EM2_G5_M1_TE_23B_971796_Updated 05.23

5 A Story of Units® Fractions Are Numbers

TEACH ▸ Module 1 ▸ Place Value Concepts for Multiplication and Division with Whole Numbers

What does this painting have to do with math?

Color and music fascinated Wassily Kandinsky, an abstract painter and trained musician in piano and cello. Some of his paintings appear to be “composed” in a way that helps us see the art as a musical composition. In math, we compose and decompose numbers to help us become more familiar with the number system. When you look at a number, can you see the parts that make up the total?

On the cover

Thirteen Rectangles, 1930

Wassily Kandinsky, Russian, 1866–1944

Oil on cardboard

Musée des Beaux-Arts, Nantes, France

Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musée des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/ Art Resource, NY

Great Minds® is the creator of Eureka Math® , Wit & Wisdom® , Alexandria Plan™, and PhD Science®

Published by Great Minds PBC. greatminds.org

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ISBN 978-1-64497-179-6

Module

Fractions

1 Place Value Concepts for Multiplication and Division with Whole Numbers

2 Addition and Subtraction with Fractions

3 Multiplication and Division with Fractions

4 Place Value Concepts for Decimal Operations

5 Addition and Multiplication with Area and Volume

6 Foundations to Geometry in the Coordinate Plane

Before This Module

Overview

Grade 4 Module 1

Students read, write, compare, and round multi-digit whole numbers to millions in standard, expanded, word, and unit forms. They describe the relationship between a digit in one place and the digit in the next larger place by using the multiplicative comparison 10 times as much as.

Grade 4 Modules 2 and 3

Students multiply and divide whole numbers of up to four digits by one-digit numbers (including expressing quotients with whole-number remainders) and multiply 2 two-digit numbers. They use methods based on the place value chart, area models, the commutative and associative properties of multiplication, and the distributive property.

Place Value Concepts for Multiplication and Division with Whole Numbers

Topic A

Place Value Understanding for Whole Numbers

Students use multiplicative comparison statements to explain that a digit in one place represents 10 times as much as what it represents in the place to the right. Students notice how digits of a number shift when they multiply or divide by a power of 10 and express a power of 10 in exponential form. Then students find products and quotients by using powers of 10 and convert metric measurements from larger to smaller units.

Topic B

Multiplication of Whole Numbers

Students build fluency with multiplying multi-digit numbers by using the standard algorithm. They use place value understanding to visualize the decomposition of factors while they multiply a single digit at a time by another single digit in the standard algorithm.

Topic C Division of Whole Numbers

Students use methods based on place value to find quotients of whole numbers with up to four-digit dividends and two-digit divisors. They estimate quotients, then use tape diagrams, area models, and vertical form to record quotients and remainders.

Topic D

Multi-Step Problems with Whole Numbers

Students move between written, pictorial, and numeric representations of mathematical statements. They use tape diagrams to determine when parentheses are needed in expressions and evaluate expressions containing grouping symbols. 3 times the sum of 15 and 25

There are 26 people at the park. 8 people go home. The rest of the people make 2 equal groups to play a game. How many people are in each group?

After This Module

Grade 5 Module 4

Students use place value knowledge and times as much as language to learn about decimal numbers. Students see how the strategies they use for whole-number operations extend to operations with decimal numbers. They convert metric measurements from smaller units to larger units.

Grade 6 Modules 2 and 4

There are 9 people in each group.

In module 2, students learn to divide whole numbers with any number of digits by using the standard algorithm. In module 4, students build upon grade 5 knowledge by writing and evaluating numerical expressions with terms that have whole-number bases and exponents.

to multiply and divide by powers of 10

Estimate products and quotients by using powers of 10 and their multiples.

Convert measurements and describe relationships between metric units.

Solve multi-step word problems by using metric measurement conversion.

Multiplication of Whole Numbers

Lesson 8

Multiply two- and three-digit numbers by two-digit numbers by using the distributive property.

Lesson 9

Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm.

Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm.

Multiply two multi-digit numbers by using the standard algorithm.

Division of Whole Numbers

Divide two-digit numbers by two-digit numbers in problems that result in one-digit quotients. Lesson 14

Divide three-digit numbers by two-digit numbers in problems that result in one-digit quotients. Lesson 15

Divide three-digit numbers by two-digit numbers in problems that result in two-digit quotients.

Divide four-digit numbers by two-digit numbers.

Multi-Step Problems with Whole Numbers

Write, interpret, and compare numerical expressions.

Create and solve real-world problems for given numerical expressions.

multi-step word problems involving multiplication and division.

Why Place Value Concepts for Multiplication and Division with Whole Numbers

Why does multiplication and division of whole numbers come first?

After much consideration of our students’ learning, teachers’ input, and a review of the research around how students learn and how mathematical concepts progress, we decided it makes the most sense to put place value concepts and operations with whole numbers first. Why?

1. The major emphasis of grade 5 standards involves understanding the place value system, performing operations with multi-digit whole numbers, and applying and extending knowledge of whole-number operations to fractions and decimals.

Beginning the year with a focus on place value and whole-number operations sets up students for success as they move into operations with fractions in modules 2 and 3, then with decimals in module 4.

2. Beginning the year with learning how to multiply multi-digit numbers provides an opportunity for students to develop fluency with using the standard algorithm throughout the year, as required by the standards.

3. Multiplying and dividing multi-digits numbers gives rise to developing estimation skills and to introducing powers of 10 in a meaningful way. Powers of 10 are not just the numbers on a place value chart, rather they are powerful tools for making estimates of products and quotients and for checking the reasonableness of answers.

When do students learn about decimals? Why?

Grade 5 module 4 addresses work with decimals and parallels the content of module 1. Students begin by relating adjacent place value units and use comparison language, such as 1 tenth is 10 times as much as 1 hundredth, just as they did with whole numbers. It makes sense mathematically to position decimals in module 4 after an in-depth study of whole numbers in module 1 and then fractions in modules 2 and 3. This move also makes sense pedagogically because students can use 1 10 , 1 100 , and 1 1,000 to describe relationships between numbers on the place value chart and to perform operations on decimals.

I notice students only convert from larger metric units to smaller metric units in this module. Why?

Metric conversions are limited to moving from larger units (such as kilometers) to smaller units (such as meters) in module 1 because conversions that move from smaller to larger units are best performed by using fractions or decimals. Students learn to multiply fractions in module 3 and they learn to multiply decimals in module 4. The remaining part of the metric conversion standard is fully met in module 4 as an application of decimals.

Achievement Descriptors: Overview

Place Value Concepts for Multiplication and Division with Whole Numbers

Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module.

Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year.

ADs and their proficiency indicators support teachers with interpreting student work on

• informal classroom observations,

• data from other lesson-embedded formative assessments,

• Exit Tickets,

• Topic Quizzes, and

• Module Assessments.

This module contains the 12 ADs listed.

5.Mod1.AD1

Write whole-number numerical expressions with parentheses.

5.Mod1.AD2

Evaluate whole-number numerical expressions with parentheses.

5.OA.A.1

5.Mod1.AD5

Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers.

5.NBT

5.Mod1.AD9

Multiply two multi-digit whole numbers by using the standard algorithm.

5.Mod1.AD6

5.OA.A.1

Explain the relationship between digits in multi-digit whole numbers.

5.Mod1.AD3

Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions.

5.OA.A.2

5.Mod1.AD7

Explain the effect of multiplying and dividing whole numbers by powers of 10.

5.Mod1.AD4

Compare the effect of each number and operation on the value of a whole-number numerical expression.

5.OA.A.2

5.Mod1.AD8

Express whole-number powers of 10 in exponential form, standard form, and as repeated multiplication.

5.NBT.B.5

5.Mod1.AD10

Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

5.Mod1.AD11

Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models.

5.NBT.B.6

5.Mod1.AD12

Convert among whole-number amounts within the metric measurement system to solve problems.

5.MD.A.1

The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance.

5.NBT.A.1

5.NBT.A.2

5.NBT.A.2

5.NBT.B.6

An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.

ADs have the following parts:

• AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 5 module 1 is coded as 5.Mod1.AD1.

• AD Language: The language is crafted from standards and concisely describes what will be assessed.

• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.

• Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models.

RELATED CCSSM

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Partially Proficient Proficient

Determine the quotient for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using a provided model.

Use the model shown to help you divide.

Create models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

Use the expression to answer part A and part B.

4,102 ÷ 14

The quotient is

Part A

Draw a model for the expression.

Part B

Use your model to determine the quotient and remainder.

Highly Proficient

Interpret models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

What values could be represented by the letters in the model? Explain your thinking.

429 Copyright © Great Minds PBC

Topic A Place Value Understanding for Whole Numbers

In topic A, students apply their understanding of place value to multiply and divide by powers of 10 and their multiples.

Prior to grade 5, students use place value understanding to round multi-digit whole numbers to any place. They compare quantities through multiplicative comparison and recognize that in a whole number, a digit in one place represents 10 times as much as what it represents in the place to the right.

The topic opens with students using place value charts to show that when two adjacent digits in a given number are the same, the digit to the left is 10 times as much as the digit to the right and the digit to the right is 10 times as small as the digit to the left. Students use dot models to understand what happens when they multiply or divide a number by 10 . Next, students apply what they learn from the dot models to conclude that when they multiply a number by 10 , it causes each digit of the number to shift one place value to the left, and when they divide a number by 10 , it causes each digit of the number to shift one place value to the right. Building on this understanding, they notice how the digits shift when they multiply or divide a number by 100 and by 1,000 .

Students find products and quotients of expressions composed only of powers of 10, such as 10,000 × 100, by using what they learn about how digits in a number shift. When students find products and quotients of expressions composed only of 10s, it gives rise to learning about exponents with a base of 10. They write powers of 10 in standard form, expanded form, and exponential form. Students extend their understanding of the shifts they notice when they multiply or divide by 10 to multiplying and dividing by 102 or 103.

Students estimate products and quotients of multi-digit numbers by rounding factors, divisors, and dividends to multiples of powers of 10 . By comparing estimates and analyzing estimation strategies, they understand what may cause an underestimate or an overestimate. Students then estimate products and quotients in real-world situations. The topic culminates with students using observations about how digits shift when they multiply by powers of 10 to convert metric measurements.

By combining multiplicative comparison language with their understanding of powers of 10 , students describe relative sizes of units of metric length, weight, and capacity. They convert between units and express larger units in terms of smaller units by using powers of 10 . Students solve multi-step word problems involving metric conversions and apply their estimation skills from previous lessons to determine whether answers are reasonable.

In topic B, students apply their understanding of place value to multiply multi-digit whole numbers.

Progression of Lessons

Lesson 1

Relate adjacent place value units by using place value understanding.

Lesson 2

Multiply and divide by 10, 100, and 1,000 and identify patterns in the products and quotients.

Lesson 3

Use exponents to multiply and divide by powers of 10.

I can represent multiplication and division by 10 on a place value chart. I notice when two adjacent digits are the same number, the digit to the left is 10 times as much as the digit to the right and the digit to the right is 10 times as small as the digit to the left.

When I multiply a number by 10, 100, or 1,000, the digits shift to the left. When I divide a number by 10, 100, or 1,000, the digits shift to the right. For example, if I multiply 4 tens by 1,000, the 4 shifts three units to the left, which is 4 ten thousands, or 40,000. If I divide 4 thousands by 100, the 4 shifts two units to the right, which is 4 tens, or 40.

I can write powers of 10 in standard form and exponential form. I can use what I know about how many 10s are in a number to efficiently multiply or divide by shifting digits to the left or the right.

Lesson 4

Estimate products and quotients by using powers of 10 and their multiples.

129 ÷ 4 ≈ 12 0 ÷ 4 = 30

I can estimate products and quotients by rounding numbers to multiples of 10. For example, I can estimate the product of 47 and 61 by finding 50 × 60. I can estimate the quotient of 316 and 45 by finding 300 ÷ 50.

Lesson 5

Convert measurements and describe relationships between metric units.

kilometer, meter, centimeter, millimeter longestshortest

I can convert larger metric units to smaller metric units by using multiplication. I can use prefixes to remind me of the relationship between metric units.

Lesson 6

Solve multi-step word problems by using metric measurement conversion.

6 m 40 cm or 64 0 cm

80 cm

I can use the Read–Draw–Write process to make sense of and to solve word problems. Models help me see different ways to solve a problem. I can solve word problems that have different metric units by converting larger units to smaller units.

Relate adjacent place value units by using place value understanding.

Lesson at a Glance

With a partner, students organize and count a collection of bills that requires them to use their place value understanding. Students model 10 times as much for each unit on the place value chart up to 1 million and determine that when two adjacent digits are the same, the digit to the left is 10 times as much as the digit to the right. Then students model division by 10 on the place value chart and find that when two adjacent digits are the same, the digit to the right is 10 times as small as the digit to the left. They compare the same digit in different places and describe the relationship between the numbers by using what they know about multiplication and division. This lesson introduces the academic verb consider.

Key Question

• How are place value units related to each other?

Achievement Descriptor

5.Mod1.AD6 Explain the relationship between digits in multi-digit whole numbers. (5.NBT.A.1)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Organize and Count Bills to Compare

• Compare and Relate the Same Digit with Different Values

• Problem Set

Land 10 min

Teacher

• Computer or device*

• Projection device*

• Teach book*

• Money Counting Collection (in the teacher edition)

• Place Value Chart to Millions (in the teacher edition)

Students

• Dry-erase marker*

• Learn book*

• Pencil*

• Personal whiteboard*

• Personal whiteboard eraser*

• Organizational tools

• Place Value Chart to Millions (in the student book)

* These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

Lesson Preparation

• Print or copy Money Counting Collection and cut out the collections of paper money. Prepare one collection per student pair.

• Consider whether to remove Place Value Chart to Millions from the student books in advance or have students remove them during the lesson.

• Provide tools for students to choose from to help organize their counts. Tools may include cups, paper clips, whiteboards, bags, rubber bands, or graph paper.

Fluency

Choral Response: Rename Place Value Units

Students use unit form to identify a number modeled with place value disks, and then compose and rename to prepare for relating adjacent place value units.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display the 10 ones disks on the chart.

What value is represented on the chart? Say the answer in unit form.

10 ones

Display 10 ones = ten.

10 ones is equal to how many tens?

1 ten

10 ones = ten 1

Teacher Note

Use hand signals to introduce a procedure for answering choral response questions. For example, cup your hand around your ear for listen, lift your finger to your temple for think, and raise your own hand to remind students to raise theirs.

Teach the procedure by using the following general knowledge questions:

• What grade are you in?

• What is the name of our school?

• What is your teacher’s name?

Differentiation: Support

Consider having place value disks available during the activity for students who need additional support.

Display the answer and the disks bundled as a ten on the chart.

Continue the process with the following sequence:

Whiteboard Exchange: Place Value

Students identify a place value and the value of a digit in a multi-digit number, and then write the number in expanded form to prepare for relating adjacent place value units.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 2,518.

When I give the signal, read the number shown. Ready?

2,518

What digit is in the thousands place?

2

2,518

2,000 + 50 0 + 10 + 8

Teacher Note

Establish a signal (e.g., show me your whiteboards) to introduce a procedure for showing whiteboard exchange responses. Practice with basic computations such as the following until students are accustomed to the procedure:

• What is 10 + 8?

• What is 500 + 18?

Establish a procedure for providing feedback on whiteboard exchanges. Consider circulating and giving hand signals—thumbs-up or try again.

Display the underlined 2.

What value does the 2 represent in this number?

2,000

Write 2,518 in expanded form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the number in expanded form.

Continue the process with the following sequence:

Launch

Students convert among different measurements and analyze their multiplicative relationships.

Introduce the Which One Doesn’t Belong? routine. Present four statements and invite students to study them.

A

1 foot = 12 inches

C 1 L = 1,000 mL

B

1 meter is the same length as 100 centimeters.

D

1,000 grams = 1 kilogram

Teacher Note

Consider asking students to express each number in expanded form differently. For example, ask students to use only addition for some numbers and incorporate multiplication for others as in the following examples:

Language Support

Consider using strategic, flexible grouping throughout the module.

• Pair students who have different levels of mathematical proficiency.

• Pair students who have different levels of English language proficiency.

• Join pairs to form small groups of four.

As applicable, complement any of these groupings by pairing students who speak the same native language.

Give students 2 minutes to find a category in which three of the items belong, but a fourth item does not.

When time is up, invite students to explain their chosen categories and to justify why one item does not fit.

Highlight responses that emphasize reasoning about the factors and multiples of 10 among the metric units.

Ask questions that invite students to use precise language, make connections, and ask questions of their own.

Use the following sample questions and prompts.

Which one doesn’t belong?

A doesn’t belong because it is the only choice that does not use metric units.

B doesn’t belong because it is the only one that uses words instead of an equal sign.

C doesn’t belong because it is the only one with abbreviated units.

D doesn’t belong because it is the only choice where 1 unit is to the right of the equal sign.

Complete this statement: × 1 mL = 1 L.

1,000 × 1 mL = 1 L

1 liter is 1,000 times as much as 1 milliliter.

Complete this statement: 1 meter = × 1 centimeter.

1 meter = 100 × 1 centimeter

1 meter is 100 times as much as 1 centimeter.

Complete this statement: 1 kilogram = × 1 gram.

1 kilogram = 1,000 × 1 gram

1 kilogram is 1,000 times as much as 1 gram.

We expressed each relationship by using multiplication.

Transition to the next segment by framing the work.

Today, we will use our place value understanding to describe the relationship between place value units by using multiplication and division.

Learn

Organize and Count Bills to Compare

Materials—S: Money counting collection, organizational tools

Students use self-selected strategies to organize and count a collection and record their process.

Partner students and distribute a counting collection to each pair.

Direct students to the recording page in their books. Briefly orient students to the materials and procedure for the counting collection activity:

• Partners collaborate to count a collection.

• Partners make their own recordings to show how they counted.

• Partners may use the place value chart and other organizational tools. Organizational tools may include readily available classroom items such as cups, paper clips, personal whiteboards, etc.

Before they begin, invite partners to work together to estimate how many dollars are in their collections. Have them write down their estimates. Then encourage them to talk about how they will organize their collections to count.

Promoting the Standards for Mathematical Practice

Students use appropriate tools strategically (MP5) as they discuss and self-select counting strategies and organizational tools for counting their collection.

Ask the following questions to promote MP5:

• What strategies or tools can help you count your collection?

• Which tool would be the most helpful to count your collection? Why?

• Why did you choose this strategy to count your collection? Did your strategy work well?

Then invite students to select organizational tools they would like to use, with the understanding that tools may be exchanged as plans are refined.

Ask partners to begin counting their collections. Circulate and notice how students engage in the following behaviors:

Organize: Strategies may include grouping bills of the same unit, making groups of 10 of the same unit, organizing bills on the place value chart, and writing expressions or equations. Students may also organize their collections by using attributes that do not support counting efficiently, such as mixing units to make equal groups of bills.

Count: Students may count subgroups and then add to find the total, or they may use a place value chart and write the digits that represent the number of each unit. Other students may use a combination of multiplication and addition to find the total.

Record: Recordings may include drawings, numbers, expressions, equations, and written explanations.

Use questions and prompts such as the following to assess and advance student thinking as they organize and count their collection:

• Show and tell me what you did.

• How can you organize your collection to make it easier for you to count?

• How does the way you organized your collection make it easier for you to count?

• How did you keep track of what you already counted and what you still needed to count?

• How did you name the larger units? Why?

• How did you know how to write your total?

• How close was your estimate to your actual count?

Teacher Note

The counting collections vary in levels of complexity. Partner students and strategically assign each pair a counting collection.

• Counting Collection A does not require composing units.

• Counting Collection B requires composing units in one place value.

• Counting Collection C requires composing units in two place values.

• Counting Collection D requires composing units in three place values.

UDL: Action & Expression

Consider offering sticky notes for labeling to support students in organizing their collections. For example, if students organize their bills like a place value chart, they can use the sticky notes to label each place. This allows flexibility as students organize and keep track of their count.

For this counting collection, I am partners with .

We are counting .

We think they have a value of .

This is how we organized and counted the collection:

We counted altogether.

An equation that describes how we counted is: .

Self-Reflection

Write one thing that worked well for you and your partner. Explain why it worked well. It was helpful to bundle when we had 10 of a unit because then we could rename it as the next largest unit. That helped us find the total.

Write one challenge you had. How did you work through the challenge?

We were not sure what some of the place value units were. We used the numbers on the bills to help us.

Gather the class and facilitate a brief discussion about how students chose to organize and count the bills.

How did you organize your bills?

We put like units together.

We put our bills into groups of 10.

We organized the bills like a place value chart.

How did you find the total?

We skip-counted by each unit.

We bundled to make larger units when we could. We found the total by writing how many of each unit we had.

We counted how many bills of each unit we had. Then we multiplied to find the amount for each unit. We added the amounts for each unit to find the total.

How did you decide when to compose a larger unit?

When we had 10 of a smaller unit, we composed them to make 1 of the next larger unit.

When we had a group of 10, we bundled it with a paper clip. Then we placed the bundle into the next larger unit on our chart.

When we had 10 thousands, we bundled them to make 1 ten thousand.

Invite each group to share the total amount of money in the collection they counted. Record the totals so they can be referred to later in the lesson.

Compare and Relate the Same Digit with Different Values

Materials—T/S: Place Value Chart to Millions

Students determine that the same digits in different places do not represent the same value and articulate how the digits in different place values are similar and different.

Direct students to Place Value Chart to Millions in their books. Have students remove the chart and insert it into their whiteboards.

Ask them to write 1,731,225 in standard form as you do the same.

Underline the 2 in the hundreds place and the 2 in the tens place. Point to them as you ask the following questions.

Do these 2s represent the same amount?

No, they represent different amounts.

Let’s write the number in expanded form so we can see more clearly how much each 2 represents.

Direct students to write 1,731,225 in expanded form as you do the same.

Gesture to the 2 in the hundreds place.

How much does this 2 represent?

200

Gesture to the 2 in the tens place.

How much does this 2 represent?

20

Gesture to the 2 in the hundreds place.

The first 2 represents 200.

Gesture to the 2 in the tens place.

The other 2 represents 20. Consider, or think about, how 2 hundreds is similar to or different from 2 tens.

Pause to allow students time to think, then invite students to respond.

Both show 2 of a unit.

2 hundreds is greater than 2 tens.

Invite students to turn and talk about whether they would rather have 2 hundred-dollar bills or 2 ten-dollar bills and why.

Direct students to show 2 tens on the place value chart.

Let’s think some more about the relationship between 2 tens and 2 hundreds. What do we need to multiply 2 tens by to get 2 hundreds? 10

Using Place Value Chart to Millions, draw two dots in the tens column. Draw an arrow, labeled × 10 from the 2 tens in the tens place to the hundreds place and draw 2 hundreds.

Display the comparison statement:

200 is  times as much as 20.

Complete the statement: 200 is times as much as 20.

200 is 10 times as much as 20.

Record 200 = 10 × 20 and direct students to do the same.

Direct students to erase.

Let’s show the relationship between 200 and 20 by using division.

Direct students to draw 2 hundreds.

Write 200 ÷ = 20. Gesture to the statement 200 = 10 × 20.

We know that 200 is 10 times as much as 20. Let’s use that to complete the statement:

200 ÷ = 20.

200 ÷ 10 = 20

Language Support

This segment introduces the term consider. Consider previewing the meaning of the term before students are asked to consider how the numbers are similar. Relate the term to thinking about the weather as they decide what to wear or thinking about reasons for choosing a recess activity.

Teacher Note

The digital interactive Place Value Chart helps students represent and compare the sizes of numbers.

Consider allowing students to experiment with the tool individually or demonstrate it for the whole class.

Draw two dots in the hundreds column. Draw an arrow, labeled ÷ 10 from the 2 hundreds in the hundreds place to the tens place and draw 2 tens.

Write the comparison statement: 20 is times as small as 200.

Complete the statement: 20 is times as small as 200. 20 is 10 times as small as 200.

When we have the same digit in adjacent places, or right next to each other, the digit on the left is 10 times as much as the digit on the right.

Let’s look at other relationships between digits in this number.

Circle the 1 in the millions place and the 1 in the thousands place.

Consider, or think about, how 1 million is similar to or different from 1 thousand.

Invite students to think–pair–share to compare the two digits. Both show 1 of a unit.

1 million is greater than 1 thousand. The 1s are in different places.

1 thousand is 10 hundreds. 1 million is 10 hundred thousands.

Direct students to the expanded form recording.

Gesture to the 1 in the millions place.

How much does this 1 represent? 1,000,000

Gesture to the 1 in the thousands place.

How much does this 1 represent?

1,000

Language Support

Consider reviewing the familiar term adjacent with students. Adjacent angles are angles that are next to each other and share a side. Angles that are nonadjacent do not share a side. Make connections to place value by discussing which places are next to each other and which are not. Highlight that the prefix non- means not to help students understand that nonadjacent means not adjacent. Create a visual you can use to highlight examples of adjacent and nonadjacent digits, such as in the following chart:

Is 1 million 10 times as much as 1 thousand? Why?

No, because the millions place is not adjacent to the thousands place. 10 times as much as 1 thousand is 1 ten thousand, not 1 million.

Let’s see how many times as much 1 million is as 1 thousand.

Draw 1 thousand on the place value chart and multiply by 10 (by using the arrow to show movement), until you reach 1 million. Label each arrow × 10.

How many times do we have to multiply by 10 to get from 1,000 to 1,000,000?

We have to multiply by 10 three times.

What is the value of 10 × 10 × 10?

1,000

Complete this statement: 1,000,000 is times as much as . 1,000,000 is 1,000 times as much as 1,000.

Record 1,000,000 = 1,000 × 1,000 and direct students to do the same.

Invite students to think–pair–share about how digits that are the same and in adjacent places are similar to or different from digits that are the same but not in adjacent places.

A digit that is the same as a digit in an adjacent place is 10 times as much as the same digit directly to its right.

Digits that are the same but not adjacent are a multiple of 10 times as much as the same digit in other place values to the right.

Let’s show the relationship between 1,000,000 and 1,000 by using division.

Direct students to draw 1 million on the place value chart while you do the same.

Write 1,000,000 ÷ = 1,000. Gesture to the statement 1,000,000 = 1,000 × 1,000.

We know that 1,000,000 is 1,000 times as much as 1,000. Let’s use that to complete this statement: 1,000,000 ÷ = 1,000.

1,000,000 ÷ 1,000 = 1,000

Differentiation: Support

Help students understand that 10 times as much as 1 thousand is 1 ten thousand by showing and bundling physical place value disks on the place value chart until their understanding of the pictorial representation is firm.

For students who need additional support, consider offering them calculators to confirm the relationship of 10 times as much and 10 times as small. Support students with entering 1 on the calculator and making the direct connection to the ones place before they begin to multiply by 10. Have students multiply by 10 and connect the tens place. Continue multiplying by 10 to millions, pointing out that each time students multiply by 10, the product is one place to the left in the place value chart. Repeat the process of dividing by 10 until students return to the ones place.

Repeatedly draw an arrow, labeled as ÷ 10, from the 1 million in the millions place to the thousands place and draw 1 thousand.

Write the comparison statement: 1,000 is times as small as .

Complete this statement: 1,000 is times as small as .

1,000 is 1,000 times as small as 1,000,000.

Invite students to turn and talk about whether they would rather have a $1,000 bill or a $1,000,000 bill and why.

Let’s see whether this works with other totals that we counted. Refer to the list of counting collection values and direct students to the number 2,988,396. Have them write the number in standard form as you do the same.

Underline the two 8s and circle the two 9s.

Use a similar sequence to guide students to describe the relationship between the 8 in the ten thousands place and the 8 in the thousands place and the relationship between the 9 in the hundred thousands place and the 9 in the tens place.

Consider using the following questions to guide students’ analysis:

• Are these two 8s equal? How do you know?

• How is 8 ten thousands similar to or different from 8 thousands?

• How is 9 hundred thousands similar to or different from 9 tens?

Gesture to the circled 9s in standard form.

If we divide 9 hundred thousands by 10, will we get 9 tens? Why?

No, we will not. The tens place is not adjacent to the hundred thousands place.

What will we get if we divide 9 hundred thousands by 10? Why?

We will get 9 ten thousands because the ten thousands place is adjacent to the hundred thousands place.

Teacher Note

Ask students to think about the relationship between the millions place and the ten millions place, and places beyond. Or ask students to think about the relationship between the tens place and the ones place. Note that the pattern continues, even as the place value units become greater or less.

Invite students to turn and talk about how they know 9 hundred thousands divided by 10 is 9 ten thousands.

Is 9 hundred thousands 10 times as much as 8 ten thousands? Why?

No, because the digits are not the same. 10 times as much as 8 ten thousands is 8 hundred thousands, not 9 hundred thousands.

Two digits that are not the same do not have the 10 times as much relationship. Display the equations.

Differentiation: Challenge

Present students with a number such as 2,458,136 and invite them to rearrange the digits to produce the number with the greatest possible value. Then ask students to choose any digit and describe its value before and after rearranging by using 10 times as much or 10 times as small language and by showing their thinking on a place value chart.

What is 100 ÷ 10?

What is 1,000 ÷ 10?

Invite students to turn and talk to predict the quotients for the remaining equations based on the pattern they see.

Invite students to think–pair–share to complete this statement: When we divide by 10, the quotient .

When we divide by 10, the quotient moves one place value unit to the right.

When we divide by 10, the quotient is 10 times as small as the dividend.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Relate adjacent place value units by using place value understanding.

Facilitate a class discussion about relating adjacent place value units by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Display the number with the digits circled and underlined.

How does 3 hundred thousands relate to 3 ten thousands?

3 hundred thousands is 10 times as much as 3 ten thousands.

Is it correct to say that 3 ten thousands is 10 times as much as 2 thousands? How do you know?

No, it is not correct. 10 times as much as 2 thousands is 20,000, not 30,000.

The digits have to be the same to be 10 times as much.

How are place value units related to each other?

There is a 10 times as much relationship from one place value unit to the next when you start at the ones place and move left.

When the digits in a number are the same and adjacent, then the digit to the left is 10 times as much as the digit next to it.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Differentiation: Support

If students need support to complete problems 6–12 in the Problem Set, model how they can continue to use their place value charts as needed. See the following example for problem 5.

Language Support

Scaffold the questions for English learners by asking them to complete the following statements:

• When I see two of the same digit in a number, I know .

• For a digit to represent 10 times as much as the next digit to its right, it must be .

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

7.

13. Consider the number shown. 8 7 7, 4 8 7

a. Complete the equation to represent the number in expanded form. 877,487 = ( 800,000 ) + ( 70,000 ) + ( 7,000 ) + ( 400 ) + ( 80 ) + ( 7 )

b. Draw a box around the digit that represents 10 times as much as the underlined digit.

c. Complete the equations to show the relationships between the boxed and underlined digits.

70,000 = 10 × 7,000

70,000 ÷ 10 = 7,000

d. Explain how the digit in the hundred thousands place is related to the digit in the tens place. 8 hundred thousands is 10,000 times as much as 8 tens.

14. Kayla and Blake both write a number. Kayla’s

a. Kayla says, “The 3 in my number is 10 times as much as the 3 in Blake’s number.” Do you agree with Kayla? Explain.

No, I do not agree with Kayla. The 3 in Blake’s number represents 3,000. The 3 in Kayla’s number represents 300,000. So the 3 in Kayla’s number represents 100 times as much as the 3 in Blake’s number, not 10 times as much. The value of the 3 in Kayla’s number is 300,000, not 30,000.

b. Write a division equation to relate the 8 in Kayla’s number to the 8 in Blake’s number. 8,000 ÷ 1,000 = 8

7.

Multiply and divide by 10 , 100 , and 1,000 and identify patterns in the products and quotients.

Lesson at a Glance

2

Students use a place value chart to multiply by 10. They notice that multiplying by 10 causes each digit to shift one place to the left. They relate multiplying by 100 or 1,000 to multiplying by 10. Students use a place value chart to divide by 10 and notice that when dividing by 10 each digit shifts one place to the right. They relate dividing by 100 or 1,000 to dividing by 10.

Key Questions

• How does multiplying or dividing by 100 or 1,000 relate to multiplying or dividing by 10?

• How is multiplying by 10, 100, and 1,000 similar to or different from dividing by 10, 100, and 1,000?

Achievement Descriptors

5.Mod1.AD6 Explain the relationship between digits in multi-digit whole numbers. (5.NBT.A.1)

5.Mod1.AD7 Explain the effect of multiplying and dividing whole numbers by powers of 10. (5.NBT.A.2)

Agenda Materials

Fluency 10 min

Launch 10 min

Learn 30 min

• Multiply by 10, 100, and 1,000

• Divide by 10, 100, and 1,000

• Problem Set

Land 10 min

Teacher

• Place Value Chart to Millions (in the teacher edition)

Students

• Place Value Chart to Millions (in the student book)

Lesson Preparation

Consider whether to remove Place Value Chart to Millions from the student books and place inside of whiteboards in advance or have students prepare them during the lesson.

Fluency

Choral Response: Rename Place Value Units

Students use unit form to identify a number modeled with place value disks and then decompose and rename to maintain place value understanding from grade 4.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display the 1 tens disk on the chart.

What value is represented on the chart? Say the answer in unit form.

1 ten

Display 1 ten = ones.

1 ten is equal to how many ones?

10 ones

Display the answer and the disk unbundled as 10 ones on the chart.

Continue the process with the following sequence:

1 ten 2 ones = 12 ones

1 hundred = 10 tens

1 hundred 4 tens = 14 tens

1 thousand = 10 hundreds

1 thousand 5 hundreds = 15 hundreds

1 ten thousand = 10 thousands

1 ten thousand 7 thousands = 17 thousands

1 hundred thousand = 10 ten thousands

1 hundred thousand 9 ten thousands = 19 ten thousands

Whiteboard Exchange: Place Value

Students identify a place value and the value of a digit in a multi-digit number and then write the number in expanded form to maintain place value understanding from grade 4.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 48,359.

When I give the signal, read the number shown. Ready?

48,359

What digit is in the ten thousands place?

4

Display the underlined 4.

What value does the 4 represent in this number?

40,000

Write 48,359 in expanded form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the number in expanded form.

Continue the process with the following sequence:

Launch

Students apply their understanding of 10 times as much to solve a problem involving 100 times as much.

Present the following problem and use the Math Chat routine to engage students in mathematical discourse.

Tara has 54 nails in her toolbox. She needs 100 times as many nails to build a tree house.

How many nails does she need?

Give students 5 minutes to discuss their thinking and to solve the problem with a partner. Allow students to self-select solution strategies. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies.

Then facilitate a class discussion. Invite students to share their thinking with the whole group and then record their reasoning.

Validate a range of ideas and support students in making connections among strategies, but focus the discussion on the following response: We know that 100 is equivalent to 10 × 10. We multiplied 54 by 10 and shifted the 5 and the 4 one place to the left. Then we multiplied by 10 one more time, so we shifted the 5 and the 4 one place to the left again. 54 × 100 = 5,400, so Tara needs 5,400 nails.

Transition to the next segment by framing the work.

Today, we will look for patterns in products and quotients when we multiply and divide by 10, 100, and 1,000 so we can do that work mentally.

UDL: Action & Expression

Consider making place value disks available for students who would benefit from a concrete representation of the problem.

If more than one set of partners would like to use place value disks to represent the problem, encourage them to work together. This teamwork will help ease the demand for many disks and will help students solve the problem in the allotted time.

Learn

Multiply by 10, 100, and 1,000

Materials—T/S: Place Value Chart to Millions

Students multiply by 10, 100, and 1,000 and notice patterns that help them multiply mentally.

Direct students to problems 1–3 in their books.

1. 5 × 10 = 50

2. 5 × 100 = 500

3. 5 × 1,000 = 5,000

Point to each problem and record the product as you ask each of the following questions.

What is 5 × 10?

50

What is 5 × 100?

500

What is 5 × 1,000?

5,000

Have students record the products for problems 1–3. Then direct students to study the problems and invite them to think–pair–share about what pattern they see.

I notice that each time I multiply by 10, there is another zero at the end of the product.

I notice that each product is 10 times as much as the previous one.

Have students remove Place Value Chart to Millions from their books and insert it into their whiteboards.

Let’s see how these patterns work for other numbers.

Teacher Note

Some students may notice that the number of zeros at the end of the product matches the number of zeros at the ends of the factors. Though that pattern of zeros does always work when multiplying whole numbers by powers of 10 , the purpose of this lesson is to focus on how the digits shift places. In module 4, when students multiply with decimals such as 5.2 × 10 , they see there are no zeros in the product, but the pattern of shifting digits still holds. To avoid misconceptions, encourage students to focus on how the digits shift when they multiply by 10, 100, and 1,000.

Direct students to problem 4. Have them represent 50 by drawing the fewest number of dots possible. Ask students to show you their whiteboards so you can check that they have five dots in the 10s column.

4. 50 × 10 = 500

How many tens is 5 tens times 10?

50 tens

Display the place value chart that shows 50 tens.

How can we rename 50 tens?

We can rename 50 tens as 5 hundreds because every time we have 10 tens, we can bundle and rename as 1 hundred.

Display the place value chart that shows 50 tens renamed as 5 hundreds.

In standard form, what is 50 × 10?

500

Record the product 500 in the blank for problem 4 and have students do the same.

When you draw dots, bundle, and rename to show 10 times as many, it can take a long time. Watch as I show 10 times as much more efficiently.

Teacher Note

Problems 1–6 are intentionally sequenced so that students notice patterns in products. As appropriate, consider removing some of the scaffolding and allowing students to work individually or in pairs.

Using Place Value Chart to Millions, draw five dots in the tens column. Draw an arrow, labeled as × 10, to the hundreds place. Then draw 5 hundreds. Direct students to do the same.

When we multiply 5 tens by 10, we can see the units shift. In what direction do the 5 units shift?

To the left

How many places do the 5 units shift?

One place

When we multiply by 10 once, the units shift to the left once.

Have everyone erase their whiteboards and begin a new problem.

Direct students to problem 5. Have them represent 50 again by drawing the fewest number of dots possible.

5. 50 × 100 = 50 × 10 × 10 = 5,000

How does 50 × 100 relate to 50 × 10?

100 is 10 times as much as 10, so 50 × 100 is 10 times as much as 50 × 10.

The product of 50 and 100 is 10 times as much as the product of 50 and 10.

Because 100 is equivalent to 10 × 10, we know 50 × 100 has the same value as 50 × 10 × 10.

Record the expression 50 × 10 × 10 in the first blank for problem 5 and have students do the same.

Invite students to think–pair–share to predict the product based on the pattern that they notice.

I notice when we multiply by 10, the units shift to the left. In this problem, we have to multiply by the factor 10 two times, so 5 tens will shift to the left two times.

If the units shift to the left two times, then the tens go to the thousands place. I think, 50 × 100 = 5,000.

Because I know 50 × 10 = 500, I think, 50 × 100 is 10 times as much as 500, which is 5,000.

What is 50 × 100?

5,000

Record 5,000 in the last blank for problem 5 and have students do the same.

Let’s show this on our place value chart. Draw five dots in the tens place. Draw an arrow to the hundreds place, labeled as × 10, and draw 5 hundreds. Draw a second arrow, labeled as × 10, and draw 5 thousands. Direct students to do the same.

In what direction do the 5 units shift?

The 5 units shift to the left.

How many places do the 5 units shift?

The 5 units shift two places.

Why do the 5 units shift two places this time?

The 5 units shift two places this time because we multiply by the factor 10 twice.

UDL: Representation

To solidify understanding about what happens when repeatedly multiplying by 10, consider displaying a place value chart with the product represented by dots alongside a place value chart such as the following, with the product represented by digits.

Direct students to problem 6.

6. 50 × 1,000 = 50 × 10 × 10 × 10 = 50,000

What expression can we write to show 1,000 by using only 10 as a factor?

50 × 10 × 10 × 10

Invite students to turn and talk to predict the product.

What is 50 × 1,000? 50,000

Display the completed place value chart.

Confirm that the place value chart shows 50 × 1,000 = 50,000. Then have students fill in the blanks for problem 6.

Direct students to study the equations in problems 1–6 and invite them to think–pair–share about the patterns they see.

I notice that the dots on the place value chart shift to the left once for each factor of 10 that we multiply by. So when we multiply by 10 twice, the dots shift to the left twice. When we multiply by 10 times as much as we did before, the product is also 10 times as much as the previous product.

Do you think we will see the same shifts if we multiply by 30 instead of 10, 100, or 1,000 like we saw in the last few problems? Turn and talk to your partner.

Direct students to problem 7.

7. 48 × 30 = 48 × 3 × 10 = 144 × 10 = 1,440

Let’s write 30 so we can see 10 as a factor. 30 is equal to 3 × 10, so we can write this problem as 48 × 3 × 10.

Record 48 × 3 × 10 in the first blank.

Differentiation: Support

If partners have difficulty finding 48 × 3, use the following questions and prompts:

• What do you know about this problem?

• Can you make a similar problem that you can do in your head?

• Model with a similar problem: If the first factor was 98, I would round it to 100 because that is an easier number. Could you use a strategy like that to round 48 to a number that is easier to work with?

Teacher Note

Students may have only thought about product as an answer to a multiplication problem. Support students with understanding that product can also refer to a multiplication expression with two or more factors.

Have students work with a partner to find 48 × 3.

What is 48 × 3?

144

Have we found the product 48 × 30?

No, we have only found the product 48 × 3.

What do we still need to do?

We still need to multiply by 10.

Record 144 × 10 in the second blank.

What happens to each unit when we show multiplying by 10 on the place value chart?

Each unit shifts one place to the left.

We know 48 × 3 = 144. What do you think 48 × 30 is?

48 × 30 = 1,440

Display the completed place value chart.

Confirm that the place value chart shows 48 × 30 = 1,440. Then have students fill in the last blank for problem 7.

Direct students to problem 8.

8. 48 × 300 = 48 × 30 × 10 = 1,440 × 10 = 14,400

Invite students to think–pair–share about how 48 × 30 can help them find 48 × 300.

100 is 10 times as much as 10, so 48 × 300 is 10 times as much as 48 × 30.

We can multiply the product of 48 and 30 by 10 again. The dots will each shift to the left one more place than they did in the last problem.

Language Support

Consider supporting student responses with the Talking Tool. Invite students to use the Share Your Thinking section to explain their strategy.

What is 48 × 300?

48 × 300 = 14,400

Display the place value chart showing 48 × 300.

Confirm that the place value chart shows 48 × 300 = 14,400. Then have students fill in the blanks for problem 8. Repeat the process used with problem 8 for problem 9.

9. 48 × 3,000 = 48 × 300 × 10 = 14,400 × 10 = 144,000

Invite students to think–pair–share about how writing expressions so that one factor is 10, 100, or 1,000 can help them find products.

If one of the factors is 10, then I know each digit in the other factor shifts one place to the left.

If one of the factors is 100, then I know each digit in the other factor shifts two places to the left.

If one of the factors is 1,000, then I know each digit in the other factor shifts three places to the left.

Divide by 10, 100, and 1,000

Students divide by 10, 100, and 1,000 and notice patterns that help them divide mentally.

Direct students to problem 10.

10. 270,000 ÷ 10 = 27,000

Promoting the Standards for Mathematical Practice

Students look for and express regularity in repeated reasoning (MP8) as they find patterns in related multiplication (and division) problems for multiplying (or dividing) by 10, 100, or 1,000. They apply these patterns to help them find products (and quotients) for more complex problems, such as 48 × 300

Ask the following questions to promote MP8:

• What patterns do you notice when you multiply by 10, 100, or 1,000? How can that help you multiply more efficiently?

• Will this pattern always work? Explain.

Differentiation: Support

Students first learn division by viewing it as an unknown factor problem. Consider writing the related multiplication problem next to each division problem to help students find the unknown.

When we multiply by 10, the units shift to the left. In what direction do you think the units shift when we divide by 10?

To the right

Direct students to represent 270,000 on the place value chart.

What is 27 ten thousands ÷ 10?

27 thousands

Draw an arrow, labeled as ÷ 10, from the 2 hundred thousands to the ten thousands place. Draw 2 ten thousands. Draw another arrow, labeled as ÷ 10, from the ten thousands place to the thousands place. Draw 7 thousands. Direct students to do the same.

In what direction do the 2 and the 7 shift?

The 2 and the 7 each shift to the right.

