h ≤ 120 be that height or shorter but you cannot be taller. a x ≤ 20

Students: Page 252

Create a strategy Virginia SOL Grade 6 252 Mathspace

Apply the idea

mathspace.co We can think of x as an object that can contain up to 20 Example 3 pieces of another object.

A sample scenario would be “A group of students is preparing boxes for toy donation. Each box can hold up to 20 small toys”.

Write a real-world scenario for each inequality. a x ≤ 20

Create a strategy

Apply the idea

We can think of x as an object that can contain up to 20 mathspace.co pieces of another object.

A sample scenario would be “A group of students is preparing boxes for toy donation. Each box can hold up to 20 small toys”.

252

Mathspace

Virginia SOL Grade 6

Purpose252 Mathspace Virginia SOL Grade 6 mathspace.co To help students understand and apply inequalities in real-world scenario.

Students: Page 253

Purpose To help students understand and apply inequalities in real-world scenario.

500

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 253

Purpose To help students understand and apply inequalities in real-world scenario.

Students: Page 253

Purpose To help students understand and apply inequalities in real-world scenario.

Students: Pages 253–254

6.05 Write inequality statements mathspace.co

501

Reﬂect and check Let’s check s ≤ 4 on a number line. The number line shows that all values to the left of 4, and including 4 are all possible values of s. 1

However, in the given context, s represents the number of garments, which means that s should have a minimum value of 0, because it is not possible to try on a negative number of garments.

Idea summary Purpose Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the >, ≤, and ≥.and apply inequalities in a real-world context. Show studentssymbols: how to<,interpret Symbol

Meaning

Example

Reflecting with< students less than, fewer than, under 3<6 Challenge advanced learners, orthan, anyexceeds, students who are ready, to create6their > greater more than > 3 own fitting room policies lessallowing than or equal to, to at most, no more up to 4 ≤expressions 6 represented by≤ inequalities, them explore howthan, mathematical model real-world greater than or equal to, at least,like no less 6≥5 constraints. For≥ example, they could design a policy s ≥than 2, where customers must try on at least two garments, or 2 ≤ s ≤ 6, setting both minimum and maximum limits. Prompt students to consider the implications of their policies and how changing the inequality signs or values affects the possible number of garments. By designing their own problems, students delve deeper into Practice understanding inequalities and their applications, enhancing their critical thinking and problem-solving skills.

What do you remember? Use math talk sentence frames to support language comprehension

use with Example 4

Support students with disabilities 1

Write the symbol (<, >, ≤, ≥ ) that best matches the description. Symbols may be used more than once.

Introduce sentence to help students articulate their understanding a Is at frames least b Is more than c Is fewer than of inequalities d Is atand mosthow they relate to real-world esituations. frames such as “An exampleg ofDoes a value that makes htheIsinequality ⬚ true is ⬚ Is no lessProvide than f Is below not exceed not less than because 2⬚ orFor“Ifeach ⬚ represents ⬚, then the inequality is ⬚.” description, determine if the scenario represents an equation or an inequality. Explain your answer. a A theme park these ride requires riders to be at least 48 inches Encourage students to use frames when converting verbaltall. statements into inequalities and identifying To qualify for aframes marathon, needposter to complete a qualifying race in under 4 hours. solution sets.bDisplay these onparticipants a classroom or give out personal reference cards for easy c Jamie is baking cookies and the recipe calls for 2 cups of sugar. reference during activities.

Reﬂect checkgoalie aims to allow at most 10 goals scored in a season. d and A soccer

Begin byLet’s modeling use requires these with to practical examples, check ≤how 4 on to aproject number line. frames e Asscience a bean plant grow to exactly 15 cm guiding students to express their reasoning verbally andthat in writing. this help students become more Theboth number line shows all values Over to the time, left of 4, andwill including 4 are all possible values of s.confident in explaining inequality statements and improve their ability to interpret and create them in various contexts. Let’s practice 1 2 3 4 5 6 3

Write the following inequalities in words:

in the given context, s represents the number of garments, which means that s should have a minimum Students:However, Page a 254 17 > x b p<4 c a ≥ 43 d k≤8 value of 0, because it is not possible to try on a negative number of garments. e b < 2.23 f m ≥ 24 g s ≤ 50 4

r>4

Write the inequality described by the following statements: aIdea n is summary greater than 9.

n is greater than or equal to 9.

c n is less than 10. d n is less than or equal to 6. Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the esymbols: a is positive. <, >, ≤, and ≥. Symbol Meaning < less than, fewer than, under 254 Mathspace Virginia SOL Grade 6 > greater than, exceeds, more than mathspace.co ≤ ≥

less than or equal to, at most, no more than, up to greater than or equal to, at least, no less than

Example 3<6 6>3 4≤6 6≥5

Practice 502

What do you remember?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 1 Write the symbol (<, >, ≤, ≥ ) that best matches the description. Symbols may be used more than once. a Is at least b Is more than c Is fewer than d Is at most e Is no less than f Is below g Does not exceed h Is not less than

Practice Students: Pages 254–255

What do you remember? 1

Write the symbol (<, >, ≤, ≥ ) that best matches the description. Symbols may be used more than once. a

Is at least

Is more than

Is fewer than

Is at most

Is no less than

Is below

Does not exceed

Is not less than

For each description, determine if the scenario represents an equation or an inequality. Explain your answer. a

A theme park ride requires riders to be at least 48 inches tall.

To qualify for a marathon, participants need to complete a qualifying race in under 4 hours.

Jamie is baking cookies and the recipe calls for 2 cups of sugar.

A soccer goalie aims to allow at most 10 goals scored in a season.

A science project requires a bean plant to grow to exactly 15 cm

Let’s practice 3

Write the following inequalities in words: a

17 > x

p<4

a ≥ 43

k≤8

b < 2.23

m ≥ 24

s ≤ 50

r>4

Write the inequality described by the following statements: n is greater than 9.

n is greater than or equal to 9.

n is less than 10.

n is less than or equal to 6.

a is positive.

A movie theater only allows entry if there are fewer than 120 tickets sold to ensure everyone has a seat. Which inequality represents this situation? A

x > 120

x ≥ 120

x ≤ 120

Jaime wants to buy a bike that costs $200. He must save at least this amount to purchase it. Which inequality represents this situation? A

x < 120

x < 200

x > 200

x ≤ 200

x ≥ 200

Write an inequality to represent each of the following situations: a

The width of a particular road is 5 meters or greater. Let w be the width of the road.

The elevator in a building should carry less than 13 people at one time. Let p be the number of people in the elevator.

Michael lives where it takes him at most 15 minutes to get to work. Let x be the time Michael takes to get to work.

In an amusement park, children below 5 years old can enter free of charge. Let c be the age of the a child that can enter the park free of charge.

Frasier is expecting that at least 50 guests will attend his party. Let g be the number of guests in Frasier’s party.

In a gift exchange party, the attendees must spend at most $50 for their gifts. Let x be the amount of a gift in the gift exchange party.

6.05 Write inequality statements mathspace.co

503

Write in a real-world scenario for each inequality: a

x≤5

y ≥ 12

a < 33

The library has a rule on how many books you can borrow. The rule can be represented by n ≤ 5, where n represents the number of books one can borrow. Which options show the number of books you can borrow? Select all correct options. A

Mitch and Nicole are having a mathematical debate. Mitch states that x > 4 is the same relationship as 4 > x. Nicole states that x > 4 is the same relationship as 4 < x. Who is correct? Explain why.

Let’s extend our thinking 11

Write the inequality described by the following statements: a

The product of 5 and x is less than or equal to 20.

Seven more than the value of x is at least 9.

Eight is greater than the result of taking 7 away from x.

Jack is saving up to buy a smartphone that is selling for $510. He has in his bank account and expects some money for his birthday next week. If the amount he is about to receive for his birthday is represented by x, write an inequality that models the situation where he is able to afford the smartphone.

504

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

5 A 6 D

6.05 Write inequality statements

7 a w ≥ 5 g g ≥ 50

What do you remember? 1 a ≥ e ≥

b >

c <

d ≤

g ≤

h ≥

2 a Inequality, the riders can be a range of heights b I nequality, the race times can be a range of times to qualify c Equation, the recipe requires an exact amount d Inequality, a range of goals will satisfy the goalie’s aim e Equation, the project requires an exact height

3 a 17 is greater than x. b p is less than 4. c a is greater than or equal to 43. d k is less than or equal to 8. e b is less than 2.23. m is greater than or equal to 24.

c n < 10

x ≤ 50

8 a A baker is preparing small boxes of cookies. Each box can hold no more than 5 cookies to maintain quality. b A park requires that all groups reserving a picnic area must have at least 12 members to ensure eﬃcient use of space. c T he speed limit in a residential area is less than 33 miles per hour. 9 A, B, C, D, E

11 a 5x ≤ 20

h r is greater than 4. b n≥9

d c<5

Let’s extend our thinking

g s is less than or equal to 50. 4 a n > 9

c x ≤ 15

10 Nicole is correct in this mathematical debate. The inequality x > 4 means that x is greater than 4. To write this inequality with 4 ﬁrst, we need to reverse the inequality symbol, so it becomes 4 < x, which also means 4 is less than x. The two inequalities x > 4 and 4 < x convey the same relationship: x is a number greater than 4.

Let’s practice

b p < 13

d n≤6

b x+7≥9

c 8>x−7

12 x + 210 ≥ 510

e a > 0

Answers mathspace.co

505

6.06 Solutions to inequalities Subtopic overview Lesson narrative In this lesson, students will learn to identify and graph solutions to inequalities. They will explore the concept of a solution set, which includes all numbers that make the inequality true. The lesson includes practical examples and exercises where students graph inequalities on a number line, using open and closed circles to indicate whether endpoints are included. Students will also practice writing inequalities based on real-world scenarios and identify values that are part of the solution set. By the end, students should confidently graph and interpret solutions to inequalities.

Learning objectives Students: Page 256

Key vocabulary 

graph



number line



scenario

 solution set

Essential understanding Inequalities have an infinite number of solutions so their solution sets are often represented on a number line.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal by providing students with real-world problems that require the application of the concepts of number lines and inequalities. They can guide students in the process of creating linear inequalities from contextual situations and how these inequalities can be represented on a number line. This will help students apply mathematical concepts and skills to solve problem situations of varying complexities.

506

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can support students in identifying and graphing solutions to inequalities by emphasizing the reasoning behind each step. Begin with contextual problems and guide students to write the corresponding inequality. Encourage students to justify each step by explaining how they determined the inequality and why they chose specific boundary points and direction of shading on the number line. Use common misconceptions as teaching points, such as misunderstanding “less than” vs. “less than or equal to,” by providing counterexamples and discussing why certain values satisfy the inequality while others do not. Encourage students to verbalize their thought processes and reason through multiple examples to build a deeper understanding of inequalities and their graphical representations. This method helps students make sense of inequalities, relate them to real-world contexts, and accurately represent solutions graphically.

Teachers can effectively teach students how to identify and graph solutions to inequalities by using a multirepresentation approach. Begin by providing a verbal description of an inequality and ask students to translate it into an inequality statement. Use a four-part graphic organizer where students write the inequality, explain their reasoning, and graph the inequality on a number line. Encourage discussions about comparison vocabulary to ensure students understand terms like “greater than” or “less than.” Use examples and non-examples to highlight differences and help students visualize solutions. Additionally, incorporate real-world contexts to make the inequalities meaningful and relevant, and ask students to justify their graphing decisions to reinforce their understanding of the relationship between the inequality statement and its graphical representation. This approach ensures that students make connections between different mathematical representations and solidify their understanding of inequalities.

Content standards 6.PFA.4 — The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

6.PFA.4c — Given a linear inequality in one variable, create a corresponding contextual situation or create a number line graph.

6.PFA.4a — Given the graph of a linear inequality in one variable on a number line, represent the inequality in two equivalent ways (e.g., x < −5 or 5 > x using symbols. Symbols include <, >, ≤, ≥.

6.PFA.4d — Use substitution or a number line graph to justify whether a given number in a specified set makes a linear inequality in one variable true.

6.PFA.4b — Write a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line.

6.PFA.4e — Identify a numerical value(s) that is part of the solution set of a given inequality in one variable.

Prior connections 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

6.PFA.3 — The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable.

Future connections 7.PFA.4 — The student will write and solve one-and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable.

7.PFA.3 — The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

6.06 Solutions to inequalities mathspace.co

507

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 6.03 One-step equations with addition and subtraction Grade 6 — 6.05 Write inequality statements

Tools You may find these tools helpful: • Number lines

Student lesson & teacher guide Solutions to inequalities Students are introduced to the concept of a solution set and what the solution set to an inequality looks like on a number line.

Students: Pages 256–257

508

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Filled or unfilled endpoints Targeted instructional strategies A number line can visually represent the solutions of an inequality by showing where the solutions begin, whether that value is included, and a ray that indicates where more solutions can be found. A useful exercise is to give students already solved and isolated inequalities, such as x > 4, y ≤ 1, x < 6, and y ≥ 0. Begin by having students discuss values that would satisfy these inequalities, and whether or not the number in the inequality itself could be included. Have students plot the numbers they found on a number line, having them draw a ray through the points they plotted, and an unfilled circle around the starting point if the “or equal to” line is missing. Connect these representations to the general rules of how to graph solutions on a number line. For x > 4, a hollow circle indicates that 4 is not part of the solution: −4 −3 −2 −1

For y ≤ 1, a filled circle indicates that 14 is part of the solution: −5

−4

−3

−2

−1

6.06 Solutions to inequalities mathspace.co

509

Collect and display English language learner support Lead a discussion with small groups or whole group of how to graph solutions on a number line. Collect and display steps and vocabulary to keep visible as students work through the process. Vocabulary should include: solution, included, number line, greater than, less than, equal to, satisfy, positive infinity, negative infinity, maximum, minimum. The following can be used as a guide to graphing solutions of inequalities on a number line. • For a “greater than” inequality (e.g., x > 3), draw an open circle at the value of the number (3 in this case) and draw a ray pointing to the right to represent all values greater than 3. The direction of the ray points towards positive infinity. • For a “less than” inequality (e.g., x < −2), draw an open circle at the value of the number (−2 in this case) and draw a ray pointing to the left to represent all values less than −2. The direction of the ray points towards negative infinity. • For a “greater than or equal to” inequality (e.g., x ≥ 5), draw a closed circle at the value of the number (5 in this case) and draw a ray pointing to the right to represent all values greater than or equal to 5. The direction of the ray points towards positive infinity. • For a “less than or equal to” inequality (e.g., x ≤ −1), draw a closed circle at the value of the number (−1 in this case) and draw a ray pointing to the left to represent all values less than or equal to −1. The direction of the ray points towards negative infinity.

Ray direction and inquality symbol Student with disabilities support Provide a number line for students to use as they complete problems to help them determine which side of the number line to shade in relation to the critical value. Students can mark integer solutions on the number line that satisfy the inequality, then join those points to determine the shaded region. Associate the inequality sign to the direction of the ray on the number line. When the variable is on the left side and the number is on the right side of the inequality sign, such as x > 5, the inequality symbol > looks like the arrowhead of a ray that points to the right direction. The graph of x > 5 is: −2

−1

The graph of x < 5 will have a ray with an arrowhead towards the left direction: −2

−1

Students can use this method to determine the shaded region, then check the reasonableness of their solution region by substituting values on that side of the number line to ensure they satisfy the inequality.

Maximum and minimum Address student misconceptions It’s possible that students grasp the definition of maximum as the greatest value, but struggle to comprehend that all other values are consequently smaller. Similarly, they may also have difficulty comprehending the concept of minimum value. To address this, engage these students in a brief discussion about real-world scenarios that incorporate minimum and maximum values.

510

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Examples Students: Page 257

Purpose To check that students can use a number line to identify whether a value is in the solution set of an inequality.

Students: Page 257

Purpose To ensure students can apply the symmetric property to flip an inequality.

6.06 Solutions to inequalities mathspace.co

511

Examples Students: Page 258

Purpose To ensure students can graph an inequality with a filled endpoint.

Students: Page 258

Purpose To ensure students can graph an inequality with an unfilled endpoint.

Advanced learners: Extend to explore compound inequalities

use with Example 2

Targeted instructional strategies Encourage students to deepen their understanding by exploring how individual inequalities can combine to form compound inequalities. After graphing x ≥ 1 and x < 7 separately, prompt students to consider the values that satisfy both conditions simultaneously. Challenge them to graph the overlap of the two solution sets, highlighting the interval where both inequalities are true. 0

Discuss the concept of intersection and how it results in the compound inequality 1 ≤ x < 7. For further extension, show students how to express this solution using interval notation [1, 7), discussing the significance of the closed bracket at 1 and the open parenthesis at 7. This approach not only reinforces their understanding of inequalities but also connects graphical representations with symbolic language, enhancing their overall mathematical reasoning.

512

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Pages 258–259

Purpose To ensure students can substitute values to identify which are in the solution set, and understand that for an inequality with “equal to” the value in the inequality statement is included in the solution set. Reflecting with students Encourage students to represent the inequality x ≤ 5 on a number line to visualize the solution set. Have them plot the given values—2, −3, 5, 7.3, and −5 −4 −3 −2

—to see where each one falls in relation to the inequality. −1

This visual aid can help them understand why certain numbers satisfy the inequality while others do not. Highlight the importance of the “equal to” part in the inequality and how it affects the inclusion of the boundary value 5 in the solution set. Ask students to consider additional numbers, such as 0 and 6, and discuss whether these values are included in the solution set. This opens up a discussion about the range of values that satisfy the inequality and reinforces the concept that inequalities represent a continuum of solutions, not just discrete points.

Students: Page 259

6.06 Solutions to inequalities mathspace.co

513

Purpose To ensure students can substitute values to identify which are in the solution set, and understand that for an inequality without “equal to” the value in the inequality statement is not included in the solution set.

Students: Page 259

Purpose To ensure students can write an inequality from a given context. Expected mistakes Students might misunderstand the phrase “at least 16” and interpret it as “less than or equal to 16.” As a result, they may write the inequality a ≤ 16 instead of a ≥ 16. To address this misconception, encourage students to think about real-world examples to clarify the meaning of “at least.” Ask them whether someone who is 15 years old meets the age requirement to swim during adult swim times. This will help them realize that only individuals who are 16 or older are eligible. Use a number line to visually represent ages that are “at least 16,” shading the region from 16 upwards to show all ages greater than or equal to 16. 12

Reinforce the concept that “at least” means “greater than or equal to” by relating it to familiar contexts such as minimum age limits or required minimum amounts. 514

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 260 Example 5 Create a scenario that could ﬁt the following inequality: x < 48

Create a strategy

Apply the idea

Think about a scenario where something has to be under 48.

A local carnival has rides based off of heights. Some rides are designed for children under the height of 48 inches. So, you must be under 48 inches to ride the kids train ride at the carnival.

Idea summary

Purpose To graph an inequality, start by determining which direction the line will be shaded, right or left. This can be To ensure students can by write a context a given determined making sure thatfrom the shaded lineinequality. covers all the values that make the inequality true. The end point of the line will be an open circle if the inequality has a < or >.

Use capturing quantities to support creating contexts The end point of the line will be a closed circle if the inequality has a ≤ or ≥.

use with Example 5

Targeted instructional strategies The solution set of an inequality is made up of all values that make the inequality true. Solutions lie in the shaded region on a number line. When asking students to create a real-world scenario for the inequality x < 48, encourage students to identify possible quantities that x could represent, such as age, height, weight, or number of items, and discuss how these quantities relate to the inequality. Have students draw diagrams or visuals to represent their scenarios— Example 5 Practice for example, a height chart showing the maximum height for a children’s ride. Create a scenario that could ﬁt the following inequality: Facilitate a class discussion where students share their diagrams and explain how their scenarios illustrate What you remember? x < 48 x < 48. Reminddo students to pay attention to the direction of the inequality symbol, as they often confuse “less than” with1 “greater than.” By capturing and visualizing the quantities involved, students can better understand Match the inequality sign with the correct description. Create a strategy Apply the idea how inequalities i <represent real-world a constraints Less than and apply them accurately in context.

Think about has to be ii > a scenario where something b Less than or equal toA local carnival has rides based off of heights. Some rides under 48. are designed for children under the height of 48 inches. iii ≤ c Greater than So, you must be under 48 inches to ride the kids train ride Students: Page iv 260 ≥ d Greater than or equal to at the carnival. 2

When writing an inequality from a graph, which symbol(s) do we use for the following endpoints: a

Closed

Open

Ideathesummary Write inequality described by the following statements: aTo graph You must be at leaststart 60 inches tall to ridewhich a carnival ride. the Let line h represent the height. an inequality, by determining direction will be shaded, right or left. This can be by more making sure that theLet shaded line covers all the values that make the inequality true. bdetermined Gianna has than 6 books. b represent books. cTheThe at most 30 students. Letcircle s represent students.has a < or >. endclass pointhas of the line will be an open if the inequality

The end the point of the line willand be ainequality: closed circle if the inequality has a ≤ or ≥. Consider given equation The• solution set of an inequality is made up of all values that make the inequality true. Solutions lie in the x=5 shaded on a number line. • x ≤region 5 a

Select all values that make the equation x = 5 true: A 5

Practice b

−5

Select all values that make the inequality x ≤ 5 true: A 5

−5

What do you remember? c

260

Match the inequality sign with the correct description.

Mathspace Virginia SOL Grade 6 i < a mathspace.co

Less than or equal to

iii

≤

Greater than

≥

Greater than or equal to

When writing an inequality from a graph, which symbol(s) do we use for the following endpoints: Closed

Open

Write the inequality described by the following statements: b

6.06 Solutions to inequalities You must be at least 60 inches tall to ride a carnival ride. Let h represent the height. mathspace.co Gianna has more than 6 books. Let b represent books.

The class has at most 30 students. Let s represent students.

Less than

a 3

Describe the difference between the solutions of equations and inequalities.

Consider the given equation and inequality:

515

Practice Students: Pages 260–263

What do you remember? 1

Match the inequality sign with the correct description. i

Less than

Less than or equal to

iii

≤

Greater than

≥

Greater than or equal to

When writing an inequality from a graph, which symbol(s) do we use for the following endpoints: a

Closed

Open

Write the inequality described by the following statements: a

You must be at least 60 inches tall to ride a carnival ride. Let h represent the height.

Gianna has more than 6 books. Let b represent books.

The class has at most 30 students. Let s represent students.

Consider the given equation and inequality: • x=5 • x≤5 a

Select all values that make the equation x = 5 true: A 5

−5

Select all values that make the inequality x ≤ 5 true: A 5

−5

Describe the difference between the solutions of equations and inequalities.

Determine whether or not each of the following is a solution of x > 5: a

x=1

x=3

x=2

x = 10

x=5

x = 0.3

x=6

Determine whether or not each of the following is a solution of k ≤ 17: a

k = 16.9

k = 17

k = 16

k = 18

k = 25

k = 10

x = 5.2

Let’s practice 7

Consider the inequality x ≥ 4. a

Determine whether the following are solutions to the inequality: i

516

x=1

x=4

What is the smallest possible value of x?

Will the graph have a closed or open endpoint?

Now, graph the inequality on a number line.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

iii

Select the values that are part of the solution set of x ≤ −2. −5

x = −2

x = −1

−4

−3

x=3

−2

−1

x = −3

x ≤ −10

x < −10

x ≥ −10

−10 ≤ x

c e

−10 ≥ x

d 0 1 2 3 4 5 6 7 8 9 10

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

−5 −4 −3 −2 −1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Match the inequalities to the number lines. i

x>7

7>x

iii

x≥7

7≥x

0 1 2 3 4 5 6 7 8 9 10

Use the number line graph to justify whether each value is a solution of the inequality x > 5: 0

x=2

x = 10

x=5

x=6

Identify two inequality statements that represent the graph. 0

a < 11 11 < a 14

−10 < x

0 1 2 3 4 5 6 7 8 9 10

x = −4

Write an inequality for x that is represented on the following number lines: a

Which inequality represents the number line graph? Select all correct answers. −15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

a > 11 11 > a

a ≤ 11 11 ≤ a

−2

−1

a ≥ 11 11 ≥ a

Write an inequality that represents the graph in two different ways. −10

−9

−8

−7

−6

−5

−4

−3

6.06 Solutions to inequalities mathspace.co

517

Select the number lines for which x = −3 is a solution. A

−6 −4 −2 0

−10 −8

−6

−4

−2

−5−4−3−2 −1 0 1 2 3 4 5

Graph the following inequalities on a number line: a

x≥0

2.5 > x

x≥1

−3 ≤ x

x≤2

−5 ≥ x

x > −1

2>x

Consider the following statement: x is no more than 3 a

Write an inequality that represents the statement.

Is x = −4 in the solution set? Use a number line graph to justify your answer.

Cadence needs to make at least 4 baskets in her next basketball game to break her record. a

Write an inequality that represents the number of baskets she needs to score to break her record. Let b represent baskets made or points scored.

Graph the inequality on a number line.

A building must be more than 160 meters tall to be considered a skyscraper. Let h be the height of the building. a

Write an inequality representing this situation.

Would a building 152 meters tall be considered a skyscraper?

The temperature inside a freezer is always below 5 °F. Let T be the temperature inside the freezer. a

Write an inequality representing this situation.

Will the temperature inside the freezer ever be 2 °F?

Let’s extend our thinking 21

For each of the following, ﬁnd the largest whole number value q can have. a

q<2

q≤4

q < 27

q ≤ 1.56

q < 95.01

q < 49.5

q≤

For each of the following, ﬁnd the smallest whole number value n can have. a

n ≥ 88

n > 15

n > 27

n > 58.32

n ≥ 77.2

n ≥ 13.98

q≥

Olivia and Lucy graph the inequality that represents the condition “the largest value of x that does not satisfy the inequality is 5.5”. Their plots are shown: Olivia’s plot

Lucy’s plot

0 1 2 3 4 5 6 7 8 9 10

518

Identify the error in Olivia’s plot.

Identify the error in Lucy’s plot.

Now, graph the solution to the inequality.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

0 1 2 3 4 5 6 7 8 9 10

For a food to be labeled high protein, it must contain at least 20 grams of protein per serving. a

Write and graph an inequality to represent the amount of protein in a high-protein serving.

Write and graph an inequality to represent the amount of protein in a serving that does not qualify as high protein.

The Nutrition Facts label provided shows a serving of Chicken Nuggets with 14.3 g of protein. Does this food qualify as a high-protein food? Explain.

−10 −9

(1149kJ)

Chicken Nuggets (6 piece) % Daily Value1 Total Fat Sat. Fat Cholesterol

17.3 g 3g

27% 15%

42 mg

Sodium

597 mg

14% 25%

Total Carbs. Dietary Fiber Sugar

16.1 g 0g 0g

Protein

14.3 g

Calcium

11.9 mg

Potassium

0 mg

5% 0%

Select the number line that shows the solution set of the inequality 2x < 18. A

Nutrition Facts Calories 275

−8 7

−7 8

−6 9

−5 10

−10 −9

−8

−7

−6

−5

Xander is on a hiking trip and is allowed to pick at most 2 types of wildﬂowers. Each type of wildﬂower must have no less than 10 petals. a

Write and graph an inequality to represent the number of types of wildﬂowers Xander is allowed to pick.

Write and graph an inequality to represent the number of petals on each type of wildﬂower Xander is allowed to pick.

Is Xander allowed to pick a sunflower, coneflower, and blanket flower that each have more than 10 petals? Explain your answer.

Lila needs to save at least $75 for a school trip. She can save $15 per week. This can be represented by the inequality 15x ≥ 75. Is 4 weeks enough time for Lila to save enough money for the trip?

6.06 Solutions to inequalities mathspace.co

519

Answers

c 0 1 2 3 4 5 6 7 8 9 10

6.06 Solutions to inequalities

What do you remember?

1 a <

b ≤

2 a ≤ or ≥

b < or >

3 a h ≥ 60

b b>6

c >

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0

0 1 2 3 4 5 6 7 8 9 10

d ≥ f

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0

c s ≤ 10

g −5 −4 −3 −2 −1 0 1 2 3 4 5

4 a A

b A, B, C, D c E quations have a ﬁnite number of solutions, but inequalities have an inﬁnite number of solutions. 5 a Not a solution

b Not a solution

c Not a solution

d Not a solution

e A solution

g Not a solution

h A solution

6 a A solution

Not a solution

−2 −1

17 a x ≤ 3 b Yes because − 4 is not more than 3. When graphed, x = −4 lies in the shaded region. 0 1 2 3 4 5 6 7 8 9 10

18 a b ≥ 4

b A solution

c A solution

d Not a solution

e Not a solution

g A solution

h A solution

A solution

b 0 1 2 3 4 5 6 7 8 9 10

19 a h > 160

b No

20 a T < 5

b Yes

Let’s practice 7 a i No

ii Yes

iii No

iv Yes

Let’s extend our thinking 21 a 1

b x = 4

b 4

e 95

c Closed d

22 a 88

0 1 2 3 4 5 6 7 8 9 10

e 78

c 26

d 1

g 1

h 4

b 16

c 28

d 59

g 7

h 5

8 A, C, D

23 a Example answer:

9 A, D

In Olivia’s plot, a filled circle is used for x = 5.5, which means she considered 5.5 as a solution. But the condition is “the largest value of x that does not satisfies the inequality is 5.5”, which means that 5.5 is not a solution to the inequality. Olivia should have used a hollow endpoint instead of a filled endpoint.

10 a x < 4 e x < −8 11 a i

b x>0 f

x ≤ −1.5

b iv

c x≤6

d x > 5.5

g x ≥ −5

h x > −4

c ii

d iii

12 a N ot a solution because it does not lie in the shaded region of the number line b A solution because it lies in the shaded region of the number line c N ot a solution because the circle at x = 5 is open, indicating this value does not satisfy the inequality d A solution because it lies in the shaded region of the number line 13 a < 11, 11 > a 14 x > −6, −6 < x 15 A, D 16 a 0 1 2 3 4 5 6 7 8 9 10

b 0 1 2 3 4 5 6 7 8 9 10

520

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b Example answer: In Lucy’s plot, the endpoint starts at 5.5 using an unﬁlled circle then extends to the left. This means that Lucy considered the values less than 5.5 as solution. But the condition is “the largest value of x that does not satisﬁes the inequality is 5.5”, which means that 5.5 does is not a solution, as well as any number 5.5. Lucy shoud have plot the solution going to the right of 5.5 instead of the left. c 0 1 2 3 4 5 6 7 8 9 10

25 D

24 a x ≥ 20

26 a Let n is the number of types of wildﬂowers: n ≤ 2

2 x

3 2

5 10 15 202530354045

1 −5 −1

−4 −3 −2 −1 −1

−2 −3

−3

b Let p is the number of petals on a wildﬂower: p ≥ 10

3 2 1 −5 −1

2 3 4

−4

b x < 20 4

x 1

−2

−4

3 2

5 10 15 202530354045

1 −4−2 −1

−2 −3

x 2 4 6 8 10 12 14 16 18

−2

−4

c A ccording to the Nutrition Facts label provided for the serving of Chicken Nuggets, which contains 14.3 grams of protein, the food does not qualify as a high-protein food. To qualify, it must contain at least 20 grams of protein per serving, and 14.3 grams is less than this required amount.

−3 −4

c N o. Although the ﬂowers satisfy the inequality from part (b), the number of types of ﬂowers does not satisfy the inequality from part (a). 27 No

Answers mathspace.co

521

Topic 6 Assessment: Equations & Inequalities 1

Given the equation 3x − 2 = 4x + 1, identify the following: a

Variable

Expression

Term

Coeﬃcient

Write an equation for each of the following sentences: a

y is equal to six.

y is 2 more than x.

4 less than x is y.

y is 18 times k.

The cost of a cricket ball is 136 cents more than the cost of a rugby ball. A cricket ball costs 836 cents. Let x be the cost of a rugby ball in cents.

Write an equation to represent this.

Solve the equation to ﬁnd the cost of the rugby ball.

Write an equation to represent the following: a

There were 3 rows of trees in a ﬁeld, with x trees in each row. There were a total of 30 trees in the ﬁeld.

4 friends share the prize of x dollars. Each one received $5.

Wax and Shine washed x cars each day for 7 days. They washed a total of 35 cars.

Sandy has y candies and gave 9 of them to her friend. She is left with 6 candies.

Write the algebraic equation that represents each of the following sets of tiles: a

b +1

−1

c =

5=x−2

522

Use algebra tiles to represent the following equations: a

x=3

2x = 10

1+x=8

Consider the equation represented by the balance scale. Use it to explain whether x = −8 is a solution to the equation +x

−1

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Solve: a

x+4=9

17.8 = 20.6 − x

20 = 4x

2x =

Create a story that the following equation could represent: x−3=5

Write an inequality to represent each of the following situations: a

The width of a particular road is 5 meters or greater. Let w be the width of the road.

The elevator in a building should carry less than 13 people at one time. Let p be the number of people in the elevator.

Michael lives where it takes him at most 15 minutes to get to work. Let x be the time Michael takes to get to work.

A building must be more than 160 meters tall to be considered a skyscraper. Let h be the height of the building.

To get a grade of C, Luke must obtain a total score of at least 140 over two exams. In the ﬁrst exam he achieved a score of 80. Let x represent what he must score on the last exam to get a C or better. a

Write an inequality for x.

What is the minimum score he can get a grade of C in the class?

For each of the following inequalities: i

Find 3 values for the variable that satisy the inequality.

Graph the inequality on a number line.

iii

Create a context for each inequality.

x > 14

x ≤ −2

Consider the following inequalities represented on number lines. i

Using the symbols <, >, ≤, and ≥, state the inequality in 2 ways.

Explain whether 0 is in the solution set.

b 0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

SOL

d 0 1 2 3 4 5 6 7 8 9 10

Which word describes 3 in the number sentence shown? 3x + 5 = 23 A

Variable

Equation

Coeﬃcient

For each of the following equations, solve for x and state which property of real numbers can be used to justify your answer. a

SOL

Term

x + (−7) = 0

⋅x=6

What is the value of p for this equation?

3 B

48 C

−4

6.06 Solutions to inequalities mathspace.co

523

Performance Task 17

Tim grows a vegetable garden from seeds. But not all of his seeds start to grow. He has found that for every 50 seeds he sows, at most 40 start to grow. Martina is in charge of harvesting the plants. To determine how quickly she can harvest, she recorded her harvest throughout the week. The most plants she ever picked from the harvest was 120 plants a day.

524

Tim sows 150 seeds. How many would you expect to grow? Explain your reasoning.

Write an inequality for how many plants Martina can harvest vegetables from each day.

Tim and Martina want to optimize their farming so nothing goes to waste. Assume every plant will be ready for harvest exactly 75 days after sowing. How many seeds should Tim sow each day so Martina is able to pick all those that are mature each day?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

11 a x ≥ 60

Answers

b 60

6.PFA.4b, 6.PFA.4e

Topic 6 Assessment: Equations & Inequalities 1 a x

12 a i Answers vary. One set would be 15, 16, 17. ii

b 3x − 2 or 4x + 1

iii Marta has more than $14 in their bank account.

c Answers vary. 3x is a possible answer.

b i Answers vary. One set would be −2, −4, −6.

d Answers vary. 4 is a possible answer.

6.PFA.3a 2 a y=6

b y=x+2

c x−4=y

d y = 18k

6.PFA.3a, 6.PFA.3e 3 a 836 = x + 136

b x = 700

13 a i x < 4 and 4 > x c 7x = 35

d y−9=6

ii 0 is in the solution set since 0 < 4.

d x−1=1

ii 0 is not in the solution set since the endpoint is not ﬁlled in.

6.PFA.3e 5 a x=8

−5

6.PFA.4c, 6.PFA.4e

6.PFA.3c, 6.PFA.3e 4 a 3x = 30

−10

iii James needs to dive lower than 2 feet below sea level.

b 5 = 2x

c x+4=5

b i x > 0 and 0 < x

c i x ≤ 1.5 and 1.5 ≥ x

6.PFA.3b

ii 0 is in the solution set since 0 ≤ 1.5.

6 a Example answer: +x

d i x ≥ 5 and 5 ≤ x

ii 0 is not in the solution set since 0 ≥ 5.

b Example answer: +x =

6.PFA.4a, 6.PFA.4d +1

14 D 6.PFA.3a

c Example answer:

15 a x = 7 Inverse property of addition b x = 48 Inverse property of multiplication

−1

6.PFA.3c 16 x = 48

d Example answer: +1

B =

6.PFA.3b 7 No, x = −8 is not a solution to the equation. If x = −8, the left hand side of the scale would be −16 while the right hand side would be −8 which is not equal. 6.PFA.3d 8 a x=5

9 Answers may vary. Iliana cut off a length of rope. She then cut 3 ft of it to use for a project. She has 5 ft of rope remaining. 6.PFA.3f

6.PFA.4b

b p < 13

c x ≤ 15

17 a A nswers may vary to be more speciﬁc with reasoning. We can use ratios to create an inequality. Since 3 ⋅ 50 = 150, out of the 150 seeds, at most 3 ⋅ 40 = 120 seeds will grow. As an inequality, if we let s represent the seeds that grow, we know s ≤ 120.

d x=

6.PFA.3c

10 a w ≥ 5

Performace Task

b Let p represent the plants she can harvest from each day.

b x = 2.8

c x=5

6.PFA.3c

d h > 160

The inequality would be p ≤ 120.

c M artina can only harvest from 120 plants or less a day, so we do not want more plants than that to grow from a single day of sowing. Since 40 out of 50 seed will grow, let’s have Tim plane 3 ⋅ 50 = 150 seeds. The number of seeds that will grow into plants is at most 3 ⋅ 40, given by the inequality s ≤ 120. Tim should plant 150 seeds to ensure Martina can harvest them all each day. 6.PFA.4b, 6.PFA.4d, 6.PFA.4e, MP1, MP4

Answers mathspace.co

525

7 Polygons Big ideas • The relationships between the sides, angles, and diagonals of a polygon can be used to classify the polygon and solve problems. • Physical objects can be modeled with 2D and 3D geometric figures whose properties can be applied to solve real-world problems. • The properties of polygons can be applied to solve problems involving other polygons. • The position in space of a geometric figure can be represented in the coordinate plane. Using coordinate algebra, the properties of that figure can be uncovered and applied to solve problems.

Chapter outline 7.01 7.02 7.03 7.04 7.05 7.06

Congruent figures (6.MG.4) Regular polygons and symmetry (6.MG.4) Perimeter of triangles and parallelograms (6.MG.2) Area of parallelograms (6.MG.2) Area of triangles (6.MG.2) Polygons in the coordinate plane (6.MG.3)

530 552 567 583 596 608

Did you know? In a perfectly cut diamond, each facet is congruent to the others. This symmetry gives diamonds their dazzling sparkle!

7. Polygons Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 5.MG.2 — The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

5.MG.3 — The student will classify and measure angles and triangles, and solve problems, including those in context.

6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

Big ideas and essential understanding The relationships between the sides, angles, and diagonals of a polygon can be used to classify the polygon and solve problems. 7.01 — Congruent polygons have the same size and same shape. In other words, two polygons are congruent if their corresponding sides and angles are congruent. 7.02 — A regular polygon has congruent sides and congruent interior angles. The number of lines of symmetry of a regular polygon is equal to the number of sides of the polygon. Physical objects can be modeled with 2D and 3D geometric figures whose properties can be applied to solve real-world problems. 7.03 — Perimeter is a linear measurement to measure the distance around the outside of a 2D figure.

7.04 — A parallelogram has the same area as a rectangle with the same base and height. The properties of polygons can be applied to solve problems involving other polygons. 7.05 — A parallelogram can always be created by combining two congruent triangles. This allows us to find the area of a triangle by taking half of the area of a parallelogram with the same base and height. The position in space of a geometric figure can be represented in the coordinate plane. Using coordinate algebra, the properties of that figure can be uncovered and applied to solve problems. 7.06 — The coordinates of the vertices of a polygon in the coordinate plane can be used to find the side lengths of the polygon.

Standards 6.MG.2a — Develop the formula for determining 6.MG.2 — The student will reason mathematically to solve problems, including those in context, that involve the area of parallelograms and triangles using the area and perimeter of triangles, and parallelograms. pictorial representations and concrete manipulatives (e.g., two-dimensional diagrams, grid paper). 7.04 Area of parallelograms 7.05 Area of triangles

528

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.MG.2b — Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms. 7.03 Perimeter of triangles and parallelograms 7.04 Area of parallelograms 7.05 Area of triangles 7.06 Polygons in the coordinate plane 6.MG.3 — The student will describe the characteristics of the coordinate plane and graph ordered pairs. 6.MG.3e — Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers. 7.06 Polygons in the coordinate plane

6.MG.4 — The student will determine congruence of segments, angles, and polygons. 6.MG.4a — Identify regular polygons. 7.02 Regular polygons and symmetry 6.MG.4b — Draw lines of symmetry to divide regular polygons into two congruent parts. 7.02 Regular polygons and symmetry 6.MG.4c — Determine the congruence of segments, angles, and polygons given their properties. 7.01 Congruent figures 6.MG.4d — Determine whether polygons are congruent or noncongruent according to the measures of their sides and angles. 7.01 Congruent figures

6.MG.3f — Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. Ordered pairs will be limited to coordinates expressed as integers. Apply these techniques in the context of solving contextual and mathematical problems. 7.06 Polygons in the coordinate plane

Future connections

8.MG.3 — The student will apply translations and reflections to polygons in the coordinate plane.

G.TR.2 — The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

8.MG.5 — The student will solve area and perimeter problems involving composite plane figures, including those in context.

G.PC.2 — The student will verify relationships and solve problems involving the number of sides and angles of convex polygons.

7.MG.4 — The student will apply dilations of polygons in the coordinate plane.

G.RLT.3 — The student will solve problems, including contextual problems, involving symmetry and transformation.

7. Polygons mathspace.co

529

7.01 Congruent figures Subtopic overview Lesson narrative In this lesson, students will learn about congruent figures, focusing on congruent segments, angles, and polygons. They will explore properties of congruence, understanding that congruent figures have equal corresponding sides and angles. Students will engage with interactive applets: manipulating and measuring segments to identify congruent segments with hash marks, exploring intersecting lines to identify congruent angles with arc markings, and comparing polygons to verify congruence by matching sides and angles. By the end, students should confidently identify, represent, and verify congruent figures in various contexts.

Learning objectives

7.01 Congruent figures

Students: Page 266

After this lesson, you will be able to... • determine the congruence of segments, angles, and polygons given their properties. • write congruence statements for segments, angles, and polygons given their properties. • state whether polygons are congruent or noncongruent using the measures of their sides and angles.

Congruent segments

Key vocabulary

Segment  congruent polygons angle A segment starts at one point and stops at the other.  corresponding angles  corresponding sides 



polygon

B A



congruent segments



hatch (hash) marks

 vertex segmentThis segment can be named using its endpoints. We call this . The line over the letters means segment, so we say “segment AB”.

Essential understanding

We can also use the reverse order to name the same segment. or “segment BA”.

When talking abouthave the length of the segment or the distance between points and B weare cancongruent say AB without Congruent polygons the same size and same shape. In other words twoApolygons if theirthe line over top. corresponding sides and angles are congruent. It’s important that when discussing a segment, we use the line over the two letters to indicate segment. For example, or is a segment, while EF or FE is a distance or length.

Standards

Interactive exploration

This subtopicExplore addresses the following Virginia 2023 Mathematics Standards of Learning standards. online to answer the questions

Mathematical process goals mathspace.co MPG2 — Mathematical Communication Use can the interactive exploration in 7.01 to answer these to questions. Teachers achieve this goal by encouraging students articulate their understanding of congruence verbally and in writing. why polygons are then congruent non-congruent, correct 1. SetThis the could slidersinvolve for eachexplaining segment to be two the same number overlaporthem. What do youusing notice? mathematical vocabulary notation. Teachers also facilitate mathematical discussions where students share 2. Set the slidersand for each segment to be could different numbers then overlap them. What do you notice? and compare their strategies for identifying congruent figures. 3. Repeat both with new numbers. Do you notice the same things each time?

530

Mathspace Virginia SOL Grade 6 Teacher Edition Congruent segments mathspace.co

Line segments that have the same length. The symbol ≅ is used to represent congruence.

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can enhance students’ understanding of congruent figures by having them compare and contrast the figures or their components directly and justify their comparisons. Provide opportunities to overlay shapes and compare the measurements of corresponding sides and angles. Utilize manipulatives such as patty paper to help students place one figure over another to check for congruence. Encourage students to color-code corresponding sides and angles to visualize relationships. Create word wall cards and anchor charts with geometric markings for congruence. Facilitate activities that explore congruent polygons, reinforcing spatial relationships and geometric properties.

Teachers can implement this goal by guiding students to represent congruence in various ways. For instance, students could use geometric markings to denote congruent sides and angles in polygons. They could also use diagrams, drawings, physical models, or digital tools to represent congruent figures. Furthermore, teachers could show how to interpret these representations to determine congruence, thus enabling students to see representation as both a process and a product.

Content standards 6.MG.4 — The student will determine congruence of segments, angles, and polygons.

6.MG.4c — Determine the congruence of segments, angles, and polygons given their properties.

6.MG.4d — Determine whether polygons are congruent or noncongruent according to the measures of their sides and angles.

Prior connections 5.MG.3 — The student will classify and measure angles and triangles, and solve problems, including those in context.

Future connections G.TR.2 — The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

Lesson Preparation Tools You may find these tools helpful: • Calculator

• Ruler

• Protractor

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Create physical representations Targeted instructional strategies To help visualize congruent segments and angles, have students create several examples using a ruler and a protractor. Students can create lines that are the same length, discuss how they know they are congruent, and mark each segment with congruent marks. Students can use the protractor to create congruent angles, discuss how they know they are congruent, and add congruent marks.

7.01 Congruent figures mathspace.co

531

7.01 Congruent figures Student lesson & teacher guide After this lesson, you will be able to...

• determine the congruence of segments, angles, and polygons given their properties. • write congruence statements for segments, angles, and polygons given their properties. • state whether polygons are congruent or noncongruent using the measures of their sides and angles.

Congruent segments

Students are introduced to the definition and notation for a line segment, learning that it is a part of a line with two endpoints, named using notations like for the segment and AB for its length or distance.

Congruent segments

Students: Pages 266–667

7.01 Congruent figures 7.01 7.01 Congruent Congruent figures figures Segment

A segment starts at one point and stops at the other.

This segment can be named using its endpoints. We call this . The line over the letters means segment, so we say “segment AB”. A •After determine the congruence segments, angles, and polygons given their properties. this lesson, you will beofable to... also use the reverse given order their to name the same segment. this lesson, you will be able to... We canangles, •After write congruence statements segments, polygons properties. • determine the congruence of for segments, angles, andand polygons given their properties. or “segment BA”. determine the polygons congruence segments, andand polygons given their properties. •• write state whether areof congruent orangles, noncongruent using the measures of their sides and angles. congruence statements for segments, angles, polygons given their properties. •• write congruence statements for segments, angles,between and polygons properties. When talking about the length of theare segment or the distance points A andtheir B we can say ABand without the state whether polygons congruent or noncongruent using the given measures of their sides angles. • state whether polygons are congruent or noncongruent using the measures of their sides and angles. line over top. After this lesson, you will be able to...

It’s important that when discussing a segment, we use the line over the two letters to indicate segment. For example, Congruent segments or is a segment, while EF or FE is a distance or length.

Congruent segments Congruent segments Segment

Interactive exploration

Segment A segment starts at one point and stops at the other. Explore online to answer the questions x y Segment x y A segment starts at one point and stops at the other. A segment starts at one point other. mathspace.co B and stops at theThis segment can be named using its endpoints. We call this . The line over the means segment, so we say B This segment canletters be named using its endpoints. We“segment call this AB”. . A B in 7.01 to answer This segment canletters be named using its endpoints. We“segment call this AB”. . Use the interactive exploration these questions. The line over the means segment, so we say We can also use the reverse order to name the same segment. A The line over the letters means segment, so we say “segment AB”. A sliders for each segment to be the or “segment 1. Set the same number then overlap them. What do notice? We can also use BA”. the reverse order to name theyou same segment. We can also use BA”. the reverse order to name the same segment. or “segment When2.talking about the length of the segment or the distance between points A and B we can say AB without the Set the sliders for each segment to be different numbers then overlap them. What do you notice? or “segment BA”. line over top. When3.talking about thewith length the segment or the distance between points and B we can say AB without the Repeat both newofnumbers. Do you notice the same things eachAtime? When talking about the length of the segment or the distance between points A and B we can say AB without the line over top. that It’s important when discussing a segment, we use the line over the two letters to indicate segment. For example, line over top. or is athat segment, while EF or is a distance length. It’s important when discussing a FE segment, we useor the line over the two letters to indicate segment. For example, It’s important that when discussing a segment, we use the line over the two letters to indicate segment. For example, or is a segment, while EF or FE is a distance or length. Congruent segmentswhile EF or FE is a distance or length. or is a segment,

Interactive exploration Interactive exploration Interactive Explore online to exploration answer the questions

Line segments that have the same The symbol ≅ is used to represent congruence. Explore online to answer thelength. questions We place small markings onanswer segments when we want to show that they are equal in length. Explore online to the questions

mathspace.co 2 in C A The small identical markings on each segment, called ‘hash’ or mathspace.co ‘hatch’ tell us they are equal in length, or congruent. Use themathspace.co interactive exploration in 7.01 to answer thesemarks, questions. Wethese also see theoverlap same length, 2 in as . We can say Use interactive exploration in 7.01 to to answer 1. theSet the sliders for each segment be the samequestions. numberhas then them. What do you notice? Use the interactive exploration these questions. 2 in in 7.01 to answer AB same = ACnumbers when discussing length, butWhat we must say notice? segment 2. then overlap overlap them. What do you you notice? 1. Set the sliders for each segment to be different the number then them. do 1. Set the sliders for each segment to be the same number then overlap them. What do you notice? congruent to segment . 3. with youdifferent notice the same things each time? 2. Repeat Set the both sliders fornew eachnumbers. segmentDo to be numbers then overlap them. What do you notice? 2. Set the sliders for each segment to be different numbers then overlap them. What notice? We can write a congruency statement usingdo ouryou congruence B new numbers. Do you notice the same things each time? 3. Repeat both with 3. Repeat both with new numbers. Do yousymbol: notice the same things each time?

Mathspacesegments Virginia SOL Grade 6 266Congruent Lengths and distances are said to be equal while segments are congruent. mathspace.co

Congruent segments Line segments that have the same length. The symbol ≅ is used to represent congruence. segments ThisCongruent does not mean that the two segments are made up of the same points - only that they have the same length. Line segments that have the same length. The symbol ≅ is used to represent congruence. Sometimes we willthat use have morethe than one length. kind of The marking to show that to some segments are equal to others. Line segments same symbol ≅ is used represent congruence. In this diagram we use both single hatch markings and double hatch markings. The segments with single hatch markings are congruent to each other. The segments with double hatch markings are congruent to each other. B

1 in

The length of AB is equal to the length of BC, so to .

is congruent

The length of CD is equal to the length of DE, so to .

is congruent

2 in A E

2 in

Mathspace Virginia SOL Grade 6 mathspace.co 266 Mathspace Virginia SOL Grade 6 Mathspace 266 mathspace.co Example 1 Virginia SOL Grade 6 Mathspacemathspace.co Virginia SOL Grade 6 Teacher Edition 266

532

Use the diagram to answer the following: mathspace.co

W Z

B A

This segment can be named using its endpoints. We call this . The line over the letters means segment, so we say “segment AB”. We can also use the reverse order to name the same segment. or “segment BA”.

Exploration When talking about the length of the segment or the distance between points A and B we can say AB without the line over top.

Students:It’sPage 266 important that when discussing a segment, we use the line over the two letters to indicate segment. For example, or

is a segment, while EF or FE is a distance or length.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.01 to answer these questions. 1.

Set the sliders for each segment to be the same number then overlap them. What do you notice?

Set the sliders for each segment to be different numbers then overlap them. What do you notice?

Repeat both with new numbers. Do you notice the same things each time?

Congruent segments

SuggestedLine student grouping: segments that haveIn thepairs same length. The symbol ≅ is used to represent congruence. Students will use a GeoGebra applet to explore two line segments of varying length. The aim is for students to realize that two segments are congruent if and only if their lengths are equal. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Set the sliders for each segment to be the same number then overlap them. What do you notice? When the lengths of the segments are the same and they are overlapped, they match exactly. It looks like there’s only one line because they are the same length. 2. Set the for Virginia each SOL segment Mathspace Grade 6 to be different numbers then overlap them. What do you notice? 266 sliders mathspace.co If the lengths of the segments are different and they are overlapped, one segment is longer than the other. Part of the longer segment extends beyond the shorter one. 3. Repeat both with new numbers. Do you notice the same things each time? Yes, every time the segments have the same length and are overlapped, they match exactly. When their lengths are different, the longer segment always extends beyond the shorter one. Purposeful questions • When you set the sliders so that segments AB and CD have the same length and then overlap them completely, can you see two different lines or do they appear to be the same line? • When you repeat the experiment with new lengths for AB and CD, do you notice any patterns in your observations? Are the results consistent with what you noticed previously? Possible misunderstandings • Students may misunderstand the purpose of the sliders, thinking that they control something other than segment length and position. For example, they might think the sliders change the thickness or orientation of the segments. • Students might think that if the lengths of segments AB and CD are the same and overlapped, they are indistinguishable. They might not understand that they are still two distinct segments, even though they appear as one. Students are introduced to congruent segments, which are line segments with the same length, represented using the symbol ≅. They learn to identify congruence through markings, distinguish between equality of lengths and congruence of segments, and use appropriate notation such as .

7.01 Congruent figures mathspace.co

533

In this diagram we use both single hatch markings and double hatch markings. The segments with single hatch 2 in C A The small identical markings on each segment, called ‘hash’ or markings are congruent to each other. The segments with double hatch markings are congruent to each other. ‘hatch’ marks, tell us they are equal in length, or congruent. 1 in B The length is equal to the length2of congruent C We also seeof ABhas the same length, in BC, as so. We is can say to . 2 in is AB = AC when discussing length, but we must say segment 1 in 2 in congruent to segment .

Examples

2 in

Students: Page 267

The length of aCD is equal tostatement the lengthusing of DE, is congruent We can write congruency oursocongruence to . symbol:

Lengths and distances are said to be equal while segments are congruent.

Example 1 mean that the two segments are made up of the same points - only that they have the same length. This does not Sometimes we will use more than one kind of marking to show that some segments are equal to others. Use the diagram to answer the following: W In this diagram we use both single hatch markings and double hatch markings. The segments with single hatch markings are congruent to each other. The segments with double hatch markings are congruent to each other.

1 in

The length of AB is equal to the length of BC, so to .

is congruent

The length of CD is equal to the length of DE, so A Choose the segment with the same marking as to . .

is congruent

a What segment is congruent to 1 in

2 in

Create a strategy E

2 in

Apply the idea Segment

Example 1

Reﬂect and check

has the same marking as

is congruent to

This shows us that

Use the diagram to answer the following:

Since segments are named using their endpoints, the order of letters does not matter. We could have also said segment is congruent to . W

b Write a congruence statement

Purposea What segment is congruent to ? Create a strategy Check for understanding of congruent segments.

Use the congruence symbol ≅ to write a congruence statement for the two segments.

Create a strategy

Reflecting with students Choosethe theidea segment with the same marking as . Apply Reﬂect and check Segment XZ and segment XY do not have markings. Since both are blank, ask students why they are not The congruence statement is also true. congruent. Make students aware that we cannot assumeReﬂect anything Apply the idea andabout checkthe lengths of these segments since we haveSegment no additional about has information the same marking as them. . Since segments are named using their endpoints, the This shows us that

Students: Page 267

is congruent to

order of letters does not matter. We could have also said 7.01 segment is congruent to . Congruent figures 267 mathspace.co

b Write a congruence statement

Create a strategy Use the congruence symbol ≅ to write a congruence statement for the two segments.

Apply the idea

Reﬂect and check The congruence statement

is also true.

7.01 Congruent figures mathspace.co

Purpose Show students how to write a congruence statement using the congruence symbol.

534

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

267

Students: Page 268 Example 2 A

Identify any congruent segments in the ﬁgure and write a congruence statement for each.

4 ft

3 ft 3 ft

4 ft

Create a strategy

Apply the idea

Remember, congruent segments are equal in length.

Since the length of can say .

is equal to the length of

, we

Since the length of can say .

is equal to the length of

, we

Idea summary

Purpose 1 in B • Congruent segments are equal in length. Show students how to identify Ccongruent segments a geometric figure and the write congruence statements. • Wein place small markings to show segments that are equal in length.

1 in

2 in Expected mistakes • We use our congruency symbol ≅ to write congruency statements. Students might not include the line above the segments when and writing the congruency statements. Remind A students that we use “congruent” when referring to two segments that have the same length, and we draw a D 2 in E show line above the letters to that they are segments.

Print out and angles color-code the diagram Congruent

use with Example 2

Support students with disabilities

To supportAngle students with visual-spatial processing difficulties, provide them with a copy of the diagram and Example 2isuse An angleto formed when two rays, or segments, areof joined at their endpoints. encourage them color-coding to lines, identify segments equal length. Have students assign a specific Angles are in degrees. A C This visual color to each distinctmeasured length—for example, color each 3 ft segment red and each 4 ft segment 4 ft blue. Identify any congruent segments in the ﬁgure and write a congruence statement 3 ft distinction helps students easily see which segments are congruent. for each. 3 ft of turn4or Whenever two lines, rays, or segments pass through the same point, we can describe the amount ft rotation it

D Additionally, offer suchusing as pieces takes to get manipulatives from one to the other an angle.of string or strips of paper cut to match the lengths of the B segments in the diagram. Students can physically compare these pieces to find segments Here are two rays drawn from the samecongruent point forming an angle. by aligning B them side by side. This hands-on approach reinforces the concept that congruent segments equal The vertex of an angle is the angle formed by two linesare or rays thatin length Create a strategy Apply the idea and aids in their understanding of congruence statements. intersect at a point.

Remember, congruent segments are equal in length. Since the length of is equal to the length of , we In this image, canour sayvertex angle . is ∠A. We use the letter at the vertex 85° to name the angle. is equal to the length of , we Since the length of A C Students: Page 268 vertex can say . Y 40° Idea summary B

1 in

2 in 65°

75° A 2 in

E 268

Let’s name the three angles of the triangle: • ∠X • ∠Y • ∠Z • Congruent segments are equal in length. identify the measure of each • Now, We let’s place small markings to show the angle: segments that are equal in • length. m∠X = 75° m∠Y 40°congruency symbol ≅ to write congruency statements. • • We use=our • m∠Z = 65° and

Z D

Mathspace Virginia SOL Grade 6 mathspace.co

Congruent angles Angle An angle is formed when two rays, lines, or segments, are joined at their endpoints. Angles are measured in degrees. Whenever two lines, rays, or segments pass through the same point, we can describe the amount of7.01 turnCongruent or rotation figures it mathspace.co takes to get from one to the other using an angle. B

Here are two rays drawn from the same point forming an angle. The vertex of an angle is the angle formed by two lines or rays that

535

Since the length of can say .

is equal to the length of

, we

Idea summary

Congruent angles 1 in B C

• •

Congruent segments are equal in length. We place small markings to show the segments that are equal in This section discusses the concept of angles, which length. are formed by two rays, lines, or segments joined at their 1 in 2 inStudents learn angle notation, naming angles using their vertex (e.g., ∠A), and endpoints and measured in degrees. • We use our congruency symbol ≅ to write congruency statements. and identifying angle measures within geometric figures. A D

2 in E Students: Pages 268–269

Congruent angles Angle An angle is formed when two rays, lines, or segments, are joined at their endpoints. Angles are measured in degrees. Whenever two lines, rays, or segments pass through the same point, we can describe the amount of turn or rotation it takes to get from one to the other using an angle. Here are two rays drawn from the same point forming an angle.

The vertex of an angle is the angle formed by two lines or rays that intersect at a point.

85° A C Interactive exploration

In this image, our vertex angle is ∠A. We use the letter at the vertex to name the angle.

vertex

Explore online to answer the questions

Let’s name the three angles of the triangle: • ∠X • ∠Y 40° Use the interactive exploration in 7.01 to answer• these ∠Z questions. Y

mathspace.co

1. 2. 3.

identify the measure of each angle: Set the sliders for each angle to be theNow, samelet’s degree measure then overlap them. What do you notice? • m∠X = 75° Set the sliders for each angle to be different degree measures then overlap them. What do you notice? • m∠Y = 40° Repeat both with new numbers. Do you notice the same things each time? • m∠Z = 65° 65° 75° X

Similar to segments, we must be careful with our notation when discussing angles. Congruent angles have the same 268 angle Mathspace measure. Virginia SOL Grade 6 mathspace.co

40°

Since the measure of ∠B is equal to the measure of ∠Q, we know ∠B is congruent to ∠Q. Congruence Statement:

∠B ≅ ∠Q Since ∠B and ∠Q have the same angle marking, this also tells us they are equal in measure and therefore congruent.

Just like with segments, we can use additional markings to show that two angles are congruent. We draw multiple arcs to show that different angles are congruent to each other. B F 1 A

3 C

In this diagram the two angles drawn with double arcs have equal measure. Since the measure of the angles is equal, we know the angles are congruent. Congruence Statement:

∠1 ≅ ∠4 D

There are no angles that are congruent to ∠2 and ∠3 because there are no other angles with the same number of markings.

Example 3 Use the image to answer the questions.

A B E

536

a Which anglesSOL are congruent? Mathspace Virginia Grade 6 Teacher Edition mathspace.co

Create a strategy Look for angles with identical angle markings showing the angle measures are equal.

Exploration Students: Page 269

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.01 to answer these questions. 1.

Set the sliders for each angle to be the same degree measure then overlap them. What do you notice?

Set the sliders for each angle to be different degree measures then overlap them. What do you notice?

Repeat both with new numbers. Do you notice the same things each time?

Similar to segments, we must be careful with our notation when discussing angles. Congruent angles have the same angle measure.

Suggested student grouping: Small groups Since the measure of ∠B is equal to the measure of ∠Q, we know In this exploration, students will use a GeoGebra∠B applet to manipulate the sizes of two angles and observe the is congruent to ∠Q. 40° 40° results when they overlap. The aim is for students to realizeStatement: that two angles are congruent if and only if their Congruence B Q measures are equal. ∠B ≅ ∠Q Ideal student responses

Since ∠B and ∠Q have the same angle marking, this also tells us they are equal in measure and therefore congruent.

These ideal responses may differ from other correct student responses. Less formal responses can be Just like with segments, we can use additional markings to show that two angles are congruent. We draw multiple connected with the more precise mathematical language presented here. arcs to show that different angles are congruent to each other.

1. Set the sliders for each angle to be the same degree measure then overlap them. What do you notice? B In this diagram the two angles drawn with double arcs have equal When both angles are set and overlapped, they of perfectly align. F to the same degree measure. Since the measure the angles is equal, we know the angles are congruent. E different 2. Set the sliders for each angle to be degree measures then overlap them. What do you notice? 2 3 1 Congruence Statement: When the angles are set to different degree measures and overlapped, one angle appears to be bigger than 4 A C ∠1 ≅ ∠4 the other. We can see both angles. Therethe aresame no angles that are congruent 3. Repeat both with new numbers. Do things each time? to ∠2 and ∠3 because D you notice there are no other angles with the same number of markings. Yes, the same things happen each time. If the angles are the same size, they align perfectly when overlapped. If they’re different sizes, we can see both when they overlap. Example 3 Purposeful questions Use the image to answer the questions.

• If A and B have different degree measures, how does the difference in their measures affect the overlap and B the appearance of the angles? E • When you repeat the experiment with new measures for angles A and B, do you notice any patterns in your observations? Are the results consistent with what you noticed previously? Which angles are congruent? Possiblea misunderstandings

• Students might think that if angles A and B are set to the same measure and overlapped, they are Create a strategy indistinguishable as separate angles. They might not understand that they are still two distinct angles even Look for angles with identical angle markings showing the angle measures are equal. though they appear as one. Apply the idea The two angles marked as having equal measure are ∠C and ∠A. So, we know ∠C is congruent to ∠A.

7.01 Congruent figures mathspace.co

269

7.01 Congruent figures mathspace.co

537

Compare and connect Interactive exploration English language learner support

Explore online to answer the questions

Provide students with two diagrams: one showing congruent angles marked with arc markings on equal angles, and another withmathspace.co the same figures labeled with actual angle measures. For example, display a pair of angles in the first diagram with congruent markings, and in the second diagram, show the same angles with angles Use the interactive exploration in 7.01 to answer these questions. labeled in degrees. Encourage students to compare the diagrams and discuss how the markings relate to the 1. Set the sliders for each angle to be the same degree measure then overlap them. What do you notice? measurements. 2.

Set the sliders for each angle to be different degree measures then overlap them. What do you notice?

Ask guiding 3. questions like, “What similarities and differences do you notice between the two diagrams?” and Repeat both with new numbers. Do you notice the same things each time? “How do the markings help us identify congruent parts without measurements?” This will help students connect the visual symbols to the numerical values and understand that the markings represent equal lengths and angles. Similar to segments, we must be careful with our notation when discussing angles. Congruent angles have the same angle measure.

Have students explain how they can determine ifSince the the angles are congruent using both approaches. Use measure of ∠B is equal to the measure of ∠Q, we know sentence stems such as, “The arc markings indicate these angles are congruent, which we can see by their ∠B is congruent to ∠Q. 40°or “Since 40° equal measures Bof ⬚,” these angles have the same arc markings, they are congruent, and their Congruence Statement: Q measures are both ⬚ degrees.” By articulating these observations, students practice using key vocabulary like ∠B ≅ ∠Q “congruent,” “measure,” and “corresponding angles.” This activity supports their language development and Since ∠B and ∠Q have the same angle marking, this also tells us deepens their understanding of congruence by they making explicit connections between different representations. are equal in measure and therefore congruent. Just like with segments, we can use additional markings to show that two angles are congruent. We draw multiple arcs to show congruent that different angles, angles are congruent eachsame other. measure, and are introduced to proper notation for Students learn about which havetothe B In this diagram the two angles drawn with are double arcsto have equal indicate indicating angle congruence, such as ∠B ≅ ∠Q. They also explore how arc markings used visually F measure. Since the measure of the angles is equal, we know the congruence between angles in geometric diagrams.

Examples A

3 C

Congruence Statement:

Students: Page 269

angles are congruent.

∠1 ≅ ∠4 D

There are no angles that are congruent to ∠2 and ∠3 because there are no other angles with the same number of markings.

Example 3 Use the image to answer the questions.

A B E

a Which angles are congruent?

Create a strategy Look for angles with identical angle markings showing the angle measures are equal.

Apply the idea The two angles marked as having equal measure are ∠C and ∠A. So, we know ∠C is congruent to ∠A.

Purpose 7.01 Congruent figures 269 mathspace.co Show students how to identify congruent segments in a geometric figure and to write congruence statements.

538

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 270 b Write a congruence statement

Create a strategy Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a).

Apply the idea

Reﬂect and check

∠C ≅ ∠A

The congruence statement ∠A ≅ ∠C is also true.

PurposeExample 4 Show students how to identify statements. Use the image to answer thecongruent questions. segments in a geometric figure and to write congruence S b Write a congruence statement

Reflecting with students 42° Create a strategy G 42° Encourage students to use precise mathematical notation when writing congruency statements. A precise Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a). response includes the angle symbol in front of the letter representing the vertex, and an imprecise response may exclude Applythe the angle idea symbol, such as C ≅ A. Reﬂect and check 34°

∠CWhich ≅ ∠A angles are congruent? a

The congruence statement ∠A ≅ ∠C is also true.

Create a strategy Create any a strategy Identify angles that have the same degree measure.

