A Story of Ratios® Proportions and Linearity
LEARN ▸ Rational and Irrational Numbers
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
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ISBN 979-8-88588-153-1
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
Student Edition: Grade 7–8, Module 1, Contents
Rational and Irrational Numbers
Topic A 5
Add and Subtract Rational Numbers
Lesson 1
Adding Integers and Rational Numbers
Lesson 2
KAKOOMA® with Rational Numbers
Lesson 3
Finding Distances to Find Differences
Lesson 4
Subtracting Integers
Lesson 5
Subtracting Rational Numbers
Topic B 81
Multiply and Divide Rational Numbers
Lesson 6
Multiplying Integers and Rational Numbers
Lesson 7
Exponential Expressions and Relating Multiplication to Division
Lesson 8
Dividing Integers and Rational Numbers
Lesson 9
Decimal Expansions of Rational Numbers
83
99
Lesson 12
More Properties of Exponents
Lesson 13
Making Sense of Integer Exponents
Lesson 14
Writing Very Large and Very Small Numbers in Scientific Notation
Lesson 15 213
Operations with Numbers Written in Scientific Notation
Lesson 16 231
Applications with Numbers Written in Scientific Notation
Lesson 17
Get to the Point
Topic D
Rational and Irrational Numbers
Lesson 18
Solving Equations with Squares and Cubes
Lesson 19
115
Topic C 139
Properties of Exponents and Scientific Notation
Lesson 10
Large and Small Positive Numbers
Lesson 11
Products of Exponential Expressions with Positive Whole-Number Exponents
141
The Pythagorean Theorem
Lesson 20
Using the Pythagorean Theorem
Lesson 21
Approximating Values of Roots
Lesson 22
Rational and Irrational Numbers
Lesson 23
Revisiting Equations with Squares and Cubes
Resources
Mixed Practice 1
Mixed Practice 2
Fluency Resources
Lesson 21 Number Lines 0 To 10
Sprint: Apply Properties of Positive and Negative Exponents
Sprint: Addition and Subtraction of Integers
Sprint: Integer Multiplication and Division
Sprint: Scientific Notation with Positive and Negative Exponents
Sprint: Squares
Bibliography
Credits
Acknowledgments
Student Edition: Grade 7–8, Module 1, Topic A
The Worst Game Show Ever
That’s correct! You lose $500! But... I got it right! Shouldn’t I gain money, rather than lose it?
Hmm... judges?
My apologies! You’re quite right! You win -$500!
Thank you. 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.
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1
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
2. −3 + (−5)
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.
4. The temperature rises 2°F from 0°F. Then the temperature drops 6°F
5. Consider the expressions.
+ (−9)
+ (−5)
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.
12 + (−10)
−30 + 24
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. 3 4 7 + 2 1 7
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1
Name Date
For problems 1 and 2, use the number line to model the expression and determine the sum.
1. −6 + 6
2. 3 + (−7)
3. Consider the following expression.
−62 + 23
a. Use the number line to model the expression.
b. Determine the sum.
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1
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.
8 + (−13)
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.
The endpoint of the second directed line segment is the value of the sum. In this case, the sum is −5.
8 + (−13) = −5
Start at 0. Because the first addend is 8, draw the directed line segment to the right and make it 8 units long.
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.
6. 2 7 8 + 1 1 8
When the addends are opposite values, they sum to 0.
The two addends have opposite signs, so using decomposition and additive inverses is an effective strategy.
To find the additive inverse of 1 1 8 , decompose −2 7 8 into −1 6 8 and −1 1 8
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.
−13 + 14 6 + (−17)
−2.4 + (−0.7)
For problems 5–7, use the number line to model the expression and determine the sum.
5. 3 + (−6)
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.
9. −60 + (−15) + 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.
3
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)
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.
1st Sum
Find the number that is the sum of 2 others.
Final Answer
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
Final Answer
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2
Name Date
For problems 1 and 2, estimate, and then find the sum.
1. 2 3 8 + ( 5 2 )
3. Noor uses decomposition to find 3 4 9 + 7 5 9 . 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
Estimate the sum by rounding each addend to an integer. Then add the integers.
