EM2_G7-8_M1_Learn_23A_881531_Updated 08.23

What does this painting have to do with math?

The French neoimpressionist Paul Signac worked with painter Georges Seurat to create the artistic style of pointillism, in which a painting is made from small dots.

Signac’s Vue de Constantinople, La Corne d’Or Matin shows the Golden Horn, a busy waterway in Istanbul, Turkey. How many dots do you think were used to make this pointillist painting? How would you make an educated guess?

On the cover

Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907

Paul Signac, French, 1863–1935

Oil on canvas

Private collection

Paul Signac (1863–1935), Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907. Oil on canvas. Private collection. Photo credit: Peter Horree/Alamy Stock Photo

LEARN

Proportions and Linearity ▸ 7–8

Module 1 Rational and Irrational Numbers

2 One- and Two-Variable Equations

3 Two-Dimensional Geometry

4 Graphs of Linear Equations and Systems of Linear Equations

5 Functions and Three-Dimensional Geometry

6 Probability and Statistics

Add and Subtract Rational Numbers

Student Edition: Grade 7–8, Module 1, Topic A The Worst Game Show Ever

TOPIC A

T

hat’s corect! You lose $500! But... I got it right! Shouldn’t I gain money, rather than lose it?

Hm... judges

y apologies! You’re quite right!

Thank you.

You win -$500!

This is the worst game show ever.

It’s easy to get confused when you see something like “− 500.” That’s because the “−” symbol has two meanings: “subtraction” and “negative.”

Are we dealing with the operation “subtract 500,” as in 150 − 500?

Or are we dealing with the number “negative 500,” as in 150 + (−500)?

Do not be deceived by words like giving or adding. Offering someone a gift of − $500 is not a generous thing to do. In fact, it is the same as stealing $500 right out of the person’s pocket.

Name Date

Adding Integers and Rational Numbers

Adding Integers

For problems 1 and 2, use the number line to model the addition expression.

1. 3 + 5

109876543210–10–9–8–7–6–5–4–3–2–1

109876543210–10–9–8–7–6–5–4–3–2–1

For problems 3 and 4, write the addition expression that represents the situation. Then use the number line to model the addition expression and determine the sum.

3. Abdul earns $6. He then spends $2.

109876543210–10–9–8–7–6–5–4–3–2–1

4. The temperature rises 2°F from 0°F. Then the temperature drops 6°F

109876543210–10–9–8–7–6–5–4–3–2–1

5. Consider the expressions.

a. Determine the absolute values of the addends in each expression.

b. Use the number line to model each expression and determine the sum.

c. What do you notice about the distance on the number line between the first addend and the sum in each expression?

d. How can absolute value be used to help determine the sign of the sum? Write a conjecture.

6. Complete the table.

7. Determine the sum.

−40 + 25 + (−5)

Adding Rational Numbers

For problems 8–11, determine the sum.

8. 2.4 + (−3.4)

9. 5 3 + 8 ( 3 1 8 )

10. −3.4 + 9.8 11.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1

For problems 1 and 2, use the number line to model the expression and determine the sum.

3. Consider the following expression.

−62 + 23

a. Use the number line to model the expression.

b. Determine the sum.

Name Date

Adding Integers and Rational Numbers

In this lesson, we

• identified additive inverses and used a number line to model them.

• used a number line to model addition expressions.

• predicted the sign of the sum of addition expressions.

• used strategies to determine sums of rational numbers.

Examples

Terminology

The additive inverse of a number is a number such that the sum of the two numbers is 0. The additive inverse of a number x is the opposite of x because x + (−x) = 0

1. Use the number line to model the expression and determine the sum.

Start where the first directed line segment ends: 8. Because the second addend is −13, draw the directed line segment to the left and make it 13 units long.

8 + (−13)

Start at 0. Because the first addend is 8, draw the directed line segment to the right and make it 8 units long.

The endpoint of the second directed line segment is the value of the sum. In this case, the sum is −5.

2. Last month, Ethan withdrew $53 from his savings account. This month, he withdrew another $40.

a. Write an addition expression to represent the changes in the balance of Ethan’s savings account.

−53 + (−40)

Withdrawing money means taking money out of an account, so the amounts Ethan withdrew are represented by negative numbers in the expression.

b. Use the number line to model your addition expression.

Use a blank number line to draw a model that represents the addition expression. Mark the location of 0 and have the lengths of the directed line segments relate to the absolute value of the numbers they represent.

For problems 3–5, predict whether the sum of the expression is a positive number, 0, or a negative number. Explain your reasoning.

3. −15.3 + (−5.4)

The sum is a negative number because both addends are negative.

4. 11 3 7 + ( 5 1 7 )

The sum is a positive number because the absolute value of 11 3 7 is greater than the absolute value of −5 1 7 .

When the addends have the same sign, the sum also has that sign.

When two addends have opposite signs, the sum has the same sign as the addend with the greater absolute value.

5. 7 8 + ( 7 8 )

The sum is 0 because 7 8 and 7 8 are additive inverses. For problems 6 and 7, determine the sum.

7. −5.2 + (−4.5)

When the addends are opposite values, they sum to 0.

−5.2 + (−4.5) = −5 + (−0.2) + (−4) + (−0.5)

= (−5 + (−4)) + (−0.2 + (−0.5))

= −9 + (−0.7)

= −9.7

The two addends have opposite signs, so using decomposition and additive inverses is an effective strategy. To find

The two addends have the same sign, so decomposing the addends into integer and non-integer parts is an effective strategy to determine the sum.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1

Name Date

1. Sara gains 10 points.

a. Describe an opposite action.

b. Write an addition expression that represents both actions.

c. Use the number line to model the addition expression you created in part (b). Then determine the sum.

2. On Monday, the temperature in Chicago was 0°F at 1:00 a.m. The temperature decreased 7°F. Then the temperature increased 10°F

a. Write an addition expression to represent the changes in temperature.

b. Use the number line to represent the addition expression you created in part (a).

c. What is the temperature after these temperature changes?

3. What is the additive inverse of −4.3? Explain how you know.

4. Predict whether the sum of each expression is a positive number, zero, or a negative number.

For problems 5–7, use the number line to model the expression and determine the sum.

5. 3 + (−6)

109876543210–10–9–8–7–6–5–4–3–2–1

6. −6 + (−3)

109876543210–10–9–8–7–6–5–4–3–2–1

7. −6 + 3

109876543210–10–9–8–7–6–5–4–3–2–1

8. Consider the following addition expression.

−73 + 18

a. Use the number line to model the expression.

b. Determine the sum.

For problems 9–12, determine the sum.

+ 35

Remember

For problems 13–16, add.

17. If 4 people share 9 cups of popcorn equally, how many cups of popcorn does each person get?

18. Use each location to plot its corresponding point on the number line.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

Name Date

KAKOOMA® with Rational Numbers

Estimate and Evaluate

For problems 1–5, estimate, and then find the sum.

1. −3.45 + (−1.52)

LESSON

4. −9.35 + 9.75

5. 7.63 + (−10.42)

6. Vic has $52.13 in his checking account. At midnight, a debit of $3.06 and a debit of $18.24 will post to his account.

a. Write an addition expression to represent the situation. Then estimate the sum.

b. What is the value of the expression in part (a)?

c. What is the amount of money in Vic’s checking account after the debits post?

Solving a KAKOOMA®

7. In each four-number square, find the one number that is the sum of two other given numbers. Transfer all four sums to the puzzle below to create one final puzzle, and then solve.

© Greg Tang

Final

Creating a KAKOOMA®

8. Create your own KAKOOMA® puzzle by using the blank puzzle shown. Refer to problem 7, and make sure that your KAKOOMA® puzzle follows this structure. Use rational numbers between −10 and 10. Each four-number square must include at least two non-integer rational numbers and at least one number less than 0. The same number cannot be used more than once in a square.

Find the number that is the sum of 2 others.

© Greg Tang

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

Name

For problems 1 and 2, estimate, and then find the sum.

1. 2 3 8 + ( 5 2 )

Date

2. 7.29 + (−12.74)

3. Noor uses decomposition to find

. Her work is shown. How does she use properties of operations in her work?

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

Name Date

KAKOOMA® with Rational Numbers

In this lesson, we

• estimated the sums of addition expressions.

• used decomposition and properties of operations to add rational numbers.

• evaluated addition expressions to solve and create puzzles.

Examples

For problems 1–3, estimate, and then find the sum.

1. −5 2 3 + 2 1 3

Estimation: −6 + 2 = −4

Sum:

The addends have opposite signs, so an effective strategy is to decompose one of the addends and use additive inverses.

Estimate the sum by rounding each addend to an integer. Then add the integers.

The additive inverse of 2 1 3 is −2 1 3 . Decompose the rational number −5 2 3 into 1 −3 3 and −2 1 3 to use additive inverses.

2. −15.6 + (−3.25)

Estimation: −16 + (−3) = −19

Sum:

−15.6 + (−3.25) = −15 + (−0.6) + (−3) + (−0.25)

= −15 + (−3) + (−0.6) + (−0.25)

= (−15 + (−3)) + (−0.6 + (−0.25))

= −18 + (−0.85)

= −18.85

Apply the associative property of addition to group the integer parts together and the non-integer parts together.

3. −21.4 + 18.45

Estimation: −21 + 18 = −3

Sum:

The addends have opposite signs.

The addends have the same sign, so decomposing each addend into its integer part and non-integer part is an effective strategy.

Apply the commutative property of addition to rearrange the integer parts and the non-integer parts.

The addends have opposite signs, so finding the difference between the absolute values of the two addends is an effective strategy.

The negative addend has the greater absolute value, so the sum is negative. The sum is −2.95.

4. In the five-number pentagon, find the one number that is the sum of two other numbers.

It is not necessary to find the sum of every pair of numbers. Because 2 and 1 1 2 are the only two positive numbers and the sum of two positive numbers is a positive number, their sum is not one of the remaining numbers.

© Greg Tang

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

1. Jonas uses decomposition to find 7

. His work is shown. How does Jonas use properties of operations in his work?

2. Ava uses decomposition to find 8.14 + (−8.34). Her work is shown. How does Ava use properties of operations in her work?

8.14 + (−8.34) = 8.14 + (−8.14) + (−0.2)

= (8.14 + (−8.14)) + (−0.2)

= 0 + (−0.2)

= −0.2

For problems 3–11, estimate, and then find the sum.

9. 5.15 + (−3.78)

11. 10 1 6 + (−8 2 5 )

12. Yu Yan has $35.27 in her checking account. At midnight, debits of $12.82 and $4.93 will post to her account.

a. Write an addition expression to represent the situation. Then estimate the sum.

b. What is the value of the expression in part (a)?

c. What is the amount of money in Yu Yan’s checking account after the debits post?

13. Ethan has $147.89 in his savings account. On the last day of the month, the bank posted a monthly service charge of $5.00 and interest earned of $0.49

a. Write an addition expression to represent the situation. Then estimate the sum.

b. What is the value of the expression in part (a)?

c. What is the amount of money in Ethan’s savings account after the transactions post?

14. Shawn evaluated the following expression but got an incorrect answer. Find and describe any errors Shawn made in his work.

15. In each five-number pentagon, find the one number that is the sum of two other numbers. Use all five sums to create one final puzzle and solve.

© Greg Tang

Remember

For problems 16–19, add.

20. Lily walks 60 feet in 10 seconds. Maya walks 25 feet in 5 seconds. Who walks at a faster rate? Explain how you know.

21. Which equation correctly models the following statement?

−30 is 30 units from 0 on the number line.

A. (−30) = 30

B. −30 = 30

C. |30| = −30

D. |−30| = 30

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3

Name Date

Finding Distances to Find Differences

1. Write a conjecture about how to subtract integers by using the pattern you see in the displayed table. Explain how the pattern you see supports your conjecture.

Relating the Distance

For problems 2–9, write the related unknown addend equation. Then use the number line to find the unknown addend. Record your answers in the columns labeled Unknown Addend Equation and Unknown Addend.

Reasoning About Integer Subtraction

10. Consider the following sample work.

32 − (−45) = 32 − 45 = −13

a. Explain an error in the sample work.

b. Correctly evaluate the expression.