Apply the idea

Reﬂect and check

∠C ≅ ∠A b Write a congruence statement

The congruence statement ∠A ≅ ∠C is also true.42°

Students:b Page 270 Write a congruence statement

The two angles with an equal measure are ∠G and ∠S Example 4 sinceforthey measure 42°. So,aswe know thatin∠G Use the congruence symbol ≅ to write a congruence statement theboth angles we identiﬁed congruent partis(a). congruent to ∠S. Use the image to answer the questions. S G

42°

Create a strategy Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in 34° part (a). Example 4 W

a Which angles are congruent?

Apply idea Use thethe image to answer the questions.

Reﬂect and check

∠G ≅ ∠Sa strategy Create

The congruence Apply the idea statement ∠S ≅ ∠G is also true.

Identify any angles that have the same degree measure.

G ∠G and ∠S The two angles with an equal measure are 42° since they both measure 42°. So, we know that ∠G is congruent to ∠S.

Idea summary

S 42°

a are congruent? b Which Write aangles congruence statement

34°

40°

B PurposeCreate a strategy Apply the idea Q Create a strategy Show students how to identify segments in a figure formulate accurate congruence Identify any angles that havecongruent the same degree measure. The twoand angles with an equal measure are ∠G and statements. ∠S

Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a). • Congruent angles have the same measure. since they both measure 42°. So, we know that ∠G is • 270 Congruent angles are marked using an arc or multiple arcs to congruent to show ∠S. congruence. Students:Apply Page the idea Reﬂect and check • An angle is named using ∠ symbol followed by letter of vertex angle. We use our congruence symbol, ≅ to ∠G ≅ ∠S write a congruence statement. ∠B ≅ ∠Q means angle The congruence statement ∠S B is congruent to angle Q.≅ ∠G is also true. b Write a congruence statement

Create a strategy

Idea summary

Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a). 270 Mathspace Apply the idea Virginia SOL Grade 6 mathspace.co

∠G ≅ ∠S •

40°

Reﬂect 40° and check The Q congruence statement ∠S ≅ ∠G is also true.

Congruent angles have the same measure.

• Congruent angles are marked using an arc or multiple arcs to show congruence. Idea summary •

An angle is named using ∠ symbol followed by letter of vertex angle. We use our congruence symbol, ≅ to write a congruence statement. ∠B ≅ ∠Q means angle B is congruent to angle Q. B

270

• •

40°

40° Q

Congruent angles have the same measure. Congruent angles are marked using an arc or multiple arcs to show congruence. Virginia SOL Grade 6

Mathspace

7.01 Congruent figures mathspace.co

539

Create a strategy

Apply the idea

Identify any angles that have the same degree measure.

The two angles with an equal measure are ∠G and ∠S since they both measure 42°. So, we know that ∠G is congruent to ∠S.

Purpose Show students how to identify congruent segments in a figure and formulate accurate congruence statements. b Write a congruence statement Reflecting with students Create a strategy Have a discussion with students about the valid ways to show angles are congruent. Angles can be shown to Use the congruence symbol ≅ to write a congruence statement for the angles we identiﬁed as congruent in part (a). be congruent with either identical angle markings or the same degree measures. Apply the idea

Reﬂect and check

Students:∠G Page 270 ≅ ∠S

The congruence statement ∠S ≅ ∠G is also true.

Idea summary

• • •

40°

40° Q

Congruent angles have the same measure. Congruent angles are marked using an arc or multiple arcs to show congruence. An angle is named using ∠ symbol followed by letter of vertex angle. We use our congruence symbol, ≅ to write a congruence statement. ∠B ≅ ∠Q means angle B is congruent to angle Q.

Congruent polygons Polygon

A polygon in is a closed plane ﬁgure composed of at least three line segments that do not cross. Congruence polygons 270

Mathspace

Virginia SOL Grade 6

Students review thearedefinition of a polygon how on to its name polygons using the vertices before engaging in an Whenmathspace.co we naming a polygon, we useand the labels vertices. exploration about congruent polygons. B

Students: Pages 271–272

Congruent polygons A

Polygon

Rectangle EFGH △ABC A polygon is a closed plane ﬁgure composed of at least three line segments that do not cross.

Interactive exploration When we Explore are naming a polygon, wethe use the labels on its vertices. online to answer questions

mathspace.co B

E C

Use the interactive exploration in 7.01 to answer these questions. 1.

Overlap several different pairs of triangles. What do you notice about the triangles that overlap perfectly?

A What do you notice about the triangles that do not overlap perfectly?

△ABC

Rectangle EFGH

Interactive exploration

Congruent polygons

Explore online to answer the questions

Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent.

mathspace.co

Corresponding anglesexploration are a pair of angles that questions. are in the same spot in two different shapes. Use the interactive in matching 7.01 to answer these Corresponding sides are a pair of matching sides that are in the same spot in two different shapes. 1. Overlap several different pairs of triangles. What do you notice about the triangles that overlap perfectly? 3 in the wenot have an image with two triangles. A you notice M L trianglesHere 2. What do about that do overlap perfectly? 3 in

60°

2 in

50° 70° C B Congruent polygons 4 in

540

2 in

60° 50° 70° 4 in

Since the corresponding angles and corresponding sides are equal, we know these two triangles are congruent.

Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are Here we have an image of two polygons. congruent. 2 m 10 m 4m 5m The corresponding angles are congruent, however, the Mathspace Virginia SOL Grade 6 Teacher Edition corresponding sides are not congruent. Corresponding angles are a pair of matching angles that are in the same spot in two different shapes. 3 m 1m 6m mathspace.co 2m This thesespot two polygons are not congruent. Corresponding sides are a pair of matching sides thatmeans are in that the same in two different shapes. If two polygons are congruent, 3we with awe congruence statement. When writing congruence statements in canLshow that Here have an image with two triangles. A M for polygons, the letters must be in the correct order according to the corresponding angles and sides.

A 3 in

60° 50° 70°

Since the corresponding angles and corresponding sides are equal, we know these two triangles are congruent.

10 m

Here we have an image with two triangles.

60° 50° 2 in 2 in 70° 4 in

4 in

3 in

Here we have an image of two polygons.

The corresponding angles are congruent, however, the corresponding sides are not congruent.

This means that these two polygons are not congruent.

If two polygons are congruent, we can show that with a congruence statement. When writing congruence statements for polygons, the letters must be in the correct order according to the corresponding angles and sides. Here is an example of two congruent triangles. Let’s identify our corresponding angles and sides. P 8 G

5 K 10

For the angles, the markings on image tell us: • ∠G ≅ ∠R • ∠S ≅ ∠K • ∠P ≅ ∠F

For the sides, we see from the labeled lengths 7.01 that: Congruent figures

271

mathspace.co

• • •

Congruent polygons This shows us that vertex G corresponds with vertex R, S with K and P with F so we can write the congruence statement: Polygon

△GSP ≅ △RKF A polygon is a closed plane ﬁgure composed of at least three line segments that do not cross. The statements △SPG ≅ △KFR and △GPS ≅ △RFK are also true because they match the corresponding angles. There are several more true congruence statements we could write as long as we make sure the order matches up When we are naming a polygon, we use the labels on its vertices. the congruent angles. B

Example 5

Exploration Determine if the two polygons are congruent. Students: Page 271

△ABC

E G

Rectangle EFGH F

M G

Interactive exploration online to answer the questions Create a Explore strategy

Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent.mathspace.co We see these triangles have an equal number of sides. So, we need to check all corresponding sides and angles to determine if they are congruent. Use the interactive exploration in 7.01 to answer these questions.

Apply idea several different pairs of triangles. What do you notice about the triangles that overlap perfectly? 1. theOverlap Let’s 2. ﬁrst start corresponding angles. Whatbydoidentifying you noticethe about the triangles that do not overlap perfectly? One arc marking is used with ∠E and ∠L, which tells us ∠E ≅ ∠L. A double arc marking is used for ∠F and ∠M, showing that ∠F ≅ ∠M. A triple arc marking is used for ∠G and ∠N, telling us that ∠G ≅ ∠N. All corresponding angles are congruent. Now let’s check the corresponding sides. Congruent polygons . Both side EG and side LN are Side EF is marked with one hatch mark and so is side LM. This tells us Suggested studentare grouping: In pairs Polygons if they haveus an equal number of hatch sides,marks and allare theused corresponding and angles are marked with two congruent hatch marks, showing . Three for side FGsides and side MN, showing In this exploration, students will use a GeoGebra applet to explore the side lengths and angle measures of two congruent. us .

triangles. Thisallactivity is designed toangles help students discover thepolygons properties of congruent triangles. Since corresponding sides and are congruent, these two are congruent. Corresponding angles are a pair of matching angles that are in the same spot in two different shapes.

Ideal student responses Corresponding sides are a pair of matching sides that are in the same spot in two different shapes. Reﬂect and check

These ideal responses may differ from other correct student responses. Less formal responses can be We can write a congruency 3 in Here we have an image with two triangles. A M statement: L connected with the more precise mathematical language presented here. △EFG ≅ △LMN 60° 50°

Since the corresponding angles and corresponding sides are equal,

2 in

3 in

in 70° of4 triangles. we knowdo these triangles are congruent. 1. Overlap several 60° different2 in pairs What youtwo notice about the triangles that overlap 50° 70° perfectly? B C 4 in N When two triangles overlap perfectly, they are congruent, meaning they have the same sides and angles. Here we have an image of two polygons. The side lengths and angle measures are also equal.

1m 272

Mathspace

10 m

Virginia SOL Grade 6

The corresponding angles are congruent, however, the corresponding sides are not congruent. This means that these two polygons are not congruent.

If two mathspace.co polygons are congruent, we can show that with a congruence statement. When writing congruence statements for polygons, the letters must be in the correct order according to the corresponding angles and sides. 7.01 Congruent figures Here is an example of two congruent triangles. Let’s identify our corresponding angles and sides.

mathspace.co

541

2. What do you notice about the triangles that do not overlap perfectly? The triangles that do not overlap perfectly are not congruent. The side lengths and angle measures do not match exactly. Purposeful questions • What can we say about the side lengths and angles of two triangles if they overlap perfectly? • What happens to the side lengths and angles when two triangles do not overlap perfectly? Possible misunderstandings R are congruent For the angles, thejust markings on imageThey tell us:need to understand that for • Students may think Pthat two triangles if they look similar. 10 • ∠G ≅ ∠R triangles to be 8congruent, 5 K they must have exactly the same three sides and three angles. G

5 F

• ∠S ≅ ∠K • ∠P ≅ ∠F

For the sides,allwecorresponding see from the labeled lengths Students explore the concept of congruent polygons, where sides and that: angles must be congruent. • how to identify them in polygons. Using examples, students They learn about corresponding sides and angles and distinguish between congruent and non-congruent •polygons based on their corresponding measurements. • Additionally, they practice writing congruence statements for polygons, emphasizing the correct order of vertices to This shows usparts. that vertex G corresponds with vertex R, S with K and P with F so we can write the congruence match corresponding statement:

△GSP ≅ △RKF

Examples The statements △SPG ≅ △KFR and △GPS ≅ △RFK are also true because they match the corresponding angles. are several more true congruence statements we could write as long as we make sure the order matches up Students:There Page 272 the congruent angles.

Example 5 Determine if the two polygons are congruent.

N M

Create a strategy Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. We see these triangles have an equal number of sides. So, we need to check all corresponding sides and angles to determine if they are congruent.

Apply the idea Let’s ﬁrst start by identifying the corresponding angles. One arc marking is used with ∠E and ∠L, which tells us ∠E ≅ ∠L. A double arc marking is used for ∠F and ∠M, showing that ∠F ≅ ∠M. A triple arc marking is used for ∠G and ∠N, telling us that ∠G ≅ ∠N. All corresponding angles are congruent. Now let’s check the corresponding sides. . Both side EG and side LN are Side EF is marked with one hatch mark and so is side LM. This tells us marked with two hatch marks, showing us . Three hatch marks are used for side FG and side MN, showing us . Since all corresponding sides and angles are congruent, these two polygons are congruent.

Reﬂect and check We can write a congruency statement: △EFG ≅ △LMN

Purpose Show students that to determine if two polygons are congruent, they need to verify that all corresponding sides and angles are congruent by observing the angle and side markings. 272

542

Mathspace

Virginia SOL Grade 6

Mathspacemathspace.co Virginia SOL Grade 6 Teacher Edition mathspace.co

Expected mistakes Students may incorrectly identify corresponding sides or angles since the triangles are oriented in different ways. Encourage them to use the geometric markings to determine corresponding sides and angles, rather than relying on visual estimates as diagrams are not always drawn to scale. Reflecting with students Challenge students to write a congruence statement for the triangles. This can lead to a discussion about how the corresponding angles should be listed in the same order when writing congruence statements for polygons.

Concrete-Representational-Abstract (CRA) approach

use with Example 5

Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives of triangles. Provide each student with two sets of triangle cut-outs labeled EFG and LMN. Make sure these triangles have sides and angles that match the ones in the problem. Have students physically overlay one triangle onto the other to see if they can match all corresponding sides and angles. Encourage them to use rulers to measure side lengths and protractors to measure angles. This hands-on activity allows students to explore congruence by directly comparing the physical shapes. Representational: Discuss with students how the physical triangles they manipulated can be represented on paper. Explain that the measurements they took can be used to draw accurate diagrams. Emphasize that the markings they observed, like side lengths and angle measures, can be shown in drawings using specific symbols. Guide students to draw the two triangles EFG and LMN on graph paper. Have them use the measurements from the concrete activity to ensure accuracy. Instruct them to label corresponding sides and angles, using hatch marks for congruent sides and arc marks for congruent angles. For example, they can draw one arc on angles E and L to show they are congruent, two arcs on angles F and M, and three arcs on angles G and N. Encourage students to visually compare the diagrams to confirm congruence. This step helps students translate their concrete experience into a visual representation. Abstract: Explain to students how the symbols and markings in their drawings represent mathematical ideas. Point out that the hatch marks and arc marks are visual ways to show congruent sides and angles. Discuss how these markings can be expressed using mathematical notation and symbols. Introduce students to the formal notation for congruence. Show them how to write congruency statements like △EFG ≅ △LMN. Teach them to write that side , side , and side . Similarly, they can note that angle E is congruent to angle L, angle F is congruent to angle M, and angle G is congruent to angle N. Encourage them to use symbols like ≅ to represent congruence. This abstract notation solidifies their understanding in mathematical language.

Students: Page 273

Example 6 Determine if the two polygons are congruent. J

44 mi

26 mi

O 26 mi

44 mi

G 26 mi H

54 mi 54 mi

26 mi Q

Create a strategy Polygons are congruent if they have an equal number of sides, and all corresponding sides and angles are congruent. These polygons both have 4 sides, so we need to check all corresponding sides and angles to determine if they are congruent. 7.01 Congruent figures

Apply the idea Since all of the angles are marked as right (90°) angles, we know all corresponding angles are congruent.

mathspace.co

543

44 mi

54 mi 54 mi

G 26 mi H

26 mi Q

Apply the idea Since all of the angles are marked as right (90°) angles, we know all corresponding angles are congruent. Let’s check if the corresponding sides are congruent. Side JI and side OP, are both 26 mi long, so

Side JG in the first figure has a length of 44 miles and has no corresponding side of equal length in the second figure. These two polygons are not congruent.

Example 6

Reﬂect and check

Determine if the two polygons are congruent. O 26 mi Notice that we did not need to check all pairs of corresponding sides. As soon as one side that did not P I 26 we mi found J have a corresponding side of equal length we were able to say the ﬁgures are not congruent. 44 mi

44 mi

54 mi 54 mi

Idea summary

Purpose R G 26 mi H 3 in M L Q and angles. Show students how to determine if two polygonsA are congruent by comparing the measure 26 of mi sides 3 in

60°

2 in

60° 50°

2 in

70°

4 in

Advanced learners: Construct and their own congruent polygons use with Example 6 50° justify 70° Create a strategy C B Targeted instructional strategies 4 in of sides, Polygons are congruent if they have an equal number andNall corresponding sides and angles are

congruent. These polygons both have 4 sides, so we need to check all corresponding sidesby and angles them to determine Encourage your advanced students to deepen their understanding of congruence having create their if they are congruent. Congruent polygons: Polygons are congruent if they have an equal number of sides, and all the own pairs of congruent polygons using tools like rulers and protractors. Allow them to choose their own shapes corresponding sides and angles are congruent. and dimensions promote creativity. After constructing the polygons, ask students to justify why the two Apply When thetoidea writing a congruency statement, we must make sure we are putting the letters in the correct order figures are congruent by explaining how alland corresponding sides and angles are equal. according to corresponding angles sides. Since all of the angles are marked as right (90°) angles, we know all corresponding angles are congruent. . Let’sstudents check if thetocorresponding sides are congruent. Side JI and side OP,using are both 26 mipaper long, so Encourage include detailed, labeled diagrams—possibly graph or geometric drawing Side JG in the first figure has aaccurately. length of 44 miles and hasalso no corresponding side of equal length in the secondand figure. software—to illustrate their work You might have them present their constructions Theseto twothe polygons not congruent. justifications class, are fostering mathematical communication and peer learning.

Reﬂect and check that273 we did not need to check all pairs of corresponding sides. As soon as we found one side that did not Students:Notice Page have a corresponding side of equal length we were able to say the ﬁgures are not congruent.

Idea summary

7.01 Congruent figures mathspace.co

A 3 in

60°

2 in

3 in

273

60° 50° 2 in

70°

4 in

50° 70° B

4 in

Congruent polygons: Polygons are congruent if they have an equal number of sides, and all the corresponding sides and angles are congruent. When writing a congruency statement, we must make sure we are putting the letters in the correct order according to corresponding angles and sides.

544

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

7.01 Congruent figures mathspace.co

273

Practice Students: Pages 274–279

What do you remember? 1

Describe each of the following geometric objects, using the points A and B: a

b A

c A

B B

Look at the rhombus shown. Which side is parallel to

Look at the rectangle shown. Which pair of lines are perpendicular to

Consider the following triangle: a

Name the angle that is directly opposite the segment

Name the segment that is directly opposite ∠1.

3 F

7.01 Congruent figures mathspace.co

545

For each diagram, name the sides and angles that have been marked congruent: a

D S

Q R

Y L

B V

Select all of the following that must be true for two polygons to be congruent. A

All congruent sides

Unequal number of sides

Same number of sides

All interior angles are congruent

Let’s practice 7

In the following diagrams, name the congruent segments: a

P 4.5 in

4.5 in 5 in

X c

7 Z

E Y

AB = 4.5 in. If

Determine whether or not the two angles are congruent given their measurements. a

37°

, what is the length of

b 50°

51°

37°

90° 90°

546

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

135°

130°

Use the triangles below and complete the statements:

• ∠M ≅ ∠⬚

• • ∠Q ≅ ∠⬚ •

Are the two triangles congruent? Explain your reasoning. K

Which two parallelograms are congruent? Parallelogram 1

Parallelogram 2

A B

Parallelogram 3

Parallelogram 4

J T V L

K U

For each of the following polygons, state whether they are congruent or noncongruent. Explain your reasoning. a

28°

20 cm 20 cm

20 cm

63° H

N 63°

40 cm

28°

20 cm 40 cm 10 cm 10 cm

8 20 cm

20 cm

10 cm 10 cm

7.01 Congruent figures mathspace.co

547

D 7

4 25 in

10 in M 10 in

7 4

5 N

Are the two ﬁgures congruent? Explain your reasoning.

R Q

K V

T L

J M

Explain why these two ﬁgures are not congruent.

20 15

Alex and Jamie are observing the following polygons: • Alex claims that Polygon X and Polygon Y are congruent because they have the same side lengths and their angles are also the same, just arranged in a different order. • Jamie says that even though Polygon X and Polygon Y have sides and angles that are the same size, they’re not exactly the same shape. He thinks it’s because the order of the sides and angles makes the shapes look different.

5 cm

7 cm

7 cm 7 cm 5 cm Polygon X

Polygon Y

Who is correct? Explain your thinking. 16

Statement: Polygon I and Polygon J are congruent polygons. Analyze the descriptions to determine if this statement is true or false: a

Polygon I: A hexagon with sides of 4 cm each, and all angles measuring 120°.

Polygon J: A hexagon with sides of 4 cm, 4 cm, 4 cm, 4 cm, 3 cm, and 4 cm, and angles of 120°, 120°, 120°, 120°, 110°, and 130°. b

Polygon I: A square with sides of 5 cm each, and all angles measuring 90°. Polygon J: A square with sides of 10 cm each, and all angles measuring 90°.

Polygon I: A triangle with sides of 3 cm, 4 cm, and 5 cm, and angles that correspond to a right triangle.

Polygon J: A triangle with sides of 5 cm, 3 cm, and 4 cm, and angles that correspond to a right triangle. d

Polygon I: An octagon with sides of 2 in each, and all angles measuring 135°. Polygon J: An octagon with seven sides of 2 in each, but one side is 3 in.

Polygon I: A rectangle with sides of 6 cm and 4 cm, and angles of 90°. Polygon J: A rectangle with sides of 4 cm and 6 cm, and angles of 90°.

548

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Given that PQRST ≅ ABCDE

What are the measures of ∠P and ∠R?

95° 80° 70° 110° 105°

C R

For each ﬁgure below, identify all congruent sides: a

Q 4 mm R

24 m

11 cm

18 cm E 8 cm

8 cm

18 m

B 4 cm C

18 m

4 mm S 6 mm

8 mm

T 2 mm

K 30 m

10 mm

30 m

S 12 m

10 cm

7 cm

8 cm

9 cm

9 cm I

11 cm

16 m

U V 10 m

6 cm N O

31 m

R Y

Z Q

7 cm

Let’s extend our thinking 19

Which two triangles are congruent? Triangle A

Triangle B

31°

73°

73° 7

Triangle C

31°

31° 73° 7

Triangle D 7 31° 73°

Consider a park garden. a

The dimensions of the garden are 22 ft, 18 ft, 22 ft, and 18 ft. All angles are congruent and have measures of 90°. What is the shape of the garden?

If a fountain in the garden is in the shape of an equilateral triangle where one side is 75 inches, what are the lengths of the other sides? Explain your thinking.

7.01 Congruent figures mathspace.co

549

If a bookshelf within the community library is designed as an isosceles triangle, will there be any congruent sides or angles? Explain your thinking.

Consider two 4-sided polygons. ABCD ≅ RSTU.

Which segment in ABCD is congruent to

Which angle in RSTU is congruent to ∠C?

Look at the quadrilateral: B

5.8

3.3

98.6°

113°

C 97°

7.8 8.5

51.5°

Construct a shape that is congruent to the quadrilateral. Explain how you made sure your constructed shape was congruent to this quadrilateral.

550

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

e N o, they are not congruent. The sides of the other are longer and markings show that they are not the same length to another ﬁgure.

7.01 Congruent figures

f What do you remember? 1 a The line through A and B.

g N o, they are not congruent. The markings show that two of the sides are not congruent to the other ﬁgure.

b The segment between A and B. c The ray from B through A. in the rhombus is

2 The side parallel to

h Y es, all the corresponding sides of the other triangle are equal in length to the sides of the other triangle.

3 The lines perpendicular to and .

in the rectangle are

4 a ∠3

b The segment

b ∠B and ∠C. and

. ∠D and ∠E

d , and are all congruent. ∠A, ∠B and ∠C are all congruent. and

e f

13 The two ﬁgures are congruent because they have the same side lengths as indicated by the geometric markings. 14 The two ﬁgures are not congruent because they do not have the same side lengths.

5 a ∠P and ∠Q

es, they are congruent. The markings of the two Y ﬁgures show that the sides have equal lengths for both pentagons.

and

, ∠W and ∠Y, ∠X, ∠V and ∠Z

, and KL are all congruent; ∠M, ∠N, ∠P, and ∠K are all congruent; ∠L and ∠O are congruent

15 Jamie is correct about Polygon X and Polygon Y not being congruent. Even though the polygons have the same side lengths and angle measures, congruent polygons must have their corresponding sides and angles not only equal in measure but also arranged in the same sequential order around the polygon. 16 a False

b False

c True

d False

e True

6 A, C, D

17 Since the ﬁgures are congruent, the corresponding angles must be equal. Therefore, ∠P = 80°, and ∠R = 110°.

Let’s practice

18 a

7 a

and

b and

and

8 ST = 4.5 in

9 a Congruent

b Not congruent

c Congruent

d Not congruent

10 • ∠M ≅ ∠T • • ∠Q ≅ ∠L •

d ,

Let’s extend our thinking 19 Triangle C and D 20 a Rectangle b T he length of the other sides is both 75 in. An equilateral triangles has all the three sides congruent.

The two triangles are congruent. The corresponding sides and corresponding angles are all congruent.

21 Yes, two of the sides will be congruent and the base angles that are opposite those sides will also be congruent, by deﬁnition of an isosceles triangle.

11 Parallelogram 1 and 4

22 a ST ≅ BC

12 a Y es, because two corresponding sides and an angle are congruent which makes third side of the triangle also congruent.

23 Answers will vary. If constructed on paper, the answer should describe measuring the sides with a ruler and the angles with a protractor to make sure all sides and angles were congruent to the given shape.

b Not congruent.

b ∠D ≅ ∠T

c Y es, two corresponding angles and one side are congruent. This makes the third angle of the two triangles congruent and the other two sides. d N o, they are not congruent. The sides of the two ﬁgures are not the same length.

Answers mathspace.co

551

7.02 Regular polygons and symmetry Subtopic overview Lesson narrative In this lesson, students will learn about regular polygons and symmetry. They will explore the properties of regular polygons, which have congruent sides and angles. The lesson includes examples to differentiate between regular and irregular polygons. Students will also explore symmetry through an interactive applet, understanding that a line of symmetry reflects a shape onto itself, dividing it into two congruent parts. They will identify the number of lines of symmetry in regular polygons and practice drawing these lines. By the end, students should confidently identify regular polygons and their lines of symmetry.

7.02 Regular polygons and Learning objectives symmetry Students: Page 280

After this lesson, you will be able to... • identify regular polygons. • draw lines of symmetry to divide regular polygons into two congruent parts.

Regular polygons

Key vocabulary

Regular Polygon  congruent parts asymmetric A regular polygon has congruent sides and congruent interior angles.  regular polygon  symmetry 



line of symmetry



transformation

Here are some examples of regular polygons.

Essential understanding A regular polygon has congruent sides and congruent interior angles. The number of lines of symmetry of a regular polygon is equal to the number of sides of the polygon.

Standards

6 cm

This subtopic addresses the following Virginia 20234Mathematics Standards of Learning All squares are regular polygons. They have Markings showing all angles standards. are congruent and sides congruent sides and all angles measure 90°.

Mathematical process goals

are congruent for this regular pentagon.

MPG3 — Mathematical Reasoning Teachers can integrate this goal by assisting students in developing reasoning skills related to polygons and symmetry. They can guide students to make conjectures about the relationships between the number of sides in a polygon and its lines of symmetry and then justify these conjectures using logical reasoning. Teachers can also ask students to analyze and evaluate different arguments about classifying polygons.

552

Equilateral are Grade regular polygons because all Mathspace triangles Virginia SOL 6 Teacher Edition mathspace.co sides and angles are congruent.

This is a regular hexagon.

Now, let’s take a look at some examples that are not regular polygons, but rather irregular polygons.

MPG4 — Mathematical Connections In teaching about polygons and symmetry, teachers can integrate this goal by connecting students’ prior knowledge of triangles and quadrilaterals to the new concepts of regular polygons and lines of symmetry. Teachers could also show the connection between mathematical concepts and real-world applications, such as how symmetry is used in architectural design or how understanding polygons can aid in tasks like tiling a floor. MPG5 — Mathematical Representations Teachers can enhance students’ understanding of symmetry in regular polygons through mathematical representatiosn by incorporating hands-on experiences and visual aids. Use activities like paper folding to help students draw lines of symmetry, which divide the polygons into congruent parts. Provide opportunities for students to rotate and manipulate shapes to see different lines of symmetry. Encourage the use of tools such as tracing paper to compare congruent shapes. Discuss the geometric markings and concepts of symmetry, and create anchor charts to solidify these ideas. They can ask students to draw regular polygons and their lines of symmetry, use geometric software to model polygons, or represent the relationship between the number of sides and lines of symmetry in a table or graph. Teachers can emphasize how these different representations can all convey the same mathematical ideas.

Content standards 6.MG.4 — The student will determine congruence of segments, angles, and polygons.

6.MG.4b — Draw lines of symmetry to divide regular polygons into two congruent parts.

6.MG.4a — Identify regular polygons.

Prior connections 5.MG.3 — The student will classify and measure angles and triangles, and solve problems, including those in context.

Future connections G.PC.2 — The student will verify relationships and solve problems involving the number of sides and angles of convex polygons.

G.RLT.3 — The student will solve problems, including contextual problems, involving symmetry and transformation.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 7.01 Congruent figures

7.02 Regular polygons and symmetry mathspace.co

553

Student lesson & teacher guide Regular polygons Students are introduced to the concept of regular polygons and learn to identify them based on their congruent sides and angles. They are also shown examples of irregular polygons that do not have congruent sides and angles.

Students: Page 280

7.02 Regular polygons and symmetry After this lesson, you will be able to... • identify regular polygons. • draw lines of symmetry to divide regular polygons into two congruent parts.

Regular polygons Regular Polygon A regular polygon has congruent sides and congruent interior angles. Here are some examples of regular polygons.

6 cm

All squares are regular polygons. They have 4 congruent sides and all angles measure 90°.

Markings showing all angles are congruent and sides are congruent for this regular pentagon.

Equilateral triangles are regular polygons because all sides and angles are congruent.

This is a regular hexagon.

Now, let’s take a look at some examples that are not regular polygons, but rather irregular polygons.

This polygon has sides and angles of different measures, so it is irregular.

280

554

Mathspace

Virginia SOL Grade 6

Mathspacemathspace.co Virginia SOL Grade 6 Teacher Edition mathspace.co

In this rectangle, all angles are congruent but the sides are not, so it is irregular.

Understand congruence with physical models Student with disabilities support When teaching about regular polygons, it’s important to stress the concept of congruence. A regular polygon is defined by its congruent sides and angles - this means all sides have the same length and all interior angles have the same measure. One way to illustrate this concept is by using physical models of different polygons (e.g., square, triangle, pentagon, hexagon). Have students measure the sides and angles of these shapes and note their observations. This allows students to see that in regular polygons, the length of all sides and the measure of all interior angles are equal. Finally, provide students with examples and non-examples of regular polygons. This can help clarify the definition and properties of regular polygons. For example, a square is a regular polygon because all its sides and angles are congruent, while a rectangle is not because while its angles are congruent, its sides are not all the same length. These hands-on activities not only help students understand the concept of regular polygons and congruence but also help them to see the practical application of these concepts.

Examples Students: Page 281

Purpose Show students how to identify regular polygons. Expected mistakes Students may believe option C is a regular shape becuase it is symmetric. Remind them that regular shapes must have both congruent sides and congruent angles.

7.02 Regular polygons and symmetry mathspace.co

555

Stronger and clearer each time

use with Example 1

English language learner support Begin by asking students to individually write or draw their understanding of what makes a polygon regular. Give them time to think about the characteristics that define regular polygons, such as congruent sides and congruent angles. Next, pair students up to share their definitions and drawings with each other. Encourage them to discuss specific properties and to ask clarifying questions like, “Do all sides need to be the same length?” or “Must all angles be equal?” Prompt them to provide constructive feedback and to use mathematical vocabulary such as “congruent,” “polygon,” “sides,” “angles,” and “symmetry.” After the first discussion, have students revisit and refine their initial definitions, incorporating new insights and clearer language. Then, have them switch partners for another round of sharing and refining, further strengthening their understanding and use of key terms. Finally, bring the class together and invite volunteers to share their final definitions. As a group, highlight how their explanations have become stronger and clearer through collaboration and precise language.

Students: Page 281

Symmetry Students review the definitions of symmetry and line of symmetry before engaging in an exploration.

Students: Pages 281–282

A line of symmetry divides a ﬁgure into two congruent parts. Notice in the diagram, the side and angle markings show us that the two halves of the ﬁgure are congruent.

A shape can have 0, 1 or many lines of symmetry.

A square has four lines of symmetry.

3 4

For a regular polygon, the number of lines of symmetry is equal to the number of sides. One way to picture lines of symmetry is with reﬂections. Another way is to picture folding the shape along a line. Whether reﬂecting or folding, the resulting shape should perfectly overlap the original.

556

Mathspace Grade 6 TeacherisEdition A shapeVirginia that hasSOL no lines of symmetry called asymmetric. Neither of these shapes have a line of symmetry. mathspace.co

A square has four lines of symmetry.

3 4

Example 2

Exploration Use the image shown to answer the following: Students: Page 281

a How many lines of symmetry does this regular polygon have?

Create a strategy

Apply the idea

In a regular polygon, we know the number of sides is equal to the number of lines of symmetry.

Since this regular polygon has 10 sides, we know it has 10 lines of symmetry.

Suggested 282 student Mathspacegrouping: Virginia SOL Small Grade 6 groups mathspace.co In this exploration, students will use a GeoGebra applet to observe different shapes and their lines of symmetry. They will use pattern analysis to draw conclusions about which shapes have at least one line of symmetry, no lines of symmetry, or two or more lines of symmetry, and discuss the characteristics of these shapes. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Which shapes had at least one line of symmetry? What similarities do you notice between those shapes? The shapes that had at least one line of symmetry were the isosceles triangle, equilateral triangle, square, rectangle, regular pentagon, regular hexagon, and isosceles trapezoid. 2. Which shapes had no lines of symmetry? What did you notice about those shapes? The scalene triangle and the parallelogram had no lines of symmetry. These shapes do not have equal sides or equal angles. 3. Which shapes had more than two lines of symmetry? What did you notice about those shapes? The shapes that had more than two lines of symmetry were the equilateral triangle, the square, the regular pentagon, and the regular hexagon. These shapes are regular polygons, which means they have all sides and angles equal. The number of lines of symmetry is the same as the number of sides the shape has.

7.02 Regular polygons and symmetry mathspace.co

557

Purposeful questions • What is a line of symmetry and how can you determine if a shape has a line of symmetry? • How does the number of lines of symmetry relate to the properties of the shape? • What makes a shape symmetrical? And why are some shapes not symmetrical? • Can a shape have more than one line of symmetry? If so, give an example. Possible misunderstandings • Students might not recognize that a line of symmetry can be seen in the applet when the two halves overlap perfectly. Show them how, if they imagined unfolding the shape rather than rotating it, the line would divide the original shape into two equal, mirrored halves. • Students might assume that changing the orientation of the shape (rotating or flipping it) changes whether or not the shape has lines of symmetry. They might not understand that symmetry is a property of the shape itself, independent of its orientation.

Not all diagonals are lines of symmetry Address student misconceptions

A line of symmetry divides a ﬁgure into two congruent parts.

in the diagram, the side and angle markings show us that Students may incorrectly believe that any line thatNotice divides a polygon into two equal halves by size is a line of the two halves of the ﬁgure are congruent. symmetry, not understanding that a line of symmetry must divide a shape into two congruent parts, or mirror images.

To address this misconception, provide a clear definition and examples of lines of symmetry. Emphasize that a line of symmetry divides a shape into two parts that are mirror images of each other. Using a rectangle, demonstrate that while the diagonals do indeed divide the rectangle into two equal halves by size, the two halves are not mirror images of each other, therefore, the diagonals are not lines of symmetry. A shape can have 0, 1 or many lines of symmetry.

A square has four lines of symmetry.

3 4

Additionally, provide students with tracing to physically the concept of symmetry, allowing For a regular polygon, the number of linespaper of symmetry is equal todemonstrate the number of sides. studentsOne to see how thelines shape folds onto itself along the line way of symmetry. way to picture of symmetry is with reﬂections. Another is to picture folding the shape along a line. Whether reﬂecting or folding, the resulting shape should perfectly overlap the original.

Students are giventhat diagrams of of lines of symmetry, identifyingNeither how they divide shapes congruent A shape has no lines symmetry is called asymmetric. of these shapes have ainto line of symmetry. parts. They explore examples like triangles with one line of symmetry and squares with four, while also discovering that regular polygons have lines of symmetry equal to their number of sides. They are introduced to asymmetric shapes, which have no lines of symmetry.

Examples Students: Page 282 Example 2 Use the image shown to answer the following:

a How many lines of symmetry does this regular polygon have?

558

Mathspace Virginia SOL Grade 6 Teacher Edition Create a strategy mathspace.co In a regular polygon, we know the number of sides is equal to the number of lines of symmetry.

Apply the idea Since this regular polygon has 10 sides, we know it has 10 lines of symmetry.

a How many lines of symmetry does this regular polygon have?

Create a strategy

Apply the idea

In a regular polygon, we know the number of sides is equal to the number of lines of symmetry.

Since this regular polygon has 10 sides, we know it has 10 lines of symmetry.

Purpose282 Mathspace Virginia SOL Grade 6 mathspace.co Show students the relationship between the number of sides in a regular polygon and its lines of symmetry. Reflecting with students Reflect with students on what they could do to show this is a regular polygon. Since there are no angle or side markings or measures, students could show all the side lengths are equal using a ruler and show all the angle measures are equal using a protractor.

Students: Page 283 b Draw the lines of symmetry

Create a strategy

Apply the idea

We know from above part that this polygon has 10 lines of symmetry.

Draw in the lines of symmetry: 1

3 4 5 6 7 8

Idea summary

Purpose A line that reﬂects a shape onto itself is called a line of symmetry. Challenge students to accurately draw the lines of symmetry for a regular polygon. Asymmetric shapes are shapes without lines of symmetry.

In a regular polygon, the number of sides is equal to the number of lines of symmetry.

Advanced learners: Inquiry-based discovery of symmetry in regular polygons use with Example 2

Targeted instructional strategies

Provide students with a set of regular polygons—such as equilateral triangles, squares, regular pentagons, Practice and hexagons—and ask them to explore by drawing all possible lines of symmetry for each shape. Encourage advanced learners to work individually or in small groups to find and record the lines of symmetry they identify. What do you remember? Facilitate a discussion where students share their findings and look for patterns between the number of sides and the number ofthe symmetry in each questions like, “What and do you notice 1 i Sides are not all congruent interior anglesabout are the Match following lines terms with their polygon. deﬁnitions: Ask guiding number of sides and the number of lines of symmetry?” and “Can predict the number of lines of symmetry not you all congruent a Regular polygon ii Has all congruent sides and congruent interior in a regular polygon withpolygon more sides?” b Irregular c

angles

Congruent ﬁgures

This inquiry-based approach allows students to discover foriiithemselves that the number of lines of symmetry in Divides a ﬁgure into two congruent parts, each of d Line of symmetry a regular polygon is equal to the number of its sides, deepeningwhich theirare understanding mirror images ofthrough each otherexploration and e Congruent parts pattern recognition. iv Parts of a ﬁgure that are identical in shape and size v 2

Arrange the polygons in ascending order according the number of sides they have. • Heptagon • Quadrilateral • Hexagon 7.02 Regular polygons and symmetry mathspace.co

• Pentagon • Triangle • Octagon 3

Have exactly the same shape and size

Which image shows a line of symmetry? A

559

5 6 7 8

Students: Page 283

Idea summary A line that reﬂects a shape onto itself is called a line of symmetry. Asymmetric shapes are shapes without lines of symmetry. In a regular polygon, the number of sides is equal to the number of lines of symmetry.

Practice What do you remember? Practice 1

Match the following terms with their deﬁnitions:

Sides are not all congruent and interior angles are not all congruent

Has all congruent sides and congruent interior angles

iii

Divides a ﬁgure into two congruent parts, each of which are mirror images of each other

Parts of a ﬁgure that are identical in shape and size

Sides are the notsame all congruent and interior angles are Have exactly shape and size

a Regular polygon Students: Pages 283–287 b

Irregular polygon

Congruent ﬁgures

Line of symmetry What do youd remember? e

Congruent parts

Match the following terms with their deﬁnitions: a

Regular polygon

Irregular polygon 2

not all congruent

Arrange the polygons in ascending order according the number of sides they have.

Congruent ﬁgures • Pentagon

• Triangle Line of symmetry

Congruent parts

• Octagon

Has all congruent sides and congruent interior • Heptagon angles • Quadrilateral

iii

• Hexagon Divides a ﬁgure into two congruent parts, each of which are mirror images of each other

Parts of a ﬁgure Cthat are identical in shape and size

Have exactly the same shape and size

Which image shows a line of symmetry? A

Arrange the in ascending order according Thepolygons image that shows a line of symmetry is image ⬚.the number of sides they have.

• Heptagon • Quadrilateral 7.02 Regular polygons and symmetry mathspace.co • Hexagon

• Pentagon • Triangle • Octagon

Which image shows a line of symmetry? A

560

The image that shows a line of symmetry is Image ⬚.

Which image shows a line that does not divide the polygon into two congruent parts? A

Do all shapes have a line symmetry? Give some examples to explain why or why not.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

283

Use geometric markings to mark the congruent angles and sides on the regular polygons.

Let’s practice 7

Identify the regular pentagon. A

Determine whether the polygons are regular or irregular: a

5 in 5 in

5 ft c

5 in

9 ft

5 in

11 ft

12 ft

6.6 ft

5 in 5 in

7.02 Regular polygons and symmetry mathspace.co

561

Sort the shapes below as regular or irregular polygons. Regular polygon

a 3 in

Irregular polygon

3 in 3 in

4m 2m

2m 4m

SOL

Circle all the shapes that are regular polygons.

Explain how you determined which shapes to circle. 11

562

Determine whether each description is of a regular polygon. Explain. a

A rhombus with all sides equal to 7 inches, 2 angles equal to 40 degrees, and 2 angles equal to 140 degrees.

A rectangle with sides measuring 4 cm and 6 cm, and all angles measuring 90 degrees.

A right triangle.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

If ABFGH and BCDEF are regular pentagons, what do we know about the side lengths of both pentagons?

C B

SOL

Draw and count all lines of symmetry. Fill in the blank with the number of lines of symmetry you drew.

Count the number of sides.

What do you notice?

For these regular polygons, determine how many lines of symmetry they have: b

22-gon

Draw as many lines of symmetry as you can in the following images: a

Hexagon

How many lines of symmetry do these irregular polygons have? a

For each regular polygon:

Complete each picture in these diagrams so that the dotted line is a line of symmetry. a

7.02 Regular polygons and symmetry mathspace.co

563

For each polygon, determine whether the line divides the ﬁgure into two congruent parts, where each part is the mirror image of the other. Explain your reasoning. a

Let’s extend our thinking 19

Henrique says the line drawn on the parallelogram is a line of symmetry because it divides the polygon into two congruent triangles. Jacqueline says it is not a line of symmetry because the triangles are not mirror images of each other. Who is correct? Explain your answer.

Complete the following tasks: a

Create your own regular polygon.

Draw all the lines of symmetry it has.

Label the sides and angles to show congruency.

Consider a rhombus. The points B, D, F and H represent the midpoints of the sides of the rhombus. a

Write down the pairs of points that can be joined to form a line of symmetry for the rhombus.

Is a rhombus a regular polygon? Explain.

564

If possible, draw an example for each of these polygons. If it is not possible, explain why not. a

A regular polygon with no lines of symmetry.

A regular polygon with 1 line of symmetry.

A regular polygon with 2 lines of symmetry.

A new community center is being designed in the shape of a regular polygon. Inside the building, there will be a circular-shaped auditorium. The outside of the building needs to have multiple, evenly spaced entrances, but the shape of the building has not yet been designed. a

Propose a solution for designing the shape of the outside of the building using a regular polygon.

Explain your choice based on the properties of symmetry and congruence, and how it would contribute to the eﬃciency and beauty of the construction.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

11 a T his shape is not a regular polygon because, even though all sides are equal in length, the angles are not all equal. A regular polygon must have all sides and all angles equal.

7.02 Regular polygons and symmetry What do you remember? 1 a ii

b i

c v

d iii

e iv 2 Triangle, quadrilateral, pentagon, hexagon, heptagon, octagon 3 C

b T his shape is not a regular polygon because, despite all angles being equal, the sides are not all the same length (a regular polygon must have sides and angles that are all equal). c A right triangle is not a regular polygon because it does not have all sides nor all angles equal. A regular polygon must have all sides and all angles equal. 12 The side lengths of both pentagons must be the same.

4 Image B 5 Not all shapes have line symmetry. For example, a rectangle has line symmetry, but a scalene triangle does not. A rectangle can be divided into two identical halves by a vertical line through its center, while a scalene triangle cannot be divided into two identical halves by a line.

13 a

5 sides

3 sides

4 sides

Let’s practice 7 D 8 a Regular e Regular Irregular

m Regular 9

b Irregular

c Irregular

d Irregular

Irregular

g Irregular

h Regular

Irregular

k Irregular

Irregular

n Irregular

Regular polygon

Irregular polygon

c F or regular polygons, the number of lines of symmetry is equal to the number of sides. 14 a 4

b 22

c 8

d 7

15 a 2

b 1

c 1

d 1

16 a

3 in

3 in 3 in

4m 2m

2m 4m

17 a

18 a N o. One part is bigger than the other and the parts are not mirror images. A regular polygon is one where all sides and all angles are equal.To determine which shapes to circle, I looked for polygons with sides of equal length and angles of equal measure.

b N o. One part is bigger than the other and the parts are not mirror images of each other. c N o. The line divides the trapezoid into two trapezoids but one is bigger than the other. d Y es. The line divides the pentagon into two congruent parts.

Answers mathspace.co

565

e N o. The parts are not congruent and not of the same shape. One is a triangle and the other is a trapezoid. f

Yes. The parts are mirror images of each other.

g Y es. The hexagon is divided into two congruent trapezoids. h No. One part is bigger than the other. Let’s extend our thinking 19 Jacqueline is correct. A line of symmetry in a polygon should divide the ﬁgure into two congruent parts that are mirror images of each other. Even if the line divides the polygon into two congruent triangles, they must also be mirror images of each other across the line for it to be considered a line of symmetry. 20 a

22 a T his is impossible. A regular polygon has the same number of lines of symmetry as sides of the polygon. We cannot draw a regular polygon with 0 sides. b T his is impossible. A regular polygon has the same number of lines of symmetry as sides of the polygon. We cannot draw a regular polygon with 1 sides. c T his is impossible. A regular polygon has the same number of lines of symmetry as sides of the polygon. We cannot draw a regular polygon with 2 sides. 23 a A nswers will vary based on the regular polygon chosen. One possible solution is a square: Door

Door

Auditorium

Door

21 a AE and CG b N o. The angles are not all congruent. In addition, a regular polygon has the same number of lines of symmetry as sides of the polygon. A rhombus has 4 sides but 2 lines of symmetry.

566

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b A possible design for the outside of the building might be a square. A square has 4 equal sides and four lines of symmetry. This design would allow for one door on every side of the building for eﬃcient access.

7.03 Perimeter of triangles and parallelograms Subtopic overview Lesson narrative In this lesson, students will learn to calculate the perimeter of triangles and parallelograms. They will explore the concept of perimeter as the total distance around a figure, using formulas for triangles (P = a + b + c), parallelograms (P = 2(a + b)), rectangles (P = 2(l + w)), and squares (P = 4l). Students will solve problems involving finding perimeters, determining unknown side lengths, and applying formulas in real-world contexts, such as fencing a garden. By the end, students should confidently calculate the perimeter of various shapes.

7.03 Perimeter of triangles and Learning objectives parallelograms Students: Page 288

After this lesson, you will be able to... • solve mathematical and contextual problems involving the perimeter of triangles and parallelograms.

Perimeter of triangles and parallelograms

Key vocabulary

Perimeter  equilateral triangle baseThe measure of the distance around a ﬁgure.  perimeter  rectangle



isosceles triangle

 parallelogram



slant height

 triangle

Interactive exploration Explore online to answer the questions Essential understanding

Perimeter is mathspace.co a linear measurement to measure the distance around the outside of a 2D figure. Use the interactive exploration in 7.03 to answer these questions.

Standards 1. How are the perimeter formulas similar? How are they different? 2. Which formulas be simplified? Why do you think that is? This subtopic addresses thecould following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals To ﬁnd the perimeter of any shape we can add up all of the side lengths, however the properties of some shapes MPG1 — in Mathematical Problem Solving result special perimeter formulas. Teachers can embed this goal into their instruction byperimeter, giving students The P, of a various triangle problems is given byto solve that involve finding the perimeter of bsquares, rectangles, triangles, and parallelograms. The problems should vary in complexity, ranging a P=a+b+c from simple calculations to more challenging real-world context problems. For example, teachers might present a where a,ofb,aand c are the lengths of each problem where students need to calculate the perimeter triangular garden given theside. lengths of the sides, or c a problem where students need to determine the length of a missing side of a rectangle when given its perimeter and the length of the other side. Teachers should model how to apply the four step process for solving contextual problems related to perimeter and area: determining which application should be with usedtwo (when stated A parallelogram is a quadrilateral pairsnot of explicitly parallel sides, its to a opposite sides congruent. find the perimeter and area), writing the formula, substituting theare values, and solving including proper units. b

We will use the base, b, and slant height, a, to ﬁnd its perimeter. 7.03 Perimeter of triangles and parallelograms Perimeter =b+a+b+a

Perimeter = 2a + 2b = 2 (a + b)

mathspace.co

567

MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can foster mathematical reasoning by encouraging students to use logical reasoning to solve problems and justify their solutions. For example, when exploring the perimeter of parallelograms, teachers can stress that opposite sides of a parallelogram have equal lengths and ask students to reason why this is so. When solving problems, teachers can also ask students to explain why their solution is correct based on the properties of the shapes involved, such as why the sides of a rectangle are equal.

Teachers can show students the mathematical connections by linking the concept of perimeter to realworld situations. For instance, teachers can present realworld problems that require calculating the perimeter, such as fencing a rectangular field or triangular park. Furthermore, teachers can connect the concept of the perimeter of squares and rectangles to the perimeter of triangles and parallelograms, emphasizing that in all cases, the perimeter is the sum of the lengths of all sides.

Content standards 6.MG.2 — The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.2b — Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

Prior connections 5.MG.2 — The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

Future connections 8.MG.5 — The student will solve area and perimeter problems involving composite plane figures, including those in context.

Lesson Preparation Tools You may find these tools helpful: • Scientific calculator • Formula sheet

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

568

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Comparing sums to compare perimeters Targeted instructional strategies Provide students with three or four labeled rectangles and triangles and ask students to add the sides of each shape. An example of these shapes is included below: 8 cm

4 cm

10 cm

8 cm 8 cm

9 cm

9.5 cm

9 cm

9.5 cm

Ask students if they could combine the sides to get the perimeter using any other methods, emphasizing students who note that the square’s side could be multiplied by 4 or the equilateral triangle’s side length could be multiplied by 3. Students should be shown the different ways of finding the perimeter of each shape prior to the corresponding sections being introduced.

Student lesson & teacher guide Perimeter of triangles and parallelograms Students are reminded of the definition of perimeter before engaging in an exploration.

Students: Page 288

7.03 Perimeter of triangles and parallelograms After this lesson, you will be able to... • solve mathematical and contextual problems involving the perimeter of triangles and parallelograms.

Perimeter of triangles and parallelograms Perimeter The measure of the distance around a ﬁgure.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.03 to answer these questions.

7.03 Perimeter of triangles and parallelograms mathspace.co

569

Solve a perimeter problem with the STEAM cycle Targeted instructional strategies Use the STEAM cycle to explore the perimeter of different shapes in a real-world context. Ask: Pose the question: “How can we determine the amount of fencing needed to enclose a garden?” Encourage them to define perimeter in the context of a garden, consider different possible shapes for the garden, and discuss factors that might influence the amount of fencing required. Imagine: Prompt students to brainstorm various shapes the garden could take—such as rectangles, squares, triangles, or parallelograms—and imagine how each shape would affect the perimeter. Have them predict which shapes might require more or less fencing, using sketches or manipulatives like string to outline different garden layouts. Plan: Guide students to develop a plan for calculating the fencing needed for each garden shape they imagined. Assist them in selecting appropriate tools (rulers, grid paper) for measuring side lengths, setting criteria for accurate calculations, and determining how they will record and compare the data for different shapes.

7.03 Perimeter of triangles and parallelograms

Create and test: Support students as they construct models of their proposed garden shapes using materials like cut-out shapes or digital drawing tools. Help them measure the side lengths and calculate the total amount of fencing needed. Encourage them to test their calculations by comparing with peers or checking against different models. Improve: Facilitate a reflection session where students analyze their results, ask questions about any unexpected findings, andlesson, collaborate to able refine After this you will be to...their garden designs or calculations. Encourage them to justify any changes they make and discuss how different shapes or dimensions the perimeter and the required • solve mathematical and contextual problems involving the perimeteraffect of triangles and parallelograms. amount of fencing.

Perimeter of triangles and parallelograms

Exploration Perimeter

Students: Page 288 The measure of the distance around a ﬁgure.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.03 to answer these questions. 1.

How are the perimeter formulas similar? How are they different?

Which formulas could be simplified? Why do you think that is?

To ﬁnd the perimeter of any shape we can add up all of the side lengths, however the properties of some shapes result in special perimeter formulas.

Suggested student grouping: Small groups The perimeter, P, of a triangle is given by In this exploration, students will a investigate the different formulas for calculating the perimeter of various shapes b P=a+b+c - rectangle, triangle, parallelogram, and square. Students will use the interactive tool to change the shape and where a, b, and c are the lengths of each side. observe the correspondingcformula and computation of the perimeter. They will analyze the similarities and differences among these formulas and discuss which ones could be simplified and why. A parallelogram is a quadrilateral with two pairs of parallel sides, its opposite sides are congruent.

We will use the base, b, and slant height, a, to ﬁnd its perimeter.

Perimeter = b + a + b + a Perimeter = 2a + 2b = 2 (a + b)

570

l Consider a rectangle, a special type of parallelogram. Its opposite Mathspace Virginia SOL Grade 6 Teacher Edition sides are congruent. mathspace.co If the width is w and the length is l, the perimeter is: w

Perimeter = l + w + l + w

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. How are the perimeter formulas similar? How are they different? The perimeter formulas for all the shapes involve adding up the lengths of all sides. However, they’re different because each shape has a different number of sides and the lengths of the sides can vary. For example, in a square, all sides are equal, while in a rectangle, opposite sides are equal. A triangle has three sides of different lengths and a parallelogram has opposite sides of equal length. 2. Which formulas could be simplified? Why do you think that is? The formula for the square’s perimeter could be simplified to 4⋅ side length, because all sides of a square are equal. The formula for a rectangle could also be simplified to 2 ⋅ (length + width) as opposite sides are equal. The formula for the parallelogram’s perimeter could be simplified to 2(a + b), where a and b are the side lengths. This is because opposite sides are equal. Purposeful questions • Why are the variables in the triangle’s formula all different, while the variables in the square’s formula are all the same? • What do we know about the lengths of opposite sides in a rectangle and parallelogram? • What does the perimeter represent in real-world situations? • Why do we use different formulas for calculating the perimeter of different shapes? Possible misunderstandings • Students might not realize that the same variable is used for sides that have the same length. This might prevent them from understanding that perimeter is the sum of the length of all sides of a shape. • Students might believe that the animation only demonstrates how to calculate the perimeter of a specific shape, rather than illustrating the concept of perimeter for various shapes. They might not generalize the understanding to apply it to other shapes.

Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Begin by engaging your students with physical manipulatives to explore perimeter. Provide them with cut-out shapes of triangles and parallelograms made from cardboard or foam. Give each student a ruler or a measuring tape to measure the sides of these shapes. Encourage them to physically trace the perimeter by moving the ruler along each side, recording the lengths as they go. You can also use string to wrap around the shapes and then measure the length of the string to find the total distance around the figure. Representational: Have students draw the triangles and parallelograms they worked with, using graph paper for accuracy. Ask them to label each side with the measurements they recorded during the concrete stage. Encourage them to add the side lengths on their drawings to calculate the perimeter visually. Use diagrams to reinforce this concept—for example, display an image of a triangle with sides labeled 5 cm, 7 cm, and 10 cm, showing the calculation 5 cm + 7 cm + 10 cm = 22 cm next to it. This visual representation helps solidify the connection between the physical shape and its measurements. Abstract: Guide students to recognize patterns in their calculations that lead to generalized formulas. Discuss how the sums they computed can be expressed using mathematical symbols. Introduce the perimeter formulas for triangles and parallelograms using abstract notation. Explain that for a triangle, the perimeter P is calculated using P = a + b + c, where a, b, and c are the side lengths. For a parallelogram, the formula is P = 2(a + b) because opposite sides are equal. Provide examples where students substitute actual measurements into these formulas to find the perimeter. Encourage them to solve problems that involve finding missing side lengths when given the perimeter, reinforcing their understanding of the formulas.

7.03 Perimeter of triangles and parallelograms mathspace.co

571

The measure of the distance around a ﬁgure.

Interactive exploration Explore online to answer the questions

Students explore the concepts of perimeter in relation to various shapes such as triangles, parallelograms, mathspace.co rectangles, and squares. While the general formula involves summing all side lengths, specific shapes have unique formulas dueUse to their properties. the interactive exploration in 7.03 to answer these questions. 1.

How are the perimeter formulas similar? How are they different?

Which formulas could be simplified? Why do you think that is?

Students: Pages 288–289

To ﬁnd the perimeter of any shape we can add up all of the side lengths, however the properties of some shapes result in special perimeter formulas. The perimeter, P, of a triangle is given by a

P=a+b+c where a, b, and c are the lengths of each side.

A parallelogram is a quadrilateral with two pairs of parallel sides, its opposite sides are congruent.

We will use the base, b, and slant height, a, to ﬁnd its perimeter.

Perimeter = b + a + b + a Perimeter = 2a + 2b = 2 (a + b) Consider a rectangle, a special type of parallelogram. Its opposite sides are congruent.

If the width is w and the length is l, the perimeter is:

Perimeter = l + w + l + w Perimeter = 2l + 2w = 2 (l + w) Consider a square, a special type of rectangle. All of its sides are congruent.

288

Mathspace Virginia SOL Grade 6 l mathspace.co

If each side has length l, the perimeter is: Perimeter = l + l + l + l Perimeter = 4l

Example 1 Find the perimeter of the isosceles triangle shown. Examples

The following supports may be useful for the examples in this section.

12 cm

Templates of shapes and perimeters for word problems Student with disabilities support

8 cm

For word problems in particular, provide pictures of the shapes with spots for labeling the sides and setting up Createformula. a strategy the perimeter Examples are shown below: Use the perimeter of the triangle formula.

Square

Equilateral Triangle

Rectangle

Apply the idea Since the triangle is isosceles, we can let the opposite sides a = 12 and b = 12. The base is c = 8. Perimeter = a + b + c

Write the formula

= 12 + 12 + 8

Substitute a = 12, b = 12, c = 8

= 32 cm

Evaluate

P=4×

P=3×

P=2×

Example 2 Find the perimeter of an equilateral triangle with a side length of 5 mm.

Create a strategy 572

Mathspace Virginia SOL Gradetriangle 6 Teacher All 3 sides in an equilateral are Edition congruent. If we call the side length s, we can write the perimeter forumula: mathspace.co P=s+s+s or P = 3s

Consider a square, a special type of rectangle. All of its sides are congruent. If each side has length l, the perimeter is:

Perimeter = l + l + l + l

Students: Page 289

Perimeter = 4l

Example 1 Find the perimeter of the isosceles triangle shown.

12 cm

Consider a square, a special type of rectangle. All of its sides are congruent. If each side has length l, the perimeter is:

Perimeter = l + l + l + l

8 cm

Perimeter = 4l

Create a strategy Use the perimeter of the triangle formula.

Example 1

Apply the idea Find the isosceles triangle shown. Sincethe theperimeter triangle isofisosceles, we can let the opposite sides a = 12 and b = 12. The base is c = 8. Perimeter = a + b + c

Write the formula

= 12 + 12 + 8

Substitute a = 12, b = 12, c = 8

= 32 cm

Evaluate

12 cm

8 cm Example 2 Purpose strategy Find the aperimeter of antoequilateral triangle a side length 5 mm. Use theCreate perimeter formula determine the with perimeter of anofisosceles triangle given two of the side lengths.

Use the perimeter of the triangle formula.

Create strategy Reflecting witha students Apply the All 3 sides inidea an equilateral triangle are congruent. we the call the side length weisosceles can write the perimeter forumula: Ask students whether they could create a formulaIffor perimeter of s,an triangle, such as Since the triangle we canwhether let the opposite sidesof a = 12 and bneeds = 12. The is c = or 8. whether knowing two of P = 2(common length) is + isosceles, base. Discuss thisPtype a base formula, = s + s +triangle s Perimeter = ais+enough. b+c Write the formula or P = 3s the sides being congruent

Students:Apply Page the289 idea

= 12 + 12 + 8

Substitute a = 12, b = 12, c = 8

= 32 cm

Evaluate

We are given that side length = 5. Perimeter = 3s

Example 2

=3⋅5

Write the formula Substitute s = 5

15equilateral mm Evaluate Find the perimeter of=an triangle the withmultiplication a side length of 5 mm.

Create a strategy All 3 sides in an equilateral triangle are congruent. If we call the side length s, we can write the perimeter forumula: P=s+s+s or P = 3s

Apply the idea

7.03 Perimeter of triangles and parallelograms mathspace.co

We are given that side length = 5. Perimeter = 3s

289

Write the formula

=3⋅5

Substitute s = 5

= 15 mm

Evaluate the multiplication

Purpose Show students how to calculate the perimeter of an equilateral triangle using its side length.

7.03 Perimeter of triangles and parallelograms mathspace.co

289

7.03 Perimeter of triangles and parallelograms mathspace.co

573

Students: Page 290 Example 3 Find the side length indicated on the diagram if the perimeter of the shape is 69 cm.

23 cm

19 cm

Create a strategy Use the formula for the perimeter of a triangle and substitute the known values.

Apply the idea The triangle has P = 69. We can let the side lengths a = 23, b = 19, and the missing side c. P=a+b+c

Write the formula

69 = 23 + 19 + c

Substitute P = 69, a = 23, b = 19

69 = 42 + c

Evaluate the addition

27 = c

Subtract 42 from both sides of the equation

c = 27 cm

Rewrite with c on the left

PurposeExample 4 Show students how toABCD, find the of a triangle given the perimeter andA two side lengths. In the rectangle sidemissing AB has a side lengthlength of 6 units. 6

B Reflecting with students Emphasize the importance of stating the final answer with appropriate units. Remind students that including units is a critical aspect of conveying their mathematical reasoning accurately. An imprecise response might be “c = 27,” whereas a precise response accurately conveys the length of the side in centimeters. By consistently D using units, students develop precision in their mathematical communication and deepen their comprehension C of the concepts. a State the other side of ABCD which must have a length of 6 units.

Create a strategy Advanced learners: Create problems involving missing side lengths

use with Example 3

The opposite sides of a rectangle are congruent. Targeted instructional strategies

Encourage advanced Apply the idea learners to design their own polygons (beyond just triangles) with one side length missing and a specified perimeter. to consider howtells different side lengths thehave overall perimeter and Since has a length Prompt of 6 units,them properties of rectangles us the opposite side, affect , will also a length of explore 6various units. shapes. Once they have created their custom problems, have them swap with classmates to solve for the missing side lengths. This strategy not only theirmeasures grasp of perimeter b If another sidereinforces of the rectangle 5 units, ﬁnd theconcepts perimeter. but also promotes creativity and critical thinking. By engaging in this peer-to-peer exchange, students can analyze diverse problem structures and a strategyenhancing their problem-solving skills and appreciation for geometric relationships. methodsCreate of reasoning, Use the formula for the perimeter of a rectangle, P = 2 (l + w). The fourth side must also be 5 units since the opposite sides of a rectangle are congruent.

290

574

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

69 = 23 + 19 + c

Substitute P = 69, a = 23, b = 19

69 = 42 + c

Evaluate the addition

27 = c

Subtract 42 from both sides of the equation

c = 27 cm

Rewrite with c on the left

Students: Page 290

23 cm

19 cm

Create a strategy Example 4 Use the formula for the perimeter of a triangle and substitute the known values. In the rectangle ABCD, side AB has a length of 6 units.

Apply the idea

The triangle has P = 69. We can let the side lengths a = 23, b = 19, and the missing side c. P=a+b+c

Write the formula

69 = 23 + 19 + c

Substitute P = 69, a = 23, b = 19

69 = 42 + c

Evaluate the addition

D C

27 = c Subtract 42 from both sides of the equation a State the other side of ABCD which must have a length of 6 units. c = 27 cm Rewrite with c on the left

Create a strategy The opposite sides of a rectangle are congruent.

Example 4

Apply the idea In the rectangle ABCD,ofside AB has a lengthofofrectangles 6 units. tells us the opposite side, Since has a length 6 units, properties 6 units.

, will alsoAhave a length of 6

b If another side of the rectangle measures 5 units, ﬁnd the perimeter.

Purpose D Create a strategy Show students that properties of rectangles can be used to determine the length of sides.

C Use the formula for the perimeter of a rectangle, P = 2 (l + w). The fourth side must also be 5 units since the opposite a State the other side of ABCD which must have a length of 6 units. sides of a rectangle are congruent.

Expected mistakes StudentsCreate mightanot use proper mathematical notation when naming the side. They might state their answer as strategy 290. Remind Mathspace Virginia SOL Grade CD or The students that the6are order of the vertices should match the order in the name of the rectangle, opposite sides of a rectangle congruent. mathspace.co is ABCD. Additionally, we are referring to a segment, so the letters should have a line above them. Thus, Apply the idea the correct notation. Since has a length of 6 units, properties of rectangles tells us the opposite side, 6 units. Students: Pages 290–291

, will also have a length of

b If another side of the rectangle measures 5 units, ﬁnd the perimeter.

Create a strategy Use the formula for the perimeter of a rectangle, P = 2 (l + w). The fourth side must also be 5 units since the opposite sides of a rectangle are congruent.

Apply the idea 290

Mathspace

Virginia SOL Grade 6

The length is 6 units and the width is 5 units. mathspace.co Perimeter = 2(l + w)

Formula for perimeter of a rectangle

Perimeter = 2(6 + 5)

Substitute the length and width

= 2 ⋅ 11

Evaluate the addition

= 22 units

Evaluate the multiplication

PurposeExample 5 Challenge students to apply the formula for the perimeter of a rectangle and properties of rectangles to find the Jacob is helping his father build a garden with a fence around it, and they decide to make the garden in the shape perimeter. of a parallelogram. The garden has one side that is 15 feet long and the side adjacent to it is 10 feet long. What is the total perimeter of the garden fence that Jacob and his father need to prepare?

Create a strategy Opposite sides of a parallelogram are equal in length. Use formula for perimeter of a parallelogram: P = 2 (a + b).

Apply the idea The base is 15 feet and the slant height is 10 feet. Using the formula to ﬁnd perimeter. 7.03 Perimeter of triangles and parallelograms Perimeter = 2(a + b) Formula for perimeter of a parallelogram mathspace.co Perimeter = 2(10 + 15) Substitute the slant height and base = 2 ⋅ 25

Evaluate the addition

575

The length is 6 units and the width is 5 units. Perimeter = 2(l + w)

Formula for perimeter of a rectangle

Perimeter = 2(6 + 5)

Substitute the length and width

Students: Page 291

= 2 ⋅ 11

Evaluate the addition

= 22 units

Evaluate the multiplication

Example 5 Jacob is helping his father build a garden with a fence around it, and they decide to make the garden in the shape of a parallelogram. The garden has one side that is 15 feet long and the side adjacent to it is 10 feet long. What is the total perimeter of the garden fence that Jacob and his father need to prepare?

Create a strategy Opposite sides of a parallelogram are equal in length. Use formula for perimeter of a parallelogram: P = 2 (a + b).