How many places do they shift?

They each shift one place.

Confirm that the place value chart shows 270,000 ÷ 10 = 27,000. Then have students fill in the blank for problem 10.

Direct students to problem 11.

11. 270,000 ÷ 100 = 270,000 ÷ 10 ÷ 10 = 2,700

How does dividing by 100 relate to dividing by 10?

When we divide by 10, we shift digits to the right once, so when we divide by 100, we shift digits to the right one more time because 100 = 10 × 10.

Because 100 = 10 × 10, we can show that dividing by 100 is equivalent to dividing by 10 once and then dividing by 10 a second time.

Record 270,000 ÷ 10 ÷ 10 in the first blank and have students do the same.

Teacher Note

Problems 10–12 are intentionally sequenced so students notice patterns in quotients. As appropriate, consider removing some of the scaffolding and allowing students to work individually or in pairs.

Teacher Note

Students’ previous experience in grades 3 and 4 with seeing how digits in a number shift when they multiply or divide by 10 should support their understanding of how to find the value of the expression with repeated division by 10 . However, some students may divide 10 ÷ 10 first in the expression 270,000 ÷ 10 ÷ 10 , leading them to 270,000 ÷ 1 = 270,000 .

Support students with understanding why this is incorrect by returning to the original expression and comparing it to the expression in problem 10.

Ask students:

• In problem 10, we divided 270,000 by 10. Is our quotient less than, equal to, or greater than 270,000?

• Does it make sense that the quotient is equal to 270,000 in problem 11 when dividing it by 100?

Invite students to think–pair–share to predict whether 270,000 ÷ 100 is greater than or less than 270,000 ÷ 10.

I think 270,000 ÷ 100 is less than 270,000 ÷ 10 because we are dividing by 10 twice instead of just once.

I think 270,000 ÷ 100 is less than 270,000 ÷ 10 because the units shift farther to the right.

We already know 270,000 ÷ 10 = 27,000. What can we do to find 270,000 ÷ 100?

We can divide by 10 again. We can divide 27,000 by 10.

Draw an arrow, labeled as ÷ 10, from the 2 ten thousands to the thousands place. Draw 2 thousands. Draw another arrow, labeled as ÷ 10, from the thousands place to the hundreds place. Draw 7 hundreds. Direct students to do the same.

What is 270,000 ÷ 100?

2,700

In what direction do the 2 and the 7 shift?

The 2 and the 7 shift to the right.

How many places do the 2 and the 7 shift to show division by 100? Why?

They each shift two places because we divide by 10 two times.

Confirm that the place value chart shows 270,000 ÷ 100 = 2,700. Then have students fill in the second blank for problem 11.

Direct students to problem 12.

12. 270,000 ÷ 1,000 = 270,000 ÷ 10 ÷ 10 ÷ 10 = 270

Repeat the process to find the quotient and ask the following questions:

• How does dividing by 1,000 relate to dividing by 100?

• In what direction did the 2 and the 7 shift?

• How many places did the 2 and the 7 shift to show division by 1,000? Why?

Have students fill in the blanks for problem 12.

Direct students to study problems 10–12 and invite them to think–pair–share about what pattern they notice. As students share, record equations to highlight the pattern.

I notice that as the divisor increases, the quotient decreases.

I notice that each time we divide by 10, the units shift one place to the right.

Direct students to problem 13.

13. 270,000 ÷ 30 = 270,000 ÷ (10 × 3) = 270,000 ÷ 10 ÷ 3 = 9,000

Invite students to think–pair–share about how 270,000 ÷ 30 is similar to or different from 270,000 ÷ 10.

Our total 270,000 is the same.

We are dividing by 30 now, not 10, but I know 30 is 10 × 3.

In problem 10, we found 270,000 ÷ 10 = 27,000. Let’s use that fact to help us find 270,000 ÷ 30 by writing 30 as 10 × 3.

Record 270,000 ÷ (10 × 3) in the first blank and have students do the same.

We can rename 30 as either 10 × 3 or 3 × 10. That means we have a choice about which number we want to divide by first. Because we have already done 270,000 ÷ 10, let’s show in the divisor that we are dividing by 10 first, then dividing by 3 second.

Teacher Note

Watch for the student misconception that dividing by 30 means dividing by 3 tens, which would be interpreted as 270,000 ÷ 10 ÷ 10 ÷ 10. Dividing by 10 three times is equal to dividing by 1,000, which is not what students should divide by in problem 13.

If students need support with understanding how ÷ (10 × 3) becomes ÷ 10 ÷ 3 , consider asking them to think about what they did when dividing by 100 and 1,000 . When dividing by 100 , they thought about 100 as 10 × 10 . When dividing by 1,000 , they thought about 1,000 as 10 × 10 × 10. The only difference in problem 13 is that one factor is a number other than 10.

Record 270,000 ÷ 10 ÷ 3 in the second blank and have students do the same. Then direct students to work with a partner to find the quotient. Confirm the quotient is 9,000 and have students record the quotient in the last blank. Have students complete problems 14 and 15 individually or with a partner. Circulate as students work and support their understanding by asking any of the following questions:

• How is this problem similar to or different from the previous problem?

• What can you use from the previous problem to help you with this problem?

• Do you expect the units to shift to the left or to the right? Why?

• How many times do you expect the units to shift? Why?

When most students are finished, invite students to think–pair–share about how renaming the divisor helps them find the quotient.

Renaming the divisor by using 10, 100, or 1,000 helps me divide because I can shift the digits to the right however many places are needed for the problem. It helps us divide mentally because when I divide by 10, I know the units will shift to the right.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Multiply and divide by 10, 100, and 1,000 and identify patterns in the products and quotients.

Facilitate a class discussion about multiplying and dividing by 10, 100, and 1,000 by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Display the picture of a student thinking.

Sana says 450 ÷ 50 is equivalent to 450 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 because 50 is 5 tens. Do you agree? Why?

I disagree. Dividing by 10 five times is equivalent to dividing by 100,000.

I disagree. Sana only needs to divide by 50, which is equivalent to dividing by 10 and then dividing by 5 because 50 = 10 × 5.

How is multiplying by 10, 100, and 1,000 similar to or different from dividing by 10, 100, and 1,000?

When you multiply by 10, 100, or 1,000, the digits shift one, two, or three places to the left.

When you divide by 10, 100, or 1,000, the digits shift one, two, or three places to the right.

How does multiplying or dividing by 100 or 1,000 relate to multiplying or dividing by 10?

When you multiply by 100, it is equivalent to multiplying by 10 twice. When you multiply by 1,000, it is equivalent to multiplying by 10 three times.

When you divide by 100, it is equivalent to dividing by 10 twice. When you divide by 1,000, it is equivalent to dividing by 10 three times.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

1. Complete the statement.

The 8 in 58,701 represents 10 times as much as the 8 in 5,870

2. Write a multiplication equation to relate the 7 in 58,701 to the 7 in 587,019

700 × 10 = 7,000

3. Write a division equation to show the relationship between the value of the 5 in 587,019 and the value of the 5 in 5,870

500,000 ÷ 100 = 5,000

Multiply.

4. 62 × 1 ten = 62 tens

62 × 10 = 620

5. 62 × 1 hundred = 62 hundreds

62 × 100 = 6,200

6. 62 × 1 thousand = 62 thousands

62 × 1,000 = 62,000

Copyright © Great Minds PBC 19

Use the number of zeros in the product to explain why Toby’s answer is not correct.

Toby multiplies by 1,000, which means each digit in the number should shift three places to the left. The zero in the ones place should shift to the thousands place, and then there should be three more zeros. Toby needs to shift the digits one more place, and then he would have four zeros in the product.

29. A banker has a total of $54,200, all in one hundred-dollar bills. How many one hundred-dollar bills does the banker have?

54,200 ÷ 100 = 542

The banker has 542 one hundred-dollar bills.

Use exponents to multiply and divide by powers of 10 .

Lesson at a Glance

Students repeatedly multiply by 10 and relate the number of factors to exponents of exponential expressions. They practice writing multiple factors of 10 in standard form, expanded form, and exponential form. Students apply this understanding to multiplication and division problems involving powers of 10. This lesson introduces the terms exponent, exponential form, and power of 10.

Key Questions

• Does knowing about 10 as a factor help us to multiply and divide by powers of 10? How?

• Why might it be helpful to write powers of 10 in exponential form?

Achievement Descriptors

5.Mod1.AD7 Explain the effect of multiplying and dividing whole numbers by powers of 10. (5.NBT.A.2)

5.Mod1.AD8 Express whole-number powers of 10 in exponential form, standard form, and as repeated multiplication. (5.NBT.A.2)

Agenda Materials

Fluency 10 min

Launch 10 min

Learn 30 min

• Determine Patterns in Powers of 10

• Whiteboard Exchange: Exponential Form and Standard Form

• Multiply with Powers of 10

• Divide by Powers of 10

• Problem Set

Land 10 min

Teacher

• Powers of 10 Charts

(in the teacher edition)

Students

• Powers of 10 Charts (in the student book)

Lesson Preparation

• Display or recreate Powers of 10 Charts so all students can see it.

• Consider whether to remove Powers  of 10 Charts from the student books in advance or have students remove them during the lesson.

Fluency

Whiteboard Exchange: Unit to Standard Form

Students write the standard form of a four-digit number given in unit form to maintain fluency with writing numbers within 1,000,000 from grade 4. Display 1 thousand 9 hundreds 4 tens 3 ones = .

When I give the signal, read the number shown in unit form. Ready?

1 thousand 9 hundreds 4 tens 3 ones

Write the number in standard form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the answer.

Repeat the process with the following sequence:

Teacher Note

Consider providing students with a place value chart, or prompting them to draw one, to support place value understanding while they write the numbers in standard form.

Whiteboard Exchange: Round Multi-Digit Numbers

Students round a three-digit number to the nearest hundred and nearest ten to prepare for estimating products beginning in lesson 4.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display 192 ≈   .

What is 192 when rounded to the nearest hundred?

Display the rounded value, and then display 192 ≈   .

What is 192 when rounded to the nearest ten?

Display the rounded value.

Repeat the process with the following sequence:

Launch

Students examine representations that show factors of 10.

Introduce the Which One Doesn’t Belong? routine. Present four representations and invite students to study them.

Give students

1 minute to find a category in which three of the items belong, but a fourth item does not.

When time is up, invite students to explain their chosen categories and to justify why one item does not fit.

Highlight responses that emphasize reasoning about 10 as a factor.

Ask questions that invite students to use precise language, make connections, and ask questions of their own.

Sample questions include the following:

Which one doesn’t belong?

A doesn’t belong because it is the only one that shows 10 times as much as 1,000 in an equation.

B doesn’t belong because it is the only one that shows 10 as a factor four times.

C doesn’t belong because it is the only one with 100 as a factor two times.

D doesn’t belong because it is the only one that shows division.

How are A and B similar or different?

Both show the product 10,000, but A shows it as 10 times as much as 1,000 and B shows it as 10 × 10 × 10 × 10.

How are B and D similar or different?

They both show units in the 10s, 100s, 1,000s, and 10,000s columns.

B shows repeated multiplication by 10.

D shows repeated division by 10.

How are A and C similar or different?

A and C both have products of 10,000.

Both equations have 10 as a factor four times. If you decompose each factor into 10s, they each have 10 as a factor four times.

If you decompose each factor into 10 s, you get the same equation for A and C, just with the 10 s grouped differently: 10 × (10 × 10 × 10) = 10,000 and (10 × 10) × (10 × 10) = 10,000 .

The only difference is how the 10s are grouped.

Both show equations with 2 factors and do not show a model.

You saw 10 as a factor in many ways.

Transition to the next segment by framing the work.

Today, we will multiply and divide with 10s.

Learn

Determine Patterns in Powers of 10

Materials—T/S: Powers of 10 Charts

Students interpret an exponent as the number of times 10 is a factor. Invite students to remove Powers of 10 Charts from their books. Have students complete each chart with you as you ask the following questions.

Our goal is to complete this chart by using only 10s as factors. Look at the first equation. How can we write a product that equals 100 by using only 10s?

10 × 10

Record 10 × 10 to complete the equation.

Let’s represent 10 × 10 on the place value chart by using as few dots as possible. Draw a single dot in the tens place. Then draw an arrow, labeled as × 10, and another dot in the hundreds place.

Where do you see the two 10s represented on your model?

One of the 10s is represented by the dot in the tens place. The other 10 is represented by the arrow that shows we multiplied 10 by 10.

We can use an exponent to represent how many times we use the same number, in this case 10, as a factor. How many 10s did we multiply to get 100?

Two 10s

UDL: Action & Expression

The recording chart requires students to write several equations and to write the exponent as a smaller-size numeral when compared to its base. To minimize fine motor demands, consider making copies of the answers for the Equation, Representation, and Exponential Form columns of the recording sheet. Offer students the option to select and glue the correct responses on the recording sheet.

To support students in gaining comfort with writing the exponent, provide a template and time to practice. 10 10

When we write very large numbers by using only 10s as factors, it takes less space on the page when we use exponential form. When we write a number with an exponent, we are writing the number in exponential form.

100 is not a very large number, but let’s use it to practice using exponential form. Because we used 10 two times to get 100, what exponent should we use?

Language Support

To support students’ understanding of the new terms exponent and exponential form, consider creating an anchor chart for student reference.

Let’s write 102 in the Exponential Form column.

Point to 102 in the chart.

Now we see that 10 × 10 , 100 , and 10 to the second power are just different ways to represent 100 .

Direct students to the next equation on the chart.

1,000 is how many times as much as 100?

1,000 is 10 times as much as 100.

100 is how many times as much as 10?

100 is 10 times as much as 10.

How can we write 1,000 by using only 10s? 10 × 10 × 10

Record 10 × 10 × 10 to complete the equation.

Ask students to represent 10 × 10 × 10 by using dots and arrows on the place value chart, as you do the same.

1,000 = ten thousands (10,000) thousands (1,0 00) hundreds (10 0) tens (10) ones (1)

How many 10s do we multiply to get 1,000?

We multiply three 10s to get 1,000.

How can we write 1,000 in exponential form?

In addition, provide students with a clear page protector for the Powers of 10 Chart and prompt them to refer to it as needed.

How does the number of zeros in the product relate to the exponent and the number of 10s as factors?

The number of zeros in each product matches the exponent in exponential form.

The number of zeros in each product is equal to the number of 10s that are multiplied together.

Have students complete the chart to 1,000,000 with a partner by using the pattern they just noticed.

Gesture to 103.

How do we read this out loud?

10 to the third power

What is 103?

It equals 1,000.

It equals 10 × 10 × 10.

A number that can be written as a product of 10s, or as a 10 with an exponent, is called a power of 10. So 100 is a power of 10. What is another power of 10?

10,000 is a power of 10.

How do you know that a number is a power of 10? Give an example.

I know a number is a power of 10 if I can write it as a product with only 10s as factors. For example, 10,000 is a power of 10 because I can write it as 10 × 10 × 10 × 10 and 104.

When you have a number in exponential form such as 10 to the fourth power, how many zeros are in the product?

There are four zeros in the product.

Can you use the number of zeros to help you write a power of 10, such as 10,000, in exponential form? How?

Yes. I can count the number of zeros in a power of 10 and use that number as the exponent.

Is counting the number of zeros the only way to determine the exponent to use? Explain.

No. I can first write the number as an expression so I can see how many factors of 10 are in the number. Then I can count the number of factors of 10 to determine the exponent to use.

Let’s return to the chart and use what we know to complete the last row. How many 10s do we multiply to equal 10?

We only need one 10 to equal 10.

How do we show 10 on the place value chart?

We can draw one dot in the tens place.

What does the exponential form of 10 look like? Why?

The exponential form of 10 has 1 as an exponent because we only need one 10 to equal 10.

Whiteboard Exchange: Exponential Form and Standard Form

Students write powers of 10 as equations, in standard form, and in exponential form.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. As students work, prompt them to write the full equation including the product and not just an expression.

Display each of the following expressions one at a time.

Write an equation that shows each power of 10 equal to a multiplication expression that uses only 10s.

• 102 = 10 × 10

• 103 = 10 × 10 × 10

• 105 = 10 × 10 × 10 × 10 × 10

• 106 = 10 × 10 × 10 × 10 × 10 × 10

• 104 = 10 × 10 × 10 × 10

• 101 = 10

Write an equation that shows each number rewritten in exponential form.

• 1,000 = 103

• 100,000 = 105

• 100 = 102

• 10,000 = 104

• 10 = 101

• 1,000,000 = 106

Invite students to turn and talk about how they determined how each number is represented in exponential form.

Multiply with Powers of 10

Students multiply powers of 10 by using a variety of strategies. Direct students to problems 1 and 2 in their books. Have students complete the problems with a partner. Circulate to identify students who multiply by using different methods and express the product in different forms.

Teacher Note

Ensure that students write the full equation and not only an expression. This practice will help them connect the standard form with the exponential form.

For this Whiteboard Exchange check for the following:

When students see 102, ensure they write the equation 102 = 10 × 10 and not only the expression 10 × 10.

When students see 1,000, ensure they write the equation 1,000 = 103 and not only the expression 103

Multiply.

1. 10,000 × 100 = 1,000,000

2. 1,000 × 103 = 106

Invite identified students to share their methods and responses with the class. Consider using any of the following questions during the discussion:

• I notice you wrote the product in standard form. What is the product in exponential form?

• I notice you wrote the product in exponential form. What is the product in standard form?

• How did you know which exponent to use when you wrote the product in exponential form?

• Did you express each power of 10 by using exponents first, or did you use another method?

• Where do you see your answer in your classmate’s work?

Direct students to problem 3. Then ask the following questions.

How is this problem similar to or different from the previous problems we solved? This problem has a factor that is not a power of 10.

How can we write 7 × 102 by using 10 as a factor?

7 × 10 × 10

Have students complete problems 3 and 4 with a partner.

Multiply.

3. 7 × 102 = 700

4. 300 × 103 = 300,000

Promoting the Standards for Mathematical Practice

Students construct viable arguments (MP3) as they think about and share a logical progression of statements, based on their new understanding of exponents, that explain how they found each product.

Ask the following questions to promote MP3:

• Is 106 a guess, or do you know for sure? How do you know?

• Is this step in your work true? How do you know?

• What questions can you ask your classmate to make sure you understand their thinking?

When most students have finished, confirm their answers, then ask the following question. Both problems had a factor that was not a power of 10. Were you able to express the product by using exponential form? Why?

No. We could not write the product in exponential form because there were factors other than 10 in the problem.

For now, we can only write numbers in exponential form when the number is a power of 10, meaning its product can be written with only 10s.

Divide by Powers of 10

Students divide by powers of 10 by using a variety of strategies and share their work to compare.

Direct students to problem 5. Then ask the following question.

How is this problem similar to or different from the previous problems?

This problem is division. The other problems we’ve done so far were multiplication. This problem has powers of 10 in it, just like all the other problems we’ve done so far.

Have students complete problems 5 and 6 with a partner. Circulate to identify students who divide by using different methods and express the quotients in different forms.

Divide.

5. 10,000 ÷ 102 = 102

6. 1,000,000 ÷ 103 = 103

Invite identified students to share their methods and responses with the class. Consider using any of the following questions during the discussion:

• I notice you wrote the quotient in standard form. What is the quotient in exponential form?

• I notice you wrote the quotient in exponential form. What is the quotient in standard form?

Teacher Note

In grade 6, students extend their understanding of exponential form to include any whole number base. In grade 8, students learn to express very large and very small numbers in scientific notation. Then grade 8 students perform operations on numbers written in scientific notation. Grade 5 sets the foundation for this understanding by focusing only on powers of 10.

• How did you know which exponent to use when you wrote the quotient in exponential form?

• Did you express each power of 10 by using exponents first, or did you use another method?

• Where do you see your answer in your classmate’s work?

Direct students to problems 7 and 8.

As before, these problems are a little different. What do you notice?

I notice that 9,000 and 360,000 are not powers of 10. Have students complete problems 7 and 8 with their partners.

Divide.

7. 9,000 ÷ 103 = 9

8. 360,000 ÷ 104 = 36

When most students have finished, confirm their answers, then ask the following questions.

Were you able to express either quotient by using exponential form? Why?

No. We could not write the quotients in exponential form because we did not multiply by any 10s to get 9 or 36.

How did you find the quotient for each problem?

I used the place value chart and thought about how many times the digits would shift to the right.

I realized the exponent shows me how many times the digit 9 shifts to the right. For 103 it means that 9 in the thousands place shifts three times to the right.

I put the digit 3 in the hundred thousands place and the digit 6 in the ten thousands place on the place value chart to represent 360,000. Then I shifted each digit four times to the right to represent dividing by 10 four times.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Use exponents to multiply and divide by powers of 10. Gather the class with their Problem Sets. Facilitate a class discussion about using exponents to multiply and divide by powers of 10 by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the multiple representations of 10,000. Allow students 1–2 minutes to study the representations.

Differentiation: Support

If students need support transitioning from drawing dots to writing digits in problems 14-19 in the Problem Set, consider providing a concrete experience. Have students move digits on the place value chart by using digits written on sticky notes or strips of construction paper. Students can physically shift the digits on the place value chart to show multiplication or division.

What number should we put in the middle to show the value of all these representations? Why?

We should write 10,000 because each representation is equal to 10,000.

The one thousand-dollar bills show 1,000 ten times, which is 10,000.

The place value chart and the expression 100 × 10 × 10 both show 100 multiplied by 10 twice. 100 is 10 × 10, so 10 × 10 × 10 × 10 is another expression equal to 100 × 10 × 10, which is 10,000.

The exponent 4 in 104 represents the number of times 10 is a factor in the number. 10 × 10 × 10 × 10 = 10,000

Does knowing about 10 as a factor help us to multiply and divide by powers of 10? How?

Yes. We know when we multiply by 10, digits shift to the left, and when we divide by 10, digits shift to the right. So we can use the number of 10s to help us find the product or the quotient just by shifting the digits.

Direct students to problems 1–5 in the Problem Set.

Why might it be helpful to write powers of 10 in exponential form?

When you write a power of 10 in exponential form, it is faster and takes up less space. For example, when you write 106, it is faster and takes up less space on the page than when you write 1,000,000.

Direct students to problem 15.

How did you find the quotient?

I thought about dividing by 105 as shifting the digits in 500,000 five places to the right on a place value chart. When the digits in 500,000 shift five places to the right, that means the quotient is 5.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

11. Use words and equations to explain how 105 is different from 10 × 5

105 = 10 × 10 × 10 × 10 × 10 = 100,000

10 × 5 = 50

Complete the table to represent each number in three different forms. The first one is done for you.

10 to the fifth power means 10 is used as a factor 5 times. 10 times 5 is equal to 10 groups of 5.

Rewrite each expression by using an exponent. Then find the product or quotient and write it in standard form.

How does the exponent help you think about shifting the digits in the first factor to find the product?

The exponent of 3 helps me know that I need to shift the digits three places. Since it’s a multiplication expression, I know I need to shift the digits three places to the left.

20. Explain how you found the quotient in problem 16. I thought about dividing by 102 as shifting the digits in 39,000 two places to the right on a place value chart. When the digits in 39,000 shift two places to the right, the quotient is 390

21. Yuna finds 300 × 103. Explain Yuna’s strategy.

First Yuna rewrote 300 as 3 × 10 × 10. Then she could see that 10 is a factor 5 times because there are two 10s and 103. She found that 3 × 105 = 300,000

Equation Representation

Form

Equation Representation Exponential Form

Equation Representation

Form

Estimate products and quotients by using powers of 10 and their multiples.

Lesson at a Glance

A large helicopter can carry 25,000 pounds. The average weight of a car is 4,110 pounds. If there is enough space, about how many cars can the helicopter carry at one time? Explain how you know. The helicopter can carry about 6 cars. If each car weighs about 4,000 pounds, then 6 cars weigh about 24,000 pounds because 4,000 × 6 = 24,000, and 7 cars weigh about 28,000 pounds because 4,000 × 7 = 28,000 The helicopter can carry about 6 cars at one time because the helicopter can carry 25,000 pounds, and 7 cars would be too heavy.

Students estimate products and quotients by applying their understanding of multiplication and division by powers of 10. Students compare estimates and discuss estimation strategies. After they discuss why estimation is a valuable skill, students apply strategies to estimate products and quotients in real-world situations. This lesson introduces the term analyze.

Key Questions

• What is important to keep in mind when you use estimation?

• Why might you round a number to a power of 10, or to a multiple of a power of 10, before you estimate a product or quotient?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD7 Explain the effect of multiplying and dividing whole numbers by powers of 10. (5.NBT.A.2)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Estimate Products

• Estimate Quotients

• Estimate Products and Quotients in the Real World

• Problem Set

Land 10 min

Lesson Preparation

Fluency

Whiteboard Exchange: Unit to Standard Form

Students write the standard form of a five-digit number given in unit form to maintain fluency with writing numbers within 1,000,000 from grade 4.

Display 1 ten thousand 3 thousands 7 hundreds 2 tens 9 ones =    .

When I give the signal, read the number shown in unit form. Ready?

1 ten thousand 3 thousands

7 hundreds 2 tens 9 ones

Write the number in standard form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the answer.

Repeat the process with the following sequence:

Whiteboard Exchange: Round Multi-Digit Numbers

Students round a four-digit number to the nearest thousand and nearest hundred to prepare for estimating products.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display 1,832 ≈   .

What is 1,832 when rounded to the nearest thousand?

Display the rounded value and then display 1,832

What is 1,832 when rounded to the nearest hundred?

Display the rounded value.

Repeat the process with the following sequence:

Launch

5 Teacher Note

Students estimate the number of days that someone has been alive. Ask students to think–pair–share about the following situation.

On his 11th birthday, Blake wants to know how many days he has been alive. How can Blake determine this?

Blake can multiply 11 by 365 to determine how many days he has been alive.

Blake does not have a pencil or paper, so he wants to estimate how many days he has been alive. With your partner, use mental math to estimate how many days Blake has been alive. Be prepared to share your thinking with the class.

Invite pairs to share their estimates and reasoning.

Blake has been alive about 3,700 days because 10 × 370 = 3,700. We rounded 11 to 10 and 365 to 370 and multiplied 10 by 370.

Blake has been alive about 3,650 days because 10 × 365 = 3,650. We rounded 11 to 10 and multiplied 10 by 365.

As students share, record their estimates so the class can see them, and ask the following questions:

• Which numbers did you choose to round? Why?

• How did you decide which place value to round to?

If students do not provide the following estimates, add them to the list.

• 10 × 365 = 3,650

• 10 × 360 = 3,600

• 10 × 370 = 3,700

• 10 × 400 = 4,000

• 11 × 400 = 4,400

Multiplying 11 by 365 does not give the exact number of days Blake has been alive because doing this does not account for leap years. Consider discussing how to adjust the answer for leap years or in what cases students might choose to ignore leap years.

Language Support

To support students’ understanding of the word rounding, consider cocreating and posting a list of reasons why numbers are rounded. Include the words about and estimate to help students when they encounter these words in problems or directions.

Highlight that the word round has multiple meanings (circular in shape and to estimate numbers) and emphasize the specific meaning of round in this lesson.

Then use the following prompts to facilitate a brief discussion.

Rounding can help us turn a challenging problem into a mental math problem that we can use to estimate when a precise answer is unnecessary or hard to find.

Why are the estimates different for the number of days that Blake has been alive? They are different because we chose different ways to round Blake’s age and to round the number of days in a year.

What do you notice about the factors in each of these estimates?

I notice at least one of the factors in each of these estimates is a power of 10.

I notice many of the factors that represent the number of days in a year can be written as a product with at least one 10.

I notice many of the estimates use a factor that is 10, a multiple of 10, or both.

Numbers such as 400 or 370 are called multiples of a power of 10 because we can write them as a number times a power of 10. How can we show 400 is a multiple of a power of 10?

Because 400 = 4 × 100, we can show it as a multiple of a power of 10 by writing 400 as 4 × 100 or as 4 × 102.

Why do you think we round factors to powers of 10 or multiples of a power of 10?

I think we round to powers of 10 or multiples of a power of 10 so we can find products by using mental math. Otherwise you might need pencil and paper.

Does it create an estimate that is closer to the actual product when you multiply 10 by 365 or multiply 10 by 360? Why?

When you multiply 10 by 365, it creates an estimate that is closer to the actual product because only one factor is rounded and not both.

For an estimate to be reasonable, it should be close to the actual value. The actual product of 11 and 365 is 4,015. Do you think the estimates are reasonable? Explain. Yes. All the estimates are close to 4,015 days.

When we make an estimate in math class, it can help us find quotients and products faster because we can use mental math. Estimates can also help us catch mistakes. If an answer is far from our estimate, we might have made a mistake when we multiplied or divided.

Estimation comes up in the real world too. For example, you might estimate how much a trip to the movies will cost to make sure you have enough money. Or you might estimate how much time it will take to finish your homework.

Invite students to turn and talk with a partner about a real-world situation in which they think estimation is useful.

Transition to the next segment by framing the work.

Today, we will use our experience with multiplying and dividing by powers of 10 to estimate products and quotients.

Learn Estimate Products

Students estimate products by using powers of 10 and their multiples.

Write 28 × 79.

We can calculate the product of 28 and 79 by using a pencil and paper. We can estimate the product by using mental math. Why might we estimate the product of 28 and 79 before we calculate it?

If we estimate, it will help us to decide whether our answer is reasonable when we find the actual product.

We can estimate a product by rounding the factors and then multiplying them. How would you round 28 and 79 so you can use mental math to estimate the product? Explain.

I would round both numbers to the nearest 10 because then I am multiplying two multiples of 10. I would round 28 to 30 and 79 to 80.

UDL: Representation

Before you begin the Learn segment of the lesson, consider activating prior knowledge by highlighting important aspects of rounding that students learned in grade 4.

• There is a difference between rounding to the nearest number, rounding to a greater number, and rounding to a smaller number.

• Rounding helps you make estimates. Estimates are not usually exact.

• Depending on the situation, you can decide the most useful way to round.

What is 30 × 80? How do you know?

30 × 80 = 2,400 because 30 × 80 = 3 × 10 × 8 × 10. That equals 24 × 102, which is 2,400.

Do you think the estimate of 2,400 is greater or less than the actual product of 28 and 79? Explain.

2,400 is greater than the actual product because we rounded both factors to the next 10.

28 × 79 = 2,212. Our estimate is greater than the actual product because we rounded both factors to the next 10, but it is still reasonable because 2,400 is close to 2,212.

Write 278 × 31. Invite students to think–pair–share to estimate the product.

The product is about 9,000 because 300 × 30 = 9,000. The product is about 9,300 because 3 × 100 × 31 = 93 × 100 = 9,300.

Highlight different reasoning that students might have. Consider making the following points:

• We could estimate the product by multiplying 280 by 30, but that product might be challenging for some people to calculate mentally.

• We could estimate the product by multiplying 300 by 30, but we could also multiply 300 by 31, which is more accurate.

Do you think your estimate is greater or less than the actual product of 278 and 31? Explain.

I think my estimate is greater than the actual product because I rounded 278 to a greater number, 300.

278 × 31 = 8,618

Write 308 × 24. Invite students to think–pair–share to estimate the product. Students’ answers are likely to include the ones shown. Explain that each of these three different estimates is reasonable.

310 × 20 = 6,200

3 × 100 × 25 = 75 × 100 = 7,500

300 × 20 = 6,000

Teacher Note

Consider showing the work for 30 × 80 by using properties of multiplication:

30 × 80 = 3 × 10 × 8 × 10 = 3 × 8 × 10 × 10 = 24 × 100 = 2,400

In line 2, the commutative property is used to change the order of the factors. In line 3, the associative property is used to multiply the 3 and the 8 separately from the other factors.

Promoting the Standards for Mathematical Practice

Students look for and make use of structure (MP7) as they round and reason about numbers in a given expression to estimate its value mentally.

Ask the following questions to promote MP7:

• What is another way you can think of 308 × 24 that will help you estimate the product?

• How can what you know about multiplying (or dividing) with powers of 10 help you multiply (or divide) with other numbers?

Do you think your estimate is greater or less than the actual product of 308 and 24? Explain.

I think my estimate is less than the actual product because I rounded both factors to smaller numbers.

I am not sure whether my estimate is greater or less than the actual product because I rounded one factor to a greater number and the other factor to a smaller number.

308 × 24 = 7,392. When we round factors differently, such as when we round one factor to a greater number and the other factor to a lesser number, it can be difficult to know whether our estimate is greater or less than the actual product. What matters is that you make thoughtful choices about how to round so you can decide whether the actual product is correct or you made a mistake.

Display the problem and sample work.

Then ask students to think–pair–share about the following prompt.

Describe what Tara did to estimate the value of 4,598 × 7.

Tara rounded 7 to 10 and rounded 4,598 to 4,000. Then she multiplied to get her estimate of 40,000.

4,598 × 7 = 32,186. Do you think Tara’s estimate is reasonable? Why?

Tara’s estimate is reasonable because she rounded one of the factors to a smaller number and the other factor to a greater number. One of the factors is a power of 10, so she was able to find the product mentally.

Tara’s estimate is not reasonable because it is too high compared to the actual product.

Can Tara make her estimate more reasonable? How?

Yes. Tara could round 4,598 to 5,000 and then find the product of 5,000 and 7, which is 35,000. The estimate 35,000 is much closer to the actual product than Tara’s estimate of 40,000.

There are many ways to make a reasonable estimate. Rounding to a number that is a power of 10 is helpful because we can use what we know about shifting digits to the left when multiplying by a power of 10. In this case, rounding 4,598 to 5,000 and then multiplying by 7 gave us a better estimate than rounding both factors.

Direct students to problem 1 in their books. Read the directions aloud, and then use the following prompts to model how to find and show an estimate for the product.

Estimate each product. Show your thinking.

1. 7,114 × 20 7,114 × 20 ≈ 7,000 × 20 = 7 × 1,000 × 2 × 10 = 7 × 10 × 10 × 10 × 2 × 10 = 14 × 104 = 140,000

7,114 × 20 ≈ 140,000

For 7,114 × 20, I notice that 7,114 is close to 7,000, which is a multiple of a power of 10.

Because 20 is the other factor, I won’t round it, but I notice it is also a multiple of a power of 10.

Record 7,114 × 20 ≈ 7,000 × 20 and have students do the same.

Because I am rounding and not finding the actual product, I need to show that by using the approximately equal to symbol.

Gesture to the symbol in 7,114 × 20 ≈ 7,000 × 20.

I know that multiplying by 10s shifts digits to the left, so I will write each factor so it shows how many 10s are in each.

Record = 7 × 1,000 × 2 × 10 and have students do the same.

Notice this time I used the equal sign. I used the equal sign because now I am finding the actual product of 7,000 and 20 and not an estimate for it. Now I will show exactly how many 10s are in this estimate.

Record = 7 × 10 × 10 × 10 × 2 × 10 and have students do the same.

Now I rearrange the factors and see the estimate 14 × 104. Turn and talk to a partner about where you see 14 and 104.

Record = 14 × 104 and = 140,000 and the statement 7,114 × 20 ≈ 140,000 and have students do the same.

To estimate the product, we used what we know about powers of 10 and their multiples, as well as what we learned about how digits shift when we multiply by 10s. Practice showing your thinking this way, with the equal sign and the approximately equal to symbol, so it is clear when you are estimating and when you are finding an actual product.

Have students complete problems 2–4. Highlight that students should show the numbers they use to find the estimate but should be able to use mental math to do the calculations. Circulate to support students and to identify those who use a power of 10 in their work.

2. 1,009 × 51

1,009 × 51 ≈ 1,000 × 50

= 10 × 10 × 10 × 5 × 10

= 5 × 104 = 50,000

Teacher Note

It is not necessary for students to replicate the work shown in problems 2 and 3 for every estimate. The intentional use of powers of 10 and their multiples in these problems is to demonstrate the usefulness of seeing 10s as factors in any problem so that calculations can be made mentally.

1,009 × 51 ≈ 50,000

3. 92 × 396,285

92 × 396,285 ≈ 90 × 400,000

= 9 × 10 × 4 × 100,000

= 9 × 10 × 4 × 10 × 10 × 10 × 10 × 10 = 36 × 106 = 36,000,000

92 × 396,285 ≈ 36,000,000

4. Which number is the best estimate of 976 × 52?

A. 4,500

B. 45,000

C. 50,000

D. 500,000

When students are finished, invite them to share their estimates for problems 2 and 3. If any students used a power of 10 or a multiple of a power of 10, invite them to share their work with the class. Emphasize that there are multiple correct answers because each person has different ideas for how to turn a challenging calculation into one that can be done by using mental math.

For problem 4, highlight that even if students did not get 50,000 for their estimate, it should be closer to 50,000 than to any of the other choices. For example, if a student estimated by multiplying 1,000 by 52, their estimate is 52,000. In this case, rounding both factors leads to a better estimate than rounding only one of the factors. It is not necessary to emphasize this with students, but the situation may arise naturally as students continue to round numbers to make estimates.

Ask students to turn and talk with a partner about how they round factors when they estimate a product.

Estimate Quotients

Students estimate quotients by using powers of 10 and their multiples.

Write 118 × 7 and 118 ÷ 7. Then use the following prompts to introduce estimating quotients.

Analyze, or look carefully at, the expressions. What expression can we use to estimate the product of 118 and 7? 120 × 7

Would you use 120 ÷ 7 to estimate the quotient of 118 and 7? Why?

No. It is difficult for me to divide 120 by 7 because 120 is not a multiple of 7.

Which numbers can we round to so we can mentally estimate the quotient of 118 and 7?

We could divide 140 by 7 to get 20.

We could divide 120 by 6 to get 20.

How do we know that we can mentally divide 140 by 7?

140 is a multiple of 7 since 140 is equal to 14 tens.

Differentiation: Support

For students who need additional support with estimating quotients, consider providing a multiplication table to use for identifying totals and divisors.

For example, a student can look in the 7 column of a multiplication table to see that 12 is not divisible by 7, but 14 is. Therefore, 140 is divisible by 7.

Language Support

This segment introduces the term analyze. Consider previewing the meaning of the term before students are asked to analyze the expressions. Relate the term to its uses in other subject areas such as analyzing the plot of a story or the results of a science experiment.

How do we know that we can mentally divide 120 by 6?

I know that 12 tens divided by 6 is 2 tens because 12 ÷ 6 = 2. 120 is a multiple of 6.

Why might it be more challenging to estimate quotients than to estimate products? We may have to work harder to determine rounded numbers that work well for mental math division.

Write 5,247 ÷ 73 and ask students to work with a partner to estimate the quotient. Anticipate that students will create estimates of 4,900 ÷ 70 = 70 or 5,600 ÷ 70 = 80. Invite students to share their estimates and reasoning. Then ask students the following question.

Did you round to any numbers that did not work well? Explain.

Yes, I rounded 5,247 to 5,200 and 73 to 70, but I can’t divide 5,200 by 70 without paper and pencil.

Write 376 ÷ 45. Invite students to think–pair–share to estimate the quotient. Students’ answers are likely to include the ones shown. Highlight that each of these three different estimates is reasonable. Ask students to share their reasoning and highlight any reasoning that involves identifying multiples or using multiplication or division facts. For example, students might choose to divide 360 by 4 because they know 360 is a multiple of 40 or because they know 36 ÷ 4 = 9.

360 ÷ 40 = 9

400 ÷ 40 = 10

400 ÷ 50 = 8

Display the problem and sample work.

Leo is making an estimate for the value of 26,215 ÷ 56. Analyze his work. Do you think Leo’s estimate is reasonable? Why?

No. Leo’s estimate is not reasonable. 56,000 is more than double 26,215, so his estimate probably will not be anywhere close to the actual quotient.

UDL: Engagement

Emphasize that identifying and understanding when strategies do not work well can help students learn and use that information to be successful in the future.

How can Leo make his estimate more reasonable?

Leo can divide 25,000 by 50.

Leo can divide 24,000 by 60.

Direct students to problems 5–7. Read the directions aloud. Explain that students should show the numbers they use to find the estimate but should be able to use mental math to do the calculations. Circulate and support students as they work.

Estimate each quotient. Show your thinking.

When students are finished, select one or two problems from problems 5–7 and invite students to share their estimates. Explain that each problem has multiple correct answers because each person has different ideas for how to turn a challenging calculation into a simpler one. Highlight student reasoning that involves thinking about multiples when choosing numbers to use to calculate an estimate.

Ask students to turn and talk with a partner about how to choose the numbers to use when they estimate a quotient.

Estimate Products and Quotients in the Real World

Students estimate products and quotients in real-world situations. Direct students to complete problems 8 and 9 with a partner.

8. Miss Baker buys 327 hats for students at her school. Each hat costs $18. About how much do the hats cost in total? Show your thinking.

327 × 18 ≈ 330 × 20 = 6,600

The total cost of the hats is about $6,600.

9. A runner climbs 1,276 stairs in 11 minutes. Estimate the number of stairs the runner climbs in 1 minute. Show your thinking.

1,276 ÷ 11 ≈ 1,300 ÷ 10 = 130

The runner climbs about 130 stairs per minute.

When students are finished, invite them to compare their answers for problems 8 and 9 with another pair. Encourage pairs to discuss which estimates they think are more reasonable and why. Then ask the following question.

How do you know that the answer to problem 8 should be an estimate and does not have to be exact?

The question asks “about how much” instead of “how much.” Invite students to turn and talk with a partner about why Miss Baker might need only an estimate of the cost.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Differentiation: Challenge

For students who need an additional challenge, consider providing the following problem.

A toilet paper company advertises that their toilet paper squares are 4 1 2 inches long and that each roll has 426 squares. Approximately how many feet long is a roll of this company’s toilet paper?

Students should reason that the answer is between 4 × 420 and 5 × 430. The first product is a low estimate and the second product is a high estimate.

Land

Debrief 5 min

Objective: Estimate products and quotients by using powers of 10 and their multiples.

Facilitate a class discussion about estimating products and quotients by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Why do we estimate?

We estimate to find an approximate answer mentally before we compute.

We estimate to determine whether an answer or a number is reasonable.

We estimate in the real world when we do not need an exact answer.

What is important to keep in mind when you use estimation?

You should round to numbers that you can calculate mentally.

You should round appropriately.

There is more than one correct estimate for any product or quotient.

Why might you round a number to a power of 10, or to a multiple of a power of 10, before you estimate a product or quotient?

I can multiply and divide mentally when I round a number to a power of 10.

When I multiply or divide by 10, 100, 1,000, or any other power of 10, it makes it simpler for me and leads to a reasonable estimate.

When I round to a multiple of a power of 10, it is helpful because I can think about how many 10s I need to equal the number I’m rounding. Then I can multiply or divide mentally and use the number of 10s to help me shift digits to the right or left.

Exit Ticket 5

min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

8. Kelly and Adesh each write an expression to show how to estimate 1,846 × 7

Estimate each product. Show your thinking.

7. Scott started to make an estimate for 718 × 41 but did not finish.

a. Complete the equations to finish Scott’s estimate.

b. Is Scott’s estimate greater or less than the actual product of 718 and 41? Explain how you know without calculating the actual product.

Scott’s estimate is less than the actual product because he rounded both factors to a smaller number.

Whose estimate is closer to the actual product? Explain your answer.

Kelly’s estimate is closer to the actual product because she only rounded one factor to a greater number. Adesh rounded both factors to greater numbers, which makes his estimate greater than Kelly’s estimate.

Estimate each quotient. Show your thinking.

15. Tim makes a mistake when he estimates 3,714 ÷ 94. What mistake does Tim make? 3,714 ÷ 94 ≈ 3,600 ÷ 90 = 400

Tim has too many zeros in his estimated quotient. Tim’s estimated quotient should be 40.

16. The table shows the cost of tickets for a concert.

Adult Ticket Child Ticket $27 $18

a. There are 8,309 adults at the concert. About how much was spent on adult tickets?

About $240,000 was spent on adult tickets.

b. The total amount spent on children’s tickets was $6,288. About how many children are at the concert?

About 300 children are at the concert.

Convert measurements and describe relationships between metric units.

Lesson at a Glance

Students identify real-world objects with given metric lengths, weights, or capacities. They use multiplicative comparison language to describe relative sizes of units. Students use whole-number multiplication to express larger-size units as smaller-size units. This lesson formalizes the terms millimeter, centiliter, kiloliter, milligram, and centigram.

