SC 6th Grade (Volume 2) by acceleratelearning

VOLUME 2

Grade 6 SOUTH

CAROLINA

ISBN: 979-8-893539-04-2

Math Nation 6-8 was originally developed by Illustrative Mathematics®, and is copyright 2019 by Illustrative Mathematics. It is licensed under Creative Commons Attribution 4.0 International License (CCBY 4.0).

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Unit 6, Lesson 1: Diagrams and Equations

Warm-Up: Exploring Diagrams

1. Two diagrams are shown. One represents 2 + 5 = 7. The other represents 5 ⋅ 2 = 10. Which is which? Label the length of each diagram.

2. A diagram is shown.

Using the diagram, identify:

a. One thing that must be true.

b. One thing that could be true or false.

c. One thing that cannot possibly be true.

Exploration Activity: Match Equations and Tape Diagrams

Here are two tape diagrams. Match each equation to one of the tape diagrams.

1. 4 + �� = 12

2. 12 ÷ 4 = ��

3. 4 ⋅ �� = 12

4. 12 = 4 + ��

5. 12 – �� = 4

6. 12 = 4 ⋅ �� 7. 12 – 4 = �� 8. �� = 12 – 4

9. �� + �� + �� + �� = 12

Collaborative Activity: Draw Tape Diagrams for Equations

For each equation, draw a diagram and ﬁnd the value of the unknown that makes the equation true.

18 = 3 + ��

18 = 3 ⋅ ��

Collaborative Activity: Match Equations and Hanger Diagrams

1. Match each hanger to an equation. Complete the equation by writing ��, ��, ��, or �� in the empty box.

a. + 3 = 6

b. 3 ⋅ = 6

c. 6 = + 1

d. 6 = 3 ⋅

2. Find a solution to each equation. Use the hangers to explain what each solution means.

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Lesson Summary

Diagrams can help with understanding relationships between quantities and how operations describe those relationships.

One such diagram is a tape diagram.

• Diagram A has 3 parts that sum to 21. Each part is labeled with the same letter, indicating the 3 parts are equal. The equations shown represent diagram A.

Notice that the number 3 is not seen in the diagram. The 3 comes from counting 3 boxes representing 3 equal parts in 21. Either the diagram or any of the equations can be used to reason that �� = 7.

• Diagram B has 2 parts that sum to 21. The equations shown represent diagram B. �� + 3 = 21 �� = 21 – 3 3 = 21 – ��

Either the diagram or any of the equations can be used to reason that �� = 18.

Another diagram that can be used to understand relationships in equations is a hanger diagram.

A hanger diagram stays balanced when the weights on both sides are equal. The weights can be changed and the hanger will stay balanced as long as both sides are changed in the same way. For example, adding 2 pounds (lb.) to each side of a balanced hanger will keep it balanced. Removing half of the weight from each side will also keep it balanced. This way of thinking can be used to ﬁnd solutions to equations. Instead of checking diﬀerent values, think about subtracting the same amount from each side or dividing each side by the same number.

• Diagram A can be represented by the equation 3�� = 12. When 12 breaks into 3 equal parts, each part will have the same weight as a block with an ��. Splitting each side of the hanger into 3 equal parts is the same as dividing each side of the equation by 3, resulting in the solution �� = 4.

• Diagram B can be represented by the equation 12 = �� + 5. A weight of 5 can be removed from each side of the hanger to keep it in balance. Removing 5 from each side of the hanger is the same as subtracting 5 from each side of the equation, resulting in the solution �� = 7.

Practice Problems

1. Diego is trying to ﬁnd the value of �� in 5 ⋅ �� = 35. He draws this diagram but is not certain how to proceed.

a. Complete the tape diagram so it represents the equation 5 ⋅ �� = 35.

b. Find the value of ��.

2. Match each equation to one of the two tape diagrams.

a. �� + 3 = 9

b. 3 ⋅ �� = 9

c. 9 = 3 ⋅ ��

d. 3 + �� = 9

e. �� = 9 – 3

f. �� = 9 ÷ 3

g. �� + �� + �� = 9

3. Select all the equations that represent the hanger.

□ �� + �� + �� = 1 + 1 + 1 + 1 + 1 + 1

□ �� ⋅ �� ⋅�� = 6

□ 3�� = 6

�� + 3 = 6

4. Write an equation to represent each hanger.

5. Andre says that �� is 7 because he can move the two 1s with the �� to the other side.

Do you agree with Andre? Explain your reasoning.

Review Problem

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6. The daily recommended allowance of calcium for a sixth grader is 1,200 milligrams (mg). One cup of milk has 25% of the recommended daily allowance of calcium. How many mg of calcium are in a cup of milk? If you get stuck, consider using the double number line.

Unit 6, Lesson 2: Truth and Equations

Warm-Up: Three Letters

1. The equation �� + �� = �� could be true or false.

a. If �� is 3, �� is 4, and �� is 5, is the equation true or false?

b. Find new values of ��, ��, and �� that make the equation true.

c. Find new values of ��, ��, and �� that make the equation false.

2. The equation �� ⋅ �� = �� could be true or false.

a. If �� is 3, �� is 4, and �� is 12, is the equation true or false?

b. Find new values of ��, ��, and �� that make the equation true.

c. Find new values of ��, ��, and �� that make the equation false.

Exploration Activity: Story Time

Here are three situations and six equations. Which equation best represents each situation? If you get stuck, consider drawing a diagram.

1. After Elena ran 5 miles (mi.) on Friday, she had run a total of 20 mi. for the week. She ran �� mi. before Friday.

2. Andre’s school has 20 clubs, which is ﬁve times as many as his cousin’s school. His cousin’s school has �� clubs.

3. Jada volunteers at the animal shelter. She divided 5 cups (c.) of cat food equally to feed 20 cats. Each cat received �� c. of food.

+ 5 = 20

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+ 20 = 5

= 20 + 5

= 20

= 5

Collaborative Activity: Using Structure to Find Solutions

Several equations that contain a variable are shown alongside a list of values. Think about what each equation means, and ﬁnd a solution in the list of values. If you get stuck, consider drawing a diagram. Be prepared to explain why your solution is correct.

Equation

List of Values

Lesson Summary

An equation can be true or false. An example of a true equation is 7 + 1 = 4 ⋅ 2. An example of a false equation is 7 + 1 = 9.

An equation can have a letter in it, such as �� + 1 = 8. This equation is false if �� is 3, because 3 + 1 does not equal 8. This equation is true if �� is 7, because 7 + 1 = 8.

In �� + 1 = 8, the variable is ��. In �� + 1 = 8, the solution is 7.

A variable is a letter that represents a number.

A solution to an equation is a number that can be used in place of the variable to make the equation true.

When a number is written next to a variable, the number and the variable are being multiplied. For example, 7�� = 21 means the same thing as 7 ⋅ �� = 21. In this case, 7 is called the coeﬃcient of ��.

A coeﬃcient is the number or constant that multiplies a variable in an algebraic expression.

If no coefficient is written, the coefficient is 1. For example, in the equation �� + 3 = 5, the coefficient of �� is 1.

Practice Problems

1. Select all the equations that are true.

5 + 0 = 0

2. Mai's water bottle had 24 ounces (oz.) in it. After she drank �� oz. of water, there were 10 oz. left. Select all the equations that represent this situation.

□ 24 ÷ 10 = ��

□ 24 + 10 = ��

□ 24 – 10 = ��

□ �� + 10 = 24

□ 10�� = 24

3. Priya has 5 pencils, each �� inches (in.) in length. When she lines up the pencils end to end, they measure 34.5 in. Select all the equations that represent this situation.

□ 5 + �� = 34.5

□ 5�� = 34.5

□ 34.5 ÷ 5 = ��

□ 34.5 – 5 = ��

□ �� = (34.5) ⋅ 5

4. Match each equation with a solution from the list of values.

1 8 �� = 3

�� ÷ 8 5 = 1

5. The daily recommended allowance of vitamin C for a sixth grader is 45 milligrams (mg). 1 orange has about 75% of the recommended daily allowance of vitamin C. How many mg are in 1 orange? If you get stuck, consider using the double number line.

6. There are 90 kids in the band. 20% of the kids own their own instruments, and the rest rent them.

a. How many kids own their own instruments?

b. How many kids rent instruments?

c. What percentage of kids rent their instruments?

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Unit 6, Lesson 3: Practice Solving Equations and Representing Situations with Equations

Warm-Up:

Subtracting

from 5 Find the value of each expression mentally.

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Collaborative Activity: Solving Equations Practice

With your partner, determine who will be partner A and who will be partner B. Solve the equations in your column only.

Check in with your partner after you ﬁnish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to ﬁnd the error and correct it.

Collaborative Activity: Choosing Equations to Match Situations

Four scenarios are given. In each scenario, identify the equation which cannot be used to represent the situation. If you get stuck, consider drawing a diagram. Then, ﬁnd the solution for each situation.

1. Olivia has 8 fewer books than Mia. If Mia has 26 books, how many books does Olivia have?

Which equation cannot represent the situation?

A. 26 – �� = 8

B. �� = 26 +8

C. �� + 8 = 26

What is the solution?

D. 26 – 8 = �� = ___________

2. A coach formed teams of 8 from all the players in a soccer league. There are 14 teams. How many players are in the league?

Which equation cannot represent the situation?

A. �� = 14 ÷ 8

B. �� 8 = 14

C. 1 8 �� = 14

What is the solution?

D. �� = 14 ⋅ 8 �� = ___________

3. Kylie scored 223 more points in a computer game than Jason did. If Kylie scored 409 points, how many points did Jason score?

Which equation cannot represent the situation?

A. 223 = 409 – ��

B. 409 – 223 = ��

C. 409 + 223 = ��

What is the solution?

D. 409 = 223 + �� = ___________

4. Chris ran 28 miles (mi.) last week, which was 4 times as far as Morgan ran. How far did Morgan run?

Which equation cannot represent the situation?

A. 4�� = 28

B. �� = 1 4 ⋅ 28

C. �� = 28 ÷ 4

What is the solution?

D. �� = 4 ⋅ 28 �� = ___________

Lesson Summary

Writing and solving equations can be helpful when answering questions about situations.

Suppose a scientist has 13.68 liters (L) of acid and needs 16.05 L for an experiment. How many more L of acid does she need for the experiment?

• This situation can be represented with the equation 13.68 + �� = 16.05.

• When working with hanger diagrams, it was possible to ﬁnd the solution by subtracting 13.68 from each side. This results in new equivalent equations that also represent the situation, as shown.

�� =16.05 – 13.68

�� = 2.37

Equations that have the exact same solutions are equivalent equations.

Finding a solution this way leads to a variable on 1 side of the equal sign and a known value on the other side. The solution, 2.37, is easily identiﬁed from an equation with a letter on 1 side and a known value on the other. Solutions are often written in this way.

Suppose a food pantry takes a 54 pound (lb.) bag of rice and splits it into portions that each weigh 3 4 of a lb. How many portions can they make from the bag?

• This situation can be represented with the equation 3 4 �� = 54.

• The value of �� can be found by dividing each side of the equation by 3 4 . This results in new equivalent equations that also represent the same situation.

�� = 54 ÷ 3 4

�� = 72

• The solution is 72 portions.

Practice Problems

1. Select all the equations that describe each situation and then ﬁnd the solution.

a. Kiran’s backpack weighs 3 lb. less than Clare’s backpack. Clare’s backpack weighs 14 lb. How much does Kiran’s backpack weigh?

□ �� + 3 = 14

□ 3�� = 14

□ �� = 14 – 3

□ �� = 14 ÷ 3

b. Each notebook contains 60 sheets of paper. Andre has 5 notebooks. How many sheets of paper do Andre’s notebooks contain?

□ �� = 60 ÷ 5

□ �� = 5 ⋅ 60

□ �� 5 = 60

□ 5�� = 60

2. Solve each equation.

2�� = 5

�� + 1.8 = 14.7

6 = 1 2 ��

3 1 4 = 1 2 + ��

e. 2.5�� = 10

Review Problems

3. For each equation, draw a tape diagram that represents the equation.

a. 3�� = 18

b. 3 + �� = 18

c. 17 – 6 = ��

4. Find the product of (21.2) ∙ (0.02).

5. For a science experiment, students need to ﬁnd 25% of 60 grams.

• Jada says, “I can ﬁnd this by calculating 1 4 of 60.”

• Andre says, “25% of 60 means 25 100 ⋅ 60.”

Do you agree with either of them? Explain your reasoning.

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Unit 6, Lesson 4: A New Way to Interpret Fractions

Warm-Up: Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

1. 0.07 = 10��

10.1 = �� + 7.2

Exploration Activity: Interpreting Fractions

Solve each equation.

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1. 35 = 7��

2. 35 = 11��

Collaborative Activity: Story Time Again

Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what the quantity �� describes, and solve the equation. If you get stuck, consider drawing a diagram.

1. 0.7 + �� = 12

2. 1 4 �� = 3 2

Lesson Summary

A fraction such as 4 5 can be thought of in a few ways.

• 4 5 is a number that can be located on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.

• 4 5 is the share that each person would have if 4 wholes were shared equally among 5 people. This means that 4 5 is the result of dividing 4 by 5.

This meaning of a fraction as a division expression can be extended to fractions whose numerators and denominators are not whole numbers. For example, 4.5 pounds (lb.) of rice divided into portions that each weigh 1.5 lb. can be represented as 4.5 1.5 , which is the same as 4.5 ÷ 1.5 = 3.

Fractions that involve non-whole numbers, written as decimals or fractions, can also be used when solving equations.

Suppose a road under construction is 5 8 ﬁnished and the length of the completed part is 7 4 miles (mi.). How long will the road be when it’s fully completed?

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• Write the equation 5 8 �� = 7 4 to represent the situation, and solve the equation. This work is shown.

• The completed road will be 2 4 5 or about 2.8 mi. long.

Practice Problems

1. Select all the expressions that equal 3.15 0.45 .

□ (3.15) ⋅ (0.45)

□ (3.15) ÷ (0.45)

□ (3.15) ⋅ 1 0.45

□ (3.15) ÷ 45 100

□ (3.15) ⋅ 100 45 □ 0.45 3.15

2. Which expressions are solutions to the equation 3 4 �� = 15? Select all that apply.

3. Solve each equation.

a. 4�� = 32

b. 4 = 32��

c. 10�� = 26

d. 26 = 100��

4. Write as many mathematical expressions or equations as you can about the image. Include a fraction, a decimal number, or a percentage in each.

5. In a lilac paint mixture, 40% of the mixture is white paint, 20% is blue, and the rest is red. There are 4 cups (c.) of blue paint used in a batch of lilac paint.

a. How many c. of white paint are used?

b. How many c. of red paint are used?

c. How many c. of lilac paint will this batch yield?

If you get stuck, consider using a tape diagram.

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6. Triangle P has a base of 12 inches (in.) and a corresponding height of 8 in. Triangle Q has a base of 15 in. and a corresponding height of 6.5 in. Which triangle has a greater area? Show your reasoning.

Unit 6, Lesson 5: Writing Algebraic Expressions

Warm-Up: Sum, Product, Quotient, or Diﬀerence

Complete each statement.

1. The sum product quotient diﬀerence of 4 and 6 is 10 because 4 + 6 = 10.

2. The sum product quotient diﬀerence of 8 and 3 is 5 because 8 – 3 = 5.

3. The sum product quotient diﬀerence of 32 and 2 is 16 because 32 2 = 16.

4. The sum product quotient diﬀerence of 7 and 9 is 63 because 7 × 9 = 63.

Guided Activity: Making Sense of Algebraic Expressions Using Models

Algebraic expressions, like numeric expressions, can include sums, diﬀerences, products, and quotients.

An algebraic expression is a mathematical statement containing numbers, operators, and at least 1 unknown value represented by a variable.

1. Complete the statement.

An expression does does not include an equal sign or inequality symbol.

Bar models can be used to connect the action of a problem to an algebraic expression.