Apply the idea The base is 15 feet and the slant height is 10 feet. Using the formula to ﬁnd perimeter. Perimeter = 2(a + b)

Formula for perimeter of a parallelogram

Perimeter = 2(10 + 15)

Substitute the slant height and base

= 2 ⋅ 25

Evaluate the addition

= 50 feet

Evaluate the multiplication

Reﬂect and check If we couldn’t remember the formula we could have just written down all of the side lengths and added them together: 10 + 10 + 15 + 15 = 50 feet

Idea summary

Purpose To ﬁnd the perimeter of a ﬁgure, add up all of its side lengths. Show students how to calculate the perimeter of a parallelogram using the formula and understanding of its The perimeter of a triangle, with sides a, b, and c has the formula: properties. Ptriangle = a + b + c

Expected mistakes The perimeter of a parallelogram, with slant height, a, and base, b, has the formula: Students might get confused about what it means for a side to be adjacent to another. Inform them that Pparallelogram = 2 (a + b) “adjacent” sides share a vertex, and encourage them to draw a diagram to understand the problem more The perimeter of a rectangle, with length, l, and width, w, has the formula: clearly. Prectangle = 2 (l + w)

perimeter of a square, with side length, l, has the formula: ThreeThe reads

English language learner support

use with Example 5

Psquare = 4l

Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students, “What do you think is happening in this question?” or “Can you explain what 7.03 thisPerimeter question is about?” of triangles and parallelograms 291 mathspace.co

On the second read, students should aim to interpret the problem by anwering questions like, “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: • The garden is shapes like a parallelogram. • One side is 15 feet long. • The connected side is 10 feet long. Students can be prompted by framing these as questions like “How can we calculate the perimeter of the garden?” or “Which pieces of information can you use to draw a diagram?”

576

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

= 50 feet

Evaluate the multiplication

Reﬂect and check If we couldn’t remember the formula we could have just written down all of the side lengths and added them together:

Students:10Page 291 + 10 + 15 + 15 = 50 feet

Idea summary To ﬁnd the perimeter of a ﬁgure, add up all of its side lengths. The perimeter of a triangle, with sides a, b, and c has the formula: Ptriangle = a + b + c The perimeter of a parallelogram, with slant height, a, and base, b, has the formula: Pparallelogram = 2 (a + b) The perimeter of a rectangle, with length, l, and width, w, has the formula: Prectangle = 2 (l + w) The perimeter of a square, with side length, l, has the formula: Psquare = 4l

7.03 Perimeter of triangles and parallelograms mathspace.co

Practice

291

Students: Pages 292–295

What do you remember? 1

Find the perimeter of each of these rectangles: a

18 m

4 cm

21 m 6m

6m 21 m 31 m

13 m

7.03 Perimeter of triangles and parallelograms mathspace.co

577

Identify the following triangles as either scalene, isosceles, or equilateral: a

f 5

7 6

h 5

Complete the steps to ﬁnd the perimeter of this triangle:

55 cm

P = 55 + ⬚ + ⬚

48 cm

= ⬚ cm 4

73 cm

Complete the steps to ﬁnd the perimeter of this parallelogram:

P = ⬚ ⋅ (⬚ + ⬚)

= ⬚ cm

6 cm

10 cm

Are these statements always, sometimes, or never true? a

Adjacent sides of a rectangle are equal in length.

The diagonals of a rectangle are congruent.

A rectangle with two equal connected sides is a square.

A rhombus is a parallelogram.

A rectangle with adjacent sides perpendicular is a square.

All sides of a square are different lengths. A

In the parallelogram ABCD, side AB has a length of 6 units. a

Find the other side of ABCD which must have a length of 6 units.

If another side has a length of 4 units, what is the perimeter of the parallelogram?

D C

Let’s practice 7

Find the perimeter of a parallelogram given the dimensions: a

578

Base: 20.5 ft; Slant Height: 7 ft

Base: 2 in; Slant Height: 13 in

Find the perimeter of each triangle: a

A scalene triangle with side lengths 13 in, 17 in and 18 in.

An equilateral triangle with a side length of 4 yd.

An isosceles triangle where the 2 equal side lengths are 13 ft each and the third side measures 4 ft.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Find the perimeter of each triangle: a

9 mm

15 cm 8 cm

9 mm

15 cm

9 mm

12.6 cm

13.5 cm

20.3 cm

10 mm

10 cm

8 cm 10

Find the perimeter of each quadrilateral. Explain your reasoning. a

Base: 4 cm 91°

15 97°

83°

8 97°

83° 15

Which parallelogram has the greater perimeter? Explain how you know. Parallelogram A

Parallelogram B

5m 7.25 m 14.33 m 9m

7.03 Perimeter of triangles and parallelograms mathspace.co

579

Your local community park is creating a triangular picnic area. Find the perimeter of the picnic area.

7 ft

6 ft 8 ft

Nadia wants to build a thin wire frame for a photo that is in the shape of a parallelogram. The frame has a base of 13 cm and a slant height of 6 cm. How much wire will she need to go around the entire photo?

A triangular garden bed has side lengths of garden bed?

You are planning to add a decorative border around your new oﬃce desk, which has a parallelogram shape. The desk has a base measuring 5 ft and a slant height of 3.5 ft. Calculate the total length of the border needed to go around the desk.

The Grand Park in Centerville forms a triangular area with its community center, water fountain, and playground serving as vertices.

inches,

inches, and 20 inches. What is the perimeter of the

The distance between the community center and the water fountain is approximately 3.25 miles. The distance between the water fountain and the playground is about 4.80 miles. The distance between the playground and the community center is roughly 6.25 miles. Estimate the perimeter of the triangular park area. Water Fountain

3.25 mi

Community Center

4.80 mi Playground 6.25 mi

The client wants the four kitchen windows to be framed with a special molding. If the windows are 18 inches by 20 inches, how many inches of molding are needed for the kitchen window?

Hebah is tiling her bathroom with equilateral triangle tiles. She knows the perimeter of each tile, but she wants to ﬁnd the length of each side. If one tile has a perimeter of 36 cm, how long is each side?

Describe and correct the error in ﬁnding the perimeter of the polygon. Perimeter = 12 yd ⋅ 18 yd ⋅ 24 yd = 5184 yd

12 yd 18 yd 24 yd

580

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s extend our thinking 20

A square has the same perimeter as an equilateral triangle. If the triangle has sides of length 8 yd, ﬁnd the side length of the square.

Given the perimeter, P, of the each triangle, ﬁnd the value of the variable: a

21 cm 7m P = 61 cm

P = 29 m

17 cm

10 m

d 5 cm P = 16 cm

d P = 35 cm

7 cm

Find the side length of a triangle that has a perimeter of 18 ft and two side lengths that both measure to be 5 ft.

Find the side length of one of the equal sides of an isosceles triangle that has a perimeter of 22 in and a third side length that measures to be 4 in.

Find the value of x if the perimeter of this parallelogram is 48 m. 8 8x

A quilt is made by sewing together 4 identical parallelograms as shown ﬁgure: If the total perimeter of the quilt is 194 cm calculate the slant height of each parallelogram piece.

54 cm

7.03 Perimeter of triangles and parallelograms mathspace.co

581

Answers 7.03 Perimeter of triangles and parallelograms

12 The perimeter of the triangular picnic area is 6 ft + 8 ft + 7 ft = 21 ft

What do you remember? 1 a 72 m

b 16 cm

c 54 m

2 a Isosceles

b Scalene

c Equilateral d Isosceles

Isosceles

g Equilateral h Isosceles

e Scalene

d 88 m

3 P = 55 + 73 + 48

= 32 cm 5 a Sometimes true

b Always true

c Always true

d Always true

e Sometimes true

+ 20 = 48.25 inches.

17 304 in 18 12 cm

Never true

b 20 units

b 30 in

Let’s extend our thinking

b 12 yd

c 30 ft

20 6 yd

b 38 cm

c 46.4 cm

Let’s practice 7 a 54 ft

30 mm

10 a T he quadrilateral is a rhombus. A rhombus has 4 congruent sides. The perimeter is 4 ⋅ 4 = 16 cm. b T he quadrilateral is a parallelogram. The perimeter is 2 ⋅ (15 + 8) = 2 ⋅ (23) = 46 units.

582

the side lengths: 12.5 +

19 The error in ﬁnding the perimeter is in the operation used. The perimeter is found by adding the lengths of all the sides, not multiplying them. The correct perimeter is 12 yd + 18 yd + 24 yd = 54 yd.

6 a CD

e 28 cm

14 To ﬁnd the perimeter of the triangular garden bed, add all

16 The perimeter of the triangular park area is about 14.30 miles.

4 P = 2 ⋅ (6 + 10)

9 a 27 mm

13 38 cm

15 2 ⋅ (5 ft + 3.5 ft) = 17 ft

= 176 cm

8 a 48 in

11 Parallelogram A has the greater perimeter. The perimeter of Parallelogram A is 2 ⋅ (14.33 m + 5 m) = 38 m, while the perimeter of Parallelogram B is 2 ⋅ (9 m + 7.25 m) = 32 m.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

21 a 12 m 22 8 ft 23 9 in 24 x = 2 cm 25 21.5 cm

b 23 cm

c d = 14 cm d k = 6 cm

7.04 Area of parallelograms Subtopic overview Lesson narrative In this lesson, students will learn to calculate the area of parallelograms. They will start by understanding that a parallelogram is a quadrilateral with two pairs of parallel sides. The lesson explains how the area of a rectangle (A = l × w) relates to the area of a parallelogram (A = b × h), where b is the base and h is the perpendicular height. Using an interactive applet, students will rearrange a parallelogram to form a rectangle, helping them visualize the relationship between the shapes. They will practice using the area formula for parallelograms to solve various problems, including real-world applications. By the end, students should confidently calculate the area of parallelograms using the formula.

Learning objectives Students: Page 296

Key vocabulary 

area



base



perpendicular height



rectangle



parallelogram

Essential understanding A parallelogram has the same area as a rectangle with the same base and height.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal into their instruction by providing students with practice problems that require them to find the area of different types of parallelograms. They can encourage students to apply their understanding of the properties of parallelograms and the area formula to solve these problems. Real-world examples, such as calculating the area of a parallelogram-shaped field, can help them see the practical applications of these mathematical concepts.

7.04 Area of parallelograms mathspace.co

583

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can create mathematical connections by reminding students of their prior knowledge on the classification of quadrilaterals, specifically parallelograms, and their properties. They can also link the concept of area to students’ prior knowledge of finding the area of rectangles and squares and triangles. Further, they can establish the connection between rectangles and parallelograms by emphasizing that rectangles are a specific type of parallelogram. Teachers can help students derive the formula for the area of parallelograms by using concrete manipulatives and visual aids. Start with a rectangle to illustrate the area formula A = lw or

Teachers can integrate this goal by using various methods to represent and describe the mathematical ideas and relationships involved in finding the area of parallelograms. This can include using manipulatives like grid paper or two-dimensional diagrams to help students visualize the area formula, and verbal explanations to describe the process of finding the area. Teachers can guide students to understand that different representations can convey the same mathematical idea, such as the formula for the area of parallelograms.

Content standards 6.MG.2 — The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms. 6.MG.2a — Develop the formula for determining the area of parallelograms and triangles using

pictorial representations and concrete manipulatives (e.g., two-dimensional diagrams, grid paper). 6.MG.2b — Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

Prior connections 5.MG.2 — The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

Future connections 8.MG.5 — The student will solve area and perimeter problems involving composite plane figures, including those in context.

Lesson Preparation Tools You may find these tools helpful: • Scissors • Paper parallelograms

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

584

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Perimeter and area confusion Address student misconceptions A common misconception students may have is confusing the concepts of perimeter and area. They might think that these two measurements are the same or related in a way that they are not. To address this misconception, make sure to clearly define both terms and explain the difference between them. Use visuals and real-world examples to reinforce the distinction. For example, explain that the perimeter of a garden is the distance around it (like a fence), while the area is the space inside it (like the ground where you can plant).

Student lesson & teacher guide Area of parallelograms Students recall the definition of a parallelogram and the formula for the area of rectangle.

Students: Page 296

7.04 Area of parallelograms mathspace.co

585

Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Engage students with physical manipulatives to explore the area of parallelograms. Provide each student with a paper cut-out of a parallelogram. Guide them to cut along a straight line from one corner perpendicular to the base, creating a right-angled triangle. Have them move this triangle to the opposite side of the parallelogram, forming a rectangle. This hands-on activity helps students see that a parallelogram can be rearranged into a rectangle with the same base and height, demonstrating that their areas are equal. Use rulers to measure the base and height of both shapes to reinforce the concept. Representational: Transition to drawing the transformation they performed with the manipulatives. Have students sketch a parallelogram on grid paper, clearly marking the base and the perpendicular height. Instruct them to draw the cut line and use arrows to show how the triangle is shifted to form a rectangle. Encourage them to shade or color-code the corresponding parts to emphasize that the area remains the same. This visual representation bridges the gap between the concrete activity and the abstract concept. Abstract: Introduce the formula for the area of a parallelogram: A = bh, where b is the base and h is the height. Explain that since the parallelogram can be rearranged into a rectangle with the same base and height, the area is calculated the same way. Provide problems where students use the formula to calculate the area of different parallelograms. Present real-world scenarios, like finding the area of a parallelogram-shaped park, to apply the formula in context.

Exploration Students: Page 296

Suggested student grouping: Individual In this exploration, students will use a slider to rearrange a parallelogram into a rectangle. This visual modification will help them understand the relationship between the base and height in a parallelogram and how it relates to the area. They will have to observe the changes and derive a formula for the area of a parallelogram. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What relationships do you notice? I notice that the base and the height of the parallelogram remain constant even when it’s rearranged into a rectangle. This implies that the area remains the same. 2. Write a formula that could be used to find the area of the parallelogram. The formula for the area of a parallelogram is the base times the height, or b ⋅ h.

586

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purposeful questions • When you use the slider to transform the parallelogram into a rectangle, what do you notice about the base and height of both shapes? • What does the rearrangement of the parallelogram into a rectangle tell you about the area of these two shapes? Possible misunderstandings • Students might think that changing a parallelogram into a rectangle necessarily alters its area. They might not understand that rearranging the shape without changing the base, b, and height, h, keeps the area constant. • Students might confuse the base and the height of the parallelogram, thinking that any side or diagonal can be considered as the base or height. They might not understand that the base is a specific side, and the height is the perpendicular distance from this base to the opposite side. After the exploration, students discover that a parallelogram can be decomposed into a rectangle. Since the area of a rectangle is equal to its length multiplied by its width, students will see that the area of the rectangle will be equal to the area of the parallelogram. This means that a parallelogram with the same base and height as a rectangle will have the same area as the rectangle.

Students: Pages 296–297

We know the area of a rectangle is length times width. 4 cm 7 cm 4 cm

We can take the rectangle, cut off one side, and move it to the other side to create a parallelogram. This does not change the size of the ﬁgure so both the rectangle and the parallelogram have the same area. In this case, both the rectangle and the parallelogram have an area of:

7 cm

7 cm ⋅ 4 cm = 28 cm2 We can use the base and perpendicular height of the parallelogram to ﬁnd its area, just like we do for a rectangle.

height base

The area of a parallelogram is found by

A=b⋅h

height

base

height

base

is the base

is the height

The height is always measured perpendicular to the base (at a right angle). Every parallelogram has two base height pairs.

Since we are ﬁnding the product of two lengths, area is always measured in square units.

Example 1 Find the area of this parallelogram.

7.04 13 Area m of parallelograms mathspace.co

587

A=b⋅h

height

Examples

base

is the base

is the height

The height is always measured perpendicular to the base (at a right angle).

height

base

Every parallelogram has two base height pairs.

Students: Page 297

Since we are ﬁnding the product of two lengths, area is always measured in square units.

Example 1 Find the area of this parallelogram.

13 m

Create a strategy Use the formula for the area of a parallelogram: A = bh.

Apply the idea We know b = 13 and h = 8. A=b⋅h

Formula for area of a parallelogram

= 13 ⋅ 8

Substitute b = 13 and h = 8 2

= 104 m

Evaluate

Purpose Show students how to find the area of a parallelogram using the formula.

7.04 Area of parallelograms mathspace.co

297

Expected mistakes Students may use a side length as the height, instead of a perpendicular segment. Remind students that the height must create a right angle with the base. Reflecting with students Emphasize the importance of stating the final answer with appropriate units. An imprecise response might be “A = 104,” whereas a precise response states that the area is measured in square meters. Encourage students to include units of measurement every time they write a numerical value in their calculations to help them see how they have multiplied meters by meters to get square meters. By consistently using units, students develop precision in their mathematical communication and deepen their comprehension of the concepts.

Use color-coding to support visual-spatial processing

use with Example 1

Student with disabilities support To assist students with visual-spatial processing difficulties, use color-coding to highlight the key components of the parallelogram and the area formula. Begin by providing students with a diagram of the parallelogram, labeling the base in one color (e.g., blue) and the height in another color (e.g., green). This visual distinction helps students identify and differentiate these dimensions. 13 m

13 m

588

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

When writing the area formula, match the colors of the variables to the corresponding parts of the diagram—write b in blue and h in green. Substitute the numerical values using the same colors. This consistent color-coding creates a visual link between the diagram and the calculation, aiding comprehension. Encourage students to use colored pencils or markers to replicate this process in their work. This hands-on activity not only reinforces the visual connections but also engages them kinesthetically, enhancing memory retention.

Students: Page 298 Example 2 Find the area of this parallelogram.

11 mm

3 mm

Create a strategy

Example 2 for the area of a parallelogram: A = bh. Use the formula Find the area of this parallelogram.

Apply the idea

We have b = 3 and h = 11. A=b⋅h

Formula for area of a parallelogram

= 3 ⋅ 11

Substitute b = 3 and h = 11

= 33 mm2

Evaluate

11 mm

3 mm

PurposeExample 3 Create a strategy Show students to apply the formula foristhe area of a parallelogram where is measured outside A school how is adding a new athletic field that parallelogram-shaped. The longest side ofthe theheight field measures Use the formula for the area of a parallelogram: A = bh. the polygon. 100 yards and the shortest distance (height) from this side to its opposite side is 60 yards. Calculate the area of the athletic field.

Apply the idea

Expected mistakes We have b = 3 and h = 11. a strategy StudentsCreate may have a hard time recognizing that the dotted line is the height. Remind students that the height is A=b⋅h Formula for area of a parallelogram Start by drawing afrom diagram thentouse the formula for the area of a parallelogram: bh. parallelogram. always perpendicular oneand base another. It does not need to be a side Aof= the = 3 ⋅ 11

Substitute b = 3 and h = 11

= 33 mm the298 idea Students:Apply Page

Evaluate

60 yards

Example 3 A school is adding a new athletic field that is parallelogram-shaped. The longest side of the field measures 100 yards 100 yards and the shortest distance (height) from this side to its opposite side is 60 yards. Calculate the area of the athletic We know b field. = 100 and h = 60. A=b⋅h

Create a strategy

Formula for area of a parallelogram

= 100 ⋅ 60 Substitute b = 100 and h = 60 Start by drawing a diagram2 and then use the formula for the area of a parallelogram: A = bh. Evaluate = 6000 yards

Apply the idea

60 yards

298

Mathspace Virginia SOL Grade 6 mathspace.co

We know b = 100 and h = 60.

100 yards

7.04 Area of parallelograms mathspace.co

589

A school is adding a new athletic field that is parallelogram-shaped. The longest side of the field measures 100 yards and the shortest distance (height) from this side to its opposite side is 60 yards. Calculate the area of the athletic field.

Create a strategy Start by drawing a diagram and then use the formula for the area of a parallelogram: A = bh.

Apply the idea

60 yards

100 yards

We know b = 100 and h = 60. A=b⋅h

Formula for area of a parallelogram

= 100 ⋅ 60

Substitute b = 100 and h = 60

= 6000 yards2

Evaluate

Purpose Students demonstrate that they can apply the formula for the area of a parallelogram to a real-world context. 298

Mathspace

Virginia SOL Grade 6

Reflecting with students mathspace.co Ask advanced learners to consider whether the parallelogram-shaped athletic field could actually be a rectangle. Have students justify their reasoning by comparing the properties of parallelograms and rectangles and determining that the field could indeed be a rectangle. Students should use the definitions (a parallelogram is a quadrilateral with both pairs of opposite sides parallel, and a rectangle is a special type of parallelogram with four right angles) to justify that the field could be a rectangle since a rectangle is a type of parallelogram. Guide them to realize that if all angles are right angles, the parallelogram becomes a rectangle with sides measuring 100 yards and 60 yards. This exercise promotes deeper thinking about geometric definitions and helps students understand how specific shapes relate within broader categories.

Three reads

use with Example 3

English language learner support Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students, “What do you think is happening in this question?” or “Can you explain what this question is about?” On the second read, students should aim to interpret the problem by anwering questions like, “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: • The field is shaped like a parallelogram • The longest side is the base of the parallelogram, which is 100 yards. • The perpendicular height is 60 yards. Students can be prompted by framing these as questions like “How can we find the area of the athletic field?” or “Which pieces of information can you use to draw a diagram?”

590

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 299

Idea summary The area of a parallelogram is found by:

A=b⋅h A is the area of a parallelogram b

is the base of a parallelogram

h is the height of a parallelogram The height is measured perpendicular to the base.

Practice What do you remember? Practice 1

Draw the perpendicular height and mark the right angle for each parallelogram:

a Students: Pages 299–302

What do you remember? 1

Draw the perpendicular height and mark the right angle for each parallelogram: a

Are these statements true or false? a

The area of a parallelogram can be found by multiplying the base by the parallel height.

Parallel sides of a parallelogram are always equal in length.

A rectangle and a parallelogram with the same base and height measurements will always have the same area.

Identify the base and the height needed to ﬁnd the area of each parallelogram: a

4.2 in Are these statements true or false? 3.6 in

10 cm 3 cm

The area of a parallelogram can be found by multiplying the base by the parallel height.

Parallel sides 2ofina parallelogram are always equal in length.

4 cm

A rectangle and a parallelogram with the same base and height measurements will always have the same area. c

2.6 ft

Identify the base and the height needed to ﬁnd the area of each paralleogram: a

3.6 ft

10 cm 10 mm

8 mm

4.2 in 1.8 ft

2 in

3.6 in

3 cm

5.5 mm

4 cm 7.04 Area of parallelograms mathspace.co

299

7.04 Area of parallelograms mathspace.co

591

2.6 ft

3.6 ft 8 mm

1.8 ft

10 mm

5.5 mm

Complete the steps to ﬁnd the area of this parallelogram:

A = ⬚ ⋅ ⬚

= ⬚ yd2

8 yd

6 yd

10 yd

Let’s practice 5

This parallelogram with a base of 9 cm and a height of 6 cm is formed into a rectangle: a

Find the length of the rectangle.

Find the width of the rectangle.

Is the area of the rectangle greater than, less than or equal to the area of the parallelogram? Explain your thinking.

9 cm 6 cm

Redraw this parallelogram as a rectangle.

Find the area of the rectangle.

Find the area of the parallelogram.

If the parallelogram is formed into a rectangle, determine: i

The length of the rectangle

The width of the rectangle

7 cm

iii The area of the rectangle

592

Find the area of the parallelogram.

State the length of the base.

State the length of the height.

What is the size of the angle formed by the base and the height?

Find the area of the parallelogram.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

11.5 cm 8m

Find the area of each parallelogram: a

11 ft

8 ft

d 5 mm 10 mm

13 cm

9 cm

15 cm

18 in

24 cm

15 in

8 cm

19 cm 14 cm

12 cm

Find the area of these parallelograms with these dimensions: a

The base is 5 ft and the perpendicular height is 2 ft.

The base is 15 in and the perpendicular height is 7 in.

Determine whether these could be the dimensions of a parallelogram with an area of 28 mm2: a

Base = 1 mm, height = 28 mm

Base = 7 mm, height = 4 mm

Base = 4 mm, height = 7 mm

Base = 2 mm, height = 28 mm

7.04 Area of parallelograms mathspace.co

593

Find the value of the base or height in each of these parallelograms: a

Area = 36 cm2

Area = 35 m2

b cm 5m 3 cm

Find the base length of a parallelogram whose area is 108 ft2 and perpendicular height is 9 ft.

Complete this table of base and height measurements for three different parallelograms with an area of 60 in2: Base (in) Height (in) Area (in2)

12 60

A parallelogram-shaped sign is being designed for a park with a base of 4.5 feet and a height of 2 feet. Find the area of the sign in square feet.

The ﬂoor of a room measures 8 meters by 10 meters. What is the area of the ﬂoor in square meters if the room is in the shape of a parallelogram?

Kwame is building a ramp for the local skateboard park. The ramp is in the shape of a parallelogram, with a base of 12 feet and a height of 6 feet. What is the area of the ramp?

Let’s extend our thinking 18

A parallelogram has parallel sides of length 7 cm and 5 cm. A rectangle also has parallel sides of length 7 cm and 5 cm. Draw the two quadrilaterals and explain which one will have the larger area.

If you are given two parallelograms with the same base but different heights, explain how you would determine which parallelogram has the greater area.

A parallelogram has a base of 10 cm and a height of 20 cm. a

If 2 cm is added to the base and the height, is the area increased by 2 cm2?

If the base and height are doubled, is the area doubled?

If the base and height are doubled, by what factor has the area increased?

Explain why this number is the increase factor.

Two parallelograms have the same area. If one parallelogram has a base of 8 cm and a height of 4 cm, what are the possible dimensions of the other parallelogram, given the base and height are whole numbers?

A roof in the shape of a parallelogram is being designed for a new house. The roof will be covered with tiles, which cost $10 per square meter. The roof has a base length of 20 meters and a height of 8 meters. a

Find how many square meters of tiles are needed to cover the roof.

How much will it cost to tile the roof?

A jewelry maker is creating parallelogram-shaped pendants for a new collection. Each pendant has a base length of 3 cm and a height of 40 mm. The pendants will be covered with a thin layer of gold that costs $0.50 per square centimeter. If the jewelery maker needs to make 500 pendants, ﬁnd the total cost of covering them with gold.

594

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

8 a Base = 8 m

d Area = 40 m2

c 90°

7.04 Area of parallelograms

9 a 20 m2 2

e 270 mm

What do you remember? 1 a

b Height = 5 m

b 88 mm2

c 117 cm2

d 50 mm2

h 266 cm2

360 cm

g 96 cm

10 a 10 ft2

b 105 in2

11 a Yes

c Yes

b Yes

12 a 12 cm

d No

b 7m

13 12 ft 14

2 a False

b True

c True b • Base : 10 cm

3 a • Base : 2 in • Height: 3.6 in

Base (in)

Height (in)

Area (in2)

15 9 ft2

• Height: 3 cm

16 80 m2

c • Base : 2.6 cm

d • Base : 10 mm

• Height: 1.8 cm

• Height: 8 mm

4 A = 10 ⋅ 6 = 60 yd2

17 The area of the ramp is 12 ft ⋅ 6 ft = 72 ft2. Let’s extend our thinking 18

Let’s practice

7 cm

5 a 9 cm 5 cm

b 6 cm

5 cm

c T he area of the paralellogram is 9 ⋅ 6 = 54 cm while the area of the rectangle is 9 ⋅ 6 = 54 cm2. So the area of the rectangle is equal to the area of parallelogram. 6 a

The rectangle will have the largest area as the height of the parallelogram must be less than 5 cm as the hypotenuse of a right triangle is always the longest side. 19 The formula is A = b ⋅ h. If the base is the same then the larger height will determine the larger area. 20 a No, the area is not increased by 2 cm2.

b No, the area is not doubled. 12

c The area has increased by a factor of 4. d Each side is multiplied by 2 and 4 is equal to 2 ⋅ 2.

b 72 units

21 2 cm ⋅ 16 cm, 1 cm ⋅ 32 cm

c 72 units2 7 a i 11.5 cm ii 7 cm b 80.5 cm2

iii 80.5 cm

22 a 160 m2

b $1600

23 $3000

Answers mathspace.co

595

7.05 Area of triangles Subtopic overview Lesson narrative In this lesson, students will learn to calculate the area of triangles. They will start by understanding that the area of a triangle is half the area of a parallelogram, leading to the formula A = 21 × b × h, where b is the base and h is the height. Using an interactive applet, students will visualize this relationship by rearranging a parallelogram to form two congruent triangles. Students will solve problems involving finding the area of triangles given base and height, real-world applications like calculating areas for construction projects or kites, and determining missing dimensions when given the area and one dimension. By the end, students should confidently calculate the area of triangles using the given formula.

Learning objectives Students: Page 303

Key vocabulary 

area



base



height

 triangle

Essential understanding A parallelogram can always be created by combining two congruent triangles. This allows us to find the area of a triangle by taking half of the area of a parallelogram with the same base and height.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal by guiding students to identify the base and height in different types of triangles and use the formula to solve for the area. Teachers can provide real-world problems such as finding the area of a triangular field, thereby allowing students to apply their mathematical problem-solving skills in a real-world context.

596

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can establish connections with students’ prior knowledge of the area of rectangles and squares to introduce the concept of area for triangles. Additionally, teachers can connect students’ learning from the previous lesson to make connections between the formula for finding the area of a parallelogram and the formula for finding the area of a triangle. They can also connect this learning to real-world contexts, such as showing how the area of triangular shapes like sails or architectural structures can be found using the same formula.

Teachers can guide students to represent the concept of area of triangles using various methods such as sketches, diagrams, and symbolic notation (A = (1/2)bh). They can also demonstrate how these different representations can help in solving real-world problems, such as designing a triangular garden or calculating the area of a triangular plot of land.

Content standards 6.MG.2 — The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.2b — Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

6.MG.2a — Develop the formula for determining the area of parallelograms and triangles using pictorial representations and concrete manipulatives (e.g., two-dimensional diagrams, grid paper).

Prior connections 5.MG.2 — The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

Future connections 8.MG.5 — The student will solve area and perimeter problems involving composite plane figures, including those in context.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 7.04 Area of parallelograms

Tools You may find these tools helpful: • Scissors • Paper rectangles

7.05 Area of triangles mathspace.co

597

Student lesson & teacher guide Area of triangles Exploration Students: Page 303

Suggested student grouping: In pairs Students will be using an applet to manipulate the dimensions of a triangle and its corresponding parallelogram. They will explore the relationship between the area of a triangle and the area of a parallelogram and derive the formulas for the area of a triangle and a parallelogram. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Create several different types of triangles and look at the parallelogram that is created. What do you think the relationship between the area of the triangle and area of parallelogram is? The area of the triangle is always half the area of the parallelogram.

598

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

2. What is the formula for the area of a parallelogram? The area is the base times the height, or A = bh. 3. What do you think the formula for the area of a triangle might be? Since the area of the triangle is always half the area of the parallelogram, the formula for the area of a triangle should be half the base times the height, or Atriangle =

bh.

Purposeful questions • What patterns do you notice about the relationship between the area of a triangle and the area of the resulting parallelogram? • How could the area of a triangle and the area of a parallelogram be related? Possible misunderstandings • Students may confuse the base and height of the triangle with other dimensions or sides of the shape. They might not realize that the height must be perpendicular to the base, leading to incorrect area calculations.

Use code to find the area of a triangle Targeted instructional strategies Incorporate an opportunity for students to practice coding by having them develop a simple program that calculates the area of a triangle. Introduce them to a user-friendly coding platform like Scratch or Python’s beginner modules. Guide them to write code that prompts the user to input the base and height values, performs the calculation, and outputs the area. Encourage students to test and refine their programs by inputting different values and verifying the results. This activity not only reinforces their understanding of the area formula but also demonstrates how coding can automate mathematical calculations and solve problems efficiently. Here’s an example of Python code that calculates the area of a triangle: 1

# Prompt the user to enter the base and height of the triangle

base = float(input(“Enter the base length of the triangle: “))

height = float(input(“Enter the height of the triangle: “))

# Calculate the area using the formula

area = 0.5 * base * height

print(“The area of the triangle is:”, area)

Alternative to using an applet Student with disabilities support Have students start by taking a paper rectangle and discussing how to find the area (base times height). Have the students create two triangles with the rectangle by cutting the rectangle at a diagonal from one vertex to another. Ask students what they think the formula should be for the area of one of the triangles if the area for the rectangle is base times height. Help them make the connection that because the triangle is rectangle, then the area formula for the triangle should be height divided by two.

of the

times the base and height or the base times the

Students may struggle to multiply the base and height by . You can explain to students that multiplying by

the same as dividing by 2. So they can find the area of a triangle by dividing the product of the base and height by 2.

7.05 Area of triangles mathspace.co

599

External height Address student misconceptions Students may think that the perpendicular height will always be inside of the triangle or be a side length of the triangle. Work through several examples where the height of the triangle cannot be found within the triangle or as a side length. Students explore how to derive the formula for the area of a triangle by realizing that cutting a parallelogram along a diagonal creates two congruent triangles, each with half the parallelogram’s area. This leads to the formula A = where b is the base and h is the height of the triangle.

Students: Page 303

Examples Students: Page 304 Example 1 Find the area of the triangle shown.

7 cm

10 cm

Create a strategy Use the area of a triangle formula.

Apply the idea Use the formula Substitute b = 10 and h = 7 Evaluate

600

Example 2 SOL Grade 6 Teacher Edition Mathspace Virginia mathspace.co Alex is making a kite shaped like a triangle. He plans to use a stick that is 15 cm long for the height, and he wants the base of the kite to be 20 cm. What is the minimum amount of material Alex needs to buy?

bh,

Purpose Students demonstrate that they can use the formula to find the area of a triangle. Expected mistakes Students might not include the units in their solution, or they might not square the units. Show students that they multiplied two lengths together, so the result will be squared units. Reflecting with students Ask students if the order in which they multiply the lengths matters. Help them to connect that the commutative property allows for the dimensions to be multiplied in any order with . Example 1 Find the area of the triangle shown. Critique, correct, and clarify

use with Example 1

English language learner support Present students with an incorrect solution such as: Area = base ⋅ height = 10 cm ⋅ 7 cm = 70 cm2

7 cm

Ask students to work in pairs to critique this solution by identifying any errors they notice. Encourage them to discuss why multiplying the base and height gives the area of a rectangle, not a triangle. Support students in 10 cm correcting the error by explaining that the area of a triangle is half the area of a rectangle with the same base and height, so the correct formula is Create a strategy

Area=

Use the area of a triangle formula.

⋅ base ⋅ height

Have students clarify Apply the idea the correct steps and recalculate the area together. This activity helps students understand the importance of using Use the the correct formula and reinforces the mathematical vocabulary related to formula area and geometric shapes. Substitute b = 10 and h = 7

Students: Page 304

Evaluate

Example 2 Alex is making a kite shaped like a triangle. He plans to use a stick that is 15 cm long for the height, and he wants the base of the kite to be 20 cm. What is the minimum amount of material Alex needs to buy?

Create a strategy We need to calculate the area of the kite.

Apply the idea Formula for area of a triangle Substitute b = 20 and h = 15 Evaluate the multiplication

Idea summary

Purpose The formula for the area of a triangle is: Show students how to find the area of a triangle in a real-world context. Reflecting with students Emphasize the importance of stating the final answer withof appropriate A Area the triangle units. Remind them that including units is a critical aspect of conveying their mathematical An imprecise response might be b reasoning Base of theaccurately. triangle “A = 150,” which does not accurately convey the area of the triangle. By consistently using units, students h Height of the triangle develop precision in their mathematical communication and deepen their comprehension of the concepts. 304

Mathspace Virginia SOL Grade 6 mathspace.co

7.05 Area of triangles mathspace.co

601

Substitute b = 10 and h = 7 Evaluate

Advanced learners: Design kites of different shapes and calculate the area Exampleinstructional 2 Targeted strategies

use with Example 1

Alexstudents is making atokite shaped various like a triangle. He plans to a stick that is 15 cm long for the height, he wantsor the Encourage explore kite designs byuse choosing different shapes such as and diamonds base of of thetriangles kite to be and 20 cm. What is the minimum to buy? for their kites and calculate combinations quadrilaterals. Inviteamount them of tomaterial select Alex the needs dimensions the area for each design to determine the minimum amount of material required.

Create a strategy

This exercise allows students to apply area formulas for different geometric shapes and to extend their thinking We need to calculate the area of the kite. by finding the area of composite shapes. By allowing students to incorporate their own interests and creativity the idea into the Apply problem, they engage more deeply with the mathematical concepts and see real-world applications of their learning. Formula for area of a triangle Substitute b = 20 and h = 15

Students: Page 304

Evaluate the multiplication

Idea summary The formula for the area of a triangle is:

304

Area of the triangle

Base of the triangle

Height of the triangle

Mathspace Virginia SOL Grade 6 mathspace.co

Practice Students: Pages 305–308

What do you remember? 1

Identify the base and height of each of these triangles: a

3 cm

36 mm

5.3 cm 2.7 cm

20 mm

15 mm

22 mm

6 cm

8 cm

6.4 cm

10 cm

12 cm

19.5 cm

29 cm 14.5 cm 12.5 cm

602

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Consider the following triangle: a

Which dimension of the rectangle forms the height of the shaded triangle?

What fraction of the rectangle is the shaded triangle?

width

length

Look at the model: a

Use the model to ﬁnd the area of the square.

Use the area of the square to ﬁnd the area of the right triangle.

Complete the steps to ﬁnd the area of this triangle: Area =

⋅⬚⋅⬚

4 ft

10 ft

For each of the following ﬁgures: i

Find the area of the entire rectangle.

Find the area of the shaded triangle.

5 cm 8 cm 10 cm

3 cm

7.05 Area of triangles mathspace.co

603

Let’s practice 6

Find the area of the following triangles: a

8.2 in

9.4 in

12 m

7.7 ft

6 cm

7 cm

10.8 ft

8 cm

10 m

6 cm

h 2 mm

3 mm

Find the area of a triangle which has the following dimensions: a

Height = 8 in and base = 9 in.

Height = 4 yd and base = 6 yd.

Height = 7 mm and base = 8 mm.

Height = 3 cm and base = 9 cm.

Determine whether the following could be the dimensions of a triangle with an area of 20 in2: a

Base = 8 in, height = 5 in.

Base = 1 in, height = 20 in.

Base = 5 in, height = 8 in.

Base = 2 in, height = 20 in.

A farmer wants to install a triangular shade sail in her paddock to protect her sheep from the sun. The base of the shade sail is 12 ft long, and the height of the triangle is 8 ft. What is the area of the triangular shade sail in square feet?

A civil engineer needs to design a triangular traﬃc island for a new road intersection. The base of the triangle is 20 ft long, and the height of the triangle is 15 ft. Calculate the area of the triangular traﬃc island in square feet.

604

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

An artist is painting a triangular canvas with a base of 80 cm and a height of 50 cm. Calculate the area of the canvas that needs to be painted.

Ambrose is retiling half of his kitchen floor which is in the shape of a rectangle. The remaining part of the floor is roughly in the shape of a triangle, as shown. The floor is 22 ft long, and 12 ft wide. Each square tile is 1 ft × 1 ft. How many square feet of flooring does Ambrose need to buy?

Describe and correct the error in calculating the area of the triangle shown. 15 ft

12 ft

18 ft

For each of the following triangles, ﬁnd the value of b or h given the area: a

Area = 20 mm2

Area = 120 mm2

b mm

5 mm

10 mm

b mm

Area = 12 mm2

6 mm

Area = 48 m2

b mm 4m

Area = 54 cm2

Area = 120 cm2 h cm

6 cm

h cm

10 cm

7.05 Area of triangles mathspace.co

605

Complete this table of base and height measurements for four different triangles, which all have an area of 30 m2: Base (m) Height (m) Area (m2)

10 12 30

15 30

2 30

A gutter running along the roof of a house has a cross-section in the shape of a triangle as shown: If the area of the cross-section is 40 cm2, and the length of the base of the gutter is 10 cm, ﬁnd the perpendicular height h of the gutter.

h cm

10 cm

Explain how the area of a triangle is connected to the area of a rectangle using this diagram: 5m

Let’s extend our thinking 18

The area of a triangle measures 45 m2. Find all the possible dimensions of the triangle.

You know the height and perimeter of an equilateral triangle. Explain how to ﬁnd the area of the triangle. Draw a diagram to support your reasoning.

If the base of a triangle is 16 cm and the height is two times the base, what is the area of the triangle?

A triangle has a height of 32 cm and a base that is

as long as the height. What is the area of the triangle?

A triangle has a base of 45 cm and a height that is

as long as the base. What is the area of the triangle?

Edith wants to place a triangular ﬂower garden in the front of the house to increase curb appeal.

606

What could be the dimensions of the garden if she has a space of 16 square feet to work with?

The longest side in the triangular ﬂower garden will be against the house. Edith wants to place decorative bricks around the other two sides of the garden. How many feet of brick does she need to purchase?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

7.05 Area of triangles

c b = 12 cm, h = 6.4 cm

b b = 36 mm, h = 15 mm d b = 19.5 cm, h = 12.5 cm

2 a Width

3 a Area = 36 units2

b Area = 18 units2

4 Area =

ii 25 cm2

b i 24 cm2

ii 12 cm2

Height (m)

Area (m2)

17 The rectangle can be cut into two equal right triangles. Find the area of the rectangle and halve it to ﬁnd the area of the triangle.

18 b = 1 m, h = 90 m

b = 5 m, h = 18 m b = 6 m, h = 15 m b = 9 m, h = 10 m

Let’s practice b 38.54 in2 40 m2

c 21 cm2

d 41.6 ft2

yd2

h 3 mm2

7 a 36 in2

b 12 yd2

c 28 mm2

d 13.5 cm2

8 a Yes

b No

c Yes

d Yes

19 To find the area of an equilateral triangle when you know its height and perimeter, first, use the perimeter to determine the length of one side. Since an equilateral triangle has three equal sides, divide the perimeter by 3 to find the length of one side. Next, use the formula A =

10 150 ft2 11 2000 cm2

20 256 cm2

12 132 ft2

21 128 cm2

13 The error in calculating the area of the given triangle is that slant height of 18 ft was used instead of the height of 15 ft.

22 337.5 cm2

Calculating the correct area:

e h = 18

⋅ base ⋅ height.

Here, the base is the length of one side of the triangle, which you found from the perimeter, and the height is given.

9 48 ft2

14 a b = 8 mm

b = 3 m, h = 30 m

5 a i 50 cm2

b = 2 m, h = 45 m

Area = 20

e 24 cm2

Let’s extend our thinking

⋅ 10 ⋅ 4

6 a 30 m2

16 h = 8 cm

What do you remember? 1 a b = 6 cm, h = 2.7 cm

Base (m)

b b = 24 f

h = 24

c b=4

d h = 24 m

23 a O ne possible set of dimensions for the triangular garden, given an area of 16 square feet, could be with edge lengths of 7.69 feet, 6.73 feet, and 4.81 feet. b G iven the possible set of dimensions in the previous question, adding the two shortest sides of 6.73 feet and 4.81 feet would give us the number of feet of brick to be purchased, which is 11.54 feet.

Answers mathspace.co

607

7.06 Polygons in the coordinate plane Subtopic overview Lesson narrative In this lesson, students will learn to plot and analyze polygons on the coordinate plane. They will start by plotting points and connecting them to form various polygons. The lesson includes calculating the side lengths of polygons using the distance between points and understanding the concepts of horizontal and vertical distances. Students will practice determining the perimeter and area of these polygons. By the end, students should confidently plot polygons, calculate distances, and find perimeters and areas of polygons on the coordinate plane

7.06 Polygons in the coordinate Learning objectives plane Students: Page 309

After this lesson, you will be able to... • solve mathematical and contextual problems involving the perimeter and area of parallelograms in the coordinate plane. • draw polygons in the coordinate plane given coordinates for the vertices. • use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. • solve real-world and mathematical problems involving finding the distance between two points and drawing polygons on the coordinate plane

Polygons in the coordinate plane Remember that the coordinate plane can be used to describe the location of points in a 2D space. Key vocabulary 

area



line segments



coordinate plane



coordinates

 horizontal



perimeter



polygon

 vertices

By connecting 3 or more points on the coordinate plane with line segments, we can plot polygons. Plotting polygons on the −4 −3 −2 −1 1 2 3 4 coordinate plane will allow us to easily determine lengths and −1 distances without needing ruler. to find the side lengths of the The coordinates of the vertices of a polygon in the coordinate plane can beaused 1

Essential understanding −2

polygon.

−3 −4

Standards

Using the points A(−1, 1), B(3, 1), C(3, 3), and D(−1, 3) we can draw quadrilateral ABCD. We can calculate the side This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. lengths of the ABCD using the ordered pairs.

4 3 2

608

The length of , which is horizontal, can be found by subtracting the x-coordinates of A and B. Because distance is always positive, we will take the absolute value. = ∣3 − (−1)∣ = 4

1 D C x , which is vertical, can be found by subtracting the Mathspace Virginia SOL Grade 6 Teacher Edition The length of −4 −3 −2 −1 1 2 3 4 y-coordinates of B and C and taking the absolute value. mathspace.co −1 −2 −3

= ∣3 − 1∣ = 2

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG4 — Mathematical Connections

Teachers can integrate this goal by presenting real-world scenarios related to the coordinate plane. For instance, teachers can provide a context that involves plotting vertices of polygons on the coordinate plane, such as mapping a park or a field. Students will then apply mathematical concepts learned to solve these problems. Additionally, they can create problems that require students to calculate the length of a side or the perimeter of polygons using the coordinates of the vertices.

In line with the lesson, teachers can show students how the concept of the coordinate plane is connected to their prior knowledge of graphing ordered pairs and plotting points. They can also link the idea of finding the length of a side and the perimeter of polygons to earlier lessons on calculating perimeters of triangles and parallelograms. Further, teachers can create connections to real-world situations, such as mapping out a route or designing a layout for a garden, to demonstrate the practical applications of these mathematical concepts.

MPG3 — Mathematical Reasoning Teachers can enhance students’ understanding of polygons in the coordinate plane through mathematical reasoning by addressing common misconceptions and emphasizing the correct method for determining distances. Begin by having students draw polygons using given coordinates and trace the length of line segments with colored pencils to count the units between points. This visual and hands-on approach helps students understand the concept of absolute value in determining distances. Encourage them to associate these lengths with the number of units each point is from the axis to reinforce accurate calculations of side lengths.

MPG5 — Mathematical Representations Teachers can promote this goal by teaching students to represent the vertices of polygons and their perimeters on the coordinate plane using various methods. This could include graphical (drawing on the coordinate plane), tabular (listing the ordered pairs and side lengths), or algebraic (calculating the length of a side or the perimeter). These different representations can help students understand the same mathematical idea in various ways and help them visualize and solve the contextual problems presented.

Content standards 6.MG.2 — The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.3e — Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers.

6.MG.2b — Solve problems, including those in context, involving the perimeter and area of triangles, and 6.MG.3f — Draw polygons in the coordinate plane parallelograms. given coordinates for the vertices; use coordinates to determine the length of a side joining points with the 6.MG.3 — The student will describe the characteristics same first coordinate or the same second coordinate. of the coordinate plane and graph ordered pairs. Ordered pairs will be limited to coordinates expressed as integers. Apply these techniques in the context of solving contextual and mathematical problems.

Prior connections 5.MG.2 — The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

7.06 Polygons in the coordinate plane mathspace.co

609

Future connections 8.MG.5 — The student will solve area and perimeter problems involving composite plane figures, including those in context.

7.MG.4 — The student will apply dilations of polygons in the coordinate plane. 8.MG.3 — The student will apply translations and reflections to polygons in the coordinate plane.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 4.05 Integers in the coordinate plane

Student lesson & teacher guide Polygons in the coordinate plane Students learn how to use the coordinate plane to plot points and connect them to form polygons, such as triangles and quadrilaterals. They calculate side lengths using differences between coordinates, enabling them to find the perimeter and area of polygons like rectangles. This reinforces concepts of geometry and the connection between algebraic and spatial reasoning.

Students: Pages 309–310

7.06 Polygons in the coordinate plane After this lesson, you will be able to... • solve mathematical and contextual problems involving the perimeter and area of parallelograms in the coordinate plane. • draw polygons in the coordinate plane given coordinates for the vertices. • use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. • solve real-world and mathematical problems involving finding the distance between two points and drawing polygons on the coordinate plane

Polygons in the coordinate plane Remember that the coordinate plane can be used to describe the location of points in a 2D space. 4

3 2

610

By connecting 3 or more points on the coordinate plane with 1 Mathspace Virginia SOL Grade 6 TeacherxEdition line segments, we can plot polygons. Plotting polygons on the mathspace.co−4 −3 −2 −1 1 2 3 4 coordinate plane will allow us to easily determine lengths and −1 distances without needing a ruler. −2

−3

drawing polygons on the coordinate plane

Polygons in the coordinate plane Remember that the coordinate plane can be used to describe the location of points in a 2D space. 4

3 2 1

−4 −3 −2 −1 −1

−2

By connecting 3 or more points on the coordinate plane with line segments, we can plot polygons. Plotting polygons on the coordinate plane will allow us to easily determine lengths and distances without needing a ruler.

−3 −4

Using the points A(−1, 1), B(3, 1), C(3, 3), and D(−1, 3) we can draw quadrilateral ABCD. We can calculate the side lengths of the ABCD using the ordered pairs.

The length of , which is horizontal, can be found by subtracting the x-coordinates of A and B. Because distance is always positive, we will take the absolute value.

3 2

= ∣3 − (−1)∣ = 4

−4 −3 −2 −1 −1

C 1

x 4

, which is vertical, can be found by subtracting the The length of y-coordinates of B and C and taking the absolute value. = ∣3 − 1∣ = 2

−2 −3 −4

We can use these side lengths to calculate perimeter and area from polygons on the coordinate plane. Recall, the perimeter of a rectangle can be found by adding up all of the side lengths, or using the formula P = 2l + 2w and here l = 4 units and w = 2 units. So: P = 2(4) + 2(2) = 8 + 4 = 12 units 7.06 Polygons in the coordinate plane mathspace.co

309

Examples Students: Page 310

7.06 Polygons in the coordinate plane mathspace.co

611

Purpose Students demonstrate that they can correctly identify the coordinates of the vertices of the polygon graphed on the coordinate plane. Reflecting with students Ask students to note the things that are in common between points A and D and between points B and C. Ask them to consider whether this is always true for the coordinates of vertical lines.

Advanced learners: Find the area of the quadrilateral using coordinate geometry Targeted instructional strategies

use with Example 1

Challenge advanced learners to calculate the area of the quadrilateral on the coordinate plane. Students may decompose the quadrilateral into shapes with areas they can easily find, such as rectangles and right triangles, using the grid lines to measure lengths directly. They might instead count the units along the axes to find the base and height for the parallelogram. This activity allows students to explore how coordinates relate to geometric measurements and deepens their spatial reasoning. By engaging in this extension, students will make valuable connections between coordinate geometry and area calculation, enhancing their problem-solving skills.

Enhancing visual clarity with enlarged graphs and bold gridlines

use with Example 1

Student with disabilities support For students who struggle with visual-spatial processing, providing enlarged graphs with bold, clearly marked gridlines and axes can significantly aid in identifying the coordinates of points. Use larger printouts of the coordinate plane, ensuring that the numbers on the axes are prominent and easily readable. Bold gridlines can help students visually track horizontal and vertical movements more effectively. Encourage students to use their fingers or a straight edge to follow the gridlines from the point to the axes, helping them accurately determine the x-coordinate and y-coordinate. By modifying the visual presentation in this way, you make the task of identifying coordinates more accessible, reducing confusion and enhancing comprehension for your students.

612

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 310

Purpose Students demonstrate that they can plot ordered pairs with integer coordinates on the coordinate plane to form a polygon. Reflecting with students Ask students what shape they have plotted as accurately as possible. Have students explain why the triangle is not an isosceles right triangle.

Students: Page 311 b What is the length of AB?

Create a strategy For points with the same x-coordinates, we can ﬁnd the distance by subtracting the absolute values of the y-coordinates.

Apply the idea The y-coordinate of A is 7, and the y-coordinate of B is −9. AB = ∣7 − (−9)∣

Find the difference of the two y-coordinates

= ∣16∣

Evaluate the subtraction

= 16

Evaluate the absolute value AB = 16 units

Reﬂect and check Another strategy is to count the number of spaces between two points on the coordinate plane. For points with the same x-coordinates, we can count the vertical spaces between the points. For points with the same y-coordinates, we can count the horizontal spaces between the points.

c Find the length of BC

Purpose Create a strategy Students demonstrate that they can find the length of segment of a polygon graphed on the coordinate plane. For points with the same y-coordinates, we can ﬁnd the distance by subtracting the absolute values of the x-coordinates.

Apply the idea The x-coordinate of B is 7, and the x-coordinate of C is −6. BC = ∣7 − (−6)∣

Find the difference of the two x-coordinates

= ∣13∣

Evaluate the subtraction

= 13

Evaluate the absolute value

7.06 Polygons in the coordinate plane mathspace.co

613

AB = ∣7 − (−9)∣

Find the difference of the two y-coordinates

= ∣16∣

Evaluate the subtraction

= 16

Evaluate the absolute value AB = 16 units

b What is the length of AB?

Reflecting with students Reﬂect and check Encourage students to think of another strategy that could be used to find the distance, like counting. Ask Create a strategy strategy is totell count of spaces between two points on the coordinate plane. For points with the studentsAnother how they could bythe justnumber looking at the the ordered pairs that ABof was For points with the same x-coordinates, we can ﬁnd coordinates the distance byofsubtracting the absolute values the a vertical line.

same x-coordinates, we can count the vertical spaces between the points. y-coordinates. points311 with the same y-coordinates, we can count the horizontal spaces between the points. Students:ForPage

Apply the idea The y-coordinate is 7, and the y-coordinate of B is −9. c Find the lengthofofABC AB = ∣7 − (−9)∣ Find the difference of the two y-coordinates = ∣16∣ Create a strategy

Evaluate the subtraction

16 same y-coordinates, Evaluate the absolute value by subtracting the absolute values of the For points with=the we can ﬁnd the distance x-coordinates. AB = 16 units

Apply the idea Reﬂect and check

The x-coordinate of B is 7, and the x-coordinate of C is −6. Another strategy is to count the number of spaces between two points on the coordinate plane. For points with the BC = ∣7 − (−6)∣ Find the difference of the two x-coordinates same x-coordinates, we can count the vertical spaces between the points. = ∣13∣ Evaluate the subtraction For points with the same y-coordinates, we can count the horizontal spaces between the points. = 13 Evaluate the absolute value BC = 13 units

c Find the length of BC d Find a the area of △ABC Create strategy

PurposeFor points with the same y-coordinates, we can ﬁnd the distance by subtracting the absolute values of the Create a strategy x-coordinates. Students demonstrate that they can find the distance between points that line up horizontally on the coordinate Use BC for the base and the length of AB for the height of the triangle. Then, use the area of a triangle formula plane. Apply A = ⋅ the b ⋅ h.idea The x-coordinate of B is 7, and the x-coordinate of C is −6.

Reflecting with students = ∣7 − (−6)∣ Find the difference of the two x-coordinates Apply theBC idea Ask students how they could tell by just looking at the coordinates that side BC of the polygon would be Evaluate the subtraction b = BC = 13 = ∣13∣ horizontal. h = AB = 16

= 13

Students: Page 311

Evaluate the absolute value

BC = 13 units Formula for area of a triangle Substitute b = 13 and h = 16

d Find the area of △ABC

Evaluate the multiplication

Create a strategy Use BC for the base and the length of AB for the height of the triangle. Then, use area of triangle formula 7.06the Polygons in a the coordinate plane A=

311

mathspace.co

⋅ b ⋅ h.

Apply the idea b = BC = 13 h = AB = 16 Formula for area of a triangle Substitute b = 13 and h = 16 Evaluate the multiplication

7.06 Polygons in the coordinate plane mathspace.co

Purpose Students demonstrate how to calculate the area of a triangle using given points in a coordinate plane.

614

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

311

Absolute value and distance

use with Example 2

Address student misconceptions Students may just add or subtract the coordinates of the points to find the distance between them. They may forget to use the absolute value of the coordinates or not understand why the absolute value is important. Discuss why distance is always positive and how that relates to absolute value. Students may also double-check the distance between two points on a segment by counting the spaces.

Students: Page 312 Example 3 Consider the square LMNO L

3 2 1 −4 −3 −2 −1 −1 O

x 1

2 N

−2 −3 −4

a Find the perimeter of LMNO.

Create a strategy Use the lengths LM, MN, OR, or LO to ﬁnd the side length of the square. Then either add up the four sides or use the perimeter of a square formula P = 4l.

Apply the idea Let’s use MN to ﬁnd the length of the side of the square LMNO. This is vertical line segment so we need to ﬁnd the absolute value of the difference between the y-coordinates. MN = ∣4 − (−1)∣ = 5 Now, we can calculate the perimeter of the square. P=4⋅l

Use the perimeter formula for a square

=4⋅5

Substitute l = 5

= 20

Evaluate

b Find the area of LMNO.

Purpose Create a strategy Show students how to use coordinates to find the length of a side of a square and use this to calculate the Use the side length of the square found in part (a) with the formula for area of a rectangle A = bh or the formula perimeter. speciﬁcally for a square A = s2, where s is the side length. Expected mistakes Apply the idea StudentsIn may forget to take theofabsolute when finding thewedifference between the part (a) we found the side the squarevalue is 5 units in length. Now, can calculate the area of they-coordinates. square. They may also forget 2 that the perimeter of a square is 4 times the length of one side. Formula for area of a square A=s = 52

Substitute s = 5

= 25

Evaluate the exponent

7.06 Polygons in the coordinate plane mathspace.co 312

Mathspace Virginia SOL Grade 6 mathspace.co

615

MN = ∣4 − (−1)∣ = 5 Now, we can calculate the perimeter of the square. P=4⋅l

Use the perimeter formula for a square

=4⋅5

Substitute l = 5

= 20 Students: Page 312

Evaluate

b Find the area of LMNO.

Create a strategy Use the side length of the square found in part (a) with the formula for area of a rectangle A = bh or the formula speciﬁcally for a square A = s2, where s is the side length.

Apply the idea In part (a) we found the side of the square is 5 units in length. Now, we can calculate the area of the square. A = s2

Formula for area of a square

= 52

Substitute s = 5

= 25

Evaluate the exponent

Purpose Show students how to calculate the area of a square when the length of the sides is known.

Compare and connect

use with Example 3

English language learner support 312

Mathspace

Virginia SOL Grade 6

mathspace.co Encourage students to compare the methods of finding the perimeter and area of polygons on the coordinate plane with those off the coordinate plane. Begin by presenting two similar polygons: one plotted on the coordinate plane (like square LMNO) and another drawn without a coordinate grid.

Facilitate a discussion by asking, “What steps do we take to find the side lengths of a polygon on the coordinate plane versus one off the coordinate plane?” Guide students to recognize that on the coordinate plane, they can calculate side lengths using the distance between coordinates, while off the coordinate plane, they rely on given measurements or measuring tools. Ask students to share how they calculate perimeter and area in each case and to identify any similarities or differences in their approaches. Highlight key vocabulary such as “coordinates,” “difference,” “measurement,” and “units squared.” By connecting these concepts, students can deepen their understanding of geometric measurements and enhance their mathematical language proficiency in both contexts.

Students: Page 313

Idea summary To find the distance between two points with the same x-coordinates, subtract the y-coordinates and then find the absolute value of the difference. The same is true for points with the same y-coordinates. Subtract the x-coordinates and then find the absolute value of the difference. We can also find the distance between points that share an x- or y-coordinate, by counting the number of spaces between them on the coordinate plane.

Practice What do you remember? 1

616

Match each description to the correct quadrant: a

The x-value is negative and the y-value is positive.

Quadrant I

The x-value and y-value are both positive.

Quadrant II

The x-value is positive and the y-value is negative.

iii

Quadrant III

The x-value and y-value are both negative. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 2 Describe how the point P moves on the coordinate plane from the origin.

Quadrant IV

y 5 P 4

5 4

Practice Students: Pages 313–317

What do you remember? 1

Match each description to the correct quadrant: a

The x-value is negative and the y-value is positive.

Quadrant I

The x-value and y-value are both positive.

Quadrant II

The x-value is positive and the y-value is negative.

iii

Quadrant III

The x-value and y-value are both negative.

Quadrant IV

Describe how the point P moves on the coordinate plane from the origin. a

y 5 P 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

10 8 6 4 2

x 2 4 6 8 10

5 4 3 2 1

x P 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

P −10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

Describe how you would move on the coordinate plane from (−1, 1) to plot the point (9, 10).

7.06 Polygons in the coordinate plane mathspace.co

617

For each of the following polygons, write the coordinates of the vertices: a

y 10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 Y −10

T A x

−10−8 −6 −4 −2 −2 G −4 −6 −8 −10

y A

10 8 6 4 2

2 4 6 8 10 B

y S

−10−8 −6 −4 −2 −2 −4 U −6 −8 − −10

2 4 6 8 10

10 8 6 H 4 2

10 8 6 4 2

−10−8 −6 −4 −2 −2 C −4 −6 −8 −10

2 4 6 8 10

618

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 C −10

B x

2 4 6 8 10

10 8 6 4 2

2 4 6 8 10

y A E x

−10−8 −6 −4 −2 −2 −4 −6 C −8 −10

2 4 6 8 10

Using the coordinate plane, choose the two correct statements about the points’ distances from the x or y-axis: A

Point H is 5 units from the x-axis and 11 units from the y-axis.

Point J is 8 units from the x-axis and 9 units from the y-axis.

Point K is 4 units from the y-axis and lies on the x-axis.

Point M is 7 units from the x-axis and 4 units from the y-axis.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

J H

y 10 8 6 S 4 2

Q −10−8 −6 −4 −2 −2 K −4 −6 M −8 −10

2 4 6 8 10 I R

Find the length of a

in the following polygons:

y 8 7 6 5 4 3 2 1

B −5 −4 −3 −2 −1 −1 −2

7 6 5 4 3 2 1

B −5 −4 −3 −2 −1 −1 −2 −3

y C

−10−8 −6 −4 −2 −2 −4 −6 −8 B −10

C x 1 2 3 4 5

y C

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 C −8 −10

x 1 2 3 4 5

2 4 6 8 10

y A

x 2 4 6 8 10 D

Let’s practice 7

Relate the coordinates of each point to the distance from the x-axis or y-axis: a

If point P has an x-coordinate of 0, which axis must it lie on?

If point Q has a y-coordinate of 0, which axis must it lie on?

If point R is located at (−7, 2), what is the distance of R from the y-axis?

If point S is located at (−8, −5), what is the distance of S from the x-axis?

Which point is furthest from the x-axis?

Which point is further from the y-axis?

(0, −3)

(5, 0)

(0, 6)

(−4, 0)

Consider the point R plotted on the coordinate plane. a

Find the coordinates of the point that is 3 units to the right of point R.

Find the coordinates of the point that is 4 units below point R.

Find the coordinates of the point that is 2 units to the left and 1 unit above point R.

Find the coordinates of the point that is 7 units to the right and 5 units below point R.

6 5 4 3 2 1

−6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

7.06 Polygons in the coordinate plane mathspace.co

619

Find the distance between the following pairs of points: a

A (−5, 8) and B (−2, 8)

A (7, 3) and B (−1, 3)

A (6, −5) and B (6, −1)

A (−5, 7) and B (−2, 7)

A (−9, 3) and B (−1, 3)

A (−5, −6) and B (4, −6) h

A (−6, 2) and B (−6, −7) A (−8, 5) and B (1, 5)

Consider the point plotted on the coordinate plane. a

Plot a point on the graph that has the same x-coordinate as point F. Label the point G.

Plot another point on graph that has the same y-coordinate as point F. Label the point H.

Describe the distance between point F and point G.

Describe the distance between point F and point H.

Which two points are along the same horizontal line?

Which two points are along the same vertical line?

6 5 4 3 2 1

−6−5−4−3−2 −1 −1

1 2 3 4 5 6

−2 −3 −4 −5 −6

Consider the points: A (−4, 8), B (−7, 8) and C (−7, 1) a

Plot the points on a coordinate plane.

Find the length of

Draw lines to connect the vertices of each point.

Find the length of

Jaime travels from Beach A to Beach C through Beach B. The coordinates are plotted on the following coordinate plane:

6 5 4 3 2 1

If each unit represents 10 meters, ﬁnd the total distance that Jaime traveled.

−6−5−4−3−2 −1 −1

Look at the points graphed in the coordinate plane. Describe the distance between: a

Point A and Point F

Point C and Point D

Point B and Point E

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

10 8 6 4 2

−10−8 −6 −4 −2 2 4 6 8 10 −2 F A −4 −6 −8 E B −10

620

A rectangular garden has vertices with the coordinates of (−6, 7), (7, 7), (7, −9) and (−6, −9). a

Plot the rectangular garden on a coordinate plane.

If each unit represents 1 foot, ﬁnd the perimeter of the garden.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A triangle has points A (1, 2), B (−2, −3) and C (6, −3). a

Plot the triangle ABC on a coordinate plane.

Find the perpendicular height of the triangle if

Find the length of base

is the base.

Consider the quadrilateral with the points A (−3, −4), B (5, −4), C (5, 4) and D (−3, 4). a

Plot the quadrilateral ABCD on a coordinate plane.

Find the length of the following sides: i

iii

State the type of the quadrilateral ABCD. Explain your thinking.

Let’s extend our thinking 18

The points given represent three vertices of a parallelogram. Find the coordinates of the fourth vertex if it is known to be in the second quadrant.

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

The points given represent two of the three vertices of a triangle. a

State three possible coordinates of point C if the three points form an isosceles triangle.

State four possible coordinates of point C if the three points form a right-angled triangle.

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 B −5

1 2 3 4 5

x 1 2 3 4 5

The coordinates of the three vertices of a rectangular farm in a topographic map are (4, 5), (−3, 5) and (−3, 4). a

Find the coordinates of the missing vertex.

Plot the farm on a coordinate plane.

The coordinates are measured in kilometers. Find the area of the farm.

The farmer is going to seed the farm with alfalfa grass. Each bag of seed will cover 9 square kilometers, how many bags of seed will the farmer need?

If each bag of seed costs $32.97, what is the total cost to seed the farm with alfalfa grass?

7.06 Polygons in the coordinate plane mathspace.co

621

You are a landscape architect designing a small neighborhood park. This park will be situated on a plot of land represented by a coordinate plane, with a total area of 2000 square units. Your design plan must include: • A square ﬂower garden that occupies an area of 200 square units. • A triangular bird-watching area with an area of 300 square units. • A rectangular relaxation zone covering an area of 400 square units. In the coordinate plane, outline each section with labels. Determine the area and perimeter for the ﬂower garden and relaxation zone. As you create your design, take into account both the functional use of space and the visual appeal of the park.

622

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

6 5 4 3 2 1

7.06 Polygons in the coordinate plane What do you remember? 1 a ii Quadrant II

b i Quadrant I

c iv Quadrant IV

d i Quadrant III

−6 −5 −4 −3 −2 −1 −1 −2 H −3 −4 −5 −6

2 a Move 4 units up b Move 3 units to the right c Move 8 units to the left d Move 2 units to the right and then 9 units down 3 Move 9 units to the right, and then 14 units up. 4 a X (5, 3), Y (−6, −8), Z (−7, 4) b A (5, 9), B (5, −3), C (−2, −3), D (−2, 9)

d T he distance between point F and point H is 8 units, horizontally. Points F and G are along the same vertical line.