11. Sea level is represented by 0 feet. A gannet bird is at an elevation of 28 feet. It dives straight down to an elevation of −19 feet to get a fish. How many feet does the gannet dive? Draw a model to represent the distance that the gannet dives.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3

Name Date

1. Consider the expression 7 − 12

a. Draw a number line and plot the integers in the expression.

b. What is the distance on the number line between the integers in the expression?

c. Write the expression as an unknown addend equation.

d. What is the unknown addend?

2. Consider the expression −4 − (−10).

a. Draw a number line and plot the integers in the expression.

b. What is the distance on the number line between the integers in the expression?

c. Write the expression as an unknown addend equation.

d. What is the unknown addend?

Name

Finding Distances to Find Differences

In this lesson, we

• examined subtraction methods and their constraints when subtracting integers.

• found the distance between the two numbers in a subtraction expression by plotting them on a number line.

• wrote subtraction expressions as unknown addend equations and used a number line to model the equations.

• found unknown addends by using a number line and drawing directed line segments.

Examples

1. Use the number line to plot points that represent A and B. State the distance between them. Then determine the number that should be added to A to get a sum of B.

To find the number to add to A to get B, it might be helpful to draw a directed line segment from A to B. Because the directed line segment from 6 to −4 goes to the left and has a length of 10 units, it represents −10 The number −10 can be added to A to get a sum of B

2. Consider the expression −4 − 6.

a. Write the expression as an unknown addend equation.

6 + = −4

Any subtraction expression can be written as an unknown addend equation. The value of the subtraction expression −4 − 6 is −10, which is also the unknown addend that is added to 6 to get a sum of −4 To determine the unknown addend, ask the question, What can be added to 6 to make −4?

b. What must be added to 6 to get a sum of −4? −10

Because addition and subtraction are related, the unknown addend is also the difference of the subtraction expression.

6 + (−10) = −4

−4 − 6 = −10

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3

For problems 1–6, use the number line to plot points that represent A and B. State the distance between them. Then determine what must be added to A to get a sum of B.

7. Consider the expression −3 − (−2).

a. Write the expression as an unknown addend equation.

b. What must be added to −2 to get a sum of −3?

c. What is the value of the subtraction expression −3 − (−2)? Explain how you know.

8. Consider the expression −2 − 9.

a. Write the expression as an unknown addend equation.

b. What must be added to 9 to get a sum of −2?

c. What is the value of the subtraction expression −2 − 9? Explain how you know.

9. Consider the expression 4 − (−10).

a. Write the expression as an unknown addend equation.

b. What must be added to −10 to get a sum of 4?

c. What is the value of the subtraction expression 4 − (−10)? Explain how you know.

10. In Baltimore, the temperature is 35°F. In Milwaukee, the temperature is −7°F. How much warmer in degrees Fahrenheit is it in Baltimore than in Milwaukee?

15. Use the number line to model the expression. Then determine the sum.

16. Determine the sum.

17. Complete each comparison by using <, >, or = .

a. 3 8 5 16

b.

c. 5.43 |−6.2|

d.

e. |

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4 LESSON

4

Subtracting Integers

Subtracting Negative Values

For problems 1–6, complete the table. An example is provided.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4

Name Date

Write each expression as an equivalent addition expression. Then find the sum.

1. −4 − (−20)

2. 7 − (−36)

3. −44 − 9

4. 37 − 62

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4

Name Date

Subtracting Integers

In this lesson, we

• represented subtraction expressions by using a number line and observed patterns to understand the relationship between subtraction and addition of integers.

• determined the difference when subtracting a negative integer.

• evaluated subtraction expressions by writing equivalent addition expressions.

Example

Complete the table. Write an equivalent addition expression and find the sum. Then write an unknown addend equation and find the unknown addend.

Subtraction expressions can be written as equivalent addition expressions. Change the subtraction sign to the addition sign and change the second integer to its opposite.

In this case, 16 − 24 is written as 16 + (−24).

Once a subtraction expression is written as an equivalent addition expression, use the integer addition strategies to find the sum.

When using unknown addend equations to check your work, ask this question: What number can I add to the second integer in the original subtraction expression to get the first integer?

In this case ask, What number can I add to −6 to get −21?

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4

Name

1. Use the expression −2 − 5 to complete parts (a)–(d).

a. Use the number line to model the expression.

Date

b. Write the subtraction expression as an equivalent addition expression.

c. Determine the sum of the expression from part (b).

d. Evaluate −2 − 5

2. Which expressions have the same value as −16 − 29? Choose all that apply.

A. 16 + (−29)

B. −16 + (−29)

C. −16 − (−29)

D. 16 − 29

E. (0 − 16) + (0 − 29)

For problems 3–10, write an equivalent addition expression and find the sum. Then write an unknown addend equation and find the unknown addend.

Subtraction Expression

Equivalent Addition Expression and Sum

Unknown Addend Equation and Unknown Addend

Remember

For problems 11–14, add.

+

(

)

15. Estimate, and then find the sum.

16. Which expressions have a value of 105? Choose all that apply.

10 × 5 B. 10 + 5 C. 10,000 D. 100,000

F. 10 + 10 + 10 + 10 + 10

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

Name Date

Subtracting Rational Numbers

LESSON 5

1. Use the given number lines to model each subtraction expression. Then evaluate the expression. Write and solve an unknown addend equation to check your work.

Decomposing to Subtract

For problems 2–4, evaluate the expression.

2. − 4 2 − 3 (− 2 1 3 ) 3. −

For problems 5–8, evaluate the expression.

5. −3.4 − 2.1

6. 0.4 − 1.4

7. −2.7 − 1.35 − (−6.25)

8. 1.85 − 6.45 + 4.3

How Much?

9. Liam owes his grandmother $5.49. His grandmother tells Liam that she will remove $2.50 from his debt after he cleans the bathroom.

a. Write a subtraction expression that represents this situation.

b. Evaluate the expression from part (a). Explain what the result means in this situation.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

Subtracting Rational Numbers

In this lesson, we

• used number lines and unknown addend equations to confirm that a subtraction expression involving rational numbers can be written as an equivalent addition expression.

• used decomposition and properties of operations to make subtraction expressions simpler to evaluate.

• evaluated subtraction expressions involving rational numbers.

• wrote and evaluated a subtraction expression to represent a real-world situation.

Examples

For problems 1–4, evaluate the expression.

Write the subtraction expression as an equivalent addition expression first.

Each mixed number can be decomposed into an integer part and a non-integer part.

Use the commutative and associative properties of addition to rearrange the expression and to group the integers and non-integer parts together before evaluating.

When using the associative property, consider grouping negative

One strategy that is useful in this case is to decompose − 0.46 into − 0.45 + (−0.01). That way, an additive inverse with 0.45 is created. Use the associative property of addition to group the additive inverses together.

5. Vic owes his brother $4.28 for a snack and $7.43 for a shirt.

a. Write a subtraction expression that represents this situation.

− 4.28 − 7.43

Vic owes his brother $4.28, so that can be represented by − 4.28.

b. Evaluate the expression from part (a). Explain what the result means in this situation.

− 4.28 − 7.43 = − 4.28 + (− 7.43)

= − 4 + (− 0.28) + (− 7) + (− 0.43)

= (− 4 + (− 7)) + (− 0.28 + (− 0.43))

= − 11 + (− 0.71)

= − 11.71

Vic owes his brother $11.71

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

PRACTICE 5

11. Identify and correct the error in the sample work shown.

− 1.23 − (− 5.26) = − 1.23 + 5.26

= − 1 + 0.23 + 5 + 0.26 = (−1 + 5) + (0.23 + 0.26) = 4 + 0.49 = 4.49

12. Maya goes deep-sea diving. She first dives 18.28 meters below sea level. Then she dives down another 4.71 meters.

a. Write a subtraction expression that represents this situation.

b. Evaluate the expression from part (a). Explain what the result means in this situation.

Remember

For problems 13–16, add.

17. Consider the expression − 8 − (−6).

a. Draw a number line and plot the integers in the expression.

b. What is the distance on the number line between the integers in the expression?

c. Write the expression as an unknown addend equation.

d. What is the unknown addend?

18. Complete each statement.

a. The opposite of − 6 is .

b. The opposite of the opposite of − 9 is .

c. The opposite of 0 is

d. The value of − (− 5 6 ) is .

A Confusingly Cold Week Ugh. It’s apparently going to be ‒6o for the next three days. That’s too cold. Oh no! That’s ‒18o in all!

That’s not how temperatures work. Oh, you’re right! I should *divide* by 3. That’s only ‒2o! Still no.

Where do we see negative numbers in real life? Well, if you live in a cold climate, you may see them in your winter temperatures.

How cold are you when temperatures are negative? It depends on which system you use for measuring temperature. In degrees Celsius, −1° is just below freezing—cold enough that you’ll definitely want a jacket. But in degrees Fahrenheit, −1° is absolutely frigid! You’ll need to cover your face if you’re outside for more than a few minutes.

Now can you explain the errors in the cartoon above? It may be trickier than it sounds!

Student Edition: Grade 7–8, Module 1,

Topic B, Lesson 6

Name Date

Multiplying Integers and Rational Numbers

A Positive Number Times a Negative Number

1. An American football team has 3 consecutive plays where they lose 2 yards during each play.

a. Use a number line to model the situation.

b. Write an addition expression to represent this situation.

c. Write a multiplication expression to represent this situation.

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

For problems 2–5, determine the product. 2. 2(−11)

A Negative Number Times a Positive Number

6. Complete the number sentences to make them true.

For problems 7–9, use the commutative property to determine the product.

7. −0.25(10)

8.

A Negative Number Times a Negative Number

For problems 10 and 11, complete the number sentences to make them true.

11.

2(−10) = −20

1(−10) = −10

0(−10) = 0

−1(−10) =

−2(−10) =

−3(−10) =

12. Use properties to fill in the blanks to determine whether the conjecture is true or false.

Conjecture: (−3)(−2) = 6

Justification:

Line 1: (−3)( ) = 0

Line 2: (−3)(−2 + ( )) = 0

Zero product property

Additive inverse

Line 3: (−3)(−2) + ( )( ) = 0 Distributive property

Line 4: (−3)(−2) + ( ) = 0

Line 5: + (−6) = 0

Conclusion:

Product of a positive number and a negative number

Additive inverse

Multiplication and Decomposition

For problems 13 and 14, evaluate the expression.

13. 4(− 2 1 –2 )

14. −3(1.5)

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6

Name Date

1. Ethan is scuba diving. He descends 6 feet from the water’s surface and rests. Then he descends another 6 feet and rests. Finally, he descends 6 more feet and examines a fish.

a. Use a number line to model the situation.

b. Write an addition expression to represent this situation.

c. Write a multiplication expression to represent this situation.

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

For problems 2 and 3, determine the product.

2. 4(5)

3. 1 4 ( 1 1 2 )

Name Date

Multiplying Integers and Rational Numbers

In this lesson, we

• related repeated addition to multiplication to make sense of a multiplication expression where the first factor is positive and the second factor is negative.

• analyzed patterns in tables to determine products.

• used properties of operations to determine products of rational numbers.

• applied decomposition and the distributive property to evaluate multiplication expressions.

Examples

Terminology

The zero product property states that if the product of two numbers is zero, then at least one of the numbers is zero.

This means that when ⋅ = 0, at least one of the factors is zero.

1. Lily is scuba diving. She descends 10 feet from the water’s surface and rests. Then she descends another 10 feet and rests. Finally, she descends 10 more feet to reach a reef.

a. Use a number line to model the situation.

The directed line segments point down because each of them represents descending 10 feet. There are 3 of them because Lily descends 3 separate times.

The end of the last directed line segment represents Lily’s depth below the water’s surface when she reaches the reef.

b. Write an addition expression to represent this situation.

−10 + (−10) + (−10) Because Lily descends 10 feet each time, the addends are −10

c. Write a multiplication expression to represent this situation. 3(−10) This multiplication expression describes an addition expression that has 3 groups of −10

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

−30

The product −30 means that Lily is 30 feet below the water’s surface when she reaches the reef.