The addends have opposite signs, so an effective strategy is to decompose one of the addends and use additive inverses. The
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.
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.
3. −21.4 + 18.45
Estimation: −21 + 18 = −3
Sum:
The addends have opposite signs.
The addends have opposite signs, so finding the difference between the absolute values of the two addends is an effective strategy.
21.4 − 18.45 = 2.95
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. – 3–1 4 © Greg Tang –3 2 – 1–3 4 1–1 2
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.
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2
Name Date
1. Jonas uses decomposition to find 7 2 3 + 5 1 3 . 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) 10. −5 4 5 + 2 9 10 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. 8 5 11 + ( 5 3 11 ) = 8 + 5 11 + ( 5) + 3 11
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.
Final Answer
Remember
For problems 16–19, add.
16. 1 3 + 5 3
5 2 + ( 2 2 )
1 9 + ( 7 9 )
1 10 + ( 14 10 )
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.
2. 10 − 8
3. 8 − 10 4. 10 − (−8)
8 − (−10)
−10 − 8
−8 − 10
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?
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3
Name Date
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.
Number 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
b. What must be added to 6 to get a sum of −4?
−10
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?
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
Name Date
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?
Remember
For problems 11–14, add.
15. Use the number line to model the expression. Then determine the sum. −5 + 8
16. Determine the sum.
−5.3 + (−4.5)
17. Complete each comparison by using <, >, or = .
a. 3 8 5 16
b. −82 3 −31 4
c. 5.43 |−6.2|
d. 1.7 |−4.32|
e. |−41 8 | |24 5 | f. |−3.21| |3.21|
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4
Name Date
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.
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4
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 Expression
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 Date
1. Use the expression −2 − 5 to complete parts (a)–(d).
a. Use the number line to model the expression.
b. Write the subtraction expression as an equivalent addition expression.
PRACTICE
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
Remember
For problems 11–14, add. 11. 2 3 + ( 5 6 ) 12. 5 2 + 3 5
15. Estimate, and then find the sum.
16. Which expressions have a value of 105? Choose all that apply.
A. 10 × 5
B. 10 + 5
C. 10,000
D. 100,000
E. 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10
F. 10 + 10 + 10 + 10 + 10
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5
Name Date
Subtracting Rational Numbers
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. 1 4 − ( 3 1 2 )
4. 5 1 3 − ( 2 5 6 ) − 3 2 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
Name Date
Evaluate each expression.
1. 4.7 − (− 3.7)
2. 3 4 − 4 1 8
3. 3 + ( 0.2) − 15.25
Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5
Name Date
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.
A decimal can be decomposed and written as 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 addends together.
4. 3.4 − 0.06 − (−6.45) 3.4 − 0.06 − (− 6.45) = − 3.4 + (− 0.06) + 6.45 = − 3 + (− 0.4) + (− 0.06) + 6 + 0.45 = (− 3 + 6) + (− 0.4 + (− 0.06)) + 0.45 = 3 + (− 0.46) + 0.45 = 3 + (− 0.45 + (− 0.01)) + 0.45 = 3 + (− 0.45 + 0.45) + (− 0.01) = 3 + 0 + (− 0.01) = 2.99
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
Name Date
PRACTICE
For problems 1–10, evaluate the expression.
3 1 5 − 3 5
7.2 − (−6.1)
2 1 8 − ( 3 4 )
7. 0.03 − (−2.8) − 11.97
8. 5 3 10 − 1 5 − ( 1 1 2 )
9. 3 1 2 − 10 3 8 + 4 1 4
13.42 + 7.56 − 1.2
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. 13. 5 3 + 3 5
5 3 + ( 3 5 )
8 9 + ( 1 6 )
5 8 + 4 20
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 .
Student Edition: Grade 7–8, Module 1, Topic B
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)
3. 3(−10)
4. 3 1 2 ( 1)
5. 1 2 ( 4.5)
A Negative Number Times a Positive Number
6. Complete the number sentences to make them true.
2(2) = 4
0(2) = 0
−1(2) =
−2(2) =
−3(2) = 1(2) = 2
For problems 7–9, use the commutative property to determine the product.
