CPM: Chapter 3 Math 7

3-7. Consider the expression 7 + 3 · 4 + 2. 3-7 HW eTool (CPM). Homework Help ✎

a. What movements does this represent for Cecil walking on his tightrope? Draw a diagram to show his

movements and the length of his walk.

b. How many different answers can you get by grouping differently? Add parentheses to the expression 7

+ 3 · 4 + 2 to create new expressions with as many different values as possible.

3-8. Find the distance between each pair of points if they were graphed on a number line. Represent your

work using absolute value symbols. Homework Help ✎

a. −27.1 and 53.2

b. 71.54 and −28.3

c. −38.9 and −7.3

3-9. Find the missing information from the following relationships. Homework Help ✎

a. Mark has downloaded four times as many songs on his music player as Chloe. If Mark has 440 songs,

how many songs does Chloe have?

b. Cici likes to collect shoes, but she only has half the number of pairs of shoes that her friend Aubree

has. If Cici has 42 pairs of shoes, how many pairs of shoes does Aubree have?

c. Tito walked three more miles than Danielle. If Danielle walked 2 miles, how far did Tito walk?

3-10. After a pizza party, Julia has parts of five pizzas left over, as shown below. Each pizza was originally

cut into 12 pieces, and the shaded areas represent the slices that were not eaten. Homework Help ✎

a. What fraction of pizza A is left?

b. If all of the pieces were put together, how many whole pizzas could Julia make? How many extra pieces

would she have?

c. Julia wants to write the amount of leftover pizza as a single fraction. How could she do this?

3-11. Copy and complete the generic rectangle below. What multiplication problem does it represent and

what is the product? Homework Help ✎

CPM: Chapter 3 Math 7

3-11. Copy and complete the generic rectangle below. What multiplication problem does it represent and

what is the product? Homework Help ✎

3-19. Show the numbers of + and – tiles represented by each expression below. Sketch each model and

state the number that it represents. Homework Help ✎

a. (−2) + (−6)

b. (−2) + 4 + (−2)

c. 5 + (−8)

3-20. Copy each expression below and circle each term. Then simplify each expression. Homework Help ✎

1. −8 + 2(−5) 2. 3(7.5 + 2) + 4.6 3. 4 (−2 + 1 + 7) 4. 5(6 + 2) + 4 + 2 (−5 + 8)

3-21. Justin is working with the integer tiles shown in the diagram at right. Homework Help ✎

a. What is the value of Justin’s diagram?

b. If Justin removes three positive tiles, what will the value be?

c. If Justin starts with the original diagram and removes three negative tiles, what will the value be?

d. Justin has a new arrangement of tiles shown at right. If he removes four positive tiles, what

will the value be?

3-22. Linh has a bag of beads that contains 10 glass beads, 7 metal beads, 15 plastic beads, and 3 clay

beads. For each part below, if the bead selected is replaced before the next draw, what is the probability that

Linh will pull out a. 3-22 HW eTool (CPM). Homework Help ✎

a. Metal bead?

b. Bead that is not plastic?

c. Glass or plastic bead?

3-23. Copy each portions web below and fill in the missing parts. Homework Help ✎

1. 2.

CPM: Chapter 3 Math 7

3-34. The number 6 is represented with + and – tiles at right. For each problem below, start with this

diagram and then show how to remove the appropriate number of tiles to represent the problem given. State the

final answer for each part. Homework Help ✎

a. 6 − 7

b. 6 − (−2)

c. 6 − 3

3-35. Use a number line to show the distance between the two given integers. Plot each integer as a point on

the number line, and then find the distance between the two points. Write each situation as a subtraction

problem using absolute value symbols. 3-35 HW eTool (CPM). Homework Help ✎

a. 7 and 4

b. 5 and −2

c. −1 and −3

3-36. Consider the expression 5 + 4 + 2(−3) + 7 + (−5). 3-36 HW eTool (CPM). Homework Help ✎

a. Draw the expression using + and – tiles.

b. Simplify the expression to find the number it represents.

c. Remove the last number from the expression (−5) and find the sum again. Show how this would change

your drawing. How much larger or smaller is the answer? Explain how your answer makes sense when

compared to the answer in part (b).