Key Questions

• What connections do you notice across metric units of weight, length, and capacity?

• How are metric units related to powers of 10?

Achievement Descriptor

5.Mod1.AD12 Convert among whole-number amounts within the metric measurement system to solve problems. (5.MD.A.1)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Relative Size of Metric Units

• Convert Metric Units

• Problem Set

Land 10 min

Teacher

• Meter stick

• Chart paper Students

• Meter stick (1 per student group)

Lesson Preparation

Gather 1 meter stick per group of 3–5 students.

Fluency

Whiteboard Exchange: Estimate Products

Students use rounding to estimate the product of a one-digit number and a two-digit number to prepare for assessing the reasonableness of products in topic B.

Display 18 × 3 ≈ × 3.

What is 18 when rounded to the nearest ten? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. 20

Display the rounded factor.

When I give the signal, let’s read the statement together. Ready?

18 × 3 is about 20 × 3.

Display 18 × 3 ≈ .

Write and complete the statement with the estimated product.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the estimated product.

Repeat the process with the following sequence:

Choral Response: Convert Metric Units

Students convert kilometers to meters, kilograms to grams, or liters to milliliters to prepare for converting metric measurement units.

Display the table.

How many meters are equal to 1 kilometer? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

1,000 meters

Teacher Note

In grade 4, students expressed measurements in a larger unit in terms of a smaller unit, such as kilometers in terms of meters. In grade 5, beginning in this lesson, students also express measurements in a smaller unit in terms of a larger unit, such as meters in terms of kilometers.

Display the answer.

Continue the process to complete the table for kilometers and meters. Repeat the process for the following sequence of tables:

Launch

Students discuss real-world objects with given metric measurements.

Play the Metric Length Units video. If necessary, replay the video and ask students to note any details. The video shows a runner preparing for a run and several measurements, such as the width of the runner’s shoelace and the distance the runner travels. Give students 1 minute to turn and talk about what they noticed.

Facilitate a discussion about what students noticed in the video and answer any relevant questions they have. Highlight thinking that relates the size of one unit to another. Consider the following possible sequence.

What did you notice about the measurement units in the video?

I notice they are metric units.

I notice the units go from very short to very long.

I notice the units measured the lengths of objects and distances.

What do you wonder about the measurement units?

I wonder what a millimeter is.

I wonder how many millimeters are in a meter.

I wonder how many centimeters are in a kilometer.

In what ways can you describe the sizes of the units?

I can use words like long and short.

I can describe other objects or distances that are the size of each unit. I can use one unit to describe another. I know a kilometer is the same length as 1,000 meters.

I know a meter is 100 times longer than a centimeter.

Why might we choose to measure the length of an object with centimeters instead of meters?

Centimeters are smaller than meters, so they are a good unit to use when you measure something that is not very big, like the length of a pencil. We might measure with centimeters if we have only a ruler and not a meter stick.

Transition to the next segment by framing the work.

Today, we will describe relationships between metric units and convert measurements.

Learn

Relative Size of Metric Units

Materials—T: Meter stick, chart paper; S: Meter stick

Students identify patterns between metric units.

Give a meter stick to each small group of three to five students. Tell students that the meter stick shows three metric units. Use a meter stick to draw and label three units: meter, centimeter, and millimeter.

Where have you seen or heard the prefixes centi- and milli- before?

We’ve measured length with centimeters, and we’ve measured the amount of water and other liquid with milliliters.

I have heard the word cent before when talking about money.

I have heard the words century and millennium before when talking about the number of years.

I have heard of centipedes and millipedes.

Hold up a meter stick. Tell students the length of the stick is 1 meter. Have students analyze the tick marks with their small group and note where they see centimeters and where they see millimeters.

Where do you see centimeters? How many centimeters are in 1 meter?

We see the centimeters labeled, and there are 100 centimeters in 1 meter.

Where do you see millimeters?

Millimeters are the smallest unit, so the smallest tick marks represent millimeters. Have students use the meter stick to count the number of millimeters between 0 centimeters and 1 centimeter.

How many millimeters are in 1 centimeter?

10 millimeters are in 1 centimeter.

Teacher Note

Students become familiar with prefixes in grade 3 in English language arts. Students also use common Greek and Latin prefixes in grade 5 to determine the meanings of words.

If there are 10 millimeters in 1 centimeter and there are 100 centimeters in 1 meter, how many millimeters are in 1 meter? How do you know?

There are 1,000 millimeters in 1 meter because 10 × 100 = 1,000.

Does the meter stick show kilometers? Why?

The meter stick does not show kilometers because a kilometer is much longer than 1 meter. We would need 1,000 meter sticks to represent 1 kilometer.

We have just named some metric units that measure length or distance: kilometer, meter, centimeter, and millimeter.

Display the list of metric units of length. Invite students to share statements that summarize the relative sizes of the units by using times as long language. As students name equivalencies, record matching equations on an anchor chart, calling attention to the abbreviations for each unit.

kilometer, meter, centimeter, millimeter

longestshortest

1 kilometer is 1,000 times as long as 1 meter.

Record 1 km = 1,000 m.

1 meter is 100 times as long as 1 centimeter.

Record 1 m = 100 cm.

1 meter is 1,000 times as long as 1 millimeter.

Record 1 m = 1,000 mm.

1 centimeter is 10 times as long as 1 millimeter.

Record 1 cm = 10 mm.

How can 1 meter equal 100 centimeters, but also equal 1,000 millimeters?

We can determine that 1 meter equals both 100 centimeters and 1,000 millimeters because centimeters and millimeters are different sizes. Millimeters are very small, so we need more of them than centimeters to equal 1 meter.

Language Support

Use the following sentence frames to support students in describing relationships between units.

Length: is times as long as .

Weight: is times as heavy as .

Capacity: is times as much as .

UDL: Representation

Display the anchor chart throughout this and the next lesson. Metric conversions are revisited in module 4.

Encourage students to identify real-world objects that represent each measurement and add illustrations or images.

Now let’s apply what we know about metric units that measure length and discuss metric units that measure weight.

Display the list of metric units of weight. Consider asking students which units they would choose to measure the weight of different objects and why. Or if needed, provide students with the weights of the following objects: a small feather weighs about 1 milligram, a large ant weighs about 1 centigram, a paperclip weighs about 1 gram, and a pineapple weighs about 1 kilogram.

kilogram, gram, centigram, milligram

heav iest lightest

How are the metric units for weight similar to the metric units for length?

Millimeter and milligram have the same prefix. Centimeter and centigram also have the same prefix, and so do kilometer and kilogram. Meter and gram do not have a prefix because both can be represented in the ones place on a place value chart.

Invite students to share statements that summarize the relative sizes of the units by using times as heavy language. As students name equivalencies, record matching equations on the anchor chart. For example, a kilogram is 1,000 times as heavy as 1 gram, so record 1 kg = 1,000 g.

Now let’s apply what we know about metric units that measure length and weight and discuss metric units that measure capacity.

Display the list of metric units of capacity.

How are metric units of capacity similar to metric units of length and weight?

The prefixes used to describe the relative sizes of units are the same. For example, there are 1,000 milliliters in 1 liter just like there are 1,000 millimeters in 1 meter and 1,000 milligrams in 1 gram.

Teacher Note

The modules in grade 5 refer to metric weight rather than mass. Technically those units are not equivalent, but both weight and mass can be used to describe the same measure if the object being measured stays on Earth and is subject to Earth’s gravity. If students have already been introduced to the distinction between weight and mass, it may be appropriate to use the term mass instead.

Invite students to share statements that summarize the relative sizes of the units by using times as much language. As students name equivalencies, record matching equations on the anchor chart. For example, a centiliter is 100 times as much as 1 liter, so record 1 L = 100 cL.

Convert Metric Units

Students rename metric units. Have two students each hold up a meter stick, end to end.

How many meters are we showing?

2 meters

How many centimeters are we showing? How do you know?

It shows 200 centimeters because 1 m = 100 cm, so 2 m = 200 cm.

How many millimeters are we showing? How do you know?

It shows 2,000 millimeters because 1 m = 1,000 mm, so 2 m = 2,000 mm.

In your head, you might have added twice or multiplied to find the number of centimeters or millimeters in 2 meters. Either way is correct. Let’s say we have 32 meters and want to rename, or convert, it to centimeters or millimeters. Is it more efficient to add or to multiply?

It is more efficient to multiply than to add 100 or 1,000 thirty-two times.

What multiplication equation can you use to convert 32 meters to centimeters?

32 × 100 = 3,200

Let’s explore more deeply how we can convert measurements by multiplying.

Write 32 m = cm. Below that, write 32 m = 32 × 1 m.

How do you know this equation is true?

32 meters is equal to 32 units, or 32 groups, of 1 meter.

What are the factors?

32 and 1 meter

Differentiation: Support

Consider giving students a ruler with centimeters and millimeters to refer to during this segment. Students can count the number of millimeters that make a centimeter and confirm the relationship between the two units.

Language Support

Convert is a familiar term from grade 4. Consider supporting students’ understanding of this term and related terms by sharing that to convert units means to rename units. The mathematically correct way to describe renaming metric units is by using the word converting.

Teacher Note

In this lesson, students convert larger units to smaller units by using whole number multiplication. In modules 3 and 4, students convert smaller units to larger units by using multiplication of fractions less than 1 and decimals.

Highlight 1 m to make it clear that this is the starting measurement.

We are converting from meters to centimeters, so we can ask ourselves, How many centimeters are equal to 1 meter?

100 centimeters

In the equation, let’s write 1 meter as 100 centimeters. Let’s use highlighting to make it clear that 100 centimeters and 1 meter represent the same length.

Write = 32 × 100 cm. Highlight 100 cm to make it clear that it is equivalent to 1 m.

What is 32 × 100 centimeters?

3,200 centimeters

So what does the equation tell us?

32 meters is equal to 3,200 centimeters.

Write 32 m = 3,200 cm.

How can these two numbers possibly represent the same length?

The numbers can represent the same length because the units are different. We know that 32 m and 3,200 cm are equal because of the sizes of the units. A meter is a larger unit than a centimeter. A meter is 100 times as long as a centimeter, so we need 100 times as much as 32 to represent an equal number of centimeters.

Direct students to problem 1 in their books. Then ask the following questions to guide students through the conversion. Write the conversion equations and direct students to do the same.

Convert.

1. 456 kL = 456,000 L

456 kL = 456 × 1 kL = 456 × 103 L = 456,000 L

Let’s convert 456 kiloliters to liters. First let’s record 456 kiloliters as a multiplication expression. How can we write that?

We can write 456 × 1 kL.

Promoting the Standards for Mathematical Practice

When students learn the meaning of prefixes for metric units and convert between units by using powers of 10, they attend to precision (MP6)

Ask the following questions to promote MP6:

• What does cg mean in the measurement 10 cg?

• What does 103 mean in the equation 456 kL = 456 × 103 L?

• Where might it be common to make mistakes when you convert between metric units?

We want to convert kiloliters as liters. What can we ask ourselves?

How many liters are equal to 1 kiloliter?

And how many liters are equal to 1 kiloliter?

1,000 liters are equal to 1 kiloliter.

We are using the conversion 1 kL = 1,000 L, and 1,000 is a power of 10. How can we write 456 × 1,000 L with exponents?

456 × 103 L

So how many liters are equal to 456 kiloliters converted to liters? How do you know?

456,000 liters is equal to 456 kiloliters. A kiloliter is 1,000 times as much as a liter, so we need 1,000 times as many as 456 to represent an equal amount of liters. Because we are multiplying by 103, we shift each digit three places to the left, so the number of liters is 456,000.

Record 456 kL = 456,000 L.

Have students complete problems 2–5 with a partner. Encourage students to practice recording their products with exponents to reinforce their learning from prior lessons. Problem 5 includes the new complexity of mixed units within a measurement (i.e., 2 kg 300 g ). Allow time for productive struggle as partners make sense of the problem, but provide support as needed by asking students whether it might be helpful to convert 2 kg to grams to find the total weight in grams before they make the conversion to milligrams.

2. 6,985 g = 6,985,000 mg

6,985 g = 6,985 × 1 g = 6,985 × 103 mg = 6,985,000 mg

3. 30,800 cm = 308 m

308 m = 308 × 1 m = 308 × 102 cm = 30,800 cm

Differentiation: Challenge

For students who need a challenge, consider asking them to convert a unit larger than a meter (or a gram or a liter) to a unit smaller than a meter (or a gram or a liter). For example, 4 km = cm

= 4 × 1 km

= 4 × 1,000 m

= 4 × 1,000 × 1 m

= 4 × 1,000 × 100 cm

= 400,000 cm

Encourage students to use multiple conversions if they are unsure of a direct conversion from kilometers to centimeters.

4. The label on a water bottle shows the capacity of the bottle is 50 centiliters. What is the capacity of the bottle in milliliters?

50 cL = 50 × 1 cL = 50 × 10 mL = 500 mL

The capacity of the bottle is 500 milliliters.

5. The label on a bag of rice reads 2 kg 300 g. The label on a bag of beans reads 2,300 mg. Which bag is heavier?

2 kg = 2 × 1 kg = 2 × 1,000 g = 2,000 g

2,000 g + 300 g = 2,300 g

2,300 g = 2,300 × 1 g = 2,300 × 1,000 mg = 2,300,000 mg

The bag of rice is heavier because 2,300,000 mg is greater than 2,300 mg.

When students are finished, ask the following questions.

What do each of the problems have in common?

In each problem we converted among metric units.

In each problem we converted from a larger metric unit to a smaller metric unit.

What did you notice about problem 3?

I noticed problem 3 had a blank on the left side of the equal sign.

Did having the blank on the left side of the equal sign make you think any differently about the problem? Explain.

No. I did not have to think differently because I did the same kind of conversions as I did for problems 1 and 2, but with meters and centimeters this time.

The only difference was that I put my answer in the blank on the left side of the equal sign.

How is problem 5 different from the others?

Problem 5 is different because one of the measurements includes kilograms and grams.

How did you determine that the bag of rice is heavier than the bag of beans?

We had to convert kilograms to grams to find the total weight of the bag of rice in grams before we converted that weight to milligrams so we could make the comparison.

Problem Set

Differentiate the set by selecting problems for students to finish within the timeframe. Problems are organized from simple to complex.

Land Debrief 5 min

Objective: Convert measurements and describe relationships between metric units.

Gather the class with their Problem Sets. Facilitate a class discussion about relationships between metric units by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

How is a millimeter related to other metric units of length?

A centimeter is 10 times as long as 1 millimeter.

A meter is 1,000 times as long as 1 millimeter.

A kilometer is 1,000,000 times as long as 1 millimeter.

UDL: Action & Expression

Consider supporting students in evaluating their progress. Provide the following prompts and encourage partners to reflect.

• What did we do well?

• What might we do differently next time?

How are metric units related to powers of 10? Give a specific example from your Problem Set about how you used a power of 10.

We used a power of 10 to write our products when we convert between metric units. In problem 1, the power of 10 I used was 1,000 or 103 because 1 m = 1,000 mm.

What connections do you notice across the metric units of weight, length, and capacity?

The prefixes we use to describe units are the same. Those prefixes help us know how units are related to each other. For example, there are 1,000 meters in 1 kilometer just like there are 1,000 liters in 1 kiloliter and 1,000 grams in 1 kilogram.

We always multiply by a power of 10 when we convert larger metric units to smaller metric units.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Convert each measurement. Write an expression to help you convert. The first one is started for you.

runs

What is the distance Riley runs in meters?

7. Consider the expressions.

a. Circle the expression that

b. Explain your choice.

I circled the first expression because I know 1 liter = 1,000 milliliters. To find how many milliliters are equal to 600 liters, I need to multiply 600 by 1,000. The second and third expressions show 600 × 1,000 in different ways.

13. Mr. Sharma’s dog weighs 21 kg 96 g. What is the weight of Mr. Sharma’s dog in grams? Mr. Sharma’s dog weighs 21,096 grams.

Solve multi-step word problems by using metric measurement conversion.

Lesson at a Glance

Students solve multi-step word problems involving metric conversions. Students use the Read–Draw–Write process to make sense of and solve problems. Through a guided experience, students draw tape diagrams, and they analyze their diagrams to determine what information they have and what information they still need to solve the problem.

Key Questions

• Does modeling with tape diagrams help you solve problems? How?

• Is a model helpful when solving problems involving metric units? How?

Achievement Descriptor

5.Mod1.AD12 Convert among whole-number amounts within the metric measurement system to solve problems. (5.MD.A.1)

Fluency

Whiteboard Exchange: Estimate Products

Students use rounding to estimate the product of a one-digit number and a three-digit number to prepare for assessing the reasonableness of products in topic B.

Display 174 × 3 ≈ × 3.

What is 174 when rounded to the nearest hundred? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

200

Display the rounded factor.

When I give the signal, let’s read the statement together. Ready?

174 × 3 is about 200 × 3.

Display 174 × 3 ≈ .

Write and complete the statement with the estimated product. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the estimated product.

Repeat the process with the following sequence:

Choral Response: Convert Metric Units

Students convert meters to centimeters, liters to centiliters, or grams to centigrams to prepare for solving multi-step word problems involving metric measurement conversions.

Display the table.

How many centimeters are equal to 1 meter? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

100 centimeters

Display the answer.

Continue the process to complete the table for meters and centimeters. Repeat the process for the following sequence of tables:

Students convert metric units to compare the weights of two collections of gold coins.

Display the bags with labeled weights.

While Toby plays a video game, he sees two collections of gold coins. If he determines which collection of coins weighs more, he gets to keep both collections. Work with a partner to determine which collection weighs more. Be prepared to defend your answer.

Invite students to share their answers and strategies. Anticipate that while some students may convert all units to grams, other students may convert all units to milligrams or to another unit.

Collection A weighs 13,000 g + 10,000 g + 200 g = 23,200 g.

Collection B weighs 21,000 g + 2,000 g + 6 g = 23,006 g.

Then ask the following question.

Why is it an efficient strategy to convert all the measurements in one collection to the same unit?

When we convert all the measurements in one collection to the same unit, we can add the measurements together and then compare that total with the total from the other collection.

Transition to the next segment by framing the work.

Today, we will solve real-world problems that involve converting metric units.

Learn

35 Teacher Note

Solve Problems with Metric Units

Students solve multi-step problems involving metric unit conversions. Tell students they will use the Read–Draw–Write process to solve multi-step word problems. Display problem 1 and direct students to the problem in their books. Ask students to silently read the problem.

Use the Read–Draw–Write process to solve the problem.

1. Sasha has 6 meters 40 centimeters of ribbon. She plans to divide the ribbon equally to wrap 8 gifts that are the same size. How many centimeters of ribbon should Sasha cut for each gift?

Consider reviewing the Read–Draw–Write problem-solving process with students before beginning the word problems.

Read–Draw–Write (RDW)

Read the problem all the way through. Then reread a chunk at a time. As you reread, ask yourself, “Can I draw something?” then “What can I draw?”

Draw to represent the problem as you reread. Add to or revise your drawing as you uncover new information or discover what is unknown. As you draw, label what is known and what is unknown.

80 cm

6 m = 600 cm

600 cm + 40 cm = 640 cm

640 ÷ 8 = 80 Sasha should cut 80 centimeters of ribbon for each gift.

Read the problem chorally with the class and encourage students to visualize the situation as they listen. Have the class read the problem again. Guide them to stop reading and draw each time new information is provided. As you model drawing and labeling, direct students to do the same.

When you finish rereading and drawing, ask yourself, “What does my drawing show me?” Let your drawing help you find a way to solve. Write number sentences or equations to represent your thinking. Solve the problem.

Then use your solution to write a statement that answers the original question.

Language Support

After reading each problem, consider showing students pictures of the items involved in the problem. For example, show ribbon or a gift wrapped with ribbon for problem 1, and a map of the United States indicating the locations of New York City, Chicago, and Seattle for problem 2.

Read the first part of the problem with the class and stop after the first sentence.

Can we draw something? What can we draw?

Yes, we can draw a tape diagram to represent the length of ribbon Sasha has. Draw a tape diagram that shows a total of 6 meters 40 centimeters.

Read the second sentence aloud.

Can we draw something? What can we draw?

Yes, we can partition our tape diagram into 8 equal parts to represent the 8 gifts.

Partition the tape diagram into 8 equal parts.

We know that the label 6 m 40 cm represents the total length of the ribbon. What does 1 part represent?

It represents the length of ribbon needed to wrap 1 gift.

Read the third sentence aloud.

Have we learned any new information?

It says Sasha will cut the ribbon into equal pieces for gifts, and the length of those pieces will be in centimeters. Right now, the total in the tape diagram shows meters and centimeters.

When we learn new information, we can revise or add on to our model. Should we revise our model? How?

Yes. We should convert the length of the ribbon to centimeters.

How many centimeters is 6 meters 40 centimeters? How do you know?

6 meters 40 centimeters is equal to 640 centimeters because 6 × 100 + 40 = 640.

Teacher Note

Tape diagrams are used across grades beginning in grade 1. They are not the only applicable model, but tape diagrams are used consistently because they can represent many situations.

Differentiation: Support

For students who need additional support with tape diagrams or who have never used tape diagrams, consider highlighting the following aspects of tape diagrams:

• Each box on the tape diagram is called a part.

• Each part has the same value when the tape diagram is partitioned equally.

• You can use brackets or arms to show the total value of the tape diagram, or the value of a section.

• You can use a question mark or a letter to represent an unknown value.

Label the total length 640 centimeters. Direct students back to the third sentence.

What does the question ask us to find?

The question asks us to find the length of ribbon Sasha needs for each gift.

Let’s show what we are asked to find by using a question mark to represent the unknown.

Label one part of the tape diagram with a question mark.

Look at our model. What do we know?

We know Sasha’s ribbon has a total length of 640 centimeters.

?

We know the length of ribbon for each gift will be less than 640 centimeters.

How can we find the length of ribbon needed to wrap each gift?

We can divide 640 by 8.

How many centimeters of ribbon should Sasha cut for each gift?

80 centimeters

Write your answer in a sentence. Sasha should cut 80 centimeters of ribbon for each gift. Have students turn and talk about whether the answer is reasonable.

Promoting the Standards for Mathematical Practice

Students reason quantitatively and abstractly (MP2) as they draw tape diagrams and use metric measurement conversions to solve real-world problems.

Ask the following questions to promote MP2:

• What does the problem ask you to find?

• Does the tape diagram represent the problem? How?

• Does your answer make sense?

Display problem 2 and ask students to silently read the problem.

Use the Read–Draw–Write process to solve the problem.

2. A family takes a road trip from New York City to Seattle, and they stop in Chicago on the way. The distance from New York City to Chicago is 1,963 kilometers less than the distance from Chicago to Seattle. The distance from Chicago to Seattle is 3,288 kilometers. If the family travels the same route to Seattle and back, how many total meters do they travel?

+

9,226 × 1,000 = 9,226,000

The family travels 9,226,000 meters.

Read the problem chorally with the class and encourage students to visualize the situation as they listen. Have students think–pair–share about where they could stop reading and draw something.

We could stop after the third sentence because that tells us something exact about the distance between two of the cities.

We could stop after the second sentence because we know we need two tapes and that one should be shorter than the other.

We could stop after the first sentence because we know we need two tapes and we could label them New York City to Chicago and Chicago to Seattle.

Remind students that as they go through the Read–Draw–Write process, it is up to them to determine when to stop and draw. Students might decide to stop at different places in the text, and that is okay.

We could stop after the first, second, or third sentence here. Let’s stop after the second sentence and use what we know to draw.

Reread the first two sentences aloud.

What can we draw?

We can draw a tape and label it New York City to Chicago. Then we can draw another tape and label it Chicago to Seattle.

Which tape should be shorter?

The tape that we label New York City to Chicago should be shorter because the story says that the distance is some kilometers less than the distance from Chicago to Seattle.

To save ourselves a little time and space, let’s use the abbreviation NYC for New York City when we label our tape diagram.

Model how to draw the tapes so they are lined up vertically. When the tapes are aligned vertically, it shows the relationship between them.

Can we label anything else?

We know the distance between New York City and Chicago is 1,963 kilometers less than the distance between Chicago and Seattle. So the part of the tape diagram that represents the difference between the distance from Chicago to Seattle and New York City is equal to 1,963 kilometers.

Label as students do the same.

Read the third sentence.

Can we draw anything? What can we draw?

There is nothing else to draw, but we can label the tape to show that the distance from Chicago to Seattle is equal to 3,288 kilometers.

Label as students do the same.

Read the fourth sentence.

What does the question ask us to find?

We need to find the total distance in meters if the family goes from New York City to Chicago to Seattle and back again.

Can we see that unknown in our model? What does our model show?

No. Our model shows the distance from New York City to Chicago and the distance from Chicago to Seattle. We know the distance from Chicago to Seattle. We do not know the distance from New York City to Chicago.

Have students independently determine the distance, in kilometers, from New York City to Chicago by using what they know from the model. Once students determine the distance, label it in the tape diagram.

Emphasize that the model now shows the distance between cities. The question asks to find the total distance both to Seattle and back. Have students think–pair–share about a model they could draw to represent the total distance the family travels.

We could make each tape twice as long.

We could show a part-whole tape with two 1,325s and two 3,288s.

Display the following tape.

What do you notice about this tape?

The tape has a question mark. The question represents the unknown of the word problem.

Why did we need to draw this tape?

Our first model only shows one part of the problem. We need a second model to represent the second part and the unknown of the problem.

Share that sometimes students may learn as they draw that the unknown of the question is not in that model and they must draw a second tape diagram to help represent the problem. Direct students to draw the tape diagram.

What is a reasonable estimate for the total distance the family travels?

10,000 km is a reasonable estimate because 1,325 ≈ 1,500 and 3,288 ≈ 3,500. Two 1,500s and two 3,500s is equal to 10,000.

Have students work with a partner to solve the problem and record their final answer statement. Then gather the class and discuss.

How did you solve the problem?

We found 1,325 + 3,288 + 1,325 + 3,288.

We found 1,325 + 3,288 and then multiplied by 2.

We found 1,325 × 2 and 3,288 × 2, then added the products.

Notice how different students took different routes to compute and still found the same answer.

Is the distance we found, 9,226 kilometers, reasonable?

Yes, because we estimated the distance would be about 10,000 kilometers.

Did you include 9,226 kilometers in your final answer sentence?

No, because the question asked for the distance in meters.

How did you convert 9,226 kilometers to meters?

We know 1 kilometer = 1,000 meters. So we found 9,226 × 1,000. When we multiplied by 1,000 we knew each digit of 9,226 should shift three places to the left, so the number of meters is 9,226,000.

Invite students to turn and talk about how tape diagrams helped them find a path to solve the problem.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

UDL: Action & Expression

Consider posting the exemplar tape diagrams for students to refer to as they work independently. ? ?

Land

Debrief 5 min

Objective: Solve multi-step word problems by using metric measurement conversion.

Gather the class with their Problem Sets. Facilitate a class discussion about solving word problems involving metric conversions by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Does modeling with tape diagrams help you solve problems? How?

Modeling helps me see what I know, what I do not know, and the unknown.

It helps me better understand the problem because I am drawing only one part at a time.

It helps me decide whether I can add, subtract, or multiply.

Modeling helps me see when I need multiple steps to find the answer.

If the answer I find makes sense for the unknown in the tape diagram, then I know I answered the question.

Is a model helpful when solving problems involving metric units? How? Give an example from your Problem Set.

It was helpful to draw a model for problem 1 because I wasn’t sure whether I needed to multiply or divide to answer the question. I drew a tape diagram with 8 equal parts to represent the 8 beakers, then I labeled each part with 750 mL. Then I realized I needed to multiply to find the total.

It was helpful to draw a model for problem 4. After I drew a tape diagram to show how much blue ribbon and how much green ribbon Eddie had, I realized that I could not divide the blue ribbon into 9 cm lengths until I converted 4 m 23 cm to centimeters.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Use the Read–Draw–Write process to solve each problem.

1. Mr. Perez pours water into 8 beakers. He pours 750 milliliters of water into each beaker.

a. About how many milliliters of water are in the beakers altogether?

8 × 750 ≈ 8 × 800 = 6,400

There are about 6,400 milliliters of water in the beakers.

b. Exactly how many milliliters of water are in the beakers altogether?

8 × 750 = 6,000

There are 6,000 milliliters of water in the beakers.

c. How do you know your answer in part (b) is reasonable?

My answer is reasonable because 6,000 milliliters is close to my estimate of 6,400 milliliters.

2. A newborn lion cub weighs 1 kg 736 g. The lion cub weighs 8 times as much as a newborn puppy.

a. Convert the weight of the lion cub to grams.

1 kg = 1,000 g

1,000 + 736 = 1,736

The lion cub weighs 1,736 grams.

b. About how many grams does the puppy weigh?

1,736 ÷ 8 ≈ 1,600 ÷ 8 = 200

The puppy weighs about 200 grams.

c. Exactly how many grams does the puppy weigh?

1,736 ÷ 8 = 217

The puppy weighs 217 grams.

d. How do you know your answer in part (c) is reasonable?

My answer is reasonable because 217 grams is close to my estimate of 200 grams.

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

3. Leo uses oil and vinegar to make a bottle of salad dressing. He uses 12 centiliters of vinegar. He uses 3 times as much oil as vinegar. How many milliliters of salad dressing does Leo make?

12 × 3 = 36

12 + 36 = 48

48 × 10 = 480

Leo makes 480 milliliters of salad dressing.

4. Eddie has a blue ribbon that is 4 m 23 cm long and a green ribbon that is 756 cm long. He cuts each ribbon into pieces that are 9 cm long. How many more pieces of green ribbon than blue ribbon does Eddie have?

4 m 23 cm = 423 cm

756 − 423 = 333

333 ÷ 9 = 37

Eddie has 37 more pieces of green ribbon than blue ribbon.

5. A farmer puts apples into 36 crates. Each crate has 25 kilograms of apples in it. She sells 486,235 grams of apples. How many grams of apples does she have left?

36 × 25 = 900

900 kg = 900,000 g

900,000 − 486,235 = 413,765

She has 413,765 grams of apples left.

Topic B Multiplication of Whole Numbers

In topic B, students explore the question, How can I multiply whole numbers efficiently?

Students dive into the learning of the topic by applying familiar methods from grades 3 and 4 to solve a word problem involving multiplication. This way, they can show what they already know, explore how those methods work with numbers that have more digits, and make connections between the various methods. Students discover that each method requires them to decompose at least one factor, whether they do it on paper or mentally.

Students continue to relate various methods of multiplication, but the number of digits in the factors increases to two- and three-digit numbers multiplied by two-digit numbers. They consider the factors in the product and learn to designate a unit that can lead to fewer partial products or to partial products that can be found by using mental math. As the number of digits in factors increases, students recognize the need to decompose both factors as opposed to only one, so they can continue to apply mental math to find the product.

Students advance to multiplying three- and four-digit numbers by three-digit numbers by using the standard algorithm while solving one-step word problems. They continue to use the area model as a tool to support their thinking and to make connections between the area model and the number of partial products in the standard algorithm. Students multiply by using the standard algorithm and by visualizing how factors can be decomposed instead of by using the area model. Throughout their multiplication work in the topic, students are prompted to estimate before finding the product and are asked to assess whether their answers are reasonable.

In topic C, students divide whole numbers.

Progression of Lessons

Lesson 7

Multiply by using familiar methods.

Lesson 8

Multiply two- and three-digit numbers by two-digit numbers by using the distributive property.

Lesson 9

Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm.

I can apply multiplication methods that I previously learned to multiply numbers with more digits. I can decompose one factor to make multiplication simpler for me.

I can make connections across the break apart and distribute method, the area model, and vertical form. In each of these methods, I can apply the distributive property to find the product, so the partial products are the same. I can also multiply the factors in any order because of the commutative property of multiplication.

I can decompose one or both factors to make it simpler for me to multiply. I can use the same thinking when I multiply by using the standard algorithm as I do when I multiply by using an area model. I can make connections between the number of rows in an area model and the number of partial products in the standard algorithm, and I can use that to decide which factor I designate as the unit and which factor represents the number of groups.

Lesson 10

Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm.

Lesson 11

Multiply two multi-digit numbers by using the standard algorithm.

I can multiply by using the standard algorithm, and I can use the area model to help me, if needed. When I find partial products, I can use unit form thinking or standard form thinking, or I can multiply single digits while I hold place value in my head.

I can multiply by using the standard algorithm. I can be strategic about which factor I designate as the unit, based on how many partial products that might make. I can analyze other students' mistakes and offer advice.

Multiply by using familiar methods.

Lesson at a Glance

Students use prior knowledge to multiply a five-digit factor by a one-digit factor. This lesson is an opportunity for informal assessment, to gauge students’ prior knowledge of multiplication, and to determine what methods, models, or representations students rely on from grade 4. Student-driven discussions occur as students explore multiplication methods.

Key Question

• Is it helpful to decompose a factor when you multiply? How?

Achievement Descriptors

5.Mod1.AD1 Write whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm. (5.NBT.B.5)

Agenda Materials

Fluency 10 min

Launch 10 min

Learn 30 min

• Select a Method to Multiply

• Share, Compare, and Connect

• Apply a Different Recording Method to Multiply

• Problem Set

Land 10 min

Lesson Preparation

Fluency

Whiteboard Exchange: Word Form to Standard Form

Students write the standard form of a four- or five-digit number given in word form to maintain fluency with writing numbers within 1,000,000 from grade 4.

Display three thousand, eight hundred nineteen =      .

When I give the signal, read the number shown in word form. Ready?

Three thousand, eight hundred nineteen

Write the number in standard form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the answer.

three thousand, eight hundred nineteen = 3,819

Repeat the process with the following sequence:

two thousand, four hundred sevent y = 2,470

four thousand, eighty-t wo = 4,082

seven thousand, seven = 7,007 four teen thousand, two hundred ninety-five = 14,295

fif ty thousand, one hu ndred three = 50,103 t wenty-five thousand, six hundred four = 25,604

eighty-six thousand, t wenty = 86,020

Copyright © Great Minds PBC

Whiteboard Exchange: Estimate Products

Students use rounding to estimate the product of a one-digit number and a four-digit number to prepare for assessing the reasonableness of products beginning in lesson 8.

Display 1,832 × 3 ≈ × 3.

What is 1,832 when rounded to the nearest thousand? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

2,000

Display the rounded factor.

When I give the signal, let’s read the statement together. Ready?

1,832 × 3 is about 2,000 × 3.

Display 1,832 × 3 ≈  .

Write and complete the statement with the estimated product.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the estimated product.

Repeat the process with the following sequence:

Launch

Students represent a five-digit number by using models and expressions.

Direct students to problem 1 in their books. Present the following number and use the Math Chat routine to engage students in mathematical discourse.

1. Write the following number in as many ways as you can.

28,741

Give students 3 minutes of silent work time to represent the number in as many ways as they can. Have students give a silent signal to indicate they are finished.

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between methods.

Then facilitate a class discussion. Invite students to share their thinking with the whole group and to record their reasoning.

As students discuss, validate the many ways in which they express 28,741, but emphasize thinking that uses place value. If any of the following forms or models are not shared by students, share it as your own.

Ask questions that invite students to make connections and allow them to ask questions of their own.

What is similar about how 28,741 is decomposed in each example?

It is decomposed by place value each time. In each representation, we can see the number of ten thousands, thousands, hundreds, tens, and ones.

Gesture to each expanded form example.

What do you notice about Expanded Form and Expanded Form with Multiplication?

Both expanded forms represent the same number.

One form has only addition, and the other form has addition and multiplication. In the form with only addition, each number is written as a multiple of a power of 10. In the form with addition and multiplication, each number is written as a product of a number and a power of 10.

An expression can be a number, such as 8,000, and an expression can also be a product of factors, such as 8 × 1,000.

Display the Expanded Form with Multiplication example that is out of place value order.

Does this example also represent 28,741? How do you know?

Yes, it does. I know because it has the same number of ten thousands, thousands, hundreds, tens, and ones. They are added in a different order.

Transition to the next segment by framing the work.

Today, we will decompose, or break apart, numbers to multiply five-digit numbers by using methods we already know.

Learn Select a Method to Multiply

30 UDL: Representation

Students multiply a one-digit number by a five-digit number.

Direct students to problem 2 and chorally read the problem with the class. Have students work independently to use the Read–Draw–Write process to solve the problem. Encourage students to self-select their tools and methods.

Circulate and observe student work. Select two or three students to share in the next segment. Look for work samples that help advance the lesson’s objective of exploring familiar multiplication methods. Use the following questions and prompts to elicit student thinking:

• Tell me about how your drawing connects to the story.

• Tell me about your method.

• What does this number represent? (Gesture to a number shown in the student’s work.)

• How does your work match your number sentence/expression/equation?

When you speak with students, focus on eliciting student thinking as you informally assess their understanding and select students to share.

Select two or three students to share in the next segment. Purposefully choose work that allows for rich discussion about connections between student work samples.

Consider inviting students to brainstorm the decomposition methods they already know. Create a web graphic organizer for students to reference as they plan to solve problem 2.

Differentiation: Support

For students who need support getting started, ask the following questions:

• What can you draw to represent 5 friends? Do you need to represent anyone else?

• What do you know about the number of breaths a grade 5 student takes? What can you label in your drawing?

• What equation can you write to find the number of breaths 6 grade 5 students take in one day?

• Which method will you use to find the total number of breaths 6 grade 5 students take in one day?

The student work samples shown demonstrate the application of the distributive property of multiplication over addition.

Break Apart and Distribute

Teacher Note

The sample student work shows common responses. Look for similar work from your students and encourage authentic classroom conversations about the key concepts.

If your students do not produce similar work, choose one or two pieces of their work to share and highlight how it shows movement toward the goal of this lesson. Then select one work sample from the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student solved the problem. What do you think this student did?”

Teacher Note

Each of these methods for multiplication is familiar from grade 4 module 3.

Teacher Note

The complete name of the property is the distributive property of multiplication over addition because both operations must be present to apply the property. In grade 5 it is acceptable for students to refer to the property as the distributive property.

Use the Read–Draw–Write process to solve the problem.

2. On a typical day, a grade 5 student takes 24,165 breaths in one day. How many breaths will you and 5 friends take in one day?

?

6 × 24,165 = 144,990

In one day, 5 friends and I will take 144,990 breaths.

Share, Compare, and Connect

Students share and compare solutions and reason about their connections.

Gather the class and invite the students you identified in the previous segment to share their solutions, one at a time. Consider purposefully ordering student work that shows solution paths from least efficient to most efficient.

As each student shares, ask questions to elicit their thinking and to clarify how they found the product. Ask the class questions to help students make connections between the demonstrated solutions and their own work. Encourage students to ask questions of their own.

The sample discussion demonstrates questions that elicit thinking and invites connections.

Break Apart and Distribute (Jada’s Way)

Class, what expression did Jada use to find the answer?

6 × 24,165

Jada, why did you use that expression?

I know each grade 5 student takes 24,165 breaths in one day, and between my 5 friends and me, that is 6 people, so I multiplied 6 by 24,165 to determine the number of breaths we would take in one day.

Language Support

Consider directing partners to the Agree or Disagree section of the Talking Tool to support students in discussing the similarities and differences in their work and the work of their classmates throughout this segment.

Class, examine Jada’s work. How did she find the product?

Jada decomposed 24,165 into 20,000 + 4,000 + 100 + 60 + 5. Then she multiplied each part by 6.

Jada added each partial product to find the total: 144,990.

Jada, did decomposing 24,165 help you multiply? How?

Yes. It helped me think about the units I was multiplying in each part. When I wrote 6 × 20,000, I thought about how I know 6 × 2 = 12, so 6 × 2 ten thousands is 12 ten thousands, or 120,000.

Area Model (Noah’s Way)

Class, examine Noah’s work. How did he find the product?

Noah used an area model. He decomposed 24,165 into five parts: 20,000, 4,000, 100, 60, and 5.

Noah multiplied each part by 6 and added the partial products together.

Noah, tell us why you showed your work this way. I wanted to multiply each part by 6, so I drew an area model to help me organize my thinking. I didn’t want to make a mistake by doing too much mental math when I added the partial products, so I showed my thinking vertically for the addition.

How is Noah’s work similar to and different from Jada’s work?

Noah and Jada both decomposed 24,165 and multiplied each part by 6. They both started by multiplying 6 by the largest unit. I can see they both started with 6 × 20,000.

They both added the products after they multiplied.

Noah showed the decomposed factor by using an area model, but Jada used expanded form with multiplication. They had similar thinking but recorded their work differently.

Partial Products (Sasha’s Way)

Sasha, how did you find the product?

I multipled 6 by the value of the digit in each place. So I got 6 ones × 5 ones = 30 ones, 6 ones × 6 tens = 36 tens or 360, and so on, up to 6 ones × 2 ten thousands = 12 ten thousands or 120,000.

I recorded each of the partial products below and lined them up by place value. Then I added to find the total.

Class, how is Sasha’s work similar to and different from Noah’s and Jada’s work?

Sasha, Noah, and Jada decomposed 24,165 into place value parts and then multiplied each part by 6.

Sasha, Noah, and Jada composed all the partial products to find the whole.

Sasha started by multiplying the smallest unit, ones, but Noah and Jada started by multiplying the largest unit, ten thousands.

Class, even though Sasha started by multiplying by a different unit than Noah and Jada, she still got the same product. Why?

Because when we add the partial products, the order does not matter. The product is the same because we are all using the distibutive property to get the partial products, and we can add the partial products in any order we want to because of the commutative property of addition.

Direct students to think–pair–share about whether decomposing a factor into multiple parts helps them multiply and how it helps.

When we decompose a factor into parts, we can multiply each part by the other factor by using multiplication facts.

Invite students to turn and talk about which recording method they plan to use for the next problem and how using that method can help them to be efficient.

Differentiation: Challenge

Direct students to a work sample that finds the product by using the standard algorithm. Ask students whether they can see the partial products in the standard algorithm. Then ask them to explain.

Promoting the Standards for Mathematical Practice

Students use appropriate tools strategically (MP5) when they choose among different methods to show their thinking, such as expanded form, expanded form with multiplication, or an area model, when they determine the product of two factors.

Ask the following questions to promote MP5:

• What methods can help you solve this problem?

• Which method is the most efficient for you to find the product of the two factors? Why?

• How can you estimate the product? Does your estimate seem reasonable?

Apply a Different Recording Method to Multiply

Students select a different recording method to multiply and to reason about efficiency.

Direct students to problem 3. Give students 3 minutes to represent the problem by using a different recording method than they used in the previous problem. Encourage students to show their thinking.

Circulate as students work. Identify one or two students to share their thinking. Purposefully choose work that allows for students to reflect on different ways they might record or think about partial products.

Multiply. Show or explain your work.

3. 4 times as much as 32,157

32,157 × 4 = 128,628

4 times as much as 32,157 is 128,628.

Invite students to think–pair–share and reflect on the recording method they chose and how that affected their work and thinking.

For problem 2, I used the break apart and distribute method, but for this problem, I showed my thinking in the area model. I was able to find the product more efficiently with the area model because I didn’t rewrite the expression. Last time, I added the partial products mentally but made a silly mistake, so this time I lined up all my partial products to help me add.

For problem 2, I used the break apart and distribute method, and I multiplied beginning with the largest unit. For this problem, I showed partial products. Using partial products, I started by multiplying by the smallest unit, ones, first.

Think about the two methods you tried. Which one was more efficient for you?

Break apart and distribute was more efficient because I could do a lot of the multiplication and addition mentally.

The area model was more efficient because it helped me see each of the partial products better.

Partial products was more efficient because all the partial products were already lined up by place value, so I didn’t have to rewrite them when I added.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Multiply by using familiar methods.

Facilitate a class discussion about multiplication methods by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Is it helpful to decompose a factor when you multiply? How?

Yes. It helps me multiply because I can use mental math to find each partial product since I am thinking about multiplication facts.

Do you think these methods will work if we multiply with numbers that have more digits, such as 234 × 4,567? Explain.

I think they will still work for numbers with more digits. I can use these methods to multiply numbers with fewer digits, so I think they will work for numbers with more digits too.

Would you think any differently when you multiply by a factor with more digits than when you multiply by a one-digit factor?

No. I would still multiply each part of the factor by each part of the other factor. Then I would add all the partial products together to find the final product.

My thinking would not be different, but there would probably be more partial products.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Use the Read–Draw–Write process to solve each problem.