Part-Part-Whole Bar Model

The part-part-whole model can be used to represent the following actions. Action Equation

Combining or joining

Taking away or separating

Part A + Part B = Whole

Whole – Part A = Part B

Whole – Part B = Part A

2. “ The sum of a number and three” can be shown as the length of the whole bar where one part is �� and the other part is 3.

The sum of a number and three

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Write an algebraic expression for “the sum of a number and three.”

3. “ The diﬀerence of ten and a number ” can be shown where the whole bar is 10 and one part is a number represented by a variable, ��. The diﬀerence is the other part.

The difference of ten and number x 10

Write an algebraic expression for “the diﬀerence of ten and a number.”

4. “Five subtracted from a number ” is shown with the whole bar as a number represented by a variable, ��, and one part is 5. The other part is the result of 5 being subtracted from the unknown number, ��.

The result of five subtracted from a number 5 m

Complete the statement. The algebraic expression for “ﬁve subtracted from a number, ��,” is _____________.

Addition or Subtraction Comparison Bar Model

Comparison models are much like part-part-whole models. They show how much more or less a value is than another.

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This model can be used to show how much more Value A is than Value B or how much less Value B is than Value A.

5. A comparison bar model is shown.

Eight less than a number

Write an algebraic expression for the comparison bar model.

Equal Parts–Whole Bar Model

The equal parts–whole bar model shows the whole sectioned into equal parts.

The equal parts–whole model can be used to show the actions described in the table.

Action

Equation

Repeated addition (multiplication) (Equal part) + . . . + (equal part) = whole (Equal part) × (number of parts) = whole

Equal sharing Whole ÷ (number of parts) = equal part

Partitioning Whole ÷ (equal part) = number of parts

6. “The product of nine and a number” can be shown as 9 equal parts with size of “a number” represented with the variable ��.

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Write an algebraic expression for “the product of nine and a number.”

7. “The quotient of a number and two” could be shown as “a number ” represented by the variable �� sectioned oﬀ into 2 equal parts. The quotient is the size of each part.

A number, p Quotient

What is the algebraic expression for “the quotient of a number and two”?

Collaborative Activity: Identifying Parts of an Algebraic Expression

Use terms from the word bank to complete the statements. Words may be used more than once, in singular or plural form.

Word Bank

term(s)

sum(s) diﬀerence(s) product(s) factor(s) quotient(s) variable(s) coeﬃcient(s) constant(s)

1. The result of 99 + �� is the __________, where 99 is the __________.

2. The expression 3.12 – �� contains two __________. The result of 3.12 – �� is the __________.

3. The result of �� 13.5 is the __________, where �� is the __________.

4. In the expression 5��, 5 and �� are the __________, where the result of 5�� is the __________.

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5. In the expression 1 2 ��, 1 2 is the __________ of ��, and �� is the __________.

Collaborative Activity: Relating Models and Expressions

1. Four bar models are shown.

Work with your partner to match each written description given in the table to the appropriate bar model, where the pink portion is the result of the written description.

Write the letter of the appropriate bar model. Then, write the algebraic expression that matches the word description. Some models will be used more than once.

The diﬀerence of a number and four

Four decreased by a number

Four more than a number

A number subtracted from four

A number increased by four

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Four less than a number

The sum of four and a number

2. Three bar models are shown.

Work with your partner to match each written description given in the table to the appropriate bar model, where the pink portion is the result of the written description.

Write the letter of the appropriate bar model. Then, write the algebraic expression that matches the word description. Some models will be used more than once.

Written Description

A number tripled

A number equally shared into three parts

Three times a number

The quotient of three and a number

The product of three and a number

A number divided by three

Bar Model Algebraic Expression

Bar Model H

Bar Model T

Lesson Summary

When working with algebraic expressions, it is helpful to know some key deﬁnitions. Algebraic expressions, equations, and inequalities are made up of terms.

A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together.

The expression 6.5 + 4�� has 2 terms.

• The ﬁrst term is 6.5.

• The second term is 4��.

An expression is a mathematical statement containing numerals, operators, grouping symbols, and symbols or variables for unknown values. An expression does not contain an equal sign or inequality symbol.

The expression 6.5 + 4�� contains a variable term and a constant term.

In an expression like 5�� + 2, the number 2 is called the constant term because it doesn’t change when �� changes.

In the expression 6.5 + 4��, 4�� is the variable term, and 6.5 is the constant term.

Variable terms have coeﬃcients, while constant terms do not.

• In the expression 4��, 4 is the coeﬃcient.

• In the expression ��, 1 is the coeﬃcient.

• In the expression 8��, 8 is the coeﬃcient.

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Practice Problems

For questions 1-4, sketch a bar model and write an algebraic expression for each written description.

1. The product of a number and ﬁve

Algebraic Expression Bar Model

2. The quotient of a number and twelve

Algebraic Expression

Model

3. A number doubled Algebraic Expression

Model

4. The diﬀerence of ﬁfteen and a number Algebraic Expression

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Model

5. Use the algebraic expression �� + 5 4 to answer the following questions.

a. How many terms are in the expression?

b. Identify the variable term(s), contant term(s), and coeﬃcient(s), if any.

Variable term(s): _________

Constant term(s): _________

Coeﬃcient(s): _________

Review Problem

6. Lin needs to save up $20 for a new game. How much money does she have if she has saved the following percentages of her goal. Explain your reasoning.

a. 25%

b. 75%

c. 125%

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Unit 6, Lesson 6: Writing and Evaluating Algebraic Expressions

Warm-Up: Algebra Talk: When the Unknown Is 6

Determine the value of each expression if �� is 6.

Exploration Activity: Lemonade Sales and Heights

1. Lin set up a lemonade stand. She sells the lemonade for $0.50 per cup.

a. Complete the table to show how much money she would collect if she sold each number of cups.

Lemonade Sold (Number of Cups) 12 183 ��

Money Collected (Dollars)

b. How many cups did she sell if she collected $127.50? Be prepared to explain your reasoning.

2. Elena is 59 inches (in.) tall. Some other people are taller than Elena.

a. Complete the table to show the height of each person.

Person's Height (in.)

b. If Noah is 64 3 4 in. tall, how much taller is he than Elena?

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Collaborative Activity: Building Expressions

1. Clare is 5 years older than her cousin.

a. How old would Clare be if her cousin is:

i. 10 years old?

ii. 2 years old?

iii. �� years old?

b. Clare is 12 years old. How old is Clare’s cousin?

2. Diego has 3 times as many comic books as Han.

a. How many comic books does Diego have if Han has:

i. 6 comic books?

ii. �� books?

b. Diego has 27 comic books. How many comic books does Han have?

3. Two ﬁfths of the vegetables in Priya’s garden are tomatoes.

a. How many tomatoes are there if Priya’s garden has:

i. 20 vegetables?

ii. �� vegetables?

b. Priya’s garden has 6 tomatoes. How many total vegetables are there?

4. A school paid $31.25 for each calculator.

a. If the school bought �� calculators, how much did they pay?

b. The school spent $500 on calculators. How many did the school buy?

Lesson Summary

Samari shares a birthday with her neighbor, but her neighbor is 3 years older than her. When Samari was 1, her neighbor was 4. When Samari was 9, her neighbor was 12. When Samari is 42, her neighbor will be 45.

If �� represents Samari’s age at any time, her neighbor’s age can be expressed as �� + 3.

Often, a letter such as �� or �� is used as a placeholder for a number in expressions. These letters are called variables, just like the letters used in equations. Variables make it possible to write an expression that represents a calculation even when not all of the numbers in the calculation are known.

How old will Samari be when her neighbor is 32? Since her neighbor’s age is calculated with the expression �� + 3, write the equation �� + 3 = 32. When her neighbor is 32, Samari will be 29 because �� + 3 = 32 is true when �� is 29.

Practice Problems

1. Instructions for a craft project say that the length of a piece of red ribbon should be 7 in. less than the length of a piece of blue ribbon.

a. How long is the red ribbon if the length of the blue ribbon is:

i. 10 in.?

ii. 27 in.?

iii. �� in.?

b. How long is the blue ribbon if the red ribbon is 12 in.?

2. Tyler has 3 times as many books as Mai.

a. How many books does Mai have if Tyler has:

i. 15 books?

ii. 21 books?

iii. �� books?

b. Tyler has 18 books. How many books does Mai have?

3. A bottle holds 24 ounces (oz.) of water. It has �� oz. of water in it.

a. What does 24 – �� represent in this situation?

b. Write a question about this situation that has 24 – �� for the answer.

Review Problems

4. Write an equation represented by this tape diagram using each of the following operations.

a. Addition

b. Subtraction

c. Multiplication

d. Division

5. Select all the equations that describe each situation and then ﬁnd the solution.

a. Han's house is 450 meters (m) from school. Lin’s house is 135 m closer to school. How far is Lin’s house from school?

□ �� = 450 + 135

□ �� = 450 – 135

□ �� – 135 = 450

□ �� + 135 = 450

b. Tyler's playlist has 36 songs. Noah’s playlist has one quarter as many songs as Tyler's playlist. How many songs are on Noah’s playlist?

□ �� = 4 ⋅ 36

□ �� = 36 ÷ 4

□ 4�� = 36

□ �� 4 = 36

6. You had $50. You spent 10% of the money on clothes, 20% on games, and the rest on books. How much money was spent on books?

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7. A trash bin has a capacity of 50 gallons (gal.). What percentage of its capacity is each amount? Show your reasoning.

a. 5 gal.

b. 30 gal.

c. 45 gal.

d. 100 gal.

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Unit 6, Lesson 7: Revisit Percentages

Warm-Up: Percentages

Solve each problem mentally.

1. Bottle A contains 4 ounces (oz.) of water, which is 25% of the amount of water in Bottle B. How much water is there in Bottle B?

2. Bottle C contains 150% of the water in Bottle B. How much water is there in Bottle C?

3. Bottle D contains 12 oz. of water. What percentage of the amount of water in Bottle B is this?

Guided Activity: Representing a Percentage Problem with an Equation

1. Answer each question and show your reasoning.

a. Is 60% of 400 equal to 87?

b. Is 60% of 200 equal to 87?

c. Is 60% of 120 equal to 87?

2. 60% of �� is equal to 87. Write an equation that expresses the relationship between 60%, ��, and 87. Solve your equation.

3. Write an equation to help you ﬁnd the value of each variable. Solve the equation.

a. 60% of �� is 43.2.

b. 38% of �� is 190.

Collaborative Activity: Puppies Grow Up, Revisited

1. Puppy A weighs 8 pounds (lb.), which is about 25% of its adult weight. What will be the adult weight of Puppy A?

2. Puppy B weighs 8 lb., which is about 75% of its adult weight. What will be the adult weight of Puppy B?

3. If you haven’t already, write an equation for each situation. Then, show how you could ﬁnd the adult weight of each puppy by solving the equation.

Lesson Summary

If 455 students are in school today and that number represents 70% attendance, an equation can be written to ﬁgure out how many students are enrolled at the school.

The number of students in school today can be summarized in the 2 ways shown.

• 70% of the students in the school

• 455 students

If �� represents the total number of students who are enrolled at the school, then 70% of ��, or 70 100 ��, represents the number of students who are in school today, which is 455.

Therefore, the equation 70 100 �� = 455 represents this situation. The equation can be solved as shown.

There are 650 students enrolled at the school.

In general, equations can help solve problems in which 1 amount is a percentage of another amount.

The equivalent equation 0.7�� = 455 could have also been used.

Practice Problems

1. A crew has paved 3 4 of a mile (mi.) of road. If they have completed 50% of the work, how long is the road they are paving?

2. 40% of �� is 35.

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a. Write an equation that shows the relationship of 40%, ��, and 35.

b. Use your equation to ﬁnd ��. Show your reasoning.

3. Priya has completed 9 exam questions. This is 60% of the questions on the exam.

a. Write an equation representing this situation. Explain the meaning of any variables you use.

b. How many questions are on the exam? Show your reasoning.

4. Answer each question. Show your reasoning.

a. 20% of �� is 11. What is ��?

b. 75% of �� is 12. What is ��?

c. 80% of �� is 20. What is ��?

d. 200% of �� is 18. What is ��?

Review Problems

5. For the equation 2�� – 3 = 7

a. What is the variable?

b. What is the coeﬃcient of the variable?

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c. Which of these is the solution to the equation? 2, 3, 5, 7, ��

6. Which of these is a solution to the equation 1 8 = 2 5 ⋅ ��?

A. 2 40

B. 5 16

C. 11 40

D. 17 40

7. Find the quotients.

a. 0.009 ÷ 0.001

b. 0.009 ÷ 0.002

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Unit 6, Lesson 8: Equal and Equivalent

Warm-Up: Tape Diagrams

1. Two diagrams are shown. One represents 3 + �� = 8. The other represents �� ⋅ 3 = 15.

Complete the table by writing the equation that matches each diagram in the space provided and then labeling the length of each diagram.

Equation

Diagram

Exploration Activity: Using Diagrams to Show That Expressions Are Equivalent

Here is a diagram of �� + 2 and 3�� when �� is 4. Notice that the two diagrams are lined up on their left sides.

In each of your drawings below, line up the diagrams on one side.

1. Draw a diagram of �� + 2, and a separate diagram of 3��, when �� is 3.

2. Draw a diagram of �� + 2, and a separate diagram of 3��, when �� is 2.

3. Draw a diagram of �� + 2, and a separate diagram of 3��, when �� is 1.

4. Draw a diagram of �� + 2, and a separate diagram of 3��, when �� is 0.

5. When are �� + 2 and 3�� equal? When are they not equal? Use your diagrams to explain.

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6. Draw a diagram of �� + 3, and a separate diagram of 3 + ��.

7. When are �� + 3 and 3 + �� equal? When are they not equal? Use your diagrams to explain.

Collaborative Activity: Identifying Equivalent Expressions

A list of expressions is shown. Determine any pairs of expressions that are equivalent. If you get stuck, try reasoning with diagrams.

Guided Activity: Identity and Inverse Properties of Operations

Two properties of operations that can be used to generate equivalent expressions are the identity property and the inverse property.

The identity property states that when any number, ��, is combined with an identity, the end result is ��.

Complete each statement.

1. The number 0 1 is the addition identity of �� because �� + 0 1 = ��.

2. The number 0 1 is the multiplicative identity of �� because �� × 0 1 = ��.

The inverse property states that when any number, ��, is multiplied by its inverse, or reciprocal, the end result is 1.

Complete the statement.

3. The multiplicative inverse of �� is –�� 1 �� because �� × –�� 1 �� = 1.

4. Complete the table by identifying a pair of expressions from the Collaborative Activity that illustrates each property of operations indicated.

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Lesson Summary

Diagrams showing lengths of rectangles can be used to see when expressions are equal. For example, the expressions �� + 9 and 4�� are equal when �� is 3, but are not equal for other values of ��.

Sometimes 2 expressions are equal for only 1 particular value of the variable. Other times, they seem to be equal no matter what the value of the variable.

Expressions that are always equal regardless of the value of their variable(s) are called equivalent expressions.

Equivalent expressions are always equal to each other. If the expessions have variables, they are equal whenever the same value is used for the variable in each expression.

However, it would be impossible to test every possible value of the variable. Properties of operations can be applied to determine whether expressions are equivalent. Some example properties are listed.

• �� + 3 is equivalent to 3 + �� by the commutative property of addition.

• 0 + �� is equivalent to �� by the identity property of addition.

• �� ∙ 1 �� is equivalent to 1 by the inverse property of multiplication.

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• �� + �� + �� + �� + �� is equivalent to 5 ⋅ �� because adding 5 of something is the same as multiplying it by 5.

• �� ÷ 3 is equivalent to �� ⋅ 1 3 because dividing by a number is the same as multiplying by its reciprocal.

In the coming lessons, the distributive property of multiplication over addition will be explored. This property can also be used to show that expressions are equivalent.