7. −0.25(10)
8. 3 4 (7)
9. 9 2 (1 3 )
A Negative Number Times a Negative Number
For problems 10 and 11, complete the number sentences to make them true. 10.
2(−2) = −4
1(−2) = −2
0(−2) = 0
−1(−2) =
−2(−2) =
−3(−2) =
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
Line 3: (−3)(−2) + ( )( ) = 0
Line 4: (−3)(−2) + ( ) = 0
Line 5: + (−6) = 0
Conclusion:
Zero product property
Additive inverse
Distributive property
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 )
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6
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)
c. Write a multiplication expression to represent this situation.
3(−10)
Because Lily descends 10 feet each time, the addends are −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.
2. 4(−6) −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.
3. (−4)(−6) 24
The product of two negative numbers is a positive number.
4. (−0.43)(6)
−0.43 can be decomposed into −0.4 and −0.03
(−0.43)(6) = (−0.4 + (−0.03))(6) = (−0.4)(6) + (−0.03)(6) = −2.4 + (−0.18) = −2.58
After decomposing, use the distributive property.
5. (−5 1 4 )(−20) ( 5 1 4 )( 20) = ( 5 + ( 1 4 ))(−20) = ( 5)( 20) + ( 1 4 )( 20) = 100 + 5 = 105
−5 1 4 can be decomposed into −5 and 1 4
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6
Name Date
PRACTICE
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.
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
Name Date Exponential Expressions and Relating Multiplication to Division
Math Chat
For problems 1–20, evaluate the expression.
11. (1 3 )2 12. ( 4 5 )(5 7 )(3 8 )
15. (−1.5)2 16. (1.5)(−1.5)2 17. (−3)3
( 1 3 )3
19. (1 3 )3
20. ( 1 3 )3
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 Date
For problems 1 and 2, evaluate the expression.
For problems 3 and 4, write the unknown factor equation related to the expression and then determine the unknown factor.
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7
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. ( 2 _ 3 )4 ( 2 3 )4 = ( 2 3 )( 2 3 )( 2 3 )( 2 3 ) = 4 9 ( 2 3 )( 2 3 ) = − 8 27 (− 2 3 ) = 16 81
5. (−0.4) 3 (−0.4) 3 = (−0.4)(−0.4)(−0.4) = 0.16(−0.4) = −0.064
6. −0.3 2 −0.3 2 = −1 ⋅ 0.3 2 = −1 ⋅ 0.09 = −0.09
7. ( 1 5 )2 ( 1 5 )2 = ( 1 5 )( 1 5 ) = ( 1 25 ) = 1 25
The negative sign outside the parentheses represents taking the opposite of ( 1 5 ) 2
The exponent is 4, so there are 4 factors of 2 3
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.
For problems 8 and 9, write the unknown factor equation related to the expression and then determine the unknown factor.
8. −80 ÷ 4
4 ⋅ = −80
−20
9. −60 ÷ (−15)
−15 ⋅ = −60
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.
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7
Name Date
PRACTICE
For problems 1–7, determine whether the value of the expression is positive or negative. Explain your answer.
6. ( 7) 5 7. (1 3 )4
For problems 8–18, evaluate. 8. ( 1 4 )( 1 4 )
3
For problems 19–22, write the unknown factor equation related to the expression and then determine the unknown factor.
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
Name Date
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.
p = 12 q = 3 p = − 18 q = − 2 p = 10 q = − 5 p = − 3 q = 4
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
Name Date
1. Which numbers shown are equivalent to 5 7 ? Circle all that apply.
For problems 2 and 3, divide.
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8
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 a b = (−5) 4 = 5 4
For problems 2–6, divide.
2. 0 ÷ (− 15) 0 15 = 0
3. 90 ÷ 0 Undefined 4. 5 ÷ − 1 1 4 5 ÷ − 1 1 4 = − 5 ÷ − 5 4 =
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.