12 a

e E (−4, 5), F (−8, 2), G (−8, −3), H (0, 4)

A (1, 9), B (−6, 4), C (−6, −6), D (7, −6), E (7, 4)

4 2

6 a 7 units

b 12 units

c 5 units

d 17 units

C −8 −6 −4 −2 −2

2 4 6 8

−4 −6

Let’s practice

−8

7 a y-axis b x-axis

c 7 units away from the y-axis

C b (−3, 1)

c (−5, 6)

d (4, 0)

10 a 3 units

b 8 units

c 4 units

d 9 units

8 units

g 9 units

h 9 units

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

9 a (0, 5)

6 5 4 3 2 1

ii B

d 5 units away from the x-axis

11 a

8 6

5 B and D

e 3 units

c T he distance between point F and point G is 6 units, vertically.

d S (8, 9), T (−5, 4), U (−7, −3), V (6, −6)

8 i

x 1 2 3 4 5 6

e Points F and H are along the same horizontal line.

c A (6, 2), B (−9, 3), C (−9, −7), D (6, −8)

C −8 −6 −4 −2 −2

x 2 4 6 8

−4 −6 −8

c 3 units

d 7 units

13 190 meters

14 a The distance between Point A and Point F is 8 units. x 1 2 3 4 5 6 F

b The distance between Point C and Point D is 7 units. c The distance between Point B and Point E is 13 units. 15 a y 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

b 58 feet

x 2 4 6 8 10

Answers mathspace.co

623

16 a

5 4 3 2 1

20 a (4, −4) b

−4 −3 −2 −1 −1 −2 −3 B −4 −5

1 2 3 4 5 6

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

b 5 units c 8 units 17 a D

c 63 km2 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 A −5

b i 8 units

x 1 2 3 4 5

d 7 bags

e $230.79 G

25 K

x 1 2 3 4 5

ii 8 units

iii 8 units

iv 8 units

c F rom part (a), we see that the quadrilateral ABCD is a rectangle. If this rectangle has equal lengths of all sides, then it is a square. From part (b), we calculated that all sides have equal length of 8 units. Now, the quadrilateral ABCD is a square. Let’s extend our thinking 18 (−2, 2) 19 a Point C could be at any location along a horizontal line except at x = −3, to form an isosceles triangle with points A and B. Speciﬁc examples could include C1 (x, 2), C2 (x, −4), and C3 (x, −1) where x ≠ −3. b F our possible coordinates of point C to form a right-angled triangle are: C1 = (1, 2), C2 = (−5, 2), C3 = (2, −4), and C4 = (−6, −4). Points C1 and C2 are horizontally aligned with A, while points C3 and C4 are horizontally aligned with B.

624

5 4 3 2 1

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

20 J

15 10 B 5

−30−25−20 −15 −10 −5 −5 A −10

x F 5 10 15 20 25 30 E

−15 −20 −25 −30

Example Answer: Flower garden area = 200 square units, Flower garden perimeter 56 units, Relaxation zone area = 400 square units, Relaxation zone perimeter = 80 units. Ensure the bird-watching area ﬁts within its designated space.

Topic 7 Assessment: Polygons 1

For each of the following triangles, ﬁnd the value of b or h given the area: a

Area = 24 mm2

Area = 72 cm2

6 mm 6 cm

h cm

b mm

Consider the triangle at right. a

Calculate the perimeter.

Calculate the area.

25 cm

20 cm

30 cm

Kryzzie has purchased a rectangular piece of fabric measuring 12 m in length and 5 m in width. Find the area of the largest triangular piece she can cut out from it.

The given parallelogram is formed into a rectangle by rearranging one of the blue triangles:

8 cm 13 cm

Find the length, l, of the rectangle.

Find the area of the parallelogram.

Are the area of the paralleogram and rectangle the same or different? Explain how you know.

Are the perimeter of the paralleogram and rectangle the same or different? Explain how you know.

5 cm

12 cm

l cm

A rhombus is a speciﬁc type of parallelogram with all sides congruent. The given rhombus can be split into two triangles:

11 mm

Find the area of one triangle.

Find the area of the rhombus.

Explain how you could use the area of the triangles to ﬁnd the area of the rhombus.

11 mm

6 mm

11 mm

7.06 Polygons in the coordinate plane mathspace.co

625

Which of the following represent regular polygons? A

Identify as many line(s) of symmetry as you can ﬁnd in the following shapes. a

60°

Abbey is planning out her garden on a coordinate grid. She plans for the corners to lie on the points (1, 2), (1, 5), (4, 5), and (4, 2). a

Plot the the corners of Abbey’s garden on a coordiante plane and connect them to form a polygon.

What type of polygon is the shape of Abbey’s garden? Explain your reasoning.

Karl travels from Beach A to Beach C through Beach B. The coordinates are plotted on the following coordinate plane:

6 5 4 3 2 1

If each unit represents 5 meters. Find the total distance that Karl traveled.

−6−5−4−3−2 −1 −1

Figure STUVWX is shown. T

S X

U W

626

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

−2 −3 −4 −5 −6

x 1 2 3 4 5 6

Which ﬁgure appears to be congruent to ﬁgure STUVWX? A

Are the following triangles congruent? Explain your reasoning. 6 cm

3 cm

60° 3 cm

6 cm

60° 60° B

3 cm

60°

Consider the isosceles triangle below. a

Which leg in △ABC is congruent to leg

Which angle in △ABC is congruent to ∠C?

Calculate the horizontal or vertical distance between the following pairs of points: a

A (−3, 9) and B (2, 9)

C (4, −2) and D (4, 5)

E (10, 4) and F (10, −1)

G (−7, 3) and H (−7, −4)

Performance Task 14

Janie has been exploring the relationship between area and perimeter. Janie claims: “When you increase the perimeter of a rectangle, the area always increases.” This is called a hypothesis. a

Draw two rectangles that conﬁrm Janie’s hypothesis. That is, draw a rectangle with a larger perimeter than another and a larger area.

We cannot prove Janie’s hypothesis with examples, but we can disprove the hypothesis with a single counter-example. Find a counterexample, which is a rectangle with a larger perimeter than another but a smaller area.

Janie’s hypothesis is also not true for triangles. Is Janie’s hypothesis true for any shape? Explain your reasoning.

7.06 Polygons in the coordinate plane mathspace.co

627

Answers

10 B 6.MG.4c

Topic 7 Assessment: Polygons 1 a b=8

11 No, they are not. While the angles are the same, the side lengths must also be the same to be congruent.

b h = 24

6.MG.2b

6.MG.4d

2 a 80 cm

b 300 cm

6.MG.2b

b ∠B

12 a 6.MG.4c

3 30 m

13 a 5 units

6.MG.2b

b 7 units

c 5 units

d 7 units

6.MG.3e

4 a l = 13 cm b 156 cm2

Performance Task

c The same. They occupy the same amount of space.

14 a Answers will vary. Possible answer: 5

d Different. The side lengths have changed. 6.MG.2a, 6.MG.2b 4

5 a 16.5 mm2 b 33 mm2 c T he rhombus consists of exactly the two congruent triangles. Thus, the area of the rhombus must be twice the area of one of the described triangles.

Rectangle A

Rectangle A: Width = 4, Height = 5

Perimeter: 2 ⋅ (4 + 5) = 18

Area: 4 ⋅ 5 = 20 8

6.MG.2a, 6.MG.2b 6 D 6.MG.4a

Rectangle B

7 a

60°

Rectangle B: Width = 6, Height = 8

Perimeter: 2 ⋅ (6 + 8) = 28

Area: 6 ⋅ 8 = 48

Rectangle B has a larger perimeter and a larger area than Rectangle A, conﬁrming Janie’s hypothesis.

6.MG.4b

b Answers will vary. Possible answer:

8 a

5 2

4 3 2 1 −1 −1

x 1

Rectangle C

Rectangle C: Width = 2, Height = 9

Perimeter: 2 ⋅ (2 + 9) = 22

Area: 2 ⋅ 9 = 18

b T hese points form a square since all angles are right angles and all four sides are the same length.

Rectangle D

6.MG.3f, 6.MG.4a 9 95 meters 6.MG.3f

628

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Rectangle D: Width = 4, Height = 5

Perimeter: 2 ⋅ (4 + 5) = 18

Area: 4 ⋅ 5 = 20

Rectangle C has a larger perimeter but a smaller area than Rectangle D, disproving Janie’s hypothesis. c J anie’s hypothesis is true for squares since they rely only on a single side length as a variable. Let’s call the side length, s. The are of a square is given by A = s ⋅ s = s2 and the perimeter is P = s + s + s + s = 4s. When the perimeter, or value of 4s increases, it must mean s increased. This would also cause the area which has a value of s2 to increase. Therefore, when you increase the perimeter of a square, the area always increases. 6.MG.2a, 6.MG.2b, MP1, MP2, MP3

Answers mathspace.co

629

8 Circles Big ideas The relationships between the parts of a circle can be used to find the measures of other parts.

Chapter outline 8.01 8.02 8.03

Characteristics of circles (6.MG.1) Circumference and pi (6.MG.1) Area of a circle (6.MG.1)

634 646 664

The Olympic symbol is made up of circles! Each ring represents a different continent, and together they symbolize unity and global competition.

8. Circles Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Big ideas and essential understanding The relationships between the parts of a circle can be used to find the measures of other parts. 8.01 — There is a constant relationship between the diameter and radius of any circle.

8.03 — There is a constant relationship between the radius and area of any circle. This relationship can be applied to solve many real-world and mathematical problems.

8.02 — There is a constant relationship between the circumference and diameter of any circle. The ratio of the circumference to the diameter is a constant called π which is used in a variety of real-world and mathematical situations.

Standards 6.MG.1 — The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area. 6.MG.1a — Identify and describe chord, diameter, radius, circumference, and area of a circle. 8.01 Characteristics of circles 6.MG.1bi — Investigate and describe the relationship between: 8.01 Characteristics of circles 6.MG.1bii — Investigate and describe the relationship between: 8.02 Circumference and pi 6.MG.1biii — Investigate and describe the relationship between: 8.02 Circumference and pi

632

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.MG.1c — Develop an approximation for π (3.14) by gathering data and comparing the circumference to the diameter of various circles, using concrete manipulatives or technological models. 8.02 Circumference and pi 6.MG.1d — Develop the formula for circumference using the relationship between diameter, radius, and π. 8.02 Circumference and pi 6.MG.1e — Solve problems, including those in context, involving circumference and area of a circle when given the length of the diameter or radius. 8.02 Circumference and pi 8.03 Area of a circle

Future connections 7.MG.1 — The student will investigate and determine the volume formulas for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.

G.PC.3 — The student will solve problems, including those in context, by applying properties of circles. G.PC.4 — The student will solve problems in the coordinate plane involving equations of circles.

8. Circles mathspace.co

633

8.01 Characteristics of circles Subtopic overview Lesson narrative In this lesson, students will explore the characteristics of circles. They will learn that a circle is defined as the set of all points equidistant from a central point, called the radius. The lesson covers important circle terminology, including radius, diameter, circumference, and chord. Students will understand that the diameter is twice the radius and that the circumference is the distance around the circle. They will solve problems involving finding the radius, diameter, and circumference of a circle. An interactive exploration allows students to manipulate a circle to observe and measure the relationships between the radius, diameter, and circumference, reinforcing these geometric properties. By the end, students should be proficient in identifying, describing and using the key properties of circles.

Learning objectives Students: Page 320

Key vocabulary 

area (of a circle)



center (of a circle)



chord



circumference



diameter



radius

 circle

Essential understanding There is a constant relationship between the diameter and radius of any circle.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG2 — Mathematical Communication Teachers can integrate Mathematical Communication into their instruction by encouraging students to express their understanding of the relationship between points, lines, rays, angles, and circles, using appropriate mathematical vocabulary. Students can be asked to explain in their own words the definitions of circle, radius, diameter, and chord, or to describe the relationship between diameter and radius. This can include verbal explanations, written descriptions, or symbolic notation.

634

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG3 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can incorporate this goal into their instruction to help students identify and describe the chord, diameter, radius, circumference, and area of a circle, and investigate the relationship between diameter and radius. Teachers can start by defining these terms and providing hands-on activities to explore the relationships between the characteristics of a circle. Students can use yarn to measure circumference and rulers to measure diameters and radii. By folding circles, students can identify chords and diameters, reinforcing that the diameter is twice the radius and that the diameter is also a chord. Visual aids like graph paper and dynamic geometry software can help students estimate and verify areas, while plotting points on graph paper solidifies their understanding of these geometric relationships.

Incorporating Mathematical Representations can be achieved by asking students to represent their understanding of circles, diameters, and radii visually. Teachers can ask students to sketch a circle and label its center, radius, and diameter. They can also ask students to draw intersecting chords or radii and identify the angles formed. Further, teachers can provide different representations of circles (e.g., real objects, diagrams, symbols) and ask students to identify the mathematical elements present in each representation.

Content standards 6.MG.1 — The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area.

6.MG.1a — Identify and describe chord, diameter, radius, circumference, and area of a circle. 6.MG.1bi — Investigate and describe the relationship between:

Prior connections 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Future connections G.PC.3 — The student will solve problems, including those in context, by applying properties of circles.

G.PC.4 — The student will solve problems in the coordinate plane involving equations of circles.

Lesson Preparation Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

8.01 Characteristics of circles mathspace.co

635

Visualizing the circle Targeted instructional strategies Encourage students to visualize the parts of a circle as they learn them. They can do this by drawing diagrams and annotating them with labels and definitions. chord

diameter

For example, when learning about the radius and diameter, they can draw a circle and mark the radius from the center to the edge and the diameter from one edge of the circle to the other, passing through the center. This will help them to better understand these concepts and their relationships to each other.

radius

circumference

Collect and display English language learner support As students work on exploring the characteristics of circles, listen for how they describe the terms “radius,” “diameter,” “circumference,” “chord,” and “area” in their own words. Collect these descriptions and display them in a place where all students can see and refer to them. If students are struggling to express these concepts, you can provide some examples to guide them. For instance: • Radius • The distance from the center of the circle to any point on the circle • Half of the diameter • A line segment from the center to the edge of the circle • Diameter • A line that passes through the center of the circle and touches two points on the edge • Twice the length of the radius • The longest distance across the circle • Circumference • The distance around the circle • The perimeter of the circle • How far you would travel if you walked around the circle once • Chord • A line segment connecting any two points on the circle • A “bridge” inside the circle • A straight path between two points on the circle’s edge • Area • The space contained within the circle • How much surface the circle covers • Measured in square units (like square centimeters) Be attentive to any misconceptions or descriptions that might be confusing, and gently guide students toward the accurate mathematical definitions. Use this collected language to create a visual display, such as a labeled diagram of a circle with these terms and student explanations. Refer back to this display throughout the lesson to reinforce understanding and encourage students to use the precise terminology when describing circles.

636

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Support visual-spatial processing with manipulatives and color-coding Student with disabilities support To support students with visual-spatial processing difficulties, incorporate hands-on manipulatives into your lesson on circle properties. Provide students with circular objects like plastic lids, hoops, or paper cut-outs, along with rulers and flexible measuring tapes or strings. Encourage students to physically measure the radius, diameter, and circumference of these objects. This tactile experience helps them concretely understand the relationships between these measurements. Additionally, use consistent color-coding in all your visual materials: for example, highlight the radius in red, the diameter in blue, the circumference in green, and chords in yellow. This visual distinction aids students in identifying and differentiating between each part of the circle. By combining physical manipulation with clear visual cues, you can enhance students’ comprehension of circle terminology and relationships.

Misconception: Diameter is not a chord Address student misconceptions Students may think that a diameter is not a chord because it passes through the center of the circle, and they believe chords are only line segments that do not pass through the center. They might view the diameter as a separate entity from chords due to its special properties. Clarify this misconception by explaining that a chord is any line segment with both endpoints on the circle. Emphasize that since the diameter has its endpoints on the circle, it is the longest possible chord. Use visual aids by drawing several chords of different lengths, including the diameter, and label them all as chords. Encourage students to see the diameter as a special type of chord by stating, “All diameters are chords, but not all chords are diameters.” This will help students understand the inclusive definition of a chord.

8.01 Characteristics of circles mathspace.co

637

Student lesson & teacher guide Characteristics of circle Students are introduced to the definitions of circle, radius, and diameter before engaging in an exploration on the relationship between the radius and diameter of any circle.

Students: Pages 320–321

638

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle.

A chord is a line segment connecting any two points on a circle. A chord may or may not go through the center of a circle. The diameter is the longest chord of a circle.

Example 1

Exploration For this circle: Students: Page 320 10 cm

a What is the diameter of the circle?

Create a strategy Find out how long the line is that goes through the middle of a circle and touches the edge at two points that are opposite each other.

Apply the idea Look for a straight line that goes from one side of the circle to the other, passing through the middle. This line is

Suggested student grouping: Individual called a diameter, and its length is 10 cm. Student use a GeoGebra applet to explore the relationship between the radius and diameter of a circle. By Diameter = 10 cm adjusting the radius using a slider, students can observe how changes in the length of the radius affect the lenght of the diameter. b What is the radius of the circle?

Ideal student responses Create a strategy

These ideal responses may differ from other correct student responses. Less formal responses can be Remember the radius is half the length of the diameter. connected with the more precise mathematical language presented here. 1. WhatApply do you the notice idea about the relationship between the radius and diameter of the circle? The In diameter of value a circle is diameter always twice the length part (a), the of the is provided as 10 cm.of the radius, and the radius is always half the length of the diameter.Radius = Divide 10 by 2 Purposeful questions = 5 cm

Evaluate

• How is the length of the radius related to the length of the diameter? • If the diameter of a circle is 8, what is its radius? • Can a circle have a radius of 1 in and a diameter of 1.5 in? Possible misunderstandings

8.01 Characteristics of circles

321

• Students might struggle to see the relationship if they only consider rational lengths. Encourage mathspace.co them to use integer lengths for the radius, then see if the relationship still holds for decimal lengths. After the exploration, students discover that the diameter of a circle is always twice its radius. Then, they are introduced to the definitions of circumference and chord.

8.01 Characteristics of circles mathspace.co

639

A chord is a line segment connecting any two points on a circle. A chord may or may not go through the center of a circle. The diameter is the longest chord of a circle.

Examples Students: Page 321

Example 1 For this circle:

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle. 10 cm

a What is the diameter of the circle?

A chord is a line segment connecting any two points on a circle. A chord may or may not go through the center of a circle. The diameter is the longest chord of a circle.

Create a strategy Find out how long the line is that goes through the middle of a circle and touches the edge at two points that are opposite each other.

Example 1 Apply the idea For Lookthis forcircle: a straight line that goes from one side of the circle to the other, passing through the middle. This line is called a diameter, and its length is 10 cm. Diameter = 10 cm

10 cm

b What is the radius of the circle?

PurposeCreate a strategy Show students how to identify and measure the diameter of a circle from a given diagram. a What is the of the Remember the diameter radius is half thecircle? length of the diameter. Reflecting witha students Create strategy Apply the idea Encourage students to always include appropriate units of when working on problems involving measurements. Find out is that goes through the middle a circle and touches the edge at two points In part (a),how thelong valuethe of line the diameter is provided as 10 cm. In this example, remind them to label their answers with centimeters. For instance, a precise response would that are opposite each other. Divide 10 by 2 = 10”. By consistently using units throughout the task, students Radius be “Diameter = 10 cm”=rather than just “Diameter theofidea = 5 cm Evaluatethat is crucial for accurate communication and understanding in developApply a habit mathematical precision Look for a straight line that goes from one side of the circle to the other, passing through the middle. This line is mathematics. called a diameter, and its length is 10 cm.

Students: Page 321

Diameter = 10 cm

b What is the radius of the circle?

8.01 Characteristics of circles mathspace.co

321

Create a strategy Remember the radius is half the length of the diameter.

Apply the idea In part (a), the value of the diameter is provided as 10 cm. Radius = = 5 cm

Divide 10 by 2 Evaluate

Purpose Help students understand the relationship between the radius and the diameter of a circle and apply this 8.01 Characteristics of circles 321 understanding to find the radius when given the diameter. mathspace.co

640

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Reflecting with students Ask students to consider whether a radius is a chord. Is a diameter a chord? Refer students back to the definition of a chord to help them understand that a radius is not a chord, but a diameter is.

Advanced learners: Discover and justify the relationship between radius and diameter use with Example 1

Targeted instructional strategies

Encourage students to explore the relationship between the radius and diameter of a circle through handson investigation. Provide them with various circles—either drawings or physical circular objects like jar lids or hoops—and ask them to measure both the diameter (the straight line passing through the center from one side to the other) and the radius (the line from the center to any point on the circumference) of each one. Have students record their measurements and observe the patterns that emerge. Then, challenge students to justify why this relationship holds true for all circles. For example, they might reason that since the radius extends from the center to the edge, and the diameter passes through the center connecting two points on the circle, the diameter must consist of two radii end to end. This means: Diameter = Radius + Radius = 2 ⋅ Radius By constructing this logical justification, students not only discover the consistent relationship but also strengthen their ability to reason and communicate mathematically. This approach allows advanced learners to deepen their understanding by actively engaging with the concept and articulating their insights.

Students: Page 322

Idea summary A circle has many parts: chord

diameter

radius

circumference

The diameter of a circle is double the radius:

d = 2r d

is the diameter

is the radius

Practice What do you remember? 1

Describe the difference between a line and a line segment.

Match the following terms with their deﬁnitions:

Chord

A line segment passing through the center of the circle with its endpoints on the circle

Diameter

A line segment with both endpoints on the circle

Radius

iii

A line segment with one endpoint at the center of the circle and the other on the circle

Circumference

The space inside the circle

Area

The distance around the edge of the circle

Name the indicated part of these circles: a

8.01 Characteristics of circles mathspace.co d

641

Practice Students: Pages 322–324

What do you remember? 1

Describe the difference between a line and a line segment.

Match the following terms with their deﬁnitions:

Chord

line segment passing through the center of the circle with its endpoints on A the circle

Diameter

A line segment with both endpoints on the circle

Radius

iii

line segment with one endpoint at the center of the circle and the other on A the circle

Circumference

The space inside the circle

Area

The distance around the edge of the circle

Name the indicated part of these circles: a

You want to calculate the distance a bicycle covers when its wheel completes one full turn (360-degree rotation). Which of the following best describes this distance? A

The radius of the wheel

The diameter of the wheel

The circumference of the wheel

The area of the wheel

Let’s practice 5

Use the three circles to draw and label radius, chord, and diameter: a

642

Describe a real-world example of each: a

Radius

Chord

Diameter

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Circumference

Area

If given the radius of a circle, explain how to ﬁnd the diameter.

State the diameter of these circles: a

b 14 cm

8 cm

State the radius of these circles: a

b 18 cm

14 cm

For each pair, determine if the radius and diameter measurements could be from the same circle. Explain your reasoning. a

Radius: 11 cm; Diameter: 22 cm

Radius: 17 cm; Diameter: 38 cm

Radius: 60 cm; Diameter: 30 cm

Radius: 41 cm; Diameter: 82 cm

Radius: 220 cm; Diameter: 110 cm

Radius: 66 cm; Diameter: 132 cm

Explain the concept of a chord and give two real-life examples where a chord is relevant.

M is the center of the circle, and the length of CG is 12 cm. a

Name a segment that is a radius. What is the length of the radius?

Name a segment that is a diameter. What is the length of the diameter?

Write a comparative statement about the diameter and a radius.

Write a comparative statement about a radius and a diameter.

E D C

M 12 cm

Let’s extend our thinking 13

The radius of the smaller circle in the ﬁgure is 16 cm. If CD = 4 cm, ﬁnd AD, the diameter of the larger circle. A B

8.01 Characteristics of circles mathspace.co

C D

643

Here is a circle with the center point P and some line segments. a

Identify all examples of diameters, radii, and chords and explain your reasoning. Measure the line segments to use as part of your justiﬁcations.

Explain why the diameter is the longest chord in a circle.

A E B P D C

644

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

10 a Yes, because the diameter is exactly twice the radius. b N o, because the diameter (38 cm) is not exactly twice the radius (17 cm). Twice the radius would be (34 cm).

8.01 Characteristics of circles What do you remember? 1 A line is an inﬁnite set of points extending in both directions without end, characterized by length but no thickness. A line segment is part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. 2 a ii

b i

c iii

d v

e iv 3 a Radius

b Diameter

c Circumference

d Center

e Chord

Area

4 C Let’s practice 5 a

c N o, because the diameter (30 cm) is half of the radius (60 cm) which contradicts the relationship; the diameter should be twice the radius. d Y es, because the diameter (82 cm) is exactly twice the radius (41 cm). e N o, because the diameter (110 cm) is half of the radius (220 cm), which is incorrect; the diameter should be twice the radius. f

es, because the diameter (132 cm) is exactly twice the Y radius (66 cm).

11 A chord in geometry is a straight line segment whose endpoints both lie on the circumference of a circle. Real-life examples where a chord is relevant include the strings of a musical instrument like a guitar, where each string represents a chord of the circular sound hole, and the design of bridges, where chords can be used in the arc structures to provide strength and support. 12 a Possible answer: MG is a radius. Other radii are MD, ME, MC, and MG. Its length is 6 cm. b CG is a diameter of length 12 cm.

Radius Diameter Chord

c The diameter of a circle is twice as long as the radius.

6 a T he spoke of a bicycle wheel is an example of a radius. It connects the center of the wheel (the hub) to the wheel’s edge.

d The radius of a circle is half the length of the diameter.

b h e width of a round dining table is an example of a diameter, measuring straight across the table from one edge to the other through the center. c T he distance around the edge of a circular swimming pool is an example of circumference. It’s the total length you would walk if you went around the pool once. d T he space inside a circular trampoline is an example of area. It represents the total space where you can jump on the trampoline. e T he line that you draw when you cut a slice of pie from edge to edge. 7 To ﬁnd the diameter of a circle given the radius, multiply the radius by 2. 8 a 16 cm

b 28 cm

9 a 7 cm

b 9 cm

Let’s extend our thinking 13 40 cm 14 a T he diameters are AC and BD because they are the line segments with end points on the circle, passing right through the center The radii are PA, PB, PD, PC because these are the lines that go from the center of the circle (point P ) straight to the edge of the circle at points A, B, C, and D.

The chords are CE, AC, and BD.

b W hen we measure the radius, chord, and diameter in a circle, the diameter always comes out as the longest chord. This is because the diameter stretches all the way across the widest part of the circle, passing through the center from one edge to the opposite edge.

Answers mathspace.co

645

8.02 Circumference and pi Subtopic overview Lesson narrative In this lesson, students will explore the circumference of circles and the constant pi (π ). They will learn that the circumference is the distance around the circle and that it can be calculated using the formulas C = π d or C = 2π r. Students will engage in an interactive exploration to understand the relationship between the diameter, radius, and circumference, using π as the ratio between circumference and diameter. They will solve problems involving these formulas, including real-world applications such as calculating the distance around circular objects. By the end, students should confidently use π to find the circumference of circles.

Learning objectives Students: Page 325

Key vocabulary 

approximation



circumference



diameter



pi (π )



proportional



radius

Essential understanding There is a constant relationship between the circumference and diameter of any circle. The ratio of the circumference to the diameter is a constant called π which is used in a variety of real-world and mathematical situations.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate mathematical problem solving into their instruction by having students practice calculating the circumference of a circle using the formulas they developed in class. The teacher can create a variety of problem scenarios where the diameter or radius is given, and students are required to calculate the circumference. This will help students apply the mathematical concepts and skills they have learned to solve problem situations of varying complexities. 646

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG2 — Mathematical Communication Teachers can promote mathematical communication by encouraging students to articulate their understanding of circle vocabulary, the concept of pi, and the relationship between diameter, radius, and circumference. This can be done through class discussions, presentations, or written assignments where students are required to use mathematical language and notation to express their mathematical ideas with precision. MPG4 — Mathematical Connections Teachers can integrate mathematical connections into their instruction by connecting the concept of circumference to the perimeter of polygons that students already know from previous lessons. They can also create connections to real-world contexts by providing examples of situations where understanding the circumference of a circle is important, such as tire sizes, circular tracks, and round objects like cakes or pizzas. Teachers can also integrate this goal by engaging students in hands-on activities where they measure the circumference and diameter of circles using yarn and rulers, and then calculate the ratio to discover the relationship involving π. Using tools like Desmos calculators, students can substitute known values into the formulas for circumference (C = 2π r), discussing their observations and the significance of π. This approach helps students visualize and understand the characteristics and relationships within circles. MPG5 — Mathematical Representations Teachers can integerate this goal into their instruction by providing students with real-world applications to represent these relationships. For example, have students explain why relationship would be the same or different from

= π. and discuss if the results of this proportional

= π ? Additionally, teachers could have students bring in circular

objects and measure the distance around (circumference) and across (diameter) each object. Teachers should faciliate discussions regarding which object have the largest or smallest circumferences. They can allow students to work in small groups to derive pi (π ), and to compare the circumference to the diameter. Allow students to exchange their items to check each other’s work.

Content standards 6.MG.1 — The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area. 6.MG.1bii — Investigate and describe the relationship between: 6.MG.1biii — Investigate and describe the relationship between:

6.MG.1c — Develop an approximation for π (3.14) by gathering data and comparing the circumference to the diameter of various circles, using concrete manipulatives or technological models. 6.MG.1d — Develop the formula for circumference using the relationship between diameter, radius, and π. 6.MG.1e — Solve problems, including those in context, involving circumference and area of a circle when given the length of the diameter or radius.

Prior connections 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Future connections G.PC.3 — The student will solve problems, including those in context, by applying properties of circles.

7.MG.1 — The student will investigate and determine the volume formulas for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.

8.02 Circumference and pi mathspace.co

647

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 8.01 Characteristics of circles

Tools You may find these tools helpful: • Scientific calculator • String • Ruler

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Finding the circumference of a circle Targeted instructional strategies The two equivalent formulas for the circumference of a circle are C = π d if given the diameter and C = 2π r if given the radius. Depending on the calculator or mental math, π can be used as part of the answer, 3.14 can be used as an approximate value, or more digits can be used as technology allows. The steps of finding the circumference of a circle can be summarized as follows: 1. Determine the radius or diameter of the circle. 2. Substitute the value in for r if given radius, or d if given diameter. 3. Simplify using the parameters specified in the problem. A useful exercise would be to go through several problems and identify the radius or diameter of the circle before solving.

Student lesson & teacher guide Circumference and pi Before the exploration, students review the definition of circumference.

648

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Pages 325–326

Concrete-Representational-Abstract (CRA) approach Targeted instructional strategies Concrete: Begin by engaging students with physical circular objects of various sizes, such as lids, plates, or hoops. Provide measuring tapes or pieces of string and rulers so students can measure the circumference, radius, and diameter of each object themselves. Encourage them to record their measurements in a table, noting the circumference and diameter side by side. This hands-on activity allows students to physically see and feel the circles, helping them understand the concepts of circumference, diameter, and radius. As they work, ask guiding questions like, “How many times do you think the diameter fits around the circle?” to prompt exploration of the relationship between these measurements. Representational: After the measurements are complete, help students transition from the physical objects to visual representations. Have them draw each circle they measured, labeling the diameter, radius, and circumference on their drawings. Encourage them to create a chart or graph plotting the diameter on one axis and the circumference on the other. Guide students to analyze their charts and graphs to notice that the circumference is always a little more than three times the diameter. Use diagrams to illustrate how the diameter can be “wrapped” around the circle’s edge approximately three times. Introduce the concept of the constant ratio π by showing that dividing the circumference by the diameter for each circle yields a similar value. Encourage students to draw conclusions from their visual data and discuss what this ratio means.

8.02 Circumference and pi mathspace.co

649

Abstract: Prepare students to move from their visual findings to mathematical expressions. Show them how their graphs represent a linear relationship between circumference and diameter, indicating a constant rate of change. This connection helps students see how the visual patterns translate into mathematical formulas. Then, introduce the formulas for circumference: C = π d and C = 2π r. Explain that π is the constant ratio they observed. Provide practice problems where students use these formulas to calculate the circumference when given the diameter or radius, without relying on physical objects or drawings. Challenge them with real-world scenarios, such as finding the distance around a circular track or the rim of a wheel. Encourage them to explain their reasoning using mathematical terms and to recognize π as an essential constant in these calculations. Connecting the stages: Throughout the lesson, help students make connections between the concrete, representational, and abstract stages. Refer back to the physical circles when discussing the formulas, reminding them of how they measured and observed the relationships themselves. Use the drawings and graphs to reinforce how the abstract formulas are based on real-world measurements and visual patterns. Encourage students to reflect on how each stage built upon the previous one, deepening their understanding of circumference and π.

Exploration Students: Pages 325

Suggested student grouping: Small groups Students use a GeoGebra applet to explore the relationships within a circle, particularly between the diameter, radius, and circumference. By adjusting a point on the circle’s circumference and observing changes, students can examine the lengths. Using pattern analysis, students can make predictions and conclude that the diameter and circumference are mathematically related to the constant π.

650

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Complete the table for various values of diameter and circumference. Recall that the circumference is the length of one full rotation around the circle. Example responses are shown: diameter: 5, circumference: 15.71,

1.8, 5.64,

2, 6.28,

2.61, 8.21,

6.9 21.66

2. Try out some calculations between the diameter and circumference. Do you notice a relationship between the diameter and circumference? Explain. The circumference is always slightly more than three times the diameter. It is approximately 3.14 times the diameter. 3. Do you think there could be a relationship between circumference and radius? If so, describe the relationship. Since the diameter is twice the radius, the circumference is always slightly more than six times the radius. The circumference is approximately 3.14 times twice the radius, or 6.28 times the radius. Purposeful questions • What do you notice about the ratio between the circumference and diameter? • Can you approximate the ratio to one or two decimal places? • What is the relationship between the radius and diameter? Can you use that to find a relationship between the radius and circumference? Possible misunderstandings • Students will not see the relationship if they do not divide the circumference by the diameter. After trying some calculations on their own, let students know that there is a relationship between the ratio of the circumference and diameter, and ask them to find it. Students discover that the ratio of the circumference to the diameter is approximately 3.14159, which is an approximation of the irrational number π. They learn that when finding the circumference of a circle, the formulas C = π d or C = 2π r depend on whether the diameter or radius is given.

Students: Pages 325–326

8.02 Circumference and pi mathspace.co

651

Examples Students: Page 327

652

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Assess student understanding of π, namely its exact value versus its approximate value. Expected mistakes Students might choose options A, D, or F if they are familiar with using these approximations for calculations in previous grades. Make them aware that these are approximations, not exact values for pi, and the statements are about the exact value.

Students: Page 327

Purpose Students demonstrate that they can find the circumference of a circle given a radius. Expected mistakes Students may use the formula for diameter instead of radius. Alternatively, they might round pi to 3.14 before performing the calculations, leading to a less precise answer. Reflecting with students The formula for circumference given a radius is 2 ⋅ π ⋅ r. Ask students whether we would get the same result if we took out the 2 and made the formula π ⋅ π ⋅ r instead. If it’s different, how would we write this instead? Ensure students understand that 2π = π + π rather than π ⋅ π, which is π 2. 8.02 Circumference and pi mathspace.co

653

Students: Page 328

Example 3 Find the circumference of the circle shown, correct to two decimal places.

13 cm

Create a strategy The circumference of the circle can be found using the formula: C = π d.

Example 3

Apply idea Find thethe circumference of the circle shown, correct to two decimal places. C = π ⋅ 13 Substitute d = 13 = 40.84 cm

Evaluate 13 cm

Example 4 Purpose Students demonstrate thatisthey theits circumference of a circle given If the radius of a circle equalcan to 17find cm ﬁnd circumference correct to onewhen decimal place.a diameter. Create a strategy

Reflecting witha students Create strategy The circumference of the circle can be found using the formula: C = π d. The circumference of a of circle can can be written depending The circumference the circle be founddifferently using the formula: C = 2π r.on the radius or diameter being given. Ask students whether there are benefits to having two formulas or whether they are really different. What are Apply the idea Apply theofidea the advantages both? C = π ⋅ 13 Substitute d = 13 C = 2 ⋅ π ⋅ 17 = 40.84 cm

Students: Page 328 = 106.8 cm

Substitute r = 17 Evaluate Evaluate

Example 4 Example 5 If the radius of a circle is equal to 17 cm ﬁnd its circumference correct to one decimal place. Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of 5 m What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place. Create a strategy The circumference of the circle can be found using the formula: C = 2π r.

Create a strategy

The distance around the outside of a circle is its circumference. Apply the idea C = 2 ⋅ π ⋅ 17

Substitute r = 17

= 106.8 cm C = 2π r

Evaluate Write the formula

=2⋅π⋅5

Substitute r = 5

= 31.4 m

Evaluate

Apply the idea

Example Lisa will walk531.4 m around the pool.

Purpose Students demonstrate they find the circumference a is circle withshape givenand terms values Lisa is cleaning thethat leaves outcan of the pool in her backyard. The of pool a circular has aand radius of 5 mbut no diagram.What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place. Mathspace Virginia SOL Grade 6 328 with Reflecting Create a students strategy mathspace.co Encourage students to always include appropriate units when working on problems involving measurements. The distance around the outside of a circle is its circumference. In this example, remind them to label their answers with centimeters. For instance, a precise response would the idea is about 106.8 cm.” rather than just “C = 106.8”. By consistently using units throughout be “TheApply circumference C =develop 2π r the formula precision that is crucial for accurate communication and the task, students a habit ofWrite mathematical π ⋅ 5 Substitute r=5 = 2 ⋅ understanding in mathematics.

= 31.4 m

654

Evaluate

Lisa will walk 31.4SOL m around pool. Edition Mathspace Virginia Gradethe 6 Teacher mathspace.co

The circumference of the circle can be found using the formula: C = 2π r.

Apply the idea C = 2 ⋅ π ⋅ 17 = 106.8 cm

Students: Page 328

Substitute r = 17 Evaluate

Example 5 Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of 5 m What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place.

Create a strategy The distance around the outside of a circle is its circumference.

Apply the idea C = 2π r

Write the formula

=2⋅π⋅5

Substitute r = 5

= 31.4 m

Evaluate

Lisa will walk 31.4 m around the pool.

Purpose328 Mathspace Virginia SOL Grade 6 mathspace.co that they can find the circumference of a circular-shaped item, and write the answer with Students demonstrate the units of the problem. Expected mistakes Students may use incorrect units or approximate π to incorrect digits. Reflecting with students Ask students to consider of the process if finding the circumference would change if the problem specified it was a different item. This problem mentions a pool, but would the process change if it was something else? What would need to change for the process to change?

Three reads

use with Example 5

8.02 Circumference and pi mathspace.co

655

Students: Page 329 Example 6 Carl is performing an experiment by spinning a metal weight around on the end of a nylon thread. How far does the metal weight travel if it completes 40 revolutions on the end of a 0.65 m thread? Give your answer correct to one decimal place.

Create a strategy The total distance traveled by metal weight can be found using the formula: Total distance traveled = circumference ⋅ number of revolutions The radius is equivalent to the length of thread.

Apply the idea Find for circumference: C = 2π r

Write the formula for circumference

= 2 ⋅ π ⋅ 0.65

Substitute r = 0.65

= 1.30π m

Evaluate

Total distance traveled = 1.30π ⋅ 40 = 163.4 m

Substitute circumference and number of revolutions Evaluate

The metal weight traveled a total of 163.4 m

Idea summary

Purpose π is the ratio between the circumference and diameter, which we approximate as 3.14. Students demonstrate that they can describe real-life situations using the circumference of a circle and use the The formula for circumference of a circle is : circumference to solve a real-life situation.

C = πd

Expected mistakes C Circumference Students may not know whether the given thread length is a diameter or a radius. Use the image in the support d Diameter below to help them understand what that length represents. and because the diameter is twice the radius, we can also write the formula as

Reflecting with students C = 2π r The number of rotations of the weight is used in the problem as a factor. Challenge advanced learners to Circumference describe a situation where we would use division orCa fraction instead of a whole number in the calculation of r Radius circumference. An example might be that they want to know the distance around the edge of a semicircle or the length of the crust of a pizza after two slices have been consumed.

Draw a diagram to support visual-spatial processing

use with Example 6

Student with disabilities support Encourage students to draw a diagram of the scenario described in the problem. Have them sketch a circle to represent the metal weight, with the center labeled and a radius drawn to show the 0.65-meter thread. By visualizing the circle and the revolutions of the weight, students can better grasp the concept of circumference and how it relates to the distance traveled.

8.02 Circumference and pi mathspace.co

656

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

329

Total distance traveled = circumference ⋅ number of revolutions The radius is equivalent to the length of thread.

Apply the idea Find for circumference:

This visual aid can bridge the gapCbetween the abstract formulas and the concrete physical Writemathematical the formula for circumference = 2π r situation, making it easier for students visual-spatial processing π ⋅ 0.65 Substitute r = 0.65 difficulties to understand and solve the = 2 ⋅with problem. Evaluate = 1.30π m Total distance traveled = 1.30π ⋅ 40 = 163.4 m

Students:The Page 329 metal weight traveled a total of 163.4 m

Substitute circumference and number of revolutions Evaluate

Idea summary π is the ratio between the circumference and diameter, which we approximate as 3.14. The formula for circumference of a circle is :

C = πd C d

Circumference Diameter

and because the diameter is twice the radius, we can also write the formula as

C = 2π r C r

Circumference Radius

Practice Students: Pages 330–334

What do you remember? 1

What term refers to the perimeter of the circle?

For this circle, identify the letter that corresponds to: a

Radius

Diameter

Circumference

Chord

Area

8.02 Circumference and pi mathspace.co

329

b e c d

8.02 Circumference and pi mathspace.co

657

Look at the table below. For each circle, analyze the relationship between the radius and circumference. Then answer true or false for each of the following questions.

Circle 1

Circle 2

Circle 3

Circle 4

Circle 2

Circle 3

Circle 4

The radius of a circle is equal to two times π times the circumference.

4π

The circumference of a circle is equal to two times π times the radius.

The circumference of a circle is double the radius.

The circumference of a circle is equal to double π times the radius.

The radius of a circle is equal to double π times the circumference.

2 cm

8 cm

41 cm

170 cm

16π

82π

170

340π

4 cm

16 cm

82 cm

340 cm

Radius (cm)

Circumference (cm)

12.57

50.27

340

The diameter of a circle is approximately one-third of the circumference.

The circumference of a circle is approximately π times the diameter.

The circumference of a circle is triple the diameter.

The circumference of a circle is 2π times the radius, which is half the diameter.

The diameter of a circle is equal to the circumference divided by π.

257.61

1068.14

Determine whether these statements are true or false:

π is a whole number. π is a number that cannot be written as a fraction using whole numbers and is irrational. π is a number that goes on forever and can’t be written exactly.

π is exactly equal to

π is the ratio between the diameter and the radius of a circle.

a b

658

Circumference (cm)

Examine the table below. Analyze the relationship between the diameter and circumference for each circle. Then, determine whether each of the following statements is true or false.

Circle 1

Radius (cm)

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice 6

Find the circumference of each circle. Use 3.14 for π and round your answers to two decimal places: a

8 cm 18 cm

14 cm

10 cm

Find the circumference of the following circles. Use 3.14 for π and round your answers to two decimal places. a

A circle of radius 7 ft

A circle with a diameter of

units

A circle of radius 11 in

A circle with a diameter of 9.5 cm

If the radius of a circular swimming pool is 6.7 yards, how far would you travel if you swam around the pool’s edge exactly once? Explain your reasoning. Use 3.14 to approximate π.

If the diameter of a wheel is 18 units, how far will the wheel roll in one full revolution? Explain your thinking. Use 3.14 for π.

A medium pizza has a circumference of 40 in. a

Describe what the radius of the pizza would be, rounded to two decimal places.

Describe what the diameter of the pizza would be, rounded to two decimal places.

Explain your thinking.

8.02 Circumference and pi mathspace.co

659

SOL

A trampoline has a circumference of 44.5 ft.

Describe what the radius of the trampoline would be, rounded to two decimal places.

Describe what the diameter of the trampoline would be, rounded to two decimal places.

Explain your thinking.

For each pair, determine if the diameter and circumference measurements could be from the same circle. Explain your reasoning.

π cm

Diameter: 14 cm; Circumference: 44 cm

Diameter:

Diameter: 30 cm; Circumference: 90 cm

Diameter: 20 cm; Circumference: 62.8 cm

Diameter: 16 cm; Circumference:

Diameter: 8 cm; Circumference: 25.12 cm

cm; Circumference:

Samantha measured a circular table and noted that d, the diameter, was 12 inches and C, the circumference, was approximately 38 inches. C = 38 in

d = 12 in

Write an expression using the values of the circumference and diameter that represents an approximate value for π. 14

After measuring several circles, a student recorded the following circumferences and diameters:

Circle 1

Circle 2

Circle 3

10 cm

15 cm

20 cm

Circumference (cm)

Diameter (cm)

31.4

47.1

62.8

Calculate the ratio of circumference to diameter for each circle.

660

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

For each circle in the previous question, what do you notice when circumference is divided by the diameter? What conclusions can you draw about the approximate measurement of π from this data?

Write down the equation for the circumference of each circle if:

you know the radius of the circle is equal to 27 cm

you know the diameter of the circle is equal to 12.5 cm

Alex says, “To ﬁnd the circumference of a circle, you just multiply the radius by π. So, if the radius is 4 inches, the circumference is 4 × π inches.” a

Identify the error in Alex’s statement and correct it by providing the correct formula.

Explain the relationship between the diameter, radius, and circumference in simple terms.

Caitlin and David calculate the circumference of this circle using different formulas. David uses the formula C = 2π r to calculate the circumference, where r is the radius. Caitlin uses the formula C = π d to calculate the circumference.

10 cm

Explain why they will get the same result.

SOL

A pie has a radius of 12 inches. Which is closest to the circumference of this pie? Use 3.14 for π to calculate the circumference. A

37.68 in

75.36 in

24.56 in

50.24 in

Jordan has a circular pool with a diameter of 15 ft. Jordan plans to install a safety barrier around the pool. Complete the statement about Jordan’s circular pool. To install the safety barrier around the pool, Jordan needs a minimum of ⬚ feet of barrier.

The bottom of a ﬂower pot has a radius of 16 cm. What is the circumference of the bottom of the ﬂower pot? Use 3.14 for π and round your answers to one decimal place.

A scooter tire has a diameter of 34 cm. What is the circumference of the tire? Use 3.14 for π and round your answer to one decimal place.

What is the length of the strip of seaweed around the outside of the sushi? Use 3.14 for π and round your answers to one decimal place.

Find the circumference of the Ferris wheel:

15 mm

Use 3.14 for π and round your answers to one decimal place. 8m

8.02 Circumference and pi mathspace.co

661

Let’s extend our thinking 25

Describe what happens to the circumference when the radius of a circle is doubled. Use examples to explain your thinking.

A shallow circular wading pool has a diameter of 4 yards. A deeper circular swimming pool has a radius of 16 yards. How many times greater is the circumference of the swimming pool than the wading pool?

A circular running track has a diameter of 23 m. How many laps must be completed to run 1600 m? Round your answer to one decimal place.

Think of a real-life situation that involves the distance around a circle. Write a story problem about it. Use what you know about circumference to ﬁnd out something interesting in your story. Solve the problem and show your solution.

662

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

12 a Y es, because the circumference is close to π × diameter, which is approximately 3.14 × 14 cm. b N o, because the circumference is just the product of the diameter and π, which does not match the given circumference.

8.02 Circumference and pi What do you remember?

b c

c b

d e

c N o, because the circumference should be π × 30 cm, which does not equal 90 cm. The correct circumference should be closer to 94.2 cm.

b True

c false

d True

d Y es, because the circumference is close to 3.14 × 20 cm, which is approximately 62.8 cm.

1 Circumference 2 a a e d 3 a false e False Radius (cm)

Diameter (cm) 4

12.57

50.27

257.61

170

340

1068.14

b True

c False

d True

b True

c True

d False

e True 5 a False

should give a circumference of 16π and not

Circumference (cm)

4 a False

e No, because when you multiply the diameter by π, it

es, because the circumference is calculated by Y π × diameter, which is approximately 25.12 cm

13 C ÷ 12 14 For each circle, the ratio

Let’s practice 6 a 50.24 cm

b 113.10 cm

c 31.40 cm

d 43.96 cm

7 a 43.96 ft

b 69.08 in

c 2.51 units

d 74.93 cm

8 The distance around the pool, or the circumference, is given by the formula C = 2π r, where r is the radius. Plugging in the given radius: C = 2 ⋅ 3.14 ⋅ 6.7 yards. Therefore, the distance traveled would be 2 ⋅ 3.14 ⋅ 6.7 yards, which simpliﬁes to 42.08 yards. 9 The distance the wheel will travel in one full revolution, or the circumference, is given by the formula C = π d, where d is the diameter. Plugging in the given dimeter:

is 3.14, which suggests the

value of π is approximately 3.14. 15 For each circle: C ÷ d = π 16 a C = 2π r

e False

cm.

b C = πd

17 a T he correct formula to ﬁnd the circumference of a circle is not just the radius multiplied by π. The actual formula is C = 2π r or C = π d. b T he diameter is twice the radius. The circumference can be found by multiplying the diameter by π or by multiplying r by 2 and by π. 18 They will get the same result because the diameter is always equal to twice the radius, so π d and 2π r are equivalent formulas. 19 B 20 47.1 ft 21 100.5 cm

C = 3.14 ⋅ 18 units.

22 106.8 cm

Therefore, the distance traveled would be 3.14 ⋅ 18 units, which simpliﬁes to 56.52 units.

23 94.2 mm 24 25.1 m

10 a 6.37 cm b 12.74 cm

Let’s extend our thinking

c T he circumference can be calculated using the formula C = 2π r. To ﬁnd the radius, r, divide the circumference C by 2, then divide again by 3.14. To ﬁnd the diameter, we can just divide the circumference by 3.14 or just multiply the radius by 2.

25 The circumference is also doubled. For example a circle with a radius of 2 in has a circumference of approximately 12.56 in. If we double the radius to 4 in, the circumference is 25.12 in which is also double the original circumference of 12.56 in

11 a 7.09 ft

26 8

b 14.18 ft

c T he circumference can be calculated using the formula C = 2π r. To find the radius, r, divide the circumference C by 2, then divide again by 3.14.

27 22.2 laps 28 Answers may vary.

To find the diameter, we can just divide the circumference by 3.14 or just multiply the radius by 2.

Answers mathspace.co

663

8.03 Area of a circle Subtopic overview Lesson narrative In this lesson, students will learn to calculate the area of a circle using the formula A = π r2, where r is the radius. They will start by understanding the space inside a circle and explore the relationship between the radius and the area. The lesson includes an interactive exploration where students unravel segments of a circle to visualize how the formula is derived. Students will solve problems involving finding the area given the radius or diameter, and apply these concepts to real-world situations, such as finding the area of a pizza. By the end, students should confidently use the formula to calculate the area of circles.

Learning objectives

8.03 Area of a circle

Students: Page 335

After this lesson, you will be able to... • solve mathematical and contextual problems involving the area of a circle when given the length of the diameter or radius.

Area of a circle We already know that area is the space inside a 2D shape. We can ﬁnd the area of a circle, but we will need a special Key vocabulary 

rule.  circle area a circle) circumference Let’s(of look at what happens when we unravel segments of a circle.

 radius

Interactive exploration

Essential understanding Explore online to answer the questions There is a constant relationship between the radius and area of any circle. This relationship can be applied to solve mathspace.co many real-world and mathematical problems. Use the interactive exploration in 8.03 to answer these questions. 1. Explain how the width of the shape relates to the circumference of the circle. Standards

2. What figure is formed? Explain how the area of this figure relates to the area of the circle. This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical We can calculateprocess the area ofgoals a circle using the formula: MPG1 — Mathematical Problem Solving

A = π r2

Teachers can integrate this goal into their lesson by providing students with various real-world problems that require A Area of the circle the calculation of the area of a circle, such as determining the amount of paint needed to cover a circular surface, or r Radius of the circle the area of a circular garden plot. Teachers can encourage students to apply the formula for the area of a circle and to select appropriate strategies to solve the problems. Students can be guided to create their own problems using real-world data, Example 1 which they can then solve using their mathematical skills. Find the area of the circle shown, correct to one decimal place.

664

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6 cm

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can show students how the concept of area of a circle is connected to their prior knowledge of geometric shapes and their properties. Teachers can also create mathematical connections by linking the concept of area to other real-world topics and situations. For instance, teachers could discuss how the area of a circle is used in determining the capacity of a cylindrical container, or the amount of material needed for a round tablecloth, thereby establishing a connection between mathematics and the real world.

Teachers can integrate this goal into their instruction by teaching students how to represent the concept of the area of a circle in various forms. For instance, they can show how to represent the formula A = π r2 graphically, using a diagram of a circle with the radius marked. Teachers can also encourage students to use different types of numbers (whole numbers, decimals, fractions) in their calculations, thus demonstrating that different representations can convey the same mathematical idea. Furthermore, teachers can guide students to interpret these representations in real-world contexts, such as calculating the area of a circular garden or a pizza, enabling students to see representation as both a process and a product.

Content standards 6.MG.1 — The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area.

6.MG.1e — Solve problems, including those in context, involving circumference and area of a circle when given the length of the diameter or radius.

Prior connections 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Future connections G.PC.3 — The student will solve problems, including those in context, by applying properties of circles.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 8.01 Characteristics of circles Grade 6 — 8.02 Circumference and pi

Tools You may find these tools helpful: • Scientific calculator

8.03 Area of a circle mathspace.co

665

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Use algorithmic thinking to create steps for finding the area of a circle Targeted instructional strategies Encourage students to develop an algorithmic approach to calculating the area of circles, accounting for different given measurements or, as an extension, partial circles. Guide them to create a step-by-step procedure that handles cases where the diameter is provided instead of the radius, and for circles that are halved or quartered. Have them practice this algorithm with various examples to test and refine their procedure. Here is an exemplar set of steps they might develop: 1. Write out the given measurements • If the radius is given, use as is • If the diameter is given, determine radius using: r = 2. Write out the formula • If a whole circle is not given, then rewrite the area formula with an appropriate fraction in front. 3. Substitute all the known values into the formula. 4. Calculate the area by evaluating the expression 5. State the final area, including appropriate units and rounding if necessary.

Student lesson & teacher guide Area of a circle Students recall that area is the space inside a 2D shape. They will begin the lesson by using an applet to explore the area of a circle.

Students: Page 335

8.03 Area of a circle After this lesson, you will be able to... • solve mathematical and contextual problems involving the area of a circle when given the length of the diameter or radius.

Area of a circle We already know that area is the space inside a 2D shape. We can ﬁnd the area of a circle, but we will need a special rule. Let’s look at what happens when we unravel segments of a circle.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.03 to answer these questions.

666

1. Explain how the width of the shape relates to the circumference of the circle. Mathspace Virginia SOL Grade 6 Teacher Edition 2. What figure is formed? Explain how the area of this figure relates to the area of the circle. mathspace.co

We can calculate the area of a circle using the formula:

After this lesson, you will be able to... • solve mathematical and contextual problems involving the area of a circle when given the length of the diameter or radius.

Area of a circle Exploration We already know that area is the space inside a 2D shape. We can ﬁnd the area of a circle, but we will need a special

Students:rule. Page 335

Let’s look at what happens when we unravel segments of a circle.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.03 to answer these questions. 1.

Explain how the width of the shape relates to the circumference of the circle.

What figure is formed? Explain how the area of this figure relates to the area of the circle.

We can calculate the area of a circle using the formula:

A = π r2 Suggested student grouping: In pairs A Area the circle Students will manipulate the applet to explore how theofarea of a circle can be approximated by slicing it and Radius of theformula circle rearranging the pieces. The aim is for students rto derive the for the area of a circle. Ideal student responses Example 1 These ideal responses may differ from other correct student responses. Less formal responses can be Findwith the area the circle shown, correct to one language decimal place. connected theofmore precise mathematical presented here.

8.03 Area of a circle

1. Slide the slider to unravel the circle. Explain how the width of the shape relates to the circumference of the circle. The width of the shape is half the circumference of the circle, which is π r. 2. What figure is formed? Explain how the area of this figure relates to the area of the circle. 6 cm this lesson, you will be able to... The figure is After a parallelogram. The area of the parallelogram is equal to the area of the circle. That is, the area • solve mathematical and contextual2 problems involving the area of a circle when given the length of the of the parallelogram and circle are A = π r . diameter or radius.

Purposeful questions • WhatCreate is the aarea of a parallelogram? strategy Apply the idea of a of circle • WhatArea is the this The areabase of a circle can parallelogram? be found using theWhat is the height? A = π ⋅ (6)2 Substitute r = 6 2 that area is the space inside a 2D shape. We can ﬁnd the area of2 a circle, but we will need a special • HowWe doalready we Afind formula: =know π rthe . area of a circle using the area of the shape?= 113.1 cm Evaluate rule.

PossibleLet’s misunderstandings look at what happens when we unravel segments of a circle. • Students may not identify the shape as a parallelogram, or they might not remember that the area of a 8.03the Arearelationship of a circle 335 Interactive parallelogram is found by Aexploration = bh. Use the questions above to help them understand mathspace.co Explore online to answer the questions between the characteristics of the circle and the area of the parallelogram.

mathspace.co

Following the exploration, students are formally introduced to the formula for finding the area of a circle, A = π r2, the interactive exploration in 8.03 to answer these questions. where r is theUse radius of the circle. 1.

Explain how the width of the shape relates to the circumference of the circle.

Students: Page 335figure is formed? Explain how the area of this figure relates to the area of the circle. 2. What We can calculate the area of a circle using the formula:

A = π r2 A Area of the circle r

Radius of the circle

Example 1 Find the area of the circle shown, correct to one decimal place.

8.03 Area of a circle mathspace.co 6 cm

667

Explain how the width of the shape relates to the circumference of the circle.

What figure is formed? Explain how the area of this figure relates to the area of the circle.

We can calculate the area of a circle using the formula:

Examples

A = π r2 A Area of the circle

Students: Page 335

Radius of the circle

Example 1 Find the area of the circle shown, correct to one decimal place.

6 cm

Create a strategy

Apply the idea A = π ⋅ (6)2

The area of a circle can be found using the formula: A = π r2.

= 113.1 cm2

Purpose Students demonstrate that they can find the area of a circle given a radius.

Substitute r = 6 Evaluate

8.03 Area of a circle mathspace.co

335

Expected mistakes Students may multiply by π before squaring the radius or try to multiply r = 6 by 2 instead of by itself. Remind students that the order of operations states that exponents must be evaluated before multiplication or that an exponent represents repeated multiplication. Reflecting with students Encourage students to consistently use appropriate units throughout their calculations to promote mathematical precision. When substituting the radius into the area formula A = π r2, prompt them to include the units, writing A = π (6 cm)2. This shows that they are squaring both the number and the unit, leading to an area in square centimeters.

Students: Page 336

Example 2 If the diameter of the circle is 24 cm, ﬁnd its area correct to one decimal place.

Create a strategy Remember that the radius of a circle is half its diameter, r =

Apply the idea r=

Divide the diameter by 2

= 12

Evaluate

A = π ⋅ 122

Substitute r

= 452.4 cm2

Evaluate

Example 3 668

Carlo and his friends ordered a pizza on a Saturday night. Each slice was 10 cm in length. Mathspace Virginia SOL Grade 6 Teacher Edition Find the area of the pizza that Carlo and his friends ordered. Use 3.14 to approximate π. mathspace.co

Create a strategy The length of the pizza is the radius of the pizza. Use the formula of the area of the circle to ﬁnd the area of the pizza.

Purpose Students demonstrate that they can find the area of a circle given its diameter. Expected mistakes Students may not divide the diameter in half, and instead use the diameter in the formula for the area. Ask them which characteristic of the circle is given, and what relationship it has to the radius, which is the value needed for the formula. Reflecting with students Challenge advanced learners to find a way to rewrite the formula for the area of a circle using d instead of r. Since r =

, the formula would become A =

d2.

Critique, correct, and clarify

use with Example 2

English language learner support 2 the following incorrect solution to the problem: Present Example students with 2 If the diameter of the circle is 24 cm, ﬁnd itsA area decimal × 242to≈one 1809.6 cmplace. = πcorrect

Ask students individually or in pairs to analyze the solution and identify any errors. Encourage them Createtoa work strategy to think about the meanings of “radius” and “diameter” and how they are used in the formula for the area of a Remember that the radius of a circle is half its diameter, r = . circle. Guide students to recognize that the number 24 was mistakenly used in place of the radius. theexplain idea StudentsApply should that the radius is 12 centimeters, not 24 centimeters. Have students correct the error by recalculating the area usingDivide the correct radius: A = π × 122. This activity helps students clarify their r= the diameter by 2 understanding of key vocabulary and reinforces the proper use of the area formula for circles. = 12

Evaluate

A = π ⋅ 122 = 452.4 cm Students: Page 336

Substitute r 2

Evaluate

Example 3 Carlo and his friends ordered a pizza on a Saturday night. Each slice was 10 cm in length. Find the area of the pizza that Carlo and his friends ordered. Use 3.14 to approximate π.

Create a strategy The length of the pizza is the radius of the pizza. Use the formula of the area of the circle to ﬁnd the area of the pizza.

Apply the idea A = 3.14(10)2 = 314

Substitute the value of the π and the radius Evaluate

The area of the pizza is 314 cm2.

Idea summary Purpose Area a circle r2 Students demonstrate that they can find the area of aofcircle given = theπradius. r

Radius of the circle

Reflecting with students Ask students how their answer would differ if they were asked for the exact area. Then, ask them if the answer would be different if they had used more decimal places in their approximation for pi. Remind students that Practice using more decimal places in the approximation of pi will make the answer closer to the exact value of the area.

What do you remember? 1

Describe the difference between area and circumference of a circle.

What does each part in the formula A = π r2 stand for?

8.03 Area of a circle mathspace.co

669

Divide the diameter by 2

= 12

Evaluate

A = π ⋅ 122

Substitute r

= 452.4 cm2

Evaluate

Draw a diagram to support visual-spatial processing

use with Example 3

Student with disabilities support Example 3 Encourage students to draw a diagram of the scenario described in the problem. Have them sketch a circle to represent theand pizza, with the center labeled and diameters to 10 represent pizza slices. Instruct students to Carlo his friends ordered a pizza on a Saturday night. Eachdrawn slice was cm in length. label theFind 10 the cmarea length of a pizza slice and ask what characteristic of the circle this of the pizza that Carlo and his friends ordered. Use 3.14 to approximate π. length represents. By visualizing the circle and the length of the slices, students can better grasp that the radius is given, and they a strategy can useCreate that length with the formula to find the area. This visual aid can bridge the gap between the abstract The length of the pizza is the radius of the pizza. Use the formula of the area of the circle to ﬁnd the area of the pizza. mathematical formulas and the concrete physical situation, making it easier for students with visual-spatial processing difficulties to understand and solve the problem. Apply the idea

A = 3.14(10)2

Substitute the value of the π and the radius

= 314

Evaluate

Students: Page 336

The area of the pizza is 314 cm2.

Idea summary

Area of a circle = π r2 r

Radius of the circle

Practice What do you remember?

Practice 1

Describe the difference between area and circumference of a circle.

Students: 2Pages What 336–339 does each part in the formula A = π r2 stand for?

What do you remember? 1

Describe the difference between area and circumference of a circle. 336 Mathspace Virginia SOL Grade 6

What does each part in the formula A = π r2 stand for?

Find the area of the following circles to one decimal place:

mathspace.co

8 cm

670

c 10 cm

d 14 cm

Find the area of the following circles, correct to one decimal place. Use 3.14 for π. a

A circle of radius 5 in.

A circle of diameter 8 in.

A circle has a radius of 4 yd.

A circle has a radius of 12 ft.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6 cm

Let’s practice 5

SOL

Match each item with its correct area. Use π = 3.14. a

Radius = 6 in

314 in2

Diameter = 14 in

113.04 in2

Radius = 10 in

iii

153.86 in2

Consider each circle. Use 3.14 for π in the calculation. i

Calculate the radius, correct to two decimal places.

Now, calculate the area, correct to two decimal places.

A circle with diameter of 12 in.

A circle with circumference of 14π ft.

A circle with circumference of 18 yd.

A circle with diameter of 22 mm.

The diameter of a circular baking tray is 10 in. Find its area, correct to two decimal places.

10 in

A gardener is creating a circular ﬂower bed. The radius of the bed will be

feet. She plans to lay mulch over the bed and install edging around it.

How much mulch will the gardener need to cover the entire ﬂower bed?

What is the minimum length of edging required to go around the ﬂower bed?

Aria is designing a round patio for her backyard. She wants the patio to ﬁt in a circular area with an area of 30 square feet. If the diameter of the patio is 6 feet, will her patio ﬁt? How do you know?

A city planner is mapping out a circular park with a diameter of 12 feet. If each square foot of park area requires 1.5 pounds of grass seed, how much grass seed is needed to cover the entire park? (Use π ≈ 3.14)

By how much greater is the coverage area of the Tawas Point Lighthouse compared to the coverage area of the Au Sable Pierhead Lighthouse?

13 mi

Round your answer to the nearest whole number. Au Sable Pierhead Lighthouse 18 mi Tawas Point Lighthouse

8.03 Area of a circle mathspace.co

671

Two students calculated the area of this circle differently. Who is correct? Justify your thinking. Diana’s method:

Daniel’s method:

A = π r

A = π r2

= π (11)2

= 3.14(11)2

= 121π

= 3.14(121)

11 cm

= 379.94

Danielle is deciding between buying a large pizza with a diameter of 46 centimeters for $18, or two medium pizzas with a diameter of 31 centimeters for $9 each. She thinks that buying two medium pizzas will give her more pizza for the same price. Is her decision justiﬁed? Explain your reasoning.

The engineering team at Rocket Surgery are building a rocket for an upcoming Mars mission. A critical piece is the circular connective disk that connects the booster rocket to the rest of the spacecraft. This disk must completely cover the top of the booster rocket. The booster rocket has a diameter of precisely 713.5 centimeters. Answer the following correct to two decimal places. a

Find the exact value of the required area of the connective disk, by using the π button on your calculator.

Instead of using the exact value, an engineer uses the approximation 3.14 for π. Find the area using the engineer’s approximation for π.

If the connective disk is more than 100 cm2 too small, the disk will malfunction, resulting in catastrophic launch failure.

Will the disk malfunction if it is built according to the engineer’s calculation? Explain your answer. 15

Find the area, rounded to one decimal place: a

5 mm

3 cm

11 m

d 7 cm

A cat is leashed to the pole in the middle of a yard. If the leash is 5 ft long, how much area can the cat roam? Explain how you found your answer using the formula for the area of a circle. 5 ft

672

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s extend our thinking 17

Jake has a bicycle. The area of its wheel is 144π in2. Find the radius, x, of the wheel. x

Find the radius of each circle, correct to two decimal places. Use 3.14 for π. a

A circle with area of 64π cm2.

A circle with area of 36 cm2.

A circle with area of 25 mm2.

A circle with area of 36π cm .

Find the diameter of each circle, correct to two decimal places. Use 3.14 for π. a

A circle has an area of 81 ft2.

A circle has an area of 144 mm2.

A ﬂowerpot has a circular base with an area of 88.25 in2. Find the radius of the base, correct to two decimal places.

A wind turbine has blades that are R m long which are attached to a tower 60 m high. When a blade is at its lowest point (pointing straight down), the distance between the tip of the blade and the ground is 20 m. a

Calculate the value of R.

Find the distance traveled by the tip of the blade during one full revolution, correct to two decimal places.

A factor in the design of wind turbines is the amount of area covered by their blades. The larger the area covered, the more air can pass through the blades.

60 m

Find the area inside the circle deﬁned by the rotation of the blade tips, correct to two decimal places.