Evaluating Algebraic Expressions An algebraic expression consists of one or more variables, or a combination of numbers and variables

connected by mathematical operations. Examples of algebraic expressions include 4x, 3(x − 5), and 4x −

3y + 7.

To evaluate an algebraic expression for particular values of the variables, replace the variables in the

expression with their known numerical values and simplify. Replacing variables with their known

values is called substitution. An example is provided below.

Evaluate 4x − 3y + 7 for x = 2 and y = 1.

Replace x and y with their known values of 2 and 1, respectively, and simplify.

CPM: Chapter 3 Math 7

3-37. Read the Math Notes box in this lesson. Then use the information to answer the following questions.

Jorge was thinking about using variables to represent lengths of a tightrope walker’s many tricks. Homework

Help ✎

a. Jorge wrote the expression x + x + x + x + 3 to represent the sequence shown in the diagram at

right. Does his expression make sense? Explain.

b. Write an expression to represent the sequence shown in the diagram at right.

c. In part (a), if x = 5 feet, how long is the tightrope?

d. In part (b), if j = 3 feet and k = 2 feet, how long is the tightrope?

3-38. Simplify the following fraction expressions. Show all of your work. Homework Help ✎

1. 2. 3. 4.

3-46. Find the value of each expression below. Change any subtraction problem to an equivalent addition

problem. Draw a diagram with + and – tiles to justify your answer. Homework Help ✎

a. 5 − 7

b. −5 + (−7)

c. −5 + 7

d. −5 − (−7)

3-47. Rewrite each of the following expressions using only addition. Then simplify the expression.

Homework Help ✎

a. 5(−2) − 3

b. 7.69 − (−2.5)(−4)

c. −7 + −3 − (−5)

d. (−4)(−25) − 300

3-48. Each of the diagrams below represents tightrope moves for an acrobat. For each diagram: Homework

Help ✎

Describe what the diagram shows about the routine and the length of the tightrope.

Figure out the length of the acrobat’s trick.

1.

2.

CPM: Chapter 3 Math 7

3-49. Copy each expression below and simplify it. Be sure to show the steps you use to get the

answer. Homework Help ✎

a.

b.

c.

3-50. Sketch the rectangle at right on your paper. Calculate the perimeter and area for the given x-values. 3-

50 HW eTool Homework Help ✎

a. x = 5

b. x = 9

c. x = 4.6

3-59. Copy and simplify each expression below. Homework Help ✎

a. 6 + (−18)

b. c. −9 + (−9)

d. −12.2 + 6.1 + 15.8

3-60. Find each of the following products or quotients without using a calculator. Draw a diagram or use

words to explain how you know your product makes sense. Homework Help ✎

a. 6(−3)

b. −6(3)

c. −8(−3)

d. −8(0)

e. −20 ÷ 5

f. −36 ÷ (−9)

3-61. Simplify each of the following expressions without using a calculator. Draw diagrams or use words to

show your thinking. Homework Help ✎

a. b. −8 + −2(3)

c. 4 + −5(−3) + (−7)

d. (−4.25)(2) + (−4.25)(−2)

3-62. Simplify the following fraction expressions. Show all of your work. Homework Help ✎

1. 2. 3. 4.

CPM: Chapter 3 Math 7

3-63. Nathan wants you to solve this puzzle: “I am thinking of a number. If you divide my number by 3 and

add –3, you will get 4. What is my number?” Show all of your work. Homework Help ✎

3-73. Use what you learned during today’s lesson to answer the following questions. Homework Help ✎

a. Show (2.3)(5.06) as a fraction multiplication problem and explain why the answer is in thousandths

(three decimal places).

b. Show (0.004)(3.42) as a fraction multiplication problem and explain why the answer is in hundred-

thousandths (five decimal places).

3-74. Mentally calculate the following products. Use the rule for decimal multiplication to write an equation

in which the decimal point is written in the correct location. Homework Help ✎

a. (−0.04)(−0.1)

b. (0.03)(−0.02)

c. (0.7)(0.4)

d. Stacey said, “Stephanie, look at my answer to the last problem, 0.7 · 0.4 = 0.28. Usually

when I multiply, I get a bigger answer than the numbers I start with. Twenty-eight

hundredths, 0.28, is less than either 0.4 or 0.7. I must have made a mistake.”