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
5. 4 9 10 ÷ 1.4 4 9 10 ÷ 1.4 = − 4.9 ÷ 1.4 = − 3.5
6. 4 7 2 3 (4 7 ÷ ( 2 3 )) = − (4 7 ( 3 2 )) = − ( 12 14 ) = 12 14
Write 4 9 10 in decimal form, 4.9, and evaluate the expression by using decimal division.
In the expression 4 7 2 3 , 4 7 is the numerator and 2 3 is the denominator, so divide 4 7 by 2 3 .
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.
52 ÷ 6 1 2 = − 52 ÷ 6.5 = − 8
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
PRACTICE
1. Which expressions are equivalent to 10 ÷ 5? Choose all that apply.
A. 10 5
B. 10 5
C. 10 5
D. 10 5
E. 10 5
2. Evaluate each expression for a = − 3 and b = 6.
For problems 3–10, divide. 3. 6 ÷ 1 1 3
5. 3 8 ÷ 0
11. Evaluate.
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.
+ (8 − 5)3 ÷ 3 + 6
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9
Name Date
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.
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
Write the number as a decimal. Use bar notation where appropriate.
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9
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. Write 12 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 of 14 80 by using prime factorization. Write 14 80 as a decimal fraction and then as a decimal.
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.
Divide the numerator and the denominator by any common factors so that their only common factor is 1.
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.
Because the remainders begin to repeat, the quotients are repeating decimals.
The block of digits 2 and 7 repeats, so the bar is over both 2 and 7
3 is the only digit that repeats, so the bar is only over 3
The quotient is a negative number because 17 15 is interpreted as the opposite of the quotient that 17 15 produces.
Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9
Name Date
PRACTICE
For problems 1 and 2, write the decimal in fraction form.
For problems 3–5, write the number as a decimal fraction. Then write it as a decimal.
For problems 6 and 7, factor the denominator by using prime factorization. Write the number as a decimal fraction and then as a decimal.
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.
24. 9 20 − 1 2
7 16 − 3 4
For problems 28 and 29, evaluate the expression.
28. ( 5 6 )3
29. ( 9 10 )2
30. Which expressions are equivalent to 2 3 ( 4)? Choose all that apply.
Properties of Exponents and Scientific Notation
Student Edition: Grade 7–8, Module 1, Topic C
Million, Billion—What’s the Difference, Really?
Thousand
Seconds
About 17 minutes
Feet
About 12 days
The rough length of a New York block The rough distance from New York to Boston
Dollars
About $0.03 every day of your life
People
A mid-sized high school
About $35 every day of your life
San Jose, CA (approximately)
The rough distance from New York to the moon
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
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 (meters)
2. Complete the table. The table shows an approximate measurement of objects seen in the demonstration.
Approximate Measurement (meters) Objects
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 C, Lesson 10
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 Date
Large and Small Positive Numbers
In this lesson, we
• 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.
Approximate Number of Stacked Pennies
Single Digit Times a Power of 10 (expanded form)
Single Digit Times a Power of 10 (exponential form)
Tower
Use place value units when writing numbers in unit form.
2 hundred thousand 2 × 100,000 2 × 105
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.
Each number in the Sea Star row is an equivalent form of 0.01.
A unit fraction has a numerator of 1.
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?
212,954 ≈ 200,000 = 2 × 105 3,854,083 ≈ 4,000,000 = 4 × 106
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 4 × 106 by 2 × 105
=
= 20
Write the factors of 10 in the numerator and in the denominator. Then pair the factors of 10 to create quotients of 10 10 , which is 1
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
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
Form Unit Form Single Digit Times a Power of 10 (expanded form) Single Digit Times a Power of 10 (exponential form)
2. Complete the table. The table shows the typical speed of the given animals.
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 is 2 × 103 times as much as what number?
For problems 8 and 9, use the values in the table to answer the questions. Location Russia
Approximate
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. 11. 8 15 − ( 4 5 )
For problems 15 and 16, divide.
15. 3
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.