8.03 Area of a circle mathspace.co

673

Answers

10 169.56 pounds 11 The coverage area of the Tawas Point Lighthouse is approximately 487 mi2 greater than the coverage area of the Au Sable Pierhead Lighthouse

8.03 Area of a circle What do you remember? 1 The area of a circle refers to the amount of space contained within its boundary and is calculated using the formula A = π r2, where A is the area and r is the radius of the circle. The circumference, on the other hand, is the distance around the circle’s edge and is determined by the formula C = 2π r, where C is the circumference and r again represents the radius. Essentially, the area measures the inside space of the circle, while the circumference measures the perimeter or the length of the circle’s boundary. 2 A represents the area of the circle and r represents the radius. 3 a 201.1 cm2

b 78.5 cm2

c 153.9 cm2 d 113.1 cm2

4 a 78.5 in2

b 50.2 in2

c 50.2 yd2

d 452.4 ft2

13 No. The area of the large pizza is: 1661.9 cm2 The area of two medium pizza is: 1508.3 cm2 Therefore, the large pizza has a greater area than the two medium pizzas combined. So, if Danielle is looking to get more pizza for the same price, she should choose the large pizza. 14 a 399 832.26 cm2 b 399 629.57 cm2 c Y es, because the approximate area is 202.69 cm2 smaller than the actual area. 15 a 14.1 cm2

Let’s practice 5 a ii

12 Both are correct; however, Diana’s method provides an exact symbolic representation of the area, using π while Daniel’s gives a numerical approximation and used π = 3.14.

b iii

c i

6 a i 6.00 in

ii 113.04 in2

b i 7.00 ft

ii 153.86 ft2

c i 2.87 yd

ii 25.80 yd2

d i 11.00 mm

ii 379.94 mm2

he amount of mulch needed to cover the entire ﬂower T bed is 7.068 58 ft2, approximately.

ii T he minimum length of edging required to go around the flower bed is 9.424 78 feet, approximately. 9 Yes, Antima’s patio will fit in the circular area. The area of a circle with a diameter of 6 feet is 28.2743 ft2, approximately. Since the area she has is 30 square feet, which is slightly larger than the patio’s area, the patio will fit within the designated area.

674

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

c 95.0 m2

d 115.5 cm2

16 The cat can roam an area of approximately 78.54 ft2 around the pole. The length of the leash corresponds to the radius of the circle. A = π r2 = π(5)2 = 25π ≈ 78.54 ft2

7 78.54 in2 8 i

b 19.6 mm2

Let’s extend our thinking 17 12 in 18 a 8.00 cm

b 3.39 cm

19 a 10.16 ft

b 13.54 mm

c 6.00 cm

d 2.82 mm

20 5.30 in 21 a 40 m

b 251.33 m

c 5026.55 m2

Topic 8 Assessment: Circles 1

Name the indicated part of each circle. a

Find the measurements for each circle. Use 3.14 for π and round your answers to two decimal places: i

Find the area.

Find the circumference.

13 cm

6 cm

Find the circumference of each circle. Use 3.14 for π and round your answers to one decimal place. a

A circle of diameter 47 in

The radius of a circular baking tray is 9 cm. Find its area, use 3.14 for π correct to two decimal places.

Consider a circle with a radius of 3 cm. a

SOL

A circle of radius 30 yd

What is the diameter of the circle?

What is the circumference of the circle?

Consider a circle with an unknown radius of x cm. a

What is the diameter of the circle? Express in terms of x.

What is the circumference of the circle? Express in terms of x and π.

Elijah measured a circular lid and found d, the diameter, was 6 in and C, the circumference, was 20 in.

C = 20 in

Which expression represents an approximate value for π?

SOL

20 + 6

20 ÷ 6

20 ⋅ 6

20 − 6

Write down the equation for the circumference of a circle if you know the radius.

Write down the equation for the circumference of a circle if you know the diameter.

Amy measured the circumference and diameters of some circular objects she found. She was also interested in the ratio of circumference to diameter for the different circles. a

Using her data, what can you say about the ratio of the circumference to the diameter of a circle?

Amy lost the data for the last row of the table. Use patterns to help her ﬁnd the missing values.

d = 6 in

Circumference (C)

Diameter (D)

30.2 20 18.4 14.5

10 16.5 5.9

Topic 8 Assessment: Circles mathspace.co

3.2 3.2 3.1

675

Performance Task 10

A local middle school wants to build a track around their football ﬁeld. They have drafted a blueprint with the dimensions of the track. The ﬁve-lane track shown is made up of a rectangle and two semicircles. 580 ft

60 ft

300 ft

1 2345

280 ft

676

What is the area of the inner ﬁeld, including all the space within the track? Round your answer to the nearest whole foot.

What is the area of the track, including only the lanes and not the inner ﬁeld? Round your answer to the nearest whole foot.

What is the perimeter of the outside of the entire track? Round your answer to the nearest whole foot. Explain your reasoning.

Today, many tracks are made from polyurethane. Every 10 square feet of polyurethane costs $25. How much will it cost to buy the polyurethane for the entire track?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers Topic 8 Assessment: Circles 1 a radius

b diameter

c circumference

d chord

6.MG.1a

8 a C = 2π r

b C = πd

6.MG.1d 9 a T he ratio of of circumference to diameter is estimating π and is very close to 3.14. b

Circumference (C)

Diameter (D)

2 a Area: 113.04 cm2

30.2

3.2

16.5

3.2

18.4

5.9

3.1

14.5

4.6

3.14

Circumference: 37.68 cm 2

b Area: 132.67 cm

Circumference: 40.82 cm

6.MG.1e

6.MG.1biii, 6.MG.1c

3 a 188.4 yd

b 147.6 in

6.MG.1e

10 a 75 834 ft2 2

b 78 839 ft2

4 254.34 cm 6.MG.1e 5 a 6cm

b 18.84 cm

6.MG.1bi, 6.MG.1bii, 6.MG.1biii 6 a 2x cm

Performance Task

b 2π x cm

c T he perimeter is 1502 ft. You can ﬁnd it by ﬁnding the perimeter of the curved side of both semicircles and adding the outside edge of each straightaway. d $197 097.50 6.MG.2b, 6.MG.1e, MP1, MP4

6.MG.1bi, 6.MG.1bii, 6.MG.1biii 7 B 6.MG.1c

Answers mathspace.co

677

9 Statistics Big ideas • Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. • Different representations of data highlight different characteristics of the data. • Many patterns can be found in sets of data. These patterns can be useful in making inferences but any inferences made from a set of data should not be taken as fact.

Chapter outline 9.01 9.02 9.03 9.04 9.05 9.06 9.07

Formulate questions and collect data (6.PS.1) Create and interpret circle graphs (6.PS.1) Compare representations of data (6.PS.1) Review: measures of center and spread Mean as a balance point (6.PS.2) Changing data values and measures of center (6.PS.2) Outliers (6.PS.2)

682 704 730 755 771 788 806

The oldest tree, a bristlecone pine, is over 4,800 years old– an outlier in tree ages!

9. Statistics Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 3.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.

5.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on line plots (dot plots) and stem-and-leaf plots.

5.PS.2 — The student will solve contextual problems using measures of center and the range.

Big ideas and essential understanding Many patterns can be found in sets of data. These patterns can be useful in making inferences but any inferences made from a set of data should not be taken as fact. 9.04 — Measures of center provide useful information when interpreting data. Different measures approximate 9.06 — Adding, removing, or changing a single value in the “center” of a data set in different ways; each measure a data set can significantly impact the measures of center. provides different insights into a set of data and is appropriate in different situations. Different representations of data highlight different characteristics of the data. 9.05 — Measures of spread provide useful information 9.02 — In a circle graph, all the data is combined to make when interpreting data. Different measures approximate a single whole with the different sectors representing the “spread” of a data set in different ways; each different categories. The larger the sector, the larger measure provides different insights into a set of data and percentage of the data points that category represents. is appropriate in different situations. Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. 9.01 — Representative samples are crucial if a data set will be used to make predictions and decisions.

9.03 — Data displays can provide information but also make it possible to draw conclusions and make generalizations.

9.07 — An outlier is an abnormal distance from the rest of the values in a data set. An outlier can sometimes give insight into the variable being explored but it can sometimes cause incorrect conclusions to be drawn from the data.

Standards 6.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs.

680

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

6.PS.1a — Formulate questions that require the collection or acquisition of data with a focus on circle graphs. 9.01 Formulate questions and collect data 9.02 Create and interpret circle graphs 9.03 Compare representations of data

6.PS.1b — Determine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments). 9.01 Formulate questions and collect data 9.02 Create and interpret circle graphs 9.03 Compare representations of data 6.PS.1c — Determine the factors that will ensure that the data collected is a sample that is representative of a larger population. 9.01 Formulate questions and collect data 9.03 Compare representations of data 6.PS.1d — Organize and represent data using circle graphs, with and without the use of technology tools. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20,

, or 35%).

9.02 Create and interpret circle graphs 9.03 Compare representations of data 6.PS.1e — Analyze data represented in a circle graph by making observations and drawing conclusions. 9.02 Create and interpret circle graphs 9.03 Compare representations of data

6.PS.1f — Compare data represented in a circle graph with the same data represented in other graphs, including but not limited to bar graphs, pictographs, and line plots (dot plots), and justify which graphical representation best represents the data. 9.03 Compare representations of data 6.PS.2 — The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed. 6.PS.2a — Represent the mean of a set of data graphically as the balance point represented in a line plot (dot plot). 9.05 Mean as a balance point 6.PS.2b — Determine the effect on measures of center when a single value of a data set is added, removed, or changed. 9.06 Changing data values and measures of center 6.PS.2c — Observe patterns in data to identify outliers and determine their effect on mean, median, mode, or range. 9.07 Outliers

Future connections 7.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on histograms.

8.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.

A2.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on univariate quantitative data represented by a smooth curve, including a normal curve.

9. Statistics mathspace.co

681

9.01 Formulate questions and collect data Subtopic overview Lesson narrative In this lesson, students will learn to formulate questions and collect data, exploring the data cycle which involves formulating questions, collecting data, displaying data, and explaining data. They will distinguish between categorical and numerical data and practice identifying whether data is discrete or continuous. Students will use methods such as surveys, observations, and experiments to gather data. An exploration activity will guide them through formulating a question, identifying data types, describing data collection processes, collecting and representing data visually, and interpreting the results. By the end, students should confidently create well-formulated questions and collect representative data effectively.

Learning objectives Students: Page 342

Key vocabulary 

attribute



categorical data



data cycle



discrete numerical data



experiment



measurement



numerical data



observation



population



representative sample



sample



secondary data



survey



well formulated question

Essential understanding Representative samples are crucial if a data set will be used to make predictions and decisions.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

682

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Mathematical process goals MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can achieve the goal of Mathematical Communication by encouraging students to communicate their reasoning through verbal explanation and written presentation. For example, when students are exploring the data cycle, they can be asked to present their work to the class, explaining the steps they took in formulating questions and collecting data. This not only reinforces the students’ understanding but also fosters a collaborative learning environment where students learn from each other’s explanations.

Teachers can incorporate Mathematical Connections into their instruction by linking the new concept of circle graphs to students’ prior knowledge of line plots and stem-and-leaf plots. Teachers can discuss the similarities and differences between these representations, illustrating how different mathematical concepts are interconnected. Additionally, teachers can draw connections between the use of circle graphs in mathematics and their application in other subjects such as science or geography, further reinforcing the interdisciplinary nature of mathematics.

6.PS.1b — Determine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).

6.PS.1a — Formulate questions that require the collection or acquisition of data with a focus on circle graphs.

6.PS.1c — Determine the factors that will ensure that the data collected is a sample that is representative of a larger population.

Prior connections 3.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.

Lesson Preparation Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

9.01 Formulate questions and collect data mathspace.co

683

Understanding the data cycle Targeted instructional strategies Start with a brief discussion about the data cycle, which includes planning, collecting, processing, and interpreting data. Ask students to list each part of the data cycle and describe in their own words what each part entails. This image could be posted publicly as a reminder. Formulate questions

Analyze and communicate results

The data cycle

Collect or acquire data

Organize and represent data

Possible responses could be: • Formulate questions • Finding a context we want to investigate • Write a statistical question for the investigation • Collect or acquire data • Deciding whether we need categorical or numerical data • Write survey questions and give it to a random sample of people • Measure objects or quantities • Plan an observation strategy • Design a scientific experiment • Research online • Organize and represent data • Draw a graph or diagram • Analyze and communicate results • Summarize findings • Answer the statistical question

Collect and display English language learner support As students are working, note how students describe the concepts of “categorical data”, “discrete numerical data”, “sample” and “population” and how they relate this to “representative sample.” Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: Categorical data: • Data is grouped by characteristics • The frequencies of characteristics are counted Discrete numerical data: • Data values are a list of numbers • Data can only be specific numbers, not just any number

684

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Population: • Everyone we wish we could survey • Who or what the survey is about • Can be people, animals, objects, or cases Sample: • Some people or objects from the population • Subset of the population • Represents the population • Who we can actually access to survey Representative sample • Part of a population that accurately reflects the members of the entire population • A small quantity that accurately reflects the larger entity

Student lesson & teacher guide Formulate questions Students will explore the data cycle—a systematic approach to formulating questions, collecting, organizing, and analyzing data to draw meaningful conclusions. Students will learn to differentiate between categorical and numerical data, using various graphical representations to enhance understanding and communication of results.

Students: Pages 342–343

9.01 Formulate questions and collect data mathspace.co

685

This column graph shows the fruits that students have in their lunch one day.

Fruit for Lunch 10 9 8 7 6 5 4 3 2 1 0

Number

Notice that we have four categories Banana, Apple, Mandarin, and Pineapple and the graph helps us count how many items fall in each category.

rin

e in

Fruit

Discrete numerical data Data that can only take certain values and has a limited range of values. It can be displayed in line plots, step-and-leaf-plots, and line graphs. Example: Shoe size or number of siblings This line plot shows the shoe sizes of a group of students.

Notice that responses are restricted to possible shoe sizes, which are discrete numerical. We can see how many people wear each size from the line plot.

A clear question helps us know what kind of data to gather and who to collect it from. The type of question we ask can lead us to collect different data. The group we are hoping to answer the question about is called the population. When we write a question, it should be about the population we want to learn about and have more than one possible answer. Non-examples of questions How old is my neighbor? What brand is my computer? What is your favorite color?

Examples of well formulated questions What ages are the people in my neighborhood? What is the most popular brand of computer among my classmates? What colors are preferred by students in my neighborhood?

It needs to be clear which attributes we are exploring with our question. An attribute is a speciﬁc characteristic or feature of a given subject.

686

For example, if we want to learn more about pets in our class, we need to be clear which attribute we are interested in. These could include: • Number of pets • TypeVirginia of pets SOL Grade 6 Teacher Edition Mathspace • Size of pets mathspace.co • Age of pets

possible answer. Non-examples of questions How old is my neighbor? What brand is my computer? What is your favorite color?

It needs to be clear which attributes we are exploring with our question. An attribute is a speciﬁc characteristic or feature of a given subject. For example, if we want to learn more about pets in our class, we need to be clear which attribute we are interested in. These could include: • Number of pets • Type of pets • Size of pets • Age of pets

Examples and non-examples of statistical questions Targeted instructional strategies To formulate effective statistical questions, students may benefit from a discussion of what makes a statistical question “good” or “bad.” Using the examples and non-examples from the lesson: Non-examples of questions

9.01 Formulate questions and collect data mathspace.co

343

Examples of well formulated questions

How old is my neighbor?

What ages are the people in my neighborhood?

What brand is my computer?

What is the most popular brand of computer among my classmates?

What is your favorite color?

What colors are preferred by students in my neighborhood?

Discuss with students why these are good or bad examples. Some discussion points might be: • The first statistical question makes the intent of the investigation clear; we want to investigate the ages of people in the neighborhood. To answer the question, we will need to collect numerical data from various people, then represent and analyze the data. • The second statistical question makes the intent of the investigation clear; we want to determine the most popular brand of computer. To answer the question, we will need to collect categorical data from various students, represent the data, and analyze the trend in the data. • “What is your favorite color?” is not a statistical question because there is a single response to this question. This is actually a survey question because it would help us collect data, but it does not lead us to investigate the data. After the discussion, encourage students to create their own examples and non-examples of statistical questions.

Examples Students: Page 344

9.01 Formulate questions and collect data mathspace.co

687

Purpose Show students how to identify discrete numerical data in real-world situations. Expected mistakes Students might select option C because the data would be numerical. Discuss with students how time can be represented by any number, so the data would be continuous instead of discrete.

Students: Page 344

Purpose Show students how to identify categorical data in practical scenarios.

688

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 345 Example 2 Is each question well formulated for the data cycle? Explain why or why not. a Who was the ﬁrst president?

Create a strategy A well formulated question should have a variety of possible answers and relate to a speciﬁc population.

Apply the idea

Example 2

Reﬂect and check

There is one answer and no clear population, so this is A related question we could use the data cycle for noteach a well formulated question forfor the data cycle. “What is the most common term length for a US Is question well formulated the data cycle? Explain is why or why not. president?” a Who was the ﬁrst president?

Create b How adostrategy the shoe sizes of 5th and 6th graders at my school compare? A well formulated question should have a variety of possible answers and relate to a speciﬁc population.

PurposeCreate a strategy Example 2 Show students how to question evaluateshould whether a question is well-formulated for the data cycle, emphasizing the Apply the idea A well formulated have a variety of possibleReflect answersand andcheck relate to a specific population. importance of having a variety of possible answers and a clear population. Is each question well formulated for the data cycle? Explain why or why not. There is one answer and no clear population, so this is A related question we could use the data cycle for not a well formulated question for the data cycle. Apply the idea a Who was the first president?

Students:This Page 345 is a well formulated question as shoe size is a clear

is “What and is thecheck most common term length for a US Reﬂect president?” This data would be discrete numerical.

attributeawith different possible answers. Create strategy A well formulated question should have variety of answers and relate to a speciﬁc population. b How do the shoe sizes of 5th and 6tha graders at possible my school compare? c How much money do professional athletes in the US make?

Apply the idea Create a strategy

Reﬂect and check

There is one answer and no clear population, so this is A related question we could use the data cycle for A well formulated question should have a variety of possibleApply answers relate to a specific population. Create a strategy theand idea not a well formulated question for the data cycle. is “What is the most common term length for a US There is a clear population of professional athletes in the Different athletes will make different amounts, so this president?” Apply the Reflect and US. Now weidea need to check if data could be collected to something wecheck could find data on. This is a well formulated give is a variety of answers.question as shoe size is a clear question. This a well formulated This data would be discrete numerical. attribute with different possible answers. b How do the shoe sizes of 5th and 6th graders at my school compare?

Create a strategy

c HowIdea muchsummary money do professional athletes in the US make? A well formulated question should have a variety of possible answers and relate to a speciﬁc population. We use the data cycle to formulate questions, then collect, show, and explain information. Depending on the PurposeCreatequestion being asked, the data may be categorical data or numerical a strategy Apply the idea data.

Apply the idea Reﬂect andwhen check conducting a statistical investigation. Show students the importance of asking well-formulated questions formulated question should have more possible answer relate to a population. There isAawell clear population of professional athletes in than the one Different athletes willand make different amounts, so this

This is a well formulated question as shoe size is a clear US. Now we need to check if data could be collected to attribute with different possible answers. Students:give Page 345 a variety of answers.

This data would be discrete numerical. something we could ﬁnd data on. This is a well formulated question.

Collect data

c How much money do professional athletes in the US make? When we have questions, we use different ways to collect data to find answers: Idea summary • Observation: Watching and noting things as they happen. CreateWe a strategy Apply show, the idea use the data cyclebirds to formulate questions, then collect, and explain • For example, watching at a feeder to see which type comes most often. information. Depending on the There isquestion a clear population of professional athletes inhow the long, Different athletes make different amounts, so this being asked, the dataout may bemuch, categorical data orhow numerical data. • Measurement: Using tools to find how or heavy will something is. US.•Now we need to check if data be collected to of asomething could find data on. This is a well formulated For example, using a ruler to could measure themore growth plant over we several A well formulated question should have than one possible answerweeks. and relate to a population. give a variety of answers. question. • Survey: Asking people questions to get information. • For example, asking classmates about their favorite school subject and recording the answers.

Collect data Idea summary

PurposeWhen we have questions, we use different ways to collect data to ﬁnd answers: We use the data cycle to formulate questions, then collect, show, and explain information. Depending on the 9.01 Formulate questions and data population 345 Observation: Watching and noting thingsaaswell-defined they happen. Students• demonstrate thatasked, they can identify question a collect specific question being the data may be categorical dataresearch or numerical data. that targets mathspace.co • For example, watching birds at a feeder to see which type comes most often. and can be answered with available data. have more than one possible answer and relate to a population. A well formulated question should • Measurement: Using tools to ﬁnd out how much, how long, or how heavy something is. • For example, using a ruler to measure the growth of a plant over several weeks. • Survey: Asking people questions to get information. • For example, asking classmates about their favorite school subject and recording the answers.

Collect data

When we have questions, we use different ways to collect data to ﬁnd answers: • Observation: Watching and noting things as they happen. • For example, watching birds at a feeder to see which type comes most often. Formulate questions 9.01 9.01 Formulate questions and collectand datacollect 345 data • Measurement: Using tools to ﬁnd out how much, how long, or how heavy something is. mathspace.co mathspace.co • For example, using a ruler to measure the growth of a plant over several weeks. • Survey: Asking people questions to get information. • For example, asking classmates about their favorite school subject and recording the answers.

689

c How much money do professional athletes in the US make? b How do the shoe sizes of 5th and 6th graders at my school compare?

Create a strategy Create a strategy

Apply the idea

Reﬂect and check

This is a well formulated question as shoe size is a clear attribute with different possible answers.

This data would be discrete numerical.

There is a clear population of professional athletes in the Different athletes will make different amounts, so this A well formulated have abe variety of possible answers and to a data speciﬁc US. Now we need question to check should if data could collected to something we relate could ﬁnd on. population. This is a well formulated Students:give Page 345 a variety of answers. question.

Idea summary

We use the data cycle to formulate questions, then collect, show, and explain information. Depending on the c Howquestion much money professional in the US make? beingdo asked, the dataathletes may be categorical data or numerical data. A well formulated question should have more than one possible answer and relate to a population.

Create a strategy

Apply the idea

There is a clear population of professional athletes in the US. Now we need to check if data could be collected to Collect give a varietydata of answers.

Different athletes will make different amounts, so this something we could ﬁnd data on. This is a well formulated question.

we have questions, we use different ways to collect data to ﬁnd answers: CollectWhen data • Observation: Watching and noting things as they happen. • For example, at a feeder to see which typeincluding comes mostobservation, often. Students are introduced towatching variousbirds methods of data collection, measurement, surveys, Idea summary • Measurement: Using tools to ﬁnd out how much, how long, or how heavy something is. experiments, •and the use of existing secondary data. The lesson discusses choosing realistic and ethical approaches Weexample, use the data to to formulate then show, and explain For usingcycle a ruler measurequestions, the growth of acollect, plant over several weeks.information. Depending on the to gather data that isAsking representative ofdata a broader populationdata to or ensure findings question being asked, the be categorical numerical data. are valid and unbiased. • Survey: people questions tomay get information. • For example, askingquestion classmates about their favorite subjectanswer and recording thetoanswers. A well formulated should have more than school one possible and relate a population.

Students: Pages 345–346

Collect data

9.01 Formulate questions and collect data

345

mathspace.co When we have questions, we use different ways to collect data to ﬁnd answers: • Observation: Watching and noting things as they happen. • For example, watching birds at a feeder to see which type comes most often. • Measurement: Using tools to ﬁnd out how much, how long, or how heavy something is. • For example, using a ruler to measure the growth of a plant over several weeks. • Survey: Asking people questions to get information. • For example, asking classmates about their favorite school subject and recording the answers.

9.01 Formulate questions and collect data mathspace.co

690

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

345

Exploration Students: Page 346

Suggested student grouping: Small groups This exploration guides students through the data cycle using their classroom as the population. Students will formulate a relevant question, decide whether to collect categorical or numerical data, and describe a realistic method for gathering this data. They will then collect, visually represent, and analyze the data to draw conclusions about their initial question, applying practical data handling skills learned in previous grades. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Formulate a question that you could easily collect data on. A simple and easily collectible question for a classroom setting might be: “What is your favorite type of snack?” This question is straightforward, allows for categorical responses, and is relevant to all students, ensuring active participation and easy data collection. 2. What type of data would be collected: categorical or discrete numerical? The type of data collected depends on the question formulated. If the question is about preferences or types (e.g., favorite subject), categorical data will be collected. If the question involves counting or measuring (e.g., number of books read in a month), then discrete numerical data will be collected. 3. Describe a realistic process for collecting the data. A realistic process for collecting data could involve creating a survey that includes questions relevant to the formulated query. This survey could be distributed either on paper during class or digitally through a platform like Google Forms. Ensure all students in the class participate to maintain representation. 4. Collect the data. After distributing the survey, collect the responses from all students. Ensure that every student has submitted their answers to achieve a complete dataset that reflects the entire classroom. 5. Represent the data visually. Depending on the type of data collected, use appropriate graphical representations. For categorical data, use bar graphs or pie charts. For discrete numerical data, line plots or histograms can be effective. Tools like spreadsheet software can aid in creating these visual representations.

9.01 Formulate questions and collect data mathspace.co

691

6. What does this data tell you about your original question? Analyze the visual representations to draw conclusions about the original question. For example, if the question was about favorite subjects, the graph will show which subjects are most popular, highlighting trends and preferences within the classroom. This analysis helps understand the broader interests or behaviors of the class. Purposeful questions • What are some characteristics of a question that is easy to collect data on? • Consider the different ways to collect data. Why do some methods work better than others for different scenarios?

Using real-world examples Targeted instructional strategies When teaching about data collection and representation, it can be beneficial to use real-world examples that students can relate to. This can make the abstract concept more concrete and relatable, therefore enhancing students’ understanding. For instance, you could discuss how businesses use data to make decisions or how scientists use data to draw conclusions in their studies.

Further decompose the data cycle into manageable chunks Student with disabilities support To support students in organizing their progress through the data cycle, provide a simple checklist that breaks down each step of the process. Consider that each step of the data cycle can be further broken down and scaffolded. The following could be given as a checklist or reworded as purposeful questions to help guide students through the data cycle. 1. Formulate questions • Select a context that you want to explore • Identify a particular characteristic of the population you want to focus on • Formulate a statistical question 2. Collect or acquire data • Choose the most appropriate method of data collection. • Acquire reliable secondary data • Survey • Measurement • Experiment • Observation • If using secondary data, find a reliable source. If collecting primary data, choose a representative sample. • Create a table, spreadsheet, online form, paper form, etc to collect the data. • Perform the research to collect the data. 3. Organize and represent data • Identify the most appropriate type(s) of data displays • Set up the axes for the data display • Create the data display by hand or using technology • Calculate measures of center (mean, median, mode) and range if the data is numerical

692

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

4. Analyze and communicate results • Identify any trends, patterns, or characteristics in the data like clusters, peaks, center, or outliers • Interpret the characteristics of the data in the context of the statistical question • Present the information needed to answer the statistical question • Identify any challenges with this round of the cycle and highlight any issues with the data like the sample being too small or not representative enough • Write an overall conclusion or prediction based on the data • Explain next steps and what another round of the data cycle could invovle.

Examples Students: Page 347 Example 3 Aditya wants to investigate the social media habits of the students in her grade. a Formulate a question to help her complete this investigation.

Create a strategy She could choose to explore which platforms are used, how much they are used, reasons for usage, and the impact on their academic and social lives.

Apply the idea

Reﬂect and check

A potential question could be, “How many hours per week do students in my grade spend on social media?”

The question can be answered by collecting data, allows for a variety of answers, and could be organized in a data display like a line plot answers are rounded to the nearest hour, so is a well formulated question.

Example 3

Aditya wants to investigate theneed social habits of the students in her grade. b What attributes would you tomedia measure to answer the question? a Formulate a question to help her complete this investigation.

PurposeCreate a strategy Create ahow strategy This question from part (a) is well written as itthat clearly states attribute. by collecting and analyzing data. Show students to formulate a question could bethe answered She could choose to explore which platforms are used, how much they are used, reasons for usage, and the impact

Apply the idea and social lives. Reﬂect and check on their academic Reflecting with students For each person,toshe wouldthe need to identify the time they Whenformulated collecting thewith data,the Aditya could collect the data Encoucage students share various questions that they class. Discuss how different Apply the idea Reﬂect and check spend across all social media platforms per day. would lead rounded to the nearest hour it discrete to or could people can formulate different questions which to different results, sotoitmake is important make the leavequestion it open-ended more options when displaying or potential question could be, “How many hours per The can be for answered by collecting data, allows intentionA of the investigation clear.

summarizing for a variety ofdata. answers, and could be organized in a data display like a line plot answers are rounded to the nearest Students: Page 347 hour, so isto a well formulated c Should she use observation, measurement, survey, or experiment collect the data?question. Explain. week do students in my grade spend on social media?”

Create strategywould you need to measure to answer the question? b Whataattributes She should choose a method that is practical, ethical, and will give reliable results.

Create a strategy Apply the idea Reﬂect and check This question from part (a) is well written as it clearly states the attribute. A survey would be the most appropriate method for this Apply the idea investigation.

The population is relatively small, so it would be possible Reﬂect andallcheck to ask them a single question about their social media For each person, she would need to identify the time they use. When collecting the data, Aditya could collect the data spend across all social media platforms per day. rounded toher theclassmates nearest hour make it discrete or could Observing allto day to make conclusions leave open-ended for more options when about ittheir social media use would likely notdisplaying be ethicalor or summarizing easy to do. data. c Should she use observation, measurement, survey, or experiment to collect the data? Explain.

Example 4

Create a strategy Collect datachoose that cana be used that to answer the question many cousins do students in my school have?” She should method is practical, ethical, “How and will giveﬁrst reliable results.

9.01 Formulate questions and collect data mathspace.co This situation would require survey. Depending on this the sizeThe of your school,isitrelatively could be small, done with sample you A survey would be the most aappropriate method for population so it awould be or possible could survey the whole population. investigation. to ask them all a single question about their social media use. It is not possible to do observation, measurement, or an experiment for this question.

Create a strategy Apply the idea

Reﬂect and check

693

Apply the idea Create a strategy

Reﬂect and check

A potential question could be, “How many hours per The question can be answered by collecting data, allows This question from part (a) is well written as it clearly states the attribute. week do students in my grade spend on social media?” for a variety of answers, and could be organized in a data display like a line plot answers are rounded to the nearest Apply the idea Reﬂect and check hour, so is a well formulated question. Purpose For each person, she would need to identify the time they When collecting the data, Aditya could collect the data Show students how to identify relevant attributes when attempting to answer a research question. spend across all social media platforms per day. rounded to the nearest hour to make it discrete or could b What attributes would you need to measure to answer the question? leave it open-ended for more options when displaying or Students: Page 347 summarizing data.

Create a strategy

This question from part (a) is well written as it clearly states the attribute. c Should she use observation, measurement, survey, or experiment to collect the data? Explain.

Apply the idea Create a strategy

Reﬂect and check

For each person, she would need to identify the time they When collecting the data, Aditya could collect the data She should choose a method that is practical, ethical, and will give reliable results. spend across all social media platforms per day. rounded to the nearest hour to make it discrete or could leave it open-ended for more options when displaying or Apply the idea Reﬂect and data. check summarizing A survey would be the most appropriate method for this The population is relatively small, so it would be possible investigation. to ask them all a single question about their social media c Should she use observation, measurement, survey, or experiment to collect the data? Explain. use. Observing her classmates all day to make conclusions about their social media use would likely not be ethical or She should choose a method that is practical, ethical, and will give reliable results. easy to do.

Create a strategy

Apply the idea

Reﬂect and check

A survey would be the most appropriate method for this Example 4 investigation.

The population is relatively small, so it would be possible to ask them all a single question about their social media Purpose use. Collect data that can be used to answer the question “How many ﬁrst cousins do students in my school have?” Students demonstrate that they can choose a practical and ethicalher method for data Observing classmates all daycollection. to make conclusions about their social media use would likely not be ethical or Create a strategy Students: Pages 347–348 easy to do. This situation would require a survey. Depending on the size of your school, it could be done with a sample or you could survey the whole population. It is not possible to do observation, measurement, or an experiment for this question.

Example 4

Collect data that can be used to answer the question “How many ﬁrst cousins studentsquestions in my school have?” 9.01do Formulate and collect data

347

mathspace.co

Create a strategy This situation would require a survey. Depending on the size of your school, it could be done with a sample or you could survey the whole population. It is not possible to do observation, measurement, or an experiment for this question.

9.01 Formulate questions and collect data mathspace.co

694

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

347

Purpose Show students how to collect data by conducting a survey and ensuring a representative sample for accurate results. Reflecting with students Prompt advanced students to extend the data collection activity by analyzing the collected data using statistical measures, including calculating the mean, median, mode, and range. Encourage them to represent their findings using various graphs like a stem-and-leaf plot to interpret distributions and identify any patterns. Finally, have them present their conclusions and justify their reasoning, fostering critical thinking and allowing them to connect their findings to broader societal or familial patterns.

Students: Page 348

Purpose Demonstrate the importance of representativeness in data collection and how it might be compromised. Expected mistakes Students might select option B because the sample is not representative of the entire population. While this is true, the statement says the adults were not randomly selected, which is not true. The sample was random, as Georgia did not influence the adults that came to drop off or pick up.

9.01 Formulate questions and collect data mathspace.co

695

Students: Page 349

Purpose Challenge students to identify potential populations from which a given sample could be drawn.

Students: Page 349

696

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice Students: Pages 349–353

What do you remember? 1

Match each term with its deﬁnition. a

Population

Sample

Categorical data

Discrete numerical data

Data that can be put in categories and may not have a speciﬁed order.

The entire group being considered in a statistical analysis.

iii

A selection of some of the population.

Data that is made up of numbers, can only take certain values, and has a limited range of values.

When formulating a question, is each statement true or false? a

The question is expected to have similar answer from all responders.

The answers to the question may vary from one person to another.

The answers to the question must have numerical values.

State whether or not each of the following questions are well formulated questions. a

How old is your principal?

How tall are the basketball players in your school?

Do you prefer burger or pizza?

Which city is the capital of the Philippines?

How does the proportion of the students at your school that prefer Math compare to the proportion that prefer Science?

Match each procedure with the type of method that was used. a

Observation

Isaiah asks each student in his dance class questions about their food preferences.

Harper uses a ruler to record data on the foot lengths of some of the students at her school.

iii

Faisal asks all of the teachers in his school to complete some tasks and counts how many they can get done. He plays music while some are doing the tasks and not for others and compares the differences between listening to music and not.

Lucas watches and counts how many people go into different stores at different times of day.

Measurement

Survey

Experiment

Hannah has chosen to collect information using a sample instead of surveying the whole population. State whether each statement is an advantage to doing a sample. a

It is cheaper to conduct.

We don’t need to worry about who we survey.

It’s more accurate.

It takes less time.

9.01 Formulate questions and collect data mathspace.co

697

Determine whether the following uses a sample or a population: a

Lucy has asked everyone in her oﬃce what snacks should be provided in the oﬃce.

James asks a few of his friends how they did in the test to see if he is above average in his class.

Joanne ﬁnds the height of the entire class to try to ﬁnd the average height of 15-year old students in America.

Justin has determined the age of 10% of houses from each suburb in Virginia.

Valentina tests every engine that the factory produces.

Oprah checks every dog brought to her vet to assess the treatment of dogs in the city.

Asking all the teachers at your school whether they approve of a new class timetable.

A taste test of a large batch of cookies Michael just baked.

A study is being done on physical activity in a small town. a

Would this characteristic be important to consider when selecting the sample? ii

Gender

iii Age

Favorite sport

Preferred method of transportation

Name

i v

Favorite color

Does the number of people in the sample matter?

Does how we select the people who will be the surveyed matter?

Classify each data set as discrete numerical or categorical. a

Types of vegetables

Brands of tablets

Daily UV index

Types of dogs

Number of siblings

Number of languages spoken at home

Let’s practice 9

698

For each well formulated question, will the collected data be categorical data, discrete numerical data, or neither? a

What sports are played by students at my school?

Precisely how long do runners take to complete a 5K race?

What attendance can be expected at a local baseball game?

What types of cats are the most popular?

How many televisions to people have in their homes?

Is each question well formulated for the data cycle? Explain why or why not. a

What are the favorite movies of sixth graders in my school?

How tall are the science teachers in your school?

Is your mom taller than your dad?

How far away is the moon from the Earth, right now?

How much do puppies weigh?

Have you ever visited Paris, France?

How old are US presidents on the day they are elected?

For each scenario, formulate a question to help them complete their investigation. a

Morgan wants to investigate the physical activity habits of the students in his grade.

Gunnar wants to investigate the impact of listening to music while doing homework.

Cadence wants to investigate the modes of transportation of the adults at her school.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A math class wants to answer the following question: “Are the 12-year olds in the class taller than the 11-year olds in the class?” a

Is this a well formulated question?

Which of the following attributes are needed to answer the question?

• Height • Age • Name • Physical activity level

Which of the following units could be used to measure someone’s height?

• Decibels • Degrees • Feet • Inches

Consider the question: “What is the expected cost of owning a dog? Does it vary by breed?” a

Is this a well formulated question?

What attributes would you need to measure to answer the question?

Determine whether the following attributes are represented by the given graph: a

Number of children who travel to school and their mode of transport

Number of people that prefer each mode of transport

Number of teachers who travel to school and their mode of transport

Key 1 graphic = 54 children

Bike

Car

Walk

Train

Bus

For each of the following samples, give an example of a population that the sample could have been chosen from. The samples may not be well selected. a

A sample containing 50 people who drive white cars.

A sample of 50 people drawn from a population. In this sample, the youngest is 4 years old, and the oldest is 18.

A sample containing the ﬁrst 50 people to enter a train station on a given day.

Beth is interested in which students from her school use public transportation. Determine whether or not the following sampling methods would result in samples that are representative of the population.

Selecting every 10th person on the bus she takes.

Selecting every 10th person on the student list.

Selecting the ﬁrst 50 students that arrive in the morning.

Selecting by having a computer randomly choose student numbers.

Petra is interested in the types of ﬂowers that are grown in her neighborhood. She formulates the question “What types of ﬂowers are most popular in my neighborhood?” a

Design a simple plan to collect data for the question using each method. i

Observation

Measurement

iii

Survey

Formulate a question involving ﬂowers that would require an experiment to collect data.

9.01 Formulate questions and collect data mathspace.co

699

Let’s extend our thinking 18

Change the following questions to make them well formulated questions for the data cycle. a

How many books does your teacher have?

How many points did the grade school basketball team score in its last game?

What is your grade in Mathematics during the ﬁrst term?

Nuala’s mom has started playing pickleball and Nuala is interested in learning more about different sports, especially ones she hasn’t heard of before. a

Formulate a question that Nuala could use to explore popularity of different sports.

Identify which attributes she would need to collect data on.

Identify the population.

Describe how she could collect data to help answer the question.

Georgia wants to know how the people in America are going to vote in an upcoming election. She selects 50 random people from her city to interview. Suggest reasons why this sampling might give poor results.

The government is thinking about increasing the minimum retirement age, but they want feedback on the change, so they plan a survey. The sample they selected was mostly people who are already retired because they were free during oﬃce hours of 9 am to 4 pm. Explain why this sample is not representative of the population and describe a better sample.

700

A political polling company calls 1000 people at home between 4 pm and 7 pm on weeknights to ﬁnd out who they are most likely to vote for in an upcoming election. They publish their numbers based on the responses of only the 410 people who answered their call. a

How could they have used random sampling to choose the 1000 people to call?

Explain why the company’s published results will not be accurate.

A random sample of 98 students were asked about their favorite sports: a

Using the sample data, which sport can we conclude is liked most by the students?

If there are 240 students at the school, about how many would you expect to have swimming as their favorite sport?

Sports Soccer Basketball Baseball Bowling Swimming

Number of students 15 35 10 8 30

There is a large apartment tower going up in Roxie’s neighborhood. She is curious about what types of homes like detached houses, semi-detached townhouses, and apartments are the most common now compared to 10 years ago. a

Formulate a question that relates to types of housing in the US.

Collect secondary data from a reliable source.

Create a data display, such as a bar graph, to organize the data.

Compare the displays for now and 10 years ago.

Make a conclusion about the types of housing that is available.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers 9.01 Formulate questions and collect data

c N o, this is not a good question for the data cycle. There is not enough variety in the possible answers, and you do not need to use any calculations to ﬁnd out the answer. d No

What do you remember? 1 a i i: Population is the entire group being considered in a statistical analysis.

e Yes

b iii: Sample is a selection of some of the population

g Yes

c i : Categorical data is data that can be put in categories and may not have a speciﬁed order d i v: Discrete numerical data is data that is made up of numbers, can only take certain values, and has a limited range of values. 2 a False

b True

c False

3 a No

b Yes

c No

d No

e Yes 4 a i v: Observation: Lucas watches and counts how many people go into different stores at different times of day. b i i: Measurement: Harper uses a ruler to record data on the foot lengths of some of the students at her school. c i : Survey: Isaiah asks each student in his dance class questions about their food preferences. d i ii: Experiment: Faisal asks all of the teachers in his school to complete some tasks and counts how many they can get done. He plays music while some are doing the tasks and not for others and compares the differences between listing to music and not. 5 a Yes

b No

6 a Population b Sample e Population f

Sample

7 a i No

ii Yes

v Yes

vi No

c No

d Yes

c Sample

d Sample

g Population h Sample iii Yes

iv Yes

11 a Answers vary. For example: What physical activity habits do students in my grade have? or How much time do student spend sitting per day? b Answers vary. For example: How does listening to music affect students’ productivity when doing homework? c Answers vary. For example: What modes of transportation do the adults at my school use? 12 a Yes

b Height and Age

c Feet and Inches 13 a Yes b T otal cost of the dog over its life (includes food, vet bills, toys, etc.) and which breed of dog 14 a Yes

b No

c No

15 a Example answers:

• Drivers at a drive-in theater.

• People stuck in traﬃc due to roadworks.

b Example answers:

• Students in a classroom.

• Kids registered for gymnastics

c Example answers:

• All people who catch the train in the morning on a given day.

• All people who catch public transport on a given day.

b Yes, the sample size needs to be big enough. c Y es, a sample should be randomly selected and represent the population.

b Categorical

16 a Not representative

b Representative

c Discrete numerical

d Categorical

c Not representative

d Representative

e Discrete numerical

8 a Categorical

Discrete numerical

Let’s practice 9 a Categorical c Discrete numerical

b Neither d Categorical

e Discrete numerical 10 a T his is a well formulated question because to answer this question, you would collect data by asking students about their favorite movies, and there would be variability in the data. The favorite movie would not be the same for every student.

17 a i M any possible answers. For example, going around the neighborhood to using an app to identify and record the types of flowers in each front yard or garden. ii Many possible answers. For example, measuring the area of flowerbeds for different types of flowers in the neighborhood. iii Asking a sanple of the neighborhood what their favorite types of flowers are. b F or the soil found in my neighborhood, what impact does fertiziler have on Hydrangeas?

b Yes

Answers mathspace.co

701

called: S2504 Physical Housing Characteristics for Occupied Housing Units and the data for 2022: ACS 5-Year Estimates Subject Tables and 2012: ACS 5-Year Estimates Subject Tables, we get this data:

Let’s extend our thinking 18 a A nswers will vary. A possible answer is How many books do teachers in our school have? b A nswers will vary. A possible answer is What is a typical number of points scored by the grade school basketball team in its games this season?

Data for 2022 Type of housing

c A nswers will vary. What are the grades of primary students in Mathematics during the ﬁrst term? 19 a What are the most popular sports in our community? b P ossible attributes include: sport name, number of participants or players this season, frequency of events or competitions, or attendance at ﬁrst match/event of the season. c The local community she is a part of. d N uala could conduct a survey within her local community about the sports they play and how frequently they attend events or competitions.

Percent

Total

Single detached

62.5%

78 642 189

Single attached

6.3%

7 879 434

2 apartments

3.3%

4 162 151

3 or 4 apartments

4.2%

5 325 871

5 to 9 apartments

4.5%

5 657 985

10 or more apartments

13.8%

17 382 021

Mobile home or other type of housing

5.3%

6 686 702

Data for 2012 Percent

Total

Single detached

Type of housing

63.2%

47 706 306

Single attached

5.9%

4 453 595

2 apartments

3.6%

2 717 448

• The sample size is too small for such a big population.

3 or 4 apartments

4.3%

3 245 840

21 The sample is not a representative of the population because retired individuals may have different concerns and priorities compared to the general working population. Their views on retirement age policies may not be the same as the views of those who are still working or who are not yet of retirement age. The sample may as not be large enough depending on how many people gave feedback.

5 to 9 apartments

4.6%

3 472 294

10 or more apartments

12.4%

9 360 098

Mobile home or other type of housing

6.1%

4 604 564

20 • The people of in one city are not a good representation of America.

c Percent estimate

Here is an example of a better sample. A sample where the population is divided into age groups and employment status, and then people are randomly selected from each group to ensure representation matches the population.

b 74

24 a W hat types of housing are most common now and 10 years ago? b W e can use the Census data which collects data on the type of structure. As of 2024, the most up to date data is for 2022, so 10 years before that is 2012. The categories are single detached house, single attached house (townhouse), buildings with 2 apartments, buildings with 3 or 4 apartments, buildings with 5 to 9 apartments, buildings with 10 or more apartments, and Mobile home or other type of housing. Using table

702

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

60 40 20

gle

sin

ta de

r r s d ts ts he ment eo en 10 o ts en n hompe ofg rtm rtm rt e a a a m ile r ty sin p ap ap art 2a r4 ap Mobothe hou o9 e r 3o 5t mo Type of housing

tt ea

sin

2012 Housing trends across the US 100 Percent estimate

23 a Basketball

22 a E xample answer: Assign a number to everyone who is eligible to vote and use a random number generator to decide who to call. b T he sample will not include anyone who was out for the evening for work or social reasons. And the results are based on less than half of the people they tried to contact, so is not representative of the sample’s opinions.

2022 Housing trends across the US 100

80 60 40 20 0

r r ts ts ts ed ed eo f en 10 o ts en en ch ch m m m a a n t t t t t ompe o g t r r r e e h a a a a d m p p p y e t a a il r t sin ar gle gle 2a r4 ap Mobothe hou o9 sin sin e r 3o 5t mo Type of housing

2022 and 2012 Housing trends across the US 2022 2012

Percent estimate

100 80 60 40 20 0

sin

d ts ts ts or or ed he en en en 10 nts me of ch tac o pe g tta rtm rtm rtm e h a a a a m y p p p e t a a il r t sin ar gle 2a r4 ap Mobothe hou o9 sin e r 3o 5t mo

e ed

Type of housing

d T he data displays for 2022 and 2012 show little difference in housing trends. Single detached housing remains above 60%, while all other housing types are below 20%. There is a slight decrease in single attached and mobile home or other types of housing, but a slight increase in 10 or more apartment buildings in 2022 compared to 2012. e T he most common type of housing 10 years ago and now is single detached housing, but these seem to be decreasing as large apartment building are increasing.

Answers mathspace.co

703

9.02 Create and interpret circle graphs Subtopic overview Lesson narrative In this lesson, students will learn to create and interpret circle graphs. They will explore how circle graphs, also known as pie charts, represent data as parts of a whole. Students will understand that each sector of a circle graph corresponds to a category, with the size of each sector proportional to the percentage of the total data. They will engage in an exploration activity where they analyze data on favorite board games, noting what they observe and infer from the graph. Students will practice creating circle graphs by hand and using technology, and interpret various examples to draw conclusions about the represented data. By the end, students should be proficient in both creating and interpreting circle graphs to represent and analyze data.

Learning objectives Students: Page 354

Key vocabulary 

certain



circle graph (pie chart)



equally likely



impossible



likely



probability



proportion



sector (of a circle)



unlikely

Essential understanding In a circle graph, all the data is combined to make a single whole with the different sectors representing different categories. The larger the sector, the larger percentage of the data points that category represents.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

704

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Mathematical process goals MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can incorporate this goal into their instruction by faciliating discussions among students when they are creating circle graphs. Teachers canask students questions about the title, labels, totals and percentages when creating their data displays and encourage students to check and make sure their circle graph addresses the question they are trying to answer.

Teachers can make mathematical connections by relating the creation and interpretation of circle graphs to students’ prior knowledge of data collection and representation. They can also connect this concept to real-world applications by providing examples of situations where circle graphs can be used to represent and analyze data. Teachers can also emphasize the importance of using representative samples in interpreting circle graphs, linking it back to Lessons 9.01 and 9.02, thus integrating different areas of mathematics.

MPG3 — Mathematical Reasoning To incorporate mathematical reasoning, teachers can guide students in making observations and drawing conclusions from data represented in a circle graph. Students can be encouraged to use logical reasoning to analyze the data, identify patterns, and justify their conclusions. Another idea for meeting this goal is to provide students with real-life examples of circle graphs in magazines, newspapers, or online sources. Students could analyze the data and write a paragraph explaining the information they learned from analyzing the different graphs and any insights they gained from looking at the data. Students at this grade level should be able to hypothesize about the relationship between two variables based on a circle graph, design an experiment to test it, and draw conclusions from the data presented. They should explain how the data supports their conclusions, evaluate the effectiveness of using a circle graph for the data set, and justify their evaluation.

MPG5 — Mathematical Representations Teachers can address the goal of mathematical representations by guiding students in representing data using circle graphs. Students will create their own circle graphs based on given data sets, understanding that this representation is both a process and a product. Teachers can demonstrate how different representations, such as percentages, fractions, and the size of the sectors, all convey the same data in different ways. They can also guide students in interpreting these representations, enabling students to understand the mathematical ideas they represent.

Content standards 6.PS.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs. 6.PS.1a — Formulate questions that require the collection or acquisition of data with a focus on circle graphs. 6.PS.1b — Determine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).

6.PS.1d — Organize and represent data using circle graphs, with and without the use of technology tools. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20,

, or 35%).

6.PS.1e — Analyze data represented in a circle graph by making observations and drawing conclusions.

9.02 Create and interpret circle graphs mathspace.co

705

8.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.

Tools You may find these tools helpful: • Compass • Graph paper

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Visual supports for understanding circle graphs Student with disabilities support For students with visual-spatial processing challenges, circle graphs can be difficult to interpret. To support these students, consider providing additional visual supports. For example, you could provide graph paper with pre-drawn circles to help students create their own circle graphs. You could also provide tools like compasses and straightedge to help students accurately draw a circle and divide it into sectors. To help students interpret circle graphs, consider using manipulatives or interactive digital tools that allow students to explore how changing the size of one sector affects the sizes of the others.

706

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide Circle graphs Exploration Students: Page 354

9.02 Create and interpret circle graphs mathspace.co

707

Suggested student grouping: In pairs Students will investigate a circle graph that represents the favorite styles of board games among a certain group of students. They will make observations and draw conclusions about the data presented in the chart. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice? Answers may vary. Possible answer is: The circle graph is divided into different sectors of different sizes corresponding to the type of board game preferred by students. The larger the percentage, the larger the area of the sector. 2. What do you wonder? Answers may vary. Possible answers are: What are the number of students surveyed to get this data? Are the sizes of the sectors calculated using the percentages? Would the popularity of these games change if the survey was taken in a different group of students. 3. What type of board game do you think is the most popular? Explain. Card games are the most popular because it has the largest area and percentage (30%). 4. What type of board game do you think is the least popular? Explain. Role playing games are the least popular because they have the smallest area and percentage (6.7%). Purposeful questions • What do the different sections of the graph represent, and how do you know? • Why do you think some sections are larger than others? • How might the numbers on the graph relate to the size of each section? • What could the total of all the percentages add up to, and why? Possible misunderstandings • Students may not realize that the percentages on the sectors are related to the size of the sectors. They also may not realize that the percentages add up to 100%, representing the full circle. Students discover that circle graphs represent data proportions as parts of a whole and are divided into labeled sectors proportional to each category’s percentage. They are helpful for visualizing relationships within a whole, such as showing probabilities or comparing categories, but they are less effective with many categories. Students learn to interpret circle graphs, ensuring the percentages add up to 100%, and use them in data analysis stages, such as determining likelihood or probability.

Students: Pages 355–356 A circle graph is different from a bar chart or line plot, because it does not show the count or frequency of each category. Instead, it shows the proportion of the data that is in a category as parts of a whole. A circle graph is sometimes called a pie chart because each sector could be a piece of pie. A circle graph: • Is a circle broken into pieces called sectors. • Has a key or labels to show what each sector represents. • Has a title that tells you what the graph is about. • Will have sectors that are proportional to the percentage of the data that is in each category. For example, half of the circle is red, so 50% of the ﬁsh in the tank would be red.

Fish colors in a tank

Orange

Blue

Red

Yellow

Fish colors in tank Fraction of total

708

Percentage

12.5% MathspaceOrange Virginia SOL Grade 6 Teacher Edition mathspace.co 50% Red Blue

25%

If we look at how much of the circle each sector takes up, we can identify what percentage of the total ﬁsh are of each color. The sum of the percentages should always be 100%, because they represent parts of a whole. We may not always be able determine the percentages just by looking at it.

of the data that is in each category. For example, half of the circle is red, so 50% of the ﬁsh in the tank would be red. Orange

Blue

Red

Yellow

Fish colors in tank Fraction of total

Percentage

Orange

12.5%

Red

50%

Blue

25%

Yellow

12.5%

We often label the percentages on each sector, so that we can compare more easily and do calculations. For example: Fruits purchased from the grocery

12% 27%

Apples Grapes Blueberries

13%

40%

Oranges Limes

Sometimes, we will show the number labels instead of percentages, for example, if 300 people were surveyed, this circle graph shows the same information as the one before: Fruits purchased from the grocery Apples

36 81

Grapes Blueberries

120

Oranges Limes

It is important that we always check that the percentages on the graph add up to 100% since a circle graph always represents the whole of the data points. 9.02 Create and interpret circle graphs

355

Circle graphs are not helpful for representing data with large numbers of categories because they get hard to read mathspace.co with too many sectors.

We can use circle graphs in the “Organize and Represent” stage of the data cycle. They can be helpful for questions that ask about a relationship of the parts of a whole. Circle graphs can show us the probability of the events they represent. Recall that Probability of an event = In a circle graph the favorable outcomes are represented by sectors of the graph and the total outcomes are the entire circle. So, the percentage of the circle that the sector(s) makes up is the probability of that event occurring. Probabilities of an event can be described as: • Impossible - if no sectors represent that event • Unlikely - if the sector(s) for that event make up much less than half of circle • Equally likely - if the sector(s) for the event make up half of the circle • Likely - if the sector(s) for the event make up much more than half of the circle • Certain - if the sector(s) for the event make up the entire circle Reason for being at the arena 5% 10% 15%

50%

20%

Watching as a spectator

Skating lessons

Playing hockey

Playing ringette

Working

For example, this circle graph represents the reasons why people are at the arena. If we randomly select one person from the arena, the probability is: • Impossible that they are there for soccer • Unlikely that they are working • Equally likely that they are a spectator • Likely that they are not playing ringette • Certain that they are in the arena

9.02 Create and interpret circle graphs mathspace.co

709

Probabilities ofaan event can of bethe described that ask about relationship parts ofas: a whole. • Impossible - if no sectors represent that event Circle graphs can show us the probability of the events they represent. Recall that • Unlikely - if the sector(s) for that event make up much less than half of circle Probability of anmake eventup = half of the circle • Equally likely - if the sector(s) for the event • Likely - if the sector(s) for the event make up much more than half of the circle In a circle graph the favorable outcomes are represented by sectors of the graph and the total outcomes are the • Certain - if the sector(s) for the event make up the entire circle entire circle. So, the percentage of the circle that the sector(s) makes up is the probability of that event occurring. for being at described the arena as: For example, this circle graph represents the reasons why Probabilities ofReason an event can be people are at the arena. If we randomly select one person • Impossible - if no sectors 5% represent that event from arena, thecircle probability is: • Unlikely - if the sector(s) for that event make up much lessthe than half of 10% • Impossible • Equally likely - if the sector(s) for the event make up half of the circlethat they are there for soccer • Unlikely that theycircle are working • Likely - if the sector(s) for the event make up much more than half of the 15% 50% • Equally • Certain - if the sector(s) for the event make up the entire circle likely that they are a spectator • Likely that they are not playing ringette Reason20% for being at the arena For example, this circle graph represents the reasons why • Certain that they are in the arena people are at the arena. If we randomly select one person 5% from the arena, the probability is: 10% • Impossible that they are there for soccer Watching as a spectator Skating lessons • Unlikely that they are working 15% Playing hockey 50% Playing ringette • Equally likely that they are a spectator Working • Likely that they are not playing ringette 20% • Certain that they are in the arena

Example 1

ExamplesWatching as a spectator

Skating lessons

For each of the following questions for the data cycle, determine if the data can be well represented using a circle Playing ringette type of display and explain why you chose it. hockey why. If not, suggest IfPlaying yes, explain a different Students:graph. Page 356 Working

a How much time do students spend on homework, in hours?

Create a strategy Example 1 If we are comparing parts of a whole then a circle graph would be a good choice. We don’t to use a circle graph if For each the following for the data cycle, determine if the dataquestion can be well represented using a circle there are of a large number questions of categories or possible answers. Decide if this would lead to seeing how big one graph. If yes, explain why. If not, suggest a different type of display and explain why you chose it. part is compared to the others. a How much time do students spend on homework, in hours?

Apply the idea Create a strategy No, a circle graph isn’t the best choice. There are too

Reﬂect and check

A line graph shows changes over time, so total amount of many possible answers and need then to compare timebe inaagood day would notWe bedon’t displayed in a line graph. If we are comparing parts of no a whole a circleparts graphofwould choice. to usewell a circle graph if a whole. there are a large number of categories or possible answers. Decide if this question would lead to seeing how big one part compared thea others. A barisgraph wouldtobe good choice as it can have more categories for the different possible answers.

Apply the idea

Reﬂect and check

No, a circle graph isn’t the best choice. There are too many possible answers and no need to compare parts of 356 Mathspace Virginia SOL Grade 6 a whole.

A line graph shows changes over time, so total amount of time in a day would not be displayed well in a line graph.

mathspace.co

A bar graph would be a good choice as it can have more categories for the different possible answers.

356

Mathspace

Virginia SOL Grade 6

Purpose mathspace.co Show students that selecting the appropriate type of graph for displaying data depends on the nature of the data and what it’s intended to represent. Expected mistakes Students might say a circle graph would be a good representation, especially if they are thinking that the time would be rounded to the nearest hour. Remind students that hours can be rational too, like 1.5 hours or 0.8 hours.

Students: Page 357

710

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Show students that selecting the appropriate type of graph for displaying data depends on the nature of the data and what it’s intended to represent. Expected mistakes Students might say a circle graph would not work well because they think a bar graph would be better. Although a bar graph would also represent the data well, a circle graph is an additional data display that could help us easily determine the most and least popular kind of candy.

Students: Page 357

Purpose Show students that selecting the appropriate type of graph for displaying data depends on the nature of the data and what it’s intended to represent.

Stronger and clearer each time

use with Example 1

English language learner support Have students first individually write their reasoning for each question about whether a circle graph is suitable, and why or why not. Encourage them to use specific mathematical vocabulary such as “circle graph,” “categories,” “parts of a whole,” and “data representation.” After they have written their initial responses, pair them up to share their explanations with a partner. Instruct students to listen carefully and provide constructive feedback to each other, focusing on clarity of ideas and accurate use of mathematical terms. Prompt them to ask questions like, “Can you explain what you mean by ‘comparing parts of a whole’?” or “Why do you think a bar graph is a better choice in this case?” After the discussion, allow students to revise their original explanations, incorporating new insights and vocabulary they gained from their partner. If time permits, repeat the process with a new partner to further refine their explanations. This routine gives students multiple opportunities to articulate and clarify their thinking, enhancing both their mathematical understanding and their academic language proficiency.

Students: Page 357

9.02 Create and interpret circle graphs mathspace.co

711

Purpose Check that students are capable of accurately identifying the largest piece in a circle graph. Reflecting with students Ask students how they determined which part is larger. Make students aware that they weren’t given any specifics (such as how many students were surveyed or what percentage of students preferred each type of food), but they were still able to effectively compare the categories in the data display.

Students: Page 357

Purpose Check that students are capable of accurately identifying equal parts in a circle graph.

Students: Page 358

712

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Check that students are capable of accurately identifying the largest piece in a circle graph.

Students: Page 358

Purpose Check for understanding of the percentage parts of a circle graph. Reflecting with students Ask the students how they determined the percentage. Discuss with students if they could find the percentage of each piece individually. The percentages would be: • History: 12.5% • Languages: 6.25% • Physical education: 6.25%

Students: Page 358

9.02 Create and interpret circle graphs mathspace.co

713

Purpose Check for understanding of percentage parts of a circle graph. Reflecting with students Challenge advanced learners to determine the number of students that chose each subject. • Languages: 12.5 • Math: 25 • English: 75 • History: 25 • Physical education: 12.5 They can check their answers by ensuring the total number of students sums to 200. Since half a student is not possible, it is likely that 12 students chose either Languages or Physical education, and 13 students chose the other subject.

Interpret circle graphs

use with Example 3

Address student misconceptions A common misconception students might have is that the size of the sectors in a circle graph represents the actual number of data points in each category, rather than the proportion or percentage. To address this misconception, emphasize to students that the entire circle represents the whole data set or 100%, and each sector represents a part of the whole. Provide opportunities for students to practice converting between raw data, proportions, and percentages, and representing these in a circle graph.

Students: Page 359

Idea summary Circle graphs represent the data as parts of a whole. Each sector of a circle graph represents a different category. The larger the sector, the larger the percentage of data in that category. A circle graph should include: • • •

A title to explain what the graph is about A key to explain how to read the graph Percents or number labels for each category

Circle graphs are good for representing categorical or countable numerical data with only a few categories.

Ideacircle summary Create graphs

Circle graphs represent data parts of a whole.Some Eachprograms sector of alike circle graph represents a different CreateCircle circle graphs graphs can be created bythe hand or as using technology. Excel or Google Sheets refer to circle The larger the sector, the larger the percentage of data in that category. graphs category. as pie charts.

Students learn how to create and interpret A circle graph should include: circle graphs (also known as pie charts) using both technology and manual Using technology, we can: methods. They discover the steps for inputting • the A title the graph isdata aboutinto programs like Google Sheets to create graphs. The lesson • Enter datatoasexplain a list orwhat a frequency table walks through how to calculate percentages and fractions by hand to divide a circle into proportional segments using • A key to explain how to read the graph • Highlight the data • aPercents or number labels for each category an example• ofInsert students’ favorite seasons. chart and select pie chart or circle graph Circle graphs are good for representing categorical or countable numerical data with only a few categories. Suppose we formulate the question “What proportion of students at my school prefer each season?” and collect this data from a sample Winter 5 of 60 students. Spring 15

Season 359–360 Number of students Students: Pages

Summer circle graphs 30 Create 10 by hand or using technology. Some programs like Excel or Google Sheets refer to circle CircleFall graphs can be created graphs as pie charts. We can convert each category to a Season Number of students Fraction Percent Using technology, we can: fraction or a percentage. • Enter Winterthe data as a list5or a frequency table • Highlight the data Springa chart and select 15 pie chart or circle graph 25% • Insert

714

50% Summer 30of students Suppose we formulate the question “What proportion of students at Season Number my school prefer each season?” and collect this data from a sample Winter 5 MathspaceFallVirginia SOL Grade 10 6 Teacher Edition of 60 students. Spring 15 mathspace.co Summer 30 Then we can divide up the circle by ﬁrst cutting it in half, then splitting one half into two quarters, then splitting one Fall 10 quarter into twelfth.

Circle graphs can be created by hand or using technology. Some programs like Excel or Google Sheets refer to circle graphs as pie charts. Using technology, we can: • Enter the data as a list or a frequency table • Highlight the data • Insert a chart and select pie chart or circle graph Season Winter Spring Summer Fall

Suppose we formulate the question “What proportion of students at my school prefer each season?” and collect this data from a sample of 60 students.

Number of students 5 15 30 10

Season

Number of students

Fraction

Winter

Spring

25%

Summer

50%

Fall

We can convert each category to a fraction or a percentage.

Percent

Then we can divide up the circle by ﬁrst cutting it in half, then splitting one half into two quarters, then splitting one quarter into twelfth.

Summer

One half of the circle represents summer

Spring

One quarter of the circle represents spring

9.02 Create and interpret circle graphs mathspace.co

359

Steps in creating a circle graph Targeted instructional strategies To help students understand how to create a circle graph, you can provide them with the following steps: 1. Collect the data and determine the total number of data points. 2. For each category, calculate its proportion of the total data points. 3. Convert these proportions to percentages. 4. Draw a circle and divide it into sectors corresponding to these percentages. 5. Label each sector with the category it represents and the corresponding percentage. Encourage students to practice this process with different data sets to strengthen their understanding of circle graphs.

9.02 Create and interpret circle graphs mathspace.co

715

Examples Students: Pages 360–361

Create a strategy Determine the percentage of market share for each manufacturer by dividing the number of responses for each manufacturer by the total number of responses and multiply the decimal by 100.

Apply the idea Let’s start by looking at the market share of Brand A. ⋅ 100 ≈ 51% 51% is more than half of 4000, so the sector corresponding to Brand A should take up more than half the circle. For Brand B, we have ⋅ 100 ≈ 24% which is a little under a quarter of a circle. For Brand C, ⋅ 100 ≈ 13% For others we have ⋅ 100 ≈ 11% So, the sector for Brand C and the others should be about the same size. Smartphone market share

Here is the circle graph that most accurately represents what the data would look like. So, the correct answer is option C.

716

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

⋅ 100 ≈ 51% 51% is more than half of 4000, so the sector corresponding to Brand A should take up more than half the circle. For Brand B, we have ⋅ 100 ≈ 24% which is a little under a quarter of a circle. For Brand C,

Create a strategy ⋅ 100 ≈ 13% Determine the percentage of market share for each manufacturer by dividing the number of responses for each manufacturer by the total number of responses and multiply the decimal by 100. For others we have ⋅ 100 ≈ 11%

Apply the idea

Let’s start by looking at the market share of Brand A. So, the sector for Brand C and the others should be about the same size. 100circle ≈ 51% Here is ⋅the graph that most accurately represents what the data would look like. 51% is more than half of 4000, so the sector corresponding to Brand A should take up more than half the circle. So, the correct answer is option C. For Brand B, we have Smartphone market share

⋅ 100 ≈ 24% which is a little under a quarter of a circle. For Brand C, Brand A

Brand B

Brand C

Other

For others we have

Reﬂect and check

⋅ 100 ≈ 13%

⋅ 100 ≈ 11%

We could create the circle graph using technology to check. So, the sector for Brand C and the others should be about the same size. Smartphone market share Here is the circle graph that most accurately represents what the data would look like. on the data and representation. b Write a conclusion that the marketing company could make based So, the correct answer is option C.

Purpose Create a strategy Show students how to determine the percentages of each category of a circle graph, and to accurately relate Circle graphs show the proportion of each category, so the conclusion will be about which brands are the most or the percentages to the size of a sector. least popular. Expected mistakes Apply the idea Reﬂect and check Brandoption A B the sizes of the sectors are very similar to option C. Point out that, in the Students might choose DBrand since Brand A has the majority of the market share with more We are assuming that this sample is representative of the Brand Cthe sector Other for Brand B is exactly a quarter of the circle, and the sectors for Brand C circle graph for option D, than 50% of those surveyed using Brand A. population. If the sample was not well selected, then this and Other areB almost identical. conclusion would only be valid for the sample and not the Brand has about a quarter of the market share, so a Reﬂect and check similar number of people use Brand B as all other brands, population.

could create theA. circle graph using technology to check. Students:We Page 361 not including Brand

b Write a conclusion that the marketing company could make based on the data and representation.

Create a strategy

9.02 Create and interpret circle graphs mathspace.co

361

Circle graphs show the proportion of each category, so the conclusion will be about which brands are the most or least popular.

Apply the idea

Reﬂect and check

Brand A has the majority of the market share with more than 50% of those surveyed using Brand A.

We are assuming that this sample is representative of the population. If the sample was not well selected, then this conclusion would only be valid for the sample and not the Brand B has about a quarter of the market share, so a similar number of people use Brand B as all other brands, population. not including Brand A.

9.02 Create and interpret circle graphs

361

mathspace.co Purpose Students demonstrate that they can draw conclusions and communicate results of a statistical survey.

