critical thinking grade 7

NAME _________________________________________________________________________________

Lesson 4.3 Constants of Proportionality

A unit rate can also be called a constant of proportionality. The constant of
proportionality, k, is the ratio of the output variable to the input variable.

Days of Car Rental 3 5 6
Cost of Car Rental 96.75 161.25 193.50

k  output ; k  96.75  32.25; 161.25  32.25; 193.25  32.25
input 3 5 5

The constant of proportionality is 32.25. The cost of renting a car is $32.25 per
day.

Use the information to calculate the constant of proportionality for each table. What
does the constant of proportionality mean in the context of the data given?

Gallons of Gas 3 8 15
Price 6.54 17.44 32.70



Time (hours) 1.75 0.25 2.5
21 3 30
Distance Biked
(miles)

Spectrum Critical Thinking for Math Lesson 4.3
Grade 7 Constants of Proportionality

49

NAME _________________________________________________________________________________

Lesson 4.4 Using Equations to Represent Proportions

The constant of proportionality can be used to write an equation to represent the
relationship.

A recipe calls for 1 cup of sugar for every 1 1 cups of flour. Write an equation
2 3

to calculate how much flour to use for each cup of sugar. How many cups of flour

will be used if 2 1 cups of sugar are used?
2

k  cups of flour  1 1  8  2 2
sugar 3 3 3

1

2

f  (2 2 )s f  (2 2 ) (2 1 )  ( 8 ) ( 5 )  40  6 2 cups of flour
3 3 2 3 2 6 3

Write an equation using the constant of proportionality to represent each relationship
described.

It takes 1 1 gallon of gas to mow 1 acre of grass. How much gas does it take to
2 2
1
mow 2 2 acres of grass?

There are 880 feet in 1 of a mile. How many feet are in 1 1 miles?
6 3

Spectrum Critical Thinking for Math Lesson 4.4
Grade 7 Using Equations to Represent Proportions

50

NAME _________________________________________________________________________________

Lesson 4.5 Proportions on the Coordinate Plane

A constant of proportionality can be found using the graph of a proportional
relationship. Identify an ordered pair (x,y) on the line. The constant of
proportionality is k  y/x.

Given the graph, calculate the constant of proportionality. Value ($) 0.90
What does the constant of proportionality represent 0.70
on the graph? 0.50

k  0.50  0.10 0.30
5
0.10
The constant of proportionality is 0.10. 0 12345678
It represents the value of each dime: $0.10. # of dimes

Calculate the constant of proportionality for each graph. What does the constant of

proportionality represent? 12
11
2.25 10
1.75
1.25 9
0.75 8
0.25 7
6
0 1234567 5
4
3
2
1

1 2 3 4 5 6 7 8 9101112

How much would 7 tokens be worth? How many blue marbles are there if
there are 3 red marbles?

Spectrum Critical Thinking for Math Lesson 4.5
Grade 7 Proportions on the Coordinate Plane

51

NAME _________________________________________________________________________________

Lesson 4.6 Proportions in the Real World

Answer the questions. Show your work.

Juice is sold at the grocery store in several different sizes. The prices are shown in
the table.

Size (fl. oz.) Price ($) Unit Price ($)

16 1.89

32 3.49

64 7.59

128 9.99



a. Complete the table with the unit prices of each size. Round to the nearest

hundredth.

b. If you wanted 64 fluid ounces of juice, which would be the best way to
purchase it?

Amount of Bill 19.00 35.00 72.00
Tip 3.42 6.30 12.96

Calculate the constant of proportionality shown in the table above. What does this
constant mean within the context of the data given? How much would the tip be if the
bill was $95.00?

Spectrum Critical Thinking for Math Lesson 4.6
Grade 7 Proportions in the Real World

52

NAME _________________________________________________________________________________

Lesson 4.6 Proportions in the Real World

Answer the questions. Show your work.
One half of a can of paint covers 150 square feet of a wall.
a. Create a table to represent this relationship.

b. Create a graph to represent this relationshiparea covered
(square feet)
100
900
800
700
600
500
400
300
200
100

0 1 2 3 4 5 6 7 8 9 10

cans of paint

c. What is the constant of proportionality? What does it represent?

d. Write an equation to represent the relationship between cans of paint used and

how much of the wall is covered. How much of the wall can be covered by 2 1
2

cans of paint?

Spectrum Critical Thinking for Math Lesson 4.6
Grade 7 Proportions in the Real World

53

NAME _________________________________________________________________________________

Check What You Learned

Ratios and Proportional Relationships

1. Pool A is being filled with 2 gallon per 1 minute. Pool B is being filled with 3
3 4 5
3
gallon per 5 minute. Which pool is being filled faster?