1050 ⋅ 1020
Multiplying Powers with Like Bases
2. Multiply. Write the product as a power of 10 in exponential form.
105 ⋅ 102
3. Multiply. Write the product as a power of 10 in exponential form.
1057 ⋅ 10342
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
C. ( 2)4
D. (24)
E. ( 2)4
F. ( 2)2 ⋅ (2)2
Multiplying Powers with Unlike Bases
For problems 19–21, apply the product of powers with like bases property to write an equivalent expression.
19. 93 ⋅ 94 ⋅ 42 ⋅ 4
20. (−2)2 195 197 ( 2)3 21. 4 104 105
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11
Properties of Exponents
Description
Property Example
Description
Definitions of Exponents
Definition Example
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11
Name Date
For problems 1–4, apply the product of powers with like bases property to write an equivalent expression.
1. 53 ⋅ 56
(3 5 )4(3 5 ) 4. 5a ⋅ 5b
5. Explain what the expression 43 ⋅ 46 represents and how it can be written as a single power.
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11
Name Date 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
x is any number m and n are positive whole numbers when x m . x n = x m+n
Examples
Apply a property of exponents to write an equivalent expression.
1. 94 ⋅ 92 94 ⋅ 92 = 94+2 = 96
94 is 4 factors of 9 92 is 2 factors of 9
So 94 . 92 is 4 + 2 factors of 9, which can be written as 94+2. 94 . 92 = 9 . 9 . 9 . 9 . 9 . 9 4 times 2 times
2. ( 1 6 )( 1 6 )3 ( 1 6 )( 1 6 )3 = ( 1 6 )1+3 = ( 1 6 )4
Negative bases and fractional bases have parentheses to avoid confusion.
The first factor represents 1 factor of ( 1 6 ), so it has an exponent of 1
3. 74 ⋅ 83 ⋅ 74 ⋅ 84 74 ⋅ 83 ⋅ 74 ⋅ 84 = 74 ⋅ 74 ⋅ 83 ⋅ 84 = 74+4 ⋅ 83+4 = 78 87
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
Name Date
PRACTICE
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. 42 ⋅ 42 = 42 ⋅ 2 = 44
Fill in the boxes to create an equation that shows that Sara’s claim is incorrect. 4 ⋅ 4 = 4
14. Write the area of the rectangle as a single base raised to an exponent. 102 ft 105 ft
For problems 15–18, indicate whether the value of the expression is a positive number or a negative number.
19. The product ( 1)3 ⋅ ( 1)n is a negative number. Which of the following values of n are possible? Choose all that apply.
2
B. 5
7
8
E. 10
For problems 20–23, solve for b.
20. 3b = 311 ⋅ 33 21. 7b ⋅ 74 = 712
22. 4 ⋅ 4b = 46 23. 65 ⋅ 63 = 62b
For problems 24–26, fill in the boxes with digits 1–6 to make the equation true. Each digit can be used only once.
24. x 2 ⋅ 4 x = 8 x 6
(2 x )(3x) = x 2
(x 4 y 2)(xy) = x y
Remember
For problems 27–30, multiply.
27. 1 3 ⋅ 6 28. 6( 2 3 ) 29. 10(2 5 ) 30. 4 7 ⋅ 14
For problems 31–34, write each number as a decimal. Use bar notation where appropriate. 31. 36 1,000
9 25 33. 7 9
5 11
Estimate, and then find the quotient.
35. 85.92 ÷ 12
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12
Name Date
More Properties of Exponents
1. Write an expression equivalent to 34 ⋅ 34 .
2. Write an expression equivalent to 34 · 34 ⋅ 34 .
3. Write an expression equivalent to 34 · 34 · 34 ·
Raising Powers to Powers
4. Without knowing what is under the mustard spot, what do you know about the expression? ( )5
5. Draw mustard spots to create an expression on the right side of the equal sign to make a true statement.
( )5 =
6. Assume 34 is under the mustard spot. ( )5
a. Write factors of 34 to create an expression on the right side of the equal sign to make a true statement. (34)5 =
b. How is (34)5 similar to the mustard spot expression in problem 5?
c. Explain the meaning of (34)5
d. Write an expression equivalent to (34)5. Explain your reasoning.