For problems 2–5, determine the product.

−24

The absolute value of 4 times the absolute value of −6 is 24. The product of a positive number and a negative number is a negative number, so 4(−6) = −24.

The product of two negative numbers is a positive number.

2. 4(−6)

3. (−4)(−6) 24

4. (−0.43)(6)

(−0.43)(6) = (−0.4 + (−0.03))(6)

= (−0.4)(6) + (−0.03)(6)

= −2.4 + (−0.18) = −2.58 5.

−0.43 can be decomposed into −0.4 and −0.03

−5

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6

Name

PRACTICE 6

Date

1. The temperature at 6:00 p.m. is −4°F. By 7:00 p.m., the temperature has dropped 4°F. By 8:00 p.m., the temperature has dropped another 4°F.

a. Use a number line to model the situation.

b. Write an addition expression to represent this situation.

c. Write a multiplication expression to represent this situation.

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

For problems 2 and 3, write the expression as repeated addition. Then evaluate the expression.

For problems 4–19, determine the product.

2. 4(2)

3. 4( 2)

4. 3( 6)

5. 3( 6)

6. 5( 4)

7. −4(5)

8. 4( 5)

9. (−6)(−3) 10.

11. −6(3)

Remember

For problems 20–23, subtract.

For problems 24 and 25, write the expression as an equivalent addition expression and then evaluate.

24. 12 − ( 46)

25. 14 − 49

26. Evaluate the expression.

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7 LESSON

Exponential Expressions and Relating Multiplication to Division

Math Chat

For problems 1–20, evaluate the expression.

Correcting the Errors

21. Use the given sample work to answer parts (a) and (b). (−1.5)3 = (−1.5)(−1.5)(−1.5) = −2.25(−1.5) = −2.25(−1 + 0.5) = −2.25(−1) + 2.25(0.5) = −2.25 + 1.125 = 1.125

a. Explain an error in the sample work.

b. Correctly evaluate the expression (−1.5)3.

Relating Multiplication and Division

22. Complete the table.

23. What is the process for dividing integers?

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7

Name

For problems 1 and 2, evaluate the expression.

1.

Date

For problems 3 and 4, write the unknown factor equation related to the expression and then determine the unknown factor.

3. −20 ÷ 4

4. −45 ÷ (−5)

Name

Date

Exponential Expressions and Relating Multiplication to Division

In this lesson, we

• predicted whether the value of a multiplication or exponential expression would be positive or negative.

• evaluated exponential expressions that included negative numbers.

• wrote unknown factor equations to divide integers.

Examples

For problems 1–3, determine whether the value of the expression is positive or negative. Explain your answer.

1. (−3)(0.5)(−9)

The value of the expression is positive because there is an even number of negative factors.

2. (−2) 3

The value of the expression is negative because there is an odd number of negative factors.

3. −(−4) 3

The value of the expression is positive because there is an even number of negative factors.

The base is −2, and the exponent is 3 So there are 3 factors of −2 (−2)(−2)(−2)

The base is −4, and the exponent is 3. So there are 3 factors of −4. The negative sign outside the parentheses can be thought of as multiplying by −1, so that makes 4 negative factors.

For problems 4–7, evaluate the expression. 4. (

The exponent is 4, so there are 4 factors of

3 = (−0.4)(−0.4)(−0.4) = 0.16(−0.4) = −0.064

The exponent is 3, so there are 3 factors of −0.4 .

To determine the opposite of a number, multiply that number by −1. Follow the order of operations by evaluating exponents before multiplication. So determine the value of 0.32 before taking the opposite.

The negative sign outside the parentheses represents taking the opposite of (

For problems 8 and 9, write the unknown factor equation related to the expression and then determine the unknown factor.

8. −80 ÷ 4

The quotient of −80 ÷ 4 is the number that is multiplied by 4 to get a product of −80. The unknown factor is shown as a blank in the equation. −20

4 ⋅ = −80

−15 ⋅ = −60 4

9. −60 ÷ (−15)

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7

Name

Date

For problems 1–7, determine whether the value of the expression is positive or negative. Explain your answer.

1. ( 4)( 1 2 )( 0.3)

2. ( 1 4)( 2 )(0.3)

For problems 8–18, evaluate.

For problems 19–22, write the unknown factor equation related to the expression and then determine the unknown factor. 19.

Remember

For problems 23–26, subtract.

For problems 27 and 28, evaluate.

27. 6.8 − ( 2.9)

28. 7 _ 8 3 1 12

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8

Dividing Integers and Rational Numbers

Patterns in Integer Division

1. Complete the table by calculating each quotient with the values of p and q given in that row.

Division Expressions

For problems 2–7, divide.

For problems 8 and 9, evaluate the expression.

For problems 10–12, divide.

Applications of Division

13. At 9 p.m., the temperature is 4°F. At 6 a.m. the next morning, the temperature is − 12°F. Assume the temperature decreases at a constant rate. What is the approximate rate at which the temperature changes in degrees per hour?

14. A dolphin is underwater at an elevation of − 10 feet. The dolphin swims down at a rate of − 24 feet per minute. How long does the dolphin take to descend to an elevation of − 286 feet?

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8

1. Which numbers shown are equivalent to − 5 7 ? Circle all that apply.

For problems 2 and 3, divide.

Name Date

Dividing Integers and Rational Numbers

In this lesson, we

• observed that the quotient of any two integers p and q can be written equivalently in different ways: − p q = p − q = − ( p q ) if q ≠ 0.

• wrote unknown factor equations to make sense of division expressions containing 0.

• evaluated division expressions that contained rational numbers in fraction form and decimal form.

• solved real-world problems by dividing rational numbers.

Examples

1. Evaluate − a b for a = − 5 and b = 4 − (−5) − a = = 5 b 4 4

For problems 2–6, divide.

Terminology

A rational number is any number that can be written in the form p q , where p and q are integers and q ≠ 0

A negative sign means taking the opposite. The expression − (−5) can be read as “the opposite of − 5,” which is 5.

0 divided by any nonzero number has a quotient of 0

Think about the related unknown factor equation: − 15 times what number equals 0? The unknown factor is 0. 3. − 90 ÷ 0 Undefined

2. 0 ÷ (− 15) 0 = 0 − 15

When 0 is the divisor, the quotient is undefined. Think about the related unknown factor equation: 0 times what number equals − 90? There is no such number, so the quotient is undefined.

Dividing by − 5 4 has the same result as multiplying by its reciprocal, − 4 5

7. A fish swims at a constant rate from sea level to an elevation of − 52 feet in 6 1 2 seconds. Write and evaluate a division expression to show the change in elevation of the fish in feet per second.

The elevation of the fish changes by − 8 feet per second.

To find the rate of the fish’s change in elevation, divide the change in elevation in feet by the number of seconds the fish swims. The rate is negative because the elevation is decreasing.

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8

Name Date

1. Which expressions are equivalent to − 10 ÷ 5? Choose all that apply.

PRACTICE 8

2. Evaluate each expression for a = − 3 and b = 6.

For problems 3-10, divide.

11. Evaluate.

2 3 (- 16 + 10 ÷ (- 2))

12. At 5 p.m., the temperature is 12°F. At 5 a.m. the next morning, the temperature is - 9°F. Assume the temperature decreases at a constant rate. At what rate does the temperature change in degrees per hour?

Remember

For problems 13-16, subtract.

For problems 17 and 18, determine the product.

17. - 8(7)

18.3 4 (- 1 1 3 )

19. Evaluate the expression.

Student Edition: Grade 7-8, Module 1, Topic B, Lesson 9 LESSON

Name

Decimal Expansions of Rational Numbers

Nora’s Method vs. Long Division

1. Nora says that she can use the prime factorization of the denominator to determine how to write a decimal fraction. Her work is shown. Explain Nora’s thinking.

For problems 2 and 3, use Nora’s method to write the number as a decimal fraction. Then write the decimal fraction as a decimal.

For problems 4 and 5, use the long division algorithm to verify that the decimal forms found in problems 2 and 3 are correct.

4. 3 - 8

5. 3 60

Repeating Decimals

For problems 6-8, use the long division algorithm to find the decimal form of the following rational numbers. Use bar notation when necessary.

Student Edition: Grade 7-8, Module 1, Topic B, Lesson 9

Student Edition: Grade 7-8, Module 1, Topic B, Lesson 9

Name Date

Decimal Expansions of Rational Numbers

In this lesson, we

• wrote rational numbers given in fraction form as decimals by first writing them as decimal fractions.

• wrote rational numbers given in fraction form as decimals by using long division.

• determined that a rational number can be written as a terminating decimal when the denominator has only prime factors of 2, 5, or both.

• represented repeating decimals by using bar notation.

Examples

Terminology

A terminating decimal is a decimal that can be written with a finite number of nonzero digits.

A repeating decimal is a decimal in which, after a certain digit, all remaining digits consist of a block of one or more digits that repeats indefinitely.

A common notation for a repeating decimal expansion is bar notation. The bar is placed over the shortest block of repeating digits after the decimal point. For example, 3.125 is a compact way to write the repeating decimal expansion 3.12525252525… .

1. Write12 25 as a decimal fraction. Then write it as a decimal.

A decimal fraction has a denominator that is a power of 10

To write decimal fractions as decimals, use the power of 10 in the denominator to determine the place value.

Create an equivalent fraction by multiplying both the numerator and the denominator by the same number.

Multiplying the denominator by 4 produces a power of 10.

2. Factor the denominator of14 80 by using prime factorization. Write14 80 as a decimal fraction and then as a decimal.14 80

Factor the denominator into prime factors. If the prime factors are only 2, 5, or both, then the number can be written as a decimal fraction. This means it can be written as a terminating decimal.

Because decimal fractions have a denominator that is a power of 10, decimal fractions always have an equal number of factors of 2 and 5 in the denominator.

To write the number as a decimal fraction, multiply by additional factors of either 2 or 5 to produce a power of 10 in the denominator.

For problems 3 and 4, write the number in decimal form. Use bar notation where appropriate.

3. 3 11

3 11 = 3 ÷ 11

To write fractions as decimals by using the long division algorithm, interpret the fraction as division. 0.2727 113. 00 00

Because the remainders begin to repeat, the quotients are repeating decimals.

0.27

The block of digits 2 and 7 repeats, so the bar is over both 2 and 7

4.17 1517 15 = - (17 ÷ 15)

1. 133 1517. 000 - 15 20 -15 50 -45 50 -45

3 is the only digit that repeats, so the bar is only over 3

5 -1.13

The quotient is a negative number because17 15 is interpreted as the opposite of the quotient 17 15 that produces.

Student Edition: Grade 7-8, Module 1, Topic B, Lesson 9

Name Date

For problems 1 and 2, write the decimal in fraction form.

1. 0.58

2. − 9.76

For problems 3-5, write the number as a decimal fraction. Then write it as a decimal.

3. − 7 25

4. 31 20

5. − 27 60

For problems 6 and 7, factor the denominator by using prime factorization. Write the number as a decimal fraction and then as a decimal.

6. 5 8

7. − 21 40

For problems 8-13, indicate whether the decimal form of the number terminates or repeats.

For problems 14-20, use long division to write the number in decimal form. Use bar notation where appropriate.

21. Henry says that every fraction with a denominator of 9 is a repeating decimal because the only factors of the divisor 9 are 3, not 2 or 5. Do you agree with Henry? Explain why.

22. Together, 6 friends buy a $20 gift for their other friend. How do the 6 friends share the cost of the gift evenly?

23. Is the decimal form of − 9 6 a terminating decimal or a repeating decimal? Explain how you know.

Remember

For problems 24-27, subtract.

For problems 28 and 29, evaluate the expression.

29.

30. Which expressions are equivalent to 2 (− 4) 3 ? Choose all that apply.

A. 8 3

B. − 8 3

C. − 8 12

D. − 2 3 (4)

E. 1 3 (−8)

F. 4 3 (−2)

Million, Billion—What’s the Difference, Really?