CHAPTER 4 POSTTEST

2. Graph the values in the table to see if they represent a proportional relationship.

Time 1 1 1 2
2 2

Amount Painted (square feet) 28 84 112

120
110
100

90
80
70
60
50
40
30
20
10

1 2 3 4 5 6 7 8 9 10

Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Learned

54

NAME _________________________________________________________________________________

Check What You Learned

Ratios and Proportional Relationships

3. Use the table to find the constant of proportionality.

Miles Walked 1 1 3 3 1
Calories Burned 4 4 2

25 175 350

4. Write an equation using the constant of proportionality in #3 on the previous CHAPTER 4 POSTTEST

page. How many calories would you expect to burn if you walk 3 1 miles?
5

5. Use the graph to create a table pay ($) 160
of values. Find the constant of 140
proportionality and write an 120
equation. What does it mean 100
within the context of the graph?
How much would the pay be 80
after 25 hours of work? 60
40
20
10

0 2 6 10 14 18

hours worked (h)

Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Learned

55

NAME _________________________________________________________________________________

Mid-Test Chapters 1–4

1. Yuri had a bank balance of $57 before he went shopping. After he used his debit
card twice, his account was overdrawn by $14. What was the total amount of the
debits?

2. Astrid played a board game in math class. She ended up with these cards:
3.9 4.2 3.9 4.1 6.8 8.5 10.8

a. The score is the sum of the card values. What was her score? Show your work.

CHAPTERS 1–4 MID-TEST
b. In order to win the game, the absolute value of a score must be less than 12.

Is it possible for Astrid to win? Explain why or why not.

3. A number j is positive and another number k is negative. Based on this
information, can you determine whether j  k is positive or negative? Explain.

Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Mid-Test

56

NAME _________________________________________________________________________________

Mid-Test Chapters 1–4

4. A mini-shelf in the food pantry can hold 3 3 pounds. If a can weighs 3 of a
4 8

pound, how many cans can it hold? If you add more support so the shelf can hold

5 1 pounds, how many cans can the shelf hold now? Show your work.
4

5. Mary Ellen says that the expression 5  7  8(4  2) simplifies to a
negative number because if you multiply three negative numbers, the final answer
will be negative. Is she correct? Show why or why not.

6. A number is multiplied by  3 , divided by 1 , and then divided by  7 . The CHAPTERS 1–4 MID-TEST
4 2 8

resulting number is 96. Work backward to get the original number by performing

opposite operations.

Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Mid-Test

57

NAME _________________________________________________________________________________

Mid-Test Chapters 1–4

7. Determine whether the expression 2(x  3) is equal to (4x  1 )  (8x  5 1 ).
2 2

Identify the properties you used in your solution steps.

8. A plumber charges $110 for a service call and $65 for each hour of work after
the first hour. Let h represent the hours the electrician works on a service call.
Write an expression to represent the cost. How many hours did it take the plumber
to complete a job if the total cost is $240?

CHAPTERS 1–4 MID-TEST 9. Candace has $65.25. She spent $31.50 on a new throw rug for her bedroom.
She wants to buy some matching throw pillows. How many pillows can she buy if
the pillows cost $11.25 each?

Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Mid-Test

58

NAME _________________________________________________________________________________

Mid-Test Chapters 1–4

10. A museum is keeping track of the number of visitors per day.

15,000

14,000

13,000

12,000

11,000

10,000

9,000

8,000

7,000

6,000

5,000

4,000
3,000

2,000

1,000

0 1 2 3 4 5 6 7 8 9 1011121314

a. Create a table that represent the proportional relationship.

b. What is the constant of proportionality to the nearest hundredth? What does it CHAPTERS 1–4 MID-TEST
represent on the graph?

c. Write an equation to represent the relationship.

d. Predict how many people will have visited the museum after 33 days.

Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Mid-Test

59

CHAPTER 5 PRETEST NAME _________________________________________________________________________________

Check What You Know

Geometry
1. Find the length of the missing side for the pair of similar triangles.

12 ft . 14 ft .
18 ft .

18 ft .

27 ft .

2. Can these lengths form a triangle?
Side 1: 9 cm
Side 2: 5 cm
Side 3: 11 cm

3. Name the shape that is created by each cross section.

Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Know

60

NAME _________________________________________________________________________________ CHAPTER 5 PRETEST

Check What You Know

Geometry

4. Find the circumference and area of the circle. Use 3.14 for . Round answers to
the nearest thousandth, if necessary.

4.3 m A  _______ square feet
C 5 _______ feet

5. If AGB is 120 degrees, what is the measure of HGE?

A B

H G C HGE  _______ degrees

E
D

F

6. What is the volume of a rectangular prism with a length of 10mm, a width of
8mm, and a height of 5mm?

7. What is the combined area of a rectangle with a length of 13.2 in. and a width of
4.1 in., and a triangle with a base of 13.2 in. and a height of 6.5 in.?

Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Know

61

NAME _________________________________________________________________________________

Lesson 5.1 Scale Drawings

Scale drawings are used to represent an object. Scale drawings can be smaller,
larger, or the same size as the original object. The scale factor shows the
proportional relationship between the original object and the scale drawing.

Benita makes a scale drawing so she can rearrange her room. Her actual room is
12 feet by 14 feet. Her drawing is 6 inches by 7 inches. What is the scale factor?
She wants to draw her bed in a new spot. If her bed is 4 feet by 8 feet, what size
should she draw it on her diagram?

scale factor  inches  6 in.  0.5 in.
feet 12 f t . 1 ft.

The scale factor can also be written as 0.5 in.:1 ft. 7 in.
6 in.
4 ft.  0.5 in.  2 inches
1 ft.

8 ft.  0.5 in.  4 inches
1 ft.

The bed should be 2 inches by 4 inches on the diagram.

Solve the problem. Show your work.

The scale in the drawing is 2 cm:5 m (2 cm  5 m). Find the length and width of the
actual room. What is the area? What is the perimeter?

20 cm

5 cm

Spectrum Critical Thinking for Math Lesson 5.1
Grade 7 Scale Drawings

62

NAME _________________________________________________________________________________

Lesson 5.2 Forming Triangles

The sum of the lengths of two sides of a triangle must be greater than the length of
the third side.

Jorge is planning to build a plant box in the shape of a triangle. He has 3
planks of wood that are 4 feet, 6 feet, and 3 feet long. Will he be able to build a
rectangular plant box?

463
634
346

Jorge will be able to build a triangular box using these three planks because each
sum of two sides is greater than the length of the remaining side.

Don has 3 straws with lengths of 3 cm, 4 cm, and 9 cm. He is trying to make a
triangle with the straws. Will he be successful? If not, which straw should he replace?
What is the minimum length of the replacement?

There are 3 line segments with lengths x  2, x 2 3,and x  1. What is the
minimum value of x that allows these line segments to form a triangle? Assume that x
is an integer.

Spectrum Critical Thinking for Math Lesson 5.2
Grade 7 Forming Triangles

63

NAME _________________________________________________________________________________

Lesson 5.3 Cross Sections of 3-Dimensional Figures

A cross section is the intersection of a 3-dimensional figure and a plane. Here
are some examples:

Intersect each 3-D figure with the given 2-D shape.

A rectangle A square

A quadrilateral A triangle

Spectrum Critical Thinking for Math Lesson 5.3
Grade 7 Cross Sections of 3-Dimensional Figures

64

NAME _________________________________________________________________________________

Lesson 5.4 Circles: Circumference

The perimeter of a circle is called the circumference.
C  2r (r  radius) or C  d (d  diameter), where   3.14

A plate has a diameter of 10 inches. What is the circumference of the plate?
C  3.14(10)  31.4 inches

Answer the questions. Show your work. Use 3.14 for p.
In college basketball rules, the ball can have a maximum circumference of 30 inches.
What is the maximum diameter of a basketball to the nearest hundredth?

A round stained-glass window has a circumference of 195 inches. What is the radius
of the window to the nearest inch?

Spectrum Critical Thinking for Math Lesson 5.4
Grade 7 Circles: Circumference

65

NAME _________________________________________________________________________________

Lesson 5.5 Circles: Area

The area of a circle is A  r2, where   3.14.

A circular pool has a radius of 10 feet. What is the area of the pool?
A  3.14(10)2
A  314 square feet

Answer the questions. Show your work. Use 3.14 for p.

A playground area is circular with a diameter of 32 feet. What is the area of the
playground? Round your answer to the nearest tenth.

A frying pan has a diameter of 11 inches. What is the area to the nearest square
inch of the smallest cover that will fit on top of the frying pan?

Justin just got his driver’s license. His parents are giving him permission to drive
within a 25-mile radius of his home. What is the area Justin is restricted to when
driving? Round your answer to the nearest tenth.

Spectrum Critical Thinking for Math Lesson 5.5
Grade 7 Circles: Area

66

NAME _________________________________________________________________________________

Lesson 5.6 Angle Relationships

When two lines intersect, they form angles that have special relationships.
• Vertical angles have the same measure.
• Supplementary angle are two angles with the sum of 180 degrees
• Complimentary angles are two angles with the sum of 90 degrees.

828 p

What is the value of p?
The angles are supplementary, so the sum is 180.

p  82  180
82 82

p  98 degrees

If 4 is a right angle and 5 measures 40 degrees, find the measures of the
remaining angles.