Practice Problems

a. Draw a diagram of �� + 3 and a diagram of 2�� when �� is 1.

b. Draw a diagram of �� + 3 and of 2�� when �� is 2.

c. Draw a diagram of �� + 3 and of 2�� when �� is 3.

d. Draw a diagram of �� + 3 and of 2�� when �� is 4.

e. When are �� + 3 and 2�� equal? When are they not equal? Use your diagrams to explain.

a. Do 4�� and 15 + �� have the same value when �� is 5?

b. Are 4�� and 15 + �� equivalent expressions? Explain your reasoning.

a. Check that 2�� + �� and 3�� have the same value when �� is 1, 2, and 3.

b. Do 2�� + �� and 3�� have the same value for all values of ��? Explain your reasoning.

c. Are 2�� + �� and 3�� equivalent expressions?

4. Complete each statement.

a. The identity property of mulitplication justiﬁes that �� is equivalent to ________.

b. The inverse property of multiplication justiﬁes that �� · _____ = 1.

c. The identity property of addition justiﬁes that �� is equivalent to ________.

Review Problems

5. 80% of �� is equal to 100.

a. Write an equation that shows the relationship of 80%, ��, and 100.

b. Use your equation to ﬁnd ��.

6. For each story problem, write an equation to represent the problem and then solve the equation. Be sure to explain the meaning of any variables you use.

a. Jada’s dog was 5 1 2 inches (in.) tall when it was a puppy. Now her dog is 14 1 2 in. taller than that. How tall is Jada’s dog now?

b. Lin picked 9 3 4 pounds (lb.) of apples, which was 3 times the weight of the apples Andre picked. How many lb. of apples did Andre pick?

7. Calculate 141.75 ÷ 2.5 using a method of your choice. Show or explain your reasoning.

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Unit 6, Lesson 9: The Distributive Property – Part 1

Warm-Up: Ways to Represent Area of a Rectangle

1. Select all of the expressions that represent the area of the large, outer rectangle in the ﬁgure shown. Explain your reasoning.

Exploration Activity: Distributive Practice

Complete the table. If you get stuck, skip an entry and come back to it, or consider drawing a diagram of two rectangles that share a side.

Exploration Activity: Partitioned Rectangles When Lengths Are Unknown

1. Here are two rectangles. The length and width of one rectangle are 8 and 5. The width of the other rectangle is 5, but its length is unknown so we labeled it ��. Write an expression for the sum of the areas of the two rectangles.

2. The two rectangles can be composed into one larger rectangle as shown. What are the width and length of the new, large rectangle?

3. Write an expression for the total area of the large rectangle as the product of its width and its length.

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Collaborative Activity: Areas of Partitioned Rectangles

For each rectangle, write expressions for the length and width and two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.

Area as a Product of Width Times Length

Area as a Sum of the Areas of the Smaller Rectangles

Rectangle Width Length

Lesson Summary

When performing calculations mentally, it’s common to break up the calculations to make them easier to do.

For example, when grocery shopping, to mentally calculate how much it will cost to buy 5 cans of beans at 79 cents per can, the mental calculation might be as shown to the right.

In general, when multiplying 2 numbers, or factors, 1 of the factors can be broken into parts. Multiply each part by the other factor, and then add the partial products. The result will be the same as the product of the 2 original factors.

When 1 factor is broken up and the parts are multipled, the distributive property of multiplication over addition is being used.

Distributive property of multiplication over addition �� × (�� + ��) = (�� × ��) + (�� × ��)

The distributive property also works with subtraction. Another way of ﬁnding 5 ⋅ 79 is shown.

Rectangles can be used to illustrate the distributive property. A rectangle composed of 2 smaller rectangles, A and B, is shown.

The following information can be observed about the area of the large rectangle.

• One side length of the large rectangle is 3, and the other is (2 + ��), so its area is 3(2 + ��).

• Since the large rectangle can be decomposed into 2 smaller rectangles, A and B, with no overlap, the area of the large rectangle is also the sum of the areas of rectangles A and B: 3(2) + 3(��) or 6 + 3��.

• Since both expressions represent the area of the large rectangle, they are equivalent to each other. 3(2 + ��) is equivalent to 6 + 3��.

This shows that multiplying 3 by the quantity (2 + ��) is equivalent to multiplying 3 by 2 and then multiplying 3 by �� and adding the two products. This relationship is an example of the distributive property.

3(2 + ��) = 3 ⋅ 2 + 3 ⋅ ��

Practice Problems

1. Draw and label diagrams that show these two methods for calculating 19 ⋅ 50.

a. First ﬁnd 10 ⋅ 50 and then add 9 ⋅ 50.

b. First ﬁnd 20 ⋅ 50 and then take away 50.

2. Complete each calculation using the distributive property.

a. 98 ⋅ 24

(100 – 2) ⋅ 24

b. 21 ⋅ 15

(20 + 1) ⋅ 15

c. 0.51 ⋅ 40

(0.5 + 0.01) ⋅ 40

3. Here is a rectangle.

a. Explain why the area of the large rectangle is 2�� + 3�� + 4��.

b. Explain why the area of the large rectangle is (2 + 3 + 4)��.

4. Is the area of the shaded rectangle 6(2 – ��) or 6(�� – 2)?

Explain how you know.

Review Problems

5. Andre ran 5 1 2 laps of a track in 8 minutes (min.) at a constant speed. It took Andre �� min. to run each lap. Select all the equations that represent this situation.

�5 1 2 �� = 8

5 1 2 + �� = 8

5 1 2 – �� = 8

5 1 2 ÷ �� = 8

�� = 8 ÷ �5 1 2 �

�� = �5 1 2 � ÷ 8

Unit 6, Lesson 10: The Distributive Property – Part 2

Warm-Up: The Shaded Region

A rectangle with dimensions 6 centimeters (cm) and �� cm is partitioned into two smaller rectangles.

Explain why each of these expressions represents the area, in square centimeters (sq. cm), of the shaded portion.

6�� – 24

6(�� – 4)

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Collaborative Activity: Matching to Practice Distributive Property

Match each expression in column 1 to an equivalent expression in column 2. If you get stuck, consider drawing a diagram.

Column 1

+ 2 + 3) 2(12 – 4)

+ 3��

3 (15�� – 18)

+ 10�� 0.4(5 – 2.5��)

Column 2

+ ��)

+ 5��)

+ 3)��

Collaborative Activity: Writing Equivalent Expressions Using the Distributive Property

The distributive property can be used to write equivalent expressions. In each row, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

Lesson Summary

The distributive property can be used to write a sum as a product, or to write a product as a sum. A partitioned rectangle can be used to help reason about it, but with enough practice, you should be able to apply the distributive property without making a drawing.

Some examples of expressions that are equivalent due to the distributive property are shown.

+ 18 = 9(1 + 2)

+ 4) = 6�� + 8

Practice Problems

1. For each expression, use the distributive property to write an equivalent expression.

a. 4(�� + 2)

b. (6 + 8) ⋅ ��

c. 4(2�� + 3)

d. 6(�� + �� + ��)

2. Priya rewrites the expression 8�� – 24 as 8(�� – 3). Han rewrites 8�� – 24 as 2(4y – 12). Are Priya's and Han's expressions each equivalent to 8y – 24? Explain your reasoning.

3. Select all the expressions that are equivalent to 16�� + 36.

□ 16(�� + 20)

□ ��(16 + 36)

□ 4(4�� + 9)

□ 2(8�� + 18)

□ 2(8�� + 36)

4. The area of a rectangle is 30 + 12��. List at least 3 possibilities for the length and width of the rectangle.

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Review Problems

5. Select all the expressions that are equivalent to 1 2 ��.

a. What is the perimeter of a square with side length: 3 cm? 7 cm? 8 cm?

b. If the perimeter of a square is 360 cm, what is its side length?

7. Solve each equation.

a. 10 = 4��

b. 5�� = 17.5

c. 1.036 = 10��

d. 0.6�� = 1.8

e. 15 = 0.1��

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Unit 6, Lesson 11: Applying Properties of Operations

Warm-Up: Mental Math

Ulrich ﬁnds the product of 5 ∙ 107 in his head. Ulrich wrote the following explanation of his thinking:

Because 107 is equal to 100 + 7, I can write 5 ∙ 107 as 5(100 + 7). I know that 5 ∙ 100 is 500 and that 5 ∙ 7 is 35. Then, I add 500 and 35 to get 535.

1. Use Ulrich’s reasoning to rewrite each expression in the form ��(�� + ��). Then, use that form to ﬁnd the product.

Exploration Activity: Commutative

and Associative Properties of Operations

In addition to the identity, inverse, and distributive properties explored in previous lessons, there are more properties of operations that can be used to generate equivalent expressions.

The commutative property states that the order in which an operation is performed does not change the value of the expression.

1. Use the pairs of expressions shown in the table to determine the operations for which the commutative property is true.

2. Complete the statement.

The commutative property only applies to _____________________________ and

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commutative property.

3. Rewrite �� + 9 using the

4. Rewrite 7�� using the commutative property.

The associative property states that the way in which numbers in an expression are grouped does not change the value of the expression.

5. Use the pairs of expressions below to determine the operations for which the associative property is true.

Operation Example Expressions Equivalent?

6. Complete the statement.

The associative property only applies to _____________________________ and _____________________________.

7. Rewrite (�� + 15) + 11 using the associative property.

8. Rewrite (2)(8 ∙ ��) using the associative property.

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Collaborative Activity: Using Properties of Operations to Justify Equivalence

1. Work with your partner to complete the table by selecting the property that justiﬁes the two expressions in each row are equivalent. Expression A Expression B

Associative property

Distributive property

Commutative property

Associative property

Distributive property

Commutative property

Associative property

Distributive property Commutative property

Associative property

Distributive property

Commutative property

Associative property

Distributive property

Commutative property

2. Two equivalent expressions are shown.

Taylor states that the distributive property justiﬁes the expressions are equivalent. Pablo states that the commutative property justiﬁes the expressions are equivalent. Discuss with a partner which student is correct, if either. Write a summary of your discussion.

Lesson Summary

Sets A and B show pairs of equivalent expressions.

These pairs of equivalent expressions demonstrate the distributive property of multiplication over addition (or subtraction), which is a property that applies to all rational numbers. The distributive property “distributes” a factor to all terms within a set of parentheses.

Sets C and D also show pairs of equivalent expressions.

These pairs of equivalent expressions demonstrate the commutative property of addition and the commutative property of multiplication, which is another property that applies to all rational numbers. The commutative property states that the order in which operations are performed does not change the value of the expression. Because the order matters in subtraction and division expressions, the commutative property only applies to addition and multiplication.

Commutative Property of Addition

Sets E and F show more pairs of equivalent expressions.

These pairs of equivalent expressions demonstrate the associative property of addition and the associative property of multiplication, which is another property that applies to all rational numbers. The associative property states that the way in which values are grouped does not change the value of the expression. Because the grouping of values matters in subtraction and division expressions, the associative property only applies to addition and multiplication.

Associative Property of Addition

Practice Problems

1. For each equivalent expression, select the property that justiﬁes it is equivalent to 4(�� + 7).

Equivalent Expression Property That Justiﬁes Equivalence to 4(�� + 7)

Associative property

(�� + 7) ∙ 4

4�� + 4 ∙ 7

28 + 4��

Distributive property Commutative property

Associative property

Distributive property Commutative property

Associative property

Distributive property Commutative property

2. Write an expression equivalent to 6(�� + 5) using the commutative property.

3. Write an expression equivalent to 3.5(�� ∙ 6.4) using the associative property.

4. Write an expression equivalent to 2 5 (10 – ��) using the distributive property.

Review Problem

5. A shopper paid $2.52 for 4.5 pounds (lb.) of potatoes, $7.75 for 2.5 lb. of broccoli, and $2.45 for 2.5 lb. of pears. What is the unit price of each item she bought? Show your reasoning.

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Unit 6, Lesson 12: Expressions with Exponents

Warm-Up: Notice and Wonder: Dots and Lines

What do you notice? What do you wonder?

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Guided Activity: The Genie’s Oﬀer

You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one: $50,000, or a magical $1 coin.

The coin will turn into two coins on the ﬁrst day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days.

1. The number of coins on the third day will be 2 ⋅ 2 ⋅ 2. Write an equivalent expression using exponents.

2. What do 25 and 26 represent in this situation? Evaluate 25 and 26 without a calculator.

3. How many days would it take for the number of magical coins to exceed $50,000?

4. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.

Collaborative Activity: Make 81

1. Here are some expressions. All but one of them equals 16. Find the one that is not equal to 16 and explain how you know.

A. 23 ⋅ 2

B. 42

C. 25 2

D. 82

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2. Write three expressions containing exponents so that each expression equals 81.

Collaborative Activity: Is the Equation True?

Decide whether each equation is true or false and explain how you know.

24 = 2 ⋅ 4

3 + 3 + 3 + 3 + 3 = 35

53 = 5 ⋅ 5 ⋅ 5

4. 23 = 32

5. 161 = 82

6. 82 = 43

Lesson Summary

In an expression like 2��, 2 is called the base and �� is called the exponent.

In expressions like 53 and 82, the 3 and the 2 are called exponents. They tell you how many factors to multiply.

The base (of an exponent) is the number used as a factor in exponential form.

If �� is a positive whole number, it indicates how many factors of 2 should be multiplied to ﬁnd the value of the expression. For example, 21 = 2, and 25 = 2

32.

There are different ways to read the expression 25. “Two raised to a power of five,” “two to the fifth power,” or “two to the fifth” are all acceptable ways to verbalize the expression 25.

Practice Problems

3. 45 is equal to 1,024. Evaluate each expression.

43 ⋅ 42

Review Problems

4. Find two diﬀerent ways to rewrite 3�� + 6�� using the distributive property.

5. Select all the expressions that represent the area of the shaded rectangle.

□ 3(10 – ��)

□ 3(�� – 10)

□ 10(�� – 3)

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□ 10(3 – ��)

□ 30 – 3��

□ 30 – 10��

a. 46

b. 44

Unit 6, Lesson 13: Evaluating Expressions with Exponents

Warm-Up: Revisiting the Cube

Based on the given information, what other measurements of the square and cube could we ﬁnd?

Exploration Activity: Calculating Surface Area

A cube has side length 10 inches (in.). Jada says the surface area of the cube is 600 square inches (sq. in.), and Noah says the surface area of the cube is 3,600 sq. in. Here is how each of them reasoned: Jada’s Method:

Do you agree with either of them? Explain your reasoning.

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Collaborative Activity: Expression Explosion

Evaluate the expressions in one of the columns. Your partner will work on the other column. Check with your partner after you ﬁnish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to ﬁnd the error(s).

Column A

Column B

Lesson Summary

Exponents provide a new way to describe operations with numbers, so it is important to understand how exponents work with the other operations.

There needs to be a consistent way to evaluate 6 ⋅ 42, because otherwise, some people might multiply first and others might evaluate the exponent first. This would result in different values for the same expression.

In the Exploration Activity, the expression 6 ⋅ 42 represented the surface area of a cube with side length 4 units. When calculating the surface area, evaluate 42 ﬁrst to ﬁnd the area of 1 face of the cube, and then multiply the result by 6.

In expressions like 6 ⋅ 42, the convention is to evaluate the part of the expression with the exponent ﬁrst. Two examples are shown.

To communicate that 6 and 4 should be multiplied ﬁrst and then squared, parentheses or other grouping symbols are used to group parts together.

Practice Problems

1. Lin says, “I took the number 8, and then multiplied it by the square of 3.” Select all expressions that equal Lin’s answer.

2. Evaluate each expression. a. 7 + 23 b. 9 ⋅ 31 c. 20 – 24

2 ⋅ 62

�8 ∙ 1 4 �2

1 2 ⋅ 23

3. Andre says, “I multiplied 4 by 5, then cubed the result.” Select all expressions that equal Andre’s answer.

(4 ⋅ 5)3

(4 ⋅ 5)2

8,000

4. Han has 10 cubes, each 5 in. on a side.

a. Find the total volume of Han’s cubes. Express your answer as an expression using an exponent.

b. Find the total surface area of Han’s cubes. Express your answer as an expression using an exponent.

Review Problems

5. Answer each question. Show your reasoning.

a. 125% of �� is 30. What is ��?

b. 35% of �� is 14. What is ��?