Stephanie responded, “Well, one half times one half is one fourth, and one fourth is less then

one half. I think when you multiply by a fraction or decimal less than one, you get less than you started

with.”

Write a sentence or two about who you think is correct and why.

3-75. The highest point in the United States is Mount McKinley, also called Denali, in Alaska. Its summit is

20,335 feet above sea level. Badwater, a basin located in Death Valley, is the lowest point in the United States

at 282 feet below sea level. How high is the summit of Denali above the Death Valley location? Show your

calculation using absolute value symbols. Homework Help ✎

3-76. Find the mean, range, and median of the values: 12, 4, −2, 0, 9, −2, 1, 7, 8, 2. Recall that the range is

calculated by finding the difference between the largest and smallest data values. 3-76 HW eTool (CPM).

Homework Help ✎

3-77. Simplify each multiplication problem below. Homework Help ✎

a.

b.

3-82. Simplify each expression without using a calculator. For each expression, draw a diagram or use words

to explain how you know your answer makes sense. Homework Help ✎

a. −8 ÷ −4

b. 18 ÷ −3

c. −24 ÷ 2

d. 17 ÷ −1

CPM: Chapter 3 Math 7

3-83. Follow the order of operations (use circling the terms if it is helpful) to simplify the following

expressions: Homework Help ✎

a. −7 + 4 ÷ (−2)

b. 17.5 ÷ (−7) + −8.1(2)

c. d. (4 − (−3)) + (5 ÷ (−5))

3-84. Read the Math Notes box in this lesson. Use the information to find the multiplicative inverse, or

reciprocal, of each number below. Homework Help ✎

a.

b. c. 2

d. e. Check your answers by multiplying the given number with your answer and verifying that the product of

the numbers is 1.

3-85. Write at least three expressions that use each of the numbers 2, 3, 6, and 8 only once and any operations

and grouping symbols (addition, subtraction, multiplication, division, and parentheses). Each expression should

have a different value, with one being equal to 28. Homework Help ✎

3-86. Copy and complete each problem mentally. Homework Help ✎

a.

b.

c.

d. e. What do all of the problems have in common?

3-93. Complete each of the following division problems by first writing each of them as a

fraction. Homework Help ✎

a. −6 ÷ 8

b. 6 ÷ (−8)

c. 0 ÷ 8

d. 6 ÷ 0

CPM: Chapter 3 Math 7

3-94. Use any method to divide. Homework Help ✎

1. 2. 3. 4.

3-95. One third of a rectangular playground is designed for young children. In that part of the playground, a

play structure covers of the children’s space. Homework Help ✎

a. Use a drawing to represent the portion of the playground that is the play structure.

b. Represent the problem with multiplication.

c. What fraction of the total playground is the play structure? Show all of your work.

3-96. Represent each operation below by drawing – tiles. Then simplify each expression. Homework Help

✎

a. −6 + (−2)

b. −6 −(−2)

3-97. If the line segment at right is subdivided into equal parts, what is the length of the marked portion of the

line? Homework Help ✎

3-107. Answer the questions below using your work from this lesson. Homework Help ✎

a. Find 15.75 ÷ 0.25 using the standard algorithm for division.

b. Find 15.75 ÷ 0.25 using a Giant One.

c. Which method do you prefer?

3-108. Write an equation and find the answer for each of the following problems. Give answers in decimal

form. Homework Help ✎

a. What is the product of three tenths and four hundredths?

b. What is the sum of five and thirteen hundredths plus twelve and six tenths?

c. What is forty-seven and seven tenths minus twenty-three and nineteen hundredths?

d. What is the product of twenty hundredths and sixty-five thousandths?

e. What is the sum of fourteen tenths and fourteen hundredths?

3-109. Each problem below has an error in the answer. Find the error, explain how to correct the mistake,

and correct it. Homework Help ✎

1. 2. 3. 4.

CPM: Chapter 3 Math 7

5. 6.

3-110. This problem is a checkpoint for multiplication of fractions and decimals. It will be

referred to as Checkpoint 3.

Multiply each pair of fractions or each pair of decimals. Simplify if possible. Homework Help ✎

a.

b.

c.

d. e. 3.62 · 3.4

f. 0.26 · 0.0008

Check your answers by referring to the Checkpoint 3 materials.