9.02 Create and interpret circle graphs mathspace.co

717

Use appropriate mathematical tools

use with Example 4

Targeted instructional strategies Explicitly teach students to use multiple tools including spreadsheets or graphing software to create circle graphs. For example, using Google Sheets the steps are: 1. Enter the data in a spreadsheet tool with categories in the first column, then the corresponding frequencies in the next column 1 2 3 4 5

A Brand A Brand B Brand C Brand D

B 2051 967 531 451

2. Highlight all of the data 3. Click “Insert” and then “Chart” 4. In the “Chart editor” that pops up, select “Pie chart” for the “Chart type” 5. Under the “Customize” panel, under the “Chart & axis title” add “Title text” 6. Adjust settings such a labeling the pie slices with the percentages (Pie chart → Slice label → Percentage), style, color, or legend location to have it look as desired Have students create a graphic organizer or poster showing the steps for different tools for future reference. As an extension for advanced learners, have them consider how to construct circle graphs by multiplying the percentage of data in a category by 360° in order to determine the central angle measure. Encourage them to use algorithmic thinking to create and refine a set of steps that could be used to create any circle graph.

Scaffolding calculations with step-by-step guides

use with Example 4

Student with disabilities support To help students who may have difficulties with conceptual processing or organization, scaffold the problem by breaking down the percentage calculations into clear, manageable steps. Provide a list that guides students through each part of the process: identifying the total number of responses, dividing the number of responses from each manufacturer by the total responses to find the decimal form, and then multiplying by 100% to convert it to a percentage. Include space for students to show their work for each step. Encourage students to check off each step as they complete it to promote a sense of accomplishment and keep track of their progress. For example: Manufacturer Brand A Brand B Brand C Other

Responses 2051 967 531 451

1. Total = ⬚

2. Brand A percentage: 3. Brand B percentage: 4. Brand C percentage: 5. Other percentage:

718

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

⋅ 100% = ⬚

Additionally, incorporate collaborative learning by having students pair up to discuss each step and compare their calculations. This not only supports psycho-social development but also allows students to learn from each other and clarify misunderstandings in a supportive environment. By providing structured support and clear guidance, you enable students to build confidence in their mathematical abilities and improve their understanding of how to interpret and represent data accurately.

Students: Page 362

Idea summary To create a circle graph, we can use technology, or create it by hand. To make it by hand, we can: • • • •

Create a table with the categories, their count, fraction, and percentage Draw a circle Divide the circle into segments that match the proportions for each category Check that the proportions add up to 1 or 100%

Practice What do you remember? Practice 1

Recently the Northern Lights, Aurora Borealis, were visible further south than usual. A survey was done of

people who saw them and they were asked what color of Northern Lights was their favorite. Students: Pages 362–369 Complete the table. Number of people

What do you remember? Purple 40 1

Fraction of people

Percentage of people

Green 30 Pink 10 Aurora Borealis, were visible further south than usual. A survey was done of Recently the Northern Lights,

people and theyquestions were asked colordetermine of Northern Lights was favorite. 2 who For saw each them of the following for the what data cycle, whether or not the their data can be well Completerepresented the table. using a circle graph. a

What writing utensils, like pencils or markers, do students prefer?

Number of people Fraction of people Percentage of people How many days do students typically spend at camps during the summer break? Purple c What clothing 40 material is preferred by students? Green 30 3 A snack company is looking to add a new item to their product line. They took Pink a small sample10 and found the following preferences: b

• Breaded veggie chips: 156 D For each of the following for the data cycle, determine whether or not the data can be well • Soft oat bars: questions 19 Gelatin based product: represented •using a circle graph. 32 C • Gluten free and vegan option: 74

What writing utensils, like pencils or markers, do students prefer?

How many days do students typically spend at camps during the summer break?

What clothing material is preferred by students?

Which sector represents Gelatin based products?

4 company The given circle graph shows of a class survey where students were A snack is looking to addthea results new item to their product line. They took asked to choose their favorite food: a small sample and found the following preferences: a

Which was the most popular food?

• Breaded veggie chips: 156 b Which two foods were equally popular? • Soft oat bars: 19 • Gelatin based product: 32 • Gluten free and vegan option: 74 Which sector represents Gelatin based products?

362

Mathspace Virginia SOL Grade 6 mathspace.co

Favorite Food

Nuggets

A D

Pizza

Noodles Burger B

9.02 Create and interpret circle graphs mathspace.co

719

The given circle graph shows the results of a class survey where students were asked to choose their favorite food: a

Which was the most popular food?

Which two foods were equally popular?

Favorite Food

Nuggets Pizza Noodles Burger

Are the following statements true or false? a

The larger the sector, the smaller the percentage of data in that category.

The percentages on a circle graph should always add up to 100%.

Circle graph are used to display categorical data.

The owner of a business wants to have a company-wide retreat in Virginia. He narrowed the list down to ﬁve cities, but he cannot decide which city to choose. He sends a survey to all 180 of his employees, asking them to choose the city in which the retreat should be held. Would the following data displays be appropriate for the owner to use to analyze the results of the survey? a

Dot plot (Line plot)

Bar graph

Pictograph

Circle graph

At different times of day, a sample of people were asked what they were doing at the park. Describe the probability that someone was there to go for a run using one of these words. • Likely • Certain

• Impossible • Unlikely • Equally likely a

At 5 a.m.

Reason for being in the park

At 10 a.m. Reason for being in the park

25.0% 50.0%

100% 25.0%

Running

Bird watching

Running

Bird watching

Walking dog

Playing at playground

Walking dog

Playing at playground

At 4 p.m.

Reason for being in the park 5%

At 7 p.m. Reason for being in the park 2%

15%

50%

720

13% 15%

30%

70%

Running

Bird watching

Running

Bird watching

Walking dog

Playing at playground

Walking dog

Playing at playground

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice 8

A ﬂorist recorded the roses of different colors sold in his store during the week of Valentine’s day. This is displayed in the circle graph.

Pink

Order the colors of roses from most to least popular. Orange

Red

White

Edward surveyed a large group of people and asked for their favourite milkshake ﬂavour. The results of the survey are in a circle graph. a

What fraction of people chose chocolate?

What percentage of people chose chocolate?

What fraction of people chose strawberry?

What percentage of people chose strawberry?

The following circle graph shows the results of a survey where 100 children were asked for their favorite color: a

Milkshake flavours

Vanilla

Strawberry

Chocolate

Caramel

Favorite color

Which was the least popular color?

How many children chose yellow as their favorite color?

How many children chose blue as their favorite color?

12% 13%

12% 6%

19%

8% 30%

Blue Green Red

Pink Label 4 Orange

Yellow

The following circle graph represents the results of 11th grade class president elections for the year 2024: a

Who won the elections?

What is the percentage of the votes obtained by Jess?

Harry Jess

Will

9.02 Create and interpret circle graphs mathspace.co

721

The results of a senatorial election in a European country, where 4000 people voted, are shown in the given circle graph: a

Who won the election?

What fraction of people voted for Zorg?

After the election, the winner decides to send promotional packages to everyone they suspect voted for Zorg. How many packages should they prepare?

Voting results Zorg 25%

At a dance studio, there are students who take tap, jazz, ballet, acro, and hip-hop. This circle graph shows the proportion in each class. One student wins a prize every month.

Palpatine 65%

Number of students in dance classes 10%

What is the probability that the winner this month is in:

Ballet?

Hip-hop or Jazz?

Acro?

18%

50%

20%

Ballet

Acro

Tap

Hip-hop

What does most of this money go towards?

What does he spend the least money on?

If his total income is $2200 per month, how much does he spend on clothes?

Mason has an income of $1200 and puts $300 into savings. Does Mason or Roald save a higher percentage of the monthly income?

30%

25% 5% 10%

Leisure Clothes

Lucille formulates the question “What dairy products are the most popular for Americans?”. She collects data from her classmates and makes Circle Graph A. Her friend DeShaun found data from a national survey of 10 000 people and summarized the results in Circle Graph B. Circle Graph A

Circle Graph B

722

Jazz

Roald organizes the break-down of his income on this circle graph.

Food Savings Housing

10% Umbridge

Milk

Yogurt

Milk

Yogurt

Cheese

Cottage cheese

Cheese

Cottage cheese

Was Lucille’s sample representative of the population?

Who’s data would be more reliable?

Draw a conclusion to answer Lucille’s question.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Manuela is an international student and is feeling homesick. She formulates the question, “How long are international students usually homesick for?” She does a survey of everyone currently on exchange through her program. She organizes the data in this table. Construct a circle graph to represent this data.

Time

Number of students

Less than a week

220

25%

1–2 weeks

264

30%

3–4 weeks

132

15%

1–2 months

176

20%

3–4 months

10%

Percentage

A math teacher formulates the question “Which other subject would students like to see incorporated into math class more often?” She surveys 220 students to ﬁnd their preference and needs to construct a circle graph to display the data. The survey results are shown in the following table: Subject

Number of students

Art

English History Science Other

55 33 77 11

Percentage

Complete the table to ﬁnd the percentage of students that prefer each subject.

Construct a circle graph to represent this data with or without technology.

Make a recommendation based on the circle graph.

A physiotherapist is doing a study with 1080 volunteers who have knee pain. He is trying a new technique and after six weeks asking the patients how they are feeling. Result Improvement No change Deterioration

Fraction

Number of volunteers 705 240 135

Create a circle graph to represent this data using technology.

Draw a conclusion about whether or not the physiotherapist should continue with this new technique.

Bob asked his students to choose their favorite toy. 35% students picked cars, 10% of the students picked planes, 30% picked balls and 25% picked video games. a

Decide the best type of display for the data and justify your choice.

Construct the display chosen in part (a).

9.02 Create and interpret circle graphs mathspace.co

723

Let’s extend our thinking 20

The following circle graphs show the number of students registered for different types of courses at a university. Courses at University

Courses at University (not Arts or Engineering) 20

1165

80 140

6450

6490

300

130 280

215

Engineering

Arts

Other

Philosophy

Communications

Library science

Family science

Theology

Precision production

Architecture

Which course is the most popular according to the circle graphs?

Which two courses have around half of the students registered that are not in Arts or Engineering?

How many courses have more than 100 students registered?

Elise thinks these circle graphs are confusing and says a bar graph would be better. Do you agree or disagree with Elise? Explain your answer.

This question is being explored with the data cycle: “What activities do students do after school? Does it vary by school level?” Random samples of students were taken at several elementary schools, middle schools, and high schools. Students were asked about the ﬁrst activity they do after school. These circle graphs summarize the data: Elementary after school activities

724

Middle after school activities

High after school activities

Play outside

Video games

Homework

Sport team

Artistic activity

Work for family business

Others

What conclusions can we make that are true for all three groups?

What are some of the difference between the different groups?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

If all of the data was grouped together, we get this circle graph. Is this a good overall summary, or is it better to have the graphs separate? All school levels

d 22

Play outside

Sport team

Artistic activity

Video games

Homework

Work for family business

Others

Formulate another question that is related to after school activities.

The given circle graph about the school election was displayed on poster: a

Would vote for Grylls

Describe two mathematical errors with this circle graph.

63%

Would vote for McGill 70%

60% Would vote for Flanders

Which of the following images better represents the same information? A

Would vote for Flanders Would vote for Grylls Would vote for McGill

Would vote for Grylls

Would vote for McGill

Would vote for Flanders

After looking at the data, a student concludes that the majority of people would vote for each of the three candidates. Which of the following is not an explanation for how this is possible? A The three candidates are running for different positions in government. B Three different samples were surveyed. C The survey questions said, “Would you vote for Grylls if they ran for class president? Would you vote for McGill if they ran for class president? Would you vote for Flanders if they ran for class president? D The survey question said, “If Grylls, McGill, and Flanders run for class president, who would you vote for?”

9.02 Create and interpret circle graphs mathspace.co

725

Emil was at his auntie’s house and noticed that she used a desktop computer. He had only ever seen gamers use them. He is now curious about what proportion of internet traﬃc comes from desktops/laptops, mobile phones, tablets, and consoles. a

Formulate a question that Emil could use to explore the proportion of different device types used on the internet using the data cycle.

Could he use observation, measurement, survey, experiment, or acquire secondary sources? Explain.

Explain how Emil could collect data that could be used to answer his question from part (a).

Suppose the given data shows the amount of time 30 of his relatives spend on different types of devices. Organize and represent the data using at least one circle graph. Person Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 Person 7 Person 8 Person 9 Person 10 Person 11 Person 12 Person 13 Person 14 Person 15 Person 16 Person 17 Person 18 Person 19 Person 20 Person 21 Person 22 Person 23 Person 24 Person 25 Person 26 Person 27 Person 28 Person 29 Person 30

726

Age 13 14 15 15 16 16 17 17 18 18 20 20 23 24 25 26 27 29 30 31 34 36 41 43 47 52 55 64 66 70

Mobile 5.3 4.3 4.4 4.9 1 4 4.6 3.2 5.1 5.5 4 7.8 3 4 2 2.8 4 4 5 2.1 5.2 2.9 1.7 1.8 2 2.1 0 1 2.5 0

Desktop/ Laptop/ Chromebook 2.3 1.8 2.1 2.1 1.5 2 2.1 2 1.6 1.8 8 3 4.4 6 0.5 6.4 3 6 3 2.3 1.6 1 0 5 8 3.9 0.9 3.3 1.1 1.3

Tablet 0.2 0.1 0.5 0.5 1.2 0.7 0 0 0.75 1.2 0.9 0.6 1.1 0 2.5 0.3 0.3 0.9 1 1.2 1.1 1.6 0 2 0.5 0 2.1 2.3 0 1.5

Other (ie: console/ e-reader) 0 2 0 1.2 5 0 0 0 1 0 0.5 0 0 0 0 0 0 0 0.5 0 0 0 0 0 0 0 0 0 0 0

Analyze the data to draw a conclusion about internet usage on different devices.

Formulate another statistical question that could be used to explore device types.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

How long homesickness lasted 10%

9.02 Create and interpret circle graphs

25% 20%

What do you remember? 1

Number of people Purple

Fraction of people

15%

Percentage of people

50%

Green

37.5%

Pink

12.5%

17 a

30%

Less than a week

1−2 weeks

1−2 months

3−4 months

3−4 weeks

Subject

Number of students

Art

= 20%

b T he data cannot be well represented using a circle graph.

English

= 25%

c The data can be well represented using a circle graph.

History

= 15%

Science

= 35%

Other

= 5%

2 a The data can be well represented using a circle graph.

3 Sector D 4 a Pizza

b Nuggets and noodles

5 a False

b True

c True

6 a No

b Yes

c No

7 a Certain

b Equally likely

c Unlikely

d Yes

Percentage

b Preferred subject to incorporate into math class 5%

d Likely

20%

35%

Let’s practice 8 Red, Orange, White, Pink b 50%

9 a 10 a Purple

25%

b 12 children

11 a Will

c 30 children

b 25%

Science

History

English

Arts

Other

12 a Palpatine

c 1000 packages 13 a 50%

b 38%

14 a Food and savings

c 2% b Leisure

c $220

d R oald saves a higher percentage of his monthly income at 30% compared to Mason, who saves =

15%

d 25%

= 25%.

c T he math teacher should incorporate science into math class more often. 18 a

Results of knee pain study 13% 22% 65%

15 a N o, her classmates are not a representative of the population “Americans”. b D eShaun’s data would be more reliable because it is from a representative sample. A larger sample size usually leads to more reliable data. c G iven the circle graphs, we can conclude that the most popular dairy product among Americans is cheese, which about half of Americans prefer. Cheese is followed by yogurt, which about a third of Americans prefer, then milk, and very few people prefer cottage cheese.

Improvement

No change

Deterioration

b T he physiotherapist should continue with the new technique, given that a large percentage of volunteers felt improvement. However, the physiotherapist should continue to improve it by investigating why some people are experiencing deterioration and no improvement.

Answers mathspace.co

727

19 a T his is categorical data but we don’t know the total or the counts so a circle graph will be the best graphical display.

b S ome of the difference between the different groups are:

• For elementary schools, there is a larger percentage of students who participate in sports teams after school compared to those who do homework. This is opposite for high schools, where a larger percentage of students do homework after school compared to those who participate in sports teams. Middle schools have percentages for homework and sports teams that are similar.

• The percentage of students who play outside after school differs among all three groups, with elementary school having the highest percentage, followed by middle school, and lastly high school.

• The percentage of students who do homework after school differs among all three groups, with high school having the largest percentage, followed by middle school, and lastly elementary school.

b 25% 35% 30%

10%

Cars

Planes

Balls

Video Games

Let’s extend our thinking 20 a Engineering b Philosophy and Architecture c 7 countries d Example answer 1: I agree with Elise. Since there is a large number of categories and the number of students in some courses is so small, it is not possible to ﬁt the data into one circle graph. If a bar graph was used, all the data could be on the same graph.

Example answer 2:

I disagree with Elise. Since the difference in the number of students in Engineering and Precision Production is so big, it would be difficult to choose a scale for the bar sizes. The scale would need to be small enough to show the different heights between courses other than Arts and Engineering but big enough to fit Arts and Engineering on the same scale. Although the data cannot fit on the same circle graph, the two circles clearly show the differences in the number of students across all courses. 21 a S ome conclusions that we can make that are true for all three groups are:

• A large proportion of students engage in artistic activities after school.

• A large proportion of students participate in sports teams after school.

• Only a small portion of students work for the family business after school.

728

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

c W e are exploring the question, ‘What activities do students do after school? Does it vary by school level?’ So, it would be better to have separate graphs to observe the differences between school levels. There might also be factors that affect the students’ chosen activity after school, which could vary by school level. d A nother question related to after school activities could be, “What factors affect students’ choice of after school activity?” or “How much time do students spend on after school activities?”. 22 a T he sector pieces are not the correct fraction of the circle. For example, the smallest percentage is the largest piece. Also, the sectors do not add to 100%. b B c D 23 a A possible question would be “What devices are most used to browse the internet?”. b H e would need a large representative sample in order to obtain reliable results, and this would be diﬃcult if he were to use observation. Measurement and experimentation are not applicable in this situation. So, Emil should use secondary sources, given the type and amount of data he would need. He could use an online survey, but acquiring secondary sources would be more eﬃcient. c E mil could access realiable websites that would provide data about internet and device usage. He could also use existing survey results.

Internet traffic from devices 5% 11% 45% 39%

Mobile

Desktop/Laptop/Chomebook

Tablet

Other (ie: console/ e-reader)

e B ased on the data table and circle graph, the highest internet traffic with about half of the traffic is from mobile devices. This is followed by desktops/laptops/ chromebooks which are not too far behind mobile devices. Tablets and other devices such as consoles and e-readers have much less traffic. f

e could formulate another statistical question, such W as “Does the internet traﬃc from various devices vary for different age groups?” or “Does a person owning multiple devices contribute more to internet traﬃc?”.

Answers mathspace.co

729

9.03 Compare representations of data Subtopic overview Lesson narrative In this lesson, students will learn to compare different representations of data, such as tables, pictographs, bar graphs, line graphs, line plots, and stem-and-leaf plots. They will explore the benefits and drawbacks of each type of data display and understand when each is most useful. The lesson includes an exploration where students analyze data on student birthdays across seasons using a table, circle graph, and bar graph. They will compare the effectiveness of each representation in answering specific questions about the data. By the end, students should confidently choose and interpret various data displays for different contexts.

Learning objectives Students: Page 370

Key vocabulary 

bar graph



categorical data



circle graph

 dot plot (line plot)



frequency



line graph



pictograph

 scale



stem-and-leaf plot



table

Essential understanding Data displays can provide information but also make it possible to draw conclusions and make generalizations.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning Teachers can foster mathematical reasoning by guiding students to evaluate the advantages and disadvantages of different types of graphical representation, and to make justified decisions based on their analysis. This involves logical reasoning and critical evaluation of mathematical information.

730

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can help students make mathematical connections by linking the choice of graphical representation to their prior knowledge of representative samples. Additionally, teachers can connect the mathematical skills learned in this lesson to real-world contexts, such as the importance of selecting the most effective graphical representation for communicating data in various professions.

Teachers can incorporate mathematical representations into their instruction on comparing data by having students create various types of graphs from given data sets. This approach helps students understand how different representations can affect data interpretation. Teachers can guide students to relate these representations to real-world contexts, highlighting the importance of effective data visualization. Additionally, students should compare different graphs representing the same data to identify the most effective one and determine the best graph types for various questions. This process encourages students to refine their methods for gathering and presenting data.

, or 35%).

6.PS.1e — Analyze data represented in a circle graph by making observations and drawing conclusions. 6.PS.1f — Compare data represented in a circle graph with the same data represented in other graphs, including but not limited to bar graphs, pictographs, and line plots (dot plots), and justify which graphical representation best represents the data.

8.PS.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.

9.03 Compare representations of data mathspace.co

731

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 9.02 Create and interpret circle graphs

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Review the different data displays Student with disabilities support Ensure that students have the prerequisite knowledge about the different types of data displays before beginning to analyze their appropriateness or interpreting them. • Pictograph

1 icon = 10 people

Response Excellent Average Poor

• Line graph

00 11: 00 12 :0 0 1:0 0 2: 00 3: 00 4: 00 5: 00 6: 00 7: 00 8: 00

10 :

00 9: 00

Speed (Mbps)

Internet speeds in an area 162 160 158 156 154 152 150 148

Time

• Bar graph Percent in favor

In favor of school policy 78 77 76 75 74 73 72 71 70 69 68

l oo

732

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

sta

rdi

a Gu

d Stu

Stakeholders

• Stem-and-leaf plots

Stem

Leaf

8 0

Key: 3 |0 = 30 • Dot plots (or line plots)

0 1 2 3 4 5 6 7 8 9 10 11 12 Minutes to Eat Breakfast

• Circle graph

Favorite Color

13%

12%

12% 6%

19%

8% 30%

Blue

Pink

Green

Purple

Red

Orange

Yellow

9.03 Compare representations of data mathspace.co

733

Student lesson & teacher guide Compare representations of data Students: Pages 370–374

734

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

9.03 Compare representations of data mathspace.co

735

Which season had the most birthdays? Which display did you use, the table, circle graph, or bar graph?

Which season had the fewest birthdays? Which display did you use?

How do winter and fall compare? Which display did you use?

Let’s look at when different displays are useful. A circle graph is useful for showing proportions and parts of a whole. Advantages:

Favorite Color

13%

12%

12% 6%

19%

8% 30%

Blue Green

Pink Purple

Red Yellow

Orange

• We can easily compare the proportions of different categories visually. They are commonly used. Disadvantages • It can be diﬃcult to compare similar groups if they are not labeled. We may lose the original totals if we just show the percentages.

A bar graph is useful for showing the count or frequency of different categories.

372

736

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

9.03 Compare representations of data mathspace.co

737

Exploration Students: Page 372

Exploration This table tells us how many students have birthdays in the different seasons. Seasons Winter Spring Summer Fall

Number of students 5 8 2 5

The data was also organized into these two displays. Number of students

2 8 Winter

Spring

Summer

Fall

Number of students

Number of students vs. Season 8 6 4 2 0

Winter

Spring Summer Season

Fall

Which season had the most birthdays? Which display did you use, the table, circle graph, or bar graph?

Which season had the fewest birthdays? Which display did you use?

How do winter and fall compare? Which display did you use?

Let’s look at when different displays are useful. A circle graph is useful for showing proportions and parts of a whole.

Suggested student grouping: Individual Favorite Color Advantages: Students are presented with a table and two graphs that represent the number of students having birthdays in • We can easily compare the proportions of different categories different seasons. The aim12% is for students to not onlyvisually. practice reading data from different types of displays, but They are commonly used. 12% 13% also to explain their reasoning for using one display over another. 6%

19%

Ideal student responses

Disadvantages • It can be diﬃcult to compare similar groups if they are not

30% These ideal responses may differ from other correct labeled. studentWeresponses. formal can may lose theLess original totalsresponses if we just show thebe percentages. connected with the more precise mathematical language presented here. Blue

Pink

1. Which season had the most birthdays? Which display did you use, the table, circle graph, or bar graph? Green Purple The season withRed the most birthdays Orange was Spring. Answers about which display was used will vary, but all Yellow three displays can be used to easily identify the season with the most birthdays. A bar graph ishad useful showingbirthdays? the count or Which frequency of different 2. Which season theforfewest display did categories. you use? The season with the fewest birthdays was Summer. Answers about which display was used will vary, but all three displays can be used to easily identify the season with the fewest birthdays.

3. How do winter and fall compare? Which display did you use? Winter and Fall have the same number of birthdays. Answers about which display was used will vary, but all three displays can be used to easily compare the birthdays during the two seasons. Purposeful questions • Why did you use that display to answer the question? 372 Mathspace Virginia SOL Grade 6 • Can a different display also be used to answer this question? mathspace.co • How could you identify the same information in the other two displays?

738

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Possible misunderstandings • Students might think that they should use a specific display to answer each question. Help students identify the same information from all three displays to show them that any display could be used to answer each question.

Examples Students: Page 374

Purpose Students show an understanding of deciding which display of data is best for a given situation. Expected mistakes Students might choose the circle graph because Brand A’s sector represents 35% of the sales. Point out that the question asks for a specific number of sales, while the circle graph represents a percentage of the sales.

9.03 Compare representations of data mathspace.co

739

Students: Page 375 b In which display is it easier to see the tire brand that was sold the most?

Create a strategy

Apply the idea

In the bar graph, we need to look for the bar with the greatest height, and for the circle graph, we need to look for the biggest section.

We can see this information easily from both graphs. So either A or B is a correct answer.

cb In thatwas made upthe halfmost? of the sales? In which which display display is is it it easier easier to to see see the the tire tire brands brand that sold

PurposeCreate Create a a strategy strategy Apply the idea StudentsIn show an understanding deciding which display of is best for a given situation. the bar theoftotals, thethe circle graph, wedata can the portions of a whole. In the bar graph, graph, we we can needsee to look for theand barfor with We can see see this information easily from both graphs. So greatest height, and for the circle graph, we need to look

Apply the375 idea Students:for Page the biggest section.

either A or B is a correct answer.

Reﬂect and check

The circle graph makes it easier to see fractions or Brands C and E make up one half of the sales and Brands portions, so the correct answer is option B. A, B, and D make up the other half. c In which display is it easier to see the tire brands that made up half of the sales?

Create a strategy

Example 2

In the bar graph, we can see the totals, and for the circle graph, we can see the portions of a whole. Five teams are completing a walking challenge. A circle graph and dot plot are used to show the distances each team Apply Reﬂect and check walked.the idea team The circle graph makes it easier to see fractions or Brands C andDistance E make walked up onebyhalf of the sales and Brands portions, so the correct answer is option B. A, B, and D make up the other half. 14%

16%

30%

Example 2 18% 22% Purpose Five teams are completing a walking challenge. A circle graph and dot plot are used to show the distances each team Studentswalked. show an understanding of deciding which display of data is best for a given situation. s

gle

ter

Lio

Lions Eagles Wombats Distance walked by team

H Ro Hyenas Roosters W Expected mistakes 14% than looking for multiple brands Studentsa might look for a single section of the circle labeled with 50% rather Which team walked the furthest? 30% that make up half of the sales. 16% Inform students the question says “which...tire brands” made up half of the sales, so they should look for Create a that strategy 22%where we can easily identify multipleWe brands that addthe to team half of the need to identify with thesales. greatest distance on the graphs. Select18% the graph

the highest number. s Lions Eagles Wombats ns ats enas sterdata les Integrate Advanced learners: Lio from both graphs to calculate exact sales figures ag mb o y

Apply the idea

R W Targeted instructional strategies

Hyenas

Roosters

use with Example 1

The Wombats team walked the furthest as clearly shown by the dot plot. a Which team walked the furthest? To deepen advanced learners’ understanding, challenge them to use both the bar graph and the circle graph

Reﬂectand andverify checkthe exact number of tire sales for each brand. Start by guiding students to analyze the to determine Create a strategy We canprovided also identify thatgraph walkedand the furthest using the circleactual graph, quantities but using thesuggested dot plot is ideal as it bar better percentages in the theteam circle relate them to the by the graph. We needthe to identify team with thethe greatest distance on the graphs. Select the graph where we can easily identify displays data wethe need to answer question.

For example, the circle the highest number.graph shows that Brand C and Brand E make up half the total sales, and the bar chart The part of the circle graph with the biggest section or highest percentage is the team that walked the furthest. shows that these two brands had 150 sales. Using this, students can estimate the total number of tires sold Apply the idea (300 tires). The Wombats team walked the furthest as clearly shown by the dot plot.

This exercise requires students to apply proportional reasoning and reinforces their ability to interpret and synthesize information from different types of data displays. Encourage students to explain their reasoning and Reﬂect and check verify their results by cross-checking between the two graphs. This not but only enhances their analytical skills but We can also identify the team that walked the furthest using the circle graph, using the dot plot is ideal as it better also promotes deeper ofquestion. how different graphical representations can provide complementary displays a the data weunderstanding need to answer the 9.03 Compare representations of data 375 mathspace.co information. The part of the circle graph with the biggest section or highest percentage is the team that walked the furthest. The sales figures for each brand are shown: • Brand A: 105 • Brand B: 15

740

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

• Brand C: 75

• Brand D: 30

• Brand E: 75

9.03 Compare representations of data mathspace.co

375

Create a strategy In the bar graph, we can see the totals, and for the circle graph, we can see the portions of a whole.

Apply the idea

Reﬂect and check

The circle graph makes it easier to see fractions or

Brands C and E make up one half of the sales and Brands A, B, and D make up the other half.

Students:portions, Page so 375 the correct answer is option B. Example 2

Five teams are completing a walking challenge. A circle graph and dot plot are used to show the distances each team walked. Distance walked by team 14% 30%

16% 18%

s s s ts rs gle mba yena oste Lion H Ro Wo

22%

Wombats

Eagles

Hyenas

Roosters

Lions

a Which team walked the furthest?

Create a strategy We need to identify the team with the greatest distance on the graphs. Select the graph where we can easily identify the highest number.

Apply the idea The Wombats team walked the furthest as clearly shown by the dot plot.

Reﬂect and check We can also identify the team that walked the furthest using the circle graph, but using the dot plot is ideal as it better displays the data we need to answer the question. The part of the circle graph with the biggest section or highest percentage is the team that walked the furthest.

Purpose Show students how to interpret and compare data from different types of graphs. 9.03 Compare representations of data

375

Reflecting with students mathspace.co Ask students to share which display they used to answer the question. Some students might say they used the circle graph to identify the largest sector. Inform students that both graphs show which team walked the furthest, so they can answer the question using whichever graph they find easiest to interpret.

Students: Page 376 b Which display would you use to compare the Eagles and Lions? Explain.

Create a strategy The dot plot shows us the number, but we need to count the dots. The circle graph shows the proportions.

Apply the idea In the dot plot, we can see that there are two more points for Eagles compared to the Lions, but it is a bit diﬃcult because they are not beside each other. The circle graph shows the proportion for each team, and the Eagles and Lions are beside one another which is nice. It may depend on the question we are looking to answer, but in general using the circle graph to compare the Eagles and Lions, would be a good choice.

c Complete this table that would show the same information as the two displays. Team Eagles Wombats Hyenas Roosters Lions

Distance (miles)

Percentage of miles

9.03 Compare representations of data mathspace.co

741

Create a strategy The dot plot shows us the number, but we need to count the dots. The circle graph shows the proportions.

Apply the idea In the dot plot, we can see that there are two more points for Eagles compared to the Lions, but it is a bit diﬃcult

Purposebecause they are not beside each other. The circle graph shows the proportion for each team, and the Eagles and Lions are how beside anotherand whichcompare is nice. data from different types of graphs. Show students toone interpret It may depend on the question we are looking to answer, but in general using the circle graph to compare the Eagles Lions,376 would be a good choice. Students:and Page

c Complete this table that would show the same information as the two displays. Team Eagles Wombats Hyenas Roosters Lions

Distance (miles)

Percentage of miles

Create a strategy We can get the distance from the dot plot and the percentage from the circle graph.

Apply the idea Fill in the distance column using the data from the dot plot and the percentage column using the data from the circle graph. Team Eagles Wombats Hyenas Roosters Lions

Distance (miles) 11 15 8 7 9

Percentage of miles 22% 30% 16% 14% 18%

Reﬂect and check We should always double check that the percentages add up to 100%. From this we can also see that the total distance walked by all teams was 50 miles.

Purpose Show students how to interpret and convert information between different types of data displays.

Compare and connect

use with Example 2

English language learner support Encourage students to compare and connect the circle graph and the dot plot to deepen their understanding of how the same information can be represented in different ways. Begin by displaying both graphs clearly and ask students to work inVirginia pairsSOL toGrade discuss what they notice about each display. Provide sentence stems such as 6 376 Mathspace mathspace.co “In the circle graph, I see that the Wombats have the largest section, which means...” or “The dot plot shows that the Lions have nine dots, indicating...”. As students share their observations, guide them to use key mathematical vocabulary like “percentage,” “proportion,” “frequency,” and “distance.” Create a visual chart that lists these vocabulary words alongside student explanations and examples from the graphs. Facilitate a class discussion where students explicitly connect features of one graph to the other—for example, how the number of dots for each team relates to their corresponding percentage in the circle graph. By making these connections, students will enhance their understanding of data representation while simultaneously developing their mathematical language skills. This routine supports English language learners by providing multiple entry points to grasp the concepts and vocabulary through visual aids and collaborative discussion.

742

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 377

Practice Students: Pages 377–385

What do you remember? 1

Match each name to the correct data display. i

Bar graph

4 5 6 7 Shifts per Week

Stem-and-leaf plot

iii

Circle graph

Line plot (dot plot)

Leaf 1

4 5

9.03 Compare representations of data mathspace.co

743

Methods of Traveling to School

What type of pet do you have? 8

11%

7% 6

9% 3%

4 70% 2

Car

Bicycle

Bus

Train

Fish Dog Cat Lizard

Walk

Is each statement true or false? a

A bar graph is best used for showing changes over time.

A circle graph is best used for showing proportions.

A line plot is best used for showing the frequency or count for a small number of categories or values.

A stem-and-leaf plot is best for data that is in a very small range.

Match each of the following scenarios to the type of data display that is best: i

Dot plot

Circle graph

A statistician surveyed 100 people to ﬁnd out which eye color is most common.

An athlete wants to identify patterns in their heartbeat during their workout.

A teacher asks her class of 25 students to tell her their favourite subject in school to see how many students prefer maths over science.

iii

Line graph

A grocery store created the following circle graph of fruit sold in their store: a

What does this circle graph tell us?

What is the least common fruit sold?

What is the second most common fruit sold?

Fruits purchased from the grocery 12% 8%

27%

40%

Use any graph to answer each question.

10% 30%

24%

16%

20%

Number of students

Favorite snacks of students

16 14 12 10 8 6 4 2 0

s es s ts er rro pple Chip ooki Oth A C Snack

Other

Carrots

Chips

Apple

744

13%

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Apples Grapes Blueberries Oranges Limes

Type of snack

Number of students

Apple Carrots Chips Cookies Other represents 2 students a

How many students listed carrots as their favorite snack?

What percentage of students listed cookies as their favorite snack?

What is the most popular snack?

Let’s practice 6

The table shows the team points earned by four different houses at their swimming competition: Team Points

Yellow 15

Blue 25

Red 40

Green 60

Select the best display to represent the given set of data: A 7

Stem-and-leaf plot

Pictograph

Circle graph

Line plot

Bar graph

The table shows the number of books students read in one month. Students Number of books a

Isla 3

Ozair 6

Mia 5

Mehrab 8

Ivy 7

Pictograph

Select the displays which could be appropriate. A Circle graph

Line graph

E Stem-and-leaf plot b 8

What display would you use to represent this data? Explain your choice.

The table shows the number of books Deana read in each month. Month January February March April May June

Number of books 5 2 3 0 4 3

Month July August September October November December

Number of books 8 7 6 5 2 1

What display would you use to represent this data? Explain your choice. 9

The data in the table shows summarizes the scores that students got on a quiz out of 15. a

What is the best way to display this data? Explain your choice. A Circle graph

Line plot

C Pictograph

Bar graph

Construct the display chosen in part (a).

Score 11 12 13 14 15

Number of students 12 9 10 9 6

9.03 Compare representations of data mathspace.co

745

Mr. Rodriguez recorded the number of pets of his students. He found that 15 students had no pets, 19 students had only one pet, 3 students had two pets and 8 students had three pets. a

What is the best way to display this data? Explain your choice. A Circle graph

Line graph

Pictograph

Bar graph

E Stem-and-leaf plot b 11

Construct the display chosen in part (a).

An online booking service allows customers give a rating out of 5 at the end of their transaction. Over the last month the service has been tracking the feedback. 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5 Determine if each plot type could be used to display this data and describe the advantages or disadvantages of each. a

Circle graph

Which representation makes it easier to answer the question: “What percentage of the survey respondents chose basketball as their favorite sport?” A

Favorite sport 10% 10%

35%

20%

Basketball:

Other

Soccer:

Baseball

Baseball:

Basketball

Tennis:

Tennis

Other:

Soccer

Key:

25%

Farah wants to answer the question “How does the average temperature change over the year?” After collecting the data, which representation would be appropriate to answer the question? A

Seasonal distribution of months Fall 25%

25%

Spring Summer Winter

25%

Which representation could be used to answer each question. a

What was the median score of students on the math test?

What percentage of students passed the math test?

How many students got an F on the math test?

Math test grades F

10% 30%

A 15%

B D

25%

746

Average temperature (°F)

5 students

20%

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Average temperature 80 70 60 50 40 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Line plot (dot plot)

Month

Math test grades 5

0 0

0 3 3 4 4 4 4 4 4 4 5 5 6 7 8

0 0 0

2 3 3 4 4 4 5 5 5 6 6 7 7 8 8 9 9 9 9

0 0 0 0

2 3 4 5 5 9

2 2 2 3 3 3 3 3 3 3 4 4 5 5 7 7 7 8 8 9 9 9 9

2 3 3 3 4 5 5 6 6 7 7 7 7 8 8 8 9

10 0 15

The principal at Sunny Meadows Middle School is asked “How has student enrollment changed over the years?” Which representation should they show to answer the question? A

2024 Enrollment

Student enrollment at Sunny Meadows Middle School Number of students

6th Grade 30%

37%

7th Grade

8th Grade

33%

20 Year 2015 2016 2017 2018 2019 2020 2021 2022 2023

This bar graph and circle graph show the number of times that each type of sport is played. Sports Played

Sports Played

Football

27.8%

33.3%

Rugby Tennis Basketball

11.1% 22.2%

5.6%

Swimming

Frequency

40 30 20 10 0 Football

Tennis

Rugby

Swimming Basketball

Sport

Which sport was played the most? State which graph you used to answer.

What are the beneﬁts and drawbacks of the circle graph?

This line plot and circle graph represent the data that was collected to answer the question “How do sales of tomato seeds vary by type of tomato?”. a

Describe the advantages and disadvantages of each display.

Create a circle graph that shows the number of seed packets sold on the display.

9.03 Compare representations of data mathspace.co

747

Cherry Beefsteak Roma Heirloom Grape Green Zebra tomatoes tomatoes tomatoes tomatoes tomatoes tomatoes Type of seed packets sold 2%

Type of seed packets sold 6%

8% 10% 54% 20%

Beefsteak tomatoes

Roma tomatoes

Heirloom tomatoes

Grape tomatoes

Green Zebra tomatoes

Mr. Smith asked his students to choose their favorite animals. He plotted the results in the bar graph. a

Complete this table using the bar graph. Animal Cats Dogs Hamsters Birds Total

Number of students

Frequency

Cherry tomatoes

50 45 40 35 30 25 20 15 10 5 0

cats

dogs hamsters birds Animals

How does the proportion of students that prefer dogs compare to those that prefer birds?

Convert this display to a circle graph and explain which graph makes it easier to compare between animals.

Let’s extend our thinking 19

748

The circle graph represents the popularity of each chip brand amongst high schoolers. a

Interpret the graph. List all information you know to be true.

Here is a dot plot for the same data set. Explain what we can see more clearly from the dot plot.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Tortillos

Hot Cheezos

Snakis

Wrinkles

Tortillos

Hot Snakis Wrinkles Cheezos

A snack brand which is preferred by more than 25% of the population does not need to worry about marketing to parents, which display can we more easily identify this from? Explain.

The sales of different products are shown in the following horizontal bar graph: Product A Product B Product C Product D Product E Product F 0

3 4 5 6 7 8 Sales (thousands of units)

If you wanted it to be clear that Product D is the most popular which type of display would you use. Explain.

If you wanted it to be unclear that Product D is the most popular which type of display would you use. Explain.

An amusement park recorded the number of people that visited during a week of the summer break and displayed the data in the graph shown. a

Convert this bar graph to a circle graph.

What is easier to visualize from the bar graph?

What is easier to visualize from the circle graph?

Number of Visitors

1000 900 800 700 600 500 400 300 200 100 0

Visitors to Adventure Land

Mon Tue Wed Thu Day

Fri

Sat

Sun

Explain why you can convert a bar graph into a circle graph, but may not be able to convert a circle graph to a bar graph.

9.03 Compare representations of data mathspace.co

749

The 6th graders need to choose a class captain who will go to student council meetings. Felicia was asked to predict who might win. She did a poll and asked an anonymous sample who they are planning to vote for. The results are shown in these two displays. a

If you were Constantina which display would you use? Explain.

If you were Alanna which display would you use? Explain.

Votes

Class representative poll

10 5 0

n tina iel a him an so All Bron stan Dan Ebra n Co Candidate

Alanna

Bronson

Constantina

Daniel

Ebrahim

750

Lashanda’s mom just bought an air fryer and she now uses it all the time. This makes her curious about what people use as their main cooking appliance. a

Formulate a question that Lashanda could use to explore the types of appliances that people use using the data cycle.

Could she use observation, measurement, survey, experiment, or acquire secondary sources? Explain.

Explain how Lashanda could collect data that could be used to answer her question from part (a).

Suppose the given data shows the preferred appliance for 40 people who she knows do the cooking. Organize and represent the data using at least one circle graph.

Person

Age

Preferred appliance

Number of people in home

George Matty Shaira Andrew Cleo Antonio

50 29 24 29 21 48

Air fryer Barbeque Countertop toaster oven Microwave Oven Stovetop

5 4 5 2 10 2

Number of meals cooked per week in preferred appliance 14 4 2 6 15 10

Zoe

Stovetop

Alberta Jack Alice David Bob

55 44 36 60 59

Stovetop Pressure cooker Countertop toaster oven Stovetop Stovetop

6 5 4 2 11

8 7 8 12 20

Charlie

Barbeque

Emily Felix Mark Laarni

18 23 32 19

Air fryer Oven Stovetop Air fryer

3 5 7 8

6 10 14 6

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Favorite food Chicken wings Ribs Croissants Steamed vegetables Lasagna Chicken Spaghetti with Marinara Sauce Omelettes Chicken Alfredo Pizza Steamed fish fillets Pancakes Slow-smoked beef brisket Shrimp Fillet mignon Stir-Fry Brussels sprouts

Person

Age

Preferred appliance

Number of people in home

Josh Clarisse Ollie Andrea Mozart Cecille Piolo Eddy

35 32 31 30 44 37 46 65

Countertop toaster oven Air fryer Oven Oven Oven Oven Oven Stovetop

2 9 12 7 5 4 2 4

Number of meals cooked per week in preferred appliance 7 18 20 3 7 10 14 5

Gwen

Stovetop

Hyacinth Arthur Gabby Leonora Antonina Faye Edgardo Xander Emilia Dexter Rose Ann John Lenn Maricel Chona

51 30 36 62 58 42 43 37 33 38 41 51 56 66

Oven Oven Barbeque Pressure cooker Stovetop Countertop toaster oven Air fryer Barbeque Countertop toaster oven Microwave Oven Stovetop Stovetop Stovetop

5 8 6 3 7 9 12 4 15 2 3 4 10 9

8 8 10 10 14 5 15 2 25 10 2 6 15 18

Favorite food Fries Spring rolls Fish tacos Beef Baked potatoes Steamed dumplings Beef Wellington Mushroom Risotto Spaghetti Carbonara Stuffed bell peppers Chicken Chicken Beef Brisket Risotto Nuggets Bacon Sausages Meatballs Tofu pad thai Quiche Rice Pilaf Lobster Beef Stroganoff

Analyze the data to draw a conclusion about preferred appliances.

Her friend Michael said his dad’s countertop toaster oven stopped working, so he is wants to make a recommendation to his dad about what he should get to replace it. Michael makes this bar chart to show the same data. Preferred appliances

Number of people

12 10 8 6 4 2

St o

ve n O

r Ai

e e p ue ur r to r av ss oke eq w er ste n t e b o n a ve ar r Pr co ic ou to o B M

fry

Appliances

Compare the conclusions that are easy to see from the circle graph compared to those with the bar chart. g

Formulate another statistical question that could be used to explore appliance popularity.

9.03 Compare representations of data mathspace.co

751

Answers

displays to use would be a bar graph, circle graph, or even a pictograph. b Only one graph is necessary:

9.03 Compare representations of data

What do you remember? b ii: Stem-and-leaf plot

c iii: Circle graph 2 a False

d i: Bar graph

b True

3 a Circle graph

Frequency

1 a iv: Line plot (dot plot)

c True

b Line graph

20 18 16 14 12 10 8 6 4 2 0

d False c Dot plot

4 a T he circle graph tells us how what percentage of fruit sales were from different types of fruits.

No pet 1 Pet 2 Pets 3 Pets Number of Pets

b Limes c Apples

5 a 10 students

b 16%

Number of Pets

c Apples

Let’s practice 6 C 7 a A , C, and D are all possible choices. The best choice can depend on what types of questions we want to answer from this.. b Answers vary. A bar graph would be a good choice to represent the data. This graph makes it easy to compare between students and displays the number of books each student read in a month. 8 Answers vary. A line graph would be a good choice. It would show the trends and patterns of Deana’s reading habits throughout the year and would allow us to read off the original data values. 9 a Many possible answers. For example: A line plot would be the best choice. Since we have the scores and the number of students for each, this type of data ﬁts a line plot well, as it would show the number of students that got each score. b

No Pet

1 Pet

2 Pets

3 Pets

11 a A line plot (dot plot) can be used to display the data on customer ratings. This graph allows us to see individual ratings and their frequency or count. One disadvantage of using this graph would be if the number of given ratings is too high, it would be hard to read. b A circle graph can be used to display the data on customer ratings. As this graph shows proportion or percentage, we can easily identify which rating has the highest or lowest number. A disadvantage of this is that it usually doesn’t display the actual count or numbers for the ratings. 12 A, because the circle graph shows percentages. 13 B, because the line graph shows the trend over a year. 14 a B

b A

c B

15 B, because the line graph shows the enrollment trend from 2015 to 2023. 16 a F ootball. Example explanation: I used the bar chart to and looked for the tallest bar, but used the circle graph to check which segment was the largest. 11

Score

10 a T here are many possible ways to represent this data. It can depend on what types of questions we want to answer from this. This is categorical data so the best

752

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

b O ne beneﬁt of a circle graph is that it easily shows proportions and distributions. Identifying which is the highest or lowest is easy. However, a drawback of this circle graph is that it doesn’t display exact numbers. We can compare percentages, but not exact numbers.

17 a A n advantage of using a line plot is simple, but clearly shows the number of seeds sold for each type of tomato. However, a disadvantage is that to determine the exact number of seed packets sold for each type, we need to count the number of points which can be slow to do.

b I n the dot plot, we can observe the number of students for each chip brand, represented by data points. By counting the points, we can determine how many students chose a particular brand of chip. Additionally, the side-by-side plotting allows for clearer visualization of the differences between the numbers.

A circle graph has the advantage of clearly showing the proportions or contributions of each type of seed packet sold to total sales. Percentages are helpful if we want to quickly understand the size of sales for each type of seed. However, a disadvantage is that we cannot see the actual number of each type of seed packet sold, making it diﬃcult to compare the sales numbers for each type.

c W e can easily identify percentages from circle graphs, as they display proportions, which dot plots do not.

Type of seed packets sold 1

Cherry tomatoes

Roma tomatoes

Grape tomatoes 26 10

Beefsteak tomatoes Heirloom tomatoes Green Zebra tomatoes

18 a

Animal

Number of students

Cats

Dogs

Hamsters

Birds

Total

100

b I n this case, a circle graph, without the percentages or numbers labeled on the sectors, would make it unclear that Product D is the most popular. This is because two other products very similar numbers to Product D. If shown using a circle graph, it would be a bit diﬃcult to identify which product has the highest sales, as the proportions for Products A, D, and E would be almost identical. However, we could still determine the ranking by examining the percentage labels. 21 a

Visitors to Adventure Land 8% 24%

16% 8%

24% 16%

b T he proportion of students that prefer dogs is greater by 30% compared to the students that prefer cats. c

20 a U sing this bar chart would show that Product D is the most popular, as the graph clearly displays the highest and lowest categories. Just by looking at the graph, it’s clear which product has the highest sales.

Students’ favorite animals

Mon

Tue

Wed

Thur

Fri

Sat

Sun 15%

25%

15%

Cats

Dogs

Hamsters

Birds

45%

Comparing proportions is easier with a circle graph since it displays the percentage for each animal. However, comparing actual numbers is simpler with a bar graph. Let’s extend our thinking 19 a W e can observe the popularity ranking of chips among high schoolers, with Snakis being the most popular, followed by Hot Cheezos, Tortillos, and ﬁnally Wrinkles.

b T he exact values and differences in the number of people who visited each day are easier to visualize in a bar graph. c T he proportion or percentage of the number of daily visitors to the total number is easier to visualize in a circle graph. 22 We can convert a bar graph into a circle graph because the data needed to form a circle graph is present in the bar graph. We can ﬁnd the total number of data points and divide each category by this total to get the percentage for each sector. However, we cannot convert a circle graph into a bar graph because a circle graph does not always show the numerical values required to construct a bar graph.

Snakis looks like about one third of the high schoolers and Hot Cheezos looks like about one quarter and Tortillos and Wrinkles look similar.

Answers mathspace.co

753

23 a I f I were Constantina, I would use the bar graph. The bar graph clearly displays who received the highest votes, which is Constantina, giving her an advantage. It also shows the differences between the votes.

c T o answer the question “What do most Americans use as their main cooking appliance?”, Lashanda could collect data by doing an online survey. She could design the survey to ask about participants’ primary cooking appliances, this would help her to gather the needed data. Also, Lashanda could make sure the survey reaches a diverse demographic by including some questions about age, cooking habits, location, etc. Lashanda should also include other factors in her survey that might be needed in case there would be follow-up questions upon reaching the answer to her initial question, or another round of the data cycle would be done.

754

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

12%

13% 8%

24 a A sample question would be “What do most Americans use as their main cooking appliance?”.

She could also use secondary sources if there are existing surveys or data available from reliable websites or other sources. However, doing a survey would enable her to acquire the most recent and reliable data.

17%

20%

b I f I were Alanna, I would use the circle graph. From the circle graph, we can observe that there is not much difference between the votes for the ﬁve students. Although careful inspection would show that Constantina is in the lead, the circle graph does not clearly highlight these differences.

b S he could use a survey to acquire the needed data. This not only allows her to easily collect the required information but is also ethical, as she would ﬁrst ask if the respondents are willing to participate, unlike other methods such as observation. Experiments and measurements are not applicable in this scenario.

Preferred main cooking appliance

15% 15%

Air fryer

Barbeque

Countertop toaster oven

Microwave

Oven

Pressure Cooker

Stovetop

e W e can use the circle graph we created in part d to help us with the analysis. The most preferred main cooking appliance is the stovetop, which is preferred by 20% of the respondents. The least preferred is oven with 8%. f

he proportion of the most preferred main cooking T appliance is easier to see in a circle graph, but the bar graph displays the most preferred main cooking appliance in total numbers instead of percentages which shows that the sample was quite small and may not be reliable.

g A nother statistical question that can be used to explore appliance popularity is “For Americans who cook more than 15 meals a week, what is their preferred main cooking appliance?”, “What are the preferred main cooking appliances of Americans with more than 5 people in their homes?”, or “How has cooking appliance popularity changed over time?”

9.04 Review: measures of center and spread Subtopic overview Lesson narrative In this lesson, students will review measures of center and spread, including mean, median, mode, and range. They will understand how each measure provides different insights into a data set. The mean represents the average, the median is the middle value, the mode is the most frequent value, and the range shows the spread from the highest to lowest value. Students will engage in examples and exercises to calculate these measures and interpret their significance in various contexts. By the end, students should confidently analyze data sets using these statistical measures.

9.04 Review: measures of center Learning objectives and spread Students: Page 386

After this lesson, you will be able to... • calculate the mean of a data set. • identify the median of a data set. • identify the mode of a data set. • calculate the range of a data set. • use mean, median, mode and range to solve problems.

Mean The mean is the average of the values in the data set. It is a measure of center, meaning it is an approximation of Key vocabulary

where the middle of a data set is.  measure of center  measure of spread average meanthree friends are planning Let’s think about a situation where a trip to Palm Springs. They plan to fly there, and learn  median  mode that the airline has a rule: each person can only bring 35 lbs of stuff in their bags. On the night before the flight they range weigh their luggage and find that their luggage weights from this data set:



29, 32, 37

Essential understanding One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now? Measures of center provide useful information when interpreting data. Different measures approximate the “center” it usingways; more each mathematical are sharing the into totalaluggage equally three groups. of aThinking data setabout in different measurelanguage, provideswe different insights set of data andamong is appropriate in As a mathematical expression, ﬁnd: provide useful information when interpreting data. Different measures different situations. Measures of we spread

approximate the “spread” of a data set in different ways; each measure provides different insights into a set of data and is appropriate in different situations. Each person carries 32.67 lbs. This amount is the mean of the data set. If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same. To calculate the mean, use the formula:

Standards

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Example 1 Find the mean of the scores: 6, 14, 10, 13, 5, 9, 14, 15 Give your answer as a decimal.

9.04 Review: measures of center and spread mathspace.co

755

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG2 — Mathematical Communication

Teachers can guide students to apply their knowledge of measures of center and spread in various problem-solving contexts. For example, students can be provided with different data sets and asked to calculate the mean, median, mode, and range. To extend the problem-solving experience, students could be asked to justify their selection of a particular measure of center in a given scenario.

Teachers can create opportunities for students to communicate their mathematical understanding by encouraging them to explain their process for calculating measures of center and spread. This can be done through group discussions, written reflections, or presentations. Further, the teacher can incorporate the use of appropriate mathematical language, like ‘mean’, ‘median’, ‘mode’, ‘range’, ‘distribution’, and ‘data set’, in classroom discussions to enhance mathematical communication.

MPG3 — Mathematical Reasoning Teachers can develop students’ mathematical reasoning skills by presenting problems that require them to decide which measure of center is most appropriate for a given data set. For example, students could be given a data set with an extreme outlier and asked to decide whether the mean or median provides a better representation of the data.

Prior connections 5.PS.2 — The student will solve contextual problems using measures of center and the range.

Future connections 6.PS.2 — The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.

Lesson Preparation Tools You may find these tools helpful: • Scientific calculator • Sticky notes

756

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Collect and display English language learner support As students are working on calculating the mean, median, mode, and range of data sets, listen for the language they use to describe these concepts. Collect their descriptions and display them in a visible area of the classroom for everyone to refer to. If students are having difficulty articulating their understanding or if alternative explanations would be helpful, provide some of your own. For example: • Mean • The average of all the numbers • The sum of all values divided by the number of values • Median • The middle number when the data is ordered from least to greatest • The value that splits the data set into two equal halves • If there’s an even number of data points, the average of the two middle numbers • Mode • The number that appears most frequently • The value with the highest frequency in the data set • A data point that occurs more than others • Range • The difference between the highest and lowest values • How spread out the data is • Maximum value minus minimum value Pay attention to any misunderstandings or inaccuracies in students’ descriptions, and gently correct them as needed. Use this collection of definitions to reinforce the different insights each measure provides into a data set. Encourage students to refer to the display during discussions and when solving problems to support their understanding of the concepts.

9.04 Review: measures of center and spread mathspace.co

757

Student lesson & teacher guide Mean Students review the concept of the mean as a measure of center, calculated by dividing the total sum of a data set by the number of values. Through the example of three friends redistributing luggage to equally share the weight, students see how the mean represents an average that balances the data. The section explains that replacing each value with the mean keeps the total sum of the data unchanged.

Students: Page 386

9.04 Review: measures of center and spread After this lesson, you will be able to... • calculate the mean of a data set. • identify the median of a data set. • identify the mode of a data set. • calculate the range of a data set. • use mean, median, mode and range to solve problems.

Mean The mean is the average of the values in the data set. It is a measure of center, meaning it is an approximation of where the middle of a data set is. Let’s think about a situation where three friends are planning a trip to Palm Springs. They plan to fly there, and learn that the airline has a rule: each person can only bring 35 lbs of stuff in their bags. On the night before the flight they weigh their luggage and find that their luggage weights from this data set: 29, 32, 37 One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now? Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find:

Each person carries 32.67 lbs. This amount is the mean of the data set. If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same. To calculate the mean, use the formula:

Example 1 Find the mean of the scores: 6, 14, 10, 13, 5, 9, 14, 15 Give your answer as a decimal.

Create a strategy Use the formula Mean =

758

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 386

Mathspace Virginia SOL Grade 6 mathspace.co

Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we ﬁnd:

Each person carries 32.67 lbs. This amount is the mean of the data set.

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the Examples same. To calculate the mean, use the formula:

Students: Pages 386–387

Example 1 Find the mean of the scores: 6, 14, 10, 13, 5, 9, 14, 15 Give your answer as a decimal.

Create a strategy Use the formula Mean =

Apply the idea Use the formula Add the numbers in the numerator 386

Perform the division

Mathspace Virginia SOL Grade 6 mathspace.co

Idea summary Purpose Students demonstrate that they can use the Mean = formula to find the mean of a data set. Reflecting with students Mean is the average of a data set. Have a discussion with students about the mean being a decimal in some situations when the data values are Apply the idea whole numbers. Explain that although the mean is a value that summarizes the center of the data, so there may Use the formula be times when it will not make sense in a real-world context.

Median

Add the numbers in the numerator

Students:The Page 387 median is the middle of the data set when ordered least to greatest. It is also a measure of center. Let’s say Perform the division seven people were asked about their weekly income, and their responses form this data set: $300, $400, $400, $430, $470, $490, $2900 The mean of this data set is

Idea summary

= $770, but this amount doesn’t represent the data set very well. Six out of seven

people earn much less than this. Instead, we can select the median, which is=the middle income. We remove the biggest and the smallest incomes to Mean get: $400, $400, $430, $470, $490 Mean is the average of a data set. Then the next biggest and the next smallest to get: $400, $430, $470 Then the next biggest and the next smallest to get: Median $430to greatest. It is also a measure of center. Let’s say The median is the middle of the data set when ordered least

Median There only one number and thisweekly is the median so for thisresponses data set the median is $430. seven is people were asked left, about their income,- and their form this data set: This weekly income is much closer to the other scores in the data set, and summarizes the set better.

$400, $400, $490, $2900 Students will learn about the concept of $300, the median in a $430, data $470, set, understanding it to be the middle value when the The median is the number in the middle of a numerical data set. The mean ofleast this data set is = $770, but this amount doesn’t represent themedian data set in very well.with Six out of seven data is ordered from to greatest. They will explore how to calculate the sets both odd and even • If the list has an odd number of data points, the median is the one right in the center. numbers ofpeople data points. earn much less than this. • If the list has an even number of data points, the median is the number halfway between (or the average of) the two middle Instead, we canones. select the median, which is the middle income. We remove the biggest and the smallest incomes to get: $400, $400, $430, $470, $490 Then the next biggest and the next smallest to get: $400, $430, $470 Then the next biggest and the next smallest to get: $430 There is only one number left, and this is the median - so for this data set the median is $430. This weekly income is 9.04 Review: measures of center and spread much closer to the other scores in the data set, and summarizes the set better. mathspace.co The median is the number in the middle of a numerical data set. • If the list has an odd number of data points, the median is the one right in the center. • If the list has an even number of data points, the median is the number halfway between (or the average of) the

759

Idea summary

Mean = Mean is the average of a data set.

Students: Pages 387–388

Median The median is the middle of the data set when ordered least to greatest. It is also a measure of center. Let’s say seven people were asked about their weekly income, and their responses form this data set: $300, $400, $400, $430, $470, $490, $2900 = $770, but this amount doesn’t represent the data set very well. Six out of seven

The mean of this data set is people earn much less than this.

Instead, we can select the median, which is the middle income. We remove the biggest and the smallest incomes to get: $400, $400, $430, $470, $490 Then the next biggest and the next smallest to get: $400, $430, $470 Then the next biggest and the next smallest to get: $430 There is only one number left, and this is the median - so for this data set the median is $430. This weekly income is much closer to the other scores in the data set, and summarizes the set better. The median is the number in the middle of a numerical data set. • If the list has an odd number of data points, the median is the one right in the center. • If the list has an even number of data points, the median is the number halfway between (or the average of) the two middle ones. Half the numbers in the list will be bigger than the median, and half will be smaller. Sets ordered smallest to largest Median Odd number of scores:

Even number of scores:

Median:

+ 2

9.04 Review: measures of center and spread mathspace.co

387

Example 2

Advanced learners: Generalize a method for finding the median Find the median of the scores:

Targeted instructional strategies

3, 18, 10, 19, 12, 5, 6, 20, 7

Encourage students to explore how the position of the median relates to the number of data values in a set. Prompt them and Createtoa consider strategy both odd and even-sized data sets Apply thederive idea a general method for finding the median We position. For example, ask students to investigate ordered data need to put the scores in order and ﬁnd the middle The scores in ordersets are: and determine that when there score. are an odd number of values, the median is at the

position. When3,there an 18, even number of values, the 5, 6, 7,are 10, 12, 19, 20

andit has + 14 positions. median will be a decimal, indicating that it is the average The of the values middle scoreat is the 10 because scores above it and 4 scores below it. The median of For example, if there are 9 data values, the median will be the = the 5thscores value.is If10.there are 10 data values, the

median will be the

= 5.5th data value, which means it will be the average of the 5th and 6th value.

Encourage students to express these methods algebraically and justify why they work. This will deepen their Idea summary understandingThe of median how the depends number points enhance their algebraic of median a numerical data set ison thethe data value in of thedata middle when and the data is ordered from least to reasoning skills. By deriving these formulas themselves, students can confidently apply them to any data set and greatest. appreciate theTounderlying mathematical ﬁnd the median of a data set: concepts. • •

760

If the list has an odd number of data points, the median is the one right in the center. If the list has an even number of data points, the median is the number halfway between (the average of) the two middle ones. Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Mode

Even number of scores:

Examples Median:

Students: Page 388

+ 2

Example 2 Find the median of the scores: 3, 18, 10, 19, 12, 5, 6, 20, 7

Create a strategy

Apply the idea

We need to put the scores in order and ﬁnd the middle score.

The scores in order are: 3, 5, 6, 7, 10, 12, 18, 19, 20 The middle score is 10 because it has 4 scores above it and 4 scores below it. The median of the scores is 10.

Half the numbers in the list will be bigger than the median, and half will be smaller.

Idea summary

Sets ordered smallest to largest Purpose The median of a numerical data set is the data value in the middle when the data is ordered from least to Show studentsgreatest. how to find the median of a set of numbers by ordering the numbers and identifying the middle Median value. To ﬁnd the median of a Odd datanumber set: of scores:

•

If the list has an odd number of data points, the median is the one right in the center.

Find the sticky use with • Ifmedian the list has with an even numbernotes of data points, the median is the number halfway between (the average of)Example 2 the two middle ones. Student with disabilities support Even number of scores:

To support students who struggle with visual-spatial processing or organizational skills, incorporate a handson activity using sticky notes to find the median. Write each of the given scores on individual sticky notes. Mode Ask students to arrange the sticky notes in order from least to greatest on their desks or a large surface like a The mode of a data set is the result with the greatest frequency,+or the data value that appears most often in the data Median: whiteboard or wall. 2 set. If there are multiple results that share the greatest frequency then there will be more than one mode.

Once the numbers are in order, students remove noteanswers. from each simultaneously—the Yvonne asks 15 of her friendsguide what their favorite to color is. She one writessticky down their Here end is what she wrote smallestdown: and the largest—and set them aside. Continue this process of removing the outermost sticky notes 2 together,Example progressively moving towards This Green, step-by-step elimination highlights how the data set Blue, the Pink,center. Blue, Yellow, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink narrowsFind down to the middle value. When only one sticky note remains, explain that this number is the median of the median of the scores: the scores. 3, 18, 10, 19, 12, 5, 6, 20, 7

Consider using a visual aid or diagram to reinforce the concept. For example, display an image showing the orderedCreate sticky anotes with arrows indicating the removal of the outer notes in each step. This tactile and visual strategy Apply the idea Virginia SOL Grade 6 388 Mathspace method We makes the abstract concept of the median more concrete, engaging mathspace.co need to put the scores in order and ﬁnd the middle The scores in order are: students kinesthetically and helping score. them retain the information more effectively. 3, 5, 6, 7, 10, 12, 18, 19, 20 The middle score is 10 because it has 4 scores above it and 4 scores below it.

Students: Page 388

The median of the scores is 10.

Idea summary The median of a numerical data set is the data value in the middle when the data is ordered from least to greatest. To ﬁnd the median of a data set: • •

mathspace.co Yvonne asks 15 of her friends what their favorite color is. She writes down their answers. Here is what she wrote down: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow,

761

The median of the scores is 10.

Idea summary

Mode

The median of a numerical data set is the data value in the middle when the data is ordered from least to greatest. To ﬁnd the median of a data set:

Students will review the mode of a data set, which refers to the result with the highest frequency. They will learn to • If the list has an odd number of data points, the median is the one right in the center. apply this understanding to real-world examples and identify the mode as the most commonly occurring value. •

If the list has an even number of data points, the median is the number halfway between (the average of) the two middle ones.

Students: Pages 388–389

Mode The mode of a data set is the result with the greatest frequency, or the data value that appears most often in the data set. If there are multiple results that share the greatest frequency then there will be more than one mode. Yvonne asks 15 of her friends what their favorite color is. She writes down their answers. Here is what she wrote down: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink She then counts the number of colors to see which is the most picked. Color Number of Friends Pink 6 388 Mathspace Virginia SOL Grade 6 Green 2 mathspace.co Blue 3 Yellow 4 The of the the datanumber is pink.of colors to see which is the most picked. She mode then counts Color

Number of Friends 6 Green 2 Thomas conducted a survey on the average number of hours his classmates exercised per day and displayed his Blue 3 data in a table. Yellow 4 Students: Page 389hours No. exercise Frequency The mode of the data is pink. 0 2 1 12 7 Example 23 3 5 Thomas conducted a survey on 0 the average number of hours his classmates exercised per day and displayed his 4 data in a table. 5 3

Example Pink 3

Examples

No. exercise hours Frequency What is the mode of the data? 0 2 1 12 Create a strategy Apply the idea 2 7 Choose the result with the greatest frequency in the data. 1 hour of exercise is the mode because it has the greatest 3 5 frequency. 4 0 5 3 What is the mode of the data?

Idea summary

mode of a data set is the result with the greatest frequency. If there are multiple results that share the CreateThe a strategy Apply the idea

greatest frequency then there will be more than one mode. Choose the result with the greatest frequency in the data. 1 hour of exercise is the mode because it has the greatest frequency.