Thousand

About 17 minutes

Seconds

About 12 daysAbout 32 years

Feet

The rough length of a New York block

The rough distance from New York to Boston

The rough distance from New York to the moon

Dollars

About $0.03 every day of your life

About $35 every day of your life

People

A mid-sized high school

San Jose, CA (approximately)

About $35,000 every day of your life

The rough population of North and South America combined

Million and billion sound awfully similar. Just one letter is different. Written out as powers of 10, they look similar too: it’s 106 versus 109. How different can they really be?

Extremely different, it turns out.

If a million seconds is a long vacation, then a billion seconds is longer than your life so far. If a million feet is a short flight in an airplane, then a billion feet is a journey to the moon. If a million dollars over your lifetime is a nice daily allowance, then a billion dollars over your lifetime is enough to buy a new car every day. If a million people is the population of a big city, then a billion people is the population of the whole Western Hemisphere.

Exponents let us write huge numbers by using just a few symbols. But don’t let that fool you into forgetting how huge the numbers might be and how different they are from one another.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10 LESSON

Name Date

Large and Small Positive Numbers

Writing Very Large and Very Small Positive Numbers

1. Complete the table. The table shows an approximate measurement of objects seen in the demonstration.

Approximate Measurement

2. Complete the table. The table shows an approximate measurement of objects seen in the demonstration.

Approximate Measurement (meters)

Approximating Very Large and Very Small Positive Numbers

3. The length of Rhode Island, from the northernmost point to the southernmost point, is 77,249 meters.

a. Approximate the length of Rhode Island by rounding to the nearest ten thousand meters.

b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

4. The width of a water molecule is 0.000 000 000 28 meters.

a. Approximate the width of a water molecule by rounding to the nearest ten billionth of a meter.

b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

Times As Much As

5. 9 billion is how many times as much as 3,000?

Record your work for each of the stations under the appropriate heading.

Station 1

Station 2

Station 3

Station 4

Student Edition: Grade 7–8, Module 1, Topic

Name Date

1. Consider the number 0.000 0285

a. Approximate the number by rounding to the nearest hundred thousandth.

b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

2. Bacterial life appeared on Earth about 3.6 billion years ago. Mammals appeared about 200,000,000 years ago.

a. Write when bacterial life approximately appeared on Earth as a single digit times a power of 10 in exponential form.

b. The number of years since bacterial life appeared on Earth is about how many times as much as the number of years since mammals appeared?

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10

Name

Large and Small Positive Numbers

In this lesson, we

Date

• explored very large and very small positive numbers by relating them to the sizes of real-world objects.

• analyzed equivalent forms of large and small positive numbers.

• approximated very large and very small positive numbers.

• wrote unknown factor equations to answer how many times as much as questions.

Examples

1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the Eiffel Tower.

Use place value units when writing numbers in unit form.

The expanded form of 2 hundred thousand is 2 × 100,000, where the power of 10 is written in standard form.

The exponential form of 1 hundred thousand is 105, so write 2 hundred thousand as 2 × 105

2. Complete the table. The table shows the typical speed in miles per hour of a sea star.

3. The land area of France is 212,954 square miles. The land area of Canada is 3,854,083 square miles. The land area of Canada is about how many times as large as the land area of France?

Approximate the value of each land area. Then write the approximation as a single digit times a power of 10 in exponential form.

To decide how to write the unknown factor equation, determine which quantity is being multiplied. To determine the unknown factor, divide

The land area of Canada is about 20 times as large as the land area of France.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10

Name Date

PRACTICE 10

1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the given objects.

Approximate Number of Stacked Pennies

Objects

Single Digit Times a Power of 10 (expanded form)

Single Digit Times a Power of 10 (exponential form) Mount Everest 6,000,000 Empire State

2. Complete the table. The table shows the typical speed of the given animals.

Typical Speed (miles per hour)

Objects

Galapagos Tortoise 0.2

Sloth 0.07

Single Digit Times a Unit Fraction (expanded form)

Single Digit Times a Unit Fraction (exponential form)

3. There are 907,200,000 milligrams in 1 ton.

a. Approximate the number of milligrams in 1 ton by rounding to the nearest hundred million milligrams.

b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

4. The smallest insect on the planet is a type of parasitic wasp that measures 0.000 139 meters.

a. Approximate the length of the insect by rounding to the nearest ten thousandth of a meter.

b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

5. 800,000 is how many times as much as 2,000?

6. 60,000,000 is how many times as much as 30,000?

7. 6 × 105 2 × 103 is times as much as what number?

For problems 8 and 9, use the values in the table to answer the questions.

8. Based on land area, about how many islands the size of Jamaica does it take to equal the size of Russia?

9. Which country’s land area is about 1 500 as large as the land area of the United States?

10. The Atlantic Ocean contains about 310,410,900 cubic kilometers of water. Lake Superior, which is the largest lake in the United States, contains about 12,000 cubic kilometers of water. Approximately how many Lake Superiors would it take to fill the Atlantic Ocean?

Remember

For problems 11–14, subtract.

(

)

For problems 15 and 16, divide.

17. Find the value of the expression shown.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11

Name Date

Products of Exponential Expressions with Positive Whole-Number Exponents

1. Multiply. Write the product as a power of 10 in exponential form.

Multiplying Powers with Like Bases

2. Multiply. Write the product as a power of 10 in exponential form.

3. Multiply. Write the product as a power of 10 in exponential form.

Applying a Property of Exponents

For problems 4–15, apply the product of powers with like bases property to write an equivalent expression.

17. Write three exponential expressions that are equivalent to 916.

18. Which expressions have a value of − 16? Choose all that apply.

A. − 1 ⋅ 24

B. − 24

Multiplying Powers with Unlike Bases

For problems 19–21, apply the product of powers with like bases property to write an equivalent expression.

Properties of Exponents

Definitions of Exponents

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11

For problems 1–4, apply the product of powers with like bases property to write an equivalent expression. 1.

5. Explain what the expression 43 ⋅ 46 represents and how it can be written as a single power.

Name

Products of Exponential Expressions with Positive Whole-Number Exponents

In this lesson, we

• discovered a pattern when multiplying powers with like bases.

• learned the product of powers with like bases property.

• applied a property of exponents to write equivalent expressions.

Product of Powers with Like Bases Property

Examples

Apply a property of exponents to write an equivalent expression. 1.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11 = 96

Negative bases and fractional bases have parentheses to avoid confusion.

The first factor represents 1 factorof (− 1 6 ), so it has an exponent of 1

Because there are two different bases, 7 and 8, use the commutative property of multiplication to rearrange the powers with like bases together. Then apply the property of exponents to powers with like bases separately.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11

PRACTICE 11

For problems 1–10, apply the product of powers with like bases property to write an equivalent expression.

11. Which expressions are equivalent to 53 ⋅ 55? Choose all that apply.

A. 53+5

B. 53 5

C. 58

D. 515

E. 54 ⋅ 54

F. 258

12. Which expressions are equivalent to 912? Choose all that apply.

A. 36 ⋅ 36

B. 99 ⋅ 93

C. 94 ⋅ 93

D. 9 ⋅ 912

E. 96 ⋅ 96

F. 96 ⋅ 92

13. Sara states that when two powers with the same base are multiplied, the exponents are multiplied together. She uses an example to support this claim.

Fill in the boxes to create an equation that shows that Sara’s claim is incorrect.

14. Write the area of the rectangle as a single base raised to an exponent.

105ft

102ft

For problems 15–18, indicate whether the value of the expression is a positive number or a negative number.

15. (− 3)2

16. (− 4)3

17. (− 5)133

18. (− 6)4,592

19. The product (− 1)3 ⋅ (− 1)n is a negative number. Which of the following values of n are possible? Choose all that apply. A.

For problems 20–23, solve for b.

For problems 24–26, fill in the boxes with digits 1–6 to make the equation true. Each digit can be used only once.

Remember

For problems 27–30, multiply.

For problems 31–34, write each number as a decimal. Use bar notation where appropriate.

31. − 36 1,000

32. − 9 25

33. − 7 9

Estimate, and then find the quotient.

35. 85.92 ÷ 12

34. 5 11

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12 LESSON

Name Date

More Properties of Exponents

Raising Powers to Powers

Raising Products to Powers

Raising Quotients to Powers

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12

Name Date

More Properties of Exponents

In this lesson, we

• established two new properties of exponents.

• used the properties of exponents to write equivalent expressions.

Power of a Power Property

Power of a Product Property (xy)

Apply the properties of exponents to write an equivalent expression.

Both the numerator, 9 2 , and the denominator, 10, are raised to the sixth power.

Both factors have a power of 3, so there are 3 factors of 2 and 3 factors of 5. By using the commutative and associative properties of multiplication, this expression can be written as 3 factors of (2

5), which is 10 3

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12

PRACTICE 12

For problems 1–10, apply the properties of exponents to write an equivalent expression. Assume m is nonzero.

11. Jonas says (5n 2 ) 3 is equivalent to 5n 2 3. Do you agree? Explain your reasoning.

In problems 12–16, fill in the boxes with values that make the equation true. Assume s is nonzero.

16. 6 106 = (1 2 )6

17. Which expressions are equal to 8 24 ? Choose all that apply.

A. (8 20 ) 4

B. (8 12 ) 2

C. 8 8 ⋅ 8 3

D. 8 14 ⋅ 8 10

E. 8 8 + 8 16

18. Which expressions are equal to x 12 y 8 z 4 ? Choose all that apply.

A. (x 8 y 6 z) 4

B. (x 3 y 2 z) 4

C. (x 6 y 4 z 2 ) 2

D. (x 3 y 4 z 2 )(x 4 y 2 z 2 )

E. (x 6 y 5 z)(x 6 y 3 z 3 )

19. The edge of a cube measures 4 n 2 inches. What is the volume of the cube?

20. The side length of a square measures meters. What is the area of the square? 3 x 4 y 5 Assume y is nonzero.

Remember

For problems 21–24, multiply.

25. There are 0.000 000 001 102 293 tons in 1 milligram.

a. Approximate the number of tons in 1 milligram by rounding to the nearest billionth of a ton.

b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

26. The world population is expected to reach about 9.7 billion people in the year 2050.

a. Approximate the expected world population in 2050 by rounding to the nearest billion people.

b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

c. The population of the United States is expected to reach about 400 million people in the year 2050. The expected world population in 2050 is about how many times as much as the expected United States population in 2050?

27. Consider the rational number − 5 9 .

a. Is the decimal form of 5 9 a terminating decimal? Explain how you know.

b. Write − _ 5 9 in decimal form.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson

Making Sense of Integer Exponents

1. Circle all expressions that have a value of 1. Assume x is nonzero.

Integer Exponents

For problems 2–5, use the definition of a negative exponent to write an equivalent expression with a positive exponent. Assume x is nonzero.

2. 10−7

Quotients of Powers

6.

So-hee applies the properties of exponents to write an equivalent expression for . Her work is shown.

Apply the properties and definitions of exponents to verify So-hee’s answer.

For problems 7 and 8, apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume d is nonzero.

tudent Edition: Grade 7–8, Module 1, Topic C, Lesson 13

For problems 1 and 2, apply the definition of the exponent of 0 to write an equivalent expression. 1.

3. Use the definition of a negative exponent to write 5−7 with a positive exponent.

4.

Kabir and Yu Yan each write an equivalent expression for . The table shows their work.

Explain how Kabir and Yu Yan each use the properties and definitions of exponents to get the final expression.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13

Name

Making Sense of Integer Exponents

In this lesson, we

Date

• used the product of powers with like bases property to determine that x 0 = 1.

• related negative exponents to multiplicative inverses.

• learned the definition of a negative exponent.

• applied the definition of a negative exponent to write equivalent expressions.

Definition

Definition of a Negative Exponent

Examples

1. Write an equivalent expression for

is nonzero. 5

The exponent of 0 applies only to the base c, not to the entire expression 5c

2. Without using the definition of a negative exponent, how can we show 10 1 −5 = 105 ?

By using the product of powers with like bases property and the definition of the exponent of 0, we know 105 10−5 = 105+(−5) = 100 = 1.

We also know 10 1 5 ⋅ 105 = 1

This means 10−5 and 1 105 are both multiplicative inverses of 105. So 10−5 = 1 105

Two factors that have a product of 1 are multiplicative inverses.