Raising Products to Powers
7. Assume 4 · 3 is under the mustard spot. In this case, what is repeated? How many times? ( )5
8. Vic looks back at two completed problems, which are now also covered in mustard. What do you think the original expressions are?
9. Students are discussing possible answers to the problem (4x)3. What do you think? (4x)3 = (4x) (4x) (4x) (4x)3 = 43x 3
It can’t be both, but which one is right?
Raising Quotients to Powers
10. Consider the expression ( x y )4, where y ≠ 0. Use what you know about exponents and their properties to write an equivalent expression.
11. Assume 3 4 is under the mustard spot. What equivalent expression should Vic write?
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12
Name Date
Apply the properties of exponents to write an equivalent expression. Assume y is nonzero.
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
x is any number m and n are positive whole numbers when (x m)n = x m . n
Power of a Product Property
x and y are any numbers n is a positive whole number when (xy)n = x ny n
Apply the properties of exponents to write an equivalent expression.
1. (w 2 ) 3 w 6 2. (3r 7 ) 4 3 4 r 28
(w2)3 = w2 w2 w2 = w2+2+2 = w3.2 3 times
(w 2 ) 3 is 3 factors of w 2 . This is 3 factors of 2 factors of w, which is 3 ⋅ 2 factors of w.
The entire term 3r 7 is raised to the fourth power (3r7)4 = 3r7 . 3r
4 times
3. ( 9 2 10 )6 912 106 4. 23 ⋅ 53
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 23 ⋅ 53 = (2 ⋅ 2 ⋅ 2) ⋅ (5 ⋅ 5 ⋅ 5) = (2 ⋅ 5) ⋅ (2 ⋅ 5) ⋅ (2 ⋅ 5) = (2 ⋅ 5)3 = 103
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12
Name Date
PRACTICE
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 3 x 4 y 5 meters. What is the area of the square? 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 13
Name Date
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.
Quotients of Powers
6. So-hee applies the properties of exponents to write an equivalent expression for 103 107 . 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.
7. 813 821 8. 7d 4 14d7
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13
Name Date
For problems 1 and 2, apply the definition of the exponent of 0 to write an equivalent expression.
1. (2 3 )0
2. 7 ⋅ x 0
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 89 85 . The table shows their work.
Kabir’s Work
Yu Yan’s 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 Date
Making Sense of Integer Exponents
In this lesson, we
• 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 of the Exponent of 0
Definition of a Negative Exponent
x 0 = 1
x is nonzero
x is nonzero
x –n = 1
x n
n is an integer when
Examples
1. Write an equivalent expression for 5 c 0. Assume c is nonzero.
5 c 0 = 5 ⋅ c 0 = 5 ⋅ 1 = 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 10−5 = 1 105 ?
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 105 ⋅ 1 105 = 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.
3. 17−4 1 ___ 17 4 4. 1 5−3 1 5−3 = 1 ÷ 5−3 = 1 ÷ 1 __ 53 = 1 53 1 = 53
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 r 5 r 11 as r 5 r −11. Then use the definition of a negative exponent to write the answer with a positive exponent. To make a quotient of r 5 r 5 , or 1, write r 11 as r 5 r 6
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13
Name Date
PRACTICE
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.
16. 80 ⋅ 82 = 8t 17. (3 8 )0 (3 8 )t = (3 8 )11
18. 25 ⋅ 20 = t 19. ( 0.25)0 ( 0.25)t = 1
For problems 20 and 21, write a number in the box to make the equation true.
20. (− 2)0 (− 2) = (− 2) 5 21. (1 4 )(1 4 )0 = 1
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 ?
A. 411 413
B. 4 42
C. 4−2
D. 25 29
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 Date
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.
Yes No
× 102
× 103
× 107
× 106
× 109
For problems 6–11, complete the table.