Range The range is asummary measure of the spread of a data set from the highest value to the lowest. Idea

PurposeTwo busThe drivers, how with manythe passengers board theirIf buses each day forresults a week. Their results modeKenji of a and dataBjorn, set is track the result greatest frequency. there are multiple that share the Show students howinto identify thethere mode frequency table. are displayed this table: then greatest frequency willinbeamore than onedistribution mode. T W T F Expected mistakesM Kenji 10 13 14 16 11 Students may mistakenly choose the maximum number of exercise hours as the mode, rather than the value Bjorn 2 27 13 5 17 with theRange greatest frequency. Encourage students to write out the individual values in the data set as a way to Both data sets same median and the same mean, the sets aretoquite different. To calculate the range, we Thedetermine range is a have measure of the spread of a data setfrequently. from thebut highest value the lowest. help them thethe value that occurs most start by ﬁnding highest and lowest of passengers each driver: Two bus drivers,the Kenji and Bjorn, track number how many passengers for board their buses each day for a week. Their results are displayedHighest in this table: Lowest

762

Kenji Virginia Mathspace SOLT Grade Teacher Edition M 16 W 610 T F Bjorn mathspace.co Kenji 10 27 13 14 2 16 11 Bjorn

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we

5 the data is pink. The mode of

What is the mode of the data?

Example 3 Create a strategy

Apply the idea

Thomas conducted a survey on the frequency average number of hours his classmates perbecause day anditdisplayed his Choose the result with the greatest in the data. 1 hour of exerciseexercised is the mode has the greatest Students:data Page 389 in a table. frequency. No. exercise hours Frequency 0 2 1 12 Idea summary 2 7 The mode of a data set is the result with the greatest frequency. If there are multiple results that share the 3 5 greatest frequency then there will be more than one mode. 4 0 5 3 What is the mode of the data?

Range

range is a measure of the spread of a data set from theApply highestthe value to the lowest. Create a strategy idea RangeThe Two bus drivers, Kenji and Bjorn, track how many passengers board their buses each day for a week. Their results

Choose the result with the greatest frequency in the data. 1 hour of exercise is the mode because it has the greatest

arelearn displayed thiscalculate table: Students will howin to the range of a data set and understand its significance in measuring the spread frequency. of the data. Using real-world M T examples, W T theyFwill calculate the range by subtracting the lowest data point from the 10 13 that14a smaller 16 11 highest. The Kenji lesson explains range indicates more predictability, while a larger range indicates more 27 13 5 17 variability. BjornIdea2summary Thesets mode of the a data setmedian is the result withsame the greatest frequency. If there multipleTo results that the share the we Both data have same and the mean, but the sets are quiteare different. calculate range,

Students:start Pages 389–390 frequency there will be more than one mode. bygreatest ﬁnding the highest then and lowest number of passengers for each driver: Kenji Bjorn Range

Highest 16 27

Lowest 10 2

The range is a measure of the spread of a data set from the highest value to the lowest. Two bus drivers, Kenji and Bjorn, track how many passengers board their buses each day for a week. Their results are displayed in this table: 9.04 Review: measures of center and spread 389 Kenji Bjorn

M 10 2

T 13 27

W 14 13

T 16 5

mathspace.co

F 11 17

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we start by ﬁnding the highest and lowest number of passengers for each driver: Kenji Bjorn

Highest 16 27

Lowest 10 2

Now we subtract the lowest from the highest to ﬁnd the difference, which is the range: Range 16 − 10 = 6 27 − 2 = 25

Kenji Bjorn

9.04 Review: measures of center and spread mathspace.co

389

Notice how Kenji’s range is quite small, at least compared to Bjorn’s. We might say that Kenji’s route is more predictable and that Bjorn’s route is much more variable (is more likely to change). The range of a numerical data set is the difference between the highest and the lowest data point. Range = Highest data point − Lowest data point

Example 4 Find the range of the following scores: 10, 7, 2, 14, 13, 15, 11, 4

Create a strategy Use the formula Range = Highest score − Lowest score.

Apply the idea The highest score is 15 and the lowest score is 2. Range = 15 – 2 = 13

Subtract 2 from 15 Perform the subtraction

Idea summary

9.04 Review: measures of center and spread mathspace.co

Range = Highest data point − Lowest data point

763

Now we subtract the lowest from the highest to ﬁnd the difference, which is the range: Range 16 − 10 = 6 27 − 2 = 25

Kenji Bjorn

Notice how Kenji’s range is quite small, at least compared to Bjorn’s. We might say that Kenji’s route is more Examples predictable and that Bjorn’s route is much more variable (is more likely to change). range390 of a numerical data set is the difference between the highest and the lowest data point. Students:The Page Range = Highest data point − Lowest data point

Example 4 Find the range of the following scores: 10, 7, 2, 14, 13, 15, 11, 4 Now we subtract the lowest from the highest to ﬁnd the difference, which is the range: Range Create a strategy

Kenji 16 − 10 = 6 Use the formula Range = Highest score − Lowest score. Bjorn 27 − 2 = 25

Apply how the idea Notice Kenji’s range is quite small, at least compared to Bjorn’s. We might say that Kenji’s route is more predictable that route is much more The highest and score is Bjorn’s 15 and the lowest score is 2.variable (is more likely to change). The rangeRange of a numerical the difference between the highest and the lowest data point. = 15 – 2 data set is Subtract 2 from 15 = 13

Perform subtraction Range =the Highest data point − Lowest data point

Example 4

Idea summary

PurposeFind the range of the following scores: Range = Highest point Lowestwhich datainvolves point finding the difference Helps students understand how to calculate the10,range of13, a 15, set11,of4−numbers, 7,data 2, 14, between the highest and lowest scores in the Range is aset. measure of how spread apart a data set is from its highest to lowest value.

Create a strategy

Expected mistakes Use the formula Range = Highest score − Lowest score. Students may mistake the range as simply the highest and lowest values in the set, rather than the difference betweenApply the two. It is important to emphasize that the range is calculated by subtracting the lowest value from the idea Practice the highest value. score is 15 and the lowest score is 2. The highest Range = 15 – 2

Students:What Pagedo390 you remember? = 13

Subtract 2 from 15 Perform the subtraction

Describe what the mean measures for a set of values.

Describe what the median measures for a set of values.

Find the mean of the data sets:

Idea summary

a 4

6, 4

Rangeb = 9,Highest data point data dpoint 7, 11, 4 c 4, − 10,Lowest 2, 9, 5 8, 15, 6, 27, 3, 19

Range is a measure of how spread apart a data set Yvonne asked a number of her friends what their favorite color is, and writes down their responses: is from its highest to lowest value. Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink Which color is the mode?

Practice 390

Mathspace

Virginia SOL Grade 6

Whatmathspace.co do you remember? 1

Describe what the mean measures for a set of values.

Describe what the median measures for a set of values.

Find the mean of the data sets: a

6, 4

9, 7, 11, 4

4, 10, 2, 9, 5

Which color is the mode?

764

8, 15, 6, 27, 3, 19

Yvonne asked a number of her friends what their favorite color is, and writes down their responses: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink

390

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Practice Students: Pages 390–393

What do you remember? 1

Describe what the mean measures for a set of values.

Describe what the median measures for a set of values.

Find the mean of the data sets: a

6, 4

9, 7, 11, 4

4, 10, 2, 9, 5

8, 15, 6, 27, 3, 19

Yvonne asked a number of her friends what their favorite color is, and writes down their reponses: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink Which color is the mode?

A diver measures how long she can hold her breath underwater over several dives. The median time is 3.9 minutes. Are these statements true or false? a

The longest she held her breath was 7.8 minutes.

Most of the time, she held her breath for less than 3.9 minutes.

For half the dives, she was able to hold her breath longer than 3.9 minutes.

The shortest time she held her breath was 1.95 minutes.

Most of the time, she held her breath for longer than 3.9 minutes.

Why might the mean not accurately represent the data set {1, 2, 2, 100}?

List the steps to ﬁnd the median of any data set with an even number of values. Sample Data Set: {23, 47, 58, 61, 34, 82, 82, 19, 7, 92}

Let’s practice 8

Find the mean of the data sets: a

6, 14, 10, 13, 5, 9, 14, 15

22.4, 25.4, 19.1, 24.3, 7.4

−14, 0, −2, −18, 0, −15, −1

Which two data sets have the same mean? • Data Set A: 11, 5, 14, 18, 15 • Data Set B: 8, 14, 24, 7, 12

• Data Set C: 13, 27, 9, 6, 8 • Data Set D: 13, 15, 7, 17, 9

Mrs. Tran asks her students how many bedrooms they have in their house and writes their responses below: 3, 3, 4, 4, 3, 3, 4, 5, 2, 2, 4, 4 Determine the mode number of bedrooms.

Victoria has scored 12 goals this season, with a mean of 2 goals per game. How many games did she play?

9.04 Review: measures of center and spread mathspace.co

765

Consider the scores: 3, 6, 8, 10, 2, 11

What is the largest number?

Find the range.

What is the smallest number?

Find the median of the data sets: a

6, 9, 3, 5, 4

1, 4, 6, 8

, , ,

13, 1, 10, 29, 12, 5, 16, 20, 6

7, 19, 6, 14, 18, 11

5.3, 8.4, 2.6, 4.7, 3.5

49.4, 25.4, 47.2, 34.2, 28.4, 32.2

Calculate the range of the scores: 10, 16, 6, 18, 17, 11, 9, 15, 14

For each data sets, calculate: i

The mode

Score 11 14 19 Frequency 10 13 4

22 7

29 12

The range

Score Frequency

14 9

19 14

22 12

25 1

32 10

A phone support center tracks the length of calls made each day in minutes. During a shift, one employee made calls of lengths: 2, 3, 3, 2, 2, 4, 4, 2, 5, 13 a

How many calls did the employee make during their shift?

Calculate the length of calls made: i

Mean

Median

iii

Mode

Range

Which set of data has the largest range? • Set A: 101, 105, 118, 129, 136 • Set B: 19, 23, 25, 28, 29 • Set C:

• Set D: −104.15, 107.05, 113.24, 128.33, 141.57 18

A group of ﬁve friends tracks the number of books they read over the summer for a reading competition. Their totals are as follows: 15, 18, 20, 22, and 25 books. a

Calculate the mean number of books read.

A new friend joins their group, who read 45 books over the summer.

Explain what would happen to the mean number of books read by the group if this new friend’s total is added. 19

766

Determine whether the statements are always, sometimes, or never true. Explain your reasoning. a

The median of a data set is a value in the set.

The mean of a data set is larger than all values in the set.

The range describes how far the lowest data values lies from the highest data value.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

A local amateur photographers’ club that meets monthly to share their best shots. The number of photos taken by seven club members in a month is as follows: 50, 55, 60, 65, 70, 75, and 80 a

Calculate the median number of photos taken by the members.

The following month, inspired by a spectacular meteor shower, the one member who previously took 80 photos manages to take 120 photos. The other members took the same amount of photos.

Explain how the median number of photos taken by the club members changed for the second month. 21

Consider this data set that represents the number of apps on six people’s phones: 110, 113, 117, 121, 127, 132 Explain what would happen to the range if each person downloads another 9 apps.

The list shows the number of points scored by a basketball team in each game of their previous season: 59, 67, 73, 82, 91, 58, 79, 88, 69, 84, 55, 80, 98, 64, 82

Find the range and explain what it represents in context.

Explain why the range might be important for the team’s coach to know.

Consider the performance data from two groups of athletes, Group A and Group B, over a series of events. The data represent the number of goals scored by each athlete in a season. For Group A, the goals scored are as follows: 5, 7, 7, 8, 10, and 12. For Group B, the goals scored are: 3, 5, 7, 7, 7, 11, and 14. a

Calculate the mean number of goals scored for each group.

Determine the median number of goals scored for each group.

Identify the mode of goals scored for each group.

Compute the range of goals scored for each group.

Based on these calculations, compare the performance of the two groups of athletes. Consider which group demonstrates greater consistency in performance and which group has a higher variability in the number of goals scored.

A school nurse is investigating the number of sick days taken by students in a class, and constructs a table: Determine the most common number of sick days for the class.

Student Sally Lachlan Tara Xavier Luigi Nadia Lisa Dave Bill Danielle

Sick days 0 2 1 4 2 4 0 1 3 4

CA retail company has compiled its weekly sales ﬁgures (in thousands of dollars) for the past twelve weeks as follows: 52, 55, 55, 60, 62, 65, 65, 70, 70, 75, 80, 85 a

Calculate the mean, median, mode and range of the weekly sales ﬁgures.

How consistent is the sales ﬁgures week over week?

What could cause the variability in the sales ﬁgures week over week?

9.04 Review: measures of center and spread mathspace.co

767

A student received a data set containing the following numbers:

, 4, 6, 6, 7, 7, 7,

, 11, and 13.

The assignment was to calculate the mean, median, mode, and range of the data set. The student’s reported answers were as follows: • Mean: 7 • Median: 7

• Mode: 6 • Range: 10.5

Identify and correct the errors in the student’s calculations.

Let’s extend our thinking 27

Create a data set where the mean is 7, median is 8, mode is 8, and range is 10.

Consider the yearly high temperatures (in degrees Fahrenheit) recorded in Springﬁeld City over the last twelve months: 58.24, 60, 65, 67.15, 70, 72.02, 75, 75, 78, 80.38, 82, and 85.46. As part of the city’s environmental study, you’re tasked with analyzing these temperatures to understand the city’s climate better. a

Calculate the mean, median, mode and range for the temperatures.

Based on your calculations, decide which measure (mean, median, mode, or range) best represents the city’s climate for the year and explain your reasoning.

The mean of four values is 21. If three of the values are 17, 3 and 8, ﬁnd the fourth value.

A set of 34 scores is arranged in increasing order. Between which two scores does the median score lie?

In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the ﬁrst 11 challenges are: 66, 105, 38, 108, 67, 82, 92, 43, 119, 45, 102 Determine her score in the 12th round if the median of all of her 12 scores is 83.5.

Counting numbers in increasing order such as 7, 8, 9 are called consecutive numbers. Write down ﬁve consecutive numbers whose median is 6.

768

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

higher than the original mean. Adding a number that is above the mean to the data set will increase the average.

9.04 Review: measures of center and spread

19 a S ometimes. If the data set has an odd number of values, the median is the middle value.

What do you remember? 1 It measures the center/average of the values. 2 It measures the middle value of the set of values. 3 a 5

b 7.75

c 6

d 13

b False

c True

d False

4 Pink 5 a False e False 6 The mean might not accurately represent the data set {1, 2, 2, 100} because the data set contains an outlier. An outlier is a value that is signiﬁcantly higher or lower than most of the other values in a data set. In this case, the value 100 is much higher than the other values ( 1, 2, and 2), which are close to each other. 7 1. Arrange the data in order: {7, 19, 23, 34, 47, 58, 61, 82, 82, 92} 2. Find the middle value or median:

If the data set has an even number of values, the median is between the two middle values and may not be in the data set. b N ever. The mean will always be less than or equal to the maxmimum value in the set. c A lways. The range represents the difference between the largest and smallest values, and the difference tells us how far apart the values are. 20 a Median = 65 b T o see how the median has changed, we don’t actually need to calculate the new median because the increased number (120) replaces the highest number (80) from the original set, and since the median is the middle value, it is unaffected by changes in the values of numbers outside of the middle position when the total count of numbers remains odd. The middle position remains the fourth number in the ordered list, which remains 65. 21 The range would not change. 22 a T he difference between the most points scored last season and the lowest points scored last season was 43 points.

Let’s practice 8 a 10.75

c 19.72

d −7.14

9 Data Set A and Data Set C 10 4 bedrooms 11 6 games 12 a 11

b 2

c 9

13 a

c 6

e 12.87

13.5

g 12.6

b T he range shows that there is a big difference between the most points scored and the lowest points scored during the season. This means the team is not consistent with the number of goals they score, since sometimes they score a lot and other times they score less. If the coach can ﬁgure out why they didn’t score as many points in the lower- scoring games, the coach may be able to help them play better and score more points more often. 23 a Group A: Mean = 8.17

d 8.75 h 12.2

Group B: Mean = 7.71

b Group A: Median = 7.5

Group B: Median = 7

c Group A: Mode = 7

14 12 15 a i 14

ii 18

b i 19

ii 18

Group B: Mode = 7

d Group A: Range = 7

16 a 10 calls b i 4 minutes

ii 3 minutes

iii 2 minutes

iv 11 minutes

• Group B shows greater consistency since the median and mode are the same and close to the mean, but also a higher variability in the number of goals scored (as indicated by a larger range).

• Both groups have the same mode.

17 Set D 18 a Mean = 20 b T he mean number of books read by the group will increase because the new friend’s total is signiﬁcantly

Group B: Range = 11

e • Group A demonstrates a slightly higher average performance (mean) compared to Group B.

Answers mathspace.co

769

We can still infer from the median and mode that Group B might have more consistent scores around the median value. Group A, having a higher mean and lower range, indicates less variability but not necessarily greater consistency in performance.

Let’s extend our thinking

• New median: 73.51

24 4 days

• New mode: 75

• New range: 27.22

25 a • Mean = 66.17

• Median = 65

• Mode = 55, 65, 70

• Range = 33 b G iven the points, while there may be a base level of consistency in the sales figures (as indicated by the median’s closeness to the mean), the range and the modes suggest that there are fluctuations in sales from week to week. c S everal factors can cause variability in weekly sales figures for a retail company, including: seasonal trends, marketing and promotions, economic conditions, product availability, competitor actions, and external events. Understanding and analyzing these factors can help a retail company better predict future sales trends, plan marketing strategies, and manage stock levels to minimize variability and optimize sales performance.

26 The student correctly calculated the median and range but made errors with the mean and mode. Corrections to the student’s answers: • Mean: 7.265 • Median: 7 • Mode: 7 • Range: 10.5

770

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

27 Answers may vary: {2, 3, 5, 5, 7, 8, 8, 8, 9, 10, 12} 28 a • New mean: 72.35

b G iven that the temperatures do not have signiﬁcant outliers, the mean could offer a fair representation of the overall climate. However, the median is often considered a more reliable measure for representing a typical climate condition because it is less susceptible to the effect of rare extreme weather events. Therefore, in the context of representing the climate for the year, the median would likely be the most representative measure of Springﬁeld City’s typical high temperature. It is less likely to be skewed by any unusually high or low temperatures and provides a central point around which the other temperatures vary. 29 11 30 17th and 18th scores 31 85 32 4, 5, 6, 7, 8

9.05 Mean as a balance point Subtopic overview Lesson narrative In this lesson, students will learn about the mean as a balance point of a data set. They will explore how the mean represents the central value where the sum of distances from the mean of all points below it equals the sum of distances from all points above it. Through examples and exercises, students will use line plots to find and interpret the mean, visualizing it as the balance point. An interactive exploration involves moving points on a line plot to visually balance the data set, reinforcing the concept of the mean. Students will also calculate the mean by summing all values and dividing by the number of data points, understanding its significance in different contexts. By the end, students should confidently find and interpret the mean as a balance point in various data sets.

Learning objectives

9.05 Mean as a balance point

Students: Page 394

After this lesson, you will be able to... • use line plots to find and interpret the mean of a set of data. • represent the mean of a set of data graphically as the balance point represented in a line plot/dot plot. .

Mean as a balance point

Key vocabulary

Interactive exploration



balance point line plot Explore online to answer the questions



mean

mathspace.co

Essential understanding

Use the interactive exploration in 9.05 to answer these questions. Measures of center provide useful information when interpreting data. Different measures approximate the “center” you make the mean the same value as one (or more) of the data points? How many ways? What do of a data1. setCan in different ways; each measure provides different insights into a set of data and is appropriate in you notice? different situations. 2. Can you make more than one data set with a mean of 4? 3.

Will the mean ever be outside of the data set?

4. Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the Standards mean and positive values for above the mean. What is the sum of these values? This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical goals The mean is theprocess balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean. MPG2 — Mathematical Communication 1 When a data the balance can be found by plotting Teachers can integrate this goal3into their instruction as given students areset analyzing andpoint interpreting data. They can the points on a line plot. 2 facilitate class discussions, or ask students to discuss in partners, answers to questions such as, “How does the mean Thea points to the arehelp a total 3 units away the mean, 6. of balance the distribution of a data set?” or “How can line plot (dotleft plot) us of make sense offrom this interpretation 0 1 2 3 4 5 6 7 8 9 10 the mean as a balance point?” The point on the right is 3 units away from the mean, 6. This creates a balanced distance of 3 on either side of the mean. To ﬁnd the balance point, when it is not given, points can be moved one-by-one towards the middle.

0 1 2 3 4 5 6 7 8 9 10 11

Step 1: Start by plotting the data set on the line plot.

9.05 Mean as a balance point mathspace.co 0 1 2 3 4 5 6 7 8 9 10 11

Step 2: Start with the farthest left point and move it

771

MPG4 — Mathematical Connections Teachers can make Mathematical Connections by showing students how the concept of mean as a balance point applies to real-life situations. For instance, teachers can use examples from finance, sports, or science that require data analysis. Students can be guided to see how the mathematical concepts they learn in class are not isolated, but are connected and applicable to various real-world contexts. MPG5 — Mathematical Representations Teachers can start to incorporate Mathematical Representations into this lesson by explaining the concept of the mean as the point where the total distances of data points below and above are equal. They can use hands-on activities like creating line plots with students’ data, such as the number of hours spent reading. Then, teachers can highlight examples where the mean is not a whole number to address common misconceptions.Students should be encouraged to calculate the mean and identify it on the line plot, reinforcing the idea that the mean represents the balance point of the data. This approach not only solidifies their understanding of the mean but also prepares them for more advanced topics like standard deviation. Teachers can also ask students to represent the same data set in different ways (e.g., bar graphs or pie charts) and discuss how the mean appears in these different representations.

Content standards 6.PS.2 — The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.

6.PS.2a — Represent the mean of a set of data graphically as the balance point represented in a line plot (dot plot).

Prior connections 5.PS.2 — The student will solve contextual problems using measures of center and the range.

Future connections A2.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on univariate quantitative data represented by a smooth curve, including a normal curve.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 9.04 Review: measures of center and spread

Tools You may find these tools helpful: • Scientific calculator • Blank number line • Coins or counters

772

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Student lesson & teacher guide as a balance point Mean Mean as a balance point Students: PagesInteractive 394–395 exploration Explore online to answer the questions

mathspace.co

9.05 Mean as a balance point Use the interactive exploration in 9.05 to answer these questions. 1.

Can you make the mean the same value as one (or more) of the data points? How many ways? What do you notice? After this lesson, you will be able to...

•Can useyou linemake plots more to findthan andone interpret thewith mean of a set data set a mean of of 4?data. •Will represent theever mean a set of graphically the mean beof outside ofdata the data set? as the balance point represented in a line plot/dot plot. .

3. 4.

Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values?

Mean as a balance point The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the Interactive exploration points below the mean is equal to the sum of the distances from the mean of all of the points above the mean. Explore online to answer the questions 1

mathspace.co 0 1 2 3 4 5 6 7 8 9 10

When given a data set the balance point can be found by plotting the points on a line plot. The points to the left are a total of 3 units away from the mean, 6.

Use the interactive exploration in 9.05 to answer Thethese pointquestions. on the right is 3 units away from the mean, 6. 1.

This a balanced of 3 on either side of the mean. Can you make the mean the same value as creates one (or more) of thedistance data points? How many ways? What do

notice? To ﬁnd theyou balance point, when it is not given, points can be moved one-by-one towards the middle. 2. Can you make more than one data set with a mean of 4? 3.

Will the mean ever be outside of the data set?

Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the 0 and 1 2 positive 3 4 5 values 6 7 8for9above 10 11 the mean. What is the sum0of 1these 2 3 values? 4 5 6 7 8 9 10 11 mean

Step 1: Start by plotting the data set on the line plot.

Step 2: Start with the farthest left point and move it one unit towards the center. The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean. 1 3 0 1 2 32 4 5 6 7 8 9 10 11

When given a data set the balance point can be found by plotting the points on a line plot.0 1 2 3 4 5 6 7 8 9 10 11

points to the left are a Step total 4: of Repeat 3 units away Step 3: Then go to the farthest right point andThe move on thefrom left. the mean, 6. 0 1 2 3 4 5 6 7 8 9 10 it one unit towards the center. The point on the right is 3 units away from the mean, 6. This creates a balanced distance of 3 on either side of the mean. To ﬁnd the balance point, when it is not given, points can be moved one-by-one towards the middle. 394

Mathspace 0 1 Virginia 2 3 SOL 4 5Grade 6 76 8 9 10 11 mathspace.co

0 1 2 3 4 5 6 7 8 9 10 11

Step 5: Repeat on the right.

Step 6: Repeat on the left.

0 1 2 3 4 5 6 7 8 9 10 11

Step 1: Start by plotting the data set on the line plot.

Step 2: Start with the farthest left point and move it one unit towards the center. 0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11

Mean “balance point”

Step 7: Repeat on the right.

Step 8: Stop when all points are stacked over the same value. The balance point for this data is 6.

0 1 2 3 4 5 6 7 8 9 10 11

Step 3: Then go is tolocated the farthest right two pointvalues, and move Repeat on thevalues left. by averaging If the balance point between we can ﬁnd the halfway Step point4:between those it one unit towards the center. the numbers. The balance point for the data set is located halfway between 7 and 8. 394

Mathspace 0 1 2 Virginia 3 4 5SOL 6 Grade 7 8 69 10 mathspace.co

Balance point

We can calculate the halfway point by ﬁnding the sum of the two values and dividing by two.

9.05 Mean as a balance point Adding or removing a data point might throw off the balance of the data set resulting in a new balance point. mathspace.co

Example 1

773

0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11

Mean “balance point”

Step 7: Repeat on the right.

Step 8: Stop when all points are stacked over the same value. The balance point for this data is 6.

If the balance point is located between two values, we can ﬁnd the halfway point between those values by averaging the numbers. The balance point for the data set is located halfway between 7 and 8. 0 1 2 3 4 5 6 7 8 9 10

We can calculate the halfway point by ﬁnding the sum of the two values and dividing by two.

9.05 Mean as a balance point Balance point

Adding or removing a data point might throw off the balance of the data set resulting in a new balance point. After this lesson, you will be able to...

Example 1

• use line plots to find and interpret the mean of a set of data. • represent of aofset of data graphically theresults balance represented in a line plot/dot plot. . A classroom recordedthe themean number pets for each student.as The forpoint the class

Exploration

are represented in the given line plot.

Students: Page 394

Mean as a balance point

Number of pets

a What was the total number of pets for the entire class?

Interactive exploration

online to answer the questions Create a Explore strategy

Multiply the total number of each number of pets by the number of students that have that many pets.

mathspace.co

Apply the idea

Use the interactive exploration in 9.05 to answer these questions. Total number of pets = (0 ⋅ 3) + (1 ⋅ 3) + (2 ⋅ 3) + (3 ⋅ 1) + (4 ⋅ 2) + (5 ⋅ 0) + (6 ⋅ 1) Multiply each number of 1. Can you make the mean the same value as one (or more) of the data points? Howpets many What do byways? its frequency you notice? =0+3+6+3+8+0+6 Evaluate the values in the 2.

Can you make more than one data set with a mean of 4?

parentheses

Will the mean ever be=outside of the data set? 26

Find the sum

Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values?

The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean.

Suggested student grouping: In pairs 1 When given a data set the balance point can be found by plotting 3 In this exploration, students applet pointsto onmanipulate a line plot. a set of data points on a number line. 2 will use a GeoGebrathe They will observe how changing the data valuesThe affects the mean and investigate between the points to the left are a total of 3 units the awayrelationship from the mean, 6. 0 1 2 3 4 5 6 7 8 9 10 distances of each value from the mean. The goalThe of point the activity is to build understanding asan a balance point 395 of the on the right is 3help unitsstudents away 9.05 fromMean the mean, 6. mathspace.co mean as a measure of center. This creates a balanced distance of 3 on either side of the mean. To ﬁnd the balance point, when it is not given, points can be moved one-by-one towards the middle. Ideal student responses

These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 0 1 mean 2 3 4 the 5 6same 7 8 value 9 10 11 as one (or more) of the 0 data 1 2 3points? 4 5 6 How 7 8 9many 10 11 ways? What do 1. Can you make the Step 1: Start by plotting the data set on the line plot. Step 2: Start with the farthest left point and move it you notice? unit towards center. Yes, the mean can be the same as one or more of the data points.one It depends onthe the distribution of the other points. When the mean is the same as a data point, it may mean that data point is at the center of all other data points.

2. Can you make more a mean of 4? 0 1 2 than 3 4 one 5 6 data 7 8 9set 10 with 11 0 1 2 3 4 5 6 7 8 9 10 11 Yes, by Step moving the data points around, multiple data sets can be created with aonmean of 4. The mean only 3: Then go to the farthest right point and move Step 4: Repeat the left. it one unit towards the center. depends on the sum of the data points and the number of them, so as long as the total stays the same, the mean does too.

394

774

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

3. Will the mean ever be outside of the data set? No, the mean is a measure of center, so it will always be within the range of the data set. 4. Set up the points to 4, 6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values? The mean of 4, 6 and 11 is 7. So, 4 is −3 from the mean, 6 is −1 from the mean, and 11 is 4 above the mean. The sum of these distances is 0. 0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11

Step 5: Repeat on the right.

Step 6: Repeat on the left.

Purposeful questions

• Why does the mean change when you move a data point? • Why does the mean stay the same when you move a data point along the mean line? • Why is the sum of the deviations from the mean always zero?

0 1 2 3 4 5 6 7 8 9 10 11

Possible misunderstandings 0 1 2 3 4 5 6 7 8 9 10 11

Mean “balance point”

• Students may mistake mean for right. the median and assume it is always oneare of stacked the data points. Step 7:the Repeat on the Step that 8: Stop when all points over the • Students may think that moving any data point will shift the mean in The thebalance same direction, realizing that the same value. point for thisnot data is 6. mean is dependent on all data points. If the balance point is located between two values, we can ﬁnd the halfway point between those values by averaging the numbers.

Students are introduced to the concept of the mean as a balance point of a set of data, and are shown an example The balance point for the data set is located halfway between with a set of data represented on a line plot. They are told that adding or removing data can affect this value. 7 and 8.

Examples

0 1 2 3 4 5 6 7 8 9 10 Balance point

We can calculate the halfway point by ﬁnding the sum of the two values and dividing by two.

Students: Page 395 Adding or removing a data point might throw off the balance of the data set resulting in a new balance point.

Example 1 A classroom recorded the number of pets for each student. The results for the class are represented in the given line plot. 0

Number of pets

a What was the total number of pets for the entire class?

Create a strategy Multiply the total number of each number of pets by the number of students that have that many pets.

Apply the idea Total number of pets = (0 ⋅ 3) + (1 ⋅ 3) + (2 ⋅ 3) + (3 ⋅ 1) + (4 ⋅ 2) + (5 ⋅ 0) + (6 ⋅ 1)

Multiply each number of pets by its frequency

=0+3+6+3+8+0+6

Evaluate the values in the parentheses

= 26

Find the sum

Purpose Show students how to interpret a line plot to determine the total number of a certain quantity in a population Expected mistakes Students may only include one of each value represented, or miscount the number of each type of value. Encourage students to write out the individual values represented by the line plot before finding the sum 9.05 Mean as a balance point 395 instead of using multiplication. mathspace.co

9.05 Mean as a balance point mathspace.co

775

Students: Pages 396–397 b What is the mean number of pets per student?

Create a strategy Create a line plot of the data set. Alternate moving each far left and far right point towards the center until ﬁnding the balance point of the data set, which is the mean.

Apply the idea Start on the far left and move one of the points at 0 one unit to the right.

Alternate to the far right and move the point at 6 one unit to the left.

Alternate back to the far left and move another point at 0 one unit to the right.

Alternate back to the far right and move the point at 5 one unit to the left.

Alternate back to the far left and move the point at 0 one unit to the right.

Alternate back to the far right and move one of the points at 4 one unit to the left.

396

776

Mathspace Virginia SOL Grade 6 mathspace.co

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Continuing this process, we end up with all points on 2.

This creates a balance point at 2. The mean is 2 pets per student in the class.

Reﬂect and check Alternatively, we can calculate the mean by ﬁnding the sum of all the points and dividing by the total number of data points. Multiply each number of pets by its frequency and divide by the total number of students Evaluate the multiplication and addition Evaluate the quotient for the mean

Example 2 Purpose During for a ﬁtness challenge, recorded thethe number of push-ups day.plot Theby table shows the number Demonstrate students howAlex to determine mean of a datacompleted set fromeach a line using a method of of push-ups did. balancing the dataAlex points. Day

Expected Push-ups mistakes 15 20 15 10 15 20 10 15 20 20 Students may not alternate their points as they try to find the balance point of the data set and try to move one a Create a line plot of the data. direction at a time. This could lead to an incorrect balance point. Remind them that we must alternate between the left and right points to maintain the balance. Create a strategy

The number line should include values in between 10 and 20 based upon the given data. Then, count the number

Concrete-Representational-Abstract (CRA) approach use with Example 1 of times that Alex completed each number of push-ups, this number will tell us how many dots to put on the number Targeted instructional strategies line. Concrete: Engage students with physical manipulatives to represent the data on the number of pets. Provide Apply the idea each student with counters or small objects, such as colored chips or blocks, where each counter represents For this problem, we do not need to know which days Alex completed the number of push-ups, only how many days one pet.heSet up a number completed them. line on a table or the floor labeled from 0 to 6. Have students place the appropriate number of counters above each number to show Alex howcompleted many students have that number of pets. 10 push-ups 2 days, so we will place 2 points 10. To find the total number of pets, guide students toabove physically count all the counters, combining the quantities Alex completed 15 push-ups 4 days, so we will place 4 points the above each number. To find the mean number of pets per student, demonstrate how to redistribute above 15. 10 11 12 13 14 15 16 17 18 19 20 counters equally among all the students. Have students work together to move the counters so that each of 20 push-ups 4 days, so we will place 4 points Number of push-ups the 13 students has the same number of counters.Alex Thiscompleted hands-on activity shows that when the 26 counters are above 20. evenly distributed, each student has 2 counters, representing a mean of 2 pets per student.

Representational: Transition to the representational stage by having students create a line plot on paper that 9.05 Mean balance point mirrors the manipulative model. Guide them to draw a horizontal axis labeled from 0 toas6.a Above each397 number, mathspace.co have them draw dots or Xs to represent the number of students with that many pets.

9.05 Mean as a balance point mathspace.co

777

Introduce the concept of the balance point (mean) on the line plot. Explain that the mean is the value where the data balances. Have students visually “shift” dots from the outer numbers toward the center to find this balance point. For example, they can move one dot from 0 to 1 and one dot from 6 to 5, continuing this process until the plot balances. Through this visual redistribution, students will see that the balance point is at 2, indicating the mean number of pets per student. Abstract: Move to the abstract stage by introducing numerical calculations using symbols and equations. Show students how to calculate the total number of pets using an equation: Total pets = (0 × 3) + (1 × 3) + (2 × 3) + (3 × 1) + (4 × 2) + (5 × 0) + (6 × 1) =0+3+6+3+8+0+6 = 26 Then, guide students to find the mean number of pets by dividing the total number of pets by the total number of students: Mean number of pets = 26 ÷ 13 = 2 Encourage students to perform these calculations independently, reinforcing their understanding of how to analyze Continuing data using operations. thismathematical process, we endsymbols up with alland points on 2.

Critique, correct, and clarify

use with Example 1

English language learner support Present students with the incorrect statement: “The mean number of pets per student is 3 because it is the center value on the number line.” Encourage students to identify any errors or misunderstandings in the statement, focusing on the concepts of mean and how it is calculated. Guide students to critique the reasoning by prompting them to 5consider how the mean is actually determined. 0 1 2 3 4 6 Remind them that the mean is the balance point of the data values, not the balance point of the number line. Provide This support bya reviewing key creates balance point at 2.vocabulary The mean is such 2 petsas per“mean,” student in“balance the class. point,” and “center value,” ensuring that students understand the differences between these terms. Have students work together to correct the Reﬂect and check the actual mean number of pets using the data from the line plot. statement by calculating Alternatively, we can calculate the mean by ﬁnding the sum of all the points and dividing by the total number of data

Finally, ask students to clarify the correct reasoning in their own words. For example, “The mean number of points. pets per student is 2 because 2 is the balance point of the data.” MultiplyThis eachallows numberstudents of pets by to its practice frequency using and precise mathematical language and to solidify their understanding. divide by the total number of students Evaluate the multiplication and addition Evaluate the quotient for the mean

Students: Page 397 Example 2

During a ﬁtness challenge, Alex recorded the number of push-ups completed each day. The table shows the number of push-ups Alex did. Day Push-ups

1 15

2 20

3 15

4 10

5 15

6 20

7 10

8 15

9 20

10 20

a Create a line plot of the data.

Create a strategy The number line should include values in between 10 and 20 based upon the given data. Then, count the number of times that Alex completed each number of push-ups, this number will tell us how many dots to put on the number line.

Apply the idea For this problem, we do not need to know which days Alex completed the number of push-ups, only how many days he completed them.

778

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Alex completed 10 push-ups 2 days, so we will place 2 points above 10.

10 11 12 13 14 15 16 17 18 19 20

Alex completed 15 push-ups 4 days, so we will place 4 points above 15.

Number of push-ups

Alex completed 20 push-ups 4 days, so we will place 4 points

a Create a line plot of the data.

Apply the idea For this problem, we do not need to know which days Alex completed the number of push-ups, only how many days he completed them. Alex completed 10 push-ups 2 days, so we will place 2 points above 10. 10 11 12 13 14 15 16 17 18 19 20

Alex completed 15 push-ups 4 days, so we will place 4 points above 15. Alex completed 20 push-ups 4 days, so we will place 4 points above 20.

Number of push-ups

9.05 Mean as a balance point mathspace.co

Purpose Show students how to visualize and interpret frequency data by creating a line plot.

397

Reflecting with students Encourage students to use precise mathematical language and notation throughout this example. When creating the line plot, students should label the axis clearly with “Number of push-ups” and ensure they plot each data point accurately. The number line should include all integers between 10 and 20, even though there are no data points above most of the numbers. This allows us to clearly see the gaps in the data.

Students: Page 398 b What was the total number of push-ups completed?

Create a strategy Multiply the number of completed push-ups by the frequency.

Apply the idea Total number of push-ups = (10 ⋅ 2) + (15 ⋅ 4) + (20 ⋅ 4) = 20 + 60 + 80 b What was the total number of push-ups completed? = 160

Multiply each number of push-ups by its frequency Evaluate the values in the parentheses Find the sum

Alex completed a total of 160 push-ups over the 10 days.

Create a strategy

Multiply the number of completed push-ups by the frequency. c What is the balance point of push-ups completed each day?

PurposeApply the idea a strategy Total number push-ups = (10 ⋅ 2) + the (15 ⋅ total 4) + (20 Multiply each number ofby push-ups by its frequency StudentsCreate demonstrate their of ability to compute of⋅ a4) frequency distribution multiplying each value by Use the line plot created in part (a) to alternate moving each far left and far right point towards the center until ﬁnd = 20 + 60 + 80 Evaluate the values in the parentheses its frequency and adding the results. the balance point of the data set.

= 160

Find the sum

completed a total of 160 push-ups over the 10 days. Students:Alex Pages 398–399

Apply the idea

Start on the far left and move one of the points at 10 one unit to the right. c What is the balance point of push-ups completed each day?

Create a strategy Use the line plot created in part (a) to alternate moving each far left and far right point towards the center until ﬁnd 10 11 12 13 14 15 16 17 18 19 20 the balance point of the data set. Alternate to the far right and move one of the points at 20 one unit to the left.

Apply the idea

Start on the far left and move one of the points at 10 one unit to the right.

10 11 12 13 14 15 16 17 18 19 20 10 11 at121013one 14 15 16to 17the 18 19 20 Alternate back to the far left and move the point unit right.

Alternate to the far right and move one of the points at 20 one unit to the left.

10 11 12 13 14 15 16 17 18 19 20 12points 13 14 15 16 17 18 unit 19 20 Alternate back to the far right and move one10of11the at 20 one to the left.

9.05 Mean as a balance point mathspace.co

779

Start on the far left and move one of the points at 10 one unit to the right.

10 11 12 13 14 15 16 17 18 19 20

Alternate to the far right and move one of the points at 20 one unit to the left.

10 11 12 13 14 15 16 17 18 19 20

Alternate back to the far left and move the point at 10 one unit to the right.

10 11 12 13 14 15 16 17 18 19 20

Alternate back to the far right and move one of the points at 20 one unit to the left.

10 11 12 13 14 15 16 17 18 19 20

Alternate back to the far left and move one of the points at 11 one unit to the right.

10 11 12 13 14 15 16 17 18 19 20 Mathspace Virginia SOL Grade 6 mathspace.co Alternate back to the far right and move one of the points at 20 one unit to the left. 398

10 11 12 13 14 15 16 17 18 19 20

Continue this process until all the points have been balanced at 16.

10 11 12 13 14 15 16 17 18 19 20 Balance point

The mean number of push-ups completed per day is 16 each day.

Reﬂect and check 6 1

Each point on the far left and right sides alternates and get moved one place closer towards the center until all the points are at the balance point. The 2 points at 10 move 6 units to the right. The 4 points at 20 move 4 units to the left.

10 11 12 13 14 15 16 17 18 19 20 Number of push-ups

The 4 points at 15 move 1 unit to the right. The points on the left side of the mean moved a total of 16 units. The points on the right side of the mean moved a total of 16 units.

780

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Purpose Show students how to find the balance point, or mean, of a data set by using a line plot and the concept of balance.

Finding totals and using manipulatives to find the mean

use with Example 2

Student with disabilities support Begin by guiding students to use the provided table to calculate the total number of push-ups. Encourage them to add the number of push-ups completed each day directly from the table, which can be more straightforward than interpreting a line plot. Next, reinforce the concept of the mean as a balance point through a hands-on activity using physical manipulatives. Set up a number line and use counters or blocks to represent each day’s data point by placing them at the corresponding positions on the number line. Have students move the counters toward the center, shifting them one unit at a time from the outer numbers toward the mean, to physically balance the data. This simplified manipulation helps students visualize how the data balances around the mean without overwhelming them with too many steps.

Mean is not always the same as a data value

use with Example 2

Address student misconceptions A common misconception is that the mean is always a value from the data set or that it is always the middle value when the data is ordered. For example, they might think the mean of this data set is 15 since it is between 10 and 20. Help students understand that the mean is a calculated value and may not be a value from the original data set. It is the balance point of the data set, not necessarily the middle value. The mean representing a point that the data moves towards helps students conceptualize the mean as a point separate from the data itself.

Students: Pages 400–401

Example 3 Use the given line plot to identify the mean of a group of friends’ shoe sizes.

Shoe sizes

Create a strategy Use the line plot to alternate moving each far left and far right point towards the center until you ﬁnd the balance point of the data set.

Apply the idea Start on the far left of the line plot and move the point at 4 one unit to the right.

Shoe sizes

Alternate to the far right side of the line plot and move the point at 9 one unit to the left.

Shoe sizes

Alternate back to far left side of the line plot and move one of the points at 5 one unit to the 9.05 right. Mean as a balance point

mathspace.co

781

Start on the far left of the line plot and move the point at 4 one unit to the right.

Shoe sizes

Alternate to the far right side of the line plot and move the point at 9 one unit to the left.

Shoe sizes

Alternate back to far left side of the line plot and move one of the points at 5 one unit to the right.

Shoe sizes

Alternate back to the far right side of the line plot and move one of the points at 8 one unit to the left.

Shoe sizes

Alternate back to the far left side of the line plot and move the point at 5 one unit to the right.

Shoe sizes

Alternate back to the far right side of the line plot and move the point at 8 one unit to the left. 400

Mathspace Virginia SOL Grade 6 mathspace.co

Shoe sizes

The balance point for this data is the value halfway between 6 and 7.

Balance point

To ﬁnd the value halfway between 6 and 7 we can ﬁnd the sum of these values and divide by 2. Halfway =

Find the halfway value between 6 and 7

Evaluate the sum

= 6.5

Evaluate the quotient

The mean of the shoe sizes for this group of friends is 6.5.

Reflect and check Alternatively, we can find the mean by finding the sum of all the shoe sizes and divide by the number of friends. Multiply each shoe size by its frequency and divide by the total number of friends Evaluate the multiplication and addition Evaluate the quotient for the mean

782

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Idea summary The mean is the balance point of a data set and is best when there are not any values which are far away from

Halfway =

Find the halfway value between 6 and 7

Evaluate the sum

= 6.5

Evaluate the quotient

The mean of the shoe sizes for this group of friends is 6.5.

Reflect and check Alternatively, we can find the mean by finding the sum of all the shoe sizes and divide by the number of friends. Alternate back to the far right side of the line plot and move the point 8 one left. Multiply each at shoe sizeunit by to its the frequency and divide by the total number of friends Evaluate the multiplication and addition Evaluate the quotient for the mean 4

Shoe sizes

Idea summary

point for this data is the value halfway between 6 and 7. PurposeThe balance The mean is the balance point of a data set and is best when there are not any values which are far away from Show studentsthehow rest.to use a line plot to manually find the balance point when it falls halfway between two numbers whenTothe set has anfrom even number data points. ﬁnddata the balance point a line plot weofalternate moving each point on the far left side and far right side one unit closer to the center of the line plot. Eventually, all the points should be at balance point, which is the mean oflearners: the data set.Reasoning the mean and verifying calculations Advanced use with Example 3

Targeted instructional strategies

6 7 8 1 Balance point 3 2

Encourage advanced learners to determine the mean shoe size by reasoning about the balance point of the To ﬁndperforming the value halfway between 6 Ask and 7students we of values and observe divide by 2.how the data points are data without calculations. tothe plot and 0 can 1 2ﬁnd 3 4study 5sum 6 the 7 these 8 line 9 10 symmetrically distributed them to6infer Find thevalue. halfwayGuide value between and 7that the mean is 6.5 since it represents the Halfway = around a central Thedata points located to the leftperfectly, of the balance are“weights” the same distance awayside. from the balance point as the point where the would balance withpoint equal on either points located = on the right side. Evaluate the sum

Then, prompt students to verify their conclusion using the arithmetic mean. Students should recognize that the = 6.5 Evaluate the quotient most efficient way to calculate the mean is to multiply each shoe size by its frequency, and sum these products The mean of the shoe sizes for this group of friends is 6.5. to find the total sum of all shoe sizes. Then, they can divide this total sum by the number of friends to find the 9.05 Mean as a balance point 401 mean. Reﬂect and check mathspace.co

Alternatively, we can ﬁnd the mean by ﬁnding the sum of all the shoe sizes and divide by the number of friends. the By exploring this dual approach, students deepen their conceptual understanding and can compare efficiency and applicability of different strategies for finding the mean. Multiply each shoe size by its frequency and divide by the total number of friends Evaluate the multiplication and addition

Students: Page 401

Evaluate the quotient for the mean

Idea summary The mean is the balance point of a data set and is best when there are not any values which are far away from the rest. To ﬁnd the balance point from a line plot we alternate moving each point on the far left side and far right side one unit closer to the center of the line plot. Eventually, all the points should be at balance point, which is the mean of the data set. 1 2

0 1 2 3 4 5 6 7 8 9 10

The points located to the left of the balance point are the same distance away from the balance point as the points located on the right side.

9.05 Mean as a balance point mathspace.co

401

9.05 Mean as a balance point mathspace.co

783

Practice Students: Pages 402–404

What do you remember? 1

Deﬁne “mean”. Explain why it is an important measure of center in data analysis.

Which of the following data displays shows a line plot? A

60 65 70 75 80 85 90 95 100

Find the mean of each set of scores: a

8, 15, 6, 27, 3

56, 89, 95, 71, 75, 84, 65, 83

22.4, 25.4, 19.1, 24.3, 7.4

−14, 0, −2, −18, −8, 0, −15, −1

1, 3, 7, 5, 2

Does each set of numbers have a mean of 3? a

8, 4, 2, 3, 1

3, 2, 5, 1, 4

2, 4, 5, 4, 3

The mean of this set of number is 5 : 2, 4, 5, 6, 8. If we add another 5 to the data set, what is the new mean?

Which line plot shows a set of data with a balance point of 37? A 33

784

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Let’s practice 7

Find the balance point of each given data set. a

In each game of the season, a basketball team recorded the number of ‘three-point shots’ they scored. The results for the season are represented in the line plot: a

How many games did they play during the season?

If the team scored 4 three-point shots in a game, how many points are scored from three-point shots?

What was the total number of points they scored from three-point shots during the season?

Use the dot plot to ﬁnd the mean number of points they scored per game from the three-point shots.

0 1 2 3 4 5 6 7 8 9 Number of 3 Point Shots

Orly created a line plot to represent his ﬁrst four test grades in Geometry. Where would he place the balance point to represent the mean of his four grades? 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Each dot represents one of his test grades. 10

Sephira completed eight laps during her daily cycling practice and recorded the time taken for each lap, as shown in the table. She wants to represent the mean as a balance point for this data. a

Use the data to create a line plot.

Find the balance point for the data.

Explain how you found the balance point using the line plot.

Sephira’s Time Record for Each Lap Lap 1 2 3 4 5 6 7 8

Minutes 9 12 14 9 9 10 8 9

9.05 Mean as a balance point mathspace.co

785

Mr. Rowan made a line plot showing how many hours each of his students spends at the library during the week. He organized the data into the line plot shown (each dot represents one student). What is the balance point for the data? Explain your reasoning.

Let’s extend our thinking 12

Cecille recorded the time it took for seedlings to grow into full-grown plants, in weeks, in three different planting beds. She made a line plot of the data for each type of planting bed and provided this summary: • In the ﬁrst planting bed, seedlings are growing faster compared to those in the second and third planting beds. • The growth times of the seedlings in the third planting bed are more similar to each other than those in the ﬁrst and second planting beds. Use Cecille’s summary to match each line plot to the correct planting bed. a

First planting bed

Third planting bed

iii

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12 Time in weeks

Second planting bed

Time in weeks

These six numbers have a mean of 107:

1 2 3 4 5 6 7 8 9 10 11 12 Time in weeks

102, 103, 111, 107, 108, 111

If a new number is added that is larger than 111, will the mean will be higher or lower?

Create a line plot to represent the data set after a number larger than 111 is added.

At a local fair, three different carnival games were set up side by side. Each game had its own line of participants eager to try their luck. The line plots represent the ages of the people who played each game during the afternoon. Carnival game A

Among these carnival games, one has no age restriction. Do you think it was carnival game A, B, or C? Explain your reasoning.

Which carnival game’s line plot shows ages centered around 30 years old?

What is a typical age for the people who were at carnival game A?

Carnival game B

30 35 Age

Carnival game C

786

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

Let’s extend our thinking 12 a ii

9.05 Mean as a balance point

b i

c iii

13 a Higher b The added number may vary.

What do you remember?

1 The “mean” is the average value or balance point of a dataset.

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

It is important in data analysis because it gives us a clear idea of the typical value or average within a set of data. This helps us understand the central tendency of the data, which is essential for making comparisons and drawing conclusions. 2 B 3 a 11.8

b 77.25

c 19.72

d −7.25

4 a No

b Yes

c No

d No

d 3

14 a C arnival game B has no restriction. The data has a wide range covering ages from 8 to 50. b Carnival game B c T he typical ages for the people at carnival game A are 39 and 43, as these ages occur with the highest frequency in the data.

5 The new mean is still 5. 6 C Let’s practice 7 a 3

b 4

c 23

8 a 18 games

b 12 points

c 252 points d 14 points

9 90 10 a

b 10 c T o ﬁnd the balance point using the line plot, we interpret the line plot, which displays the lap times of Sephira’s cycling practice. Each data point on the plot represents a lap time. By calculating the mean of these lap times, we determine the balance point, which is the average lap time. On the line plot, this balance point represents the central value around which Sephira’s lap times are distributed. 11 The balance point is 6.5. Each dot corresponds to a value, and the average of these values is 6.5.

Answers mathspace.co

787

9.06 Changing data values and measures of center Subtopic overview Lesson narrative In this lesson, students will learn how changing data values affect measures of center, including the mean, median, and mode. They will explore how adding, removing, or changing values impacts these measures. The lesson includes practical examples and an interactive exploration where students manipulate data values to observe changes in the mean, median, and mode. Through exercises, students will understand the sensitivity of the mean to changes in data and how the median and mode can also be affected. By the end, students should confidently analyze how changes in data values influence measures of center.

9.06 Changing data values and Learning objectives measures of center Students: Page 405

After this lesson, you will be able to... • determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Changing data values and measures of center We have previously learned about various measures of center: Key vocabulary

• mean - also called average, is the sum of values divided by the number of values.  mean  measure of center frequency • median - is the middle value when the values are sorted. • mode - the value that occurs most often.  mode 

 median

Interactive exploration Explore online to answer the questions Essential understanding

Adding, removing, or changing a single value in a data set can significantly impact the measures of center. mathspace.co Use the interactive exploration in 9.06 to answer these questions.

Standards 1. What happens to the measures of center when the blue point is removed? 2. What happens to following the measures of center the blue point is added back? standards. This subtopic addresses the Virginia 2023when Mathematics Standards of Learning 3. What happens to the measures of center when the blue point is changed?

Mathematical process goals 4. Repeat with new data sets to see how the measures of center change as the blue point is added, removed, orProblem changed.Solving Do your previous observations continue to be true? MPG1 — Mathematical

Teachers can integrate this goal into their instruction by presenting students with various real-world problems that require to apply theirthe understanding of measures of center. For set: instance, teachers can create scenarios where Recallthem we can calculate mean by ﬁnding the ‘average’ of the data students need to determine the impact of adding, removing, or changing a data point on the mean, median, and = mode of a data set. This will enable students toMean use problem-solving strategies to determine the effect of changes on the interpretation of the data. Since every data value in the set is a part of the sum, adding, removing, or changing a value can change the numerator signiﬁcantly, depending on what that value is. While the denominator will only increase or decrease by 1 (or not at all if we’ve only changed an existing value). This is why the mean is so easily affected by changing the Mathspace Virginia SOL Grade 6 Teacher Edition 788data. mathspace.co

To ﬁnd the median, we list all the numbers in order from smallest to largest and ﬁnd the middle value. Adding, removing, or changing a value in a data set can often change its median. Though this change will not be major because the data values are ordered numerically and changing a single value will only cause it to shift to a nearby

MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can foster mathematical reasoning by guiding students to analyze the impact of changes on the measures of center, and to justify their observations. Students can be encouraged to reason about why certain changes might have a more significant effect on the measures of center than others, using logical reasoning to support their conclusions.

Teachers can create mathematical connections by linking the concept of changing a single data point to other real-world situations and mathematical topics, such as the concept of outliers. Teachers can also draw connections between the measures of center and other statistical concepts previously learned, reinforcing the integrated nature of mathematical knowledge.

Content standards 6.PS.2 — The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.

6.PS.2b — Determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Prior connections 5.PS.2 — The student will solve contextual problems using measures of center and the range.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 9.04 Review: measures of center and spread Grade 6 — 9.05 Mean as a balance point

Tools You may find these tools helpful: • Statistics calculator

Student lesson & teacher guide Changing data values and measures of center Students are reminded of the measures of center they have previously learned about before engaging in an exploration.

9.06 Changing data values and measures of center mathspace.co

789

Students: Page 405

9.06 Changing data values and measures of center After this lesson, you will be able to... • determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Changing data values and measures of center We have previously learned about various measures of center: • mean - also called average, is the sum of values divided by the number of values. • median - is the middle value when the values are sorted. • mode - the value that occurs most often.

Interactive exploration Explore online to answer the questions

Use the STEAM cycle for changing data values Targetedmathspace.co instructional strategies Use the STEAM to explore the ineffect adding, or removing a data value has on measures of center. Use thecycle interactive exploration 9.06 to answer changing, these questions. Ask: Start by1.posing question students: “How does single value in a data set affect the mean, What the happens to the to measures of center when thechanging blue point isaremoved? median, and2.mode?” Encourage to consider different scenarios and share What happens to thethem measures of center when the blue point is added back? their initial thoughts.

9.06 Changing data values and measures of center

3. students What happens to the measures of center when the blue point is changed? Imagine: Invite to predict and hypothesize about the effects of adding, removing, or changing a data 4. Repeat with new data sets to see how the measures of center change blue point is added, value on measures of center, perhaps by visualizing data on a number lineasorthe using dot plots. removed, or changed. Do your previous observations continue to be true?

Plan: Guide students to develop a plan for testing their hypotheses by selecting or creating data sets, defining which values to change, and deciding how they will record their observations and results. Recall we can calculate the mean by ﬁnding the ‘average’ of the data set:

Create and test: After Facilitate hands-on activities where students manipulate data sets according to their plans, this lesson, you will be able to... Mean = on how each change impacts the mean, median, and mode. calculate the new measures center, and collect data • determine theofeffect on measures of center when a single value of a data set is added, removed, or changed. Improve:Since Encourage to set analyze discuss howorthe outcomes with every datastudents value in the is a parttheir of thefindings, sum, adding, removing, changing a valuecompare can change the their initial numerator signiﬁcantly, on whatby thatexploring value is. While the denominator willor only increase types or decrease by predictions, and refine theirdepending understanding additional data sets different of changes. 1 (or not at all if we’ve only changed an existing value). This is why the mean is so easily affected by changing the data.

Changing data values and measures of center

To ﬁnd the median, we list all the numbers in order from smallest to largest and ﬁnd the middle value. Adding, Exploration We have previously learned about various measures of center:

removing, or changing a value in a data set can often change its median. Though this change will not be major • mean - also called average, is the sum of values divided by the number of values.

data values are ordered numerically and changing a single value will only cause it to shift to a nearby Students:because Pagethe 405 • median - is the middle value when the values are sorted.

value. • mode - the value that occurs most often. The mode is the value with the highest frequency (the one that appears most often). When we add, remove, or change a value in a data set, it may affect the mode by causing a new number to become the mode, or the mode may Interactive exploration remain the same. Explore online to answer the questions

mathspace.co Use the interactive exploration in 9.06 to answer these questions.

What happens to the measures of center when the blue point is removed? 9.06 Changing data values and measures of center 405 mathspace.co What happens to the measures of center when the blue point is added back?

What happens to the measures of center when the blue point is changed?

Repeat with new data sets to see how the measures of center change as the blue point is added, removed, or changed. Do your previous observations continue to be true?

Recall we can calculate the mean by ﬁnding the ‘average’ of the data set:

790

Mathspace Virginia SOL Grade 6 Teacher EditionMean = mathspace.co Since every data value in the set is a part of the sum, adding, removing, or changing a value can change the numerator signiﬁcantly, depending on what that value is. While the denominator will only increase or decrease by 1 (or not at all if we’ve only changed an existing value). This is why the mean is so easily affected by changing the

Suggested student grouping: In pairs In this exploration, students will manipulate a data point within a data set and observe how the measures of center (mean, median, and mode) change. By removing, adding, and changing the value of the data point, students will discover how changes to individual data points affect the measures of center. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What happens to the measures of center when the blue point is removed? Answers will vary based on where the blue point is initially placed. If the blue point has a lower value than all other data points, an example answer is: When the blue point is removed, the mean and median increase, but the mode stays the same. 2. What happens to the measures of center when the blue point is added back? Answers will vary based on where the blue point is initially placed. If the blue point has a lower value than all other data points, an example answer is: When the blue point is added back, the mean and median decreases, but the mode stays the same. 3. What happens to the measures of center when the blue point is changed? If the blue point’s value is changed, the mean, median, and mode will change according to the new value. If the new value is larger or smaller than the previous value, these measures of center will increase or decrease accordingly. 4. Repeat with new data sets to see how the measures of center change as the blue point is added, removed, or changed. Do your previous observations continue to be true? Yes, the observations continue to be true with new data sets. The mean and median usually change whenever the blue point is added, removed, or changed in value. Purposeful questions • According to your answers, can a single data point impact the mean, median, and mode of the entire data set? • How does the position of the blue point within the data set (whether it’s a high or low value) impact the measures of center? Possible misunderstandings • A common misunderstanding might be that all measures of center (mean, median, mode) will always change in the same way when manipulating a single data point. However, the median and mode might remain unchanged while the mean changes.

Advanced learners: Generalizing the effect of data changes on measures of center Targeted instructional strategies Encourage students to generalize the effects of adding, removing, or changing data values on the mean, median, and mode. Ask them to derive general rules that describe how these measures of center change when a data point is modified. For example, challenge them to justify why adding a value equal to the current mean does not change the mean of the data set, or how removing the highest or lowest value impacts the median and mode. Students could fill out tables like the ones shown: Median Mean Mode

Removing a low value Might increase Will increase Typically no change

Removing a high value Might decrease Will decrease Typically no change

9.06 Changing data values and measures of center mathspace.co

791

Adding a low value Might increase Will increase Typically no change

Median Mean Mode

Adding a high value Might decrease Will decrease Typically no change

9.06 Changing data values and measures of center

By engaging in this deeper analysis, students will not only understand the computational aspects but also the underlying mathematical relationships. This approach promotes critical thinking and allows students to explore the fundamental properties of statistical measures. After this lesson, you will be able to...

After the exploration, the effects on the measures center removing, or is changing a data value are • determine the effect on measures of of center whenofa adding, single value of a data set added, removed, summarized. or changed.

Students: Page 405

9.06 Changing data values and measures of center Interactive exploration

Explore online to answer the questions

After this lesson, you will be able to...

mathspace.co

• determine the effect on measures of center when a single value of a data set is added, removed, or changed. Use the interactive exploration in 9.06 to answer these questions. 1.

What happens to the measures of center when the blue point is removed?

2. What happens to the measures of center when the blue point is added back? Changing data values and measures of center

3. What happens to the measures of center when the blue point is changed? We have previously learned about various measures of center: 4. Repeat with new data sets to see how the measures of center change as the blue point is added, • mean - also called average, is the sum of values divided by the number of values. removed, or changed. Do your previous observations continue to be true? • median - is the middle value when the values are sorted. • mode - the value that occurs most often. Recall we can calculate the mean by ﬁnding the ‘average’ of the data set:

Interactive exploration

Mean = Explore online to answer the questions Since every data value in the set is a part of the sum, adding, removing, or changing a value can change the mathspace.co numerator significantly, depending on what that value is. While the denominator will only increase or decrease by 1 (or not at all if we’ve only changed an existing value). This is why the mean is so easily affected by changing the Use the interactive exploration in 9.06 to answer these questions. data. 1. theWhat happens the measures whensmallest the bluetopoint is removed? To find median, we listtoall the numbersofincenter order from largest and find the middle value. Adding, 2. What happens to thein measures ofcan center when the blue point isThough added back? removing, or changing a value a data set often change its median. this change will not be major because data values to arethe ordered numerically and changing a single value will only cause it to shift to a nearby 3. the What happens measures of center when the blue point is changed? value. 4. Repeat with new data sets to see how the measures of center change as the blue point is added, The moderemoved, is the value with the highest frequency one that appears or changed. Do your previous (the observations continuemost to beoften). true? When we add, remove, or change a value in a data set, it may affect the mode by causing a new number to become the mode, or the mode may remain the same. Recall we can calculate the mean by finding the ‘average’ of the data set: Mean = Since every data value in the set is a part of the sum, adding, removing, or changing a value can change the numerator significantly, depending on what that value is. While the denominator will only increase or decrease by 1 (or not at all if we’ve only changed an existing value). This is why the mean is sodata easily affected by changing the 405 9.06 Changing values and measures of center mathspace.co data. To find the median, we list all the numbers in order from smallest to largest and find the middle value. Adding, removing, or changing a value in a data set can often change its median. Though this change will not be major because the data values are ordered numerically and changing a single value will only cause it to shift to a nearby value.

792

The mode is the SOL valueGrade with the highest frequency (the one that appears most often). When we add, remove, or Mathspace Virginia 6 Teacher Edition change a value in a data set, it may affect the mode by causing a new number to become the mode, or the mode may mathspace.co remain the same.

Examples Students: Page 406 Example 1 Consider the data: Scores: {39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 41, 41, 42, 42, 43, 43, 43, 43} a Find the total number of scores.

Create a strategy Add the total number of scores.

Apply the idea Total number of scores = 6 + 4 + 2 + 2 + 4 = 18

Add the total number of frequencies Evaluate the addition

There are 181total scores. Example Consider the data: b Approximate the sum of the scores. Scores: {39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 41, 41, 42, 42, 43, 43, 43, 43}

PurposeaCreate Find a the total number of scores. strategy Show students how to determine the total number of scores in a data set. Add all the scores together. Create a strategy

Expected mistakes Add thethe total number of scores. Apply idea Students might misunderstand the⋅ 2total of the scores, rather total number Sum of the scoresthe = 39question ⋅ 6 + 40 ⋅ 4and + 41 find ⋅ 2 + 42 + 43 value ⋅4 Multiply each score than by its the frequency Apply thethem idea aware that of scores. Make each number represents a score or data value. The question is asking how = 234 + 160 + 82 + 84 + 172 Evaluate the multiplication Total scores = 6 + 4 + 2 + 2 + 4 Add the total number of frequencies many data values arenumber in theofdata set. = 732 Evaluate the addition The sum of all the scores is 732 = 18

Evaluate the addition

Students:There Page are 406 18 total scores.

c Find the mean, median, and mode of the scores, correct to two decimal places. b Approximate the sum of the scores.

Create a strategy Create a strategy

For the mean, we can use the formula: Mean = Add all the scores together. The mode is the most repeated score.

Apply the idea The median is the middle score.

Sum of the scores = 39 ⋅ 6 + 40 ⋅ 4 + 41 ⋅ 2 + 42 ⋅ 2 + 43 ⋅ 4

Apply the idea

= 234 + 160 + 82 + 84 + 172

Multiply each score by its frequency Evaluate the multiplication

= 732 Evaluate the addition Mean = Divide the sum of the scores by the total number of scores The sum of all the scores is 732 = 40.67 Evaluate to two decimal places Mode: c Find theismean, median, mode offrequency the scores, correct to two decimal places. The mode the score with and the highest which is 39.

PurposeMedian: Create a strategy There are 18 scores. The median score should be the average of the 9th and 10th score. The 9th and 10th score are Demonstrate to students how to approximate the sum of scores in a dataset. For the can use the formula: Mean = both 40,mean, so thewe median is 40. The mode is the most repeated score. The median is the middle score.

Apply the idea 406

Mathspace Mean =Virginia SOL Grade 6Divide the sum of the scores by the total number of scores mathspace.co

= 40.67

Evaluate to two decimal places

Mode: The mode is the score with the highest frequency which is 39. Median: There are 18 scores. The median score should be the average of the 9th and 10th score. The 9th and 10th score are 9.06 Changing data values and measures of center both 40, so the median is 40. mathspace.co

793

Apply the idea Sum of the scores = 39 ⋅ 6 + 40 ⋅ 4 + 41 ⋅ 2 + 42 ⋅ 2 + 43 ⋅ 4

Multiply each score by its frequency

= 234 + 160 + 82 + 84 + 172

Evaluate the multiplication

= 732

Evaluate the addition

sum of all the scores is 732 Students:The Page 406

c Find the mean, median, and mode of the scores, correct to two decimal places.

Create a strategy For the mean, we can use the formula: Mean = The mode is the most repeated score. The median is the middle score.

Apply the idea Mean = = 40.67

Divide the sum of the scores by the total number of scores Evaluate to two decimal places

Purpose Teach students how to calculate the mean, median, and mode of a dataset, and how to interpret these 406 Mathspace Virginia SOL Grade 6 measures of mathspace.co central tendency.

Students: Page 407 d A new score of 10 is added. Find the new mean, median, and mode of the scores, correct to two decimal places.

Create a strategy For the mean, we can use the formula: Mean = The mode is the most repeated score. The median is the middle score.

Apply the idea Mean = = 39.05

Divide the sum of the scores by the total number of scores Evaluate to two decimal places

Mode: The mode is the score with the highest frequency which is still 39. Median: There are 19 scores. The median score should be the average of the 10th score. The 10th score is 40, so the median is 40.

Reﬂect and check Notice that the mode and median did not change. When much smaller or much larger data is added, it is more likely to have a greater impact on the mean since it measures the distance of each point from the balance point.

Example 2 Purpose Challenge students to understand the11,impact A data set consists of ﬁve numbers 13, 9, 13,of 9. adding a new score to a dataset on the mean, median, and mode. a The data set has a current mean of 11. If the data set changes to 11, 15, 9, 13, 9 will the mean be higher, lower, or remain the same?