5. Mrs. Chan takes 13,564 steps each day for 4 days. How many total steps does she take in those 4 days?

13,564 × 4 = 54,256

Mrs. Chan takes 54,256 steps in those 4 days.

6. An airplane weighs 40,823 kilograms. What is the total weight of 7 of these airplanes?

7 × 40,823 = 285,761

The total weight of 7 of these airplanes is 285,761 kilograms.

Multiply two- and three-digit numbers by two-digit numbers by using the distributive property.

Lesson at a Glance

Students multiply multi-digit numbers by using the distributive property. They record their thinking with area models, in vertical form, and the break apart and distribute method. Students determine that all of these methods are related because they all decompose one factor, find partial products, and add to find the total. Students explore designating one factor as the unit to expand their understanding of the commutative property of multiplication.

Key Questions

• Does decomposing factors help us multiply? How?

• How do you describe the distributive property?

Achievement Descriptors

5.Mod1.AD1 Write whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm. (5.NBT.B.5)

Agenda Materials

Fluency 10 min

Launch 10 min

Learn 30 min

• Relate Vertical Form and the Break Apart and Distribute Method to the Area Model

• Designate a Unit

• Problem Set

Land 10 min

•

Lesson Preparation

Fluency

Whiteboard Exchange: Word Form to Standard Form

Students write the standard form of a six- or seven-digit number given in word form to build fluency with writing numbers within 10,000,000 from topic A.

Display one hundred sixteen thousand, three hundred ninety-five = .

When I give the signal, read the number shown in word form. Ready?

One hundred sixteen thousand, three hundred ninety-five

Write the number in standard form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the answer.

Repeat the process with the following sequence:

Whiteboard Exchange: Estimate Products

Students use rounding to estimate the product of a one-digit number and a five-digit number to prepare for assessing the reasonableness of products.

Display 19,352 × 3

× 3.

What is 19,352 when rounded to the nearest ten thousand? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

20,000

Display the rounded factor.

When I give the signal, let’s read the statement together. Ready?

19,352 × 3 is about 20,000 × 3.

Display 19,352 × 3 ≈ .

Write and complete the statement with the estimated product.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the estimated product.

Repeat the process with the following sequence:

Launch

Students solve a word problem involving two-digit by three-digit multiplication. Direct students to problem 1 in their books.

Use the Read–Draw–Write process to solve the problem.

1. There are 122 cities competing in a math relay race. Each city sends 41 grade 5 students to compete. How many students compete?

?

41 . . .

122 cities

122 × 41 = 5,002

5,002 grade 5 students compete in the math relay race.

Use the Math Chat routine to engage students in mathematical discourse.

Give students 3–4 minutes of silent work time to use the Read–Draw–Write process to solve the problem. Encourage students to practice modeling with a tape diagram before they solve the problem.

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the different ways students might have shown 122 × 41. Then facilitate a class discussion. Invite students to share their thinking with the whole group and to record their reasoning. Validate a range of ideas and support students in making observations about the work samples, but focus the discussion on the following response:

I decomposed each factor into place value parts, and then I used the distributive property to multiply each part of one of the factors with each part of the other factor. I multiplied each part together to find the partial products and added the partial products to find the total.

Display a tape diagram that matches the story.

Before you multiplied, you modeled the story. This is one model you may have drawn. Looking at this model, what do you notice?

I notice the tape diagram does not show all 122 cities. It only shows one part labeled 41.

I notice the dots in the tape diagram represent all the other cities and students who compete.

In this model, what is the value of one part?

So in the expression 122 × 41, what does 122 represent?

It represents the number of cities or the number of groups.

What does 41 represent?

It represents the number of students from each city sent to the race.

We can also say the unit is 41, and we can see that unit 41 in our tape diagram. We can interpret the expression as one hundred twenty-two 41s.

Have students turn and talk about whether the product would change if the unit was 122 (meaning they would show forty-one 122s).

Transition to the next segment by framing the work. Today, we will explore the commutative property of multiplication and the distributive property.

Learn

Relate Vertical Form and the Break Apart and Distribute Method to the Area Model

Students decompose factors to multiply and relate the break apart and distribute method to the area model.

Throughout the lesson, students work and think–pair–share with a partner. Consider designating students in the partnership as partner A and partner B.

Direct students to problem 2. As you guide students through the problem by using the following prompts, have them record the work to find the product in their books.

Let’s find the product of 24 and 40 by using the distributive property. We can interpret this multiplication expression as 24 groups of 40 or as 40 groups of 24. Let’s interpret it as 24 groups of 40, which we can say in unit form is 24 forties.

Write 24 forties = 20 forties + 4 forties.

Is this a true number sentence? How do you know?

Yes, it is a true number sentence because the unit is forty and 20 + 4 = 24.

Let’s record that in standard form.

Teacher Note

Prompting students to multiply in unit form helps them recall multiplication facts. 4 ones × 4 tens is more familiar to students than 4 × 40 because they know 4 × 4 = 16.

Additionally, it helps students record the correct value when they use the standard algorithm. For example, students may think, “I know 10 × 10 = 100, so 2 tens × 4 tens = 8 hundreds.”

Record 24 × 40 = (20 + 4) × 40.

To better represent the sizes of the partial products, let’s show our thinking in the area model.

Label the length of the rectangle 40 and the width 4 and 20.

When we multiply 20 by 40 and 4 by 40, does it matter which we do first? Why?

No. We can multiply 20 by 40 first or we can multiply 4 by 40 first. The multiplication can be done in either order because the partial products we add are the same no matter the order.

That’s right, we can multiply in either order. Let’s begin by first multiplying the 4 by 40.

Gesture to each corresponding part of the area model and record the partial products as you ask the following questions.

In unit form, what is 4 ones × 4 tens?

16 tens

What is 16 tens in standard form?

160

In unit form, what is 2 tens × 4 tens?

8 hundreds

What is 8 hundreds in standard form?

800

Let’s record those partial products vertically.

Record 40 × 24 vertically in the grid to the right of the area model.

We know 24 × 40 = 40 × 24 because of the commutative property of multiplication. Here, we are recording the unit forty first. To find the value of 24 forties, we must add the partial products.

How many partial products do you see in our area model? What are they?

There are two partial products: 160 and 800.

Teacher Note

The terms length and width are generally used to indicate the longer and shorter sides of a rectangle, respectively. However, they can be used interchangeably. In some contexts, for example, length may correspond to height or to the number in each row and may be the smaller of the two dimensions. The orientation in which you draw a rectangle may influence which dimension is considered the width. When no context is given, either term is appropriate for either dimension. Encourage students to use the terms interchangeably.

Interchanging these terms also prepares students to use the word by to identify and describe rectangles. For example, consider describing a rectangle as 4 by 7 instead of as 4 feet wide and 7 feet long, as students become more comfortable interchanging the terms length and width

What does 160 represent?

4 × 40

Record 160 vertically as the first partial product.

What does 800 represent?

20 × 40

Record 800 vertically as the second partial product.

What is 160 + 800?

960

What is the completed equation?

24 × 40 = 960

Record the product in the blank of the equation.

Why are there two partial products?

Because we multiplied 40 by two parts. We multiplied 40 by 4 and by 20.

Was it helpful to decompose the factor 24? How?

It was helpful. When we decomposed 24, it broke the problem down into more manageable parts for me to find the product.

It was helpful. I was able to multiply 20 by 40 in my head because I know 2 × 4 = 8, so 2 tens × 4 tens = 8 hundreds.

It was not helpful. I would have thought about 24 × 4, which I know is 96, and then multiplied by 10 because 40 is 10 times as much as 4.

Teacher Note

Students explore three properties in this lesson:

Commutative property of multiplication

Students can choose which factor is the unit and can multiply the factors in any order.

24 × 40 = 40 × 24

Distributive property of multiplication over addition

Students multiply each part of one factor by the unit.

24 × 40 = (20 + 4) × 40 = (20 × 40) + (4 × 40)

Commutative property of addition

Students can multiply the unit by the parts of the other factor in any order.

(20 × 40) + (4 × 40) = (4 × 40) + (20 × 40)

Display the sample work. Have students think–pair–share about the observations and connections they see between this work and what they did with the area model and vertical form.

I notice that this work shows the break apart and distribute method.

24 × 40 = (20 + 4) × 40

= (20 × 40) + (4 × 40) )

= 20 × 40 + 4 × 40

= 80 0 + 16 0

= 96 0

This work matches how we multiplied with the area model. First we decomposed 24 into 20 + 4. Then we multiplied each part by 40. Once we had the partial products, we added. Display the picture of all three work samples showing the partial products highlighted.

Invite students to think–pair–share about how all three representations (the area model, vertical form, and break apart and distribute) are similar or different.

All three methods use the distributive property to decompose 24 and multiply each part by 40.

With all three models, we multiply first and then add.

All three methods have the same partial products.

The only difference is the way we record the products.

Remind students that they can show their thinking in any of these three ways because with all three methods they can apply the distributive property.

Designate a Unit

Students determine that designating a different unit does not change the product.

Direct students to problem 3. Have students work with a partner to record an estimate. In what two ways might we interpret the expression 22 × 41?

22 groups of 41 or 41 groups of 22 22 forty-ones or 41 twenty-twos

Display the two area models.

What do you notice about the models?

Model A decomposes 22 and model

B decomposes 41.

Both models represent the expression 22 × 41.

Each model will have two partial products.

Model A shows 22 groups of 41, so the choice of unit is 41. Model B shows 41 groups of 22, so the choice of unit is 22. Each model shows a different number of groups of a different unit. How does that affect the product?

It should not affect the product because 22 × 41 = 41 × 22.

Invite students to think–pair–share about which model they would choose to solve the problem and why.

I would choose model B because it is simpler for me to multiply by 1 one.

I would use model A because I can multiply 41 and 2 in my head. I can also multiply 41 and 20 in my head because 20 is 10 times as much as 2.

Teacher Note

Encourage students to record their estimates by using the following structure that is seen in the Problem Set:

22 × 41 ≈ 20 × 40

= 800

Direct students to work with a partner. One partner should multiply by using model A and one partner should multiply by using model B. Encourage students to record their thinking vertically. Circulate as students work and select one student’s work that uses model A and another student’s work that uses model B.

Promoting the Standards for Mathematical Practice

When students use different models to multiply two numbers and notice that decomposing either factor will result in the same product, they are looking for and expressing regularity in repeated reasoning (MP8).

Ask the following questions to promote MP8:

• When you use different models to find the product of two factors, does anything repeat? How can this repetition help you multiply more efficiently?

Show two pieces of correct student work: one that uses model A and one that uses model B.

• Will this pattern always be true?

What do you notice?

The partial products are different. The products are the same.

We can designate either factor as the unit and the products will still be equal. We can choose to multiply by using whichever unit makes multiplying more efficient or simpler for us.

Invite students to turn and talk about whether they preferred designating 22 or 41 as the unit.

We estimated the product is about 800. Is our product 902 reasonable?

Yes. 902 is not too far from 800. We rounded each factor down, so it makes sense that the actual product is larger than our estimated product.

As time permits, repeat the process to find the products for problems 4 and 5, allowing students to work independently as they demonstrate proficiency. Direct students to estimate before multiplying.

Language Support

To support understanding of the word designate, consider asking students what other word they might use in its place. Ensure that students understand they need to choose one factor and name it as the unit. Share these examples of designate used in familiar contexts:

• The teacher designates a different student each week to collect homework.

• The principal designates the playground as the place to gather during a fire drill.

UDL: Action & Expression

Consider offering students a template and posting an example to support students in their understanding of designating the unit. Students may highlight the unit in one color and the factor to decompose in another color.

22 × 41 =

Invite students to think–pair–share about which area model they chose to use to solve the problem and why.

I chose model B so I wouldn’t have to multiply 201 by 30 in my head.

I chose model B because it is simpler for me to multiply 200 by 32 mentally. I think about 2 hundreds × 3 tens, which is 6 thousands. Then I add the remaining 400.

Differentiation: Support

If students need support with using prior knowledge to do mental math, consider asking the following questions:

• What is 3 tens × 2 hundreds? What is 3 tens × 0 tens? What is 3 tens × 1 one? What else must we do to find 32 × 201?

• What is 32 in unit form? Can you use unit form to help you multiply?

• If you know what 3 × 201 is equal to, how can you use that to find 30 × 201?

Differentiation: Challenge

Challenge students who finish early to complete the following problem in two different ways, designating a different unit each time:

At birth, a baby panda weighs 130 grams. Two months later, the panda weighs 32 times as much as its birth weight. How much does the baby panda weigh when it is two months old?

Discuss which way is simpler for them to multiply mentally.

I chose to multiply by using model A. 2 × 201 is 201 doubled, and when I multiplied 30 and 201, I thought about 30 × 200, which is 6,000. Then I added 1 more 30.

We can use either area model to multiply the factors in any order because of the commutative property of multiplication. The product will remain the same.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Teacher Note

It is likely more familiar and comfortable for adults to write the greater factor on top in vertical form. Show students that they can write either factor on top so they can see and understand that it does not matter which factor they designate as the unit because of the commutative property of multiplication. This exploration continues in subsequent lessons.

Land Debrief 5 min

Objective: Multiply two- and three-digit numbers by two-digit numbers by using the distributive property.

Facilitate a class discussion about multiplying by using the distributive property by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Based on the methods explored today, how can you describe the distributive property?

We can decompose one or both factors and multiply each part of each factor by each part of the other factor because of the distributive property.

We use the distributive property in the area model, the standard algorithm, and the break apart and distribute method.

When we decompose one factor, does it help us multiply? How?

Yes. When we decompose, it breaks the problem down into more manageable parts for us to multiply.

We can multiply the factors mentally when we decompose one or both of them.

At the beginning of the lesson, you discussed this question: Would the product change if our unit was 122 and we instead showed forty-one 122s? Why?

Have students think–pair–share about whether they choose to repeat, revise, or add on to what they shared earlier.

I choose to repeat what I said earlier because I said the product would not change because 122 × 41 = 41 × 122.

I choose to revise what I said earlier because I said the product would be different if the other factor was the unit. Now I see that it does not matter, and the product will not change.

I choose to add on to what I said earlier. I said the product would not change, but now I would also say that the product will not change because of the commutative property of multiplication.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm.

Lesson at a Glance

9

Students multiply two- and three-digit numbers by two-digit numbers by using the area model. They relate the model to the standard algorithm and discuss which strategy is most efficient. Students explore how naming one factor as the unit affects the number of partial products shown in the standard algorithm.

Key Question

• How does the area model relate to the standard algorithm for multiplication?

Achievement Descriptor

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm. (5.NBT.B.5)

Agenda Materials

Fluency 15 min

Launch 5 min

Learn 30 min

• Relate the Area Model to the Standard Algorithm

• Multiply Three-Digit Numbers by Two-Digit Numbers

• Problem Set

Land 10 min

Teacher

• None Students

• Multiply by Multiples of 10, 100, 1,000, and 10,000 Sprint (in the student book)

Lesson Preparation

Consider tearing out the Sprint pages in advance of the lesson.

Fluency

Sprint: Multiply by Multiples of 10, 100, 1,000, and 10,000

Materials—S: Multiply by Multiples of 10, 100, 1,000, and 10,000 Sprint

Students multiply a one-digit factor by multiples of 10, 100, 1,000, or 10,000 to develop fluency with assessing the reasonableness of products. Have students read the instructions and complete the sample problems.

Sprint

Write the product.

1. 1 × 20 = 20

2. 2 × 600 = 1,200

3. 3 × 9,000 = 27,000

Direct students to Sprint A. Frame the task.

I do not expect you to finish. Do as many problems as you can, your personal best. Take your mark. Get set. Think!

Time students for 1 minute on Sprint A.

Stop! Underline the last problem you did.

I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer.

Read the answers to Sprint A quickly and energetically.

Count the number you got correct and write the number at the top of the page. This is your personal goal for Sprint B.

Celebrate students’ effort and success.

Provide about 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A. If students are provided time to complete more problems on Sprint A, reread the answers but do not have them alter their personal goals.

Lead students in one fast-paced and one slow-paced counting activity, each with a stretch or physical movement.

Point to the number you got correct on Sprint A. Remember this is your personal goal for Sprint B.

Direct students to Sprint B.

Take your mark. Get set. Improve!

Time students for 1 minute on Sprint B.

Stop! Underline the last problem you did.

I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer.

Read the answers to Sprint B quickly and energetically.

Count the number you got correct and write the number at the top of the page.

Determine your improvement score and write the number at the top of the page.

Celebrate students’ improvement.

Teacher Note

Consider asking the following questions to discuss the patterns in Sprint A:

• What do you notice about problems 1–7? How do problems 1–7 compare to problems 8–12?

• What strategy did you use for problems 1–7? Did you change your strategy during Sprint A? Why?

Teacher Note

Count forward by 200 from 0 to 2,000 for the fast-paced counting activity.

Count backward by 2,000 from 20,000 to 0 for the slow-paced counting activity.

Launch

Students discuss a real-world use of an area model.

Play the Painting a Mural video. The video shows a mural being painted in sections. Lines and labels appear around sections to outline rectangular regions. As students watch the video, replay portions of it as needed, and allow students to record information they think is important on their personal whiteboards. Then facilitate a discussion by asking the following questions about what students noticed and wondered.

What did you notice in the video?

I noticed that finding the area of the mural looks similar to finding a product by using an area model.

I noticed the area of the mural was found in parts as the painter worked.

I noticed the dimensions of the mural were decomposed by hundreds, tens, and ones.

What do you wonder?

I wonder about the total area of the mural.

I wonder whether we can use the partial products we already know to find the total area of the mural.

Have partners find 28 × 140.

Does having the factors broken apart by place value help you multiply? How?

Yes. It helped me find the product because I could use multiplication facts to find the partial products. Then I added the partial products to find the total area.

The break apart and distribute method, the area model, and the standard algorithm are related. They all decompose the factors, multiply the parts, and add the partial products to find the total.

Transition to the next segment by framing the work.

Today, we will get better at using the standard algorithm when multiplying two- and three-digit numbers.

Teacher Note

The focus of the discussion should be making connections between finding the area of a rectangle, the area model, and breaking apart factors to find partial products. Students may generate the question, Does the painter have enough paint for the whole mural? If so, consider having them briefly answer the question after they find the product.

Learn

Relate the Area Model to the Standard Algorithm

Students multiply by using the area model and relate it to the standard algorithm. Direct students to problem 1 in their books and have them read the problem silently.

1.

paints the gymnasium wall. The wall is 24 feet wide and 33 feet long. How many square feet does Mr. Perez paint?

Teacher Note

The digital interactive Area Model helps to visually connect area to this multiplication method.

Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

Mr. Perez paints 792 square feet.

Have students turn and talk to discuss what they know and what they do not know from the story. Encourage students to model with a tape diagram as needed.

What do we know and what do we need to know?

We know that Mr. Perez paints the gym wall and we know the length and width of the wall.

We need to know how many square feet Mr. Perez paints.

Mr. Perez

If we need to determine how many square feet he paints, what does the question ask us to find?

The area of the wall

What should we do to find the total area of the wall?

We need to multiply the length of the wall by the width of the wall.

About how many square feet of wall does Mr. Perez paint? How do you know? He paints about 600 square feet because 24 × 33 ≈ 20 × 30 and 20 × 30 = 600. Direct students to record their estimates.

In other problems, we used multiplication facts to find the partial products after decomposing one factor. Can we use multiplication facts to find the product 33 × 24?

Yes.

Should we decompose one or both factors? Why?

We should decompose both. Neither factor is a multiple of 10, so if we decompose both factors, it might help us use our multiplication facts to multiply different place value units.

What is the width of the wall?

24 feet

What is 24 in expanded form?

20 + 4

Label 20 and 4 along the left side of the area model and direct students to do the same.

What is the length of the wall?

33 feet

What is 33 in expanded form?

30 + 3

Record 30 and 3 along the top of the area model and direct students to do the same. Let’s multiply each part, one at a time, starting with the ones.

Gesture to each corresponding part of the area model and record the partial products in standard form as you ask the following questions and ask for a choral response.

In unit form, what is 4 ones × 3 ones?

12 ones

In unit form, what is 4 ones × 3 tens?

12 tens

In unit form, what is 2 tens × 3 ones?

6 tens

In unit form, what is 2 tens × 3 tens?

6 hundreds

What is 4 groups of 33 equal to?

132

What is 20 groups of 33 equal to?

660

What is 24 × 33?

792

Let’s use the standard algorithm to show what we did.

Gesture to each corresponding part of the standard algorithm as you ask the following questions.

In unit form, what is 4 ones × 3 ones?

12 ones

12 ones can be renamed as 1 ten 2 ones. Watch as I record 12 ones.

Record 1 ten 2 ones and direct students to do the same.

In unit form, what is 4 ones × 3 tens?

12 tens

Teacher Note

To multiply each part of the area model, start with the ones place as you would with the standard algorithm and then move to the left. When all partial products are found, repeat for the row below.

What is 12 tens plus 1 more ten?

13 tens

Record 1 hundred 3 tens and cross out the additional 1 ten to show it was added. Direct students to do the same.

In unit form, what is 2 tens × 3 ones?

6 tens

What is 6 tens in standard form?

60

Record 60 and direct students to do the same.

In unit form, what is 2 tens × 3 tens?

6 hundreds

Record 6 hundreds and direct students to do the same.

What is 132 + 660?

792

Is 792 a reasonable answer based on our estimates?

Yes. We estimated the product would be 600 square feet. Because we chose to estimate with numbers less than the actual factors, it makes sense that the actual product is a little greater than the estimated product.

Have students write the final answer statement: Mr. Perez paints 792 square feet.

Where do you see the partial products from the area model in the standard algorithm?

The sum of 4 × 30 and 4 × 3 is 132.

We see that in the first row of partial products in the standard algorithm. 132 is the product of 4 and 33.

The sum of 20 × 3 and 20 × 30 is 660.

We see that in the second row of partial products in the standard algorithm. 660 is the product of 20 and 33.

Highlight or circle the partial products in both methods and direct students to do the same.

How is multiplying by using the area model like multiplying by using the standard algorithm?

When I multiply by using either method, I think, “What is 4 ones × 3 ones ? What is 4 ones × 3 tens?” and so on.

Now that we know how the area model relates to the standard algorithm for multiplication, let’s multiply by using the standard algorithm.

Direct students to problem 2.

2. 28 × 63 = 1,764

We know we can interpret this problem as twenty-eight 63s or as sixty-three 28s. So we can practice together, let’s all interpret it as twenty-eight 63s, which means we are finding the total of 28 groups of 63.

Invite students to turn and talk to estimate the product.

Direct students to record the factors vertically.

Even though we are not showing the factors or partial products on an area model, we can still use the same thinking.

Promoting the Standards for Mathematical Practice

When students decompose factors and find partial products when they multiply a two-digit number by a two- or three-digit number by using the standard multiplication algorithm, they are attending to precision (MP6).

Ask the following questions to promote MP6:

• How can you write the partial products when you use the standard algorithm?

• Where might you make mistakes when you use the standard algorithm?

Direct students to work with a partner to multiply by using the standard algorithm to find the partial products 8 × 63 and 20 × 63. Use the following questions to support students as they multiply by using the standard algorithm:

• In unit form, what is 8 ones × 3 ones?

• In unit form, what is 8 ones × 6 tens?

• In unit form, what is 2 tens × 3 ones?

• In unit form, what is 2 tens × 6 tens? Gather students to discuss.

What does 504 represent?

Eight 63s

What does 1,260 represent?

Twenty 63s

What does 1,764 represent?

Twenty-eight 63s

If we interpret the equation as sixty-three 28s instead, would we have a different number of partial products? How do you know?

We would still have two partial products because we would find 3 × 28 and then 60 × 28.

Invite students to turn and talk to discuss why there are two partial products in both interpretations and what situations might lead to more than two partial products.

Multiply Three-Digit Numbers by Two-Digit Numbers

Students multiply by using the standard algorithm, discuss how it relates to the area model, and determine the more efficient strategy.

Direct students to problem 3.

UDL: Representation

After partners find the product, consider annotating the standard algorithm to highlight decomposing the factors, multiplying the parts, and adding the partial products.

3. Flatback turtles lay 52 eggs in a nest. How many turtle eggs would there be in 427 nests?

There would be 22,204 turtle eggs in 427 nests.

Have students turn and talk to discuss what they know and what they do not know from the story. Students can model with a tape diagram as needed.

What do we know and what do we need to know?

We know how many nests there are and we know how many eggs are in each nest. We do not know how many turtle eggs there would be in all.

Display the following tape diagram, or if a student drew a similar tape diagram, show the student’s diagram.

What expression can we use to determine the number of turtle eggs?

52 × 427

?

52 . . . 42 7 nests

Have students turn and talk with a partner to estimate the total number of turtle eggs. Encourage students to record their estimates. Display the following area models.

What do you notice? What is the same and what is different?

Both area models show 52 × 427.

Both area models have six partial products.

Model A shows 52 groups of 427, so 427 is the unit.

Model B shows 427 groups of 52, so 52 is the unit.

Gesture to the area models. Have students think–pair–share about a response to the following question.

Think back to the connection between area models and the standard algorithm. How many partial products are there in each of the models? How do you know?

Model A has two partial products because there are two rows. The standard algorithm will show 2 × 427 and 50 × 427.

Model B has three partial products because there are three rows. The standard algorithm will show 7 × 52, 20 × 52, and 400 × 52.

We know we can designate either factor as the unit. If we are using the standard algorithm, which factor should we designate as the unit? Why?

We should use 427 as the unit so we are finding 52 groups of 427. We would have only two partial products that way.

Direct students to work with a partner to multiply by using the standard algorithm to find the partial products 2 × 427 and 50 × 427. Use the following prompts to support students as they multiply by using the standard algorithm:

• What is 2 × 7? 2 × 20? 2 × 400?

• What is 50 × 7? 50 × 20? 50 × 400?

The questions here intentionally shift to standard form language so students can consider partial products by using both forms. Students decide later which form is more helpful.

Gather students to discuss.

What product did you find? Is it reasonable based on your estimate?

The product is 22,204. It is reasonable because I estimated the product as 20,000 because 400 × 50 = 20,000.

Have students record the final answer statement: There would be 22,204 turtle eggs in 427 nests.

Display the following work.

Gesture to the models on the right.

The work on the right shows how to find the product by using 52 as the unit. Notice that the product is the same in both models, but there are three partial products in the model on the right instead of the two in the model on the left. Why is the product the same although there is a different number of partial products?

The product is the same because the commutative property of multiplication says we can multiply factors in any order. We can name either factor as the unit and we will get the same product.

Have students turn and talk about why it is helpful to think about which factor to designate as the unit when they multiply.

Language Support

Consider supporting student responses with the Talking Tool. Invite students to use the Share Your Thinking section to explain how they found the product.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land Debrief 5 min

Objective: Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm.

Facilitate a class discussion about using the standard algorithm for multiplication by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

How does the area model relate to the standard algorithm for multiplication?

With both methods, we decompose factors and use the distributive property to multiply.

With both methods, we can choose which factor can be the unit and which factor can be the number of groups.

The number of rows in the area model matches the number of partial products shown in the standard algorithm.

Is the standard algorithm helpful when we multiply? How?

It is helpful because we write fewer partial products than we might write in an area model.

All the partial products are already lined up and ready to add.

Teacher Note

Beginning in grade 4, students consistently see the length decomposed to show the smallest unit first. This is intentional as it corresponds directly to rows of the standard algorithm for multiplication. Students might decompose the factor on the left side of the area model differently and still find the same partial products, but their rows might not correspond to the rows of the standard algorithm. As long as students decompose factors accurately and multiply precisely, they will still find the same product.

Is the area model helpful when you multiply, even if you know the standard algorithm? How?

Yes. It can help me decide which factor to name as the unit because the number of rows will help me see how many partial products to show in the standard algorithm.

Yes. It can help me remember what I am multiplying by in the standard algorithm.

Yes. The size of the rows helps me remember the size of the partial products. The first partial product will have a lesser value than the second or third. This can help me check for reasonableness along the way.

When we multiply numbers with even more digits, will you think about partial products in standard form or unit form? Give an example to support your choice.

I will think about the partial products in unit form because it helps me multiply mentally. When I multiply 22 by 342 , I can think about 2 ones × 2 ones , 2 ones × 4 tens , 2 ones × 3 hundreds, and so on.

I will think about partial products in standard form because I can use what I know about multiples of powers of 10 and multiplication facts. When I multiply 50 by 300 , I think about 5 × 10 × 3 × 100, which is equal to 15 × 1,000 or 15,000.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Estimate the product. Then multiply.

5. 38 × 529 ≈ 40 × 500 = 20,000

38 × 529 = 20,102

6. 63 × 804 ≈ 60 × 800 = 48,000

63 × 804 = 50,652

Use

8. A school bus travels 508 kilometers each week. How many kilometers does the school bus travel in 36 weeks?

36 × 508 = 18,288

The school bus travels 18,288 kilometers in 36 weeks.

7. Julie makes a mistake when she uses the distributive property to find 83 × 624. Look at her work.

83 × 624 = 80 × 600 + 80 × 20 + 80 × 4 = 48,000 × 1,600 + 320 = 49,920

a. What mistake did Julie make?

Julie multiplied 80 by 624 but she forgot to multiply 3 by 624

b. Find the product.

83 × 624 = 51,792

Copyright © Great Minds PBC 83

Copyright © Great Minds

Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm.

Lesson at a Glance

Students solve one-step word problems with multiplication by using the area model and the standard algorithm simultaneously. Students then transition fully to the standard algorithm and explore how they can multiply digits while they hold the concept of place value in their heads.

Key Questions

• Why would you choose to multiply by using the standard algorithm?

• How do you decide which factor to designate as the unit?

Achievement Descriptor

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm. (5.NBT.B.5)

Agenda Materials

Fluency 10 min

Launch 10 min

Learn 30 min

• Relate the Area Model to the Standard Algorithm

• Problem Set

Land 10 min

Teacher

• None Students

• None

Lesson Preparation

Review the Math Past resource to support the delivery of the lesson.

Fluency

Whiteboard Exchange: Divide by 2, 3, or 4

Students use a place value strategy to divide a two-digit number by a one-digit number to prepare for dividing multi-digit numbers by two-digit numbers in topic C.

Display 84 ÷ 2 = .

Write the quotient and the remainder. Show your method.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the quotient and the remainder.

Repeat the process with the following sequence:

96 ÷ 3 =

Quotient: 32

Remainder: 0

76 ÷ 3 =

84 ÷ 2 =

Quotient: 42

Remainder: 0

Teacher Note

In grade 4, students use a variety of models to represent place value strategies for division, including place value charts, area models, vertical form, and equations. As the numbers increase in size, especially the divisor, the place value chart becomes even more inefficient than other models. Students may select any method to solve problems in this activity but encourage them to use more efficient models if you notice they gravitate toward those that are less efficient.

Teacher Note

25

Remainder: 1

92 ÷ 4 = Quotient: 20

23

0

83 ÷ 4 =

.

With this recording, students can clearly distinguish the quotient from what is left over. Students learn to record remainders in fraction form in module 2 and in decimal form in grade 6.

Choral Response: Exponential to Standard Form

Students read a power of 10 in exponential form and say the value in standard form to build fluency with exponents from topic A.

Display 101 in the two-column chart.

When I give the signal, read the number in exponential form.

10 to the first power

What is 101 in standard form? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

10

Display the number in standard form.

Continue the process with 102, 103, 104, 105, and 106.

Divide students into two groups and have them stand on separate sides of the room, designated as sides A and B.

Each side will take turns saying a power of 10 in a different form, going from 101 to 106.

Side A will say the number in exponential form, and then side B will say the value in standard form.

Side A, say … (Gesture to side A.)

10 to the first power

Side B, say … (Gesture to side B.)

10

Continue the process to 106, or 1,000,000. Transition to using only gestures as students are ready.

Now switch roles and go from 106 to 101. Side A, start with 1,000,000. Side B, say 10 to the sixth power.

Launch

Students test an ancient method of multiplication and compare it with the standard algorithm.

We have been learning how to multiply by using the standard algorithm of multiplication. Prior to using this algorithm, many people used a different series of steps to find the product. These steps are referred to as the Ethiopian multiplication method.

The method was used in the 1900s when an Austrian colonel who was visiting Ethiopia wanted to buy 7 bulls that cost 22 Maria Theresa dollars each, but no one in the village could figure out the total cost for all 7 bulls.

To help find the total cost of the bulls, a local priest and his helper were called. They built two columns into the ground with holes in each column called houses. The column on the left was for halving, and the column on the right was for doubling. They placed 22 pebbles in the first house, or row, in the halving column and 7 pebbles in the first house in the doubling column. This method led them to find the correct product: 154.

Let’s try this method to multiply 44 by 15.

Teacher Note

The Math Past resource includes more information about the Ethiopian multiplication method. Consider inviting students to use the method to find other products.

Display the steps to multiply by using this method.

Steps of the Ethiopian Multiplication Method

1. Put one factor in each column.

2. In the halving column, repeatedly divide by 2 (and ignore any remainders) until only the number 1 remains.

3. In the doubling column, repeatedly multiply by 2 until both columns have the same number of rows filled in.

4. Cross out any rows that have an even number in the halving column.

5. Add the remaining numbers in the doubling column.

Direct students to split their personal whiteboards into two columns by drawing a line down the middle.

The left column is the halving column. Write 44 at the top of the left column.

The right column is the doubling column. Write 15 at the top of the right column.

What is half of 44? 22

Direct students to record 22 in the left column.

What is 15 doubled? 30

Language Support

Define the words halving and doubling for students.

• Halving: dividing by 2

• Doubling: multiplying by 2

Direct students to record 30 in the right column.

Continue this process until the halving column reaches the number 1.

Even numbers were considered evil, so cross out every even number in the halving column. Then cross out the number in the same house, or row, in the doubling column.

Direct students to add the remaining numbers in the doubling column.

What is the total?

660

The product of 44 and 15 is equal to 660, so this method of multiplication does work. Though we may not use this method much today, it is an interesting way to multiply.

Time permitting, allow students to try using the Ethiopian method on the following example with a partner: 36 × 12.

Transition to the next segment by framing the work.

Today, we will get better at finding products by using the standard algorithm for multiplication.

Challenge students to think about why the even numbers and the remainders might be crossed off.

Learn

Relate the Area Model to the Standard Algorithm

Students multiply by using the area model and the standard algorithm simultaneously.

Direct students to problem 1 in their books. Have students turn and talk to discuss what they know and what they do not know from the story. Students can model with a tape diagram as needed.

1. Lisa tiles a rectangular floor that is 204 inches long and 123 inches wide. How many square inches of tile does Lisa use?

Differentiation: Support

Invite students to multiply out loud in unit form to support their work. For example, a student might say,

Lisa uses 25,092 square inches of tile.

What do we know and what do we need to know?

We know Lisa tiles a floor and we know the length and width of the floor. We do not know how many square inches of tile she uses.

.”

Unit form helps students understand that when they write a 4, after multiplying 2 tens by 2 hundreds, for example, it represents 4 thousands.

What is an expression that can help us find the number of square inches of tile? How do you know?

We could multiply 204 by 123 because we need to find the area of the rectangular floor because she is covering it with tiles.

Have students think–pair–share about a response to the following question. Encourage students to create an area model that represents either scenario.

When we multiply, we know we can designate either factor as the number of groups and either factor as the unit. Which factor would you designate as the number of groups and which as the unit? Why?

I would designate 123 as the unit and 204 as the number of groups because then there would be two partial products in the standard algorithm: 123 × 4 and 123 × 200. I would designate 204 as the unit and 123 as the number of groups. In an area model, I would show 123 as the width and 204 as the length. Even though there would be more partial products in the standard algorithm, it makes sense to me to do it this way because it matches better with the story.

About how many square inches of tile does Lisa use? How do you know?

Lisa uses about 20,000 square inches of tile because 204 ≈ 200 and 123 ≈ 100. I know 200 × 100 = 20,000.

Have students find the product for problem 1. Circulate and monitor student work. Encourage students to practice using the standard algorithm but expect some students to begin with an area model. Students may choose either factor as the unit. When most students are finished, gather to discuss.

How many square inches of tile does Lisa use? Is that answer reasonable?

Lisa uses 25,092 square inches of tile. 25,092 is reasonable because the numbers we used to make an estimate were both less than the numbers given in the problem, so it makes sense that the actual area is greater than our estimate.

Have students record the final answer statement: Lisa uses 25,092 square inches of tile.

What does the highlighted 2 represent?

20,000

What two factors were multiplied to produce 20,000?

100 × 200

If we were thinking in unit form, we would ask ourselves, What is 1 hundred × 2 hundreds? If we were thinking in standard form, we would ask ourselves, What is 100 × 200? Which question is simpler for you to visualize?

100 × 200 is simpler for me to visualize because I can multiply by using the multiplication fact and shift the digits based on my understanding of powers of 10. I know 1 × 2 = 2. So I could think about 2 × 104 and shift the digit 2 four places to the left.

Let’s take a closer look at where the 2 is located in the standard algorithm. How many places from the left of the ones place is the 2 located?

Four places to the left

So the place value is already maintained because of where we recorded that 2. Let’s practice multiplying by using the standard algorithm and multiplying single digits while holding place value.

Have students watch and respond as you guide and record.

Let’s designate 123 as our unit so we will have two partial products. What does the first partial product represent?

4 × 123

What does the second partial product represent?

200 × 123

×

+

What is 4 × 3?

Record 12.

What is 4 × 2?

8

We know this 8 represents 8 tens, and we regrouped 1 ten from 12, so we now have 9 tens.

Cross out 1. Record 9.

What is 4 × 1?

4

We know 4 represents 4 hundreds. Watch how we record 4 in the hundreds place.

Record 4.

What is 4 × 123?

492

We found the partial product 492 by multiplying single-digit by single-digit while holding on to place value in our heads. Let’s find the second partial product.

What is 2 × 3?

6

What does 6 represent?

600

Let’s record it as 600 so we can hold the place value of other digits in our heads.

Record 600.

What is 2 × 2?

4

Record 4.

Why do we record 4 here?

We know 4 represents 200 × 20, which is equal to 4 × 103, so the 4 should shift three places to the left.

What is 2 × 1?

Record 2. Why do we record 2 here?

We know 2 represents 200 × 100, which is equal to 2 × 104, so the 2 should shift four places to the left.

We found the partial product 24,600 by multiplying single-digit by single-digit while holding on to place value in our heads. Moving forward, when you multiply by using the standard algorithm, you might find partial products by using unit form, standard form, or single-digit by single-digit multiplication.

Invite students to turn and talk about which of the three methods they find the most helpful and why.

Direct students to problem 2.

Have students turn and talk to discuss what they know and what they do not know from the story. Students can model with a tape diagram as needed.

2. The population of Waverly, Pennsylvania, is 604 people. The population of Scranton, Pennsylvania, is 127 times as much as the population of Waverly. What is the population of Scranton?

604 × 127 = 76,708

The population of Scranton, Pennsylvania, is 76,708 people.

What do we know and what do we need to know?

We know the population of Waverly.

We know the population of Scranton is 127 times as much as the population of Waverly. We need to know the population of Scranton. Display the following tape diagram. If students created a similar tape diagram, display a student’s model.

What expression can we use to determine the population of Scranton?

127 × 604

Invite students to turn and talk to discuss, record, and estimate.

Scranton ranton 127 times as many

In the tape diagram, 604 is the unit and 127 is the number of groups. Could you designate 127 as the unit? Why might you do that?

Yes, we could designate 127 as the unit because we can multiply the factors in any order. We might designate 127 as the unit so there are only two partial products: 4 × 127 and 600 × 127.

Have students work independently or with a partner to multiply by using the standard algorithm.

What is the population of Scranton?

76,708

Invite students to turn and talk about whether the product is reasonable. Have students record the final answer statement: The population of Scranton, Pennsylvania, is 76,708 people.

Display the student work and direct students to study it to find the mistake.

Invite students to think–pair–share about what the error is and how they would correct it.

They multiplied by 27 and not 127.

The error is the student did not multiply 604 by 100. They did not fully distribute all the parts of 127.

I would correct it by multiplying 1 hundred by 4 ones, 0 tens, and 6 hundreds to find the third partial product: 60,400.

How can we avoid making the same error?

We can check to make sure we have the correct number of partial products based on what we designated as our unit and as our number of groups.

Direct students to problem 3.

Have students turn and talk to estimate the product.

Then invite students to think–pair–share about whether they would rather designate 1,429 or 312 as the unit and why.

I would rather designate 1,429 as the unit so we have only three partial products instead of four partial products.

Promoting the Standards for Mathematical Practice

When students analyze sample work that shows the standard multiplication algorithm being used to determine the product of two three-digit numbers and identify and explain how they would correct the error made, they are attending to precision (MP6).

Ask the following questions to promote MP6:

• What details are important to think about in this work?

• When you use the standard algorithm to multiply multi-digit numbers, with what steps do you need to be extra careful? Why?

Have students complete problem 3 independently or with a partner. Circulate and monitor student work, assisting students as needed. Encourage students to multiply by using the standard algorithm without the support of the area model.

UDL: Action & Expression

Support students in monitoring their progress by encouraging self-questioning when they use the standard algorithm. Emphasize the importance of thinking through decisions and changing course if a strategy is not working. Think aloud to model self-questioning by using problem 3 as an example. Discuss how asking questions such as these may have helped the student avoid the error and work more efficiently:

• Which number should I designate as the unit?

• Do I have the correct number of partial products?

• Did I fully distribute all the parts of the other factor?

Invite students to turn and talk about how they multiplied by using the standard algorithm. Then consider posing the following questions to the class:

• Did you think about products in unit form or standard form?

• Did you multiply single-digit by single-digit while holding place value in your head?

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

• Should I do anything differently?

3. 1,429 × 312

Land

Debrief 5 min

Objective: Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm.

Facilitate a class discussion about multiplying by using the standard algorithm by using the following prompts. Encourage students to restate their classmates’ responses.

Why would you choose to multiply by using the standard algorithm?

If the factors in the problem have a lot of digits, I would choose the standard algorithm because I can multiply each digit individually. If I use an area model, it might take too much time to draw.

I can multiply a single digit by another single digit because I know my multiplication facts. I can also hold place value in my head while I use multiplication facts, so that makes the standard algorithm efficient for me to use.

Direct students to problem 2 in the Problem Set and display the area model.

Which factor is designated as the unit in this area model: 342 or 1,627? Why do you think the student chose that unit?

1,627 is designated as the unit. I think the student made that choice so there would be only three partial products. If the student designated 342 as the unit, then there would be four partial products.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Complete the area model and find the sum of the partial products. Then multiply by using the standard algorithm. Compare your answers in each part to check that the product is correct.

Draw an area model to find the partial products and find their sum. Then multiply by using the standard algorithm. Compare your answers in each part to check that the product is correct.

Use the Read–Draw–Write process to solve the problem.

7. Sana drinks from a bottle that holds 946 milliliters of water. She fills the bottle and drinks all the water in it twice each day.

a. How many milliliters of water does Sana drink each day?

2 × 946 = 1,892

Sana drinks 1,892 milliliters of water each day.

b. How many milliliters of water does Sana drink in 365 days?

365 × 1,892 = 690,580

Sana drinks 690,580 milliliters of water in 365 days.

Multiply two multi-digit numbers by using the standard algorithm.

Lesson at a Glance

Students use the standard algorithm for multiplication to multiply two multi-digit numbers. They solve problems with and without renaming and use vertical form to record the steps of the standard algorithm. Students use estimation to assess the reasonableness of their answers.

Key Questions

• Why do we estimate before we multiply?

• Is the standard algorithm for multiplication efficient and useful? How?

Achievement Descriptor

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm. (5.NBT.B.5)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Multiply Two Multi-Digit Numbers

• Critique a Flawed Response

• Pass the Whiteboard

• Problem Set

Land 10 min

Lesson Preparation

Fluency

Choral Response: Exponential to Standard Form

Students read a power of 10 in exponential form and say the value in standard form to build fluency with exponents from topic A.

Display 106 in the two-column chart.

When I give the signal, read the number in exponential form.

10 to the sixth power

What is 106 in standard form? Raise your hand when you know.

Wait until most students raise their hands, and then signal for students to respond.

1,000,000

Display the number in standard form. Continue the process with 105, 104, 103, 102, and 101.