Ideally, at this point you are comfortable working with these types of problems and can solve them

correctly. If you feel that you need more confidence when solving these types of problems, then review

the Checkpoint 3 materials and try the practice problems provided. From this point on, you will be expected to

do problems like these correctly and with confidence.

3-111. Complete the following division problems using any method. Homework Help ✎

1. 2. 3. 4.

3-120. Hilda was simplifying some numerical expressions and made each of the following sequences of

calculations. Name the mathematical property, operation, or idea that justifies how Hilda went from each step

to the next step. Homework Help ✎

1. 2.

CPM: Chapter 3 Math 7

3-121. Simplify each of the following expressions. Homework Help ✎

a. 3 + 4(10 − 8)

b. 3 + 4 · 10 − 8

c. (3 + 4)(10 − 8)

3-122. Refer to the diagram at right to answer the questions below. 3-122 HW eTool (CPM). Homework

Help ✎

1. Copy the diagram and add two tiles so that the result is a value of –2. You can add positive and/or

negative tiles. Show the tiles you added to get this answer.

2. Copy the diagram again, but this time remove four tiles (again positives and/or negatives) so that the

result is a value of 2. Show the tiles you removed to get this answer.

3-123. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems

is shown at right. Homework Help ✎

1. 2. 3. 4.

3-124. Greta is trying to determine the portion of green candies in various bags of green and yellow

candies. Using the information below, determine the portion of green candies in each bag. Homework Help ✎

a. Bag A: Two thirds of the candies are yellow. What portion of the candies is green?

b. Bag B: 29% of the candies are yellow. What portion of the candies is green?

c. Bag C: 4 out of every 9 candies are yellow. What portion of the candies is green?

3-125. Solve the following problems without paper or pencil. Just write the answer but be ready to explain

your thinking. Homework Help ✎

a. 2 ÷

b. 2 ÷

3-126. Simplify each of the following expressions. Homework Help ✎

a. 3 − 2 · 5

b. 7(4 + 3 · 2)

c. 10 − 6 ÷ 2 · 4 + 2

CPM: Chapter 3 Math 7

3-127. Four friends worked together to wash all of the cars that the Kish family owns. They received $42.36

for doing the work and agreed to divide the earnings evenly. How much money will each friend earn? Show

how you know. Homework Help ✎

3-128. Find each product. Express your answer as both a fraction and a decimal. Homework Help ✎

a.

b. c. −4.37 · (−5.2)

3-129. Cecil the Acrobat walked 3 feet on his tightrope, backed up 1 foot, then walked 6 feet to get to

the other side. Homework Help ✎

a. Write an expression to represent Cecil’s walk.

b. How far did Cecil walk?

c. Cecil started at the beginning again and walked 6 feet, backed up 1 foot, then walked 3 feet. How

far did he walk this time? How does this compare to your answer from part (b)? Explain why these

answers are the same or different in a sentence or with a drawing.

CL 3-130. At right is a representation of the number 6. Copy this diagram and use it to illustrate each of the

subtraction problems shown below.

a. 6 − 8

b. 6 − (−1)

CL 3-131. Simplify each of the following expressions.

a. (−2)(−5) + 7

b. 4.18 + 3(−4.6) − 2.36

c. (3.71 + (−5.25))(4 + (−9))

d. −3(4 + (−7) + −2(5) − 17

CL 3-132. Robert found a game spinner that was not completely labeled. The spinner

is shown at right. Help Robert figure out what fraction of the spinner is missing.

CL 3-133. What number is represented by each of the letters on the number lines

below?

1. 2.

CPM: Chapter 3 Math 7

CL 3-134. The diagram at right represents an acrobat’s sequence on a tightrope,

where m represents the distance in feet that she covers each time she does a leap.

a. How long is each of her leaps? How can you tell?

b. How long is the tightrope?

c. Write expressions to represent each of her two sequences across the tightrope.

CL 3-135. Complete a portions web for each representation of a rational number below.

a.

b. 118%

c. 0.085

d.

CL 3-136. Without a calculator, find the following products.

1. 2. 3. 4.16(0.2) 4.

2. CL 3-137. Without a calculator, find the following quotients.

1. 2. 1.2 ÷ 0.04 3. 4.