6. Which expressions are solutions to the equation 2.4�� = 13.75? Select all that apply.

□ 13.75 – 1.4

□ 13.75 ⋅ 2.4

□ 13.75 ÷ 2.4 □ 13.75 2.4

□ 2.4 ÷ 13.75

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7. Jada explains how she ﬁnds 15 ⋅ 23:

“I know that ten 23s is 230, so ﬁve 23s will be half of 230, which is 115. 15 is 10 plus 5, so 15 ⋅ 23 is 230 plus 115, which is 345.”

a. Do you agree with Jada? Explain.

b. Draw a 15 by 23 rectangle. Partition the rectangle into two rectangles and label them to show Jada’s reasoning.

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Unit 6, Lesson 14: Evaluating Multistep Expressions

Warm-Up: Evaluating Numeric Expressions

The question shown is dividing the internet.

What is the value of 8 ÷ 2(2 + 2)?

Alex @XanderTheGreat - 2h Ms. B Replying to @Warm-up_Math

Anything other than 1 is absolutely wrong hth

1. Do you agree with Alex or Ms. B? Explain or show why.

2. Find the value of the expression. Show your work. 4 – 2 × 6 ÷ 3 + 7

Exploration Activity: Modeling the Meaning of Expressions

1. Determine which mathematical model matches each numerical expression. Use arrows to indicate matches. Be prepared to explain your thinking. Numerical Expression

3 × 5 + 4

(3 + 5) × 4

3 + (5 × 4)

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2. How did you use the grouping symbols to help you determine which model matched each expression?

3. With the grouping symbols removed, which model matches each expression? Use arrows to indicate matches.

Numerical Expression

(3 × 5) + 4

Mathematical Model

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3 + (5 × 4)

Guided Activity: Applying Order of Operations

Grouping symbols such as parentheses, ( ), and brackets, [ ], are used to indicate which operations should be performed ﬁrst. The following order is used:

Step 1: Perform operations within grouping symbols.

Step 2: Multiply and divide in order from left to right.

Step 3: Add and subtract in order from left to right.

1. Three students completed the warm-up. The steps in each student’s work are shown.

a. Complete the statements.

Leila Farah Jaylah evaluated the numerical expression correctly. She ﬁrst performed all multiplication and division addition and subtraction in order from left to right and then performed all multiplication and division addition and subtraction in order from left to right.

b. With your partner, discuss the work of each student: Leila, Farah, and Jaylah. Two of these students made a mistake somewhere. Circle the mistakes that you and your partner found.

2. Omari evaluated expression A and expression B. His work is shown.

Expression A Expression B

+ 24 ÷ 6 – 2 × 2

+ 4 – 2 × 2

+ 4 – 4

– 4

12 + 24 ÷ [(6 – 2) × 2] 12 + 24 ÷ [4 × 2]

+ 24 ÷ 8

a. With your partner, discuss the 2 original expressions that Omari evaluated and his corresponding answers.

b. Omari’s partner agreed with both of his answers. Explain why Omari arrived at 2 diﬀerent values when evaluating these expressions.

Lesson Summary

The agreed upon order of operations for performing computations is to ﬁrst work within grouping symbols. Then, simplify terms with exponents, which will be practiced in the next lesson. Next, perform multiplication and division in the order in which they appear, from left to right. Finally, perform addition and subtraction in the order in which they appear, from left to right.

Two expressions are shown, each evaluated using the order of operations.

The values and operations in both expressions are the same. The addition of the grouping symbols in expression B changes the value of the expression because it changes the order in which the operations are performed.

Practice Problems

1. Evaluate 4 + 3 × [15 – (12 ÷ 3)].

2. Evaluate 1 5 + 3 × 12 4 ÷ 9.

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3. Evaluate 3 2 + [(25 – 7) ÷ 4] – 6.

4. Two expressions are shown.

Expression A

(4 + 6) ÷ 0.5 – 2

Expression B

4 + 6 ÷ 0.5 – 2

Henrie evaluated expression A and expression B and determined the answer is the same for both expressions. Evaluate the expressions, and explain whether you agree with Henrie.

Review Problems

5. Which shape has a larger area: a rectangle that is 7 inches (in.) by 3 4 in., or a square with side length of 2 1 2 in.? Show your reasoning.

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6. Diego has 90 songs on his playlist. How many songs are there for each genre?

a. 40% rock

b. 10% country

c. 30% hip-hop

d. The rest is electronica

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Unit 6, Lesson 15: Order of Operations

Warm-Up: Error Analysis

Joshua incorrectly evaluated the expression 8(2) + 3(6)[2(2) – 4]. His work is shown. 8(2) + 3(6)[2(2) – 4]

Describe Joshua’s error.

Guided Activity: Evaluating Numerical Expressions with Exponents

Writing out exponential expressions as repeated multiplication can get cumbersome and complicated, as shown.

In the rewritten expression, notice that both 33 and 26 were expressed as repeated multiplication, grouped inside parentheses.

1. Complete the statements.

In the order of operations, grouping symbols indicate that the operations inside should be done ___________ other steps. Therefore, evaluating exponents is done __________ other multiplication and division when following the order of operations.

2. Evaluate (3)3 + 26 ÷ 16 + 5.

3. Elizabeth incorrectly evaluated 17 + (4)3 ∙ 7 – 9. Her work is shown. 17 + (4)3 ∙ 7 – 9 17 + (64) ∙ 7 – 9 (81) ∙ 7 – 9 567 – 9

a. Explain the error in Elizabeth’s work.

b. Correctly evaluate 17 + (4)3 ∙ 7 – 9.

c. Compare Elizabeth’s work to the expression [17 + (4)3] ∙ 7 – 9. What do you notice?

Collaborative Activity: The Importance of Left to Right

1. Sabrina is evaluating the expression 43 ÷ 8 ∙ (32 – 5). Her ﬁrst few steps are shown.

÷ 8 ∙ (32 – 5)

÷ 8 ∙ (9 – 5)

a. Sabrina thinks that she could finish her work by finding either 64 ÷ 8 or 8 ∙ 4 first. Show the work for both possibilities.

b. Was Sabrina correct when she stated that both methods would work? Use the steps of the order of operations to justify whether Sabrina is correct.

2. Complete the following.

a. Evaluate 4 ∙ 60 ÷ 10 ÷ 2.

b. Evaluate (4 ∙ 60) ÷ (10 ÷ 2).

c. Explain how the use of parentheses aﬀected the answers to parts A and B.

3. Complete the following .

a. Evaluate 72 ÷ 6 ∙ 9 + 32 .

b. Evaluate 72 ∙ 6 ÷ 9 + 32 .

c. Explain how rearranging the operations aﬀected the answers to parts A and B.

Lesson Summary

This lesson explored how results can vary when the agreed upon order of operations is not followed, which illustrates the necessity of following the order of operations.

The agreed upon order of operations is shown.

Step 1: Perform operations inside grouping symbols.

Step 2: Evaluate expressions with exponents.

Step 3: Perform multiplication and division in order from left to right (→).

Step 4: Perform addition and subtraction in order from left to right (→).

Practice Problems

1. Evaluate 16 ÷ 24 – 1.

2. Evaluate each expression.

62 ÷ 4 ∙ 10 – 32

62 ÷ [4 ∙ (10 – 32)]

3. Evaluate 10 + 2 ∙ (52 – 6 ∙ 3).

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4. Jayce evaluated the expression 72 – 32 + 10 ÷ 2 in two diﬀerent ways. His work is shown.

1st Answer

– 32 + 10 ÷ 2

Explain to Jayce which answer is correct and why.

2nd Answer

Review Problems

5. Compare using >, =, or <.

0.7 ______ 0.70

0.03 + 6 10 _______ 0.30 + 6 100

0.9 ______ 0.12

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6. Tulip bulbs are on sale at store A, at 5 for $11.00, and the regular price at store B is 6 for $13. Is each store pricing tulip bulbs at the same rate? Explain how you know.

Unit 6, Lesson 16: Two Related Quantities – Part 1

Warm-Up: Which One Would You Choose?

Which one would you choose? Be prepared to explain your reasoning.

• A 5 pound (lb.) jug of honey for $15.35

• Three 1.5 lb. jars of honey for $13.05

Guided Activity: Painting the Set

Lin needs to mix a speciﬁc shade of orange paint for the set of the school play. The color uses 3 parts yellow for every 2 parts red.

1. Complete the table to show diﬀerent combinations of red and yellow paint that will make the shade of orange Lin needs.

2. Lin notices that the number of cups of red paint is always 2 5 of the total number of cups. She writes the equation �� = 2 5 �� to describe the relationship. Which is the independent variable? Which is the dependent variable? Explain how you know.

3. Write an equation that describes the relationship between �� and ��, where �� is the independent variable.

4. Write an equation that describes the relationship between �� and ��, where �� is the independent variable.

5. Use the points in the table to create two graphs that show the relationship between �� and ��. Match each relationship to one of the equations you wrote.

Lesson Summary

Equations are useful for describing sets of equivalent ratios. An example is shown.

A pie recipe calls for 4 green apples, ��, for every 5 red apples, ��. An input-output, or function, table shows some equivalent ratios in this scenario.

The table shows that �� is always 5 4 as large as �� and that �� is always 4 5 as large as ��. Equations can be written to describe the relationship between �� and ��.

When the number of green apples is known and the number of red apples is needed, the equation �� = 5 4 �� can be used.

• In this equation, if �� changes, �� is aﬀected by the change, so the input value, ��, is the independent variable, and the output value, ��, is the dependent variable.

The independent variable is used to calculate the value of another variable.

The dependent variable represents the output of a function.

• This equation can be used with any value of �� to ﬁnd ��. For example, if 272 green apples are used, then � 5 4 ∙ 272� or 340 red apples are used.

When the number of red apples is known and the number of green apples is needed, the equation �� = 4 5 �� can be used.

• In this equation, if �� changes, �� is aﬀected by the change, so now �� is the independent variable, and �� is the dependent variable.

• Use this equation with any value of �� to ﬁnd ��. If 275 red apples are used, then 4 5 ⋅ (275) or 220 green apples are used.

The 2 equations can also be graphed on the coordinate plane for a visual representation of the relationship between the 2 quantities.

Practice Problems

1. Here is a graph that shows some values for the number of cups of sugar, ��, required to make �� batches of brownies.

a. Complete the table so that the pair of numbers in each column represents the coordinates of a point on the graph.

b. What does the point (8, 4) mean in terms of the amount of sugar and number of batches of brownies?

c. Write an equation that shows the amount of sugar in terms of the number of batches.

2. Each serving of a certain fruit snack contains 90 calories.

a. Han wants to know how many calories he gets from the fruit snacks. Write an equation that shows the number of calories, ��, in terms of the number of servings, ��.

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b. Tyler needs some extra calories each day during his sports season. He wants to know how many servings he can have each day if all the extra calories come from the fruit snack. Write an equation that shows the number of servings, ��, in terms of the number of calories, ��.

3. Kiran shops for books during a 20% oﬀ sale.

a. What percent of the original price of a book does Kiran pay during the sale?

b. Complete the table to show how much Kiran pays for books during the sale.

c. Write an equation that relates the sale price, ��, to the original price ��.

Review Problems

4. Use the algebraic expression 1.4�� + 6.18 to answer the following questions.

a. How many terms are in the expression?

b. Identify the coeﬃcient(s) and constant term(s), if any.

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5. Find (12.34) ∙ (0.7). Show your reasoning.

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6. For each expression, write another division expression that has the same value and that can be used to help ﬁnd the quotient. Then, ﬁnd each quotient.

a. 302.1 ÷ 0.5

b. 12.15 ÷ 0.02

c. 1.375 ÷ 0.11

Unit 6, Lesson 17: Two Related Quantities – Part 2

Warm-Up: Walking to the Library

Lin and Jada each walk at a steady rate from school to the library. Lin can walk 13 miles (mi.) in 5 hours (hr.), and Jada can walk 25 mi. in 10 hr. They each leave school at 3:00 and walk 3 1 4 mi. to the library. What time do they each arrive?

Collaborative Activity: The Walk-a-thon

Diego, Elena, and Andre participated in a walk-a-thon to raise money for cancer research. They each walked at a constant rate, but their rates were diﬀerent.

1. Complete the table to show how far each participant walked during the walk-a-thon.

2. How fast was each participant walking in miles per hour (mph)?

3. How long did it take each participant to walk one mi.?

4. Graph the progress of each person in the coordinate plane. Use a diﬀerent color for each participant.

5. Diego says that �� = 3�� represents his walk, where �� is the distance walked in mi. and �� is the time in hr.

a. Explain why �� = 3�� relates the distance Diego walked to the time it took.

b. Write two equations that relate distance and time: one for Elena and one for Andre.

6. Use the equations you wrote to predict how far each participant would walk, at their same rate, in 8 hr.

7. For Diego’s equation and the equations you wrote, which is the dependent variable and which is the independent variable?

Lesson Summary

Equations are very useful for solving problems with constant speeds. Consider the example shown.

A boat is traveling at a constant speed of 25 mph.

How far can the boat travel in 3.25 hr.?

How long does it take for the boat to travel 60 mi.?

Let �� represent the time the boat travels, in hr., and let �� represent the distance the boat travels, in mi.

When the time is known and the distance is needed, the equation �� = 25�� can be used.

• In this equation, if �� changes, �� is aﬀected by the change, so �� is the independent variable, and �� is the dependent variable.

• This equation can be used to ﬁnd �� for any value of ��. In 3.25 hr., the boat can travel 25(3.25) = 81.25 mi.

When distance is known and the time is needed, the equation �� = �� 25 can be used.

• In this equation, if �� changes, �� is aﬀected by the change, so �� is the independent variable and �� is the dependent variable.

• This equation can be used to ﬁnd �� for any value of ��. To travel 60 mi., it will take 60 25 = 2 2 5 hr.

These problems can also be solved using important ratio techniques such as a table of equivalent ratios. The equations are particularly valuable in this case because the answers are not round numbers or easy to quickly evaluate.

The equations can also be graphed to illustrate the relationship between the quantities, as shown.

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Practice Problems

1. A car is traveling down a road at a constant speed of 50 mph.

a. Complete the table with the amounts of time it takes the car to travel certain distances, or the distances traveled for certain amounts of time.

b. Write an equation that represents the distance traveled by the car, ��, for an amount of time, ��.

c. In your equation, which is the dependent variable and which is the independent variable?

2. The graph represents the amount of time in hr. it takes a ship to travel various distances in mi.

a. Write the coordinates of one point on the graph. What does the point represent?

b. What is the speed of the ship in mph?

c. Write an equation that relates the time, ��, it takes to travel a given distance, ��.

Review Problems

3. Find a solution to each equation, choosing from the list of values shown. Not all values will be used.

a. 2�� = 8

b. 2�� = 2

c. ��2 = 100

d. ��1 = 7

e. ��3 = 125

f. 2�� ⋅ 23 = 27

4. Select all the expressions that are equivalent to 5�� + 30�� – 15��.

□ 5(�� + 6�� – 3��)

□ (5 + 30 – 15) ⋅ ��

□ ��(5 + 30�� – 15��)

□ 5��(1 + 6 – 3)

□ 5(�� + 30�� – 15��)

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Unit 7, Lesson 1: Positive and Negative Numbers

Warm-Up: Memphis and Bangor

What do you notice? What do you wonder?

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Exploration Activity: Above and Below Zero

1. Here are three situations involving changes in temperature and three number lines. Represent each change on a number line. Then, answer the question.

a. At noon, the temperature was 5 degrees Celsius (℃). By late afternoon, it has risen 6℃. What was the temperature late in the afternoon?

b. The temperature was 8℃ at midnight. By dawn, it has dropped 12℃. What was the temperature at dawn?

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c. Water freezes at 0℃, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7℃. What is the freezing temperature of the gallon of salt water?

2. Discuss with a partner:

a. How did each of you name the resulting temperature in each situation?

b. What does it mean when the temperature is above 0? Below 0?

c. Do numbers less than 0 make sense in other contexts? Give some speciﬁc examples to show how they do or do not make sense.