Create a strategy 794

Identify what hasSOL changed data set. One of the values of 13 has been increased to 15. Think about what Mathspace Virginia Gradein6the Teacher Edition increasing the sum of the data will do to the mean. mathspace.co

Apply the idea Remember the mean is calculated by dividing the sum of all values by the number of values in the set.

Use visual aids to understand changes in measures of center

use with Example 1

Targeted instructional strategies To help students understand how changing data values impact measures of center (mean, median, and mode), use visual aids such as bar graphs or line plots. This can help students visually see how the addition, removal, or change of a value can affect the overall data set. For example, a line plot can be used to demonstrate how adding a data point of 10 does not change the mode. The line plot highlights the mode of 39, and adding a single data d A new score of 10 is added. Find the new mean, median, andadd mode the scores, column correct towith two decimal places. point of 10 would anofadditional only one “x”.

Create a strategy 39 40 41 42

The new column would not be taller than the column at 39, meaning the mode is unaffected.

For the mean, we can use the formula: Mean =

When teaching the concept of median, show how the middle value can shift with the addition of a lower data The mode is the most repeated score. point. Explain how adding a data point with a much smaller value can affect the balance of the data, hence The median is the middle score. affecting the mean. Apply the idea Mean =

Divide the sum of the scores by the total number of scores

The measures of centers may not change = 39.05

Evaluate to two decimal places

use with Example 1

Address student misconceptions Mode:

StudentsThe might assume that when a data value is removed, added, or changed the measures of center will mode is the score with the highest frequency which is still 39. always be affected. Use this example to highlight the fact that some measures of center might change while Median: others might not change. There are 19 scores. The median score should be the average of the 10th score. The 10th score is 40, so the median 40. Remind isstudents of the importance of recalculating the measures of center to determine whether each has been affected by the change in the data.

Reﬂect and check

Notice that the mode and median did not change. When much smaller or much larger data is added, it is more likely to have a greater impact on the mean since it measures the distance of each point from the balance point.

Students: Page 407 Example 2

A data set consists of ﬁve numbers 11, 13, 9, 13, 9. a The data set has a current mean of 11. If the data set changes to 11, 15, 9, 13, 9 will the mean be higher, lower, or remain the same?

Create a strategy Identify what has changed in the data set. One of the values of 13 has been increased to 15. Think about what increasing the sum of the data will do to the mean.

Apply the idea Remember the mean is calculated by dividing the sum of all values by the number of values in the set. Increasing a data value will increase the sum (numerator) but the number of data values (denominator) will stay the same. When we divide a larger numerator by the same denominator, the result we get will be larger than the original. The mean will increase.

Reﬂect and check Find the sum of the data set Divide the sum of the values by the total number of values Evaluate to two decimal places The mean will be higher because the balance point is pulled up towards the changed, larger data value.

9.06 Changing data values and measures 9.06 Changing data values and measures of center of 407center mathspace.co mathspace.co

795

9.06 Changing data values and measures of center Purpose mathspace.co Show students how changing a data set can affect the mean, median, and mode of the set.

407

Students: Page 408 b The data set has a current median of 11. If a new number is added that is larger than 13, will the median be higher, lower, or remain the same?

Create a strategy The median is the middle value in a data set. So, adding a value could affect the location of the middle of the data set.

Apply the idea If we arrange the original data set in ascending order, the data set looks like this: 9, 9, 11, 13, 13. The median of this ordered set, which is the middle value when all the numbers are listed from smallest to largest, is 11. If we add a number larger than 13, it is added to the right side of the ordered list of data and would look like: 9, 9, 11, 13, 13, ⬚. We can see the new median will fall between 11 and 13.

Without calculating the value of the median, we can say that it will increase.

Reﬂect and check To ﬁnd the middle, organize the data from smallest to largest. 9, 9, 11, 13, 13, ⬚

There are now 6 data values so the middle falls between the 3rd and 4th value. To ﬁnd the middle average 11 and 13. Find the sum of the values being averaged Find the quotient Evaluate

c The current data set has two modes of 9 and 13. If the data set changes to 11, 9, 13, 9 will the modes remain the same?

Purpose Demonstrate Createtoa students strategy how adding a number to a data set can affect its median. Remember the mode represents the data value occurring with the highest frequency.

Apply the idea In the original data set, the values of 9 and 13 occurred twice. The new data set removed one of the values of 13, so it now only occurs once and is no longer a mode. This means, 9 is the only mode of the new data set.

Example 3 25 students took an assessment. Their scores are shown below. 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70

796

Mathspace Virginia SOL Grade 6 Teacher a A teacher calculated the mean of 25Edition students’ scores to be 64. A student who later completed the assessment mathspace.co got a score of 55. Find the new mean of the class, correct to two decimal places.

Create a strategy

There are now 6 data values so the middle falls between the 3rd and 4th value. To ﬁnd the middle average 11 and 13. Find the sum of the values being averaged Find the quotient

Students: Page 408

Evaluate

c The current data set has two modes of 9 and 13. If the data set changes to 11, 9, 13, 9 will the modes remain the same?

Create a strategy Remember the mode represents the data value occurring with the highest frequency.

Example 3 Purpose 25 students took an assessment. Their scores are shown below. Illustrate to students how changing the frequency of a value in a data set can affect the mode. 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70

Concrete-Representational-Abstract (CRA) approach

use with Example 2

a A teacher calculated thestrategies mean of 25 students’ scores to be 64. A student who later completed the assessment Targeted instructional got a score of 55. Find the new mean of the class, correct to two decimal places.

Concrete: Begin by engaging students with physical manipulatives to represent the data set. Use sticky notes toCreate write each number from the data set. Have students place the sticky notes on a board or desk in any a strategy order. Encourage them to physically We can use the formula: Mean = rearrange the sticky notes to see the numbers clearly. When the data set changes—for example, changing one of the 13 sticky notes to 15—have students replace the sticky note to reflect the new number. This hands-on activity helps students visualize how changing a number in the data set 408 Mathspace Virginia SOL Grade 6 affects measures like the mean, median, and mode. Ask students to group and count the sticky notes to begin mathspace.co exploring these concepts. Representational: Transition to drawing representations of the data. Have students create a number line on paper and plot each data point from the sticky notes onto the number line. For instance, they can place dots or marks above the numbers 9, 11, 13, 15 to represent their frequency in the data set. When the data changes, instruct students to update their number lines accordingly, such as adding a mark above 15 instead of 13. This visual representation helps students see the distribution of the data and understand how the measures of center may shift due to changes. Encourage students to note which numbers appear most often and where the middle of the data lies. Abstract: Move to abstract calculations by introducing the formulas for mean, median, and mode. Guide students in calculating the mean by adding all the numbers and dividing by the total count: • Original mean: Mean = • New mean: Mean =

= 11 = 11.4

Show them how to find the median by arranging the numbers in order and identifying the middle value. Explain that in the original data set, the median is 11, but it changes when a new number is added or altered. Discuss how to determine the mode by finding the most frequently occurring number, and how this may change with the data set. Work through these calculations with the students to solidify their understanding.

9.06 Changing data values and measures of center mathspace.co

797

To ﬁnd the middle, organize the data from smallest to largest. 9, 9, 11, 13, 13, ⬚

There are now 6 data values so the middle falls between the 3rd and 4th value. To ﬁnd the middle average 11 and 13. Find the sum of the values being averaged

Use technology to understand the effect of adding data values

use with Example 2

Find the quotient Student with disabilities support Evaluate

As students develop a deeper understanding of how changes in a data set affect measures of center abstractly, allow them to use technology to calculate the new measures of center after data values are added, removed, c The current datacalculations, set has two modes of 9 and 13. If theto data set changes to 11,findings 9, 13, 9 will modes remain theeach or changed. After these encourage them generalize their bythe comparing how measure ofsame? center changed. Using a Create statistics calculator can allow students to add data points to a set and quickly see changes in the mean, a strategy median,Remember and mode. help them focus onoccurring the effect newfrequency. data value rather than the calculations for theThis modecan represents the data value withof thethe highest finding the new mean, median, and mode. Apply the idea In the original data set, the values of 9 and 13 occurred twice. The new data set removed one of the values of 13,

Students:soPages 408–409 it now only occurs once and is no longer a mode. This means, 9 is the only mode of the new data set. Example 3 25 students took an assessment. Their scores are shown below. 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70

a A teacher calculated the mean of 25 students’ scores to be 64. A student who later completed the assessment got a score of 55. Find the new mean of the class, correct to two decimal places.

Create a strategy We can use the formula: Mean =

Apply the idea 408 Mathspace Virginia SOL Grade 6 mathspace.co

Substitute the values Evaluate the addition and multiplication Evaluate to two decimal places

The new mean of the class is 63.65 which is 0.35 lower than the original mean of 64. This makes sense because the new score of 55 being added is a value far away from the previous mean, causing the mean score to drop.

Reﬂect and check Notice the sum of all the scores was not recalculated because the old mean was already calculated. The old mean can be used in calculating the new mean as long as none of those previous data values change. The new score needs to be included in the new sum and the total number of scores needs to be increased by 1 to show that there are now a total of 26 scores in the data set.

b Find the median of the class before and after the ﬁnal student took the assessment. Did the median change?

Purpose Create a strategy Show students how to calculate the new mean of a data set when a new data point is added, and understand The median is the middle value of a data set when ordered from least to greatest. So, adding a value could affect the how thislocation new data affects the set. mean of the data set in real-world context. of thepoint middle of the data Reflecting with students Apply the idea Ask students to explain the values in the equation There were originally 25 students who took the assessment, making the median the 13th score. The median of the original data set, before the ﬁnal student took the assessment, is 64. We can list out the new list of scores with the ﬁnal student’s score included. 55, 58, 60, 60, 60,additional 61, 62, 62, 63, 63, 63, 64, For 64, 64 If students struggle to explain the values, provide assistance. example, explain that the original 66, 66, 66, 67, 68, 70, 70 Since the mean was found to be 64, the sum of original mean was calculated using Mean = 64, 64, 65, 65, .65, values must beanother equalscore to 25to×the 64. Adding data set makes 26 scores so the median falls between the 12th and 13th score. Adding the of 55 to making the previous sum64.gives us the new total in the numerator. Since we added Bothadditional the 12th andvalue 13th score is 64, the median a data value, we must add 1did in not thechange denominator. In this case the median because the score of the middle value remained 64.

798

Mathspace Virginia SOL Grade 6 Teacher Edition c Find the mode of the class before and after the ﬁnal student took the assessment. Did the mode change? mathspace.co

Create a strategy

Apply Reﬂectthe andidea check Notice the sum of all the scores was not recalculated because the old mean was already calculated. The old mean Substitute the values can be used in calculating the new mean as long as none of those previous data values change. The new score needs to be included in the new sum and the total scores needs to be increased by 1 to show that there Evaluate thenumber additionofand multiplication now a 409 total of 26 scores in the data set. Students:arePage Evaluate to two decimal places The new mean of the class is 63.65 which is 0.35 lower than the original mean of 64. This makes sense because the b theofmedian of the class after the the causing assessment. Did the median change? newFind score 55 being added is before a valueand far away fromﬁnal thestudent previoustook mean, the mean score to drop.

Reflect check Create aand strategy Notice the sum of all the scores was not recalculated because theleast old to mean was already calculated. oldaffect meanthe The median is the middle value of a data set when ordered from greatest. So, adding a valueThe could can be used in middle calculating the newset. mean as long as none of those previous data values change. The new score location of the of the data needs to be included in the new sum and the total number of scores needs to be increased by 1 to show that there are nowthe a total Apply ideaof 26 scores in the data set. There were originally 25 students who took the assessment, making the median the 13th score. The median of the original data set, before the final student took the assessment, is 64. b Find the median of the class before and after the final student took the assessment. Did the median change? We can list out the new list of scores with the final student’s score included. 55, 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70 The median is the middle value of a data set when ordered from least to greatest. So, adding a value could affect the locationanother of the middle Adding score of to the the data data set. set makes 26 scores so the median falls between the 12th and 13th score.

Create a strategy

Both the 12th and 13th score is 64, making the median 64.

Apply the idea

In this case the median did not change because the score of the middle value remained 64. There were originally 25 students who took the assessment, making the median the 13th score. The median of the original data set, before the final student took the assessment, is 64. cWe Find theout mode thelist class beforewith and the afterfinal thestudent’s final student took the assessment. Did the mode change? can list the of new of scores score included. 55, 58, 60, 60, 60, 61, 62, 62, 63, 63, 63, 64, 64, 64

PurposeCreate a strategy 64, 64, 65, 65, 65, 66, 66, 66, 67, 68, 70, 70 Show students how to calculate the new mean of a data set when a new data point is added, and understand The mode represents the data value with the highest frequency. So, we will need to count out the common scores another score to the data set makes 26 scores the median falls between the 12th and 13th score. how thisAdding new point affects mean of the data so set. and ﬁnddata the score that occursthe most frequently. Both the 12th and 13th score is 64, making the median 64.

this case the median did not change because the score of the middle value remained 64. Students:In Page Apply the409 idea The new score of 55 only occurs 1 time, so it does not affect the mode before or after the final student took the assessment. c Find the mode of the class before and after the final student took the assessment. Did the mode change? The scores of 60, 63, 65, and 66 have a frequency of 3. The score of 64 has a frequency of 5 making it the score with the highest frequency. Create a strategy Therefore, the mode ofthe thedata class before the frequency. final student is 64. The mode represents value withand theafter highest So,took we the will assessment need to count out the common scores and find the score that occurs most frequently.

Apply the idea The new score of 55 only occurs 1 time, so it does not affect the mode before or after the ﬁnal student took the 9.06 Changing data values and measures of center 409 assessment. mathspace.co

The scores of 60, 63, 65, and 66 have a frequency of 3. The score of 64 has a frequency of 5 making it the score with the highest frequency. Therefore, the mode of the class before and after the ﬁnal student took the assessment is 64.

Purpose 9.06 Changing data values and measures of center 409 Teach students to calculate the mode of a data set, and explore how the addition of a newmathspace.co data point may affect the mode, in real-world context.

Three reads

use with Example 3

9.06 Changing data values and measures of center mathspace.co

799

On the second read, students should aim to interpret the problem by answering questions like, “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: • 25 students took the exam, and another student took it later, making a total of 26 students. • The median before the final student took the exam will be the 13th score, but the new median will be between the 12th and 13th scores. • There are no other scores of 55, so the final student’s score will not affect the mode. Students can be prompted by framing these as questions like “How many students took the exam in total?” or “What information do you need to use to find the original median and the new median?”

Students: Page 410

Idea summary Adding a new data point can signiﬁcantly impact the measures of center, which include the mean, median, and mode. Mean: • • • •

The most sensitive to changing the data set Increases if an existing value is increased, a value smaller than the mean is removed, or if a value that is larger than the mean is added to the set Decreases if an existing value is decreased, if a value larger than the mean is removed, or if a value that is smaller than the mean is added to the set Stays the same if a value equal to the mean is added or removed

Median and mode are not signiﬁcantly impacted by changing a value in the data set.

Practice What do you remember?

Practice 1

Match each term with the correct deﬁnition:

a Mean Students: Pages 410–413

The average of all the values in the data set

Median

The most common value in the data set

Mode

iii

The difference between the highest and lowest values in the data set

Range

The middle value of the data set when in descending order

What do you remember? 2

Which two data sets have the same mean?

• Data Set A: 9, 14, 16,deﬁnition: 11 Match each term with the13,correct

Mean

3 The mean of a set of scores is 38.6 and the sum of the scores is 694.8. the number of scores. Median ii The most common value in the dataCalculate set

4 A set of ﬁve numbers has of 10. Two ofbetween the numbers arehighest 6 and 13.and Determine other set Mode iii a mean The difference the lowestwhether valuesthe in three the data

Rangenumbers could be in theivset:The middle value of the data set when in descending order a

15, 11, 8

The average of all the values in the data set

10, 13, 8

13, 5, 6

Mode•

Data Set C: 12,d 28, 10, 6, 7 Range • Data Set D: 13, 15, 7, 18, 8

The Let’s meanpractice of a set of scores is 38.6 and the sum of the scores is 694.8. Calculate the number of scores. 6

The ﬁve numbers 16, 16, 17, 24, 17 have a mean of 18. If a new number is added that is bigger than 24, will the mean be higher or lower?

The ﬁve numbers 9, 13, 9, 11, 8 have a mean of 10. If a number is removed that is smaller than 9, will the mean be higher or lower? Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 8 Five numbers have a mean of 7. If 4 of the numbers are 10, 10, 8 and 7 and the last number is x, ﬁnd the value of x. 7

800

10, 6, 18

Which5 twoGiven datathe sets the same mean? datahave set {4, 7, 5, 9}, calculate: • Data A: 13, 9, 14, 16, 11 b Median a Set Mean • Data Set B: 8, 24, 14, 8, 17

• Data Set C: 12, 28, 10, 6, 7 • Data Set D: 13, 15, 7, 18, 8

• Data Set B: 8, 24, 14, 8, 17

The mean of four scores is 21. If three of the scores are 17, 3 and 8, ﬁnd the fourth score.

A set of ﬁve numbers has a mean of 10. Two of the numbers are 6 and 13. Determine whether the three other numbers could be in the set: a

15, 11, 8

10, 13, 8

13, 5, 6

10, 6, 18

Mode

Range

Given the data set {4, 7, 5, 9}, calculate: a

Mean

Median

Let’s practice 6

The ﬁve numbers 16, 16, 17, 24, 17 have a mean of 18. If a new number is added that is bigger than 24, will the mean be higher or lower?

The ﬁve numbers 9, 13, 9, 11, 8 have a mean of 10. If a number is removed that is smaller than 9, will the mean be higher or lower?

Five numbers have a mean of 7. If 4 of the numbers are 10, 10, 8 and 7 and the last number is x, ﬁnd the value of x.

The mean of four scores is 21. If three of the scores are 17, 3 and 8, ﬁnd the fourth score.

The mean of a set of 41 scores is 18.6. If a score of 71.8 is removed from the set, ﬁnd the new mean. Round your answer to two decimal places.

A teacher calculated the mean of 25 students’ marks to be 64. A student who later completed the assessment got a mark of 55. What is the new mean of the class, to two decimal places?

Change the value 6 to 9 in the data set {3, 5, 6, 7} and recalculate the mean, median, and mode. Compare the results.

Given the data set {4, 5, 6, 7}, perform three operations: Add 10, remove 5, change 6 to 8. Discuss how each operation affects the mean, median, and mode.

Moris runs every morning and records the minutes in his tracker. Monday 20

Tuesday 20

Wednesday 30

Thursday 10

Friday 20

What is his average or mean number of minutes for these ﬁve days?

If Moris skips his Friday run, what is the average number of minutes for the other days?

What happens to Moris’ mean number of minutes if he doesn’t run on Friday?

Last week, Alberto earned $18, $14, $16, $19, and $17 by helping at his uncle’s bakery. a

Calculate the measures of center:

mean ⬚, median ⬚, mode ⬚

Predictions: mean ⬚ median ⬚ mode ⬚

This week, Alberto attended a sports clinic and only earned money for helping on four days. He earned $18, $16, $19, and $17. Discuss how you think each measure of center will be affected by earning for four days instead of ﬁve days.

Calculate the new measures of center and compare to your predictions. mean ⬚, median ⬚, mode ⬚

9.06 Changing data values and measures of center mathspace.co

801

Let’s extend our thinking 16

The table shows the monthly rent for various residential properties. 1950 1540

Find the mean, median, and mode of the data.

Rent on each of the residential properties raised by 5%. Find the mean, median, and mode of the data with the raise. How does this increase affect the mean, median, and mode of the data?

Use the original monthly rents to calculate the annual rents. Find the mean, median, and mode of the annual rents. How are these values related to the mean, median, and mode of the monthly rents?

You are part of a research team that analyzing the annual crop yields (in tons) from farms situated across diverse agricultural regions in your country, aiming to understand agricultural productivity ﬂuctuations and identify factors inﬂuencing harvest outcomes. The data set provided to you contains the annual total yield recorded by 15 farms over the last year. Here is the data: Yield (tons) 120 95 110 130 85

Farm F G H I J

Yield (tons) 105 500 125 100 115

Farm K L M N O

Yield (tons) 90 135 80 121 124

Calculate the mean, median, and mode of the annual yield for the 15 farms. Brieﬂy discuss your ﬁndings.

Inspect the data and identify any value(s) that seem unusually high or low compared to the rest. Explain why you consider this value(s) an outlier.

Remove the value you identiﬁed in part (b) from the data set and recalculate the mean, median, and mode of the annual yield for the remaining farms.

Discuss how the removal of the value you choose affected the mean, median, and mode.

The head coach of your school’s cycling team is analyzing the times (in minutes) of the 30-mile race results from the most recent sports meet to evaluate the team’s performance. The data set consists of the following times: Athlete 1 2 3 4 5

802

1730 1600

Farm A B C D E

Monthly Rents (dollars) 1670 1870 2200 1760 1940 1820

Time (minutes) 60.5 65.5 62.2 65.6 63.4

Athlete 6 7 8 9 10

Time (minutes) 64.1 69.7 67.8 60.2 70.1

Calculate the range of the times for all of the athletes. What does this measure tell you about the team’s performance variability?

A new athlete’s time of 70.4 minutes is added to the data set. How did adding this point affect the range?

The time of athlete 9 is determined to be incorrect due to a timing error and is removed from the data set. How did the removal of athlete 9’s time affect the range?

Why is the range a valuable measure to consider when analyzing data?

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

City Wheels, a local pre-owned car dealer, is analyzing the sale prices of pre-owned cars to understand the current market and set pricing guidelines for similar vehicles. The data set provided to you contains the sale prices (in thousands of dollars) of pre-owned cars sold in the last month. Car 1 2 3 4

Car 5 6 7 8

Sale Price (in thousands of dollars) 150 18 13 15

Calculate the mean, median, and mode of the pre-owned car sale prices. Brieﬂy discuss your initial observations about the central tendency of this data.

Discuss why the mean might not be a reliable measure of center for this data set.

Explain how the presence of the outlier (Car 5) affects the mean compared to the median and mode.

The table shows the scores of Student A and Student B in ﬁve separate tests: a

Find the mean score for Student A.

Find the mean score for Student B.

What is the combined mean of the scores of the two students.

What is the highest score overall? Which student obtained that score?

What is the lowest score overall? Which student obtained that score?

Test 1 2 3 4 5

The Stem and Leaf plot shows the batting scores of two cricket teams, England and India: a

What is the highest score from England?

What is the highest score from India?

Find the mean score of England.

Find the mean score of India.

Calculate the combined mean of the two teams.

Student A 97 87 94 73 79

Student B 78 96 92 72 86

England

India 1

Key: 1 |2|4 = 21 and 24

The column graph shows the total rainfall received during each month of the year: 12 10 Rainfall (mm)

Sale Price (in thousands of dollars) 10 21 20 22

8 6 4 2 0

Jan Fab Mar Apr Jan Jun Jul Aug Sept Oct Nov Dec Month

What measure of center would be most appropriate to measure the average rainfall per month?

Find this measure of center to the nearest whole number.

9.06 Changing data values and measures of center mathspace.co

803

Answers

Let’s extend our thinking

9.06 Changing data values and measures of center What do you remember? 1 a i. The average of all the values in the data set b i v. The middle value of the data set when in descending order c ii. The most common value in the data set d i ii. The difference between the highest and lowest values in the data set 2 A and C 3 18 4 a No

b Yes

c No

d No

5 a 6.25

b 6

c None

d 5

Let’s practice 6 Higher 7 Higher 8 x=0 9 56 10 17.27 11 63.65 12 The mean increased from 5.25 to 6. The median increased from 5.5 to 6. There’s no mode for both sets. 13 The data set {4, 5, 6, 7} has a mean of 5.5, a median of 5.5, and has no mode. Adding 10 would would increase the mean to 6.4, change the median to 6, but would not affect the mode (none). Removing 5 would increase the mean to 5.67, change the median to 6, but would still not affect the mode (none). Changing 6 to 8 increase the mean to 6, change the median to 6, but would still not affect the mode (none). 14 a 20 minutes b 20 minutes c M oris’ mean number of minutes remains at 20 minutes even if he doesn’t run on Friday. 15 a mean $16.8, median $17, mode none b P redictions: mean will increase, median will increase, mode might stay the same c mean $17.5, median $17.5, mode none The mean and median increased as predicted in part b. There is still no mode.

804

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

16 a Mean: $1808, Median: $1790, Mode: None b Mean: $1898.4, Median: $1879.5, Mode: None The percentage increase applied to each number is equal to the percentage increase in both the mean and median. Although there is no mode within the given set, it would likewise increase by the same percentage. The increase in monthly rents is proportional to the increase in mean, median and mode. c Mean: $21 696, Median: $21 480, Mode: None The mean and median of the annual rents is equal to twelve times the monthly rents. Although there is no mode within the given set, it would likewise increase by the same amount. 17 a T he mean yield is 135.67 tons. This represents the average yield per farm. The median yield is 115 tons. This indicates that half of the farms have yields equal to or less than, 115 tons, while the other half have yields equal to or greater than 115 tons. Since there are no duplicate yield values, there is no mode in this dataset. This suggests that there is no single yield value that occurs more frequently than others. b T he yield of 500 tons from farm G is unusually high compared to the rest. This would be considered an an outlier as the value is signiﬁcantly higher than the other yields, which range between 80 tons and 135 tons. c A fter removing 500 tons from the data set, the mean is 109.64 tons and the median is 112.5 tons. There is no mode. d A fter removing 500 tons from the data set, the mean decreased by ≈ 19%, and the median decreased by ≈ 2%. Removing the value has no effect on the mode, as there is no value that appears more frequently than the others. 18 a T he range of the times for all of the athletes is 9.9. This indicates that there is a signiﬁcant difference in performance levels of individual atheletes within the team. b A dding the new athelete’s time increased the range to 10.2. This is because the added time would be the highest value in the data set. c R emoving athlete 9’s time would decrease the range. This is because the removed time is the lowest value in the data set. d T he range is valuable for analyzing data as it offers a quick assessment of the spread between the highest and lowest values, aiding understanding variability within the dataset.

19 a T he mean is $33 625, the median is $19 000, and there is no mode for the pre-owned car sale prices. The mean appears to be signiﬁcantly inﬂuenced by the outlier (Car 5), causing it to be higher than expected based on the other sale prices. The median seems to be a more representative measure of the central tendency in this data set.

20 a 86

b 84.8

c 85.4

d 97, Student A

e 72, Student B 21 a 57

b 64

c 44.3

d 43.6

e 43.95 22 a Mean

b 8 mm

b S ince the mean considers all data points equally, it is heavily affected by such extreme values, causing it to be skewed. c T he outlier (Car 5’s sale price ) significantly affects the mean by pulling it towards higher values. The mean is calculated by summing all values and dividing by the total number of values. As a result, the mean is greatly influenced by outliers because it incorporates all values equally. However, the median is less affected by outliers. It only considers the middle values of the data set, regardless of extreme values. In this specific data set, there is no mode as each sale price occurs only once.

Answers mathspace.co

805

9.07 Outliers Subtopic overview Lesson narrative In this lesson, students will learn about outliers in data sets. They will explore how outliers, which are values significantly higher or lower than the rest of the data, impact measures of center (mean, median, and mode) and spread (range). The lesson includes identifying outliers in various data sets and understanding their effects on data analysis. Students will engage in an interactive exploration where they manipulate data sets to observe the effects of adding or removing outliers on mean, median, mode, and range. By the end, students should confidently identify and understand the impact of outliers on data sets.

Learning objectives

9.07 Outliers

Students: Page 414

After this lesson, you will be able to... • identify outliers in a given data set. • describe the effect of outliers on the mean, median, mode and range of a data set.

Outliers In statistics, we tend to assume that our data will ﬁt some kind of trend and that most things will ﬁt into a “normal” Key vocabulary

range. This is why we look at measures of center, such as the mean, median and mode.  median  mode mean A measure of center is a way to describe where the center of a set of data is. However, not all measures describe the  outlier  range  spread center in the same way and some measures are heavily affected by extreme data values, or outliers. 

Outlier

Essential understanding A data value that is an abnormal distance from the other data values in the set (much larger or much smaller) An outlier is an abnormal distance from the rest of the values in a data set. An outlier can sometimes give insight into the variable being explored but it can sometimes cause incorrect conclusions to be drawn from the data.

Interactive exploration

Explore online to answer the questions

Standards mathspace.co This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. Use the interactive exploration in 9.07 to answer these questions.

Mathematical process goals

1. What do you notice about the mean as you move the position of the blue point to be much larger than the data? Problem Solving MPG1 — Mathematical 2. do youthe notice the mean asproblem you move the position of the blueby point to bestudents much smaller than Teachers canWhat integrate goalabout of mathematical solving into this lesson guiding to recognize data? situation within the context of data sets. Students can be encouraged to apply their outliers as a the problem 3. What do you notice about the move position the blue point to outliers be muchimpact larger than understanding of measures of center andmedian spreadas toyou solve thisthe problem byofdetermining how these the data? measures. Real-world activities, such as analyzing data sets with and without outliers, can be used to enhance their problem-solving 4. Whatskills. do you notice about the median as you move the position of the blue point to be much smaller than the data?

Outliers are data points that lie far outside the majority of a data set and can signiﬁcantly affect the measures of (mean, median, andGrade mode)6 as well as the range. Mathspace Virginia SOL Teacher Edition 806center • mathspace.co The mean is most affected by outliers. Extreme data values cause the mean to increase or decrease signiﬁcantly. • The median is less affected by outliers because it only shifts based on how many data values are added or removed from the set, their values do not matter.

MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

To incorporate mathematical reasoning into instruction on identifying outliers and their impact on measures of center and spread, teachers can begin by explaining what outliers are and how they affect data sets. Provide students with examples, such as test score data, to calculate the mean, median, and mode, and discuss how these measures change when an outlier is added or removed. Use manipulatives to model these changes physically, facilitating hands-on learning. Encourage students to analyze different scenarios, predict outcomes, and justify their reasoning, fostering deeper understanding and critical thinking about data representation and analysis.

Teachers can help students make mathematical connections by relating the concept of outliers to their prior knowledge of measures of center and spread. They can also connect this topic to real-world situations, such as the potential presence of outliers in statistical data used in decision-making processes in various fields, thereby establishing a connection between mathematics and the real-world context.

Content standards 6.PS.2 — The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.

6.PS.2c — Observe patterns in data to identify outliers and determine their effect on mean, median, mode, or range.

Prior connections 5.PS.2 — The student will solve contextual problems using measures of center and the range.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 9.06 Changing data values and measures of center

Tools You may find these tools helpful: • Number line • Coloring materials • Blocks or cubes

9.07 Outliers mathspace.co

807

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Real-world problems Targeted instructional strategies Use a real-world scenario to model an outlier and its effects on a data set. For example, discuss a student with the following test grades: 87, 91, 84, 90, 91 Ask students to find the average of the test grades. Tell students that the student took another test but was absent for most of the unit and made a 20. Ask students if that value fits with the rest of the data. Then, ask students how adding that value to the data would affect the mean. Explain that the value is considered an “outlier” because it is extremely low compared to the data values. Ask students if it would be right or fair to include that value of 20 in the average if the student’s parents asked how they were doing overall on tests. Explain that outliers can skew the mean, which makes the mean not as good of a representation of the data as the median would. Then, proceed to work through several examples.

Outliers and averages Address student misconceptions A common misconception that students might have is that the mean and the median of a data set are always close to each other. While this can be true for many data sets, the presence of outliers can significantly affect the mean while having little to no impact on the median. To correct this misconception, it could be helpful to show students multiple examples of data sets with and without outliers and ask them to calculate the mean and median each time. Over time, they will begin to see the pattern and understand how outliers can skew the mean.

Student lesson & teacher guide Outliers This section introduces the concept of outliers in statistics, which are data points that differ significantly from the rest of the data set. It emphasizes the importance of identifying outliers and understanding how they can affect measures of central tendency and spread.

808

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 414

9.07 Outliers After this lesson, you will be able to... • identify outliers in a given data set. • describe the effect of outliers on the mean, median, mode and range of a data set.

9.07 Outliers Outliers In statistics, we tend to assume that our data will ﬁt some kind of trend and that most things will ﬁt into a “normal” thiswe lesson, will be able to... such as the mean, median and mode. range. ThisAfter is why look atyou measures of center, • identify outliers in a given data set. A measure of center is a way to describe where the center of a set of data is. However, not all measures describe the • describe the effect of outliers on the mean, median, mode and range of a data set. center in the same way and some measures are heavily affected by extreme data values, or outliers. Outlier

Outliers A data value that is an abnormal distance from the other data values in the set (much larger or much smaller) In statistics, we tend to assume that our data will ﬁt some kind of trend and that most things will ﬁt into a “normal” range. This is why we look at measures of center, such as the mean, median and mode.

exploration A measureInteractive of center is a way to describe where the center of a set of data is. However, not all measures describe the Explore to answer the questions center in the sameonline way and some measures are heavily affected by extreme data values, or outliers.

Exploration

mathspace.co Outlier Students: Page 414

A data that isexploration an abnormal from the other data values in the set (much larger or much smaller) Use thevalue interactive in distance 9.07 to answer these questions. 1.

What do you notice about the mean as you move the position of the blue point to be much larger than the data?

What doonline you notice aboutthe thequestions mean as you move the position of the blue point to be much smaller than Explore to answer the data?

What do you notice about the median as you move the position of the blue point to be much larger than mathspace.co the data?

Interactive exploration

Use interactive in 9.07 to answer these questions. 4. theWhat do youexploration notice about the median as you move the position of the blue point to be much smaller than the data? 1. What do you notice about the mean as you move the position of the blue point to be much larger than the data? 2. are What you notice the mean as you move the position the blue point affect to be much smaller than Outliers datadopoints that lieabout far outside the majority of a data set and of can significantly the measures of the data? center (mean, median, and mode) as well as the range. 3. mean Whatis do youaffected notice about the median as data you move position of theto blue point to much larger than • The most by outliers. Extreme valuesthe cause the mean increase or be decrease significantly. the data? • The median is less affected by outliers because it only shifts based on how many data values are added or removed from theirabout values domedian not matter. 4. What dothe youset, notice the as you move the position of the blue point to be much smaller • The mode least affected by an outlier because an outlier should be far away from the rest of the data so it is thanisthe data? unlikely to impact the mode which is the value that appears most often in the set. • The range is extremely affected by outliers because an outlier greatly increases the distance between the largest and smallest value. Outliers are datadata points that lie far outside the majority of a data set and can significantly affect the measures of center (mean, median, and mode) as well as the range. • The mean isgrouping: most affected outliers. Extreme data values cause the mean to increase or decrease significantly. Suggested student In by pairs • The median is less affected by outliers because it only shifts based on how many data values are added or In this exploration, students manipulate a data set by adjusting the position of an outlier (blue point) to observe removed from the set, their values do not matter. the impact on measures central specifically theshould meanbeand the median. By of making • The mode is leastofaffected bytendency, an outlier because an outlier far away from the rest the datathe so itoutlier is significantlyunlikely largertoor smaller than which the other students caninanalyze impact the mode is the data value points, that appears most often the set. patterns and draw conclusions on • The range is extremely affected because an outlier greatly increases the distance between the largest Mathspace Virginia Grade 6 by outliers how the414 values of the meanSOL and median change. mathspace.co and smallest data value.

414

Mathspace Virginia SOL Grade 6 mathspace.co

9.07 Outliers mathspace.co

809

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the mean as you move the position of the blue point to be much larger than the data? When the blue point is moved to be much larger, the mean increases. This is because the mean is the sum of all the values divided by the number of values. So, when a single value becomes much larger, it increases the total sum, and thus the mean. 2. What do you notice about the mean as you move the position of the blue point to be much smaller than the data? When the blue point is moved to be much smaller than the rest of the data, the mean decreases. This is because the mean is calculated by adding all the values together and dividing by the total number of values. So, when one value becomes much smaller, it decreases the total sum, which subsequently decreases the mean. 3. What do you notice about the median as you move the position of the blue point to be much larger than the data? When the blue point is moved to be much larger than the data, the median remains relatively unchanged. This is because the median is the middle value when all values are ordered from least to greatest, and it isn’t significantly affected by outliers. 4. What do you notice about the median as you move the position of the blue point to be much smaller than the data? When the blue point is moved to be much smaller than the other data points, the median doesn’t change significantly. This is because the median is just the middle number when all the data points are arranged in order, so it’s less affected by extreme values or outliers. Purposeful questions • Why does changing the value of a single data point have a different effect on the mean and the median? • What might be advantages or disadvantages of using the mean or median to describe the center of a data set? Possible misunderstandings • Students may assume that the mean is always the best measure of central tendency. They should realize that in the presence of outliers, the median may provide a more representative value of the dataset.

Advanced learners: Investigate outliers through real-world data sets Targeted instructional strategies Encourage advanced learners to create their own data sets, incorporating data from areas of personal interest such as sports statistics, environmental trends, or social media metrics. Allow them to include intentional outliers and analyze the effects on the mean, median, mode, and range. Ask students to manipulate these data sets by adding or removing outliers and observe how each measure of central tendency and spread changes. This exploratory approach connects the lesson to real-world contexts and promotes deeper understanding as students discover patterns and relationships themselves. Prompt them to generalize their findings by formulating rules about how outliers affect different statistical measures. Additionally, encourage them to justify their conclusions with mathematical reasoning, fostering critical thinking and analytical skills. Students explore how outliers significantly affect the mean and range but have less impact on the median and mode. Students also learn that analysts often remove outliers to better understand data trends, though outliers can provide valuable insights, such as identifying unusual events or errors. Examples of outliers in different representations like stem-and-leaf plots, dot plots, and line graphs are shown. 810

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

What do you notice about the mean as you move the position of the blue point to be much smaller than the data?

What do you notice about the median as you move the position of the blue point to be much larger than the data?

What do you notice about the median as you move the position of the blue point to be much smaller

Students: Pagesthan 414–415 the data?

Outliers are data points that lie far outside the majority of a data set and can signiﬁcantly affect the measures of center (mean, median, and mode) as well as the range. • The mean is most affected by outliers. Extreme data values cause the mean to increase or decrease signiﬁcantly. • The median is less affected by outliers because it only shifts based on how many data values are added or removed from the set, their values do not matter. • The mode is least affected by an outlier because an outlier should be far away from the rest of the data so it is unlikely to impact the mode which is the value that appears most often in the set. • The range is extremely affected by outliers because an outlier greatly increases the distance between the largest and smallest data value.

414

Mathspace Virginia SOL Grade 6 mathspace.co

Examples Students: Page 415

Purpose Show students how to identify the outlier from a data set. 9.07 Outliers mathspace.co

811

Use a number line to identify outliers

use with Example 1

Student with disabilities support Encourage students to create a visual representation of the data set by plotting each number on a number line. This approach supports students with visual-spatial processing difficulties by allowing them to see the relative positions of the numbers visually. Provide a number line ranging from 60 to 120, marked in increments of 5, and guide students in plotting the points 63, 67, 71, 76, and 111. 60

100

105

110

115

120

By seeing the data points spread out, students can easily observe that 111 is significantly distant from the other values. This visual aid helps students understand the concept of an outlier more concretely and reinforces the connection between numerical data and their graphical representations.

Students: Page 416

Purpose Show students how to identify the outlier from a stem-and-leaf plot. Reflecting with students Ask students to consider whether it would be representative to include the outlier in the summary statistics, or if it would be more representative to exclude it. Inform them that sometimes, the answer depends on the context. If the 7 hours represents someone who took 4 days off of work that week, then it is not representative. If that value represents one of the intern’s hours, then it would be representative, as there may be other interns or part-time employees in the population.

812

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Students: Page 416

Purpose Show students how to calculate the mean and median of a given data set and how these measures of central tendency could be influenced by the presence of an outlier.

Students: Page 417 b Identify the outlier.

Create a strategy

Apply the idea

Identify the game score that is much greater or smaller than most of the scores.

We can see that the dot for 300 is far away from the rest of the dots. Outlier = 300

c State the mean and median of the data without the outlier.

PurposeCreate a strategy Teach students how to identify an outlier in a data set and understand its effect on the mean and median of To find the mean, use the formula: Mean = the data. To find the median, find the middle score.

Apply the idea Add all the scores and divide by the total 19 Evaluate the division Since the number of scores is odd, then the middle score is 49. Median = 49

Reﬂect and check Notice that the mean changed from 63 to 50.53 and the median stayed the same. This is because the mean is 9.07 Outliers extremely affected by outliers while the median is much less affected. mathspace.co

Example 4

813

b Identify the outlier.

Create a strategy

Apply the idea

Identify the game score that is much greater or smaller than most of the scores.

We can see that the dot for 300 is far away from the rest of the dots.

Students: Page 417

Outlier = 300

c State the mean and median of the data without the outlier.

Create a strategy To find the mean, use the formula: Mean = To find the median, find the middle score. b Identify the outlier.

Apply the idea

Create a strategy

Apply idea19 Add all the scores and divide by the the total Identify the game score that is much greater or smaller We can see that the dot for 300 is far away from the rest Evaluate the division than most of the scores. of the dots. Since the number of scores is odd, then the middle score is 49. Outlier = 300 Median = 49 c State the mean and median of the data without the outlier.

Reﬂect and check

Notice the mean changed from 63 to 50.53 and the median stayed the same. This is because the mean is Createthat a strategy extremely affected by outliers while the median is much less affected. To find the mean, use the formula: Mean = To find the median, find the middle score.

Example 4 PurposeApply the idea The data set 6, 8, 10, 10, 12 has measures of: Challenge students to recalculate the mean median afterbyremoving Add all theand scores and divide the total 19the identified outlier and observe the • Mean = 9.2 changes •inMedian these =measures of central tendency. 10 Evaluate the division • Mode = 10 Since the number of scores is odd, then the middle score is 49. Expected • mistakes Range =6 49 StudentsSuppose might we divide by number 20, the20 original number ofMedian scores, rather than 19,outlier the number scores without the add the to the data set. Predict how =the addition of this will affectofthe mean, median,

mode, and range of the data set. outlier. Remind students thatnew they removed a data value, so they need to subtract one from the total number of scores. Reﬂect and check a Will the mean be higher, lower, or remain the same? Explain. Notice that the mean changed from 63 to 50.53 and the median stayed the same. This is because the mean is

Students:extremely Pages 417–418 Create a strategy affected by outliers while the median is much less affected.

The mean is the average of all the data values. Values signiﬁcantly higher or lower than the mean will cause the mean to shift in the direction of that high or low value.

Example 4

Apply the idea The data 8, higher. 10, 10, 12 has measures of: meanset will6,be • Mean The mean=is9.2 calculated as the sum of all values divided by the number of values. Adding a number as high as 20 • Median =increases 10 signiﬁcantly the sum of the data set (numerator) while only increasing the number of values (denominator) • 1.Mode 10 is much higher than the original mean, 9.2, the new mean must be higher. by Since=20 • Range = 6 Suppose we add the number 20 to the data set. Predict how the addition of this outlier will affect the mean, median, mode, and range of the new data set. a Will the mean be higher, lower, or remain the same? Explain.

9.07 Outliers mathspace.co

417

Create a strategy The mean is the average of all the data values. Values signiﬁcantly higher or lower than the mean will cause the mean to shift in the direction of that high or low value.

Apply the idea The mean will be higher. The mean is calculated as the sum of all values divided by the number of values. Adding a number as high as 20 signiﬁcantly increases the sum of the data set (numerator) while only increasing the number of values (denominator) by 1. Since 20 is much higher than the original mean, 9.2, the new mean must be higher.

9.07 Outliers mathspace.co

814

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

417

Reﬂect and check Let’s check our prediction by calculating the new mean. To ﬁnd the mean, use the formula: Mean = The sum of all values divided by the total number of values Evaluate the addition

Reﬂect and check

Evaluate the division Let’s check our prediction by calculating the new mean. The new mean, 11, is higher than the original mean, 9.2. To ﬁnd the mean, use the formula: Mean = b Will the median be higher, lower, or remain the same? Explain The sum of all values divided by the total number of values

PurposeCreate a strategy Evaluate the addition StudentsThe demonstrate their ofwhen howthe thedata mean of a data setleast is affected byAdding the addition of an outlier. median of a data setunderstanding is the middle value is organized from to greatest. a value could

Evaluate the division cause a shift in the location of the middle of the data. Students:The Page 418 11, is higher than the original mean, 9.2. new mean,

Apply the idea Originally, the median was 10, which is the middle value when the numbers are ordered. b Will the median be higher, lower, or remain the same? Explain After adding the number 20, the ordered data set becomes 6, 8, 10, 10, 12, 20. The median will now between, 10 and 10.

Create a strategy

The median remains the same. The median of a data set is the middle value when the data is organized from least to greatest. Adding a value could cause a shift in the location of the middle of the data.

Reﬂect and check

Let’s check our prediction by calculating the new median.

Apply the idea

First, we list the data set with the new value added. Originally, the median was 10, which is the middle value when the numbers are ordered. 6, 8, 10, 10,6,12, After adding the number 20, the ordered data set becomes 8, 20 10, 10, 12, 20. The median will now between, 10 and 10. There are now 6 values in the data set, so the median will fall between the 3rd and 4th terms. The median remains the same. The 3rd and 4th terms are both 10. To ﬁnd the average these values we need to ﬁnd the sum of the values and divide by two.

Reﬂect and check

Let’s check our prediction by calculating the new median. First, we list the data the set with The median remains samethe at new 10. value added. 6, 8, 10, 10, 12, 20 c Willare thenow mode be higher, lower, remain same?will Explain. There 6 values in the dataor set, so thethe median fall between the 3rd and 4th terms. The 3rd and 4th terms are both 10. To ﬁnd the average these values we need to ﬁnd the sum of the values and divide Create by two. a strategy The mode is the most frequently occurring value in the data set. The median remains the same at 10. Apply the idea The original mode is 10, and since the added number 20 was not in the original data set, it will not appear more frequently than 10. c Will the mode be higher, lower, or remain the same? Explain. The mode will remain the same.

PurposeCreate a strategy StudentsThe demonstrate theirfrequently understanding how thedata median mode is the most occurring of value in the set. of a data set is affected by the addition of an outlier. ApplyMathspace the idea Virginia SOL Grade 6 mathspace.co 418

The original mode is 10, and since the added number 20 was not in the original data set, it will not appear more frequently than 10. The mode will remain the same.

418

Mathspace Virginia SOL Grade 6 mathspace.co

9.07 Outliers mathspace.co

815

There are now 6 values in the data set, so the median will fall between the 3rd and 4th terms. The 3rd and 4th terms are both 10. To ﬁnd the average these values we need to ﬁnd the sum of the values and divide by two.

median remains the same at 10. Students:The Page 418

c Will the mode be higher, lower, or remain the same? Explain.

Create a strategy The mode is the most frequently occurring value in the data set.

Apply the idea The original mode is 10, and since the added number 20 was not in the original data set, it will not appear more frequently than 10. The mode will remain the same.

Purpose418 Mathspace Virginia SOL Grade 6 mathspace.co Students demonstrate their understanding of how the mode of a data set is affected by the addition of an outlier.

Students: Page 419 d Will the range be higher, lower, or remain the same? Explain.

Create a strategy The range is the difference between the largest and smallest values in the data set.

Apply the idea

Reﬂect and check

Originally, the range was from 6 to 12, which is 6 units. Let’s check our prediction by calculating the new range. By adding 20, the new range will be from 6 to 20, which The range is calculated by ﬁnding the difference of the is covering a larger spread of values than the original set. highest and lowest values. The range will be higher. 20 − 6 = 14 The new range of 14 is higher than the original range of 6.

Idea summary Purpose An outlier is a data point that varies signiﬁcantly from the rest of the data. An outlier will be a value that is Students demonstrate their understanding how the range of a data set is affected by the addition of an either signiﬁcantly larger or smaller of than other observations. outlier. Removing outliers will have the following effects on the summary statistics: A really low outlier

A really high outlier

Stronger each time The and range clearer will decrease The range will decrease

use with Example 4

English The language learner support The median might decrease median might increase The mean will increase

The mean will decrease

Begin by asking students to individually write their predictions about how adding the number 20 to the data The mode will not change The mode will not change set will affect the mean, median, mode, and range. Encourage students to use key mathematical terms such as “increase,” “decrease,” “remain the same,” and to explain their reasoning as best as they can. Next, have students pair up and share their explanations with a partner. Instruct them to listen carefully to each Practice other, ask clarifying questions, and provide constructive feedback. After the first discussion, ask students to find a new partner and repeat the process, incorporating any new ideas or vocabulary they have learned. Whatstudents do you remember? Finally, have revise their original written explanations, making them stronger and clearer by adding new insights, using precise mathematical language, and correcting any misunderstandings. This process not 1 What is an outlier? only deepens their understanding of how outliers affect statistical measures but also enhances their ability to 2 Identify the outlier ideas for eacheffectively. data set: communicate mathematical a

9, 10, 12, 9, 1

54, 52, 99, 50, 57

104, 115, 275, 109, 118, 121

4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1

816

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co 0 1 2 3 4 5 6 7 8 9 10 3

11.5

11.7

11.9

12.1

12.3

Below is a set of data representing the number of books read by students in a 6th grade class during summer

Students: Page 419

The new range of 14 is higher than the original range of 6.

Idea summary An outlier is a data point that varies signiﬁcantly from the rest of the data. An outlier will be a value that is either signiﬁcantly larger or smaller than other observations. Removing outliers will have the following effects on the summary statistics: A really low outlier The range will decrease The median might increase The mean will increase The mode will not change

A really high outlier The range will decrease The median might decrease The mean will decrease The mode will not change

Practice What do you remember? Practice 1

What is an outlier?

Students: 2Pages 419–423 Identify the outlier for each data set: a

9, 10, 12, 9, 1

54, 52, 99, 50, 57

104, 115, 275, 109, 118, 121

4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1

What do youe remember? 1

What is an outlier?

Identify the outlier 0 1 for 2 3each 4 5 data 6 7 set: 8 9 10 a c e

9, 10, 12, 9, 1

11.5

12.1

12.3

Below is a set of data representing the number of books read by students in a 6th grade class during summer d 4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1 break: f 5, 3, 7, 5, 4, 5, 6, 4 3

Calculate the mean number of books read.

Find the median number of books read.

Determine the mode of the dataset.

11.5

11.7

11.9

12.1

12.3 9.07 Outliers

419

Below is a set of data representing the number of books read by students in a 6th grade class during summer mathspace.co break: 5, 3, 7, 5, 4, 5, 6, 4 a

Calculate the mean number of books read.

Find the median number of books read.

Determine the mode of the dataset.

Calculate the range of the dataset.

When an outlier is removed from a data set, describe the effect on these statistical measures: a

11.9

104, 115, 275, 109, 118, 121

0 1d 2 Calculate 3 4 5the6 range 7 8of 9the10dataset.

11.7

54, 52, 99, 50, 57

Mode

Range

For each scenario, an outlier was removed. Was the outlier smaller or larger than the values that remain? a

The mean decreased after the outlier was removed.

The mean increased after the outlier was removed.

The median decreased after the outlier was removed.

The median increased after the outlier was removed.

9.07 Outliers mathspace.co

817

Let’s practice 6

The weight of ﬁsh caught in a “weigh and release” ﬁshing competition, in kilograms are given: 12.5, 15.1, 13, 14.2, 14.5, 14.9, 12.5, 14.3

Find the mean weight.

Find the median weight.

Recalculate the mean weight if the value of 1.5 is added to the data set.

Recalculate the median weight if the value of 1.5 is added to the data set.

Consider the given frequency table: Weight in kilograms 14 15 16 17 18 19 20

Frequency 1 0 0 3 6 4 2

What is the mode?

Which weight is the outlier?

If the outlier is removed, what is the new mode?

Compare the mode before and after removing the outlier.

Consider the given frequency table: Weight in kilograms 12 13 14 15 16 17 18

Frequency 2 5 1 2 0 0 1

Which weight is an outlier?

If the outlier is removed, will the new mean be higher or lower than the current mean?

Calculate the new mean, if the outlier is removed. Round your answer to one decimal place if needed.

For each scenario, decide if the given measure would increase, decrease, or stay the same: i

Mean

If 25 is added to the data set:

Median

iii

Mode

1, 1, 2, 3, 3, 3, 4 b

If 10 is removed from the data set: 10, 42, 55, 60, 65, 70

If 5 and 27 are removed from the data set: 5, 12, 12, 13, 15, 15, 27

If 1 is added to the data set: 24, 25, 25, 26

818

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Range

Consider the data set shown in the frequency table: Score 1 2 4 5 6 8 9

Frequency 1 2 2 1 2 3 2

Suppose one score of 8 is changed to a 15. Would any statistical measure change? a 11

Mean

Median

Mode

Range

Look at data sets a, b, and c below. For each set of data: i

Find the mean, median, mode, and range. Round your answers to two decimal places where necessary.

Which data value is an outlier?

iii

Describe how each of the four statistics may change if the outlier is removed.

Remove the outlier from the set and recalculate the values found in part (i).

27, 50, 24, 37, 47, 41, 27, 126, 44, 27

4.7, 2.8, 1.9, 0.9, 0.9, 2.2, 2.2, 1.2, 1.5, 0.9

4700, 4700, 4700, 4500, 5300, 4900, 5200, 4800, 1500, 5100

The ﬁve numbers 16, 16, 17, 24, 17 have a mean of 18 and a median of 17. Describe the effect on the mean and median if a new number is added that is larger than 24.

The number of three-pointers scored in different basketball games by a single team are shown in the table. Three-pointers 0 1 2 3 4 5 6 10

Frequency 2 4 5 3 2 2 2 1

Describe the effect on each statistical measure if the outlier is removed: a

Mean

Median

Mode

Range

Let’s extend our thinking 14

Tanya loves bird-watching and keeps a record of the number of different bird species she spots each week. One month, the mean number of different bird species she spotted per week was 21. However, three out of these four weeks, she spotted 17, 3, and 8 species respectively. a

Find the number of species Tanya spotted in the fourth week.

Describe how the outlier value of bird species spotted in the fourth week affects the mean and median of Tanya’s four-week bird-watching data.

9.07 Outliers mathspace.co

819

Below is a set of data showing the scores of 5 students on a math test: 72, 76, 80, 84, 88

Calculate the original mean and range of the scores.

Choose an outlier to add to this data set.

After adding the outlier, calculate the new mean and range.

Ted is an employee who works for a company and currently gets paid $26 000 annually. Here is the breakdown of all the employees’ salaries at the company he works for in a frequency table: Annual income $26 000 $58 000 $62 000 $150 000 $400 000

Number of employees 4 10 5 3 1

Would the owner of the company use the mean, median, or mode to describe the company’s annual pay? Why?

Would Ted use the mean, median, or mode to describe the company’s annual pay? Why?

Would a mathematician who is not involved with the company use the mean, median, or mode to describe the company’s annual pay? Why?

A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times 20 speed cameras issued a ﬁne to motorists in one month. The results were: 101, 102, 115, 115, 121, 124, 127, 128, 130, 130, 143, 143, 146, 162, 162, 163, 178, 183, 194, 977 The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which measure of center should she use in her article? Explain your answer.

The dot plot shows the temperature (°C) in a town over several weeks.

Temperature in a town

Which measure of center would represent this data best? Explain your reasoning.

Consider the given stem-and-leaf plot: Which measure of center best represents the center of the data? Explain your reasoning.

23 25 27 29

33 35

Stem Leaf 2

3 4 5 6 7

Key: 1 |2 = 12 820

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Answers

iii The mean, median and range will decrease, but the mode will stay the same. iv

9.07 Outliers What do you remember? 1 An outlier is a data point that varies signiﬁcantly from the body of the data. An outlier will be a value that is either signiﬁcantly larger or smaller than other observations. 2 a 1

b 99

e 10

3 a 4.875

c 275

d 4

4 a The mode does not change. b The range always decreases. 5 a Larger

b Smaller

d Smaller

Let’s practice 6 a 13.875 kg b 14.25 kg 7 a 18 kg

c 16.67 kg

b 14 kg

b Lower

Range:

1.9 ii 1500

4750

Mode:

4700

Range:

3800

Mean:

4877.8

Median:

4800

Mode:

4700

Range:

800

d 14.2 kg c 18 kg

d T he mode are the same before and after removing the outlier. 8 a 18 kg

0.9

iii The mean and median will increase, the range will decrease, but the mode will stay the same. iv

c Larger

1.5

Mode:

Median:

d 5.1

c 5

1.61

c i Mean: 4540

11.5

b 5

Mean: Median:

c 13.3 kg

12 The mean will be higher, but the median will stay the same. 13 a T he mean will be lower since the outlier is larger than the rest of the values.

9 a i Increase

ii Stay the same

iii Stay the same

iv Increase

b T he median will be unchanged since the middle of a large group of data points is mostly unchanged when extreme values are removed.

b i Increase

ii Increase

c The mode of the data set is unchanged.

iii Stay the same

iv Decrease

d T he range will decrease since the largest data value is being removed.

c i Decrease

ii Stay the same

iii Stay the same

iv Decrease

d i Decrease

ii Stay the same

iii Stay the same

iv Increase

10 a Yes

c Yes

b No

11 a i Mean: 45 Median:

14 a 56 d Yes ii 126

Mode:

Range:

102

iii The mean, median and range of the data will decrease, but the mode will stay the same. iv

Mean:

Median:

Mode:

Range:

Let’s extend our thinking

b T he mean is signiﬁcantly higher than the median due to the inﬂuence of the unusually high data point (56), showcasing how an outlier can skew the mean while the median remains less affected. This might suggest to Tanya that while the 56 species week was exceptional, typically her weekly spotting is around 12.5 species. 15 a The original mean is 80 and the original range is 16. b Answers may vary. Students may choose a high or a low outlier value that would still make sense for the given context of test scores. For example, a student could choose an outlier of 100. c Answers may vary.

b i Mean: 1.92 Median:

1.7

Mode:

0.9

Range:

3.8

ii 4.7

If a student had chosen an outlier of 100 the new mean would be 83 and the new range would be 28.

Answers mathspace.co

821

16 a Example answer: The mode. Since most people get paid $58 000, the owner might think it is the most representative of the company’s annual pay rate. b Example answer: The mean. If Ted wants to get a raise, the mean will show the largest difference between the average pay rate and his pay rate. c Example answer: The median. A mathematician would recognize that $400 000 is an outlier and know that outliers affect the mean but do not always affect the median.

822

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

17 She should use the mean since it is the higher measure of average because it is affected by the outlier. 18 The median would best represent the data as there is an outlier at 21 °C. The mean is not the best option, as outliers greatly affect the result. The mode is also not the best option, as there are two modes. 19 The median or the mode. The mean would be affected by the outlier, so the median or mode would be a better representation of the centre of the data.

Topic 9 Assessment: Statistics 1

A science class wants to answer the question: Do different types of soil affect the growth rate of bean plants? a

Which of the following attributes are needed to answer the question?

• Type of soil • Amount of water • Light exposure • Growth rate

Which of the following units could be used to measure the bean plants?

• Inches • Centimeters • Liters • Grams

Thomas wants to know whether online shopping or in-store shopping is more popular. He walks around a shopping mall and chooses 30 customers in the store for the survey. State whether the sample is representative of the population. Explain your answer.

SOL

Marco wants to investigate how many books his classmates read each year. a

Formulate a question to help him complete this investigation.

What attributes would you need to measure to answer the question?

How can Marco ensure that his data is representative of a larger population?

Albert collected data on the types of clubs 6th grade students would choose to join. He asked 40 students and created a circle graph. Use the graph Albert made to answer the following questions. a

How many students chose the sport club? Explain how you know.

How many more students chose the sport club than the science club? Explain how you know.

How many students chose the art club? Explain how you know.

6th grader’s club choices

25% 50% 12.5% 12.5% Sport club

Art club

Science club

Music Club

The Cozy Bakery sold 400 pies during their weekend bake sale. Pie Flavor Apple Pumpkin Cherry

Number of Pies Sold 200 120 80

Create and label a circle graph to represent the data for the number of pies sold.

Topic 9 Assessment: Statistics mathspace.co

823

SOL

Eunice asked her friends to name their favorite type of cake. She organized the data into a circle graph and pictograph. Use the graphs to answer the questions. Favorite type of cake

Favorite type of cake Red velvet

14% 40%

Chocolate Carrot

22%

Vanilla

10% 14% Chocolate Carrot Vanilla

SOL

Lemon = 2 votes

Red velvet Lemon

How many of her friends chose red velvet as their favorite cake? Which graph did you use to determine your answer?

What type of cake did the most of her friends choose as their favorite? Which graph did you use to determine your answer?

What percent of her friends chose lemon as their favorite? Which graph did you use to determine your answer?

Which graph better helps you understand how many of her friends chose each type of cake?

Which of the following line plots have a balance point of 21? A

17 18 19 20 21 22 23 24 25

SOL

Arthur goes for a walk every day and records the minutes in his log. Monday 40

17 18 19 20 21 22 23 24 25

Tuesday 40

Wednesday 50

Thursday 30

Friday 40

What is the average or mean number of minutes for these ﬁve days?

If Arthur instead skipped his walk on Friday, what would be his average for the ﬁve days?

How does skipping one day affect his average?

If Arthur had really enjoyed his walk, and instead walked for 80 minutes on Friday, what would be his average for the ﬁve days?

How does walking twice as long on Friday affect his average?

Consider the following set of data. 45, 52, 17, 63, 57, 42, 54, 58, 45, 50

824

Identify the outlier.

Find the mean, median, and mode with the outlier.

Find the mean, median, and mode without the outlier.

Describe the effect of the outlier on the mean, median, and mode.

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

SOL

For each of the following scenarios, determine whether the outlier that was removed must have had a value smaller or larger than the values that remain: a

A set of data has an outlier removed and the mean lowers.

A set of data has an outlier removed and the mean rises.

A set of data has an outlier removed and the median lowers.

A set of data has an outlier removed and the median rises.

Clyde’s teacher gave him the circle graph that represents 80 students’ choices for their favorite movie genre. Clyde made the bar graph to represent the same data. Use the graphs to answer the questions. Student’s favorite movie genre Number of students

60 50 40 30 20 10 0 Fantasy Drama

SOL

Horror Comedy

Student’s favorite movie genre

Fantasy Comedy Drama Horror Genre

What do you notice about the bar graph in comparison to the circle graph Clyde made?

Has Clyde correctly made the bar graph from the circle graph? If not, what should he change?

Write a survey question that could have been asked to produce the type of data at the two pie charts. 16.6% 33.3%

33.3%

25%

12.5%

50%

Performance task 13

Ike is planning his birthday party and is having a “Level 12 Unlocked” video game theme. He has been to a few birthday parties with video game themes and a few with unicorn themes. He is curious about what other themes are popular too. Go through the whole data cycle at least once using a context that can be represented with a circle graph and involves event themes.

Topic 9 Assessment: Statistics mathspace.co

825

Answers Topic 9 Assessment: Statistics 1 a Type of soil and Growth rate b Inches or Centimeters

d T he pictograph shows how many her friends chose each type of cake. 6.PS.1e, 6.PS.1f 7 B 6.PS.2a

6.PS.1b 8 a 40 minutes 2 This sample is not representative of the population. The population is anyone who shops, but the sample is only people who are shopping in person. People at the mall are more likely to prefer in-store shopping since those who prefer online shopping would tend to shop at home. 6.PS.1c

b 32 minutes c S kipping a walk on Friday will lower the average by 8 minutes. d 48 minutes e A dditional walking will raise the original average by 8 minutes. 6.PS.2b

3 a W hat is the typical number of books students read per year? b W e would need to measure the number of books students read each year.

9 a 17 b Mean: 48.3

Median: 51

c A nswers vary. Answer should include a wide sample from various types of students.

Mode: 45

6.PS.1a, 6.PS.1b, 6.PS.1c

Median: 52

Mode: 45

4 a 2 0 students chose the sport club. We know this because half the circle represents students who chose the sport club and half of 40 is 20. b 1 0 more students chose the sport club. We know this because a quarter of the circle represents students who chose the science club and a quarter of 40 is 10 and there were 10 more students who chose the sport club in part (a). c 5 students chose the art club. We know this because one eighth of the circle represents students who chose the art club and one eight of 40 is 5. 6.PS.1e 5

Pies sold

c Mean: 51.8

d T he outlier, when removed, reduces the mean by 3.5, drops the median by 1, but doesn’t affect the mode. 6.PS.2c 10 a Larger

b Smaller

c Larger

d Smaller

6.PS.2c 11 a T he bar graph shows number of students and the circle graph shows fractions. The bar graph represents more than the 80 total students. b C lyde did not make the graph correctly. His fractions in each sport are correct, but they represent the wrong number of students. Clyde should recalculate the numbers based on 80 total students. 6.PS.1e, 6.PS.1f

20% 50% 30%

12 Students were surveyed with the question “What is your favorite genre of book?” One pie chart represents the response from male students and the other one represents females. 6.PS.1a

Apple pie Cherry pie

Pumpkin pie

6.PS.1d 6 a E leven friends chose red velvet as their favorite. Used the pictograph. b M ost of her friends chose chocolate as their favorite. Either graph can be used. c 1 0% of her friends chose lemon as their favorite. Used the circle graph.

826

Mathspace Virginia SOL Grade 6 Teacher Edition mathspace.co

Performance task 13 We should go through each of the four stages and possibly do a second cycle. We may need to group themes if they are all unique. 1. Formulate questions There are many possible question such as “What baby shower themes are there and how does their popularity vary?” or “What type of holiday parties are preferred by students at my school?”

Let’s look at the question: “What birthday party themes are popular for kids 12 and under in my area?”

Birthday party themes

2. Collect or acquire data A survey is the best method for this question since it involves people’s opinions. We might be able to acquire secondary data from a source like a party planner or online sales of decorations, but they likely won’t be speciﬁc to my area. Using observation, measurements, or experiments does not make sense for this scenario.

5 (19.2%)

We could write the survey question “What theme was your or your child’s last birthday party?” We could get data that looks like this: Name

Age

Theme

Antwan

Soccer theme

Group Sport

Brandy

No theme

None

Celeste

Unicorn theme

Magical

Fairy theme

Magical

Football theme

Sport

Feng

Cow theme

Animal

Gabino

Gymnastics theme

Sport

Herman

Batman theme

Character

Spiderman theme

Character

Bluey theme

Character

Kelley

Ninja warrior theme

Sport

Lena

Unicorn theme

Magical

Manjit

No theme

None

Nolan

Character theme

Character

Ophelia

Unicorn theme

Magical

Preston

Ninja turtle theme

Character

Qin

Minecraft theme

Video game Magical

Rochelle

Princess theme

Samson

Laser tag theme

Sport

Thuy

Bunny theme

Animal

Urbana

No theme

None

Vijay

Farm theme

Animal

Webster

No theme

None

Xavier

Gamer theme

Video game

Yen

Fortnite theme

Video game

Zachariah

Basketball theme

Sport

5 (19.2%)

Animal

Video game

None

Magical

Superhero

Sport

Birthday party themes

4 (15.4%)

Desi

Indu

3 (11.5%)

We can also represent this data in a bar graph which allows us to see the number in each category more clearly.

Eung

Judd

3 (11.5%)

6 (23.1%)

Animal

Video game

None

Magical Superhero Sport

Number of kids 12 and under

4. Analyze data and communicate results From the circle graph and table, we can see that almost a quarter of kids 12 and under in my area prefer a sports themed birthday, which was the most popular theme. Close to 10% preferred either an animal or video game theme. This was a very small data set of only 26 people, so for more reliable results, we should use a bigger sample so that the sample would look more like the population. This might lead up to formulate another question like “Do birthday party theme vary by age?” where we split the data out by 6 and under versus 7 to 12.

3. Organize and represent data We can organize into a table ﬁrst and then use technology to make the circle graph. Theme

Number of people

Animals

Video games

None

Magical

Superhero

Sport

Answers mathspace.co

827