For problems 3 and 4, use the definition of a negative exponent to write an equivalent expression with a positive exponent.

A fraction represents division.

5. Apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume r is nonzero.

To use the product of powers with like bases property, write

5

11 as r 5 r −11. Then use the definition of a negative exponent to write the answer with a positive exponent.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13

For problems 1–15, apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume all variables are nonzero.

For problems 16–19, apply the properties and definitions of exponents to determine the value of t.

For problems 20 and 21, write a number in the box to make the equation true.

22. Choose the expression that has a value of 9,804.

A. 9 × 103 + 8 × 102 + 4 × 101

B. 9 × 104 + 8 × 103 + 4 × 101

C. 9 × 104 + 8 × 103 + 4 × 100

D. 9 × 103 + 8 × 102 + 4 × 100

23. What is the value of 6 × 104 + 3 × 102 + 2 × 100?

A. 632

B. 6,032

C. 6,320

D. 60,302

E. 60,300

24. Maya says 10−5 is equivalent to (− 10)5. Do you agree? Explain your reasoning.

25. Which expression does not have a value of 1 16 ?

26. Order the values from least to greatest.

Remember

For problems 27–30, multiply.

For problems 31 and 32, apply the product of powers with like bases property to write an equivalent expression.

31. (2 7 )4 (2 7 )8

32. 9a ⋅ 9b

33. Explain what the expression 53 57 represents and how it can be written as a single power.

34. Write 17 16 in decimal form.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14

Name

Writing Very Large and Very Small Numbers in Scientific Notation

Archimedes was a Greek mathematician whose ideas were ahead of his time. He lived in Sicily during the third century BCE and was the first to develop fundamental concepts in geometry, calculus, and physics.

Fascinated by very large numbers and living on the coast of the Ionian Sea, Archimedes set out to determine how many grains of sand are needed to fill the known universe.

Mathematicians used Ionic Greek notation in the third century BCE. So instead of numbers, they used symbols and letters from the Greek alphabet.

1. In this system, χνδ represents the number 654. What does σπε represent?

2. What does ψοζ represent?

Another Way to Represent Numbers

3. Fill in the blanks to complete the statements.

A number is written in scientific notation when it is represented as a number a multiplied by a of .

The general expression that represents a number written in scientific notation is × .

The absolute value of a must be at but than

4. Identify the first factor and the order of magnitude of the expression 8.86 × 106.

5. Indicate whether each number is written in scientific notation.

0.38 × 102 4.8 × 103 19.04 × 107 − 6.75 × 106

× 109

× 1010

For problems 6–11, complete the table.

Small Positive Numbers

12. Complete the table.

Approximate

For problems 13–18, complete the table.

Ordering Numbers in Scientific Notation

19. Order the given numbers from least to greatest.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14

Name

For problems 1 and 2, write the number in scientific notation.

1. 60,630,000

2. 0.0051

For problems 3 and 4, write the number in standard form.

3. 3 × 10−5

4. 3.4 × 1010

5. Order the numbers from least to greatest.

Date

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14

Name Date

Writing Very Large and Very Small Numbers in Scientific Notation

In this lesson, we

• defined scientific notation.

• identified examples and nonexamples of numbers written in scientific notation.

• wrote numbers in scientific notation and in standard form.

• ordered numbers written in scientific notation.

Examples

Terminology

• A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. Numbers written in scientific notation are in the form a × 10n. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10

• The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation.

1. Circle all the numbers written in scientific notation. Numbers written in scientific notation must be in the form a × 10n .

The absolute value of a, the first factor, must be at least 1 but less than 10

2. Complete the table.

The first nonzero digit is 3. The place value of the digit 3 is represented by 10−8. The first factor is 3.2, which is at least 1 but less than 10

The power of 10, 10−6, shows the place value of the first nonzero digit only, which is 2. The digits 0 and 9 are written after the decimal.

The first nonzero digit is 5. The place value of the digit 5 is represented by 10−4, which is equal to 1 104 , or 1 10,000

3. The table shows the approximate weights of animals in pounds. Order the animals from heaviest to lightest.

Blue whale, African elephant, hippopotamus, giraffe, zebra, grizzly bear, colossal squid

The blue whale is the heaviest because its weight has the greatest order of magnitude, which is 5

The orders of magnitude are equal for the weights of the African elephant, the hippopotamus, and the giraffe.

In this case, use the first factor of each weight to determine the correct order.

9.5 > 6 > 2.2

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14

PRACTICE 14

1. Circle all the numbers written in scientific notation.

2. Match each number written in standard form with its corresponding number written in scientific notation.

For problems 3–14, complete the table.

15. In 2014, the United States discarded a total of 5.08 × 109 pounds of trash. Write this number in standard form.

16. Sara believes that the number 3 × 10 is not written in scientific notation. Do you agree or disagree? Explain.

17. In 2021, the richest person in the world had a net worth totaling about $177 billion. Write this number in scientific notation.

18. The mass of a neutron is about 1.67493 × 10−27 kg. The mass of a proton is about 1.67262 × 10−27 kg. Explain which is heavier.

19. Before 2006, Pluto was considered to be one of the planets in our solar system. Many people thought it should be classified as a dwarf planet instead. The table lists planets in our solar system, the dwarf planet Pluto, and their approximate width in meters.

List the planets, including Pluto, from least to greatest based on their width.

For problems 20–23, multiply.

For problems 24–26, apply the properties of exponents to write an equivalent expression. Assume y is nonzero.

27. Choose all the true number sentences.

A. 82 = 8 ⋅ 8

B. 52 = 5 2

C. 42 = 8

D. 4 ⋅ 4 ⋅ 4 = 34

E. 7 ⋅ 7 ⋅ 7 = 73

28. Choose one false number sentence from problem 27. Explain why it is false.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15

Name Date

LESSON 15

Operations with Numbers Written in Scientific Notation

1. Liam enters 200,000 × 450,000 into his calculator. The screen shows the following display.

a. Write 200,000 in scientific notation.

b. Write 450,000 in scientific notation.

Adding and Subtracting

2. The table shows the number of views for the following online videos.

a. How many total views do the singing cat video and the science experiment video receive? Write the answer in scientific notation.

b. How many more views does the singing cat video receive than the science experiment video? Write the answer in scientific notation.

c. How many more views does the singing cat video receive than the talking parrot video? Write the answer in scientific notation.

d. How many total views do the talking parrot video and the dancing baby video receive? Write the answer in scientific notation.

e. How many more views does the science experiment video receive than the dancing baby video? Write the answer in scientific notation.

Multiplying and Dividing

3. The length of Colorado is about 611,551 meters. The width is about 450,616 meters.

a. Approximate the length and width of Colorado by rounding to the nearest hundred thousand meters. Then write the length and width in scientific notation.

Colorado

b. Approximate the area of Colorado. Write the answer in scientific notation.

c. The area of Denver, the capital of Colorado, is approximately 4 × 108 square meters. The area of Colorado is about how many times as large as the area of Denver? Write the answer in scientific notation.

Power to a Power

For problems 4 and 5, use the properties of exponents and the properties of operations to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator.

4. (8.85 × 103)2

5. (2 × 10−9)3

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15

Name

Date

1. After a series of calculations, a calculator screen displays this result.

4.399e14

Write the displayed value in scientific notation.

For problems 2–4, evaluate the expression. Write the answer in scientific notation.

2. 1.3 × 10−4 + 2.1 × 10−4

3. 4.7 × 105 − 4.1 × 105

4. (3 × 10−4)3

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15

Name Date

Operations with Numbers Written in Scientific Notation

In this lesson, we

• interpreted numbers displayed in scientific notation on digital devices.

• used the properties and definitions of exponents and the properties of operations to efficiently operate with numbers written in scientific notation.

• wrote sums, differences, products, and quotients in scientific notation.

Examples

1. A calculator displays 3.45e−4. Write this number in scientific notation and in standard form. 3.45 × 10−4 = 0.000 345

For problems 2–4, evaluate the expression. Write the answer in scientific notation.

2. 2.7 × 10−9 + 8.1 × 10−9

The addends have a common power of 10, 10−9. So the distributive property can be applied.

2.7 × 10−9 + 8.1 × 10−9 = (2.7 + 8.1) × 10−9

= 10.8 × 10−9

= (1.08 × 10) × 10−9 = 1.08 × (10 × 10−9) = 1.08 × 10−8

The number 10.8 × 10−9 is not written in scientific notation because the absolute value of the first factor is greater than 10. Use the associative property of multiplication and the property xmxn = xm+n to write 10.8 × 10−9 in scientific notation.

3. (9 × 10 −7)(3.4 × 10 2)

(9 × 10−7)(3.4 × 102) = (9)(3.4) × (10−7)(102) Use the commutative and associative properties of multiplication to rewrite the original expression. Then multiply the factors in each pair.

= 30.6 × 10−5 = (3.06 × 10) × 10−5 = 3.06 × (10 × 10−5) = 3.06 × 10−4

4. (1.6 × 10−8)3 (1.6 × 10−8)3 = 1.63 × (10−8)3

Use the properties (xy)n=xnyn and (xm)n=xm ⋅ n to evaluate (1.6 × 10−8)3 = 4.096 × 10−24

5. Use the table of approximate animal weights to complete each part.

a. About how many more pounds does a gray tree frog weigh than an aphid?

(1.6 × 10−3) − (4.4 × 10−7) = (1.6 × 10−3) − (4.4 × 10−4 × 10−3)

To determine how many more pounds the gray tree frog weighs than the aphid, find the difference of their weights.

= (1.6 × 10−3) − (0.00044 × 10−3) = (1.6 − 0.00044) × 10−3

= 1.59956 × 10−3

Write 10−7 as 10−4 × 10−3 Then apply the distributive property.

Another strategy is to use 10−7 as the common power of 10 1.6 × 10−3 can be written as 1.6 × 104 × 10−7

A gray tree frog weighs about 1.59956 × 10−3 pounds more than an aphid.

b. An emperor scorpion is about how many times as heavy as a termite?

Divide the weight of the scorpion by the weight of the termite.

= 2 × 104 = 20,000

An emperor scorpion is about 20,000 times as heavy as a termite.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15

Name

PRACTICE 15

Date

For problems 1–4, write the answer displayed on the calculator screen in scientific notation and in standard form.

1. 5.678e+17

2. 0.0000023*0.0000 5 1.15˜ -10

3. 1.386471e9

4. (1.39*109 )2 1.9321*1018

For problems 5–14, evaluate the expression. Write the answer in scientific notation.

5. 4 × 103 + 3 × 103

6. 9 × 10−8 − 2.7 × 10−8

7. (9 × 1010)(1.1 × 103)

8. 6 × 10−6 + 8 × 10−6

15. Use the table of approximate animal weights to complete each part. Check each answer with a calculator.

a. About how many more pounds does a hummingbird weigh than a monarch butterfly?

b. About how many more pounds does a wood mouse weigh than a monarch butterfly?

c. A wood mouse is about how many times as heavy as a monarch butterfly?

d. About how many more pounds does a zebra weigh than a wood mouse?

e. Which animal or insect is about 8 times as heavy as a monarch butterfly?

f. A zebra is about how many times as heavy as a hummingbird?

20. Eve and Lily each write an equivalent expression for x 7 x 3 . The table shows their work.

Eve’s Work Lily’s Work

Explain how Eve and Lily each use the properties and definitions of exponents to get the final expression.

21. Which expression is equivalent to 5 6 ⋅ 54?

A. 2524

B. 1010

C. 524

D. 510

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16 LESSON 16

Name Date

Applications with Numbers Written in Scientific Notation

Operating with Numbers in Scientific Notation

Comparing and Converting Units

How Many Times as Large and How Much Larger

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16

Name

Date

A popular television series is added to your favorite online streaming service. It will take 1.902 × 105 seconds to watch the entire series.

a. Choose a more appropriate unit of measurement to describe the amount of time needed to watch the entire series. Explain why you chose that unit.

b. Convert the unit from seconds to the unit of measurement you chose in part (a). Round your answer to the nearest tenth of a unit.

c. The entire series has 15 episodes for each of the 4 seasons. What is the approximate number of minutes per episode?