Divide students into two groups and have them stand on separate sides of the room, designated as sides A and B.

Each side will take turns saying a power of 10 in a different form, going from 106 to 101.

Side A will say the number in exponential form, and then side B will say the value in standard form.

Side A, say … (Gesture to side A.)

10 to the sixth power

Side B, say … (Gesture to side B.)

1,000,000

Continue the process to 101, or 10. Transition to only using gestures as students are ready. Now switch roles and go from 101 to 106. Side A, start with 10. Side B, say 10 to the first power.

Whiteboard Exchange: Divide by 2, 3, or 4

Students use a place value strategy to divide a three-digit number by a one-digit number to prepare for dividing multi-digit numbers by two-digit numbers in topic C.

Display 264 ÷ 2 = .

Write the quotient and the remainder. Show your method.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the quotient and the remainder.

Repeat the process with the following sequence:

26 4 ÷ 2 =

Quotient: 132

Remainder: 0

Quotient: 122 Remainder: 2

54

82

76

0

0

1

Launch

Students compare partial products with the standard algorithm for multiplication.

Display the shaded work samples that show 1,243 × 132.

In all three methods of multiplication, we see the same three partial products of 2,486, 37,290, and 124,300.

What does the partial product 2,486 represent?

2 × 1,243

2 times as much as 1,243

2 groups of 1,243

Where is 2,486 represented in sample A?

The sum of the first four rows of partial products, 6 + 80 + 400 + 2,000, represents 2,486.

Consider highlighting the difference between a partial product and a product. Tell students that they add the partial products to determine the product. Draw students’ attention to the word part in partial products.

For example, in sample B, the partial products of 1,243 × 132 are 2,486, 37,290, and 124,300. The product is 164,076. In sample B, the partial products are the numbers in the rectangles of the area model.

Where is 2,486 represented in sample B?

It is the first partial product.

Where is 2,486 represented in sample C?

The four partial products in the first row of the area model add to 2,486.

Let’s look at sample B. Why is there an 8 in the tens place of 2,486?

2 times 40 equals 80, or 8 tens.

Why is there a 4 in the hundreds place of 2,486?

2 times 200 equals 400, or 4 hundreds.

What does the partial product 37,290 represent?

30 × 1,243

30 times as much as 1,243

30 groups of 1,243

Explain why we can multiply 30 by 1,243 and record the partial product 37,290 on one line in sample B.

When the student multiplied 3 by 3 in the algorithm to get 9, they actually multiplied 30 by 3 to get 90.

The 2 represents 200 and is part of 1,200, which is the product of 30 and 40.

The 7 represents 7,000. It comes from adding 30 groups of 200, which makes 6,000, plus the 1,000 from 1,200 in the previous partial product.

The 3 represents 30,000 and comes from 30 groups of 1,000.

What does the partial product 124,300 represent?

100 × 1,243

100 times as much as 1,243

100 groups of 1,243

When we multiply by using the standard algorithm, recording can be simpler than when we use an area model because it combines several partial products into one, but we need to keep track of place value in our heads.

When do you think someone might choose to multiply multi-digit numbers with the standard algorithm instead of with partial products or the area model?

The standard algorithm is a more efficient method of recording when we have two factors with several nonzero digits.

Someone might use the standard algorithm when the factors are multi-digit numbers and none or few of the digits are zeros.

Why do you think someone might choose not to use the standard algorithm to multiply multi-digit numbers? Give an example.

Someone might use a different method if the multiplication can be done mentally with partial products. For example, someone might multiply 1,200 by 40 mentally by finding 12 × 4 × 100 × 10. Or someone might calculate (1,000 × 40) + (200 × 40).

Transition to the next segment by framing the work.

Today, we will multiply two multi-digit numbers by using the standard multiplication algorithm.

Learn

Multiply Two Multi-Digit Numbers

Students multiply two multi-digit numbers by using the standard multiplication algorithm.

Present the following problem.

What number is 111 times as much as 2,222?

What multiplication expression can we use to answer this question? 111 × 2,222

Ask students to think–pair–share to estimate the product of 111 and 2,222.

110 × 2,000 = 220,000

100 × 2,200 = 220,000

100 × 2,000 = 200,000

Tell students to work with a partner to determine the product of 111 and 2,222 by using the standard algorithm.

Invite students to share their solutions with the class. Ask whether their answers are reasonable and how they know.

Write 4,603 × 507.

How are the factors in this four-digit by three-digit multiplication problem different from the factors in the previous multiplication problem?

The digit 0 is in each factor.

Some of the digits in the factors are greater.

The factors do not have the same digit repeated.

Ask students to think–pair–share to estimate the product.

4,500 × 500 = 2,250,000

5,000 × 500 = 2,500,000

Tell students to work with a partner to determine the product of 4,603 and 507 by using the standard algorithm.

Invite students to share their answers with the class. Ask whether their answers are reasonable and how they know.

Display the work for 4,603 × 507.

Is this answer reasonable? How do you know?

Yes. 2,333,721 is close to my estimate of 2,500,000.

Differentiation: Challenge

For students who need an additional challenge, present the following problems.

Ask students the following questions: What patterns do you notice in the products? Why do you think those patterns are there? What do you predict for the value of 11 × 111,111?

Ask students the following questions: What patterns do you notice in the products? Why do you think those patterns are there? What do you predict for the value of 111 × 66?

Why are two 0s at the end of the second partial product?

There are two 0s because we multiplied 500 by 4,603.

Why are there two partial products? What does each partial product represent?

There are two partial products because we found 7 groups of 4,603 and 500 groups of 4,603.

32,221 = 7 × 4,603

2,301,500 = 500 × 4,603

Critique a Flawed Response

Students analyze sample work involving the standard algorithm for multiplication.

Introduce the Critique a Flawed Response routine and present the following problem:

A company plans to buy 112 desks that each cost $249. Julie says the total cost of the desks is about $25,000. Do you agree? Why?

Direct students to work with a partner to answer the questions.

Do you agree with Julie? Why?

I agree. 112 desks is about 100 desks, and $249 is about $250. I know 100 × 250 = 25,000. So the total cost of the desks is about $25,000.

Display the incorrect work for 249 × 112.

This work shows Toby’s incorrect calculation of the total cost of the desks. How do you know that Toby made a mistake?

Toby’s answer is not close to the estimate of 25,000.

Give students 1 minute to identify the error. Invite students to share. The last partial product should be 24,900 instead of 2,490. Toby multiplied 10 by 249 instead of multiplying 100 by 249.

Give students 2 minutes to solve the problem, based on their own understanding. Circulate and check for correctness as students work.

Teacher Note

Students may record 507 above 4,603, which is another correct way of setting up the algorithm. The product is the same, but students will have three partial products instead of two because they have to find 3 × 507, 600 × 507, and 4,000 × 507.

Language Support

As students review each other’s work, encourage them to use the Agree or Disagree section of the Talking Tool to support their discussions.

Then facilitate a class discussion. Invite students to share their solutions with the whole group.

Ask students to think–pair–share to summarize Toby’s mistake and how to correct it. Lead the class to consensus about how to best correct the flawed response.

Toby should remember which partial products he is finding before he multiplies. In this problem, we must find 2 × 249, 10 × 249, and 100 × 249.

Toby should connect the product back to the estimate and ask, Does my product make sense?

Toby can check his work by using an area model to see whether he gets the same answer.

Toby can multiply again by using the standard algorithm, but he needs to switch the order of the factors to see whether he gets the same answer.

Pass the Whiteboard

Students participate in a group activity involving the multiplication of multi-digit numbers.

Place students in groups of 4.

This is an activity called Pass the Whiteboard. Each person writes a three-digit by four-digit multiplication problem on their whiteboard. Once you write your problem, pass the whiteboard to the person on your left.

When you receive a whiteboard with a multiplication problem, estimate the product and write it on the whiteboard. Then pass the whiteboard to your left.

When you receive a whiteboard with a multiplication problem and an estimate, use the standard algorithm to multiply the two numbers. Use the estimate to check the reasonableness of your answer. When you think your answer is correct, pass the whiteboard to your left.

UDL: Engagement

The Pass the Whiteboard activity provides choice by allowing students to create their own multiplication problems.

Differentiation: Support

For students who need additional support, consider placing them in groups of 2 instead of groups of 4 for the Pass the Whiteboard activity.

Promoting the Standards for Mathematical Practice

When students estimate, calculate, and check their teammates’ work for the product of multi-digit numbers during the Pass the Whiteboard activity, they are constructing viable arguments and critiquing the reasoning of others (MP3).

Ask questions such as the following to promote MP3:

• What parts of the work shown for the standard algorithm for multiplication do you question? Why?

• What questions can you ask your teammate to make sure you understand their work?

When you receive a whiteboard with an answer, check your teammate’s work. How can you check a teammate’s work?

I can work it out myself and see whether I get the same answer. I can follow their process and check each calculation.

If you think they made a mistake, ask the teammate to explain their thinking. Suggest ways they could change their work to make it correct.

Allow students to engage in the Pass the Whiteboard activity until they need to begin their Problem Sets. Once a group has completed a full rotation with the whiteboards, they should start the cycle again. Circulate as students work, checking their work and correcting any misconceptions.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Multiply two multi-digit numbers by using the standard algorithm. Facilitate a class discussion about reasons for using the standard algorithm by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Why do we estimate before we multiply?

We estimate a product so we can determine whether our answer is reasonable.

Teacher Note

To support students’ ability to give helpful feedback in the Pass the Whiteboard activity, consider sharing the following prompts with students. They can use the prompts with one another so that their feedback is helpful and constructive.

• Be mindful of multiplication facts, so you can be precise.

• Be mindful of the sizes of the units you are multiplying by.

• Decompose the factor you are naming as the number of groups to keep track. (Here we are multiplying by 9, 10, and 100.)

• Check your product with the estimate—is it reasonable?

If a student makes a mistake in their work, consider asking them to either acknowledge what their mistake was or asking them to redo the multiplication with your help or the help of the group.

Why might we use the standard algorithm for multiplication instead of an area model or partial products?

The standard algorithm is often more efficient than other methods, particularly when you are multiplying two multi-digit numbers that may be challenging to multiply by using mental math. With the standard algorithm, you don’t have to write as many partial products.

Give an example of two multi-digit numbers you could multiply mentally. Why would you use mental multiplication with that example?

I can mentally multiply 40 by 700. It would take longer to use the standard algorithm because I can multiply 4 by 7 and then shift the digits three times to the left because 10 × 100 = 1,000.

Give an example of two multi-digit numbers you would use the standard algorithm to multiply. Why would you use the standard algorithm with that example?

I would use the standard algorithm to multiply 465 by 23. It would be difficult to keep track of using the distributive property because we have several digits that are not 0, so the standard algorithm could help me keep track and record my thinking efficiently.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

a. Is Blake’s answer reasonable? How do you know? 312

210,000

No. Blake’s answer is not reasonable because his answer of 4,050 is not close to my estimate of 210,000

b. What mistakes did Blake make? Blake multiplied 675 by 1 instead of by 10. He also multiplied 675 by 3 instead of by 300

Use the Read–Draw–Write process to solve the problem.

10. A cow weighs 712 kilograms. A blue whale is 255 times as heavy as the cow. How many kilograms does the blue whale weigh?

255 × 712 = 181,560

The blue whale weighs 181,560 kilograms.

Topic C Division of Whole Numbers

In grade 4, students learn to find whole-number quotients and remainders for problems with up to four-digit dividends and one-digit divisors by using mental math, area models, and partial quotients. In topic C, students use a variety of methods to determine quotients of multi-digit dividends and divisors.

The topic opens with students dividing two- and three-digit numbers by two-digit multiples of 10. Before they divide, students estimate quotients. They iterate units in tape diagrams to determine quotients and record their thinking in vertical form. Students then check their answers by writing equations involving multiplication and addition and determining whether the equations are true.

Next, students transition to dividing two- and three-digit numbers by two-digit numbers in problems that produce one-digit quotients. They use the process of estimate, divide, and check, and record their division work in vertical form. Students explain why division methods are the same regardless of how many digits are in the dividend or in the divisor.

To divide multi-digit numbers in problems that result in two-digit quotients, students use partial quotients and record their work in both area models and in vertical form. Students recognize that each partial quotient represents some number of groups of the divisor that fits into the dividend and that the sum of the partial quotients is the quotient. To deepen their understanding of division, students identify errors in sample work.

In topic C, students develop fluency with estimation. They explain how estimation helps them get started with division and determine whether a quotient is reasonable. Students know that if an estimate is too high, the number of groups of the divisor does not fit into the dividend, so they must remove at least one group. If an estimate is too low, there are more groups of the divisor that fit into the dividend, so they must add at least one group.

Throughout the topic, students explore meanings of quotients and remainders in both mathematical and real-world situations. They practice interpreting remainders by using context from the word problems. Students also explain why two division expressions with the same quotient and remainder are not necessarily equivalent.

In topic D, students apply their understanding of division when they solve multi-step word problems.

Progression of Lessons

Lesson 12

Divide two- and three-digit numbers by multiples of 10.

Lesson 13

Divide two-digit numbers by two-digit numbers in problems that result in one-digit quotients.

Lesson 14

Divide three-digit numbers by two-digit numbers in problems that result in one-digit quotients.

I can divide multi-digit numbers by multiples of 10 and record my work in vertical form. I can repeatedly draw units of the divisor in a tape diagram to help me see how many groups of the divisor fit in the dividend. Using multiplication and division facts helps me estimate quotients.

I can divide two-digit by two-digit numbers and record my work in vertical form. I can estimate to check whether a quotient is reasonable. I can find the quotient and then check my answer by using multiplication. I can interpret quotients and remainders in word problems.

When I look at a division problem, I know whether the quotient is less than or greater than 10. I understand I can use the same thinking to divide, whether the dividend has three digits or two digits.

Lesson 15

Divide three-digit numbers by two-digit numbers in problems that result in two-digit quotients.

Lesson 16

Divide four-digit numbers by two-digit numbers.

I can use partial quotients to divide. I understand how to record my thinking in an area model and in vertical form. I know the quotient is the sum of the partial quotients.

I can use estimation to determine whether a quotient is reasonable and I can use estimation to help find partial quotients. I know how to interpret quotients and remainders in realworld situations.

Divide two- and three-digit numbers by multiples of 10 .

Lesson at a Glance

Students divide two- and three-digit numbers by multiples of 10. They explore cases with and without remainders. Students estimate quotients before they divide, then use tape diagrams to determine quotients, record their thinking in vertical form, and use multiplication and addition to check their answers. This lesson formalizes the term dividend.

Key Questions

• Why is it helpful to estimate a quotient before you divide?

• How can we represent division with a tape diagram?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits. (5.NBT.B.6)

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models. (5.NBT.B.6)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Divide by Multiples of 10

• Division Problems

• Problem Set

Land 10 min

•

Lesson Preparation

•

Fluency

Whiteboard Exchange: Powers of 10

Students write a multiplication expression by using only 10 as a factor and the exponential form for a power of 10 given in standard form to build fluency with exponents from topic A.

Display 100.

When I give the signal, read the number shown. Ready? 100

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Write the number in exponential form.

Display the number in exponential form.

Write the number as a multiplication expression by using only 10 as a factor.

Display the multiplication expression.

Repeat the process with the following sequence:

Teacher Note

Students may hesitate when they encounter 10 in the sequence. Consider restating the explanation of an exponent and the special case for 101 from lesson 3.

An exponent represents how many times the same number is used as a factor in a multiplication expression. In the case of 10, a multiplication expression cannot be written without using another factor. This is different from the other numbers in this sequence, where 10 is the only factor in the multiplication expressions.

Counting by Multiples of 2 and 20

Students say the first ten multiples of 2 and 20 to prepare for estimating quotients.

When I give the signal, say the first ten multiples of 2. Ready?

Display each multiple one at a time as students count.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

When I give the signal, say the first ten multiples of 20. Ready?

Display each multiple one at a time as students count.

20, 40, 60, 80, 100, 120, 140, 160, 180, 200

Teacher Note

Because zero multiplied by any number is zero, it could be considered the first multiple of every number. However, to skip-count, it is typical to start with the unit that is being counted by and to think of the unit itself as the first multiple instead of zero. For example, to count by a unit of 10, the first multiple is 1 × 10, the second is 2 × 10, and so on.

Teacher Note

Consider asking students to compare and explain what they notice about the two lists of multiples. Students should notice the numbers in the second list are 10 times as much as the corresponding multiples of 2 in the first list.

Choral Response: Divide in Unit and Standard Form

Students divide tens by tens and say the equation with the numbers in standard form to prepare for dividing two- and three-digit numbers by multiples of 10.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 4 tens ÷ 2 tens = .

How many groups of 2 tens are in 4 tens?

2

Display the answer.

On my signal, say the equation with the numbers in standard form.

40 ÷ 20 = 2

Display the equation with the numbers in standard form.

Repeat the process with the following sequence:

Launch

Students identify methods to divide by multiples of 10.

Display the following problem:

Miss Song separates 360 T-shirts into boxes. She puts 40 T-shirts into each box.

Ask students to work with a partner to determine the exact number of boxes of T-shirts. Students should use at least two different methods.

Invite students to share their thoughts. Consider showing the methods that students share for all students to see.

I can divide 36 tens by 4 tens to get 9.

I can decompose 360 into 200 and 160, and then find how many 40s are in 200 and how many 40s are in 160.

How can we check that our answer is correct?

We can multiply. 9 × 40 = 360.

Ask students to turn and talk about which method they like best to divide 360 by 40 and why.

Transition to the next segment by framing the work. Today, we will divide two- and three-digit numbers by multiples of 10.

Teacher Note

Anticipate that students may refer to the following methods of division in Launch:

• Rename in unit form. (Divide 36 tens by 4 tens to get 9.)

• Use unit form with division facts. (36 tens by 4 tens can be found by using the division fact 36 ÷ 4.)

• Think about how many units are in a whole. (Decompose 360 into 200 and 160 or any other multiple of 40 that is familiar, and then determine how many 40s are in 200 and how many 40s are in 160.)

Learn

Divide by Multiples of 10

Students use estimates and partial quotients to divide by multiples of 10.

Write 360 ÷ 40.

How can we interpret this division problem?

We can think, How many groups of 40 are in 360?

We can think, 40 groups of what number make 360?

Let’s represent this division with a tape diagram. We can determine how many groups of 40 are in 360 by adding groups of 40 until we get to 360.

Draw a unit of 40.

Continue to add units, skip-counting until you get to a total of 360.

The tape diagram represents the total of 360. The divisor, 40, shows us the size of each group.

This diagram shows 360 is equivalent to 9 groups of 40. How do you know by looking at the tape diagram that the remainder of 360 ÷ 40 is 0?

The tape diagram shows 9 units of 40 have a total of 360.

We can also represent 360 ÷ 40 with a different tape diagram. What would the other tape diagram look like?

It would have 40 groups and we would not know the size of each group.

Invite students to think–pair–share about which tape diagram they would use to represent 360 ÷ 40 and why.

I would use the first tape diagram to represent groups of 40 because I do not want to draw a tape diagram with 40 units.

I would use the first tape diagram because the second diagram represents the expression, but it does not help me determine the value of 360 ÷ 40.

We can record the work for 360 ÷ 40 in vertical form.

Write 360 ÷ 40 in vertical form.

Why did I write the 9 above the 0?

9 represents 9 ones and when we write division in vertical form, the digits line up by place value. The ones digit in the quotient goes above the ones digit in the total.

What is the remainder when we divide 360 by 40?

How do you know by looking at the vertical form that the remainder is 0?

When we subtracted 360 from 360 in vertical form, the difference, or the part left over is 0.

Teacher Note

Students learn to record their work in vertical form in grade 4. This lesson reminds students of their previous experience and prepares them to use vertical form for problems involving dividends and divisors with more digits in the following lessons. While students may prefer to use other methods to find the value of 360 ÷ 40, using vertical form now lays the foundation for performing complex division, such as 3,268 ÷ 47, in later lessons.

Write 360 ÷ 40 = 9.

In this equation, what is the quotient? How do you know?

The quotient is 9 because a quotient is the number you get when you divide a number by another number.

What is the divisor? How do you know?

The divisor is 40 because a divisor is the number you divide by.

The number 360 also has a special name. It is called the dividend. In a division expression, the number that is divided by another number is the dividend.

Display the three representations of 360 ÷ 40 = 9.

Invite students to turn and talk about where they see the dividend, divisor, and quotient in each representation.

Division Problems

Students solve division problems that involve dividing by multiples of 10. Display the problem about Mr. Perez.

Mr. Perez separates 216 hats into bags. He fills each bag with 30 hats and donates the remaining hats to a raffle. How many hats does he donate?

What does 216 represent?

216 represents the total number of hats that are separated into bags or donated.

What does 30 represent?

30 represents the number of hats in each bag, or the size of each group.

Language Support

In grade 4, students refer to the dividend as the total. To support the use of the new term dividend, and other familiar division terms, consider displaying an anchor chart, such as the following, for student reference throughout this topic.

Promoting the Standards for Mathematical Practice

When students use division to solve real-world problems and interpret the meaning of the dividend, divisor, quotient, and remainder in the context of the problem, they are reasoning abstractly and quantitatively (MP2).

Ask the following questions to promote MP2:

• What does the problem ask you to do?

• What does the remainder mean in the problem about Mr. Perez?

• What real-world situations can we model with division?

What can we do to find how many bags Mr. Perez fills with hats? Why?

We can use division to find how many bags he fills with hats. He fills each bag with 30 hats, so we need to figure out how many 30s are in 216.

What is an expression you can use to find how many bags he fills?

216 ÷ 30

In the expression 216 ÷ 30, which number is the dividend and which number is the divisor? What does each represent?

The dividend is 216, and it represents the total number of hats. The divisor is 30, and it represents the size of each group.

Approximately how many bags can he fill? How do you know? He can fill approximately 7 bags. I can think about 216 as 210 because I can divide 210 by 30 mentally. 210 ÷ 30 = 7.

We can say that 216 divided by 30 is approximately 210 divided by 30. Why is 210 an appropriate number to use to estimate the quotient 216 ÷ 30? Is 220 also appropriate to use?

210 is a multiple of 30, and 210 is close to the dividend 216. Because 220 is not a multiple of 30, I think 210 is a more appropriate dividend to use to make an estimate.

Let’s build a tape diagram to determine how many 30s are in 216.

Draw a unit of 30. Continue to add units, skip-counting until you get to the estimate of 7 groups of 30.

How many 30s are in 216? Where do you see that number in the tape diagram? There are seven 30s in 216 because there are 7 groups of 30 in the tape diagram.

Differentiation: Support

For students who need additional support with dividing by powers of 10, when they make estimates, suggest that they use unit form. For example, 210 ÷ 30 is equal to 21 tens ÷ 3 tens, or 7.

Teacher Note

As with the term product, students may have only thought about the word quotient as an answer to a division expression. Support students with understanding that quotient can refer to both a division expression and its answer.

Let’s record this information in vertical form.

We made 7 groups of 30. How much of the 216 remains? Give the full subtraction sentence.

6 of the 216 remain because 216 − 210 = 6.

What does the 6 represent?

It represents the hats that remain after Mr. Perez fills 7 bags with 30 hats each.

Let’s revisit our tape diagram. We have 6 left over when we divide 216 by 30. Can we make another group of 30 with the 6 left over? Why?

No. We need 30 to make another group and 6 is less than 30.

We cannot make another group of 30, but we can show our remainder by labeling a part with 6 on our tape diagram.

Draw a part at the end of the tape diagram and label it 6.

We can write the quotient and the remainder next to the vertical form recording.

Write the quotient and remainder next to the vertical form work.

What are the quotient and the remainder of 216 ÷ 30?

The quotient is 7. The remainder is 6.

Based on our work and our tape diagram, we found that 216 is equal to 7 groups of 30 plus 6.

Write 216 = 7 × 30 + 6.

Let’s see whether this is a true equation. What is the value of 7 × 30 + 6?

216

We checked our answer by writing and evaluating an expression. The equation is true because the value of each side is 216, so we know our answer is correct.

UDL: Representation

Consider creating a two-column chart, such as the following, to compare the tape diagram method with the vertical method. Label each column and provide an example of the same division problem in each column. Discuss the similarities and differences to support students in seeing connections.

How many hats does Mr. Perez donate to the raffle? How do you know?

Mr. Perez donates 6 hats because he has 6 extra hats that do not fit in the bags.

Why might we use vertical form to record division work?

It can be more efficient and can require less writing than other methods. It clearly shows the dividend, divisor, quotient, and remainder.

Have students turn and talk with a partner to discuss whether estimation helped them determine the value of 216 divided by 30 and explain how estimation helped.

Write 546 ÷ 70.

In the expression 546 ÷ 70, what number is the dividend and what number is the divisor? How do you know? What does each mean in this expression?

The dividend is 546 because it is the number that is divided by another number. The divisor is 70 because it is the number 546 is divided by.

Let’s estimate the quotient 546 ÷ 70. What number should we use for the dividend to help us estimate? Why?

We can use 560 for the dividend because 560 is the multiple of 70 that is closest to 546.

Write 560 ÷ 70.

In unit form, how do we say 560 ÷ 70?

56 tens ÷ 7 tens

What division fact might you use to find the quotient of this expression?

56 ÷ 7

What is 56 ÷ 7?

8

What is 560 ÷ 70?

8

What does this tell us about the quotient 546 ÷ 70?

The quotient 546 ÷ 70 is about 8.

Draw a tape diagram with 8 parts and label each part 70.

8 seems like a good estimate because 560 is the multiple of 70 that is nearest to 546, but let’s take a closer look. What do you notice about our estimate?

Our estimate is too high. 8 groups of 70 is 560, so we cannot divide 546 into 8 groups of 70 because 546 is less than 560.

Sometimes we use estimates to check for reasonableness. When we divide, we also use estimates as a starting point. In this case, we overestimated. Because our estimate of 8 is too high, what should we try next?

We should try 7 groups of 70, instead of 8 groups of 70.

Draw a tape diagram with 7 parts and label each part 70.

How many 70s are in 546? Why?

There are seven 70s in 546. That is the greatest number of 70s you can fit into 546.

Where do you see 7 in the tape diagram?

There are 7 parts in the tape diagram.

Let’s record this information in vertical form.

We made 7 groups of 70. How much of the 546 remains? Give the full subtraction sentence.

56 of the 546 remain because 546 − 490 = 56.

Let’s revisit our tape diagram. We have 56 left over when we divide 546 by 70. Can we make another group of 70 with the 56 left over? No.

We cannot make another group of 70, but we can show our remainder by labeling a part with 56 on our tape diagram.

Language Support

Consider providing additional support for the terms underestimate and overestimate. Ask students what they think these terms mean. Encourage students to provide underestimates and overestimates of quantities they are familiar with.

• A student estimates the quotient 75 ÷ 5 as 10. Ask: Is that an overestimate or an underestimate? Why?

• A student estimates the quotient 87 ÷ 10 as 9. Ask: Is that an overestimate or an underestimate? Why?

Draw a part at the end of the tape diagram and label it 56.

We can write the quotient and the remainder next to our vertical form work.

Write the following information next to the vertical form work.

Quotient: 7

Remainder: 56

Based on our work, we found that 546 is equal to 7 groups of 70 plus 56, or that 546 is equal to 70 groups of 7 plus 56. Notice our divisor 70 can be the number of groups or the size of the groups.

Write 546 = 7 × 70 + 56.

Let’s see whether this is a true equation. What is the value of 7 × 70 + 56?

546

We checked our answer by writing and evaluating an expression. The equation is true because the value of each side is 546, so we know our answer is correct. Is 546 = 70 × 7 + 56 also true? How do you know?

Yes. The equation 546 = 70 × 7 + 56 is also true. The only difference between the equations is the order of the factors 70 and 7. I know 70 × 7 = 7 × 70 because of the commutative property of multiplication.

Invite students to turn and talk with a partner about whether making an estimate helped to find the answer and explain how estimation helped.

Display the following problem.

A number divided by 60 has a quotient of 3 and a remainder of 12. What is the number?

Ask students to work with a partner to find the number.

What is the number? How do you know?

The number is 192 because 60 × 3 + 12 = 192.

Differentiation: Challenge

For students who need an additional challenge, consider presenting the following prompt:

A number divided by another number has a quotient of 7 and a remainder of 12. List at least three pairs of numbers for which this statement is true.

Display the word problem.

Read the problem aloud. Ask students to turn and talk with a partner and use the Read–Draw–Write process to solve the problem.

Then direct students to complete problem 1 in their books with their partner. Use

1. Tyler wants to donate boxes of crayons to kindergarten classes. He has 347 boxes. He donates sets of 40 boxes to as many classes as he can. How many boxes remain? 40

How many sets of 40 boxes?

Differentiation: Support

For students who need additional support, encourage them to use the Estimate, Tape Diagram, Divide, Check process.

Tyler has 27 boxes remaining.

When most students are finished, ask the following questions.

To how many kindergarten classes does Tyler donate crayons? How do you know?

He donates to 8 kindergarten classes because there are eight 40s in 347.

How many boxes of crayons does Tyler donate to the kindergarten classes? How do you know?

Tyler donates 320 boxes of crayons to the kindergarten classes because 8 × 40 = 320.

How many boxes of crayons does Tyler have that remain? How do you know?

Tyler has 27 boxes that remain because that is how many he has left over after he donates an equal number of boxes of crayons to the kindergarten classes.

Invite students to turn and talk about how they used estimation to help solve this problem or how they estimated the reasonableness of their solution.

What do all the divisors we used today have in common?

They are multiples of 10.

Is it helpful to have a divisor that is a multiple of 10? How?

Yes. I only need to think about what number to use for the dividend to make an estimate.

Yes. I can use mental math when I work with multiples of 10.

Does a divisor have to be a multiple of 10?

No.

How can we estimate if our divisor is not a multiple of 10?

We can use a number for our divisor that is a multiple of 10 before we make an estimate.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Divide two- and three-digit numbers by multiples of 10.

Facilitate a class discussion about division by multiples of 10 by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Why is it helpful to estimate a quotient before you divide?

If you estimate first, it helps you figure out how to get started with division. When you estimate first, you compare the quotient to your estimate to see whether the quotient is reasonable.

How can a tape diagram represent a quotient and a remainder?

A tape diagram shows how many units of the divisor fit into the dividend and it also shows what is left over.

Why do we record our work in vertical form when we divide?

Vertical form can be more efficient and require less writing than other methods. Vertical form helps us keep track of our work. The recording clearly shows the dividend, divisor, quotient, and remainder.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect

to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Divide. Then check your work.

9. A number divided by 40 has a quotient of 6 and a remainder of 15. What is the number?

40 × 6 + 15 = 240 + 15 = 255

The number is 255

Use the Read–Draw–Write process to solve the problem.

10. A student has 174 centimeters of ribbon for making bows. Each bow is made with 20 centimeters of ribbon. The student wants to make as many bows as possible. How many bows can the student make? How many centimeters of ribbon will be left over?

174 ÷ 20

Quotient: 8

Remainder: 14

The student can make 8 bows. There will be 14 centimeters of ribbon left over.

PROBLEM SET

Copyright © Great Minds PBC 104 PROBLEM SET

Divide two-digit numbers by two-digit numbers in problems that result in one-digit quotients.

Lesson at a Glance

Students use the Co-construction routine to contextualize a statement involving division. By using the process of estimate, solve, and check, students divide two-digit numbers by two-digit numbers and record their work in vertical form. Students explain how to handle cases in which they underestimate or overestimate a quotient. Students interpret remainders in real-world situations.

Key Questions

• What is the purpose of estimation when you divide?

• What does it mean if an estimate is too low or too high?

• How do you check the answer to a division problem if there is a remainder?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits. (5.NBT.B.6)

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models. (5.NBT.B.6)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Divide Two-Digit Numbers by Two-Digit Numbers

• Division Puzzle

• Division Word Problems

• Problem Set

Land 10 min

Lesson Preparation

Fluency

Whiteboard Exchange: Powers of 10

Students write a power of 10 expressed in words in exponential form as a multiplication expression by using only 10 as a factor, and in standard form to build fluency with exponents from topic A.

Display Ten to the second power.

When I give the signal, read the number shown. Ready? Ten to the second power

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Write the number in exponential form.

Show the number in exponential form.

Write the number as a multiplication expression by using only 10 as a factor.

Show the multiplication expression.

Write the number in standard form.

Show the number in standard form.

Ten to the second power

Repeat the process with the following sequence:

Counting by Multiples of 3 and 30

Students say the first ten multiples of 3 and 30 to prepare for estimating quotients.

When I give the signal, say the first ten multiples of 3. Ready?

Display each multiple, one at a time, as students count.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

When I give the signal, say the first ten multiples of 30. Ready?

Display each multiple, one at a time, as students count. 30

Choral Response: Divide in Unit and Standard Form

Students divide tens by tens and say the equation with the numbers in standard form to prepare for dividing two-digit numbers by two-digit numbers.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 4 tens ÷ 2 tens = .

How many groups of 2 tens is in 4 tens?

2

Display the answer.

On my signal, say the equation with the numbers in standard form.

40 ÷ 20 = 2

Display the equation with the numbers in standard form.

Repeat the process with the following sequence:

Launch

Students use the Co-construction routine to contextualize a statement involving division.

Write the following information:

87 ÷ 13

Quotient: 6

Remainder: 9

When we divide 87 by 13, the quotient is 6 and the remainder is 9.

Pair students and use the Co-construction routine to have partners create a real-world situation that could be represented by this mathematical statement.

Give pairs 1 minute to compare the situations they construct with the situations other pairs of students construct. Then invite pairs to share with the class. Have students explain the relationships between the situation and the sentence.

A person sorts 87 apples into 13 baskets and each basket has 6 apples. There are 9 apples left over.

A florist uses 87 flowers to make bouquets that each have 13 flowers. She makes 6 bouquets and has 9 flowers left over.

Someone pours 87 ounces of water into cups that each hold 13 ounces of water. They fill 6 cups and have 9 ounces of water remaining.

Invite students to turn and talk about how they can create a real-world situation that is represented by a mathematical statement.

Transition to the next segment by framing the work.

Today, we will divide two-digit numbers by two-digit numbers.

Teacher Note

The quotient and remainder of 87 ÷ 13 are intentionally written as Quotient: 6 and Remainder: 9 and not expressed as 6 R 9

Both 69 ÷ 10 and 87 ÷ 13 have quotients of 6 with remainders of 9. However, these expressions are not equivalent. Writing the quotients and remainders as 6 R 9 implies that these expressions are equivalent.

In module 2, students learn to express remainders as fractions. They write 69 ÷ 10 as 6 9 10 and 87 ÷ 13 as 6 9 13 and see that 6 9 10 and 6 9 13 are not equivalent.

Learn

Divide Two-Digit Numbers by Two-Digit Numbers

Students divide two-digit numbers by two-digit numbers.

Display the following problem:

95 is 19 times as much as what number?

What unknown factor equation can we write to represent this question?

95 = 19 ×

What division expression can we write to determine the value of the unknown factor?

95 ÷ 19

Write 95 ÷ 19. As you lead students through finding an estimate, drawing a tape diagram, dividing, and checking their answer, record the work as shown.

Teacher Note

In grade 3, students represent division as multiplication with an unknown factor. This understanding can assist students as they interpret multiplicative comparison statements as division.

Let’s begin with an estimate. Which number should we look at first? Why?

We should look at 19 because it is the divisor. Once we consider the divisor, we can do the same for the dividend and then divide to get an estimate.

What number should we use for 19? 20

Differentiation: Support

Encourage students who need additional support in making estimates to continue using the multiplication or division facts they know. This way of thinking is important for success with estimation.

If we use 20 as our divisor, what should we use for the dividend? Why?

We should use 100 because 100 is a multiple of 20 that is close to 95. Both 80 and 100 are multiples of 20, but 100 is closer to the dividend.

We can say 95 divided by 19 is approximately 100 divided by 20.

Write 100 ÷ 20.

What division fact do you see in this expression?

10 ÷ 2

What is 10 ÷ 2?

5

What is 100 ÷ 20?

5

What does that tell us?

The quotient 95 ÷ 19 is about 5. There are about 5 groups of 19 in 95.

As a starting point, let’s consider the question, What is 5 groups of 19 equal to?

95

In this case, our estimate is the quotient.

We can show the division by using a tape diagram. What will the tape diagram look like?

The tape diagram will show 5 parts and each part will be labeled 19. Draw a part and label it 19. Continue to add parts, skip-counting by 19s, until you get to a total of 95.

Why might you choose not to make a tape diagram with a divisor of 19?

It is challenging for me to skip-count by 19s.

Let’s record the work in vertical form.

Record the division work in vertical form.

Teacher Note

In this lesson, students draw tape diagrams to find quotients. In the next lesson, they draw tape diagrams to make sense of problems involving division, and then they find quotients by using vertical form.

What is the remainder of 95 ÷ 19?

Based on our work, 95 is equal to 5 groups of 19.

Write 95 = 5 × 19.

Let’s see whether this is a true equation. What is the value of 5 × 19?

We checked our answer by writing and evaluating an expression. The equation is true because the value of each side is 95, so we know our answer is correct.

Let’s revisit the original question. 95 is 19 times as much as what number?

5

Turn and talk with a partner about how you can estimate a quotient by using multiplication or division facts.

Display the work showing 84 ÷ 16.

This work shows an estimate and the calculations that a student made when they divided 84 by 16. Invite students to think–pair–share about what they notice about the estimate.

I notice there is a remainder of 20, which means one more group of 16 can fit into 84.

I notice that the estimate is an underestimate because there is a remainder of 20, which is greater than the divisor of 16.

Promoting the Standards for Mathematical Practice

When students make appropriate approximations for dividends and divisors to estimate the quotient of a division expression and learn how to adjust their work if the estimates are too high or too low, they are looking for and making use of structure (MP7).

Ask the following questions to promote MP7:

• How can what you know about 80 ÷ 20 help you find 84 ÷ 16?

• How are the divisor of 16 and the remainder of 20 shown in the work related? How can that help you determine whether the student made an underestimate or an overestimate?

• H ow can what you know about underestimates help you adjust the quotient?

Is this student’s estimate reasonable? Explain.

Yes. The estimate is reasonable because the student used numbers for the dividend and divisor that are close to the actual dividend and divisor, and 80 is a multiple of 20.

Even though 4 is not the final quotient, it is part of the quotient and a reasonable estimate that gives us a starting point for the division. We can see from the sample work that 4 units of 16 is equal to 64.

The remainder is greater than the divisor. What should we do?

We can add another group of 16 to the quotient. We can change the quotient from 4 to 5.

Ask students to work with a partner to determine the quotient and remainder of 84 ÷ 16. Then ask them to check their answers. Invite students to share their thinking. 84 ÷ 16 has a quotient of 5 and a remainder of 4 because 16 × 5 + 4 = 84.

Display the work that shows 92 ÷ 13.

This work shows an estimate and the calculations that a student made when they divided 92 by 13.

Invite students to think–pair–share about the estimate of 9 and what they notice.

I notice the estimate is an overestimate because 9 groups of 13 is 117 and the dividend is only 92. 9 groups of 13 is greater than the dividend.

Did this student underestimate or overestimate? How do you know?

The student overestimated because 9 groups of 13 is equal to 117, and 117 is greater than the dividend.

What could this student do?

The student could try fewer groups because 9 groups of 13 has a total greater than the dividend.

The student could try two fewer groups than 9 because 117 is 25 more than 92 and I know 2 × 13 = 26.

Teacher Note

Students can show another group, as pictured in the following vertical recording, as four 16 s and one 16 for the quotient. (Note that all ones are recorded vertically in the ones column.) The 4 and 1 are called partial quotients and are explored more fully in subsequent lessons.

Ask students to work with a partner to determine the quotient and remainder of 92 ÷ 13. Then ask them to check their answers. Invite students to share their thinking.

92 ÷ 13 has a quotient of 7 and a remainder of 1 because 7 × 13 + 1 = 92.

Invite students to turn and talk about how to handle cases in which they underestimate or overestimate.

Division Puzzle

Students use a given tape diagram to write a division statement. Display the tape diagram that shows units of 27 and 16 and the statement.

16 27 27 27

divided by has a quotient of and a remainder of .

Ask students to work with a partner and use the tape diagram to determine what number belongs in each blank.

97 divided by 27 has a quotient of 3 and a remainder of 16.

Invite students to share their thinking and their strategies. Consider asking any of the following questions as students share:

• How did you determine the dividend is 97?

• How did you determine the divisor is 27?

• How did you determine the quotient is 3?

• How did you determine the remainder is 16?

• How can you represent this tape diagram with a number sentence that uses multiplication and addition?

Invite students to turn and talk about what strategies they can use to solve division puzzles.

Differentiation: Challenge

For students who need an additional challenge, consider using the following prompt:

Write three different division expressions that can be represented by this tape diagram. For each expression, use the tape diagram to justify your thinking. Each of the five unmarked units have the same value. 11

Division Word Problems

Students solve real-world problems involving division.

Direct students to problem 1 in their books. Give students several minutes to complete problems 1–3 with a partner.

Use the Read–Draw–Write process to solve each problem.

1. Sasha is training for a competition and plans to do 96 push-ups in one day. She plans to do these push-ups in sets of 16. How many sets of push-ups will she need to do to reach her goal of 96 push-ups? Show your thinking, including an estimate and a check.

Language Support

Consider discussing the contexts of problems 1–3 before students begin working on them. It may be helpful to do the following:

• Demonstrate a push-up or invite students to do one.

• Show students a picture of children at a summer camp.

• Show students a picture of different types of coins. 16

?

Differentiation: Support

If students need additional support to determine the division expression that represents the situation, consider moving the question to the beginning of the problem. For example, read problem 1 to students reworded as: How many sets of push-ups will Sasha need to do to reach her goal of 96 push-ups? Sasha is training for a competition and plans to do 96 push-ups in one day. She plans to do one set of 16 push-ups at a time.

Check:

16 × 6 = 96

Sasha needs to do 6 sets of push-ups.

If more support is needed to complete the problem, consider having students label three sections as Estimate, Divide, and Check (or abbreviate to E, D, C) under problems 2 and 3. Students can use problem 1 as a model to help gauge the amount of space to leave under each section. Alternatively, label the sections for the student or provide a template.

2. A camp plans to take its 92 students on a field trip. Each bus holds 21 students. How many buses does the camp need for the field trip? Show your thinking.

4 21 92

– 84 8

The camp needs 5 buses.

UDL: Action & Expression

92

? buses

3. There are 92 coins split into 21 piles. Each pile has the same number of coins and as many coins as possible. How many coins are in each pile?

92

4 21 92

– 84 8

There are 4 coins in each pile.

When students are finished, ask the following questions.

21 piles

Consider posting a chart of tape diagrams that represent equations. This will support students’ understanding of the relationships (with or without context) between the dividend, divisor, quotient, and remainder. 96 = 16 × 6 92 = 21 × 4 + 8

Was your first estimate in problem 1 the actual quotient or did you need to revise?

My estimate was too low. I estimated 5 but the quotient is 6.

Why do you think an estimate of 5 was too small?

I rounded the divisor from 16 to 20, so not as many 20s as 16s will fit in the dividend.

In problem 2, why does the camp need 5 buses instead of 4 buses?

The camp needs 5 buses to hold the remaining students. 4 buses hold 84 students, but there are 92 students.

In problem 2, we decided the camp needs 5 buses and not 4. In problem 3, why are there 4 coins in each pile and not 5?

There are not enough coins to have 5 coins in each pile.

In problems 2 and 3, the dividend, divisor, and quotient are the same. However, the way we had to think about the division was different. Explain the difference.

In problem 2, the divisor 21 represents the size of each group. We determined how many 21s fit into 92.

In problem 3, the divisor 21 represents the number of groups. We distributed 92 into 21 equal-size groups.

Invite students to turn and talk to discuss strategies to interpret remainders in real-world situations.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Divide two-digit numbers by two-digit numbers in problems that result in one-digit quotients.

Facilitate a class discussion about dividing two-digit numbers by two-digit numbers by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

What is the purpose of estimation when you divide?

With estimation, we have a starting point for division.

With estimation, we can determine the reasonableness of the quotient.

What does it mean if an estimate is too low or too high?

If our estimate is too high, the number of groups of the divisor cannot fit into the dividend. So we can remove a group.

If our estimate is too low, there are more groups of the divisor that can fit into the dividend. So we can add a group.