Small Positive Numbers
12. Complete the table.
Approximate Measurement
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 Date
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
5. Order the numbers from least to greatest.
6.0 × 108, 9.2 × 10−15, 5.1 × 109, 8.4 × 109, 7.4 × 10−10, 6.8 × 10−15
Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14
Name Date
RECAP
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.
Number in Standard Form
Number in Scientific Notation
0.0005 5 × 10−4
90,000 9 × 104
0.000 0007 7 × 10−7
8,000 8 × 103
0.000 002 09 2.09 × 10−6
0.000 000 032 3.2 × 10−8
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.
Animal
African Elephant
Blue Whale
Colossal Squid
Giraffe
Grizzly Bear
Hippopotamus
Zebra
Approximate Weight (pounds)
9.5 × 103
3.5 × 105
4.4 × 102
2.2 × 103
8 × 102
6 × 103
8.8 × 102
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
Name Date
1. Circle all the numbers written in scientific notation.
PRACTICE
2. Match each number written in standard form with its corresponding number written in scientific notation.
For problems 3–14, complete the table. Number in Standard Form
3,000
575,000
0.0006
in Scientific Notation
3 × 10−3
2 × 106
4.5 × 105
0.00045
0.000 007
× 10−6
8,095,000
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.
Width
× 106
× 107
× 107
× 106
× 108
× 108
× 107
× 107
× 106
List the planets, including Pluto, from least to greatest based on their width.
Remember
For problems 20–23, multiply.
20. 2 3 ( 7 10 ) 21. 2 5 (7 9 )
5 6 ( 1 5 )
2 9 ( 7 10 )
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 Operations with Numbers Written in Scientific Notation
1. Liam enters 200,000 × 450,000 into his calculator. The screen shows the following display. 9 e + 10
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.
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.
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.
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 x m x n = x m+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) = 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 = 4.096 × 10−24
Use the commutative and associative properties of multiplication to rewrite the original expression. Then multiply the factors in each pair.
Use the properties (xy)n = x n y n and (x m)n = x m ⋅ n to evaluate (1.6 × 10−8)3
5. Use the table of approximate animal weights to complete each part.
Animal
Aphid
Emperor scorpion
Gray tree frog
Termite
Approximate Weight (pounds)
4.4 × 10−7
6.6 × 10−2
1.6 × 10−3
3.3 × 10−6
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) = (1.6 × 10−3) − (0.00044 × 10−3) = (1.6 − 0.00044) × 10−3 = 1.59956 × 10−3
To determine how many more pounds the gray tree frog weighs than the aphid, find the difference of their weights.
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? 6.6 × 10−2 = ⋅ (3.3 × 10−6)
Divide the weight of the scorpion by the weight of the termite.
6.6 × 10−2 ________ 3.3 × 10−6 = (6.6 ___ 3.3 ) × (10−2 ____ 10−6 ) = 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 Date
PRACTICE
For problems 1–4, write the answer displayed on the calculator screen in scientific notation and in standard form. 1. . 678e+ 5 17
5 0.0000023*0.0000 1.15𝝚 -10 3. 1.386471e9
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
5 × 105 + 3 × 104
1010 + 2.2 × 1010
15. Use the table of approximate animal weights to complete each part. Check each answer with a calculator.
Animal
Giraffe
Housefly
Hummingbird
Monarch butterfly
Wood mouse
Approximate Weight (pounds)
2.2 × 103
2.6 × 10−5
8.8 × 10−3
1.1 × 10−3
4.4 × 10−2
Zebra 8.8 × 102
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?
Remember For problems 16–19, divide.
20. Eve and Lily each write an equivalent expression for x 7 x 3 . The table shows their work.
Eve’s Work
7 __ x 3 = x 7 ⋅ x −3 = x 7+(−3) = x 4
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
Name Date Applications with Numbers Written in Scientific Notation
1. The desired volume for the balloon is 3.6 × 100 cubic meters. If the volume of air in 1 breath is about 3.6 × 10−3 cubic meters, about how many breaths does it take to fill the balloon to the desired volume?