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16

Name

Date

Applications with Numbers Written in Scientific Notation

In this lesson, we

• determined an appropriate unit of measurement for a given situation.

• converted units of measurement to more appropriate units of measurement.

• operated with numbers written in scientific notation.

Examples

1. A hawk can travel about 1.552 × 107 inches per day.

a. Choose a more appropriate unit of measurement to describe the distance a hawk travels per day. Explain why you chose that unit.

A more appropriate unit of measurement is miles per day because it uses a larger unit for the distance, which makes the measurement simpler to interpret.

b. Convert the unit from inches per day to the unit of measurement you chose in part (a).

1 foot = 12 inches

1 mile = 5,280 feet

There are 12(5,280) , or 63,360 , inches in 1 mile. Divide 1.552 × 107 inches by 63,360 inches to find how many miles the hawk can travel each day.

107

A hawk can travel about 2.4 × 10 2 , or 240, miles per day.

c. The longest distance around Earth is about 4 × 104 kilometers. Using your answer from part (b), about how many weeks would it take for a hawk to travel that distance? Assume the hawk travels at a constant rate. (1 mile ≈

kilometers)

A hawk travels about 384 kilometers per day.

It would take about 104 days. There are 7 days in 1 week.

It would take about 15 weeks for a hawk to travel the longest distance around Earth. Round 14.86 to the nearest whole number to determine the approximate number of weeks.

2. Water flows from a shower head at a rate of 7.57 × 103 cubic centimeters per minute.

a. Determine the rate that water flows from the shower head in cubic meters per minute. There are 106 cubic centimeters in 1 cubic meter.

7.57 × 103 106 = 7.57 × 103(106 ) = 7.57 × 10−3

Water flows from the shower head at a rate of 7.57 × 10−3 cubic meters per minute.

Visualize a cube with an edge length of 1 meter, which is 100 centimeters.

The volume of the cube can be expressed as 1 cubic meter or 1,000,000 cubic centimeters.

1 cubic meter = 106 cubic centimeters

b. Determine the rate that water flows from the shower head in cubic meters per second. There are seconds in minute. 60 1

7.57 × 10−3 60 = (7.57 60 ) × 10−3

≈ 0.126 × 10−3

= (1.26 × 10−1) × 10−3 = 1.26 × (10−1 × 10−3)

= 1.26 × 10−4

Water flows from the shower head at a rate of about 1.26 × 10−4 cubic meters per second.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16

Name Date

PRACTICE 16

1. Use the following information to fill in the blanks. On Earth, there are about 1 quadrillion (1,000,000,000,000,000) ants and about 8 billion people.

a. On Earth, there are about more ants than people.

b. On Earth, there are about times as many ants as people.

2. It takes Earth 8.61624 × 104 seconds to complete a single rotation on its axis. Choose a more appropriate unit of measurement to report this data. Explain why you chose that unit.

3. The table shows the distances of several stars from Earth measured in light-years. A light-year is the distance that light travels in 1 year. For example, the distance from the closest star, Proxima Centauri, to Earth is 4.243 light-years because it takes light 4.243 years to reach Earth from Proxima Centauri. Light travels at a speed of approximately 9.46 × 1015 meters per year.

Star Name Distance

a. Determine which stars given in the table are between 1.3 × 1017 and 2.5 × 1017 meters from Earth.

b. Approximately how many kilometers per day does light travel?

c. Choose a more appropriate unit of measurement to report the speed in which light travels. Explain.

4. Maya bought a fish tank that has a volume of 175 liters. She read a fun fact that it would take 7.43 × 1018 fish tanks of water at that tank size to fill all the oceans in the world.

a. Use the fun fact to determine the total volume of water in the world’s oceans in liters. Write your answer in scientific notation.

b. Given that 1 liter is 1 × 10−12 cubic kilometers, find the total volume of water in the world’s oceans in cubic kilometers. Write your answer in scientific notation.

c. Use your answer from part (b) to find the total volume of water in the world’s oceans in cubic centimeters.

d. The Atlantic Ocean has a volume of about 323,600,000 cubic kilometers. You bought a fish tank that holds 75 more liters than Maya’s tank. About how many tanks like yours would it take to fill the Atlantic Ocean? Write your answer in scientific notation.

For problems 5–8, divide.

9. Write 870,000,000 in scientific notation.

10. Write 5.8 × 10−3 in standard form.

11. Order the numbers from least to greatest.

12. Which expressions are equal to 520? Choose all that apply.

A. (515)5

B. (510)2

C. 510 ⋅ 510

D. 54 ⋅ 55

E. 512 + 58

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 17 LESSON 17

Name Date

Get to the Point

Point by Point

1. Study the painting La Corne d’Or. Matin by Paul Signac.

Record some questions that come to mind.

2. Focus Question:

Guess:

3. Strategy:

Tools Used:

Estimate:

How Large?

4. Consider your answers from problems 2 and 3.

a. Which is larger: your guess from problem 2 or your estimate from problem 3?

b. How many times as large is it as your other answer?

c. How much larger is it than your other answer?

Expensive Dots

5. What is the approximate dollar value for each brushstroke in the painting? Create a model to support your response.

Model:

Estimate:

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 17

Modeling in A Story of Ratios

Read

Read the problem all the way through. Ask yourself:

• What is this problem asking me to find?

Then reread a chunk at a time. As you reread, ask yourself:

• What do I know?

Model the situation by using tools such as tables, graphs, diagrams, and equations.

Represent

Represent the problem by using your chosen model. Ask yourself:

• What labels do I use on the table, graph, or diagram?

• How should I define the variables?

• Are the known and the unknown clear in the model?

Add to or revise your model as necessary.

Solve

Solve the problem to determine whether your result answers the question. Ask yourself:

• Does my answer make sense?

• Does my result answer the question?

If your answer to either question is no, then revise your model or create a new one. Then ask yourself these questions again using your new result.

Summarize

Summarize your result and be ready to justify your reasoning.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 17

Name Date

Reflect on the lesson.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 17

Name Date

1. What assumptions did you make when you modeled the number of brushstrokes in the painting La Corne d’Or. Matin?

2. Was writing numbers in scientific notation helpful in the lesson? If so, when? If not, why not?

3. If you had more time to explore another question related to Paul Signac’s paintings, what question would you choose? What would be your plan to determine the answer?

For problems 4 and 5, use your estimations from the lesson.

4. Paul Signac completed about 500 pointillism paintings during his career as a painter.

a. Estimate the number of brushstrokes Paul Signac painted during his entire career.

b. What assumptions did you make in answering part (a)?

c. Do you think your estimation for part (a) is reasonable?

5. You want to recreate the painting La Corne d’Or. Matin. Each brushstroke takes approximately 20 seconds to paint, which includes mixing the paint, painting the canvas, and washing your brush.

a. How long does it take you to recreate the painting if you paint at a constant rate?

b. What unit of measurement did you choose to report the amount of time it takes in part (a)? Explain why you chose this unit of measurement.

Remember

For problems 6–9, divide.

For problems 10 and 11, evaluate the expression. Write the answer in scientific notation.

10. 6.7 × 106 − 6.2 × 106

11. (2.5 × 10−5)3

For problems 12–14, apply the properties and definitions of exponents to write an equivalent expression. Assume all variables are nonzero.

12. 36 3

13. (15x 2)3

14. a

b

Student Edition: Grade 7–8, Module 1, Topic D

Helo, square rot of 2? I have a package for you, but I’m having trouble finding the adres.

Oh, I’m betwen 1.4 and 1.5.

I’m here. Where are you?

Okay, now go betwen 1.41 and 1.42.

1.4 1.5

Okay, now go betwen 1.41421356 and 1.41421357.

Lok, can I just leave the package with one of your neighbors?

1.4142135

Irrational numbers are strange.

Very strange.

1.4142136

Irrational means “not rational.” In other words, an irrational number can’t be written as a nice fraction. Even writing it as a decimal is hard because the digits will continue forever, without repeating.

This makes it rather hard to find a number like √2 on a number line. No matter how far you zoom in, it always falls just between your nicely labeled intervals.

Student Edition: Grade 7–8, Module 1, Topic D, Lesso

Solving Equations with Squares and Cubes

Squares and Cubes of Fractions and Decimals 1.

Solving Equations of the Form x 2 = p

For problems 5–13, solve the equation.

Solving Equations of the Form x 3 = p

For problems 14–21, solve the equation.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18

Name Date

For problems 1–3, state whether the number is a perfect square, a perfect cube, both, or neither. Explain.

1. 121

2. 75

3. 64

For problems 4–7, solve the equation. 4.

Name Date

Solving Equations with Squares and Cubes

In this lesson, we

• classified numbers as perfect squares, perfect cubes, both, or neither.

• solved equations of the form x 2 = p, where p is either a perfect square or a rational number related to a perfect square.

• solved equations of the form x 3 = p, where p is either a perfect cube or a rational number related to a perfect cube.

Examples

Terminology

A perfect square is the square of an integer.

A perfect cube is the cube of an integer.

For problems 1 and 2, determine all numbers that square to the given number.

1. 81 9 and − 9

2. 0.64 0.8 and − 0.8

Multiplying two numbers with the same sign results in a positive number. So there are two numbers, one positive and one negative, that have the same positive square.

92 = 81 and (− 9)2 = 81

When the number given is not a perfect square, determine whether it is related to a perfect square.

For decimals, write the number as a fraction and determine whether the numerator and the denominator are perfect squares.

For problems 3 and 4, determine all the numbers that cube to the given number.

3. − 27

4. 8 125

2 5

Solutions of x 2 = p

When p is a positive number: the equation has two solutions because both positive and negative numbers have positive squares.

Solutions of x 3 = p

When p is a positive number: the equation has one solution because only positive numbers have positive cubes.

Only one integer has a cube of − 27.

33 = (3)(3)(3) = (9)(3) = 27

When p is a negative number: the equation has no solution because no numbers have negative squares.

When p is 0:

the equation has one solution because only 0 has a square of 0.

When p is a negative number: the equation has one solution because only negative numbers have negative cubes.

When p is 0:

the equation has one solution because only 0 has a cube of 0.

For problems 5–10, solve the equation. The phrase solve the equation means to find the values that, when substituted for the variable, make the equation true.

5. x 2 = 121

The solutions are 11 and − 11.

7. 25 49 = m2

The solutions are 5 7 and − 5 7 .

Use the perfect squares 25 and 49 to determine the solutions.

9. k2 = − 25 No solution

No number squared equals − 25 because no numbers have negative squares, so there is no solution.

6. − 8 = a3

The solution is − 2.

8. q3 = 0.064

The solution is 0.4.

Use a fraction to make sense of the place value.

10. 0 = n3

The solution is 0.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18

Name Date

PRACTICE 18

For problems 1–3, state whether the number is a perfect square, a perfect cube, both, or neither. Explain.

1. 1

2. 25

3. 60

For problems 4–7, determine all the numbers that square to the given number.

4. 16 81

6. 2.25

5. 9 64

7. 0.49

For problems 8–11, determine all the numbers that cube to the given number.

8. 0.008

9. 27 64

12. Liam says that when he squares or cubes any number, the result is always a greater number. Do you agree with Liam? Explain why and provide an example to support your thinking.

For problems 13–16, determine all the values that make the equation true.

For problems 17–28, solve the equation.

29. Fill in the box to create an equation that has the solutions − 13 and 13. x 2 =

30. Fill in the box to create an equation that has − 1 4 as its only solution. x 3 =

Remember

For problems 31–34, divide.

35. A used car costs 4.5 × 103 dollars. A luxury car costs 9 × 104 dollars. The cost of the luxury car is how many times greater than the cost of the used car?