What can you do if you underestimate or overestimate a quotient? Give an example.

I can still use my estimate to help me find the quotient, even if I underestimate or overestimate. For example, if I estimate a quotient is 6 and see that 6 groups of my divisor is more than the dividend, I can try 5 as the quotient.

How do you check the answer to a division problem if there is a remainder?

I multiply the quotient by the divisor and then add the remainder. That number should equal the dividend.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Divide.

9. Scott wants to find 78 ÷ 42. First, he estimates the quotient. Then he uses his estimate to divide.

a. What should Scott do next?

Scott should try 1 as the quotient because his estimate is too high. Two groups of 42 is 84, which is greater than the dividend.

b. Find 78 ÷ 42

Quotient: 1

Remainder: 36 Use the Read–Draw–Write process to solve the problem.

10. An auditorium has 25 seats in each row. How many rows are needed to seat 92 students?

92 ÷ 25

Quotient: 3

Remainder: 17

4 rows are needed to seat 92 students.

Divide three-digit numbers by two-digit numbers in problems that result in one-digit quotients.

Lesson at a Glance

Students explore and explain why two division expressions with the same remainder and quotient are not necessarily equivalent. Students explain why they can use the same methods to divide, regardless of the number of digits in the dividend. Students solve word problems in which they divide three-digit numbers by two-digit numbers and interpret remainders in context.

Key Questions

• Why is interpreting the remainder an important part of solving real-world problems?

• How might you know, even before you estimate, that a quotient is greater or less than 10?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits. (5.NBT.B.6)

Agenda Materials

Fluency 15 min

Launch 5 min

Learn 30 min

• Compare Division Expressions

• Division Word Problems

• Problem Set

Land 10 min

Teacher

• None Students

• Powers of 10 Sprint (in the student book)

Lesson Preparation

Consider tearing out the Sprint pages in advance of the lesson.

Fluency

Sprint: Powers of 10

Materials—S: Powers of 10 Sprint

Students use exponential form to write a power of 10 expressed in a variety of forms to build fluency with exponents from topic A.

Have students read the instructions and complete the sample problems.

Direct students to Sprint A. Frame the task.

I do not expect you to finish. Do as many problems as you can, your personal best.

Take your mark. Get set. Think!

Time students for 1 minute on Sprint A.

Stop! Underline the last problem you did.

I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer.

Read the answers to Sprint A quickly and energetically.

Count the number you got correct and write the number at the top of the page. This is your personal goal for Sprint B.

Celebrate students’ effort and success.

Provide about 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A. If students are provided time to complete more problems on Sprint A, reread the answers but do not have them alter their personal goals.

Lead students in one fast-paced and one slow-paced counting activity, each with a stretch or physical movement.

Point to the number you got correct on Sprint A. Remember this is your personal goal for Sprint B.

Direct students to Sprint B.

Take your mark. Get set. Improve!

Time students for 1 minute on Sprint B.

Stop! Underline the last problem you did.

I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer.

Read the answers to Sprint B quickly and energetically.

Count the number you got correct and write the number at the top of the page.

Determine your improvement score and write the number at the top of the page.

Celebrate students’ improvement.

Teacher Note

Consider asking the following questions to discuss the patterns in Sprint A:

• What do you notice about problems 1–12? 13–22?

• What strategy did you use for problems 1–6? 7–12? 18–22?

Teacher Note

Count forward by 100,000 from 0 to 1,000,000 for the fast-paced counting activity.

Count backward by 10,000 from 100,000 to 0 for the slow-paced counting activity.

Launch

Students determine why expressions with the same quotient and remainder may not have the same value.

Write the following expressions.

92 ÷ 3

122 ÷ 4

Invite students to determine the quotient and remainder of each expression. When students are finished, ask the following questions.

What do you notice? What do you wonder?

I notice the expressions have different dividends and divisors but the same quotients and remainders.

I wonder how expressions that are different can have the same quotient and remainder.

I wonder whether there are other numbers we can divide that give us the same quotient and remainder.

Invite students to think–pair–share about the following question. Allow students 1–2 minutes to reason about the question. Encourage them to draw pictures or to consider a similar situation with smaller numbers. Accept all student responses.

We know the expressions 12 ÷ 3 and 20 ÷ 5 have the same value, 4. If 92 ÷ 3 and 122 ÷ 4 have the same quotient and the same remainder, do the expressions have the same value? Explain your thinking.

I think they have the same value because they have the same quotient and the same remainder.

The expressions 5 ÷ 4 and 6 ÷ 5 both have a quotient of 1 and a remainder of 1, but I don’t think they have the same value.

We can draw a diagram to help us reason about this situation. Let’s think of splitting $92 among 3 people and splitting $122 among 4 people.

Display the diagrams that show 92 ÷ 3 and 122 ÷ 4.

In both situations, after the money is split equally, each person gets $30 and there is $2 left over to share. If we can equally split the leftover $2 between each person in each group, do the people in the group of 4 get the same amount of money as the people in the group of 3? How do you know?

No. In both situations $2 is being split, but because the number of people in each group is different, they get different amounts of money. The group of 4 people gets less money than the group of 3 people because more people have to share the same amount of money.

We know the expressions 92 ÷ 3 and 122 ÷ 4 do not have the same value because each person does not receive the same amount of money. In the future, you will learn more about how to handle remainders in situations like this one.

Transition to the next segment by framing the work.

Today, we will interpret remainders when we divide three-digit numbers by two-digit numbers.

Learn

Compare Division Expressions

310

How are these two expressions alike? How are they different?

Both expressions show division.

In both expressions the divisor is a two-digit number.

The expressions are different because one of them has a three-digit dividend and the other has a two-digit dividend.

Can we use the same process to divide that we used in previous lessons? Why?

Yes. We can still use estimation, show division in an area model or vertical form, and then check by using multiplication.

Does having more digits in the dividend affect how you will divide? Does your understanding of the dividend and divisor change?

No. We can interpret the dividend as the number we are dividing, regardless of how many digits it has. The divisor can represent the number of groups or the size of the group. So we can evenly distribute the total into equal-size groups, or we can determine how many units of the divisor fit into the total.

Invite students to turn and talk about the process they use to determine a quotient.

Division Word Problems

Students solve real-world problems involving division.

Direct students to problem 1 in their books. Invite them to complete problems 1–4 with a partner.

Circulate and encourage students to make estimates, divide, and then check their answers. Use questions and prompts such as the following to support students and advance their thinking:

• What is the dividend? What is the divisor? How do you know? What do they represent?

• How can you estimate the quotient?

• If you overestimated, why do you think your estimate was too high?

• If you underestimated, why do you think your estimate was too low?

• What is the quotient? What does it represent?

• How do you know your answer is correct?

• Is there a remainder?

• What does the remainder represent in this situation and how does it affect your answer to this question?

Use the Read–Draw–Write process to solve each problem.

1. A school activity has 301 students split into 43 equal-size groups. How many students are in each group?

301

? . . . 43 groups

Differentiation: Support

In this lesson, students draw tape diagrams to make sense of problems involving division and then find quotients by using vertical form. For students who need additional support, encourage them to continue to use tape diagrams to divide.

Language Support

Consider directing students to use their Talking Tools while they complete the division word problems. Students can use the I Can Share My Thinking section of the tool when they communicate their ideas.

Promoting the Standards for Mathematical Practice

When students make estimates, adjust their estimates, and interpret remainders to solve real-world problems by using division, they are making sense of problems and persevering in solving them (MP1).

Ask the following questions to promote MP1:

• How can you simplify the problem?

• Does your estimate work? Is there something else you could try?

• Does your answer make sense? Why?

Divide:

Check: 7 × 43 = 301

There are 7 students in each group.

2. Eddie has 34 days to read a 170-page book. If he reads the same number of pages each day, how many pages does he need to read each day to finish the book in 34 days? ?

Teacher Note

Students may check their work in a variety of ways, such as with an area model, the standard algorithm, or the break apart and distribute method. For example, some students may check their work for problem 1 like this:

7 × 43 = 7 × (40 + 3)

= 7 × 40 + 7 × 3

= 280 + 21 = 301

Checking their work is more important than the method that students choose.

UDL: Engagement

Consider providing mastery-oriented feedback that focuses attention on students’ effort and strategies. For example, recognize students for approximating the dividend and divisor to estimate the quotient, changing the quotient if they underestimated or overestimated, and checking their answer by using multiplication and addition. Acknowledge students by giving feedback, such as the following statements:

• I like how you tried 6 as your estimate, but when you realized you underestimated, you tried 7

Eddie needs to read 5 pages per day.

• I like how you tried 8 as your estimate, but when you realized you overestimated, you tried 7.

3. Miss Baker needs to order 546 pencils. If each pack has 72 pencils, what is the fewest number of packs Miss Baker should order? 54

UDL: Representation

? pack s of pencils

Check: 7 × 72 + 42 = 504 + 42 = 546

Miss Baker should order 8 packs of pencils.

Copyright © Great

These word problems include two interpretations of the divisor. In problems 1 and 2, the divisor represents the number of groups. In problems 3 and 4, the divisor represents the size of each group. The tape diagrams reflect those interpretations. Consider creating an anchor chart for students to reference. Constructing tape diagrams for word problems helps students recognize which operation they can use to solve the problem. Making sense of a context and interpreting the meaning of the divisor in a word problem helps students recognize they can use division to solve the problem.

In this tape diagram, the divisor represents the number of groups, so the quotient represents the size of each group. ?

In this tape diagram, the divisor represents the size of each group, so the quotient represents the number of groups.

4. Riley has 457 centimeters of ribbon. Each costume he makes needs 55 centimeters of ribbon. How many costumes can Riley make? 457 55 . . .

? costumes

Estimate:

457 ÷ 55 ≈ 480 ÷ 60 = 48 ÷ 6 = 8

Divide:

8

55 457 – 440 17

Check:

8 × 55 + 17 = 440 + 17 = 457 Riley can make 8 costumes.

When students are finished, bring the class together. Invite students to turn and talk with a new partner about what the dividend, divisor, quotient, and remainder represent in each problem.

Then ask the following questions.

Previously when we multiplied, we considered the factors first. Sometimes when we noticed a zero in one of the factors, we considered which factor we would designate as the unit so we would have fewer partial products. In both problems 1 and 2, one of the digits in the dividend is zero. When there is a zero in the dividend, does it change how you approach the division? Why?

Differentiation: Challenge

For students who finish problems 1–4 quickly or need an additional challenge, pose the following problem:

15 friends plan to split $126 equally. How much money does each friend receive?

Note that students may not yet know how to divide with decimals or to write remainders as decimals. But the intent with this problem is that some students may be able to determine how to split the $6 that remains after giving each friend the same number of dollars.

No. I still think about the number of groups of the divisor that can fit into the total. In which problems did you underestimate? What did you do next?

I underestimated in problem 1 because I thought about 43 as 50 and 301 as 300 and got 6 as my estimate. I added 1 more to the divisor and tried a quotient of 7 instead of a quotient of 6.

I underestimated in problem 2 because I thought about 34 as 40 and 170 as 160 and got 4 as my estimate. I added 1 more to the divisor and tried a quotient of 5 instead of a quotient of 4.

I underestimated in problem 4 because I thought about 55 as 60 and 457 as 420 and got 7 as my estimate. I added 1 more to the divisor and tried a quotient of 8 instead of 7.

Direct students to problem 4. Display the following work sample. Invite students to share what they notice.

What do you notice?

I notice two quotients. Both quotients are written vertically in the ones place.

We can call 7 and 1 partial quotients. They are part of the quotient. What does 7 represent?

7 groups of 55

What does 1 represent?

1 more group of 55

Why do you think they recorded 7 and 1 vertically like this?

They are both ones.

They underestimated and then showed another group with the 1 above the 7 they started with.

Instead of erasing 7 and writing 8, they wrote 1 to show 1 more group of 55.

When we worked with partial products, how did we find the actual product?

We added the partial products to find the actual product.

Teacher Note

Students are introduced to partial quotients in grade 4. This discussion reminds students of that learning and prepares them to use partial quotients fluently in the discussion in the next lesson.

So what could we do with our partial quotients to find the actual quotient?

We could add them because they are parts of the quotient.

What is the quotient? How do you know?

The quotient is 8 because seven 55s and one 55 is equal to 440, and that is as many equal groups as we can create from the dividend 457.

In which problems did you overestimate? What did you do?

I overestimated in problem 2 because I thought about 34 as 30 and 170 as 180 and got 6 as my estimate. So I erased 6, knew to subtract 1 from the divisor, and then tried a quotient of 5.

I overestimated in problem 4 because I thought about 457 as 500 and 55 as 50 and got 10 as my estimate. So I erased 10, knew to subtract 1 group of the divisor, and then tried a quotient of 9. That quotient was still too large, so I erased again and subtracted another group of the divisor, and then I tried a quotient of 8.

If you underestimated or overestimated, was your estimate still useful? Explain. Yes. Even if an estimate is too low or too high, it is still a good place to get started with division. We can also use our estimate to determine whether our answer is reasonable. Yes. If our estimate is too low, that means we found only part of the quotient and need more groups of the divisor.

In problem 3, the quotient is 7. Why does Miss Baker need to order 8 packs of pencils?

If Miss Baker orders 7 packs, she has 504 pencils. Because she needs 546 pencils, she must order an additional pack.

How much extra ribbon does Riley have in problem 4? How do you know?

Riley has 17 centimeters of extra ribbon because that is the remainder after he makes 8 costumes.

What do you notice about the quotients in problems 1–4?

The quotients are less than 10.

Invite students to think–pair–share about how they know whether a quotient is less than 10 or greater than 10.

If the dividend is more than 10 times as much as the divisor, the quotient is greater than 10.

If a dividend is less than 10 times as much as the divisor, the quotient is less than 10.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. Problem 9 extends the thinking that began in Launch about how two division expressions can have the same quotient and remainder but not be equivalent expressions.

Land

Debrief 5 min

Objective: Divide three-digit numbers by two-digit numbers in problems that result in one-digit quotients.

Facilitate a class discussion about division by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

How is dividing a three-digit number by a two-digit number similar to dividing a two-digit number by a two-digit number?

The thinking is the same no matter how many digits are in the problem. We can use numbers in place of the original dividend and divisor to make estimating the quotient a mental math problem. We record our work in vertical form. Then we can change our quotient if we underestimated or overestimated. We can check our answer by using multiplication and addition.

Why is interpreting the remainder an important part of solving real-world problems? Give an example.

The remainder can tell us that even though we may have found the right quotient, sometimes we need to use the next highest whole number for our answer. For example, if I want to determine how many tables I need for a group of people, and I have a remainder when I divide, then I need to add an extra table for the people who do not fit into one of the groups in the quotient.

Display the following expressions.

Ask students to work with a partner and use reasoning to sort the expressions into two groups: quotients less than 10 and quotients greater than 10. Have students record their groupings on their personal whiteboards.

How might you know, even before you estimate, that the quotient is greater or less than 10?

If the dividend is at least 10 times as much as the divisor, the quotient is greater than 10. For example, 577 ÷ 55 is greater than 10 because 55 × 10 = 550, and 577 > 550.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Estimate the quotient. Then complete the vertical form and check your work. Draw a tape diagram if it helps you divide.

Divide. Then check your work. 5.

9. Consider the division work.

a. Show another division problem with the same quotient and remainder as 284 ÷

b. Explain how you found another division problem with the same quotient and remainder as 284 ÷ 39.

I thought about 7 × + 11, which is how I usually check my division work. I picked 52 and multiplied it by 7, then added 11 more to get 375

Use the Read–Draw–Write process to solve the problem.

10. Kayla’s book has 307 pages. She plans to read 45 pages each day. How many days will it take Kayla to finish reading the book?

307 ÷ 45

Quotient: 6

Remainder: 37

It will take Kayla 7 days to finish reading the book.

Divide three-digit numbers by two-digit numbers in problems that result in two-digit quotients.

Lesson at a Glance

To prepare for representing partial quotients with an area model, students identify unknown values in an area model and write multiplication and division equations to represent the model. Students use partial quotients to divide a three-digit number by a two-digit number and explain what the partial quotients represent. They record their work in an area model and in vertical form. Students solve division problems with and without remainders.

Key Questions

• Do we all have to use the same partial quotients when we divide? Why?

• Why do both the area model and vertical form work to record partial quotients?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits. (5.NBT.B.6)

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models. (5.NBT.B.6)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Compare an Area Model and Vertical Form

• Find Quotients Without Remainders

• Find Quotients With Remainders

• Problem Set

Land 10 min

Lesson Preparation

Fluency

Whiteboard Exchange: Write and Evaluate Expressions

Students write and evaluate an expression to prepare for two-step calculations beginning in topic D.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the statement: The total of 2 and 3.

Write an expression to represent the statement.

Display the sample expression.

Write the value of the expression.

Display the answer.

Repeat the process with the following sequence:

Teacher Note

Validate all correct responses that may not be displayed on the image. For example, a student may choose to write 0.7 − 0.4 for 4 tenths less than 7 tenths, although computation with decimals has not yet been formally taught.

Counting by Multiples of 4 and 40

Students say the first ten multiples of 4 and 40 to develop fluency with estimating quotients.

When I give the signal, say the first ten multiples of 4. Ready?

Display each multiple one at a time as students count.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40

When I give the signal, say the first ten multiples of 40. Ready?

Display each multiple one at a time as students count.

Choral Response: Divide in Standard Form

Students divide in standard form to develop fluency with dividing three-digit numbers by two-digit numbers.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 140 ÷ 20 = .

How many groups of 20 are in 140? 7

Display the answer.

On my signal, say the complete equation.

140 ÷ 20 = 7

Display the complete equation.

Repeat the process with the following sequence:

Launch

Students write multiplication and division equations that are represented by area models.

Display the area model that shows 11 × 52.

Ask students to share what they notice and wonder about the model.

I notice the model is an area model.

I notice the area of each rectangle is written inside the rectangle.

I wonder why 11 was designated as the unit.

I wonder why the model is not vertical.

11

Invite students to think–pair–share about which multiplication equation and which division equation the model represents. Encourage students who have ideas to share them.

52 × 11 = 572

572 ÷ 11 = 52

Area models can be drawn vertically or horizontally. For consistency, the area models in this topic are drawn horizontally, with the divisor on the left side of the rectangle to represent its width.

Similar to how we can use area models to multiply and determine products, we can use area models to divide and determine quotients.

Display the area model puzzle from problem 1 and direct students to the problem in their books.

Read the directions for problem 1 aloud. Ask students to complete the problem with a partner.

1. Determine the unknown values in the area model. Then write a multiplication equation and a division equation that the area model represents.

When most students are finished, invite a student or two to share their methods for how they determined the unknown values. Then ask the following questions.

Does the left side of the model have to be 14 or could it be another number? How do you know?

The left side of the model has to be 14 because the only number that can be multiplied by 60 to get 840 is 14.

Where in this model do you see the dividend?

The dividend is the total area of the rectangles, which is 840 + 70 + 28.

The dividend is the sum of the partial products, which is 840 + 70 + 28.

If 14 is the divisor, where in the model do you see the quotient? How do you know it is the quotient?

The quotient is the total length of the top of the model, which is 60 + 5 + 2. I know it is the quotient because I can see that 67 groups of 14 equals 938. Invite students to think–pair–share about the following question.

How can a completed area model represent both a multiplication equation and a division equation?

It depends on how you look at the area model. You can think of the length and width as factors of a multiplication equation or as the divisor and quotient in a division equation.

The length and width of an area model represent the factors in a multiplication expression. When you multiply to get the partial products, they are inside the area model. The factors and the sum of the partial products make the multiplication equation.

An area model can represent a division equation when the divisor is on the side. The dividend is the sum of the areas inside the model. The quotient is the sum of the numbers on the top of the model. Together, they make a division equation.

Transition to the next segment by framing the work.

Today, we will divide three-digit numbers by two-digit numbers.

Learn

Compare an Area Model and Vertical Form

Students compare sample work that records partial quotients in area models with work that records partial quotients in vertical form.

Display the work samples that show three methods to find 798 ÷ 38. Use the Five Framing Questions routine to invite students to analyze the three work samples.

Notice and Wonder

The picture shows three methods for recording the work to divide 798 by 38. Kayla and Tara used an area model to determine the partial quotients and Eddie used vertical form to determine the partial quotients. What do you notice about their work? From your observations, what do you wonder?

I notice the final equations are the same.

I notice the divisor 38 is on the left side of the area models.

Language Support

To support understanding of partial quotients, consider asking students to recall what they know about partial products. Then explain that when they break a dividend into parts, they create smaller division problems, each with its own quotient. The sum of the partial quotients is the whole quotient. Discuss the similarities and differences between partial products and partial quotients.

I notice both Kayla and Tara used area models, but Kayla has two rectangles that represent 10 groups of 38 and Tara has one rectangle that represents 20 groups of 38.

Kayla has three partial quotients and Tara only has two.

I notice the same partial quotients of 20 and 1 are shown in Tara’s and Eddie’s methods.

I notice subtraction is shown inside the vertical form and addition is shown below each area model.

I wonder whether the area model could be vertical instead of horizontal.

I wonder whether I could use vertical form and have the same partial quotients as Kayla.

Organize

Kayla and Tara have different area models and different partial quotients, but they have the same quotient. How is that possible?

Kayla and Tara used a different number of partial quotients, so their area models look different, but the sum of their partial quotients is the same because 10 + 10 + 1 = 21 and 20 + 1 = 21.

Kayla and Tara used different multiplication facts to begin finding the quotient. Kayla multiplied 38 by 10 twice, but Tara multiplied 38 by 20 once. That is why the area models look different, but the partial quotients have the same sum of 21.

While Tara’s and Eddie’s methods look different, they have the same partial quotients, quotient, and remainder. Describe the thinking process that both Tara and Eddie used.

Tara and Eddie each used 20 as the first partial quotient. 20 groups of 38 is 760, so out of 798, they had 38 left over. They each added a partial quotient of 1 to fit 1 more group of 38 in 798. They fit a total of 21 groups of 38 into 798 with nothing left over, so the quotient is 21 and the remainder is 0.

Advance the discussion to focus on partial quotients and encourage student thinking that makes connections between the area model and vertical form.

Reveal

Let’s focus on how the area model and vertical form support thinking about partial quotients. How many groups of 38 did both Tara and Eddie decide would fit into 798 at first? How do you know?

At first, they both decided that 20 groups of 38 can fit into 798. Both methods show they multiplied 20 by 38 to get 760. Tara’s method shows 20 as the first partial quotient along the top of the area model, and Eddie’s method shows 20 as the first partial quotient along the top of the vertical form.

How might Tara and Eddie have known to try 20 groups of 38?

Maybe they estimated the quotient as 20 because 800 ÷ 40 = 20.

Maybe they knew the dividend 798 is at least 10 times as much as the divisor because 38 × 10 = 380, and they realized they could fit at least 10 more groups of 38 into 798 because 38 × 20 = 760.

Why did Eddie subtract 760 from 798?

Eddie wanted to see how much was left to divide after he took 20 groups of 38 away from 798.

We do not see subtraction in Kayla’s work. How might Kayla have figured out what was left to divide?

She might have used a counting up strategy or mental math. I can find the difference between 798 and 760 without showing any work. She might have done the subtraction on a whiteboard or scratch paper.

How do you know that 798 ÷ 38 has a quotient of 21 and a remainder of 0? 21 groups of 38 is exactly 798.

How can you check that the quotient and remainder for 798 ÷ 38 are correct?

We can check that the quotient and remainder are correct by using multiplication: 21 × 38 = 798.

Distill

Why might you use an area model to divide?

When you decompose the dividend into parts, you can divide each part by the divisor, which is simpler than trying to divide the dividend all at once.

An area model helps me see what the partial quotients are and what they represent.

When I draw the area model, I can make as many rectangles as I need to represent the partial quotients until I reach the total.

Know

We can record division work in an area model or in vertical form. Is it helpful to know that with both representations the quotients are the same? How?

Yes. I know I can record my thinking and my work in whichever way makes the most sense to me.

Yes. If a visual model would help me divide, then I can use the area model.

Ask students to turn and talk about how they prefer to record their work for division and why.

Find Quotients Without Remainders

Students divide three-digit numbers by two-digit numbers without remainders by using partial quotients.

Display the following problem. Ask students to read the problem silently as you read the problem aloud.

A machine makes 28 shirts in 1 hour. How many hours does it take the machine to make 672 shirts?

Have students use the Read–Draw–Write process to model the problem, either independently or with a partner. Then display the following tape diagram. If a student has created a similar tape diagram, display the student’s work.

672 ? hours

Promoting the Standards for Mathematical Practice

As students estimate and use area models throughout the lesson to divide three-digit numbers by two-digit numbers, they are looking for and making use of structure (MP7).

Ask the following questions to promote MP7:

• What is another way you can estimate 672 ÷ 28 that can help you find the quotient?

• How can what you know about multiples of 10 help you decide whether the quotient is less than 10 or greater than 10?

• Can you break 672 ÷ 28 into simpler division problems to find partial quotients?

Invite students to think–pair–share about the conclusions they make from the model. The machine makes a total of 672 shirts. The machine can make 28 shirts in 1 hour.

We do not know how many hours it takes to make 672 shirts. We are trying to find out how many 28s are in 672.

What expression we can use to determine the number of hours it will take to make 672 shirts? How do you know?

We can use 672 ÷ 28. We know the machine makes a total of 672 shirts and it makes 28 shirts in 1 hour, so we can divide 672 by 28 to find the number of hours it takes to make 672 shirts.

Write 672 ÷ 28.

Is the quotient of 672 and 28 greater than 10 or less than 10? How do you know?

The quotient is greater than 10 because 28 × 10 = 280 and 672 is greater than 280. Invite students to think–pair–share to determine a reasonable estimate for the quotient.

660 ÷ 30 = 22

690 ÷ 30 = 23

600 ÷ 30 = 20

Let’s record our work for the division in an area model. Draw a rectangle and write 28 on the left side of the area model. Continue to add on to the area model throughout the discussion as shown.

Teacher Note

Because you build the area model as you think through the division, use input from students to build the diagram. The area model you build as a class may differ from the one used in the discussion. Students should know that any area model is correct if it is mathematically correct. Other possible area models include the following:

We can label the left side of our area model 28 because it is the divisor.

You shared before that there are at least 10 groups of 28 in 672 because 10 × 28 = 280. Because we know there are at least 10 groups of 28 in 672, let’s show 10 as a partial quotient.

What is 10 groups of 28 equal to?

280

Can we fit 10 more groups of 28 into 672? How do you know?

Yes. 280 + 280 = 560, which is less than 672.

Add a part showing 10 more groups to the tape diagram.

So far, we have 20 groups of 28 for a total of 560.

How much remains of the dividend 672?

112

Can we fit 10 more groups of 28 into our area model? How do you know?

No. 10 × 28 = 280 and we have only 112 remaining.

Have students think–pair–share about a response to the following question.

How many groups of 28 can we fit into the 112 that remains? Why?

We can fit 4 groups of 28. I know 10 × 28 = 280 and 5 × 28 = 140 because 5 is half of 10.

Half of 280 is 140 and 1 less group of 28 is 112.

Since 5 groups of 28 is too much, let’s try 4 groups of 28. What is 4 groups of 28 equal to?

112

Add a part showing 4 groups to the tape diagram.

What is 280 + 280 + 112?

672

UDL: Representation

To help students make connections between an area model and vertical form, consider creating a color-coded anchor chart with the same division problem shown in both an area model and in vertical form. Highlight partial quotients from each representation in the same color.

Do we need to add another partial quotient to our rectangle? Why?

No. The total area of our rectangle is 672 and our dividend is 672.

Do we have a remainder? No.

How many groups of 28 did we fit into 672? 24

What is the quotient 672 ÷ 28? 24

How can we check our answer?

We can multiply 24 by 28 to make sure it equals 672.

Allow students time to check with multiplication. Continue to display the area model as you record the work for 24 × 28.

How long does it take the machine to make 672 shirts?

It takes the machine 24 hours to make 672 shirts.

Display the following equation. (280

Differentiation: Challenge

For students who need an additional challenge, ask them to use an area model to determine the value of 986 ÷ 34 by using as few partial quotients as possible. Ask students to show their thinking.

Invite students to think–pair–share about how this equation relates to the area model.

The expression in each set of parentheses represents the length of one rectangle from the area model.

The area of each rectangle divided by its width gives its length. The sum of the three lengths is 24.

The expression in each set of parentheses represents one partial quotient.

The expression in each set of parentheses does represent one partial quotient. Why is the divisor 28 in each set of parentheses?

Because the divisor is 28 and we made three partial quotients, it makes sense for 28 to be in each set of parentheses. 280 + 280 + 112 = 672 and the area model represents 672 ÷ 28, which is equal to (280 ÷ 28) + (280 ÷ 28) + (112 ÷ 28).

Invite students to turn and talk about how they might have constructed their area model differently.

Find Quotients With Remainders

Students divide three-digit numbers by two-digit numbers with remainders by using partial quotients.

Write 926 ÷ 23.

Invite students to think–pair–share about an estimate for the quotient.

900 ÷ 30 = 30

900 ÷ 20 = 45

1,000 ÷ 20 = 50

Why do you think there is such a wide range in our estimates?

It is difficult for me to think of division facts I can use with numbers that are close to 926 and 23.

Are any of our estimates wrong? Explain.

No. The estimates help us get started with the division and determine whether our answer is reasonable.

Invite students to turn and talk about which estimate they would use to get started with the division and why.

There is not one correct strategy to determine which estimate to use. One strategy is to use a low estimate. If we underestimate, then we can add more partial quotients.

What happens if we use an estimate that is too high for our first partial quotient?

We must try a smaller quotient.

Teacher Note

The goal for students with division of multi-digit numbers at this point in their learning is accuracy as opposed to fluency. It is acceptable for students to underestimate or overestimate quotients and adjust as needed.

As the dividend and divisor increase, estimating can become more challenging because students may not always be able to rely on multiplication or division facts. When possible, discuss estimation strategies with students. For example, in the problem 926 ÷ 23, you might not use 45 as an estimate because that number is challenging to multiply by mentally, and you might not use 50 because the product of 50 and 23 is greater than 926.

Let’s use 30 as our first partial quotient. We can record our work in an area model and in vertical form.

Draw a rectangle with the divisor labeled and set up the vertical form for 926 ÷ 23. As students answer the following questions, record the work in both the area model and in vertical form until the final work looks similar to what is shown.

Differentiation: Support

The area model used in the lesson to represent 926 ÷ 23 shows a first partial quotient of 30. For students who need additional support, consider encouraging them to start with partial quotients of 10.

Why did I label the left side of my area model 23? 23 is the divisor.

How many groups of 23 fit in 926? Why?

Our estimate was 30 and 23 × 30 = 690, so we know at least 30 groups of 23 fit in 926. 30 groups of 23 is equal to 690. What should we do next? Why?

We should subtract 690 from 926 to find out whether we can fit more groups of 23 into what is left.

How much is left after we subtract 690 from 926? 236

Are we finished dividing or can we make another partial quotient? How do you know?

We can make another partial quotient because there are more groups of 23 that fit in 236.

Can we fit 10 more groups of 23 in 236? How do you know?

Yes. 23 × 10 = 230, which is less than 236.

So far, we have 30 groups of 23 plus 10 more groups of 23, for a total of 40 groups of 23 in 926. How much is left over?

6

Can we fit another group of 23 in 6? No.

What is the quotient and what is the remainder?

The quotient is 40 and the remainder is 6.

How can we check our answer?

We can check that 40 × 23 + 6 is equal to 926.

Write 40 × 23 + 6 = 926. Ask students to use their personal whiteboards to determine whether this equation is true.

After students confirm that the number sentence is true, invite them to turn and talk about why they should get the same quotient regardless of which method they use to record their work.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Teacher Note

The remainder is not included as part of the area model because the area that would be represented by the remainder is a fraction. Students learn to interpret remainders as fractions in module 2.

Differentiation: Challenge

For students who need an additional challenge, pose the following task: Find another division expression that has a quotient of 40 and a remainder of 6.

Land

Debrief 5 min

Objective: Divide three-digit numbers by two-digit numbers in problems that result in two-digit quotients.

Gather the class with their Problem Sets. Use the following prompts to facilitate a class discussion that emphasizes how to divide strategically with partial quotients. Encourage students to restate or add on to their classmates’ responses.

Have students compare their work for problems 4 and 5 in the Problem Set with multiple partners.

Do we all have to use the same partial quotients when we divide? Why?

No, we do not all have to use the same partial quotients. We might have more or fewer tens and ones in the partial quotients compared to someone else, depending on how we think about the dividend, the divisor, and the number of groups we could create.

Why can we use both the area model and vertical form to record partial quotients?

The thinking we use to divide is the same between methods.

The quotient and remainder are the same regardless of how you choose to record the division.

In either case you can record the same partial quotients. Both methods show the dividend, the divisor, the partial quotients, the quotient, and the remainder.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Divide by using an area model. Then check your work.

7.

Divide four-digit numbers by two-digit numbers.

Lesson at a Glance

Students divide four-digit numbers by two-digit numbers. They identify errors in sample work and discuss strategies for making estimates when they work with quotients of large numbers. Working both as a class and with a partner, students solve real-world division problems with and without remainders.

Key Questions

• Does the number of digits in the dividend and in the divisor change what you think about while you divide? Why?

• Without context, which interpretation of the divisor, number of groups or size of the group, did you use while dividing? Why?

Achievement Descriptors

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits. (5.NBT.B.6)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Analyze Estimates

• Division Word Problem Without a Remainder

• Division Word Problem With a Remainder

• Problem Set

Land 10 min

•

Lesson Preparation

•

Fluency

Whiteboard Exchange: Write and Evaluate Expressions

Students write and evaluate an expression to prepare for two-step calculations beginning in topic D.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the statement: 2 times 4.

Write an expression to represent the statement.

Display the sample expression.

Write the value of the expression.

Display the answer.

Repeat the process with the following sequence:

Counting by Multiples of 5 and 50

Students say the first ten multiples of 5 and 50 to develop fluency with estimating quotients.

When I give the signal, say the first ten multiples of 5. Ready?

Display each multiple one at a time as students count.

5, 10, 15, 20, 25, 30, 35, 40, 45, 50

When I give the signal, say the first ten multiples of 50. Ready?

Display each multiple one at a time as students count.

Choral Response: Divide in Standard Form

Students divide in standard form to prepare for dividing multi-digit numbers.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 120 ÷ 20 = .

How many groups of 20 are in 120?

Display the answer.

On my signal, say the complete equation.

120 ÷ 20 = 6

Display the complete equation.

Repeat the process with the following sequence:

Launch

Students identify errors in work that shows division of a four-digit number by a two-digit number.

Write 3,618 ÷ 27.

Is the quotient 3,618 ÷ 27 greater than 10 or less than 10? Why?

I think it is greater than 10 because 10 × 27 = 270 and 270 is less than 3,618.

Is the quotient 3,618 ÷ 27 greater than 100 or less than 100? Why?

I think it is greater than 100 because 100 × 27 = 2,700 and 2,700 is less than 3,618.

Is the quotient 3,618 ÷ 27 greater than 1,000 or less than 1,000? Why?

I think it is less than 1,000 because 1,000 × 27 = 27,000 and 27,000 is greater than 3,618.

Display the incorrect work for 3,618 ÷ 27.

This picture shows Scott’s work. What quotient did Scott find for 3,618 ÷ 27? How do you know?

Scott found the quotient 1,304 because that is the sum of his partial quotients 4, 300, and 1,000.

Is Scott correct? How do you know?

No. The quotient should be less than 1,000, but this work shows a quotient greater than 1,000.

Invite students to think–pair–share about mistakes they see in the work.

The first partial quotient should be 100, not 1,000, because 3,618 ÷ 27 ≈ 3,000 ÷ 30, or 100, and 27 × 100 = 2,700.

The second partial quotient should be 30, not 300, because 918 ÷ 27 ≈ 900 ÷ 30, or 30, and 30 × 27 = 810.

How could estimation have helped Scott to identify his error?

If he estimated, he would have known the quotient 1,304 is not reasonable.

If he estimated first, it would have shown that the first partial quotient needs to be about 100, not 1,000.

How is this division problem different from others that you have worked in previous lessons?

The dividend has four digits.

Invite students to turn and talk about why it is important to estimate when they divide. Transition to the next segment by framing the work.

Today, we will divide four-digit numbers by two-digit numbers.

Learn Analyze Estimates

Students compare estimates for the quotient of a four-digit number and a two-digit number.

Write 2,792 ÷ 76.

Display the three estimates for the quotient.

Estimate A

2,400 ÷ 80 = 30

Estimate B

3,200 ÷ 80 = 40

Here are three different estimates of 2,792 ÷ 76.

Estimate C

2,800 ÷ 70 = 40

Differentiation: Challenge

For students who need an additional challenge, consider asking the following questions:

• Can a four-digit dividend divided by a two-digit divisor create a four-digit quotient? Justify your answer.

• Can a four-digit dividend divided by a two-digit divisor create a three-digit quotient? Justify your answer.

Invite students to think–pair–share about what students might have thought about when they made these estimates.

For estimate A, I think the student rounded 76 to the nearest ten, which is 80, and then chose 2,400 as the dividend because 2,400 is a multiple of 80.

Teacher Note

Consider including these strategies when discussing how to estimate quotients of large numbers.

• Round the divisor to the nearest ten or hundred, and then approximate the dividend to the nearest multiple of the rounded divisor.

3,402 ÷ 46 ≈ 3,500 ÷ 50

• Think about division facts and use a fact to choose both the divisor and the dividend at the same time.

2,792 ÷ 76 ≈ 2,800 ÷ 70

• Intentionally underestimate the first partial quotient by choosing a dividend less than the actual dividend because additional partial quotients can be added if needed.

2,002 ÷ 32 ≈ 1,800 ÷ 30

For estimate B, I think the student rounded 76 to the nearest ten, which is 80, and then chose 3,200 as the dividend because 3,200 is a multiple of 80.

For estimate C, I think the student rounded the dividend to 2,800. Then they chose 70 as the divisor because they knew 28 ÷ 7 = 4.

Are these estimates helpful? Explain.

Yes. The estimates are close to one another. We can use any of the estimates to begin dividing or to check whether our quotient is reasonable.

Yes. Together, they tell us the quotient is greater than 30 but less than 40.

Which estimate would you use to find the quotient of this division expression? Why?

I would use estimate A because I prefer to start dividing with an estimate that I think is less than the quotient. Then I can add more partial quotients if needed. I would use estimate C because I think it is closest to the actual quotient.

Tell students that the actual quotient is 36 and there is a remainder of 56.

Each of the estimates is helpful in some way. We can use other helpful estimates as well. The numbers we choose for making estimates are personal choices and vary from person to person.

The quotient 36 falls between our two estimates 30 and 40. We knew the quotient would be at least 30 but would not be greater than 40 because 40 × 76 = 3,040.

Invite students to turn and talk about the strategies they can use to make estimates of quotients.

Division Word Problem Without a Remainder

Students solve a real-world division problem without a remainder.

Display problem 1.

Ask students to silently read the problem. Then read the problem aloud. Tell students they are going to complete the problem as a class by using the Read–Draw–Write process. Ask students to follow along and record as you progress through the problem.

Teacher Note

Context videos for problems 1 and 2 are available. The videos may be used to remove language or cultural barriers and to provide student engagement. Before beginning each problem, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before they are asked to interpret it mathematically.

Use the Read–Draw–Write process to solve the problem.

1. A tree farm has 15 rows of trees. Each row has the same number of trees. If there is a total of 1,635 trees, how many trees are in each row?

1,635

. . . ?

15 rows

1,635 ÷ 15 = 109

Each row has 109 trees.

Read the first and second sentences.

Can we draw something? What can we draw?

We can draw a tape diagram and partition it into 15 parts to show the 15 rows.

We can draw a tape diagram that shows the first row and the last row because 15 rows are a lot to draw.

Draw the tape diagram.

Read the third sentence.

Can we label something? What can we label?

We know there are 1,635 trees in total, so we could label the whole tape 1,635.

Label the top of the tape diagram.

How can we find how many trees are in each row? How do you know?

? ?

We can divide 1,635 by 15 because we know 1,635 trees are in 15 rows or 15 groups. To figure out the number of trees in each row, we can divide the total number of trees by the number of rows.

Do you think the number of trees in each row is greater than 100 or less than 100?

The number of trees is greater than 100 because 15 × 100 = 1,500 and 1,635 is greater than 1,500.

Let’s record our work to divide 1,635 by 15 in vertical form.

Write the vertical form for 1,635 ÷ 15.

What does the divisor 15 represent?

15 represents the number of groups, or rows of trees.

If the divisor represents the number of groups, or rows of trees, what does the quotient represent?

The quotient represents the size of each group, or how many trees are in each row.

As we continue to divide, let’s use the context of how many rows and how many trees in each row to help us think about and answer the questions we ask ourselves.

What estimate can we use for the quotient 1,635 ÷ 15? How do you know?

We can use 100 because 1,500 ÷ 15 = 100.

Write the estimate next to the vertical model.

We think there are about 100 trees in each of the 15 rows, or groups. What can we do with that information?

We can determine what 15 groups of 100 is equal to and subtract that amount from the dividend.

Why do we subtract that total from 1,635?

The difference tells us how many trees are left to distribute into each of the 15 groups. Write the first partial quotient on the vertical model, and then write the subtraction problem and find the difference.

Why is 100 in the quotient lined up above 635 in the dividend?

When we record division in vertical form, we line up the quotient and the dividend by place value.

Differentiation: Support

For students who need additional support with problems 1 and 2, encourage them to use an area model to do the division.

Do we need another partial quotient? How do you know?

Yes. There is 135 left over and that number is greater than the divisor.

In this problem, 135 represents the number of trees that are left over after we evenly distribute 100 trees into each of the 15 rows. Now we should ask ourselves how we can evenly distribute the remaining trees. If we have 135 trees left, how many trees can we evenly distribute into 15 rows, or groups? What estimate can we use to help us?

140 ÷ 20 = 7 is a helpful estimate.

Write the estimate next to the vertical model.

We think we can evenly distribute 7 more trees into each of the 15 rows, or groups. What can we do next? Why?

Determine what 15 groups of 7 is equal to and subtract that amount from 135 to determine how much of the dividend is left to divide.

Write the partial quotient 7 in the ones place above the partial quotient 100.

What would it mean if we put the 7 in the tens place?

If we put 7 in the tens place we would have seventy 15s and not seven 15s, and seventy 15s is too many.

Write 105 and the difference on the vertical model.

We have 30 trees left. Can we evenly distribute more trees into each row, or group? If so, how many?

Yes. We can distribute 2 more trees into each of the 15 groups.

Because we can evenly distribute 2 more trees into each of the 15 groups with no trees left over, there is no need to estimate our last partial quotient.

Teacher Note

Encourage students to be flexible in how they reason about estimates and partial quotients. For example, a student could estimate 135 ÷ 15 is about 7 because 140 ÷ 20 = 7. A student could also think about multiples of 15. Because the student knows that 10 × 15 = 150, and 135 is one 15 less than 150, they can reason that 15 × 9 = 135. Reasoning that 15 × 9 = 135 allows the student to be more flexible in their thinking and to eliminate an additional partial quotient.

Write the third partial quotient in the ones place.

Are we finished dividing? How do you know?

Yes. The remainder is 0, so we are finished dividing. Yes. We distributed 1,635 evenly into each of the groups.

What is the quotient 1,635 ÷ 15?

109

How can we check our answer?

We can multiply 109 by 15.

Have students multiply to check the accuracy of the quotient.

When we check our quotient by using multiplication, we can see that 109 is the correct answer to the division problem. That means we found the number that goes where the question mark is in our model. Now we can answer the question in the problem, How many trees are in each row? What should we record as the statement to answer this question?

Each row has 109 trees.

Invite students to turn and talk about how making estimates helped them determine how many trees are in each row.

Division Word Problem with a Remainder

Students solve a real-world division problem involving a remainder.

Display problem 2. Read the problem aloud to students.

Lacy plans to ride her bike 2,900 miles, which is about the distance from San Francisco to New York. If she rides 68 miles each week, how many weeks will it take Lacy to ride 2,900 miles?

UDL: Action & Expression

Consider asking students what questions they can ask themselves as they work through division problems. Compile the list on an anchor chart for students to refer to. Sample questions include the following: •

Language Support

Consider activating prior knowledge by discussing and showing on a map the locations of San Francisco and New York.