Exploration Activity: High Places, Low Places

1. Here is a table that shows elevations, in feet (ft.), of various cities.

a. On the list of cities, which city has the second highest elevation?

b. How would you describe the elevation of Coachella, CA, in relation to sea level?

c. How would you describe the elevation of Death Valley, CA, in relation to sea level?

d. If you are standing on a beach right next to the ocean, what is your elevation?

e. How would you describe the elevation of Miami, FL?

f. A city has a higher elevation than Coachella, CA. Select all numbers that could represent the city’s elevation. Be prepared to explain your reasoning.

2. Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level, in meters (m).

a. Which point in the ocean is the lowest in the world? What is its elevation?

b. Which mountain is the highest in the world? What is its elevation?

c. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0?

d. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.

Lesson Summary

Positive numbers and negative numbers can be used to represent various real-world contexts.

A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

The meaning of a negative number in a given context depends on the meaning of zero in that context. For example, when measuring temperatures in ℃, 0℃ corresponds to the temperature at which water freezes.

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In this situation, positive temperatures are warmer than the freezing point, and negative temperatures are colder than the freezing point. A temperature of –6℃ means that the temperature is 6℃ away from 0 and it is less than 0.

The thermometer shows a temperature of –6℃.

If the temperature rises a few degrees and gets very close to 0℃ without reaching it, the temperature is still a negative number.

Another example is elevation, which is a distance above or below sea level. An elevation of 0 refers to sea level, as shown on the vertical number line. Positive elevations are higher than sea level, and negative elevations are lower than sea level.

Practice Problems

a. Is a temperature of –11° warmer or colder than a temperature of –15°?

b. Is an elevation of –10 feet (ft.) closer or farther from the surface of the ocean than an elevation of –8 ft.?

c. It was 8° at nightfall. The temperature dropped 10° by midnight. What was the temperature at midnight?

d. A diver is 25 ft. below sea level. After he swims up 15 ft. toward the surface, what is his elevation?

a. A whale is at the surface of the ocean to breathe. What is the whale’s elevation?

b. The whale swims down 300 ft. to feed. What is the whale’s elevation now?

c. The whale swims down 150 more ft. more. What is the whale’s elevation now?

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d. Plot each of the three elevations as a point on a vertical number line. Label each point with its numeric value.

3. Explain how to calculate a number that is equal to 2.1 1.5 .

4. Write an equation to represent each situation and then solve the equation.

a. Andre drinks 15 ounces (oz.) of water, which is 3 5 of a bottle. How much does the bottle hold? Use �� for the number of oz. of water the bottle holds.

b. A bottle holds 15 oz. of water. Jada drank 8.5 oz. of water. How many oz. of water are left in the bottle? Use �� for the number of ounces of water left in the bottle.

c. A bottle holds �� oz. of water. A second bottle holds 16 oz., which is 8 5 times as much water. How much does the ﬁrst bottle hold?

5. A rectangle has an area of 24 square units (sq. units) and a side length of 2 3 4 units. Find the other side length of the rectangle. Show your reasoning.

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Unit 7, Lesson 2: Points on the Number Line

Warm-Up: A Point on the Number Line

Collaborative Activity: What’s the Temperature?

1. Here are ﬁve thermometers. The ﬁrst four thermometers show temperatures in degrees Celsius, (℃). Write the temperatures in the blanks. a. _____ b. _____ c. _____ d. _____ e. _____

The last thermometer is missing some numbers. Write them in the boxes.

2. Elena says that the thermometer shown here reads –2.5 ℃ because the line of the liquid is above –2℃. Jada says that it is –1.5℃. Do you agree with either one of them? Explain your reasoning.

3. One morning, the temperature in Phoenix, Arizona, was 8℃ and the temperature in Portland, Maine, was 12℃ cooler. What was the temperature in Portland?

Exploration Activity: Folded Number Lines

Your teacher will give you a sheet of tracing paper on which to draw a number line.

1. Follow the steps to make your own number line.

• Use a straightedge or a ruler to draw a horizontal line. Mark the middle point of the line and label it 0.

• To the right of 0, draw tick marks that are 1 centimeter (cm) apart. Label the tick marks 1, 2 ,3 . . .10. This represents the positive side of your number line.

• Fold your paper so that a vertical crease goes through 0 and the two sides of the number line match up perfectly.

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• Use the fold to help you trace the tick marks that you already drew onto the opposite side of the number line. Unfold and label the tick marks –1, –2, –3 . . . –10. This represents the negative side of your number line.

2. Use your number line to answer these questions.

a. Which number is the same distance away from zero as is the number 4?

b. Which number is the same distance away from zero as is the number –7?

c. Two numbers that are the same distance from zero on the number line are called opposites. Find another pair of opposites on the number line.

d. Determine how far away the number 5 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number 5.

e. Determine how far away the number –2 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number –2.

Pause here so your teacher can review your work.

3. Here is a number line with some points labeled with letters. Determine the location of points P, X , and Y.

If you get stuck, trace the number line and points onto a sheet of tracing paper, fold it so that a vertical crease goes through 0, and use the folded number line to help you ﬁnd the unknown values.

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Lesson Summary

The number line shown is labeled with positive and negative numbers. The number 4 is positive, so its location is 4 units to the right of 0 on the number line. The number –1.1 is negative, so its location is 1.1 units to the left of 0 on the number line.

The opposite of 8.3 is –8.3, and the opposite of –3 2 is 3 2 .

Two numbers are opposites if they are the same distance from 0 and on diﬀerent sides of the number line.

Points A and B are opposites because they are both 2.5 units away from 0 and A is to the left of 0 while B is to the right of 0.

A positive number has a negative number for its opposite. A negative number has a positive number for its opposite. The opposite of 0 is itself.

All of the familiar positive numbers, both whole and non-whole, can be thought of as fractions and can be located on a number line.

To locate a non-whole number on a number line, divide the distance between 2 whole numbers into fractional parts, and then count the number of parts. For example, 2.7 can be written as 2 7 10 . The segment between 2 and 3 can be partitioned into 10 equal parts, or 10 tenths. From 2, count 7 of the tenths to locate 2.7 on the number line.

All of the fractions and their opposites are rational numbers. For example, 4, –1.1, 8.3, –8.3, – 3 2 , and 3 2 are all rational numbers.

A rational number is a real number that can be expressed as the ratio of two integers.

In addition to whole numbers and natural numbers, another subset of rational numbers is integers.

Integers are whole numbers and their opposites.

Practice Problems

1. For each number, name its opposite.

a. –5

b. 28

c. –10.4

d. 0.875

e. 0

f. –8,003

2. Plot the numbers –1.5, 3 2 , –3 2 , and –5 4 on the number line. Label each point with its numeric value.

3. Plot the following points on a number line.

• –1.5

• the opposite of –2

• the opposite of 0.5

• –2

a. Represent each of these temperatures in degrees Fahrenheit (℉) with a positive or negative number.

i. 5℉ above zero

ii. 3℉ below zero

iii. 6℉ above zero

iv. 2 3 4 ℉ below zero

b. Order the temperatures above from the coldest to the warmest.

5. Solve each equation.

a. 8�� = 2 5

b. 1 1 2 = 2��

c. 5�� = 2 15

d. 1 4 �� = 5

e. 1 5 = 2 50 ��

6. Write the solution to each equation as a fraction and as a decimal.

a. 2�� = 3

b. 5 �� = 3

c. 0.3�� = 0.009

Unit 7, Lesson 3: Comparing Positive and Negative Numbers

Warm-Up: Which One Doesn’t Belong: Inequalities

Which inequality doesn’t belong?

A. 5 4 < 2

B. 8.5 > 0.95

C. 8.5 < 7

D. 10.00 < 100

Exploration Activity: Comparing Temperatures

The low temperatures, in degrees Celsius (℃), for a week in Anchorage, Alaska, are shown in the table.

1. Plot the temperatures on a number line.

2. Which day of the week had the lowest low temperature?

3. The lowest temperature ever recorded in the US was –62℃, in Prospect Creek Camp, Alaska. The average temperature on Mars is about –55℃.

a. Which is warmer, the coldest temperature ever recorded in the US or the average temperature on Mars? Explain how you know.

b. Write an inequality to represent your answer to part A.

4. On a winter day, the low temperature in Anchorage, Alaska, was –21℃, and the low temperature in Minneapolis, Minnesota, was –14℃.

Karyna said, “I know that 14 is less than 21, so –14 is also less than –21. This means that it was colder in Minneapolis than in Anchorage.”

Explain whether you agree with Karyna’s reasoning.

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Collaborative Activity: Rational Numbers on a Number Line

1. Plot the numbers –2, 4, –7, and 10 on the number line. Label each point with its numeric value.

2. Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.

a. –2 < 4

b. –2 < –7

c. 4 > –7

d. –7 > 10

3. Andre says that 1 4 is less than –3 4 because, of the two numbers, 1 4 is closer to 0. Do you agree? Explain your reasoning.

4. Answer each question. Be prepared to explain how you know.

a. Which number is greater: 1 4 or 5 4 ?

b. Which number is farther from 0: 1 4 or 5 4 ?

c. Which number is greater: –3 4 or 5 8 ?

d. Which number is farther from 0: –3 4 or 5 8 ?

e. Is the number that is farther from 0 always the greater number? Explain your reasoning.

Lesson Summary

The phrases “is greater than” and “is less than” are used to compare numbers on a number line. For example, the integers –7, –3, 1, and 9 are shown on the number line.

Because –7 is to the left of –3, this is stated as “–7 is less than –3” and written as –7 < –3.

In general, any number that is to the left of a number n on a horiztonal number line (or below a number n on a vertical number line) is less than n.

The number line illustrates that –3 is greater than –7 because –3 is to the right of –7. This is written as –3 > –7.

In general, any number that is to the right of a number n on a horiztonal number line (or above a number n on a vertical number line) is greater than n.

The number line also shows that 1 > – 3 and 9 > – 7. In general, any positive number is greater than any negative number.

Practice Problems

1. Decide whether each inequality statement is true or false. Explain your reasoning.

a. – 5 > 2

b. 3 > –8

c. –12 > –15

2. The statement –8 < – 6 is true. Select all of the statements that are equivalent to –8 < –6.

□ –8 is farther to the right on the number line than –6.

□ –8 is farther to the left on the number line than –6.

□ –8 is less than –6.

□ –8 is greater than –6.

□ –6 is less than –8.

□ –6 is greater than –8.

Review Problems

3. Plot each of the following numbers on the number line. Label each point with its numeric value.

4. The table shows ﬁve states and the lowest point in each state.

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Put the states in order by their lowest elevation, from least to greatest.

5. Each lap around the track is 400 meters (m).

a. How many m does someone run if they run: 2 laps? 5 laps? �� laps?

b. If Noah ran 14 laps, how many m did he run?

c. If Noah ran 7,600 m, how many laps did he run?

6. A stadium can seat 16,000 people at full capacity.

a. If there are 13,920 people in the stadium, what percentage of the capacity is ﬁlled? Explain or show your reasoning.

b. What percentage of the capacity is not ﬁlled?

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Unit 7, Lesson 4: Ordering Rational Numbers

Warm-Up: How Do They Compare?

Use the symbols >, <, or = to compare each pair of numbers. Be prepared to explain your reasoning.

12 _________ 19

Collaborative Activity: Ordering Rational Number Cards

Your teacher will give you a set of number cards. Order them from least to greatest.

Your teacher will give you a second set of number cards. Add these to the correct places in the ordered set.

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Collaborative Activity: Comparing Points on a Line

A number line is shown with points M, N, P, and R.

1. Use each of the following terms at least once to describe or compare the values of points M, N, P, R. • greater than • less than • opposite of (or opposites)

• negative number

2. Tell what the value of each point would be if:

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Lesson Summary

To order rational numbers from least to greatest, list them in the order they appear on the number line, from left to right on a horizontal number line or from bottom to top on a vertical number line.

For example, the number line shows that the numbers –7, –3, 1, and 9 are listed from least to greatest because of the order they appear on the number line. 0

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To list numbers from greatest to least, list them in the opposite direction. For example, the values plotted on the number line listed from greatest to least are 9, 1, –3, and –7.

Comparison symbols such as < or > can also be used when listing a set of numbers being compared. For example, both inequalities shown are true.

Practice Problems

1. Select all of the numbers that are greater than –5.

2. The boiling points of certain elements are shown, rounded to the nearest integer, in degrees Celsius (℃).

List the elements from lowest to highest boiling point.

3. Order numbers shown from greatest to least.

Review Problems

4. Explain why zero is considered its own opposite.

5. Find the quotients.

Warm-Up: Notice and Wonder: It Comes and Goes

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Exploration Activity: The Concession Stand

The manager of a concession stand keeps records of all the supplies she buys and all the items she sells. The table shows some of her records for Tuesday.

1. Which items did she sell? Explain your reasoning.

2. Interpret –58 in this situation.

3. Interpret –10.00 in this situation.

4. Which item did she spend the most money on? Explain your reasoning.

Collaborative Activity: Drinks for Sale

A vending machine in an oﬃce building sells bottled beverages. The machine keeps track of all changes in the number of bottles from sales and from machine reﬁlls and maintenance. This record shows the changes for every 5 minute (min.) period over 1 hour (hr.).

1. What might a positive number mean in this context? What about a negative number?

2. What would a “0” in the second column mean in this context?

3. Which numbers, positive or negative, result in fewer bottles in the machine?

4. At what time was there the greatest change to the number of bottles in the machine? How did that change aﬀect the number of remaining bottles in the machine?

5. At which time period, 8:05–8:09 or 8:25–8:29, was there a greater change to the number of bottles in the machine? Explain your reasoning.

6. The machine must be emptied to be serviced. If there are 40 bottles in the machine when it is to be serviced, what number will go in the second column in the table?

Lesson Summary

Sometimes, changes in a quantity are represented with positive and negative numbers. If the quantity increases, the change is positive. If it decreases, the change is negative.

• Suppose 5 gallons (gal.) of water are put in a washing machine. The change in the number of gal. is represented as + 5. If 3 gal. are emptied from the machine, the change is represented as –3.

It’s common to represent money received with positive numbers and money spent with negative numbers.

• Suppose Cristal gets $30.00 for her birthday and spends $18.00 buying lunch for herself and a friend. To her, the value of the gift can be represented as +30.00 and the value of the lunch can be represented as –18.00.

Whether a value is considered positive or negative depends on a person’s perspective. If Cristal’s grandmother gives her $20 for her birthday, Cristal might see this as +20, because to her, the amount of money in her wallet has increased. But her grandmother might see it as –20, because to her, the amount of money in her wallet has decreased.

In general, when using positive and negative numbers to represent changes, be very clear about what it means when the change is positive and what it means when the change is negative.

Practice Problems

1. Write a positive or negative number to represent each change in the high temperature, in degrees (°).

a. Tuesday’s high temperature was 4° less than Monday’s high temperature.

b. Wednesday’s high temperature was 3° less than Tuesday’s high temperature.

c. Thursday’s high temperature was 6° greater than Wednesday’s high temperature.

d. Friday’s high temperature was 2° less than Thursday’s high temperature.

2. Decide which of the following quantities can be represented by a positive number and which can be represented by a negative number. Give an example of a quantity with the opposite sign in the same situation.

a. Tyler’s puppy gained 5 pounds (lb.).

b. The aquarium leaked 2 gal. of water.

c. Andre received a gift of $10.

d. Kiran gave a gift of $10.

e. A climber descended 550 feet (ft.).

3. Make up a situation where a quantity is changing.

a. Explain what it means to have a negative change.

b. Explain what it means to have a positive change.

c. Give an example of each.

a. On the number line, label the points that are 4 units away from 0.

b. If you fold the number line so that a vertical crease goes through 0, the points you label would match up. Explain why this happens.

c. On the number line, label the points that are 5 2 units from 0. What is the distance between these points?

5. Evaluate each expression.

a. 23 ⋅ 3 b. 42 2

c. 31

d. 62 ÷ 4

e. 23 –2

f. 102 + 52

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Unit 7, Lesson 6: Absolute Value of Numbers

Warm-Up: Number Talk: Closer to Zero

For each pair of expressions, decide mentally which one has a value that is closer to 0.