36. Which expressions are equivalent to 1.2 × 105? Choose all that apply.

A. 120,000

B. (1 × 106) + (2 × 105)

C. (2 × 103)(6 × 102)

D. 4.8 × 107 4 × 102

E. (2 × 105) − (8 × 104)

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19

Name Date

The Pythagorean Theorem

Squares and Right Triangles

LESSON 19

1. Make a conjecture about how the areas of the blue square and the red square relate to the area of the white square.

2. Test the conjecture with the following triangle. State whether the conjecture holds true and explain your reasoning.

3. Test the conjecture with the following triangle. State whether the conjecture holds true and explain your reasoning.

4. Use the Pythagorean theorem to find the length of the hypotenuse c.

5. Find the length of the hypotenuse c.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19

Name Date

1. To which type of triangle does the Pythagorean theorem apply?

2. Identify the sides of each right triangle by labeling each side as a leg or a hypotenuse.

3. Find the length of the hypotenuse c.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19

Name Date

The Pythagorean Theorem

In this lesson, we

• developed a conjecture about the relationship among the side lengths of a right triangle.

• discussed the conditions required for using the Pythagorean theorem.

• found the hypotenuse length of right triangles when the square of the hypotenuse length was a perfect square or related to a perfect square.

The Pythagorean Theorem

In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse length.

Terminology

A leg of a right triangle is a side adjacent to the right angle.

The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of a right triangle.

Examples

Find the length of the hypotenuse c.

1. The legs of a right triangle are the sides that are adjacent to the right angle.

Substitute the values of the leg lengths for a and b in either order.

The length of the hypotenuse is 25 units.

The hypotenuse is the side opposite the right angle.

6. So-hee and Dylan each make errors while finding the length of the hypotenuse c of the following right triangle.

So-hee’s work:

a2 + b 2 = c 2

122 + 92 = c 2

144 + 81 = c 2

225 = c 2

112.5 = c

The length of the hypotenuse is 112.5 units.

Dylan’s work:

a2 + b 2 = c 2

92 + 122 = c 2

18 + 24 = c 2

42 = c 2

The length of the hypotenuse is 42 units.

a. Describe all the errors So-hee makes.

b. Describe all the errors Dylan makes.

c. Find the correct length of the hypotenuse.

For problems 7–10, divide.

For problems 11 and 12, evaluate. Write the answer in scientific notation.

11. (7.5 × 109) + (4.7 × 109)

12. (6.4 × 108) ÷ 3,200

13. On a given day, the population of Earth was 7,597,175,534 people. Which number is the closest estimate of Earth’s population on that day?

A. 7 × 109

B. 8 × 109

C. 7 × 1010

D. 8 × 1010

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20 LESSON

Name Date

Using the Pythagorean Theorem

Using Square Root Notation

1. Write the square root symbol.

2. Describe why the square root symbol is needed and how it is used to find the length of the hypotenuse.

For problems 3 and 4, use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

4.

The Spiral of Theodorus

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20

Name Date

1. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know. c

For problems 2 and 3, evaluate.

4. Order the following numbers from least to greatest:

.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20

Using the Pythagorean Theorem

In this lesson, we

• used square root notation to express hypotenuse lengths that are not rational.

• determined which two consecutive whole numbers the length of a hypotenuse is between.

• found the squares of numbers written with square root notation.

• created the Spiral of Theodorus to relate square roots to length measurements.

Terminology

A square root of a nonnegative number x is a number with a square that is x. The expression √x represents the positive square root of x when x is a positive number. If x is 0, then √0 = 0

• plotted points on a number line to represent locations of square roots.

Examples

1. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

What number squared is 106? Use the square root symbol to represent the exact value.

This expression means the square root of 106

The number 106 is between 100 and 121. So the value of c is between 10 and 11 because 102 = 100 and 112 = 121

The length of the hypotenuse is √106 units, which is between 10 units and 11 units.

2. Evaluate (√70 )2 . 70

Because √70 is the number with a square that is 70, then (√70 )2 = 70

3. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

The number 8 is between 4 and 9. So the value of c is between 2 and 3 because 22 = 4 and 32 = 9.

The length of the hypotenuse is √8 units, which is between 2 units and 3 units.

4. Which point represents the approximate location of √12 on a number line? (√12 )2 = 12, and 12 is between the two perfect squares 9 and 16 913

Because 32 = 9 and 42 = 16, the number squared to get 12 must be between 3 and 4

Point B represents the approximate location of √12 on the number line.

The inequality 3 < √12 < 4 indicates that √12 is greater than and less than 4 3

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20

For problems 1 and 2, determine which two consecutive whole numbers the length of the hypotenuse c is between. 1.

For problems 3–6, state whether the expression represents a whole number. If yes, state the whole number that is equal to the expression. If no, state the two consecutive whole numbers that the expression is between. 3.

For problems 7–11, determine whether the statement is true or false.

For problems 12–17, use square root notation to express the length of the hypotenuse. If the length is not a whole number, approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between.

12. The leg lengths of a right triangle measure 1 unit and 3 units.

13. The leg lengths of a right triangle measure 4 units and 10 units.

1 unit and √5 units.

18. Consider a right triangle with hypotenuse length c and leg lengths a and b. Is it possible for c 2 to be a perfect square when a2 and b2 are not perfect squares? If so, give an example.

14. The leg lengths of a right triangle measure

15. The leg lengths of a right triangle measure √2 units and √7 units.

19. Which point represents the approximate location of √8 on a number line? Write an inequality to describe the location of √8 between two consecutive whole numbers on a number line.

20. Which numbers are between 3 and 4? Choose all that apply.

Remember

For problems 21–24, divide.

For problems 25 and 26, state whether the number is a perfect square, a perfect cube, both, or neither. Explain how you know.

25. 110

26. −8

For problems 27 and 28, solve

27. x 2 = 64

29. While working on calculations for her science homework, Eve saw the following display on her calculator. 4.1633363e-

Write this number in scientific notation to help Eve interpret her calculator’s display.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21 LESSON

Name Date

Approximating Values of Roots

Approximating Square Roots

Approximating Cube Roots

1. Write the cube root symbol.

2. Explain why the cube root symbol is needed to determine the edge length of the cube shown.

Volume: 18 cubic units

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21

Name Date

Approximate the value of √17 to the nearest tenth by using consecutive whole numbers, tenths, and hundredths. Explain your thinking.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21

Approximating Values of Roots

In this lesson, we • approximated values of square roots. • explored and defined cube root notation and approximated the values of cube roots.

Examples

Terminology

The cube root of a number x is a number with a cube that is x. The

x expression represents the cube root of x.

For problems 1 and 2, determine the two consecutive whole numbers each value is between.

For problems 3 and 4, round each value to the nearest whole number.

5. Approximate the value of 3 √46 to the nearest tenth. Explain your thinking. Because 46 is between the perfect cubes 27 and 64, the value of 3 √46 is between 3 and 4.

Next, I consider the tenths from 3 to 4. I find that 46 is between 3.53 and 3.63, or 42.875 and 46.656.

Then, I consider the hundredths from 3.5 to 3.6. I find that 46 is between 3.583 and 3.593, or 45.882712 and 46.268279. Both 3.58 and 3.59 round to 3.6, so 3 √46 ≈ 3.6.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21

Name Date

For problems 1–4, determine the two consecutive whole numbers each value is between.

1. < √17 <

3. < √6 <

2. < √88 <

4. < √ 3 30 <

For problems 5–8, round each value to the nearest whole number.

5. √2

√90 7. √150

9. Henry states that the value of √30 is between 5.5 and 5.6. Do you agree with Henry? Why?

10. Is the value of 3 √25 greater than or less than 3? How do you know?

11. Approximate the value of √23 to the nearest tenth. Explain your thinking.

For problems 12–14, approximate the value by rounding to the nearest whole number, tenth, and hundredth.

15. Find the exact length of the hypotenuse c. Then approximate the length by rounding to the nearest tenth.

For problems 16–19, divide.

20. Find the length of the hypotenuse c.

21. The total volume of fresh water on Earth is approximately 3.5 × 107 cubic kilometers. The total volume of all water on Earth is approximately 1.4 × 109 cubic kilometers. What is the approximate volume of the water on Earth that is not fresh water? Write your answer in scientific notation.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22

Name Date

Rational and Irrational Numbers

LESSON 22

For problems 1–6, identify the decimal digit that comes next in the decimal form of the number. If you cannot identify the next decimal digit, indicate that.

1. 1 = 0.333333333 3 …

2. √ 3 12 = 2.2894284…

3. 144 = 1.45454545 99

4. 3 5 = 3.41666666 12 …

5. √42 = 6.480740698…

6. 3 = 0.428571428571 7 …

Battle Cards

7. Plot and label the approximate location of each value on the number line.

Ordering Expressions

For problems 8 and 9, order the expressions from least to greatest. 8.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22

1. Indicate whether each number is rational or irrational.

10−3

2. Compare the values by using the < or > symbol. Explain your reasoning.

3. Plot and label a point at the approximate location of each value on the number line.

Name Date

Rational and Irrational Numbers

In this lesson, we

• identified real numbers as rational or irrational.

• compared and ordered rational and irrational numbers.

• plotted the approximate locations of rational and irrational numbers on a number line.

Examples

Terminology

An irrational number is a number that is not rational and cannot be expressed as p q for integer p and nonzero integer q. An irrational number has a decimal form that neither terminates nor repeats.

A real number is any number that is either rational or irrational.

For problems 1–4, identify whether the number is rational or irrational.

1. 4 3 Rational A rational number can be written as a fraction.

2. −0.24680153…

Irrational

When the decimal form of a number neither terminates nor repeats, the number cannot be written as a fraction and is irrational.

Negative numbers can be rational or irrational.

For problems 5–8, compare the values of the expressions by using the < or > symbol.

5. 3.94 < 3 √65

Because 3 √ 64 = 4, 3 √ 65 is greater than 4. So 3. 94 is less than 3 √ 65

15

Because 3 √ 64 = 4, and 3 √125 = 5, 3 √120 is between 4 and 5.

Because √9 = 3 and √16 = 4, √15 is between 3 and 4.

Because √49 = 7, √50 is greater than 7

√50 > 62 9 7.

Because 63 9 = 7, 62 9 is less than 7

The value of 3 √ 25 is between 2 and 3. Adding 4 to 3 √ 25 gives a number that is between 6 and 7. Multiplying 3 √ 25 by 2 gives a number that is between 4 and 6

9. Plot and label a point at the approximate location of each value on the number line.

17

Because 3 √ 27 = 3 and 3 √ 64 = 4, 3 √ 30 is between 3 and 4

Because √16 = 4 and √25 = 5, √17 is between 4 and 5

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22

PRACTICE 22

1. Write each value in the appropriate column of the table.

Rational

Irrational

2. Abdul can predict the next decimal digit in the number 0.010305070… , so he concludes that the number is rational. Explain the error in Abdul’s thinking.

3. Ethan states that √5 is an irrational number. Nora states that √5 is a real number. Who is correct? Explain.

For problems 4–9, compare the values by using the < or > symbol.

For problems 10 and 11, plot and label a point at the approximate location of each value on the number line.

12. Order the expressions from least to greatest.

Remember

For problems 13–16, divide.

For problems 17 and 18, evaluate.

17. (√25 ) 2

19. Order the numbers from least to greatest.

For problems 20 and 21, solve.

20. g 6 = 8

21. 40 = 3n

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23 LESSON

Name Date

Revisiting Equations with Squares and Cubes

1. Indicate whether each equation has one solution, two solutions, or no solution.

Equation

x 2 = 0

x 2 = 1

x 2 = −1

x 3 = −1

x 3 = 8

x 2 = 8

Revisiting Equations of the Form x

2

= p

For problems 2–5, solve the equation. Identify the solutions as rational or irrational.

2. x 2 = 25

3. x 2 = 8

Revisiting Equations of the Form x 3 = p

For problems 6–9, solve the equation. Identify all solutions as rational or irrational. 6.

10. Dylan solves the equation x 3 = −64 but makes some errors. His work is shown. Help Dylan correct his work. x

The solutions are −8 and 8

The Baseball Bat Problem

11. Lily wants to ship a baseball bat that is 32 inches long. She can buy a box that measures 20 inches by 20 inches by 20 inches. Will the baseball bat fit in the box? Explain your reasoning.

20inches

20inches

20inches

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23

Name Date

For problems 1–3, solve the equation. Identify all solutions as rational or irrational.