Invite students to complete the problem with a partner. Circulate as students work to identify students who use different partial quotients. Support students as you circulate by asking the following questions:

• Can you draw something? What can you draw? What can you label?

• What expression can you use to find the unknown?

• What does the dividend represent? The divisor? The quotient?

• How can you record your work?

• What can you do first?

• What estimate can you make?

• Is your answer reasonable? How do you know?

• What does the remainder represent in this situation?

2. Lacy plans to ride her bike 2,900 miles, which is about the distance from San Francisco to New York. If she rides 68 miles each week, how many weeks will it take Lacy to ride 2,900 miles?

Promoting the Standards for Mathematical Practice

When students use tape diagrams to interpret division word problems and analyze remainders in context to find the solution to the problem, they are reasoning quantitatively and abstractly (MP2).

Ask the following questions to promote MP2:

• What does your tape diagram tell you about the problem?

• What does a remainder of 44 mean in this context?

It will take Lacy about 43 weeks to ride 2,900 miles.

• Does your answer make sense in this context?

When students are finished, bring the class together. Invite students to share and compare their estimates. Ask whether any students overestimated and, if so, how they adjusted their work.

Invite students who used different partial quotients to share their work. Encourage students to explain their thinking by asking, Why did you use those partial quotients? How did the context from the story change the questions you asked yourself?

Point out that the quotient and the remainder are the same, regardless of how many partial quotients are used. For example, the sample work shows another way a student may have used partial quotients.

Why will it take Lacy 43 weeks instead of 42 weeks to ride 2,900 miles if 42 is the quotient?

The remainder is 44. If Lacy rides for 42 weeks, she still has 44 more miles left to go. She will need to ride about 1 more week for a total of about 43 weeks to ride 2,900 miles.

Display the following tape diagrams.

Invite students to turn and talk about what they notice is the same and what is different between the two tape diagrams.

Differentiation: Challenge

For students who finish problem 2 early, consider asking the following questions:

• How many miles does Lacy ride in 1 day?

• How many miles does Lacy ride in 1 hour?

• How many days does it take Lacy to ride 2,900 miles?

What is different about the two tape diagrams?

In the first tape diagram, we know the number of groups and the total. We are trying to find the size of each group.

In the second tape diagram, we know the size of each group and the total. We are trying to find the number of groups.

In the first tape, the divisor is the number of groups and the quotient is the size of each group. In the second tape, the divisor is the size of each group and the quotient is the number of groups. Does this difference affect the questions you ask yourself while you divide?

Yes. In the problem about the trees, I thought about distributing 1,635 trees evenly into groups. So I asked myself questions about how many trees I can fit into the 15 groups. For example, Can I distribute 10 trees equally into the 15 groups? 100 trees? If I have 30 trees left, how many trees can I distribute evenly into each of the 15 groups?

In the problem about the bike, I thought about how many 68s are in 2,900. So I asked myself questions about how many groups of 68 can fit into 2,900. For example, Can I fit ten 68s into 2,900? Twenty 68s? Forty 68s? If there are 136 miles left, can I fit two 68s?

When you do not have a word problem, does it matter whether you think about the divisor as the number of groups or the size of each group? Why?

No. It does not matter because we can interpret the divisor in different ways.

As we continue to practice division or to solve problems involving division, remember you can think about the divisor as the number of groups or the size of each group. Choose whichever interpretation helps you find partial quotients.

Invite students to turn and talk to discuss which interpretation they find more helpful when finding partial quotients.

Teacher Note

Students should use context to drive the questions they ask themselves as they divide. This can help students interpret the meaning of the dividend, divisor, partial quotients, and quotient.

When the divisor is the size of the group or the size of the unit, they might ask questions such as the following:

• How many fit into ?

• Can I fit 10 s into ? 100? 1,000?

When the divisor is the number of groups, students might ask questions such as the following:

• If I need to evenly distribute into groups, how many can I distribute into each group?

• Can I evenly distribute 10 into each group? 100? 1,000?

Eventually, students use more formal multiplication and division language. Using the interpretation of the divisor to drive the metacognitive work of solving problems supports students’ conceptual understanding of division.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land Debrief 5

min

Objective: Divide four-digit numbers by two-digit numbers.

Gather the class with their Problem Sets. Facilitate a class discussion about dividing four-digit numbers by two-digit numbers by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Does the number of digits in the dividend and in the divisor change what you think about while you divide? Why?

No. For any division problem, we are trying to find the quotient. It does not matter how many digits are in the problem because I still think about finding partial quotients. Not really. The only change in my thinking is that I might have more partial quotients because the numbers are larger.

Yes. I started by thinking about which powers of 10 the quotient is between. That helped me get an idea of the size of the quotient.

Direct students to choose one of the computation problems from the Problem Set to discuss with a partner. Invite students to think–pair–share about a response to the following question.

Without context, which interpretation of the divisor, number of groups or size of the group, did you use while dividing? Why?

In problem 4, I thought about the divisor as the size of each group, so I thought about how many thirty-fours are in 7,242. It helped me to think about the divisor as the unit because I could ask myself, Can I fit 10 thirty-fours? 100 thirty-fours? Those questions helped me think about multiplication, and I like to use multiplication to help me divide.

In problem 6, I thought about the divisor as the number of groups, so I thought about distributing 5,123 equally into 47 groups. I made up the context of someone sharing $5,123 equally between 47 people to help me visualize what I needed to do.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Estimate the partial quotients as you divide. The first estimate is started for you. Make as many estimates as you need to. Then check your work.

225

6 Divide. Then check your work.

Use

7. A warehouse has 1,250 video games to distribute evenly to 12 stores. If the warehouse distributes as many as possible, how many games does each store get? How many games are left over?

1,250 ÷ 12

Q: 104

R: 2

Each store gets 104 games and 2 games are left over.

Topic D Multi-Step Problems with Whole Numbers

Having further developed multiplication and division strategies in topic C, students now look to answer the question: How can operating with whole numbers help me solve challenging real-world problems?

Students begin topic D by representing a multi-part statement with a tape diagram and an expression. For example, given this statement: 3 times as much as the sum of 11 and 29, students draw a matching tape diagram and write an expression by first thinking of one unit as 11 + 29.

11 + 29 11 + 29 11 + 29 3 × (11 + 29)

Depending on the representation they are given, students generate statements, expressions, or tape diagrams. They consider how the presence or absence of parentheses changes the value of an expression. For example, students evaluate expressions such as 3 × 11 + 29 and 3 × (11 + 29) and then compare the results.

As students create and solve real-world problems, they continue to build their understanding of the importance of using parentheses to clarify the meaning of and to ensure the correct interpretation of expressions. Students write word problems that demonstrate their understanding of how operations are reflected in real-world situations. One consideration that students make is the placement of parentheses in an expression when they write a matching word problem. For example, given the expression (24 − 6) ÷ 3, students first craft a situation that represents the difference of 24 and 6, then expand the story to represent dividing the difference by 3.

Students advance to solving multi-step word problems involving multiplication and division. They determine the value of multiple unknowns by first drawing a model to make sense of the problem, and then writing an expression to find the product or quotient. Students recognize that the way they draw their tape diagram helps them determine which operation to use to solve the problem. Once students know the appropriate operation, they are empowered to choose a strategy, model, or method to solve the problem.

Topic D closes by pushing students to solve multi-step word problems by using all four operations. Students notice there are multiple ways to draw a model that represents the same problem.

In module 2, students apply their understanding of division to interpret fractions as division.

Progression of Lessons

Lesson 17

Write, interpret, and compare numerical expressions.

3 times the sum of 15 and 25

15 + 25 15 + 25 15 + 25

3 × (15 + 25)

I can draw a tape diagram to represent a statement. The tape diagram can help me write an expression that matches the statement. I understand that parentheses can change the value of an expression. I can compare expressions without evaluating them.

Lesson 18

Create and solve real-world problems for given numerical expressions.

Lesson 19

Solve multi-step word problems involving multiplication and division.

Sample:

Blake makes 96 muffins for the bake sale. He sells 33 of them and puts the remaining muffins in 3 containers to take home. If he puts the same number of muffins in each container, how many muffins are in each?

I can create real-world situations that match an expression or a tape diagram. I understand that groupings within an expression represent multiple steps within a problem.

I can use the Read–Draw–Write process to solve multi-step word problems involving multiplication and division. I can draw a model to represent a word problem, write an expression to match, and select a strategy, model, or method to solve the problem.

Lesson 20

Solve multi-step word problems involving the four operations. $ $

$ $ $ $ $ $ $ $ ?

$

I can use the Read–Draw–Write process to solve multi-step word problems involving all four operations. I can use a tape diagram to uncover the multiple steps and operations necessary to solve the problem.

Write, interpret, and compare numerical expressions.

Lesson at a Glance

Students analyze work that evaluates numerical expressions and use the Take a Stand routine to consider the correct way to evaluate an expression. They realize that parentheses change the value of the expression, depending on where the parentheses are placed in an expression. Tape diagrams support students with writing, evaluating, and matching statements and numerical expressions. Pairs of students work together to compare numerical expressions without calculating. This lesson introduces the verb evaluate.

Key Questions

• Does modeling with tape diagrams help you write statements for equations and expressions? How?

• Does modeling with tape diagrams help you make sense of where parentheses belong in equations and expressions? How?

Achievement Descriptors

5.Mod1.AD1 Write whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD3 Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions. (5.OA.A.2)

5.Mod1.AD4 Compare the effect of each number and operation on the value of a whole-number numerical expression. (5.OA.A.2)

1. Write an expression to represent the statement. Draw a tape diagram if it helps you.

times as much as the sum of 3 and

× (3 + 12)

2. Place parentheses to make the equation true.

Use >

, or < to compare the expressions.

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Represent Statements with Tape Diagrams and Equations

• Write Statements to Represent Tape Diagrams

• Match Tape Diagrams, Statements, and Expressions

• Compare Expressions

• Problem Set

Land 10 min

Teacher

• Numerical Expressions (in the teacher edition)

• Paper (3 sheets) Students

• None

Lesson Preparation

• Prepare three signs on paper. Label one sign 77, another sign 128, and the third sign Undecided. Hang the signs in different locations in the classroom.

• Print or copy Numerical Expressions and cut out each rectangle.

Fluency

Whiteboard Exchange: Interpret Tape Diagrams

Students write and evaluate an expression to prepare for relating tape diagrams, statements, and expressions.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the tape diagram.

What does the tape diagram show? Tell your partner.

Provide time for students to think and share with their partners.

The total is unknown. There are two parts. One part has a value of 30 and the other part has a value of 18.

30

a 18

30 + 18 = a 48 = a

Write an equation to represent the tape diagram. Use the letter a to represent the unknown.

Display the sample equation.

Write the value of the unknown. Display the answer.

Teacher Note

Validate all correct responses that may not be displayed on the image. For example, students may write addition or subtraction equations for tape diagrams with an unknown part.

Repeat the process with the following sequence:

Whiteboard Exchange: Write and Evaluate Expressions

Students express an addition, subtraction, multiplication, or division statement as an expression and evaluate the expression to prepare for two-step calculations.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the statement: 11 more than 73.

Write an expression to represent the statement.

Display the sample expression.

Write the value of the expression.

Display the answer.

11 more than 73

73 + 11 84

Launch

Materials—T: Signs

Students consider the correct way to evaluate an expression.

Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: 77, 128, and Undecided.

Display the following student work and invite students to stand beside the sign that best describes their thinking about whose work is correct.

When all students are standing near a sign, allow 2 minutes for groups to discuss the reasons they chose that sign.

Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group.

Have students return to their seats. As a class, reflect on how one expression can lead to two different answers.

We can see from the answers in Toby’s and Yuna’s work that they found the value of the expression differently. When we evaluate an expression, or find its value, it is important that we all get the same answer. Why do you think that might be important?

It would cause a lot of confusion if we got different answers for the same problem. It does not make sense that one problem could have two completely different answers, so it is important we all get the same number. You would never know whether your answer is right if it could be lots of different numbers.

Transition to the next segment by framing the work. Today, we will write, interpret, and evaluate expressions.

Language Support

This segment introduces the verb evaluate. Consider posting the term and reiterating the meaning: to find the value of. Ask students where they see part of the word value in evaluate . Discuss the meaning of value , highlight the letters v, a, l, and u in the term, and write the definition. Refer to the posting throughout the lesson as the term is used.

Learn

Represent Statements with Tape Diagrams and Equations

Students draw and use tape diagrams to write statements as numerical expressions. Display the problem and related tape diagrams and invite students to study them.

Leo plants 17 daisies and 15 sunflowers. Riley plants 4 times as many flowers as Leo does. How many flowers does Riley plant?

The work we saw earlier shows what Toby and Yuna did to solve this problem. Now we have more information about where their work came from and, seeing their tape diagrams, how they got different answers.

Toby's Work : ?

15 15 15 15 17

Yuna's Work :

17 15

Invite students to think–pair–share about which tape diagram correctly represents the problem.

Yuna’s tape diagram correctly represents the problem because Leo planted a total of 17 daisies and 15 sunflowers. Riley planted 4 times as many flowers as Leo, or 4 times as much as the sum of 17 + 15.

Yuna’s tape diagram is correct because it shows 4 groups of the total number of flowers Leo planted.

Toby’s tape diagram is incorrect because it shows Riley planted 4 times as much as 15 flowers and then planted another 17 flowers. which does not match the problem.

Which student correctly solved the word problem? How do you know?

Yuna correctly solved the problem. Yuna’s tape diagram shows the total of daisies and sunflowers with the units of 17 and 15 repeated 4 times.

Teacher Note

The digital interactive Tape Diagram Expressions helps visually represent and manipulate numerical expressions.

Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

It was difficult for us to determine earlier whether Toby or Yuna was correct because we did not have the context to help us understand why 17 and 15 needed to be added before multiplying by 4. What can we include in the expression 17 + 15 × 4 that makes it clear we need to add first?

We can put parentheses around 17 + 15 to show we need to add the number of daisies and sunflowers first before we multiply by 4.

Write (17 + 15) × 4 = 128.

If the expression we looked at earlier included parentheses, it would have been clear that Yuna’s work was correct. To avoid causing confusion like we experienced, it is important to include parentheses so that our expressions are evaluated correctly, even when we do not know the context or do not see a tape diagram.

Direct students to problem 1 in their books. Have students work individually or with a partner to write the expression. Circulate to ensure students place the parentheses correctly and to identify students who may write the factor 3 to the right and to the left of the sum.

Write an expression to represent the statement. Use the tape diagram to help you.

1. 3 times the sum of 15 and 25

15 + 25 15 + 25 15 + 25 ?

Teacher Note

In previous grades, students use only a single number or symbol to represent a unit in a tape diagram. Problem 1 shows a unit as the expression 15 + 25. As needed, show students that in Yuna’s work, she used a single brace for a unit in her tape diagram to represent 17 daisies and 15 sunflowers, which is the same as showing a unit with the expression 17 + 15.

Differentiation: Challenge

3 × (15 + 25)

When students finish writing the expression, invite identified students to share their expressions with the class.

What did you include in your expression to clearly show we need to find the sum before we multiply?

We included parentheses around 15 + 25.

Ask students who finish early to write and evaluate an expression to match the tape shown. Then have them find the difference between the value of the expression in problem 1 and the value of the expression for this tape diagram.

15 15 15 25

Does it matter that some of us wrote 3 × (15 + 25) and others wrote (15 + 25) × 3? Why?

It does not matter because the commutative property of multiplication says we can multiply a number times another number in any order and we will get the same product.

Evaluate your expression.

Provide students a moment to evaluate the expression. Ensure they all got 120 and then have students turn their expression into an equation by writing = 120 in problem 1.

Direct students to problem 2 and have them read the problem.

Draw a tape diagram and write an expression to represent the statement.

2. The difference between 72 and 48, then divide by 2 48

(72 − 48) ÷ 2

What is different about this statement compared to the last one? It says difference this time instead of sum, so we need to subtract. It says divided by, so we need a division symbol in our expression.

Display the tape diagrams.

Diagram A

Teacher Note

There is often more than one correct way to represent a situation with a tape diagram. Here is another way to represent (72 − 48) ÷ 2 by using a tape diagram.

Diagram B

When students draw their own tape diagrams to make sense of problems, expect some variety.

UDL: Representation

Consider creating an anchor chart as students describe how they know that the placement of parentheses affects the value of the expression. Post the chart for the remainder of the topic as an example of why parentheses are used and of the importance of showing groups in an equation or expression.

Invite students to think–pair–share to determine which tape diagram matches the statement.

Diagram A matches the statement because it shows the difference between 72 and 48 being divided by 2.

Diagram B does not match the statement because it shows 48 being divided by 2. Because diagram A matches the statement, it means the grouping in this statement is the difference between 72 and 48.

Have students write an expression to match the statement and find the quotient. Circulate and monitor student work.

Invite students to share their expressions and justify the placement of the parentheses.

I wrote (72 − 48) ÷ 2 to represent the statement because I have to find the difference between 72 and 48 to match the statement. The difference is divided by 2.

I wrote (72 − 48) ÷ 2 because we need to divide the difference into 2 parts, not the 72 nor the 48.

Does the placement of the parentheses make a difference in how you evaluate the expression?

Yes.

How do you know?

(72 − 48) ÷ 2 = 12. If we moved the parentheses and wrote 72 − (48 ÷ 2), we would get 72 − 24, which is 48, not 12.

Write Statements to Represent Tape Diagrams

Students write statements and equations to demonstrate reasoning about the groups they see in tape diagrams.

Direct students to problem 3. Consider having students work in pairs to write the statement and equation to match the tape diagram. Circulate and monitor student work. Expect a variety of equations, such as the following:

• 8 + 8 + 8 + 6 + 6 = 36

• 3 × 8 + 2 × 6 = 36

• (3 × 8) + (2 × 6) = 36

Write a statement and equation to represent the tape diagram.

3. 86 6 8 8

The sum of three groups of 8 and two groups of 6 (3 × 8) + (2 × 6) = 36

Invite students to share their equations. Then display the equation 3 × (8 + 2) × 6 = 180.

Does this equation match the tape diagram? Why?

No, it does not match because it shows a group of 8 + 2, but there is not a group of 8 + 2 on the tape diagram.

No, it does not represent the tape diagram because it shows a product of 180, but our tape diagram’s total is 36.

Imagine that a student in another class did not have the tape diagram. Where should we place parentheses to ensure the student would correctly evaluate the expression 3 × 8 + 2 × 6?

We should place parentheses around 3 × 8 and around 2 × 6.

What statement can we write to represent this tape diagram?

The sum of 3 eights and 2 sixes

The sum of three groups of 8 and two groups of 6

Direct students to record the statement.

Match Tape Diagrams, Statements, and Expressions

Materials—T: Numerical Expressions

Students determine whether expressions, tape diagrams, and statements match by analyzing the placement of parentheses.

Distribute one representation from Numerical Expressions to each student.

• If students receive an expression, direct them to write a statement and draw a tape diagram to match.

• If students receive a statement, direct them to write an expression and draw a tape diagram to match.

• If students receive a tape diagram, direct them to write an expression and a statement to match.

Have students stand and find two other students who have matching representations. For example, a student with a tape diagram that shows 2 sixes and 5 fives should look for a student who has the expression (2 × 6) + (5 × 5) and a student who has the statement: The sum of 2 sixes and 5 fives.

Invite students to turn and talk about how their representations match.

Teacher Note

Intentionally distribute each representation from Numerical Expressions to students based on their needs. Students who are beginning to build the skill may benefit from a tape diagram, whereas students who have demonstrated mastery may be most challenged by being given a statement.

Promoting the Standards for Mathematical Practice

When students determine two additional ways to model a given representation of a numerical expression and find the two other students with matching representations, they are attending to precision (MP6).

Ask the following questions to promote MP6:

• What details are important to think about in this activity?

• When you write an expression, with what do you need to be extra precise? Why?

Compare Expressions

Students apply their understanding of the role of parentheses to compare expressions without evaluating them.

Direct students to problems 4–6. Have students work with a partner to compare the expressions without multiplying.

Use >, =, or < to compare the expressions.

4. 22 × (18 + 31) < (18 + 31) × 34

5. (2 × 8) + (10 × 8) > (7 × 8) − (4 × 8)

6. 145 × 71 = (100 + 45) × (70 + 1)

Gather students to discuss.

How were you able to compare each of the expressions without multiplying?

In problem 4, I thought about the units in each expression. The unit is the same: (18 + 31). 22 times as much as the sum of 18 and 31 is less than 34 times as much as the sum of 18 and 31.

In problem 5, I know two groups of 8 plus ten groups of 8 is twelve groups of 8, and seven groups of 8 minus four groups of 8 is three groups of 8. Twelve 8s is greater than three 8s.

In problem 6, the value of the group (100 + 45) is equal to 145, and the value of the group (70 + 1) is equal to 71. Because the factors are the same in each expression, I know the values of the expressions are equal.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Differentiation: Support

For students who need support comparing the expressions, consider having students cover or look at just single parts of each expression so they can make sense of the entire expression, one part at a time.

For problem 4, have students cover the factor 22 on the left and the factor 34 on the right. Ask students what they notice about the other factor in each expression. Then ask them to compare the expressions.

For problem 5, have students cover the factor (2 × 8) on the left and the factor (4 × 8) on the right. Have students compare the remaining expressions. Then have students notice that more groups of 8 are added to the left and that groups of 8 are subtracted from the right before they compare the expressions again.

For problem 6, have students cover the factor 71 on the left and the factor (70 + 1) on the right. Ask them what they notice. Then repeat for the other factor in each expression.

Land

Debrief 5 min

Objective: Write, interpret, and compare numerical expressions.

Gather the class with their Problem Sets. Facilitate a class discussion about numerical expressions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Have students refer to their Problem Sets to explain their thinking.

Does modeling with tape diagrams help you write statements for equations and expressions? How?

Yes. When I see the equation and expression represented by a tape diagram, it is simpler for me to think of a statement to match.

It does sometimes. If it is an equation, then I can tell by the groupings which numbers need to be added or multiplied, and I know the value of the expression. If it is an expression, the tape diagram can help me make sense of the expression so I can write a statement that matches.

Does modeling with tape diagrams help you make sense of where parentheses belong in expressions? How?

Yes. Modeling with tape diagrams helps me see which parts of the expression need to be grouped.

Display the following expression: 20 − 2 × 3

One student evaluated this expression as 54 and another student evaluated it as 14. What can we do to make sure that someone who does not see a tape diagram or know the context where this expression came from evaluates it as 14? Why? We should place parentheses around 2 × 3 so when it is evaluated, students know to find the difference between 20 and 2 groups of 3.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Write

1.

Value of expression: 220

6. 9999 7 7 Statement: The sum of four 9s and two 7s

Expression: (4 × 9) + (2 × 7)

Value of expression: 50

2.

7. Evaluate.

a. 40 + (3 × 9) − 6 61

b. (40 + 3) × (9 − 6) 129

c. Why do expressions (a) and (b) have different values?

Each expression groups the numbers differently. In expression (a), (3 × 9) is a group, so we are finding the value of 40 + 27 − 6. In expression (b), (40 + 3) is a group and (9 − 6) is a group, so we are finding the value of 43 × 3

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

8. Kelly forgot to put parentheses in her equation. Write parentheses to make her equation true.

6 + 8 × 12 2 = 140

Use >, =, or < to compare the expressions. Explain how you can compare the expressions without evaluating them.

9. 35 × (12 + 28) < (12 + 28) × 70

Explain: The sum is the same in both expressions, but the second expression is multiplied by 70 instead of 35, so I know it has a greater value.

10. 225 × 81 = (200 + 25) × (80 + 1)

Explain: They are equal because the second expression shows how the numbers in the first expression can be decomposed.

11. (48 × 7) − (37 × 7) > (5 × 7) + (5 × 7)

Explain: The first expression has a greater value because it is 11 sevens, and the other expression is only 10 sevens.

12. Consider the statement.

5 times as much as the sum of 319 and 758

a. Adesh makes a mistake when he writes an expression to represent the statement. What mistake does Adesh make?

(5 × 31 9) + 758

Adesh put the parentheses in the wrong place.

b. Write an expression to represent the statement.

5 × (319 + 758)

c. Evaluate the expression you wrote in part (b).

5 × (319 + 758) = 5 × 1,077 = 5,385

Copyright © Great Minds PBC

149 PROBLEM SET

150 PROBLEM SET

Sample:

Create and solve real-world problems for given numerical expressions.

Lesson at a Glance

Given an expression or a tape diagram, students construct word problems based on real-world situations that are relevant to them. Students consider placement of parentheses and how groupings within parentheses determine multiple steps in a word problem.

Key Questions

• What is important to think about when you build word problems?

• Did parentheses in expressions help you write word problems? How?

Achievement Descriptors

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD3 Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions. (5.OA.A.2)

Agenda Materials

Fluency 15 min

Launch 5 min

Learn 30 min

• Develop Multi-Part Word Problem Situations

• Write Word Problems to Match Expressions and Tape Diagrams

• Problem Set

Land 10 min

Teacher

• Chart paper (1 sheet)

Students

• None

Lesson Preparation

Partition a chart into five columns. Label the columns Situation, Add, Subtract, Multiply, and Divide.

Fluency

Whiteboard Exchange: Place Value Relationships

Students say the value of two identical adjacent digits in a four- or five-digit number, and then write a multiplication and a division equation to build fluency with place value relationships from topic A.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 2,388 with the ones place underlined.

What is the value of the underlined digit?

8

Display the answer and then display the tens place underlined.

What is the value of this underlined digit?

80

Display the answer.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Write a multiplication equation to show the relationship between the values of the underlined digits.

Display the multiplication equation.

Write a division equation to show the relationship between the values of the underlined digits.

Display the division equation.

Repeat the process with the following sequence:

Counting by Multiples of 6

60

,

,

7, and 70

Students say the first ten multiples of 6 and 60, and then 7 and 70 to build fluency with estimating quotients.

When I give the signal, say the first ten multiples of 6. Ready?

Display each multiple one at a time as students count.

6, 12, 18, 24, 30, 36, 42, 48, 54, 60

When I give the signal, say the first ten multiples of 60. Ready?

Display each multiple one at a time as students count.

, 180, 240, 300,

Now let’s say the first ten multiples of 7. Ready?

Display each multiple one at a time as students count.

Teacher Note

Control the pace of the count as you show the numbers. Remember to listen to student responses and be mindful of errors, hesitation, and lack of full class participation.

When I give the signal, say the first ten multiples of 70. Ready?

Display each multiple one at a time as students count.

70, 140, 210, 280, 350, 420, 490, 560, 630, 700

Whiteboard Exchange: Write and Evaluate Expressions

Students write and evaluate an expression to prepare for solving real-world problems with numerical expressions.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the statement: The sum of 1 seventh and 2 sevenths.

Write an expression to represent the statement.

Show the sample expression.

Write the value of the expression.

Show the answer.

Repeat the process with the following sequence:

sum of 1 seventh and 2 sevenths

Launch

Students use parentheses as they write expressions to match a word problem context.

Display the following expression to the class:

5 + 2

Pair students and use the Co-construction routine to have partners create a real-world situation that could apply to the expression.

Give pairs 1 minute to compare the contexts they construct with other groups.

Invite students to share their ideas and to explain the relationship to the expression with the class.

There are 5 goldfish and 2 clown fish in the fish tank. How many fish are in the tank?

5 students stand at a bus stop. 2 more students join them. How many students are at the bus stop now?

Have students adjust the situations they created to match this expression:

3 × (5 + 2)

5 goldfish and 2 clown fish are in the small fish tank. There are 3 times as many fish in the large fish tank. How many fish are there in the large fish tank?

5 students wait at a bus stop. 2 more students join them. There are 3 times as many students on the bus than there are students waiting at the bus stop. How many students are on the bus?

Transition to the next segment by framing the work.

Today, we will write and solve real-world problems that match expressions.

Learn Develop Multi-Part Word Problem Situations

Materials—T: Chart paper

Students develop word problem situations to match each operation. Ask students to name situations in real life in which they use math, such as when baking cookies. Record their ideas on a chart.

One operation at a time, invite students to give an example of how they would use that operation in each situation. Repeat the process for two or three situations.

As students respond, fill in the chart. The examples below illustrate the types of responses you may hear.

UDL: Engagement

The situations presented here are only samples. Invite the class to brainstorm real-world situations that are relevant to their lives. If students drive the situations discussed, there is an opportunity to anchor instruction in contexts that are familiar and meaningful to students.

Baking cookies

Organizing pencils

Find the total number of cookies.

Find the total number of pencils.

How many cookies are left after you eat some?

How many pencils are left after another class borrows some?

There are times as many chocolate chip cookies as sugar cookies.

There are times as many purple pencils as blue pencils.

How many cookies can each person have?

How many pencils can go in each bin?

Counting money

Riding the subway

How much money do you have in all?

Find the total number of people in the subway car.

If you spend some money, how much money is left?

How many people are still on the subway car after people get off?

Find the amount of money it costs to buy sandwiches.

Find the amount of money each person gets if it is divided equally.

If there are people altogether in cars, and the same number of people are in each car, how many people are in each car?

Sometimes word problems have more than one step and each step might involve a different operation. Let’s look at some expressions that have more than one operation and write word problems to match. You may use the situations we came up with or come up with your own.

Direct students to problem 1 in their books.

1. 2 × (15 + 20)

Sample:

Scott bakes 15 peanut butter cookies and 20 chocolate chip cookies. Sana bakes twice as many cookies as Scott. How many cookies does Sana bake?

Pair students and use the Co-construction routine to have partners create a context that could apply to the expression.

Give pairs 2 minutes to compare the contexts they construct with other groups. Invite students to share their ideas.

Scott bakes 15 peanut butter cookies and 20 chocolate chip cookies. Sana bakes twice as many cookies as Scott. How many cookies does Sana bake?

Lacy spends 15 minutes on her math homework and 20 minutes on her science homework. Ryan spends twice as much time as Lacy on his homework. How many minutes does Ryan spend on his homework?

On Monday Jada walks for 15 minutes. On Tuesday she walks for 20 minutes. On Wednesday she walks 2 times as much as she did on Monday and Tuesday. How long did Jada walk on Wednesday?

Direct students to swap with a partner and solve each other’s word problems. Have students turn and talk to explain how their partner’s word problem matches the expression.

Differentiation: Support

Invite students to draw a tape diagram to represent and make sense of the expression before they write a word problem. ?

15 + 20 15 + 20

Write Word Problems to Match Expressions and Tape Diagrams

Students analyze tape diagrams and expressions to develop word problem situations that match.

Direct students to problems 2–5. Consider forming groups of students with the following options, based on students’ needs:

• Students practice writing word problems for all the problems independently, solve them, and then compare their word problems with a partner.

• Partners practice writing word problems for some or all the problems and solve them.

• Groups of three students practice writing word problems for one or two problems and complete a gallery walk to study the other problems.

Provide instructions and time for students to work. Encourage students to refer to the chart of word problem situations as needed.

2. 9

? 16 16 16 9 9

Sample:

An action figure costs $9 and a puzzle costs $16. A parent purchases 3 action figures and 3 puzzles for their children. How much money does the parent spend?

3. 33

96 ?

Sample:

Blake makes 96 muffins for the bake sale. He sells 33 of them and puts the remaining muffins in 3 containers to take home. If he puts the same number of muffins in each container, how many muffins are in each?

UDL: Engagement

Consider offering students a choice for completing problems 2–5. Instead of forming groups for students, encourage them to select the grouping arrangement that works best for them.

Promoting the Standards for Mathematical Practice

Students reason abstractly and quantitatively as they write word problems that match tape diagrams and numerical expressions (MP2).

Ask the following questions to promote MP2:

• What real-world situations are modeled by the tape diagram?

• What do the parentheses in the expression tell you about your word problem?

4. (24 − 6) ÷ 3

Sample:

Julie bakes cookies. She makes 24 cookies but 6 burn, so she throws them away. She gives the rest of the cookies to her 3 friends. If each friend gets the same number of cookies, how many cookies does each friend get?

5. (9 + 4) × 3 − 6

Sample:

On Monday a teacher bought 3 boxes of whiteboard markers. Each box had 9 black markers and 4 blue markers. By Friday, 6 of the markers had dried up. How many markers are left?

Gather the class.

Invite students to select one problem they completed and to turn and talk about how their word problem matches the corresponding expression or tape diagram.

Direct students to problem 5 and display the incorrect word problem.

Have students read the word problem chorally.

Invite students to think–pair–share about whether the word problem matches the expression (9 + 4) × 3 − 6 and how they know.

No, the word problem does not match the expression because the word problem shows 3 times as much as $4, but the expression shows 3 times as much as the sum of 9 and 4.

Noah earns $4 each time he takes out the trash. He takes out the trash

3 times. He earns another $9 for mowing the lawn. He spends $6 on a baseball card. How much money does Noah have?

We have to consider the groups as well as the order when we write word problems to match expressions.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Differentiation: Challenge

Invite students to write an expression and to draw a tape diagram to represent the incorrect word problem and compare it to their own response to problem 5.

Language Support

To support students in explaining whether the problem matches the expression, direct students’ attention to the Share Your Thinking portion of the Talking Tool.

Land

Debrief 5 min

Objective: Create and solve real-world problems for given numerical expressions.

Facilitate a class discussion about numerical expressions by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Today we practiced writing word problems to match expressions or tape diagrams. What did you think about to help you build your word problems?

I thought about whether there were any groups in parentheses and what situation I could write to match those groups.

I thought about math that I see in my life outside of school and about word problems I have solved in the past.

I thought about which operation was necessary and then made sure my story matched that operation.

Did parentheses in expressions help you write word problems? How?

When there were parentheses, it helped me think about a situation that represented that grouping. It helped me think about one part of the expression at a time.

Display the following tape diagrams:

Differentiation: Challenge

Invite students to write an expression that matches the second tape diagram and an expression that matches the third tape diagram.

Invite students to think–pair–share about their responses to the following questions. Would the word problem you create for each tape diagram be the same? Why?

The real-world situation might be the same, but the question would not. Because the question mark represents a different unknown in each tape diagram, the question asked at the end of the word problem would not be the same.

Remind students that even though numbers or tape diagrams might be the same or similar from problem to problem, the unknown makes a difference. So when they create or solve word problems, they should be careful to pay attention to what the question is asking them to find.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Name Date

1. Draw lines to match the expressions to the word problems.

a. (3 + 9 − 5) × 12

Yuna buys 3 bags of oranges. There are 9 oranges in each bag. She eats 5 oranges. Then she gives 12 oranges to her friends. How many oranges does Yuna have now?

3. Consider the expression. 4 × (15 + 8) Write a word problem that can be represented by the given expression.

Sample:

On Monday, Julie does 15 jumping jacks in the morning and 8 in the evening. On Tuesday, Julie does 4 times as many jumping jacks as she did on Monday. How many jumping jacks did Julie do on Tuesday?

b. 3 × 9 − 5 − 12

Tyler has 3 pencils. He finds 9 more pencils. Sasha has 5 times as many pencils as Tyler. Eddie has 12 fewer pencils than Sasha. How many pencils does Eddie have?

c. (3 + 9) × 5 − 12

Riley gets 3 books from the library on Monday and 9 more books on Tuesday. She reads and returns 5 books on Wednesday. Riley has 12 times as many books on her bookshelf as she still has from the library. How many books are on Riley’s bookshelf?

2. Write an expression that represents the tape diagram. Then write a word problem that can be represented by the tape diagram and expression.

12

? 12 12 17

3 × 12 + 17

Sample:

There are 3 shelves with 12 cans of soup on each shelf. A fourth shelf has 17 cans of soup. How many cans of soup are there in all?

4. Consider the expression. (26 − 8) ÷ 2

a. Write a word problem that can be represented by the given expression.

Sample:

There are 26 people at the park. 8 people go home. The rest of the people make 2 equal groups to play a game. How many people are in each group?

b. Solve your problem. There are 9 people in each group.

Solve multi-step word problems involving multiplication and division.

Lesson at a Glance

Students use the Read–Draw–Write process to solve multi-step word problems involving multiplication and division. Student-driven discussions occur as students explore how modeling with a tape diagram helps them make sense of multi-step word problems. Students also see that once they understand the problem, they can choose the pathway in which to solve the problem, such as by using the tape diagram or solving numerically by writing and evaluating expressions.

Key Questions

• Does modeling help you when you are solving problems? How?

• How does thinking about what is known and unknown help you solve multiplication and division word problems?

Achievement Descriptors

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD3 Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions. (5.OA.A.2)

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Represent Word Problems with Models and Expressions

• Share, Compare, and Connect

• Solve Multi-Step Problems

• Problem Set

Land 10 min

Teacher

• Multiplication and Division Tape Diagram Card Sort cards (in the student book)

• Envelopes (8)

Students

• Envelope of Multiplication and Division Tape Diagram Card Sort cards (1 per student group)

Lesson Preparation

Tear out and cut apart Multiplication and Division Tape Diagram Card Sort cards from the student books. Organize each set into an envelope (1 per student group). Consider whether to prepare these materials in advance or have students assemble them prior to the lesson.

Fluency

Whiteboard Exchange: Place Value Relationships

Students say the value of two identical adjacent digits in a six- or seven-digit number and then write a multiplication and a division equation to build fluency with place value relationships from topic A.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 156,629 with the hundreds place underlined.

What is the value of the underlined digit?

600

Display the answer and then display the thousands place underlined.

What is the value of the underlined digit?

6,000

Display the answer.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Write a multiplication equation to show the relationship between the values of the underlined digits.

Display the multiplication equation.

Write a division equation to show the relationship between the values of the underlined digits.

Display the division equation.

Repeat the process with the following sequence:

Whiteboard Exchange: Interpret Tape Diagrams

Students write and complete an equation to represent a tape diagram to prepare for solving multi-step word problems involving the four operations.

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the tape diagram.

What does the tape diagram show? Tell your partner.

Provide time for students to think and share with their partners.

The total is unknown. There are two parts. One part has a value of 300 and the other part has a value of 180.

Write an equation to represent the tape diagram. Use the letter shown on the tape to represent the unknown.

Display the sample equation.

Write the value of the unknown.

Encourage students to use a written method of their choice to find the value of the unknown, if needed.

Display the answer.

Continue the process with the following sequence:

Materials—S: Envelope of Multiplication and Division Tape Diagram Card Sort cards

Students sort tape diagrams into three groups: multiplication, division (number of groups known), and division (group size known).

Place students in groups of three. Distribute one Envelope of Multiplication and Division Tape Diagram Sort cards to each group. Have students take out the three cards that show the categories: Multiplication, Division (number of groups known), and Division (group size known). Invite students to sort each tape diagram into the category in which it belongs. Allow students 2 minutes to sort.

In the card sort activity, there are two tape diagrams without question marks that can belong to any of the three categories. The discussion that follows the activity explores why. Allow for productive struggle as students grapple with uncertainty around the proper placement of these tape diagrams.

Teacher Note

Circulate as students sort and informally assess students’ reasoning for why a tape diagram belongs in a particular category.

How did you know a tape diagram represented multiplication?

If I could see the number of groups and the size of each group, but not the total, I knew that tape diagram represented multiplication.

How did you know a tape diagram represented division with the number of groups known? With group size known?

If I could see the total and the number of equal-size groups, but not the size of each group, I knew that tape diagram represented division with number of groups known.

If I could see the total and the size of each group or unit, but not the number of equal groups, I knew that tape diagram represented division with group size known.

Were there any tape diagrams you were not sure where to place? Why?

I was not sure about the tape that showed a as the total because it might be showing 5 × 3 = a, or a ÷ 3 = 5, or a ÷ 5 = 3.

I was not sure about the tape that showed 20 as the total because it might be showing 4 × 5 = 20, or 20 ÷ 5 = 4, or 20 ÷ 4 = 5.

What would have helped you decide in which category those tape diagrams belonged?

It would have helped if there was a question mark in the tape diagram that showed 20 as the total, so I could clearly see what was unknown.

If I had a word problem to better understand what the numbers represented, I could have decided in which category the tape diagram belonged.

Transition to the next segment by framing the work.

Today, we will use a model to understand word problems involving multiplication and division, then choose to use a model or another method to solve the problems.

Learn

Represent Word Problems with Models and Expressions

Students reason about, represent, and solve multi-step word problems involving multiplication and division.

Direct students to problem 1 in their books. Have them use the Read–Draw–Write process to solve the problem independently. Encourage students to self-select their tools and strategies. Circulate and observe student work.

As you circulate, consider using the following prompts:

• Tell me about your method.

• Tell me about how your drawing connects to the story.

• What does this number represent?

• Have you revised your tape diagram? Why?

• Why did you use that operation?

• Does your answer seem reasonable? Why?

Teacher Note

To assist students with the Read–Draw–Write process as they work, consider asking the following questions:

• What do you know?

• Can you draw something?

• What can you draw?

• Can you label anything?

• Should you revise or add to your drawing?

• Do you have all the information you need to solve the problem?

• What conclusions can you make, based on your drawing? Use the Read–Draw–Write process to solve each problem.

1. A florist uses 2,448 flowers to make bouquets. They put 24 flowers in each bouquet and sell the bouquets for $25 each. If the florist sells all the bouquets of flowers, how much money do they earn?

2,448 ÷ 24 = 102 102 × 25 = 2,550

The florist earns $2,550.

Select one or two students to share their drawings. Look for work samples that model the story with two tapes: one that represents division and the other that represents multiplication.

Teacher Note

The sample student work shows common responses. Look for similar work from your students and encourage authentic classroom conversations about the key concepts.

Select one or two students to share their solution methods. Look for work samples that show different ways to multiply or divide. Purposefully choose work that allows for rich discussion about how drawing a model helps us reason about a problem and choose the appropriate operations to use to solve the problem.

The following student work samples demonstrate using different methods to multiply and divide.

If your students do not produce similar work, choose one or two pieces of their work to share, and highlight how it shows movement toward the goal of this lesson. Then select one work sample from the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student solved the problem. What do you think this student did?”

Promoting the Standards for Mathematical Practice

When students use the Read–Draw–Write process to create models to represent word problems and demonstrate methods for multiplication and division to solve the word problems, they are modeling with mathematics (MP4).

Ask the following questions to promote MP4:

• What key ideas in this problem do you need to include in your model?

• How do you represent the key ideas in this problem in your model?

• How can you improve your model to better represent the problem?

Share, Compare, and Connect

Students share solutions and reason about the connections.

Gather the class and invite the students identified in the previous segment to share their solutions one at a time.

As each student shares, ask questions to elicit their thinking. Ask the class questions to help students make connections between the demonstrated solutions and their own work. Encourage students to ask questions of their own.

The following sample discussion demonstrates questions that elicit thinking and invite connections.

Sasha, tell us about why you built your tape diagrams the way you did.

First, I drew a tape to show 2,448 flowers. Then I drew a unit to show 1 bouquet has 24 flowers. I did not know how many bouquets the florist made, so I stopped to find that answer.

I found the florist made 102 bouquets, so I included that information in my tape diagram. The problem asks to find how much money the florist earned, so I drew another tape to show 1 bouquet means the florist earns $25. Because I did not know how much money in all, I labeled the total with a question mark.

Sasha, what operation did you use to find the number of bouquets? Division

Looking at Sasha’s first tape, how do you think she knew to divide?

She knew the total and the size of each group, but not the number of equal groups. So she realized she could find the number of groups by using division.

Sasha, what operation did you use to find the amount of money the florist earns? Multiplication

Looking at Sasha’s second tape, how do you think she knew to multiply?

She knew the number of groups and the size of each equal group but not the total. So she realized she could find the total by multiplying.

Invite students to think–pair–share about how to write one expression that matches Sasha’s tape diagrams.

(2,448 ÷ 24) × 25

How does the expression (2,448 ÷ 24) × 25 match the story?

First, we need to find how many bouquets the florist makes by determining how many 24s are in 2,448. That number, 102, represents the total number of bouquets. To find the amount of money the florist earns, we multiply 102 by $25 because the florist earns $25 for each bouquet they sell.

Invite students to think–pair–share about a response to the following questions.

Do the tape diagrams help you write an expression that matches the story? How?

The tape diagrams help me write an expression because I can see everything about the story in the tape diagram. Then I know what to write in the expression.