1. 9 10 or 15 10

2. 1 5 or 1 8

3. 1.25 or 5 4

4. 0.01 or 0.001

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Exploration Activity: Jumping Flea

1. A ﬂea is jumping around on a number line.

a. If the ﬂea starts at 1 and jumps 4 units to the right, where does it end up? How far away from 0 is this?

b. If the ﬂea starts at 1 and jumps 4 units to the left, where does it end up? How far away from 0 is this?

c. If the ﬂea starts at 0 and jumps 3 units away, where might it land?

d. If the ﬂea jumps 7 units and lands at 0, where could it have started?

e. The absolute value of a number is the distance it is from 0. The ﬂea is currently to the left of 0 and the absolute value of its location is 4. Where on the number line is it?

f. If the ﬂea is to the left of 0 and the absolute value of its location is 5, where on the number line is it?

g. If the ﬂea is to the right of 0 and the absolute value of its location is 2.5, where on the number line is it?

2. We use the notation |–2| to say “the absolute value of –2,” which means ”the distance of –2 from 0 on the number line.”

a. What does |–7| mean and what is its value?

b. What does |1.8| mean and what is its value?

Collaborative Activity: Absolute Elevation and Temperature

1. A part of the city of New Orleans is 6 feet (ft.) below sea level. We can use “–6 ft.” to describe its elevation, and “|–6| ft.” to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe?

a. 25 ft.

b. |25| ft.

c. –8 ft.

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d. |–8| ft.

2. The elevation of a city is diﬀerent from sea level by 10 ft. Name the two elevations that the city could have.

3. We write “–5 degrees Celsius (℃)” to describe a temperature that is 5℃ below freezing point and “5℃” for a temperature that is 5° above freezing. In this context, what do each of the following numbers describe?

a. Which temperature is colder: –6∘C or 3∘C?

b. Which temperature is closer to freezing temperature: –6∘C or 3∘C?

c. Which temperature has a smaller absolute value? Explain how you know.

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Lesson Summary

Numbers are compared by comparing their positions on the number line. The number farther to the right is greater, and the one farther to the left is less.

Sometimes, the comparison is made to determine which value is closer to or farther from 0. For example, there may be a desire to know how far away a temperature is from the freezing point of 0∘C, regardless of whether it is above or below freezing. The absolute value of the number can be used in this case.

The absolute value of a number is its distance from 0 on the number line.

The absolute value of –4 is 4 because –4 is 4 units to the left of 0. The absolute value of 4 is also 4 because 4 is 4 units to the right of 0. Opposites always have the same absolute value because they are both the same distance from 0.

The distance from 0 to itself is 0, so the absolute value of 0 is 0. Zero is the only number whose distance to 0 is 0. For all other absolute values, there are always 2 numbers (a positive number and a negative number) that are that same distance from 0.

The notation |4| is used to express “the absolute value of 4.”

The true equation |–8| = 8 reads, “The absolute value of –8 is 8.”

Practice Problems

1. On the number line, plot and label all numbers with an absolute value of 3 2 .

2. The temperature at dawn is 6∘C away from 0. Select all the temperatures that are possible.

3. Put these numbers in order, from least to greatest.

Review

Problems

4. Lin’s family needs to travel 325 miles (mi.) to reach her grandmother’s house.

a. At 26 mi., what percentage of the trip’s distance have they completed?

b. How far have they traveled when they have completed 72% of the trip’s distance?

c. At 377 mi., what percentage of the trip’s distance have they completed?

5. Elena donates some money to charity whenever she earns money as a babysitter. The table shows how much money, ��, she donates for diﬀerent amounts of money, ��, that she earns.

a. What percent of her income does Elena donate to charity? Explain or show your work.

b. Which quantity, m or d, would be the better choice for the dependent variable in an equation describing the relationship between m and d? Explain your reasoning.

c. Use your choice from the second question to write an equation that relates m and d.

6. How many times larger is the ﬁrst number in the pair than the second?

a. 34 is _______ times larger than 33.

b. 53 is _______ times larger than 52.

c. 710 is _______ times larger than 78.

d. 176 is _______ times larger than 174.

e. 510 is _______ times larger than 54.

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Unit 7, Lesson 7: Comparing Numbers and Distance from Zero

Warm-Up: Opposites

1. �� is a rational number. Choose a value for �� and plot it on the number line.

a. Based on where you plotted ��, plot −�� on the same number line.

b. What is the value of −�� that you plotted?

3. Noah said, “If �� is a rational number, −�� will always be a negative number.” Do you agree with Noah? Explain your reasoning.

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Exploration Activity: Submarine

A submarine is at an elevation of –100 feet (ft.) (100 ft. below sea level). Let’s compare the elevations of these four people to that of the submarine.

• Clare’s elevation is greater than the elevation of the submarine. Clare is farther from sea level than the submarine.

• Andre’s elevation is less than the elevation of the submarine. Andre is farther away from sea level than the submarine.

• Han’s elevation is greater than the elevation of the submarine. Han is closer to sea level than is the submarine.

• Lin’s elevation is the same distance away from sea level as the submarine’s.

1. Complete the table as follows.

a. Write a possible elevation for each person.

b. Use <, >, or = to compare the elevation of that person to that of the submarine.

c. Use absolute value to tell how far away the person is from sea level (elevation 0).

As an example, the ﬁrst row has been ﬁlled with a possible elevation for Clare.

2. Priya says her elevation is less than the submarine’s and she is closer to sea level. Is this possible? Explain your reasoning.

Collaborative Activity: Info Gap: Points on the Number Line

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think about what information you need to be able to answer the question.

2. Ask your partner for the speciﬁc information that you need.

3. Explain how you are using the information to solve the problem.

4. Continue to ask questions until you have enough information to solve the problem.

5. Share the problem card and solve the problem independently.

6. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

1. Silently read your card.

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2. Ask your partner “What speciﬁc information do you need?” and wait for them to ask for information.

3. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

4. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

5. Read the problem card and solve the problem independently.

6. Share the data card and discuss your reasoning.

Collaborative Activity: Inequality Mix and Match

Here are some numbers and inequality symbols. Work with your partner to write true comparison statements.

One partner should select two numbers and one comparison symbol and use them to write a true statement using symbols. The other partner should write a sentence in words with the same meaning, using the following phrases:

• is equal to

• is the absolute value of

• is greater than

• is less than

For example, one partner could write 4 < 8 and the other would write, “4 is less than 8.” Switch roles until each partner has three true mathematical statements and three sentences written down.

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Lesson Summary

Elevation can be used to compare 2 rational numbers or 2 absolute values.

• Suppose an anchor has an elevation of –10 meters (m) and a house has an elevation of 12 m. To describe the anchor as having a lower elevation than the house, write –10 < 12, and say, “–10 is less than 12.”

• The anchor is closer to sea level than the house is to sea level, which has an elevation of 0 m. To describe this, write |–10| < |12|, and say, “The distance between –10 and 0 is less than the distance between 12 and 0.”

Similar descriptions can be used to compare rational numbers and their absolute values outside of the context of elevation.

• Absolute value can be used to compare the distances of –47.5 and 5.2 from 0. |–47.5| is 47.5 units away from 0, and |5.2| is 5.2 units away from 0, so |–47.5| > |5.2|.

• | –18| > 4 means that the absolute value of –18 is greater than 4. This is true because 18 is a greater distance from 0 than 4. To compare the values, though, –18 < 4.

Practice Problems

1. In the context of elevation, what would | –7| ft. mean?

2. Compare each pair of expressions using >, <, or =.

3. Match the statements written in English with the mathematical statements.

a. The number –4 is a distance of 4 units away from 0 on the number line.

b. The number –63 is more than 4 units away from 0 on the number line.

c. The number 4 is greater than the number –4.

d. The numbers 4 and –4 are the same distance away from 0 on the number line.

e. The number –63 is less than the number 4.

f. The number –63 is further away from 0 than the number 4 on the number line.

Review Problems

i. | – 63| > 4

ii. –63 < 4

iii. | –63| > |4|

iv. | –4| = 4

v. 4 > –4

vi. |4| = |–4|

4. Mai received and spent money in the following ways last month. For each example, write a signed number to represent the change in money from her perspective.

a. Her grandmother gave her $25 in a birthday card.

b. She earned $14 dollars babysitting.

c. She spent $10 on a ticket to the concert.

d. She donated $3 to a local charity.

e. She got $2 interest on money that was in her savings account.

5. Here are the lowest temperatures recorded in the last 2 centuries for some US cities. Temperatures are in degrees Fahrenheit (℉).

• Death Valley, CA was –45℉ in January of 1937.

• Danbury, CT was –37℉ in February of 1943.

• Monticello, FL was –2℉ in February of 1899.

• East Saint Louis, IL was –36℉ in January of 1999.

• Greenville, GA was –17℉ in January of 1940.

a. Which of these states has the lowest record temperature?

b. Which state has a lower record temperature, FL or GA?

c. Which state has a lower record temperature, CT or IL?

d. How many more degrees colder is the record temperature for GA than for FL?

6. Find the quotients.

a. 0.024 ÷ 0.015

b. 0.24 ÷ 0.015

c. 0.024 ÷ 0.15

d. 24 ÷ 15

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Unit 7, Lesson 8: Writing and Graphing Inequalities

Warm-Up: Estimate Heights of People

1. Here is a picture of a man.

a. Name a number, in feet (ft.), that is clearly too high for this man’s height.

b. Name a number, in ft., that is clearly too low for his height.

c. Make an estimate of his height.

Pause here for a class discussion.

2. Here is a picture of the same man standing next to a child.

If the man’s actual height is 5 ft. 10 inches (in.), what can you say about the height of the child in this picture?

Be prepared to explain your reasoning.

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Collaborative Activity: Stories about 9

1. Your teacher will give you a set of paper slips with four stories and questions involving the number 9. Match each question to three representations of the solution: a description or a list, a number line, or an inequality statement.

2. Compare your matching decisions with another group’s. If there are disagreements, discuss until both groups come to an agreement. Then, record your ﬁnal matching decisions here.

a. A ﬁshing boat can hold fewer than 9 people. How many people, ��, can it hold?

• Description or list:

• Number line:

• Inequality:

b. Lin needs more than 9 ounces (oz.) of butter to make cookies for her party. How many oz. of butter, ��, would be enough?

• Description or list:

• Number line:

• Inequality:

c. A magician will perform her magic tricks only if there are at least 9 people in the audience. For how many people, ��, will she perform her magic tricks?

• Description or list:

• Number line:

• Inequality:

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d. A food scale can measure up to 9 kilograms (kg) of weight. What weights, ��, can the scale measure?

• Description or list:

• Number line:

• Inequality:

Exploration Activity: How High and How Low Can It Be?

Here is a picture of a person and a basketball hoop. Based on the picture, what do you think are reasonable estimates for the maximum and minimum heights of the basketball hoop?

1. Complete the ﬁrst blank in each sentence with an estimate, and the second blank with ”taller” or “shorter.”

a. I estimate the minimum height of the basketball hoop to be _________ ft. This means the hoop cannot be _____________ than this height.

b. I estimate the maximum height of the basketball hoop to be _________ ft. This means the hoop cannot be _____________ than this height.

2. Write two inequalities—one to show your estimate for the minimum height of the basketball hoop, and another for the maximum height. Use an inequality symbol and the variable ℎ to represent the unknown height.

3. Plot each estimate for minimum or maximum value on a number line.

Minimum:

Maximum:

4. Suppose a classmate estimated the value of ℎ to be 19 ft. Does this estimate agree with your inequality for the maximum height? Does it agree with your inequality for the minimum height? Explain or show how you know.

5. Ask a partner for an estimate of ℎ. Record the estimate and check if it agrees with your inequalities for maximum and minimum heights.

Lesson Summary

Inequalities are used to communicate that a value is less than or is greater than another value.

Suppose the temperature is less than 3 degrees Fahrenheit (℉), but the exact temperature is unknown. To represent what is known about the temperature, ��, in ℉, the inequality �� < 3 can be used.

The possible temperatures can also be graphed on a number line. Any point to the left of 3 is a possible value for ��. The open circle at 3 means that �� cannot be equal to 3, because the temperature is less than 3.

Suppose a young traveler has to be at least 16 years old to ﬂy on an airplane without an accompanying adult.

If �� represents the age of the traveler, then any number greater than 16 is �� possible value of ��, and 16 itself is also a possible value of ��. This can be represented on a number line by drawing a closed circle at 16 to show that it meets the requirement, because a 16-year-old person can travel alone. From there, a line can be drawn pointing to the right to represent that all of the values greater than 16 are also included.

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Both an inequality and an equation can be used together to represent all the possible values of ��, �� > 16 and �� = 16.

Practice Problems

1. At the book sale, all books cost less than $5.

a. What is the most expensive a book could be?

b. Write an inequality to represent costs of books at the sale.

c. Draw a number line to represent the inequality.

2. Kiran started his homework before 7:00 p.m. and ﬁnished his homework after 8:00 p.m. Let ℎ represent the number of hours (hr.) Kiran worked on his homework.

Decide if each statement is deﬁnitely true, deﬁnitely not true, or possibly true. Explain your reasoning.

a. ℎ > 1

b. ℎ > 2

c. ℎ < 1

d. ℎ < 2

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3. Consider a rectangular prism with length 4 and width and height ��.

a. Find an expression for the volume of the prism in terms of ��.

b. Compute the volume of the prism when �� = 1, when �� = 2, and when �� = 1 2 .

4. Match the statements written in English with the mathematical statements. All of these statements are true.

a. The number –15 is further away from 0 than the number –12 on the number line.

b. The number –12 is a distance of 12 units away from 0 on the number line.

c. The distance between –12 and 0 on the number line is greater than –15.

d. The numbers 12 and –12 are the same distance away from 0 on the number line.

e. The number –15 is less than the number –12.

f. The number 12 is greater than the number –12.

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i. |–12| > –15

ii. –15 < –12

iii. | –15| > | –12|

iv. | –12| = 12

v. 12 > –12

vi. |12| = | –12|

5. Here are ﬁve sums. Use the distributive property to write each sum as a product with two factors.

Unit 7, Lesson 9: Solutions of Inequalities

Warm-Up: Unknowns on a Number Line

The number line shows several points, each labeled with a letter.

1. Fill in each blank with a letter so that the inequality statements are true.

a. _______ > _______

b. _______ < _______

2. Jada says that she found three diﬀerent ways to complete the ﬁrst question correctly. Do you think this is possible? Explain your reasoning.

3. List a possible value for each letter on the number line based on its location.

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Exploration Activity: Amusement Park Rides

Priya ﬁnds these height requirements for some of the rides at an amusement park.

To ride the . . . you must be . . .

High Bounce between 55 and 72 inches (in.) tall

Climb-A-Thon under 60 in. tall

Twirl-O-Coaster 58 in. minimum

1. Write an inequality for each of the three height requirements. Use ℎ for the unknown height. Then, represent each height requirement on a number line.

• High Bounce _____________________

• Climb-A-Thon _____________________

• Twirl-O-Coaster _____________________

Pause here for additional instructions from your teacher.

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2. Han’s cousin is 55 in. tall. Han doesn’t think she is tall enough to ride the High Bounce, but Kiran believes that she is tall enough. Do you agree with Han or Kiran? Be prepared to explain your reasoning.

3. Priya can ride the Climb-A-Thon, but she cannot ride the High Bounce or the Twirl-O-Coaster. Which, if any, of the following could be Priya’s height? Be prepared to explain your reasoning.

• 59 in.

• 53 in.

• 56 in.

4. Jada is 56 in. tall. Which rides can she go on?

5. Kiran is 60 in. tall. Which rides can he go on?

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6. The inequalities ℎ < 75 and ℎ > 64 represent the height restrictions, in inches, of another ride. Write three values that are solutions to both of these inequalities.

Guided Activity: More Fun at the Amusement Park

Hector was happy to realize that he is not too tall to go on the Climb-A-Thon ride. Hector’s friend, Daisy, is 2 in. taller than him.