1. x 2 = 13

2. x 3 = 36

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23

Name

Date

Revisiting Equations with Squares and Cubes

In this lesson, we

• solved equations of the forms x 2 = p and x 3 = p.

• expressed irrational solutions by using square root and cube root notation.

Examples

For problems 1 and 2, solve the equation. Identify all solutions as rational or irrational.

1. m2 = 35

m2 = 35

m = √35 or m = −√35

The solutions are √35 and −√35

Irrational

There are two numbers that equal a positive number when squared: a positive number and its opposite. So the equation has two solutions.

2. −100 = r 3

The solution is 3 √−100 .

Irrational

Notice that the variable is cubed. Although 100 is a perfect square, 100 and −100 are not perfect cubes. √ 3 −100 = r

−100 = r 3

3. A square has an area of 46 square centimeters. What is the side length of the square?

Let s represent the side length of the square in centimeters.

s2 = 46

s = √46

The side length of the square is √46 centimeters.

The area of a square can be found by squaring the side length. The side length of a square cannot be negative, so the positive solution is the only answer to the problem.

4. A cube has a volume of 12 cubic inches. What is the edge length of the cube?

Let x represent the edge length of the cube in inches.

x 3 = 12

x = √ 3 12

The edge length of the cube is √ 3 12 inches.

The volume of a cube can be found by cubing the edge length.

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23

1. Which numbers are solutions to the equation x 3 = 27? Choose all that apply.

2. The number −1 is a solution to which equations? Choose all that apply.

For problems 3–10, solve the equation. Identify all solutions as rational or irrational.

11. In the equations, n, m, p, and q are positive numbers. Order n, m, p, and q from least to greatest.

15

30

12. A cube has a volume of 58 cubic centimeters. What is the edge length of the cube?

13. A square has an area of 32 square inches. A cube has a volume of 120 cubic inches. Which is greater, the side length of the square or the edge length of the cube? Explain.

Remember

For problems 14–17, evaluate.

18. Approximate the value of √21 to the nearest tenth by using consecutive whole numbers, tenths, and hundredths. Explain your thinking.

19. Ava solves the equation 1 2 x = 26 but makes an error.

a. Identify Ava’s error. Explain what she should have done.

b. Show the correct work to solve the equation.

Student Edition: Grade 7–8, Module 1, Mixed Practice 1

Mixed Practice 1

Name

For problems 1–3, determine the unit rate for the situation.

Date

1. A car travels 105 miles in 3 hours. What is the unit rate associated with the rate of miles per hour?

2. Maya buys 6 limes for $1.50. What is the unit rate associated with the rate of dollars per lime?

3. Pedro receives 10 text messages in 2 1 _ 2 hours. What is the unit rate associated with the rate of text messages per hour?

4. Liam makes green paint by mixing 3 gallons of yellow paint with 2 gallons of blue paint. He likes the shade of green paint he makes and wants to make more in the same ratio. Complete the table to show the relationship between the number of gallons of yellow paint and the number of gallons of blue paint.

5. Ethan drives 150 miles in 3 hours. If he continues to drive at a constant rate, how many hours will it take Ethan to drive 600 miles?

6. Mrs. Kondo makes bags of school supplies for her students. Mrs. Kondo has 36 pencils and 48 erasers, and she wants to place all of them in the bags.

a. What is the greatest number of bags Mrs. Kondo can make with each bag containing the same number of pencils and the same number of erasers?

b. How many pencils and erasers will be in each bag?

7. Consider the given equation.

a. Fill in the boxes to make a true number sentence.

b. Find the sum of 4 5 and 3 10 . Choose all that apply.

8. Find the product.

9. Use the standard algorithm to find the sum.

For problems 10 and 11, divide.

12. Order the values from least to greatest.

Student Edition: Grade 7–8, Module 1, Mixed Practice 2

Mixed Practice 2

1. Evaluate (2.1)3

A. 4.41

B. 6.3

C. 8.001

D. 9.261

Name

2. Which expressions represent twice the product of 3 and g? Choose all that apply.

A. 2 + 3 + g

B. (3g) ⋅ 2

C. 2(3 + g)

D. 2 + 3g

E. 2(3g)

3. Evaluate 4x 3 + 6 − 16 when x = 2.

Date

4. The measurements of a rectangular floor are shown in the figure.

a. Find the area of the floor.

b. Find the perimeter of the floor.

5. Find the area of the triangle.

6. Two vertices of a rectangle are located at (−2, − 9) and (3, − 9). The area of the rectangle is 100 square units. Name one set of possible coordinates for the other two vertices. Use the coordinate plane as needed.

7. Use the figure shown to complete parts (a)–(f).

a. Name a line.

c. Name a line segment.

e. Name an obtuse angle.

b. Name a ray.

d. Name an acute angle.

f. Name a pair of parallel line segments.

For problems 8 and 9, determine whether the data distribution is approximately symmetric or skewed. Describe the shape of the data distribution.

9. Points Scored in Each Game

Number of Points

1.

2.

3.

4.

5.

Student Edition: Grade 7–8, Module 1, Sprint: Apply Properties of Positive and Negative Exponents

Sprint

Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.

Number Correct:

AApply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.

Number Correct:

Improvement:

BApply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.

Addition and Subtraction of Integers

Student Edition: Grade 7–8, Module 1, Sprint: Addition and Subtraction of Integers

Sprint

Add or subtract the integers.

1. 5 + (−9)

2. − 6 − (−3)

Number Correct:

Add or subtract the integers.

1. 1 + 6

2. − 1 + (−6)

3. 2 + 6

4. − 2 + (−6)

5. − 8 + (−6)

+ (−12)

− (−3)

Add or subtract the integers.

1. 2 + 7

2. − 2 + (−7)

3. 3 + 5

4. − 3 + (−5)

5. − 9 + (−5) 6. − 9 + (−8) 7.

(−28) 20. 6 − (−128) 21. 25 − (−5) 22. 5 − (−25)

Number Correct:

Improvement:

23. − 40 + (−25)

24. 40 + (−25)

25. 40 + 25 26. 65 + (−32)

Student Edition: Grade 7–8, Module 1, Sprint: Integer Multiplication and Division

Sprint

Multiply or divide the integers.

1. 4 ⋅ (−3)

2. − 42 ÷ 6

Number Correct:

Multiply or divide the integers. 1.

Number Correct:

Improvement:

BMultiply or divide the integers. 1.

36. (4)(4)(−3)

Write the number in standard form.

1. 4.25 × 103

2. 7,000 × 10−2

Number Correct:

Write the number in standard form. 1.

Number Correct:

Improvement:

BWrite the number in standard form.

Student Edition: Grade 7–8, Module 1, Sprint: Squares

Sprint

Evaluate the expression.

ANumber Correct: Evaluate

Student Edition: Grade 7–8, Module 1, Bibliograph

Bibliography

CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast .org, 2018.

Common Core Standards Writing Team. Progressions for the Common Core State Standards in Mathematics. Tucson, AZ: Institute for Mathematics and Education, University of Arizona, 2011–2015. https://www.math.arizona.edu/~ime/progressions/.

Ellis, Julie. What’s Your Angle, Pythagoras? Watertown, MA: Charlesbridge Publishing, 2004.

Heath, Thomas L., trans. The Sand Reckoner of Archimedes. London: 2008.

Heath, Thomas L., ed. The Works of Archimedes. New York: Dover Publications, 2002.

National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010.

Zwiers, Jeffrey, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Palo Alto: Stanford University, UL/SCALE, 2017. https://ul.stanford

.edu/sites/default/files/resource/2021-11/Principles%20for%20the%20Design%20of%20 Mathematics%20Curricula_1.pdf.

Student Edition: Grade 7–8, Module 1, Credits

Credits

Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this presentation.

Common Core State Standards for Mathematics Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All Rights Reserved.

For a complete list of credits, visit http://eurmath.link/media-credits.

Cover image copyright Paul Signac, Vue de Constantinople, La Corne d’OR (Gold Coast) matin (Morning), 1907, Private Collection. Photo credit: Peter Horree/Alamy Stock Photo; page 193, Courtesy Future Perfect at Sunrise. Public domain via Wikimedia Commons; page 213, EPS/ShutterStock.com; page 241, Paul Signac, Vue de Constantinople, La Corne d’OR (Gold Coast) matin (Morning), 1907, Private Collection. Photo credit: Peter Horree/Alamy Stock Photo.

All images are the property of Great Minds.

Student Edition: Grade 7–8, Module 1, Acknowledg

Acknowledgments

ments

Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Chris Barbee, Reshma P. Bell, David Choukalas, Mary Christensen-Cooper, Jill Diniz, Kelli Ferko, Levi Fletcher, Anita Geevarghese, Krysta Gibbs, Stefanie Hassan, Travis Jones, Robin Kubasiak, Connie Laughlin, Maureen McNamara Jones, Dave Morris, Ben Orlin, Brian Petras, April Picard, Lora Podgorny, Janae Pritchett, Aly Schooley, Erika Silva, Tara Stewart, Heidi Strate, Cathy Terwilliger, Cody Waters

Ana Alvarez, Lynne Askin-Roush, Trevor Barnes, Brianna Bemel, Carolyn Buck, Lisa Buckley, Adam Cardais, Christina Cooper, Kim Cotter, Lisa Crowe, Brandon Dawley, Cherry dela Victoria, Delsena Draper, Sandy Engelman, Tamara Estrada, Ubaldo Feliciano-Hernández, Soudea Forbes, Jen Forbus, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Adam Green, Dennis Hamel, Cassie Hart, Sagal Hassan, Kristen Hayes, Marcela Hernandez, Abbi Hoerst, Libby Howard, Elizabeth Jacobsen, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Tara O’Hare, Max Oosterbaan, Tamara Otto, Christine Palmtag, Laura Parker, Katie Prince, Gilbert Rodriguez, Todd Rogers, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Tracy Vigliotti, Dave White, Charmaine Whitman, Glenda Wisenburn-Burke, Howard Yaffe

Share Your Thinking

Talking Tool

I know . . . .

I did it this way because . . . .

The answer is because . . . .

My drawing shows . . . .

Agree or Disagree

I agree because . . . .

That is true because . . . .

I disagree because . . . .

That is not true because . . . .

Do you agree or disagree with ? Why?

Ask for Reasoning

Why did you . . . ?

Can you explain . . . ?

What can we do first?

How is related to ?

Say It Again

I heard you say . . . . said . . . .

Another way to say that is . . . .

What does that mean?

Thinking Tool

When I solve a problem or work on a task, I ask myself

Before

Have I done something like this before?

What strategy will I use?

Do I need any tools?

During Is my strategy working?

Should I try something else?

Does this make sense?

After

What worked well?

What will I do differently next time?

At the end of each class, I ask myself

What did I learn?

What do I have a question about?

MATH IS EVERYWHERE

Do you want to compare how fast you and your friends can run?

Or estimate how many bees are in a hive?

Or calculate your batting average?

Math lies behind so many of life’s wonders, puzzles, and plans. From ancient times to today, we have used math to construct pyramids, sail the seas, build skyscrapers—and even send spacecraft to Mars.

Fueled by your curiosity to understand the world, math will propel you down any path you choose.

Ready to get started?

ISBN

Module 1

Rational and Irrational Numbers

Module 2

One- and Two-Variable Equations

Module 3

Two-Dimensional Geometry

Module 4

Graphs of Linear Equations and Systems of Linear Equations

Module 5

Functions and Three-Dimensional Geometry

Module 6

Probability and Statistics

What does this painting have to do with math?

The French neoimpressionist Paul Signac worked with painter Georges Seurat to create the artistic style of pointillism, in which a painting is made from small dots. Signac’s Vue de Constantinople, La Corne d’Or Matin shows the Golden Horn, a busy waterway in Istanbul, Turkey. How many dots do you think were used to make this pointillist painting? How would you make an educated guess?

On the cover

Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907

Paul Signac, French, 1863–1935

Oil on canvas

Private collection

Paul Signac (1863–1935), Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907. Oil on canvas. Private collection. Photo credit: Peter Horree/Alamy Stock Photo