The tape diagrams help me see which operations I can use to solve the problem. The diagram also helps me see what might be grouped together based on the steps I took to solve the problem, so I know where to put parentheses.

Now let’s look at how some other classmates solved the problem.

Multiplication Area Model and Partial Quotients with Vertical Form (Noah’s Way)

Invite students to think–pair–share about their observations about Noah’s work.

Noah divided by using partial quotients and showed his thinking by using vertical form. He has three partial quotients.

Noah multiplied by using an area model. He has four partial products.

Noah, tell us about your choice to divide by using partial quotients in vertical form and then to multiply by using the area model.

I divided by using partial quotients because it helped me show the groups of the divisor I was making, one group at a time. I started with the largest group possible, which was hundreds. When I had 48 left, I fit two more 24s, but I created one group at a time, which made it simpler for me than trying to divide everything all at once.

I multiplied by using the area model because it helped me make sure I was using the distributive property accurately and multiplying all the parts of one factor by all the parts of the other factor.

Let’s discuss Sasha’s tape diagram model and Noah’s choice of the area model for multiplication. What is different about these two models?

Sasha’s tape diagrams helped her make sense of the story. They helped her know which operation to use and whether she had more steps to find the answer. The tape diagrams also helped her write an expression. Noah’s area model for multiplication was his method for multiplying.

Though we often model to understand a problem, we can also use a model to multiply, divide, add, or subtract. You can decide whether the model you use to understand a problem is also the model you use to solve the problem.

You might have used the same methods as Noah or different methods to divide and multiply. We heard Noah’s reasoning for the methods he chose. Now think about why you decided to divide and to multiply the way you did.

Invite students to turn and talk about the reasons for their choices of methods for division and multiplication.

Remember that once you understand the problem, the strategy, method, or model you use to solve the problem is your choice.

UDL: Action & Expression

When students compare their work with their classmates’ work and discuss the reasons for their choices of methods, they are monitoring and evaluating their progress with division and multiplication. Encourage students to think about what they did well and to try a new method next time.

Solve Multi-Step Problems

Students draw models to represent and solve word problems. Direct students to problems 2 and 3. Consider assigning students only one problem, based on their needs. Students who finish early could complete the remaining problem. Give students 5 minutes to solve their assigned problem. Encourage students to use the Read–Draw–Write process and to use the tape diagram to guide their thinking about the problem, help them discover the operations needed to solve the problem, and help them write and evaluate an expression.

2. Miss Song buys 15 boxes of fruit snacks for the school field day. Each box holds 24 packs of fruit snacks. She gives as many packs of fruit snacks as possible to 22 classrooms so that they each get the same number. How many extra packs of fruit snacks does Miss Song have?

Differentiation: Support

Consider removing one level of complexity by providing students with tape diagrams. Then have them label, write expressions, and solve the problem.

Miss Song has 8 extra packs of fruit snacks.

Language Support

Consider providing images or realia to support students’ understanding of the problem contexts.

3. A carton of eggs has 12 eggs. A box of eggs holds 12 cartons. A baker uses 5 eggs for each cake he makes. If the baker buys 3 boxes of eggs, what is the greatest number of cakes he can make?

Differentiation: Challenge

Have students consider how the answer for problem 3 would change if each carton of eggs held a baker’s dozen, or 13 eggs.

The greatest number of cakes the baker can make is 86.

Allow students 2 minutes to compare their representations, methods, and answers with a partner.

Invite students to turn and talk about how their tape diagrams helped them decide on an operation, complete each of the necessary steps, and write an equation.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Solve multi-step word problems involving multiplication and division.

Facilitate a class discussion about solving word problems by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Does modeling help you when you are solving problems? How?

Yes, modeling helps us make sense of a problem because we can draw what we understand from the story as we read it. Once we draw a model, we can see how many unknowns there are, and it helps us determine which operation to use to solve the problem.

We can also use a model to multiply or divide. For example, if our tape diagram shows us that we can multiply, we could multiply by using the area model. Or, if the tape diagram shows us that we can divide, we could divide by using the area model.

How does thinking about what is known and unknown help you solve multiplication and division word problems?

If I know the number of equal groups and the number in each group, but I don’t know the total, I can multiply to find the whole.

If I know the total and the size of each group, but not the number of equal groups, I can divide.

If I know the total and the number of equal groups, but not the size of each group, I can divide.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Name

Use the Read–Draw–Write process to solve each problem.

1. Miss Baker orders 13 cases of soup for her grocery store. Each case has 48 cans of soup. She puts all the cans on the shelves so that each shelf has an equal number. If there are 16 shelves, how many cans of soup are on each shelf?

13 × 48 = 624

624 ÷ 16 = 39

There are 39 cans of soup on each shelf.

2. Mr. Sharma bakes 732 cupcakes each week for his bakery. He puts 12 cupcakes in each box and earns $14 for each box he sells. If he sells all the boxes of cupcakes, how much money does he earn?

732 ÷ 12 = 61

61 × 14 = 854

3. There are 9,675 people at a concert. An equal number of people sit in each of the 15 sections. A ticket for a seat in section B costs $47. What is the total cost of the tickets for the seats in section B?

9,675 ÷ 15 = 645

645 × 47 = 30,315

The total cost of the tickets in section B is $30,315

4. 24 students are in each classroom at Oak Street School. There are 37 classrooms. Each row in the auditorium has 45 seats. What is the fewest number of rows needed for all the students to have a seat?

24 × 37 = 888

888 ÷ 45

Q: 19

R: 33

The fewest number of rows needed is 20

5. A box of name tags holds 18 name tags. A case of name tags has 25 boxes. Principal Song buys 17 cases of name tags. She gives an equal number of name tags to each of 42 classrooms. If she gives out as many name tags as possible, how many extra name tags does Principal Song have?

25 × 18 = 450

17 × 450 = 7,650

7,650 ÷ 42

Q: 182

R: 6

Principal Song has 6 extra name tags.

6. A farmer’s cows produce 9,548 liters of milk in 31 days. Each cow produces 28 liters of milk a day. The farmer feeds each cow 17 kilograms of hay each day. What is the total number of kilograms of hay the cows eat each day?

9,548 ÷ 31 = 308

308 ÷ 28 = 11 11 × 17 = 187

The cows eat a total of 187 kilograms of hay each day.

Solve multi-step word problems involving the four operations.

Lesson at a Glance

Students use the Read–Draw–Write process to solve multi-step word problems involving addition, subtraction, multiplication, and division. Student-driven discussions occur as students explore how modeling with a tape diagram helps them make sense of multi-step word problems and find a pathway toward the solution.

Key Questions

• Does the Read–Draw–Write process help us solve multi-step word problems? How?

• How do you know which operation to use when you solve a word problem?

Achievement Descriptors

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses. (5.OA.A.1)

5.Mod1.AD3 Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions. (5.OA.A.2)

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers. (5.NBT)

Agenda Materials

Fluency 5 min

Launch 5 min

Learn 40 min

• Solve a Word Problem

• Share, Compare, and Connect

• Problem Set

Land 10 min

•

Lesson Preparation

•

Fluency

Counting by Multiples of 8, 80, 9, and 90

Students say the first ten multiples of 8 and 80, and then 9 and 90 to build fluency with estimating quotients.

When I give the signal, say the first ten multiples of 8. Ready?

Display each multiple one at a time as students count.

8, 16, 24, 32, 40, 48, 56, 64, 72, 80

When I give the signal, say the first ten multiples of 80. Ready?

Display each multiple one at a time as students count.

80, 160, 240, 320, 400, 480, 560,

640, 720, 800

Now let’s say the first ten multiples of 9. Ready?

Display each multiple one at a time as students count.

9, 18, 27, 36, 45, 54, 63, 72, 81, 90

When I give the signal, say the first ten multiples of 90. Ready?

Display each multiple one at a time as students count.

Launch

Students match mathematical expressions with real-world situations.

Direct students to problem 1 in their books. Read the directions aloud. Have students complete the problem with a partner.

1. Match each mathematical expression with the real-world situation it represents.

Mathematical Expression

A. (18 × 4) + 5

B. 18 ÷ (4 + 5)

C. (18 × 4) − 5

D. 18 + (4 × 5)

Real-World Situation

Leo buys 4 pens. Blake buys 5 pens. The total cost of the pens is $18. If all the pens cost the same amount, what is the cost for 1 pen?

At a camp, 1 group has 18 kids, and 4 groups have 5 kids each. How many kids are at the camp?

Sana buys 4 cases of water. Each case has 18 bottles. If she also has 5 cans of juice, how many total drinks does she have?

Yuna mows 4 lawns and gets paid $18 per lawn. If Yuna spends $5, how much money does she have left over?

After students have finished, bring the class together. Share the correct answers and ask the following questions.

What is the difference between the meaning of expression A and the meaning of expression C?

Expression A shows 5 more than 18 groups of 4, and expression C shows 5 less than 18 groups of 4.

What is the difference between the meaning of expression A and the meaning of expression D?

Expression A shows 5 more than 18 groups of 4, and expression D shows 18 more than 4 groups of 5.

Direct students to the real-world situation connected to expression D.

There are usually several mathematical expressions that represent any real-world situation. Can you think of another expression to represent the total number of kids at camp?

4 × 5 + 1 × 18

4 × 5 + 18

5 + 5 + 5 + 5 + 18

Invite students to turn and talk to choose one of the other three situations and discuss another expression that represents the situation. Transition to the next segment by framing the work. Today, we will solve multi-step word problems involving the four operations.

Learn

Solve a Word Problem

Students solve multi-step word problems with multiple operations and compare their methods with other students.

Direct students to problem 2. Have them use the Read–Draw–Write process to solve the problem independently. Encourage students to self-select their tools and strategies. Circulate and observe student work.

As you circulate, consider using the following prompts:

• Tell me about your method.

• Tell me about how your drawing connects to the story.

• What does this number represent?

• Have you revised your tape diagram? Why?

• Why did you use that operation?

• Does your answer seem reasonable? Why?

• How can you check your answer?

Teacher Note

Use this lesson as a formative assessment to see how students approach multi-step word problems. The intent of this lesson is not to introduce new arithmetic skills. Highlight the similarities and differences in how students choose to make their calculations.

Teacher Note

To assist students with the Read–Draw–Write process as they work, consider asking the following questions:

• What do you know?

• Can you draw something?

• What can you draw?

• Can you label anything?

• Should you revise or add to your drawing?

• Do you have all the information you need to solve the problem?

• What conclusions can you make based on your drawing?

Use the Read–Draw–Write process to solve the problem. Show your thinking.

2. Jada is saving money for a computer that costs $1,149. That is three times as much money as she has already saved. Her parents also gave her $150 for the computer. Jada earns $14 each hour at her job. How many hours does Jada need to work to earn the remaining money she needs to buy the computer?

Differentiation: Support

Consider breaking the problem into three distinct parts. Invite students to draw tape diagrams to represent each part in the word problem.

• Part (a): Find how much money Jada has.

• Part (b): Find how much money Jada needs.

• Part (c): Find how many hours Jada needs to work.

Jada needs to work 44 hours.

Select two or three students to share their work. Purposefully choose work that allows for rich discussion about different ways to approach solving the problem. The following student work samples demonstrate how multiple tape diagrams can connect to multiple steps or multiple operations.

Teacher Note

The sample student work shows common responses. Look for similar work from your students and encourage authentic classroom conversations about the key concepts.

If your students do not produce similar work, choose one or two pieces of their work to share, and highlight how it shows movement toward the goal of this lesson. Then select one work sample from the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student solved the problem. What do you think this student did?”

Share, Compare, and Connect

Students share and compare solutions and reason about their connections.

Gather the class and invite the students identified in the previous segment to share their solutions one at a time.

As each student shares, ask questions to elicit their thinking. Ask the class questions to help students make connections between the demonstrated solutions and their own work. Encourage students to ask questions of their own.

The following sample discussion demonstrates questions that elicit thinking and invite connections.

Promoting the Standards for Mathematical Practice

As students solve multi-step word problems by using the Read–Draw–Write process to unpack the problem and find entry points, monitor their own progress, and question whether the values they calculate make sense, they make sense of problems and persevere in solving them (MP1).

Ask the following questions to promote MP1:

• What steps can you take to start solving the problem?

• Does your answer make sense? Why?

Multiple Tape Diagrams to Show Each Step (Ryan’s Way)

Invite students to think–pair–share about their observations about Ryan’s work.

Ryan showed three tape diagrams. He showed his calculations to the right of his tape diagrams.

Ryan showed partial quotients by using an area model. When he subtracted, he used vertical form.

Differentiation: Challenge

Have students who finish early use the Read–Draw–Write process to solve this problem:

A rectangular park is 3 times as long as it is wide. Noah runs 5 laps around the perimeter of the park for a total distance of 8 kilometers. What are the length and width of the park, in meters?

8 × 1,000 = 8,000

Noah runs 8,000 meters.

Each lap around the park is 1,600 meters. Half of a lap around the park is 800 meters.

Width

800 ÷ 4 = 200

The park is 200 meters wide and 600 meters long.

Language Support

Consider directing partners to the Agree or Disagree section of the Talking Tool to support them in discussing the similarities and differences in their work and the work of their classmates throughout this segment.

Ryan, tell us about your first tape diagram. Why did you draw it that way?

I drew a tape diagram to represent the cost of the computer. Then I split the tape into 3 equal parts because I knew the cost of the computer is 3 times as much as the amount of money Jada saved.

Then I divided 1,149 by 3 to determine that she already saved $383. I subtracted $383 from $1,149 to find that she still needs $766.

So, in your first tape, you used both division and subtraction. How do you think Ryan knew from the tape diagram to divide and then subtract?

The total 1,149 is partitioned into 3 equal parts, and because the groups are all the same size, he knew he could use division to find the value of one group.

Once he knew the value of one group, he did not know how much money Jada still had left to earn, so he needs to find the difference between 1,149 and 383.

Ryan, why did you make a second tape?

766 is not the answer to what the question asks. 766 represents the amount of money Jada still needs to buy the computer. The story was not over, so I drew another tape diagram to represent how much money she still needed after the gift from her parents.

I subtracted $150 from $766 to determine that she needs another $616.

Why do you think Ryan made a third tape?

The story was still not over. $616 represents the amount of money Jada needs after the gift from her parents. The question asks how many hours Jada needs to work, and the second tape does not show that.

Ryan, how did you know you could divide to find the number of hours?

I knew the total was $616, and I knew that 1 hour meant Jada would earn $14. From the tape diagram, I saw I needed to determine how many 14s are in 616. So 14 meant the size of the unit, and I needed to determine how many of those equal-size units fit into 616.

Invite students to turn and talk about the similarities and differences between Ryan’s work and their work.

Part–Whole Tape Diagrams (Lacy’s Way)

Invite students to think–pair–share to share observations about Lacy’s work and to compare it to Ryan’s work.

Lacy modeled the story with two tapes, and Ryan modeled with three.

Lacy used vertical form to show the partial quotients, and Ryan used an area model. They both used division and subtraction to find unknowns.

Lacy, tell us about your first tape diagram. Why did you draw it that way?

I drew a tape diagram to represent the cost of the computer. Then I partitioned the tape into 3 equal parts because I knew the cost of the computer is 3 times as much money as what Jada already saved. I divided 1,149 by 3 to determine she already saved $383. Then I showed the $150 Jada got from her parents. I added $383 and $150 to find that she has $533 of the $1,149 needed to buy the computer. I subtracted $533 from $1,149 to determine that she needs another $616.

So, in your first tape, you used division, addition, and subtraction. Class, how do you think Lacy knew from the tape diagram to divide, then add, and then subtract? She could see on the tape diagram the amount Jada already saved was one of three equal-size units with a total of 1,149, so Lacy could divide to find the value of one of the units.

She knew she could add 383 and 150 because that would tell her how much money Jada has after her parents gave her $150, so we could add those two parts of the tape diagram.

She knew she could subtract because 533 represents how much money Jada has, and the difference between 533 and 1,149 represents how much money she still needs.

In Ryan’s first tape, he used division and subtraction to find unknowns. In Lacy’s first tape, she used division, addition, and subtraction to find unknowns. When they divided, Ryan used the area model and Lacy used vertical form. Do we all need to do the exact same thing to find unknowns? Why?

No, we don’t because our tape diagrams might look a little different. As long as tape diagrams accurately show the story, we can make decisions about how to find unknowns based on what we see in our tape diagrams.

Lacy, why did you make a second tape?

My first tape showed how much money Jada has and how much money she needs. It did not tell me how many hours she would need to work, so I drew another tape diagram to represent the $616 she still needs to save.

Invite students to turn and talk to discuss which operations they used to find unknowns and why.

Invite students to think–pair–share to make observations about Kelly’s work and to compare Kelly’s work to Lacy’s and Ryan’s work.

Kelly has three tape diagrams like Ryan does, but her three are different.

Kelly labeled the tape diagrams and their parts, but Lacy and Ryan only labeled parts of their tape diagrams.

Kelly used division, subtraction, and addition like Lacy.

Kelly, tell us about your first tape diagram. Why did you draw it that way?

First, I drew a long tape to represent the cost of the computer and I labeled it $1,149. Then I knew the cost is 3 times as much as what Jada saved, so I partitioned the tape into three parts. I made another tape below it that was the same size as one unit from the previous tape to represent the money she saved. Once I knew that she got a gift of $150, I added a part to the second tape to show how much money she had. Then I drew a brace to show how much more money she still needs for the computer.

How is Kelly’s first tape diagram model different from Ryan’s and Lacy’s?

Kelly showed the relationship between the computer cost and the money that Jada saved differently. Instead of writing the amount of money Jada saved and the gift she got from her parents in the computer cost tape, she made a separate tape diagram for that information.

When a story uses language such as three times as much, we can show the relationship between numbers by using two tape diagrams. We call them comparison tape diagrams.

Which tape diagram did all three students show the same way?

They all showed the tape diagram that represented the $616 Jada still needs in the same way. Each showed that we needed to figure out how many 14s are in 616.

What do you notice about the length of that tape compared to earlier tapes?

The tape that shows Jada still needs $616, is the shortest tape in all three students’ work.

How does the length of the tapes help you keep track of reasonableness?

The amount of money Jada still needs should be less than $1,149 because she already saved some. If I had found that Jada still needed more than $1,149, that would not have made sense based on the model.

The word problem we solved is a multi-step problem, and you each had the opportunity to practice using the Read–Draw–Write process to solve problems. Then we made observations and connections by analyzing Ryan’s, Lacy’s, and Kelly’s work.

Have students think–pair–share about a response to the following question.

Does the Read–Draw–Write process help us solve multi-step word problems? How?

Yes. We read, draw, and write in chunks. When we learn new information, we pause to draw, then go back to reading, and draw when we learn something new. We write expressions or equations as we realize which operation we can use to find unknown information.

Yes. The model helps me decide which operation I might use to find an unknown. Once I know the operation, I can choose any method I want to calculate. Each time we find new information by calculating, we can compare to our tape diagram model and ask, Does that make sense based on what I see in the model?

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

UDL: Action & Expression

Consider providing time for students to reflect prior to beginning the Problem Set. Direct students to look at their work once more and then again at their classmates’ work. Ask questions such as the following:

• What worked well for you?

• What might you do differently next time?

10

Debrief 5 min

Objective: Solve multi-step word problems involving the four operations.

Gather the class with their Problem Sets. Facilitate a class discussion about solving multi-step word problems by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Invite students to turn and talk about one of the problems in the Problem Set. Have them discuss why they drew the tape diagrams the way they did and how they decided which operation to use to solve the problem.

How did drawing a model help you make sense of one of the problems in the Problem Set?

In problem 2, I drew tape diagrams for the bike, motorcycle, and car to help me keep track of their weights.

In problem 3, I drew a diagram to help me determine the area of the library. In problem 4, a tape diagram helped me make sense of the relationship between the weight of loads of bricks and the weight of loads of wood.

How do you know which operation to use when you solve a word problem?

I figure out which operation I need to use when I solve a problem after I label my model. For example, when I show known information and unknown information in a model, it helps me determine which operation to use.

How did your model help you see that this problem has multiple steps?

I drew a tape diagram and after I filled it in with the information from the problem, I realized I did not see what I needed to answer the question. So I made another tape diagram with the information I knew and the information I needed to find out next.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Sample Solutions

Use the Read–Draw–Write process to solve each problem.

1. Noah delivers packages 4 days per week. He is expected to deliver 115 packages each day that he works. This week, he delivers 48 extra packages. How many packages does Noah deliver this week?

115 × 4 = 460

460 + 48 = 508

Noah delivers 508 packages.

2. A motorcycle is 24 times as heavy as a bike. The motorcycle weighs 1,329 kilograms less than a car. The car weighs 1,521 kilograms. How many kilograms does the bike weigh?

1,521 − 1,329 = 192

192 ÷ 24 = 8

The bike weighs 8 kilograms.

3. The school librarian has $9,050 to spend on new carpet and chairs for the library. The library is 42 feet long and 37 feet wide. He buys carpet that costs $4 for each square foot. How much money does the librarian have to spend on chairs?

42 × 37 = 1,554

1,554 × 4 = 6,216

9,050 − 6,216 = 2,834

The librarian has $2,834 to spend on chairs.

4. A load of bricks is twice as heavy as a load of wood. The total weight of 4 loads of bricks and 4 loads of wood is 768 kilograms. What is the total weight of 17 loads of wood?

768 ÷ 12 = 64

17 × 64 = 1,088

The total weight of 17 loads of wood is 1,088 kilograms.

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

5. A driver earns $17 each hour. He earns a total of $1,224 in 4 weeks. A gardener works twice as many hours as the driver and earns $21 each hour. How much more money does the gardener earn than the driver in 4 weeks?

1,224 ÷ 17 = 72

72 × 2 = 144

144 × 21 = 3,024

3,024 − 1,224 = 1,800

The gardener earns $1,800 more than the driver.

6. Each fish tank at the pet store holds 662 liters of water. There are 9 tanks of goldfish and 4 tanks of angelfish. The aquarium at the zoo holds 78 times as many liters of water as all the tanks at the pet store. How many more liters of water does the aquarium hold than the fish tanks?

9 + 4 = 13

662 × 13 = 8,606

8,606 × 78 = 671,268

671,268 − 8,606 = 662,662

The aquarium holds 662,662 liters more of water than the fish tanks.

Standards

Module Content Standards

Convert like measurement units within a given measurement system.

5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Understand the place value system.

5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1 __ 10 of what it represents in the place to its left.

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Write and interpret numerical expressions.

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Standards for Mathematical Practice

MP1 Make sense of problems and persevere in solving them.

MP2 Reason abstractly and quantitatively.

MP3 Construct viable arguments and critique the reasoning of others.

MP4 Model with mathematics.

MP5 Use appropriate tools strategically.

MP6 Attend to precision.

MP7 Look for and make use of structure.

MP8 Look for and express regularity in repeated reasoning.

Achievement Descriptors: Proficiency Indicators

5.Mod1.AD1 Write whole-number numerical expressions with parentheses.

RELATED CCSSM

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Partially Proficient Proficient

Identify the effect of parentheses in whole-number numerical expressions.

Which expression is equal to 10 × 10 + 2 + 5 + 4?

A. (10 × 10) + 2 + 5 + 4

B. 10 × (10 + 2) + 5 + 4

C. 10 × (10 + 2 + 5) + 4

D. 10 × (10 + 2 + 5 + 4)

Create whole-number numerical expressions to equal a specified value.

Kayla forgot to write parentheses in her number sentence. Insert parentheses to make Kayla’s number sentence true.

10 × 10 + 2 + 5 + 4 = 129

Highly Proficient

5.Mod1.AD2 Evaluate whole-number numerical expressions with parentheses.

RELATED CCSSM

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Partially Proficient Proficient

Evaluate whole-number numerical expressions with a single set of parentheses.

Evaluate.

17 × (18 + 2)

Evaluate whole-number numerical expressions with two sets of non-nested parentheses.

Evaluate.

(8 − 1) × (16 + 7)

Highly Proficient

5.Mod1.AD3 Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions.

RELATED CCSSM

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Partially Proficient Proficient Highly Proficient

Translate between whole-number numerical expressions and verbal descriptions.

Write the phrase as a numerical expression.

Three times as much as the difference of nineteen and two.

Write whole-number numerical expressions to represent context-based verbal descriptions.

Write a numerical expression that can be used to solve the problem.

Kelly buys 28 cases of water and 40 cases of juice. Each case of water has 42 bottles and each case of juice has 24 bottles.

How many bottles did Kelly buy altogether?

Create and explain contexts that can be modeled by given whole-number numerical expressions.

Write a word problem that can be solved by evaluating the expression shown.

28 × 42 + 40 × 24

5.Mod1.AD4 Compare the effect of each number and operation on the value of a whole-number numerical expression.

RELATED CCSSM

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Partially Proficient Proficient Highly Proficient

Compare the values of two expressions that have at most two operations with whole numbers and at most one set of parentheses without evaluating.

Compare the expressions by using >, =, or <

65 × (63 + 39) 37 × (63 + 39)

Compare the values of two expressions that have at least two operations with whole numbers or multiple sets of parentheses without evaluating.

Compare the expressions by using >, =, or <

(2 + 6) × (63 + 39) (1 + 3) × (63 + 39)

Justify comparisons of two different expressions without evaluating.

Explain how you can tell which expression is greater without evaluating.

(2 + 6) × (63 + 39) (1 + 3) × (63 + 39)

RELATED CCSSM

5.NBT Number and Operations in Base Ten

Partially Proficient Proficient Highly Proficient

Identify the relevant operation in a context.

Mr. Evans has 4,724 comic books. He buys boxes to put them in. Each box holds 75 comic books. What is the fewest number of boxes Mr. Evans needs to hold all his comic books?

Which expression represents the problem?

A. 4,724 + 75

B. 4,724 − 75

C. 4,724 × 75

D. 4,724 ÷ 75

Solve real-world and mathematical problems that involve no more than two operations with multi-digit whole numbers.

Eddie mixes 14 cans of paint. Each can has 128 ounces of white paint and 32 ounces of green paint. How many ounces of paint does Eddie mix altogether?

Solve real-world and mathematical problems that involve three or more operations with multi-digit whole numbers.

Riley has 1,361 grams of chocolate chips, twice as many grams of raisins as chocolate chips, and 4,536 grams of nuts to make bags of trail mix for 82 people. Approximately how many grams of trail mix will each person get?

5.Mod1.AD5 Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers.

5.Mod1.AD6 Explain the relationship between digits in multi-digit whole numbers.

RELATED CCSSM

5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1 10 of what it represents in the place to its left.

Partially Proficient Proficient

Identify whole numbers where the value of a given digit is the value of the same digit in another whole number either multiplied by 10 or divided by 10.

When you divide 26,540 by 10, the digit 5 in the quotient has the same value as the digit 5 in which number shown?

A. 145

B. 1,450

C. 4,510

D. 5,410

Explain the relationship between a digit in one place and the same digit in an adjacent place in the same or different numbers.

Use the number shown to answer part A and part B.

7,6 6 4, 948

Part A

Write a number in each blank to correctly complete each statement.

The value of the underlined 6 is .

The value of the boxed 6 is .

Part B

Circle one word from A and one word from B to make the statement true.

The (A) 6 represents 10 times as much as the (B) 6.

Highly Proficient

Explain the relationship between a digit in a given place and the same digit in any other place in the same or different numbers.

Use the number shown to answer part A and part B. 724,766

Part A

Describe the value of each 7.

Part B

Describe the relationship between the values of each 7.

5.Mod1.AD7 Explain the effect of multiplying and dividing whole numbers by powers of 10.

RELATED CCSSM

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Partially Proficient Proficient

Multiply and divide whole numbers by powers of 10. Evaluate.

54,000 ÷ 102

Explain the effect of multiplying and dividing whole numbers by powers of 10.

How many zeros are in the product shown? Explain how you know.

54 × 102

Highly Proficient

5.Mod1.AD8 Express whole-number powers of 10 in exponential form, standard form, and as repeated multiplication.

RELATED CCSSM

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Partially Proficient Proficient

Express whole-number powers of 10 in exponential form, standard form, and as repeated multiplication. Write each number in exponential form.

10 × 10 × 10 =

10,000 =

Highly Proficient

5.Mod1.AD9 Multiply two multi-digit whole numbers by using the standard algorithm.

RELATED CCSSM

5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Partially Proficient Proficient

Multiply two multi-digit whole numbers by using familiar strategies.

Complete the expressions to determine the product.

669 × 52 = ( × ) + ( × )

Multiply two multi-digit whole numbers by using the standard algorithm.

Multiply.

106 × 973

Highly Proficient

Analyze representations of the standard algorithm for the multiplication of two multi-digit whole numbers.

Explain why the work shown is incorrect.

5.Mod1.AD10 Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

RELATED CCSSM

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Partially Proficient Proficient Highly Proficient

Divide multiples of 100 with up to four-digits by two-digit multiples of 10 .

Divide.

2,400 ÷ 20

Divide whole-number dividends with up to four digits by whole-number divisors with up to two digits.

Divide.

2,415 ÷ 21

Analyze and compare representations of division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

Miss Baker gives the class the problem shown.

372 ÷ 12

Sasha and Yuna find the quotient differently. Who is correct? Explain the mistake the other person made.

Sasha’s Work:

12 x 10 = 120

372 – 120 = 252

252 – 120 = 132

132 – 120 = 12

12 ÷ 12 = 1

372 ÷ 12 = 31

Yuna’s Work:

12 x 30 = 360

37 2 – 360 = 12

12 x 1 = 12

37 2 ÷ 12 = 301

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models.

RELATED CCSSM

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Partially Proficient Proficient Highly Proficient

Determine the quotient for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using a provided model.

Use the model shown to help you divide.

1,540 ÷ 14 14 1,400 140

The quotient is .

Create models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

Use the expression to answer part A and part B.

4,102 ÷ 14

Interpret models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

What values could be represented by the letters in the model? Explain your thinking.

Part A

Draw a model for the expression.

Part B

Use your model to determine the quotient and remainder.

5.Mod1.AD12 Convert among whole-number amounts within the metric measurement system to solve problems.

RELATED CCSSM

5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Partially Proficient Proficient

Convert among whole-number amounts within the metric measurement system.

Rename each measurement.

34 m = cm

79 km = m

Convert among whole-number amounts within the metric measurement system to solve real-world and mathematical problems.

Sasha has 3 kilograms 14 grams of walnuts and 1 kilogram 53 grams of macadamia nuts. How many grams of walnuts and macadamia nuts does Sasha have altogether?

Highly Proficient

Terminology

The following terms are critical to the work of grade 5 module 1. This resource groups terms into categories called New, Familiar, and Academic Verbs. The lessons in this module incorporate terminology with the expectation that students work toward applying it during discussions and in writing.

Items in the New category are discipline-specific words that are introduced to students in this module. These items include the definition, description, or illustration as it is presented to students. At times, this resource also includes italicized language for teachers that expands on the wording used with students.

Items in the Familiar category are discipline-specific words introduced in prior modules or in previous grade levels.

Items in the Academic Verbs category are high-utility terms that are used across disciplines. These terms come from a list of academic verbs that the curriculum strategically introduces at this grade level.

New centigram

A centigram is a metric unit for measuring weight. A gram is 100 centigrams. A large ant weighs about 1 cg. (Lesson 5)

centiliter

A centiliter is a metric unit for measuring capacity or liquid volume. A liter is 100 centiliters. (Lesson 5)

dividend

In a division expression, the number being divided by another number is the dividend. For example, in the expression 18 ÷ 3, the dividend is 18. (Lesson 12)

exponent

An exponent represents how many times the same number is used as a factor. (Lesson 3)

exponential form

A number with an exponent is in exponential form. (Lesson 3)

kiloliter

A kiloliter is a metric unit for measuring capacity or liquid volume. A kiloliter is 1,000 liters. (Lesson 5)

milligram

A milligram is a metric unit for measuring weight. A gram is 1,000 milligrams. A small feather weighs about 1 mg. (Lesson 5)

millimeter

A millimeter is a metric unit for measuring length. A meter is 1,000 millimeters. (Lesson 5)

power of 10

A number that can be written as a product of 10s, or as a 10 with an exponent, is a power of 10. 100 is a power of 10. (Lesson 3)

partial

Math Past

Pebbles in the Sand

What is the Ethiopian multiplication method? Why does it work? Is this method exclusive to Ethiopia?

In the 1900s, an Austrian colonel visiting a remote part of Ethiopia wanted to buy seven bulls. The cost of one bull was 22 Maria Theresa dollars, but no one in the village could figure out the total cost of all seven bulls. As the story is told in the book Excursions in Number Theory by C.S. Ogilvy and J. T. Anderson, the transaction involved pebbles, a priest, and many holes in the sand. Ask students to perform their own calculation for the total cost of the bulls by using a method with which they are familiar. Then instruct students to keep their answers for later.

To figure out the cost of the seven bulls, the local priest and his helper were called in to assist. They dug several small holes in the ground, arranged in two columns. They called the holes houses. In the first house of the first column, they placed 22 pebbles—the price of one bull. In the first house in the second column, they placed 7 pebbles—one for each bull.

The first column halved the pebbles, so 22 pebbles in the first house led to 11 pebbles in the house below it, which we might think would lead to 5 1 _ 2 pebbles in the house below that. However,

odd numbers such as 11 could not be equally halved into two groups because there could not be half a pebble! When a fraction arose, the helper would ignore the fractional amount and keep the whole number. So 5 pebbles were placed in the next house, 2 pebbles in the house below (again, the fractional part was ignored), and then 1 pebble in the house below that. The helper stopped halving when there was only 1 pebble left in the last house.

Working down the right column of houses, which was used for doubling, the helper placed 14 pebbles in the second house, 28 pebbles in the third, and then 56 pebbles, and so on, always doubling the count of pebbles placed in the previous house. The helper finished doubling when the number of houses matched the number of houses in the halving column.

7 × 22

At this point, the priest checked the halving column to see which houses had an even number of pebbles and which houses had

an odd number; an even number was considered evil, and an odd number was considered good. If any evil houses were discovered in the halving column, the pebbles were thrown out and not counted. The pebbles in the corresponding doubling column were also thrown out. All remaining pebbles in the doubling column were counted, resulting in the final answer. In this case, the 22 and 2 in the halving column were thrown out (they are even), along with the corresponding 7 and 56 in the doubling column. The priest added the remaining numbers in the doubling column and found that the total cost of the seven bulls was 154 Maria Theresa dollars. Did students find the same total with their own calculation methods?

Perhaps designating the even numbers as evil had to do with the spiritual nature of the priest who performed the calculations. Interestingly, the algorithm does not work if the odd houses in the halving column are thrown out instead of the even houses. The Ethiopian multiplication method utilizes three operations: pairing off (arranging the corresponding numbers in a chart), halving, and doubling. Ask students to calculate the following products by using this multiplication method. Encourage them to use the story to guide their steps.

1. 4 × 8 2. 13 × 5 3. 35 × 20

• Put one factor in each column. (Thanks to the commutative property of multiplication, it doesn’t matter which factor is halved and which is doubled.)

• In the halving column, repeatedly divide by 2 (and ignore any remainders) until only the number 1 remains.

• In the doubling column, repeatedly multiply by 2 until both columns have the same number of rows filled in.

• Cross out any rows that have an even number in the halving column.

• Add the remaining numbers in the doubling column.

Ask students to pick one of the three practice problems and multiply again, but this time reverse the order of the factors. The answer will still be the same, though sometimes it will take longer to solve the problem when the factors are reversed.

But why is it that the Ethiopian multiplication method works so well? In a nutshell, repeatedly halving the multiplier (the number that tells us how many times to multiply), and then crossing off the even houses in the halving column, gives us a nice way to break apart the multiplier. For example, it tells us to break 22 into 2 + 4 + 16.

To help see how 22 is broken apart, add a third column to the table used to calculate the cost of the bulls. Start with 1 and double it. Continue doubling until the third column has the same number of rows as the halving column. Cross off any numbers that align with the numbers crossed off in the halving column.

After the rows have been crossed off, the numbers 2, 4, and 16 remain. By no coincidence, these numbers sum to the original multiplier: 22.

The process can be represented by the following equation:

The fourth line adds everything up and contains the numbers from the doubling column.

Though the story about using this method to find the cost of the bulls took place in the 1900s, early versions of this multiplication technique also appear in an ancient text from Egypt called the Rhind papyrus (also known as the Ahmes papyrus), dating back to 1650 BCE. This method was also particularly popular in Russia. In fact, this technique is also known as the Russian multiplication method. Whatever you call it, the method is a handy way to multiply whole numbers. Just don’t insist on calculating your next big purchase with a bag full of pebbles!

The second line of the equation is true thanks to the distributive property. The next line shows that 2, 4, and 16 can be rewritten as doubling (i.e., repeated multiplication by 2). The 7, is doubled (7 × 2) doubled twice (7 × 4), and then doubled four times (7 × 16).

Materials

The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher.

3 Blank paper, sheets

5 Chart paper, sheets

25 Dry-erase markers

8 Envelopes

24 Learn books

9 Meter sticks, wood

Visit http://eurmath.link/materials to learn more.

25 Pencils

25 Personal whiteboards

25 Personal whiteboard erasers

1 Projection device

1 Teach book

1 Teacher computer or device

Please see lesson 1 for a list of organizational tools (cups, rubber bands, graph paper, etc.) suggested for the counting collection.

Works Cited

Boaler, Jo, Jen Munsen, and Cathy Williams. Mindset Mathematics: Visualizing and Investigating Big Ideas: Grade 3. San Francisco, CA: Jossey-Bass, 2018.

Bogomolny, Alexander. Rhind Papyrus. Interactive Mathematics Miscellany and Puzzles. 2018. https://www.cut -the-knot.org/.

Carpenter, Thomas P., Megan L. Franke, and Linda Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann, 2003.

Carpenter, Thomas P., Megan L. Franke, Nicholas C. Johnson, Angela C. Turrou, and Anita A. Wager. Young Children’s Mathematics: Cognitively Guided Instruction in Early Childhood Education. Portsmouth, NH: Heinemann, 2017.

CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018.

Clements, Douglas H. and Julie Sarama. Learning and Teaching Early Math: The Learning Trajectories Approach. New York: Routledge, 2014.

Common Core Standards Writing Team. Progressions for the Common Core State Standards in Mathematics. Tucson, AZ: Institute for Mathematics and Education, University of Arizona, 2011–2015. https://www.math.arizona .edu/~ime/progressions/.

Danielson, Christopher. Which One Doesn’t Belong?: A Teacher’s Guide. Portland, ME: Stenhouse, 2016.

Danielson, Christopher. Which One Doesn’t Belong?: Playing with Shapes. Watertown, MA: Charlesbridge, 2019.

Empson, Susan B. and Linda Levi. Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann, 2011.

Flynn, Mike. Beyond Answers: Exploring Mathematical Practices with Young Children. Portsmouth, NH: Stenhouse, 2017.

Fosnot, Catherine Twomey, and Maarten Dolk. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann, 2001.

Franke, Megan L., Elham Kazemi, and Angela Chan Turrou. Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom. Portsmouth, NH: Stenhouse, 2018.

Hattie, John, Douglas Fisher, and Nancy Frey. Visible Learning for Mathematics: What Works Best to Optimize Student Learning. Thousand Oaks, CA: Corwin Mathematics, 2017.

Hayes, Sebastian. “Russian Peasant Multiplication.” Origins of Mathematics. Retrieved from Mathematical Association of America website: https://www.maa.org/press /periodicals/convergence/russian-multiplication -microprocessors-and-leibniz, 2012.

Huinker, DeAnn and Victoria Bill. Taking Action: Implementing Effective Mathematics Teaching Practices. Kindergarten–Grade 5, edited by Margaret Smith. Reston, VA: National Council of Teachers of Mathematics, 2017.

Kelemanik, Grace, Amy Lucenta, Susan Janssen Creighton, and Magdalene Lampert. Routines for Reasoning: Fostering the Mathematical Practices in All Students. Portsmouth, NH: Heinemann, 2016.

Kolpas, Sid. “Russian Multiplication, Microprocessors, and Leibniz.” Mathematical Association of America. Accessed March 31, 2021. https://originsofmathematics.com/2012/01/26 /russian-peasant-multiplication/.

Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New York: Routledge, 2010.

National Council for Teachers of Mathematics, Developing an Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3–5. Reston, VA: National Council for Teachers of Mathematics, 2011.

National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010.

Ogilvy, C. Stanley and John T. Anderson. Excursions in Number Theory. New York: Oxford University Press, 1966.

Parker, Thomas and Scott Baldridge. Elementary Mathematics for Teachers. Okemos, MI: Sefton-Ash, 2004.

Shumway, Jessica F. Number Sense Routines: Building

Mathematical Understanding Every Day in Grades 3–5. Portland, ME: Stenhouse Publishing, 2018.

Smith, Margaret S. and Mary K. Stein. 5 Practices for Orchestrating Productive Mathematics Discussions, 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 2018.

Smith, Margaret S., Victoria Bill, and Miriam Gamoran Sherin. The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your Elementary Classroom, 2nd ed. Thousand Oaks, CA: Corwin Mathematics; Reston, VA: National Council of Teachers of Mathematics, 2020.

Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. New York: Pearson, 2004.

Van de Walle, John A., Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3–5, 3rd ed. New York: Pearson, 2018.

Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics

Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: http://ell.stanford.edu/content/mathematics-resources -additional-resources, 2017.

Credits

Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module.

Common Core State Standards for Mathematics © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

All United States currency images Courtesy the United States Mint and the National Numismatic Collection, National Museum of American History.

For a complete list of credits, visit http://eurmath.link /media-credits.

Cover, Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musee des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/Art Resource, NY.; pages 25, 39, 45, 50, 56, 57, GAlexS/Shutterstock.com; page 132, Brovko Serhii/Shutterstock.com; page 434, Shane Dial/Shutterstock.com; All other images are the property of Great Minds.

Acknowledgments

Kelly Alsup, Adam Baker, Agnes P. Bannigan, Reshma P Bell, Joseph T. Brennan, Dawn Burns, Amanda H. Carter, David Choukalas, Mary Christensen-Cooper, Cheri DeBusk, Lauren DelFavero, Jill Diniz, Mary Drayer, Karen Eckberg, Melissa Elias, Danielle A Esposito, Janice Fan, Scott Farrar, Krysta Gibbs, January Gordon, Torrie K. Guzzetta, Kimberly Hager, Karen Hall, Eddie Hampton, Andrea Hart, Stefanie Hassan, Tiffany Hill, Christine Hopkinson, Rachel Hylton, Travis Jones, Laura Khalil, Raena King, Jennifer Koepp Neeley, Emily Koesters, Liz Krisher, Leticia Lemus, Marie Libassi-Behr, Courtney Lowe, Sonia Mabry, Bobbe Maier, Ben McCarty, Maureen McNamara Jones, Pat Mohr, Bruce Myers, Marya Myers, Kati O’Neill, Darion Pack, Geoff Patterson, Victoria Peacock, Maximilian Peiler-Burrows, Brian Petras, April Picard, Marlene Pineda, DesLey V. Plaisance, Lora Podgorny, Janae Pritchett, Elizabeth Re, Meri Robie-Craven, Deborah Schluben, Michael Short, Erika Silva, Jessica Sims, Heidi Strate, Theresa Streeter, James Tanton, Cathy Terwilliger, Rafael Vélez, Jessica Vialva, Allison Witcraft, Jackie Wolford, Caroline Yang, Jill Zintsmaster

Trevor Barnes, Brianna Bemel, Lisa Buckley, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe

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What does this painting have to do with math?

Color and music fascinated Wassily Kandinsky, an abstract painter and trained musician in piano and cello. Some of his paintings appear to be “composed” in a way that helps us see the art as a musical composition. In math, we compose and decompose numbers to help us become more familiar with the number system. When you look at a number, can you see the parts that make up the total?

On the cover

Thirteen Rectangles, 1930

Wassily Kandinsky, Russian, 1866–1944

Oil on cardboard

Musée des Beaux-Arts, Nantes, France

Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musée des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/ Art Resource, NY

Module 1

Place Value Concepts for Multiplication and Division with Whole Numbers

Module 2

Addition and Subtraction with Fractions

Module 3

Multiplication and Division with Fractions

Module 4

Place Value Concepts for Decimal Operations

Module 5

Addition and Multiplication with Area and Volume

Module 6

Foundations to Geometry in the Coordinate Plane

ISBN 978-1-64497-179-6