2. Now suppose the desired volume of the balloon is the same as the volume of the moon.
a. How many breaths do you think it will take to reach this desired volume?
b. The volume of the moon is approximately 2.1968 × 1019 cubic meters. If the volume of air in 1 breath is about 3.6 × 10−3 cubic meters, about how many breaths do you think it takes to reach the desired volume of the moon-size balloon?
Operating with Numbers Written in Scientific Notation
3. Ava pumps air into a balloon 12 times in 10 seconds. The volume of air in each pump is 3.6 × 10−3 cubic meters.
a. Determine the volume of air in cubic meters that Ava pumps into the balloon in 10 seconds.
b. At that rate, determine the volume of air in cubic meters that Ava pumps into the balloon in 1 second.
c. At that rate, determine the volume of air in cubic meters that Ava pumps into the balloon in 1 minute.
d. At that rate, determine the volume of air in cubic kilometers that Ava pumps into the balloon in 1 minute.
Comparing and Converting Units
4. The volume of the moon is about 2.1968 × 1010 cubic kilometers. Consider the following automated air pumps.
× 101 cubic meters per hour
× 104 cubic centimeters per second
× 108 cubic centimeters per hour
× 1013 cubic millimeters per year
a. Choose one of the automated air pumps to inflate a moon-size balloon. Approximately how long will it take to fill a moon-size balloon with that air pump?
b. Which of the automated air pumps will fill the moon-size balloon the fastest? Explain how you know.
5. A typical 12-year-old breathes 14 times per minute.
a. Assume you have been breathing at about the same rate your whole life. How many breaths do you think you have taken so far? Write your answer in scientific notation.
b. Assume that the number of breaths in your life is used to find the volume of your life-breath balloon. If the volume of 1 breath is about 3.6 × 10−3 cubic meters, what is the approximate volume of your life-breath balloon? Write your answer in scientific notation.
How Many Times as Large and How Much Larger
For problems 6–9, consider a life-breath balloon with a volume of approximately 3.45 × 108 cubic meters.
6. The volume of the life-breath balloon is how many times as large as the volume of each of the following objects?
• Volume of blood in a typical heart: 0.00028 cubic meters
• Volume of a water molecule: 2.99 × 10−23 cubic meters
7. The volume of each of the following objects is how many times as large as the volume of the life-breath balloon?
• Volume of the moon: 2.1968 × 1010 cubic kilometers
• Volume of the sun: 1.409 × 1018 cubic kilometers
8. The volume of the life-breath balloon is how much larger than the volume of each of the following objects?
• Volume of the Great Pyramid of Giza: 2,600,000 cubic meters
• Volume of the Empire State Building: 1.04 × 106 cubic meters
9. The volume of each of the following objects is how much larger than the volume of the life-breath balloon?
• Volume of Crater Lake: 1.74 × 1010 cubic meters
• Volume of Lake Pontchartrain: 6 × 109 cubic meters
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.
1.552 × 107 12(5,280) = 1.552 × 107 63,360 = 1.552 × 107 6.336 × 104 = (1.552 6.336 ) × (107 104 ) ≈ 0.24 × 103 = (2.4 × 10−1) × 10 3 = 2.4 × (10−1 × 103) = 2.4 × 102
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 ≈ 1.6 kilometers) 240 ⋅ 1.6 = 384
A hawk travels about 384 kilometers per day.
Convert the number of miles traveled in 1 day to kilometers.
It would take about 104 days. There are 7 days in 1 week.
7 ≈ 14.86
It would take about 15 weeks for a hawk to travel the longest distance around Earth.
Divide the longest distance around Earth by the distance traveled by the hawk per day.
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 60 seconds in 1 minute.
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
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
Canis Majoris (Sirius)
Canis Minoris (Procyon)
Lyrae (Vega)
Eridani (Rana)
Pavonis
Zeta Tucanae
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.
Remember
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.
7.2 × 105, 8.1 × 10−3, 6.1 × 108, 4.7 × 108, 9.3 × 10−12, 5.6 × 10−8
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
Name Date Get to the Point
Point by Point
1. Study the
Record some questions that come to mind.
2. Focus Question:
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?