1. Explain whether Daisy is too tall to go on the ride with Hector.

2. Complete the statements.

If Hector is �� in. tall, then Daisy is �� + 2 �� – 2 in. tall. For Daisy to meet the height restriction for the Climb-A-Thon, the value of �� + 2 �� – 2 must be less than greater than

58. 60. Therefore, the solutions to the inequality �� + 2 < �� + 2 > �� – 2 < �� – 2 > 58 60 represent

Hector’s height if Daisy can also ride the Climb-A-Thon.

3. On the number line, show all the possible heights that Hector could be if Daisy also meets the ride’s height restriction.

4. What is the maximum height Hector can be in order for Daisy to be able to go on the ride as well?

5. Franny noticed the graph also represents the solutions to the inequality �� < 58. Work with your partner to explain why the solutions to �� + 2 < 60 and �� < 58 are represented the same way on a number line.

Collaborative Activity: Inequalities in Mathematical and Real-World Contexts

The solution set of an inequality includes the values that make the inequality true. An inequality has an inﬁnite number of values in the solution set. However, some parts of a solution set may not make sense, given a real-world context.

For example, to ride the the Twirl-O-Coaster at an amusement park, guests must be a minimum of 58 in. tall.

1. What does 58 represent in this situation?

2. Graph the inequality that represents this situation. 40424446485052545658606264666870

3. Explain what the arrow on your graph represents.

4. Provide 2 values that are part of the solution set but do not make sense in this context. Explain your reasoning.

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Lesson Summary

To beat the school record for the 50 yard (yd.) dash would take a time under 10 seconds (sec.). If �� represents the time it takes to run the 50 yd. dash, then �� < 10 represents the time needed to break the school record.

Any value of �� that makes the inequality true is called a solution to the inequality.

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

For example, 5 is a solution to the inequality �� < 10 because 5 < 10 is a true statement, but 12 is not a solution because 12 < 10 is not a true statement. There are an inﬁnite number of solutions that make this inequality true, such as 7 sec., 8.29 sec., 9.245 sec., and an endless list of other values. All of these values are part of the solution set of the inequality �� < 10.

Sometimes, there are values that are part of the mathematical solution set that do not make sense in the context. In those cases, the variable may have a constraint placed on it so that the solutions are realistic.

A constraint is a limitation on the possible values of variables in a model, often expressed by an equation or inequality.

If a situation involves more than one boundary or limit, more than one inequality is needed to express it.

For example, to ride the High Bounce at an amusement park, guests have to be more than 55 in. tall but less than 72 in. tall. The height restriction can be represented by the inequalities and number lines shown.

48505254565860626466687072747678

> 55

48505254565860626466687072747678

< 72

Any number greater than 55 is a solution to �� > 55, and any number less than 72 is a solution to �� < 72. But to meet the condition of “more than 55 but less than 72,” the solutions are limited to the numbers between 55 in. and 72 in., not including 55 and 72.

This solution set can be shown visually by graphing the 2 inequalities on a single number line, as shown.

48505254565860626466687072747678

Practice Problems

a. Select all numbers that are solutions to the inequality �� > 5.

□ 4

□ 5

□ 6

□ 5.2

□ 5.01

□ 0.5

b. Draw a number line to represent this inequality.

2. One day in Boston, MA, the high temperature was 60 degrees Fahrenheit (℉), and the low temperature was 52℉.

a. Write one or more inequalities to describe the temperatures, ��, that are between the high and low temperature on that day.

b. Show the possible temperatures on a number line.

3. The graph of an inequality is shown.

-8-7-6-5-4-3-2-1012345678

Circle the inequalities that can be represented by the graph.

Review Problems

4. Select all the true statements.

–5 < | –5|

| –6 | < 3

4 < | – 7|

| –7| < | –8|

a. The price of a cell phone is usually $250. Elena’s mom buys one of these cell phones for $150. What percentage of the usual price did she pay?

b. Elena’s dad buys another type of cell phone that also usually sells for $250. He pays 75% of the usual price. How much did he pay?

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Unit 7, Lesson 10: Solving One-Variable Inequalities

Warm-Up: Visual Models

An unbalanced hanger diagram is shown. It shows that the weight of 1 circle is greater than the weight of 1 pentagon.

1. Complete the table.

Hanger Diagram Description Inequality

The weight of 1 circle, ��, is greater than the weight of 1 pentagon, ��.

The weight of 1 square, ��, is __________ than the weight of 1 circle and 1 pentagon.

Exploration Activity: Is the New Inequality True?

1. Four true inequalities are given in the table. Work with your partner to complete the table by following the steps indicated for each inequality and then answering the questions.

2. Use your work from the table to compete the statements.

a. When the same value is added to both sides of a true inequality, the new inequality is true. false.

b. When the same value is subtracted from both sides of a true inequality, the new inequality is true. false.

c. When both sides of a true inequality are multiplied by the same positive value, the new inequality is true. false.

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d. When both sides of a true inequality are divided by the same positive value, the new inequality is true. false.

Collaborative Activity: Solving One-Step Inequalities with Positive Rational Numbers

The process for solving one-variable inequalities is similar to the process for solving one-variable equations.

1. Complete the table to determine the value of �� – 3 for each value of ��.

2. For which value of �� is �� – 3 = 2.2 true?

3. For which values of �� is �� – 3 > 2.2 true?

4. Represent the values from question 3 on the number line.

-8-7-6-5-4-3-2-1012345678

5. Discuss with a partner whether the solution set of �� – 3 > 2.2 includes more values than are represented in the table and on the number line.

6. Graph the solution set for �� – 3 > 2.2.

-8-7-6-5-4-3-2-1012345678

7. Complete the mathematical statement by writing a comparison symbol and a constant value to create an inequality equivalent to the inequality graphed in question 6.

8. What value can be added to both sides of the inequality �� – 3 > 2.2 to result in the inequality written in question 7?

Guided Activity: Solving and Checking One-Step Inequalities

1. Use the inequality 10 > 5 4 �� to complete the following.

a. Solve the inequality. Show your work.

b. Graph the solution set on the number line.

c. Choose a value from the solution set. Substitute the value for �� into the original inequality to verify your solution and graph. Show your work.

2. Use the inequality �� 4 > 1.25 to complete the following.

a. Solve the inequality. Show your work.

b. Graph the solution set on the number line.

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c. Choose a value from the solution set. Substitute the value for �� into the original inequality to verify your solution and graph. Show your work.

3. Michael got an estimate on the cost to repair his mountain bike. The parts will cost $16.38, and he was told that the repair will be less than $40.50 total, including labor.

The inequality 16.38 + �� < 40.50 represents this situation, where �� is the cost of labor.

a. Solve the inequality to determine the maximum amount that the labor will cost.

b. Graph the solution set on the number line.

c. Explain which solutions, if any, do not make sense for this context.

Lesson Summary

Unit 6 explored solving one-step equations. One-step inequalities are solved in the same way, by performing the same operations on both sides of the inequality to maintain the original relationship between the quantities. When solving one-step equations, there was only 1 value of the variable that made the equation true. With inequalities, however, there are an inﬁnite number of solutions that make the inequality true.

Just like with equations, performing the same operation on both sides of a true inequality results in an equivalent inequality that is also true. To solve a given inequality, choose an operation that, when applied to both sides of the inequality, will isolate the variable on one side of the comparison symbol. When the variable is isolated and compared to a constant term, the inequality describes the solution set, which can then be graphed on a number line.

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Solutions can be veriﬁed by choosing values that are in the solution set, substituting the value(s) into the original inequality, and evaluating the statement. If the statement is not true with the substituted value, there is an error in the solution. Revisit the question, correct any errors, and verify the revised solution.

Practice Problems

1. Use the inequality 7 2 5 + �� < 9 9 10 to complete the following.

a. Solve the inequality. Show your work.

b. Graph the solution set on the number line.

2. Gabriel and Lauren made errors when solving the inequality 14.8 < 3.7��. Their work is shown.

a. Explain the error(s) each student made.

b. Correctly solve the inequality 14.8 < 3.7��.

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3. To go on a water slide at a water park, a rider must be more than 48 inches (in.) tall. Brent is tall enough to go on the slide, but his friend, Sheri, is 3 1 4 in. shorter than him. The inequality �� – 3 1 4 > 48, represents Brent’s height if Sheri is also tall enough to go on the slide.

a. Solve the inequality to determine possible heights for Brent that would mean he and Sheri both get to ride the water slide.

b. Explain whether there are any values that do not make sense in this context.

Review Problems

4. A sign on the road says, “Speed limit, 60 miles per hour (mph).”

a. Let �� be the speed of a car. Write an inequality that matches the information on the sign.

b. Draw a number line to represent the solutions to the inequality.

c. Could 60 be a value of ��? Explain your reasoning.

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5. Solve each equation.

a. �� – 2.01 = 5.5

b. �� + 2.01 = 5.5

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c. 10�� = 13.71

d. 100�� = 13.71

Unit 7, Lesson 11: Writing and Interpreting One-Step Inequalities

Warm-Up: True or False: Fraction and Decimals

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State whether each equation is true or false. Be prepared to explain your reasoning.

1. 3(12 + 5) = (3 ⋅ 12) ⋅ (3 ⋅ 5)

Collaborative Activity: Basketball Game

Noah scored �� points in a basketball game.

1. What does 15 < �� mean in the context of the basketball game?

2. What does �� < 25 mean in the context of the basketball game?

3. Draw two number lines to represent the solutions to the two inequalities.

4. Name a possible value for �� that is a solution to both inequalities.

5. Name a possible value for �� that is a solution to 15 < ��, but not a solution to �� < 25.

6. Can –8 be a solution to �� in this context? Explain your reasoning.

Collaborative Activity: Relating Real-World Context to Inequalities

1. Four inequalities are shown. �� + 4 < 32 32 – �� > 4 �� 32 < 4 �� 4 > 32

Discuss with your partner which inequality correctly represents each situation, where �� represents the unknown quantity. Then, write the matching inequality in the space provided.

Situation 1

At a restaurant, the bill had to be split among 4 friends.

If each person paid more than $32, which inequality represents this context?

Inequality:

Situation 2

There are fewer than 32 objects to ﬁnd in a hidden picture puzzle.

If Martell has found 4 of the hidden objects, which inequality represents this context?

Inequality:

2. Solve the inequalities selected to answer the questions for each situation.

Situation 1

If each person paid more than $32, what was the amount of the total bill?

Solution:

Situation 2

If Martell has found 4 of the hidden objects, how many remain for Martell to ﬁnd?

Solution:

3. Interpret the solution set for each situation.

Situation 1

Intepretation:

Situation 2

Interpretation:

Collaborative Activity: Writing and Solving Inequalities for Real-World Contexts

1. Keegan collected 7 more items than his cousin Joel did for the annual toy drive. Joel collected more than 12 items.

a. Which inequality could be used to represent the number of items Keegan collected?

A. �� – 7 < 12

B. �� – 7 > 12

C. �� + 7 < 12

D. �� + 12 > 7

b. Explain what the variable �� represents in this situation.

c. How many items did Keegan collect for the toy drive?

d. Explain whether there any values in the solution set that may not make sense in this context.

2. Drake is selling candy bars for a school fundraiser. Each candy bar sells for $2.75.

a. Write an inequality using multiplication that can be used to determine how many candy bars Drake sold, ��, if he raised less than $352.

b. Compare your inequality with a partner’s. Discuss any diﬀerences in your inequalities, and make revisions if needed.

c. Solve the inequality.

d. Interpret the solution set in context.

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Lesson Summary

When representing a real-world situation with an inequality, use key words from the context to determine the comparison symbol that should be used in the inequality. Some common key words are shown.

Is Greater Than (>) Is Less Than (<)

exceeds more than over below fewer than under

When ﬁnding the solutions to an inequality, consider the context carefully. A number that is a solution to an inequality mathematically may not make sense when considered in context. Consider the 2 examples shown.

• Suppose a basketball player scored more than 11 points in a game and the number of points she scored, ��, is represented by the inequality �� > 11. Judging only by looking at �� > 11, numbers such as 12, 14 1 2 , and 130.25 are all solutions to the inequality because they each make the inequality true.

In a basketball game, however, it is only possible to score a whole number of points, so fractional and decimal scores are not possible. It is also highly unlikely that one person would score more than 130 points in a single game. In other words, the context of an inequality may limit its solutions.

• The solutions to �� < 30 can include numbers such as 27 3 4 , 18.5, 0, and –7. But if �� represents the number of minutes (min.) of rain yesterday, assuming it did rain, then the solutions are limited to positive numbers. Zero min. or negative numbers of min. would not make sense in this context.

Practice Problems

1. There is a closed carton of eggs in Mai’s refrigerator. The carton contains �� eggs and it can hold 12 eggs.

a. What does the inequality �� < 12 mean in this context?

b. What does the inequality �� > 0 mean in this context?

c. What are some possible values of �� that will make both �� < 12 and �� > 0 true?

2. One orange tree can produce more than 275 oranges. A crate holds 25 oranges.

a. Which inequality represents this situation?

A. 275 �� < 25

B. �� 275 > 25

C. 25�� < 275

D. 25�� > 275

b. Solve the inequality, and interpret the solution in context.

3. Karen’s cat weighs 5 2 5 pounds (lb.) less than Corey’s puppy. Corey’s puppy weighs less than 15 lb.

a. Write an inequality using addition that can be used to represent the weight of Karen’s cat, ��.

b. What is the possible weight range for Karen’s cat? Explain your reasoning.

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Review Problems

4. Tyler has more than $10. Elena has more money than Tyler. Mai has more money than Elena. Let �� be the amount of money that Tyler has, let �� be the amount of money that Elena has, and let �� be the amount of money that Mai has. Select all statements that are true.

5. Which is greater, –9 or –5? Explain how you know. If you get stuck, consider plotting the numbers on a number line.

Unit 7, Lesson 12: Points on the Coordinate Plane

Warm-Up: Guess My Line

1. Choose a horizontal or a vertical line on the grid. Draw 4 points on the line and label each point with its coordinates.

2. Tell your partner whether your line is horizontal or vertical, and have your partner guess the locations of your points by naming coordinates.

If a guess is correct, put an X through the point. If your partner guessed a point that is on your line but not the point that you plotted, say, “That point is on my line, but is not one of my points.”

Take turns guessing each other’s points, 3 guesses per turn.

Guided Activity: The Coordinate Plane

1. Label each point on the coordinate plane with an ordered pair.

2. What do you notice about the locations and ordered pairs of ��, ��, and ��? How are they diﬀerent from those for point ��?

3. Plot a point at (–2, 5). Label it ��. Plot another point at (3, –4.5). Label it ��.

4. The coordinate plane is divided into four quadrants, I, II, III, and IV, as shown here.

a. In which quadrant is �� located? ��? ��?

�� = (5, 2)

�� = (–1, –5)

�� = (7, –4)

b. A point has a positive ��-coordinate. In which quadrant could it be?

Collaborative Activity: Coordinated Archery

Here is an image of an archery target on a coordinate plane. The scores for landing an arrow in the colored regions are shown.

• Yellow: 10 points

• Red: 8 points

• Blue: 6 points

• Green: 4 points

• White: 2 points

Name the coordinates for a possible landing point to score.

6 points

Lesson Summary

Just as the number line can be extended to the left of 0 or below 0 to include negative numbers, the �� -axis and ��-axis of a coordinate plane can also be extended to include negative values.

The ��-axis is the horizontal axis in the coordinate system. The ��-axis divides positive ��-values from negative ��-values, and the ��-value of any point lying on the ��-axis equals zero.

The ��-axis is the vertical axis in the coordinate system. The ��-axis divides positive ��-values from negative ��-values, and the ��-value of any point lying on the ��-axis equals zero.

The ��- and ��-axes intersect at a point called the origin.

In the coordinate plane, the origin is the location where the ��-axis and ��-axis intersect. The coordinates of the origin are (0,0).

When negative values are included on the coordinate plane, all 4 quadrants are included.

A quadrant is any of the four regions separated by the axes in a coordinate plane.

The ordered pair (��, ��) can have negative ��- and ��-values. For the point �� = (–4, 1), the ��-value of –4 indicates that the point is 4 units to the left of the ��-axis. The ��-value of 1 indicates that the point is 1 unit above the ��-axis.

The same reasoning applies to the points �� and ��. The ��- and ��-coordinates for point �� are positive, so �� is to the right of the ��-axis and above the ��-axis. The ��- and ��-coordinates of point �� are negative, so �� is to the left of the ��-axis and below the ��-axis.

The quadrants are named in counterclockwise (↺) order using the Roman numerals for 1–4, which are I, II, III, and IV. Point �� is in quadrant I, point �� is in quadrant II, point �� is in quadrant III, and there is no labeled point shown in quadrant IV.

Practice Problems

a. Graph these points in the coordinate plane: (–2, 3), (2, 3), (–2, –3), (2, –3).

b. Connect all of the points. Describe the ﬁgure.

2. Write the coordinates of each point.

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3. These three points form a horizontal line: (–3.5, 4), (0, 4), and (6.2, 4). Name two additional points that fall on this line.

Review Problems

4. One night, it is 24∘ Celsius (℃) warmer in Tucson than it was in Minneapolis. If the temperatures in Tucson and Minneapolis are opposites, what is the temperature in Tucson?

A. –24∘C

B. –12∘C

C. 12∘C

D. 24∘C

5. Noah is helping his band sell boxes of chocolate to fund a ﬁeld trip. Each box contains 20 bars and each bar sells for $1.50.

a. Complete the table for values of ��.

b. Write an equation for the amount of money, ��, that will be collected if �� boxes of chocolate bars are sold. Which is the independent variable and which is the dependent variable in your equation?

c. Write an equation for the number of boxes, ��, that were sold if �� dollars were collected. Which is the independent variable and which is the dependent variable in your equation?

Unit 7, Lesson 13: Plotting and Identifying Points on the Coordinate Plane

Warm-Up: Unlabeled Points

Label each point on the coordinate plane with the appropriate letter and ordered pair.

�� (7, –5.5) �� (–8, 4) �� (3, 2)

(–3.5, 0.2)

Collaborative Activity: Signs of Numbers in Coordinates

1. Write the coordinates of each point.

2. Plot and label the points shown.

• �� (–6, –4.5)

• �� (1, 5)

• �� (–5 2, 3)

3. List all the points in each quadrant.

Quadrant I: _____________

Quadrant II: _____________

Quadrant III: _____________

Quadrant IV: _____________

Collaborative Activity: Game Score

A game board is shown on the coordinate plane, along with the number of points earned for landing in each area.

1. Several points are given in the table. Locate each point on the coordinate grid. Complete the table by writing the score value for landing on each point.

2. Calculate the game score by ﬁnding the sum of all the points.

Game score:

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Lesson Summary

The 2 numbers that name the location of a point in the coordinate plane are called the coordinates.

A coordinate is a number used to locate a point on a number line. It is one of the numbers in an ordered pair that locates a point on a coordinate plane.

Locations of points in the coordinate plane include 2 coordinates, (��, ��). For this reason, the location is also called a coordinate pair or an ordered pair. The ��-value of the ordered pair represents the location of the point along the horizontal axis (��-axis), and the ��-value of the ordered pair represents the location of the point along the vertical axis (��-axis). Sometimes, points are named with a letter, such as point ��. This is to help identify the point more easily, such as when being asked to locate or identify the coordinates of a point .

• The points in quadrant I have positive ��- and ��-values.

• The points in quadrant II have negative ��-values and positive ��-values.

• The points in quadrant III have negative ��- and ��-values.

• The points in quadrant IV have positive ��-values and negative ��-values.

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Practice Problems

1. Four points are shown on the coordinate plane.

Label each point with its coordinates.

-1-2-3-4-5-6-7-8-9-10

2. Plot and label each point on the coordinate plane based on the characteristics described.

a. Plot point �� with an ��-coordinate of 0.

b. Plot point �� with a ��-coordinate of 0.

c. Plot point �� in quadrant II with an ��-coordinate of –3.

d. Plot point �� in quadrant III with a ��-coordinate of –4.

e. Plot point �� in quadrant I with a ��-coordinate of 2.

f. Plot point �� in quadrant IV with an ��-coordinate of 1.

3. Solve each equation.

a. 3�� = 12

b. �� + 3.3 = 8.9

c. 1 = 1 4 ��

d. 5 1 2 = �� + 1 4

e. 2�� = 6.4

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Unit 7, Lesson 14: Constructing the Coordinate Plane

Warm-Up: English Winter

The following data were collected over one December afternoon in England.

1. Which set of axes would you choose to represent these data? Explain your reasoning.

2. Explain why the other two sets of axes did not seem as appropriate as the one you chose.

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Exploration Activity: Axes Drawing Decisions

1. Here are three sets of coordinates. For each set, draw and label an appropriate pair of axes and plot the points.

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a. (1, 2), (3, –4), (–5, –2), (0, 2.5)

b. (50, 50), (0, 0), (–10, –30), (–35, 40)

2. Discuss with a partner:

• How are the axes and labels of your three drawings diﬀerent?

• How did the coordinates aﬀect the way you drew the axes and label the numbers?

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Collaborative Activity: Postively A-maze-ing

Here is a maze on a coordinate plane. The black point in the center is (0, 0). The side of each grid square is 2 units long.

1. Enter the above maze at the location marked with a green segment. Draw line segments to show your way through and out of the maze. Label each turning point with a letter. Then, list all the letters and write their coordinates.

2. Choose any 2 turning points that share the same line segment. What is the same about their coordinates? Explain why they share that feature.

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Lesson Summary

The coordinate plane can be used to show information involving pairs of numbers. When using the coordinate plane, pay close attention to what each axis represents and what scale each uses.

The table shows data about the temperatures, in degrees Celsius (℃), in Minneapolis one evening in relation to the time since midnight, in hours (hr.).

To plot the temperature data, let the ��-axis represent time, in hr. since midnight, and let the ��-axis represent temperature, in ℃.

• In this case, ��-values less than 0 represent hr. before midnight, and ��-values greater than 0 represent hr. after midnight.

• On the ��-axis, the values represent temperatures above and below the freezing point, 0℃.

The data involves whole numbers, so it is appropriate that the each square on the grid represents a whole number.

• To the left of the origin, the ��-axis needs to go at least as far as –4. To the right, it needs to go to 3 or greater.

• Below the origin, the ��-axis has to go at least as far as –8. Above the origin, it needs to go to 3 or greater. These ranges will show all the data from the table.

A graph of the data with the axes labeled appropriately is shown.

On this coordinate plane, a point at (0, 0) would mean the temperature is 0℃ at midnight. The point at (–4, 3) means the temperature is 3℃ at 4 hr. before midnight, or 8 p.m.

Practice Problems

1. Draw and label an appropriate pair of axes and plot the points.

• � 1 5 , 4 5 �

• �–3 5 , 2 5 � • �–1 1 5 , –4 5 � • � 1 5 , –3 5 �

2. Diego was asked to plot these points: (–50, 0), (150, 100), (200, –100), (350, 50), (–250, 0). What interval could he use for each axis? Explain your reasoning.

a. Name 4 points that would form a square with the origin at its center.

b. Graph these points to check if they form a square.

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Review Problems

4. Which of the following changes would you represent using a negative number? Explain what a positive number would represent in that situation.

a. A loss of 4 points

b. A gain of 50 yards

c. A loss of $10

d. An elevation above sea level

5. Jada is buying notebooks for school. The cost of each notebook is $1.75.

a. Write an equation that shows the cost of Jada’s notebooks, ��, in terms of the number of notebooks, ��, that she buys.

b. Which of the following could be points on the graph of your equation?

□ (1.75, 1)

□ (2, 3.50)

□ (5, 8.75)

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□ (17.50, 10)

□ (9, 15.35)

6. A corn ﬁeld has an area of 28.6 acres. It requires about 15,000,000 gallons (gal.) of water. About how many gal. of water per acre is that?

a. 5,000

b. 50,000

c. 500,000

d. 5,000,000

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Unit 7, Lesson 15: Shapes on the Coordinate Plane

Warm-Up: Figuring Out the Coordinate Plane

1. Draw a ﬁgure in the coordinate plane with at least three of following properties:

• 6 vertices

• 1 pair of parallel sides

• At least 1 right angle

• 2 sides with the same length

2. Is your ﬁgure a polygon? Explain how you know.

Exploration Activity: Find the Polygon

1. The table lists the vertices of a polygon.

a. Without plotting the coordinates, explain what type of polygon you think will be created by plotting and connecting the points in the table.

b. Plot the polygon on the coordinate plane. Connect the points in the order listed, connecting the last coordinate back to the ﬁrst.

c. Revisit your answer to part A. Explain whether your answer matches the polygon plotted in part B.

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Collaborative Activity: Plotting Polygons

The tables list the coordinates of 4 polygons.

1. Plot each polygon on the coordinate plane, connect the points in the order they are listed, and connect the last point back to the ﬁrst one. Label each polygon with its letter name.

2. How many sides does each polygon have?

Polygon A

Polygon

Lesson Summary

Coordinates can be given in lists, as seen in previous lessons, or in tables, as seen in this lesson. Tables can be represented vertically or horizontally. Each corresponding pair of ��- and ��-values from a table can be represented by an ordered pair, (��, ��).

Coordinates can also be used to graph a polygon in the coordinate plane, given the coordinates of the polygon’s vertices. An example is shown.

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Practice Problems

1. The coordinates of a rectangle are (3, 0), (3, –5), (–4, 0), and (–4, –5). Plot the rectangle on the coordinate plane.

2. Draw a square with one vertex at the point (–3, 5) and another at the point (2, 0). Write the coordinates of the other vertices.

3. Plot and connect the points given in the table to form a polygon.

Review Problem

4. For each situation, select all the equations that represent it. Choose one equation and solve it.

a. Jada’s cat weighs 3.45 kg. Andre’s cat weighs 1.2 kg more than Jada’s cat. How much does Andre’s cat weigh?

□ �� = 3.45 + 1.2

□ �� = 3.45 – 1.2

□ �� + 1.2 = 3.45

□ �� – 1.2 = 3.45

b. Apples cost $1.60 per pound at the farmer’s market. They cost 1.5 times as much at the grocery store. How much do the apples cost per pound at the grocery store?

□ �� = (1.5) ⋅ (1.60)

□ �� = 1.60 ÷ 1.5

□ (1.5)�� = 1.60

□ �� 1.5 = 1.60 y x 0

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Unit 8, Lesson 1: Interpreting Negative Integers

Warm-Up: Using the Thermometer

Here is a weather thermometer. Three of the numbers have been left oﬀ.

1. What numbers go in the boxes?

2. What temperature does the thermometer show?

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Exploration Activity: Gauging the Temperature

Four thermometers are shown, each with a diﬀerent temperature gauged in degrees Celsius (℃).

1. What temperature is shown on each thermometer?

A: __________ B: __________ C: __________

__________

2. Which thermometer shows the highest temperature?

3. Which thermometer shows the lowest temperature?

4. Suppose the temperature outside is –4℃. Explain whether that is colder or warmer than the coldest temperature shown.

Exploration Activity: Using Integers with Elevation

A picture of some sea animals is shown. The number line to the left of the animals shows the vertical position of each animal above or below sea level, in meters (m).

1. To the nearest integer, how far above or below sea level is each animal? Measure to their eye level, and label each animal’s elevation in the image.

2. A mobula ray is 3 m above the surface of the ocean. Describe how its vertical position compares to the height or depth of each of the following.

a. The jumping dolphin

b. The ﬂying seagull

c. The octopus

3. An albatross is 5 m above the surface of the ocean. Describe how its vertical position compares to the height or depth of each of the following.

a. The jumping dolphin

b. The ﬂying seagull

c. The octopus

4. A clownﬁsh is 2 m below the surface of the ocean. Describe how its vertical position compares to the height or depth of each of the following.

a. The jumping dolphin

b. The ﬂying seagull

c. The octopus

5. The vertical distance of another dolphin above the dolphin in the picture is 3 m. What is the new dolphin’s distance from the surface of the ocean?

Lesson Summary

Positive and negative integers can be used to represent temperature and elevation.

When numbers represent temperatures, positive numbers represent temperatures that are warmer than zero, and negative numbers represent temperatures that are colder than zero. This thermometer shows a temperature of –1℃, which is expressed as –1℃.

When numbers represent elevations, positive numbers indicate positions above sea level, and negative numbers indicate positions below sea level. The number line shows the order of integers.

On a horizontal number line, a number is always less than any number to its right. On a vertical number line, a number is always less than any number above it. For example, –7 < –3.

Use absolute value to describe how far a number is from 0. The numbers 15 and –15 are both 15 units from 0, so |15| = 15 and |–15| = 15. These values, 15 and –15, are examples of opposites. They are on opposite sides of 0 on the number line, and they are the same distance from 0.

Practice Problems

1. It was –5℃ in Copenhagen and –12℃ in Oslo. Which city was colder?

a. A ﬁsh is 12 m below the surface of the ocean. What is its elevation?

b. A sea bird is 28 m above the surface of the ocean. What is its elevation?

c. If the bird is directly above the ﬁsh, how far apart are they?

3. Compare each pair of values using >, =, or <.

a. 3 _____ –3

b. 12 _____ 24

c. –12 _____ –24

d. –5 _____ 5

e. 7 _____ |–7|

f. |–7| _____ –7

4. Find the area of each shape. Show your reasoning.

5. A particular shade of orange paint has 2 cups (c.) of yellow paint for every 3 c. of red paint. On the double number line, circle the numbers of c. of yellow and red paint needed for 3 batches of orange paint.

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Unit 8, Lesson 2: Changing Temperature

Warm-Up:

Which One Doesn’t Belong: Arrows

Which pair of arrows doesn’t belong?

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Exploration Activity: Warmer and Colder

1. Complete the table and draw a number line diagram for each situation.

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2. Complete the table and draw a number line diagram for each situation.

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Collaborative Activity: Winter Temperatures

One winter day, the temperature in Houston is 8° Celsius (℃). Find the temperatures in these other cities. Explain or show your reasoning.

1. In Orlando, it is 10° warmer than it is in Houston.

2. In Salt Lake City, it is 8° colder than it is in Houston.

3. In Minneapolis, it is 20° colder than it is in Houston.

4. In Fairbanks, it is 10° colder than it is in Minneapolis.

5. Write an addition equation that represents the relationship between the temperature in Houston and the temperature in Fairbanks.

Lesson Summary

If it is 42° outside and the temperature increases by 7°, add the initial temperature and the change in temperature to ﬁnd the ﬁnal temperature.

42 + 7 = 49

If the temperature decreases by 7°, subtract 42 – 7 to find the final temperature, or think of the change as –7°. Then, add to find the final temperature, as shown.

42 + (–7) = 35

In general, a change in temperature can be represented by a positive number if it increases and a negative number if it decreases. The final temperature can be determined by adding the initial temperature and the change. If it is 3° and the temperature decreases by 7°, then add 3 + (–7) to find the final temperature, which is –4°.

Integers can be represented with arrows on a number line. Positive integers are represented with arrows that start at 0 and point to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.

Represent negative integers with arrows that start at 0 and point to the left. For example, this arrow represents –4 because it is 4 units long and it points to the left.

To represent addition on a number line, put the arrows “tip to tail” in the order presented. Two examples are shown.

Practice Problems

a. The temperature is –2℃. If the temperature rises by 15℃, what is the new temperature?

b. At midnight, the temperature is –6℃. At midday the temperature is 9℃. By how much did the temperature rise?

2. Draw a diagram to represent each of these situations. Then write an addition expression that represents the ﬁnal temperature.

a. The temperature was 80° Fahrenheit (℉) and then fell 20℉.

b. The temperature was –13℉ and then rose 9℉.

c. The temperature was –5℉ and then fell 8℉.

Review Problems

3. Han earns $33.00 for babysitting 4 hours (hr.). At this rate, how much will he earn if he babysits for 7 hr.? Explain your reasoning.

4. Without computing, decide whether the value of each expression is much smaller than 1, close to 1, or much greater than 1.