GH

3 J
L 4I 2

51

K

Spectrum Critical Thinking for Math Lesson 5.6
Grade 7 Angle Relationships

67

NAME _________________________________________________________________________________

Lesson 5.7 Area of Composite Shapes

Shapes that are composed of other shapes are called composite shapes The area
of a composite shape is equal to the sum of each shape it is made of.
Find the area of the composite shape:

3m

11.4 m

Area = area of rectangle + area of semicircle
area of rectangle  lw  (11.4)(3)  34.2 m2

Area of semicircle  1 r2  1 (3.14) ( 3 )2  3.53 m2
2 2 2
34.2  3.53  37.73 m2

Find the area of the composite shapes. Show your work. Use 3.14 for p. Round
answers to the nearest hundredth.

5 ft.
3 ft.

3.1 cm Lesson 5.7
7.4 cm Area of Composite Shapes

4.2 cm

Spectrum Critical Thinking for Math
Grade 7
68

NAME _________________________________________________________________________________

Lesson 5.8 Volume of Rectangular Prisms

Volume is the amount of space an object occupies. The volume of a rectangular
prism can be calculated using the formula V  bh, where b  area of the base
and h  height. The area of the base is b  lw, where l  length and w  width.
The volume of a rectangular prism is 210 cm3. If it has a length of 5 cm and a
width of 6 cm, what is the height?

V  210 cm3
210  (5)(6)h

210  30h
h  70 cm

Answer the questions. Show your work.

Penny is using colored sand to fill a jar that is shaped like a rectangular prism. The
bag of sand contains 150 cubic inches. The base of the prism is 6.5 inches by 7.4
inches. The height of the box is 2.2 inches. Will all the sand fit in the jar?

A rectangular prism has a volume of 966 ft3. The prism’s height is 4 feet, and its
length is 14 feet. What is its width?

Spectrum Critical Thinking for Math Lesson 5.8
Grade 7 Volume of Rectangular Prisms

69

NAME _________________________________________________________________________________

Lesson 5.9 Volume of Triangular Prisms

A triangular prism is a prism whose base is a triangle. The volume of a triangular
prism is the product of the area of the base and the height of the prism.

Volume  bh, where b  1 bh
2

The triangular base has a height of 3 cm and a

base of 8 cm. The height of the prism is 12 cm.

b 1 (8)(3)  1 (24)  12 cm2
2 2
V  (12)(12)  144 cm3
3 cm 12 cm
8 cm

Find the volume of each figure. Show your work.

6 cm

12 cm

4 cm

4 cm Lesson 5.9
Volume of Triangular Prisms
4 cm
12.5 cm

3 cm

Spectrum Critical Thinking for Math
Grade 7
70

NAME _________________________________________________________________________________ CHAPTER 5 POSTTEST

Check What You Learned

Geometry
1. At the photo lab, a customer brings in a photograph that is 4 inches wide by 6

inches high. The customer wants the photograph enlarged to 20 inches wide by
25 inches high. Can this be done? Explain your reasoning.

2. If a triangle XYZ has two sides with lengths of 5 cm and 8 cm, what is the
maximum and minimum length of the third side? Explain your answer. Assume
that the length of the third side is an integer.

3. What are two possible shapes that can be formed by a cross section of this
shape? Describe the angle of the cross section.

Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Learned

71

NAME _________________________________________________________________________________

Check What You Learned

Geometry

4. A mini pancake has a circumference of 3 centimeters. A regular pancake has a
circumference of 6 centimeters. Is the area of the regular pancake twice the area
of the mini pancake? Use 3.14 for .

CHAPTER 5 POSTTEST

5. If 4 is a right angle and 5  40°, find the measure of the remaining angles.

C

A 54
1B 2 3 D

E
F

Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Learned

72

NAME _________________________________________________________________________________

Check What You Learned

Geometry
6. Find the volume of the figure.

2 mm

10 mm 5 mm CHAPTER 5 POSTTEST
2 mm
2 mm

9 mm

7. Bill wants to fill this triangular prism 2 full of water. How much water does he
3

need?

20 m

16 m
12 m

Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Learned

73

CHAPTER 6 PRETEST NAME _________________________________________________________________________________

Check What You Know

Statistics
1. Are the following samples biased or random? Explain your answer.
a. Wendy wants to find out the favorite sports of the students at her school. She

asks 25 students who were at the basketball team tryouts.


b. Jake wants to know how many students are interested in buying a yearbook

this year. He used a random number generator to randomly select 25 students
from each grade level.

2. The graph represents a sample of football players’ heights. If there were 100Number of Players
players, how many players could be expected to be 70 inches tall?

 
Football Players’ Heights (in.)

3

2

1

0
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Heights (in.)

Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Know

74

NAME _________________________________________________________________________________ CHAPTER 6 PRETEST

Check What You Know

Statistics
3. A sample of people were asked how far they drive to work. What percentage

of people drive 6 miles to work? Round your answer to the nearest tenth of a
percent.

1 2 3 4 5 6 7 8 9 10
Distance to Work from Home

4. What can you infer from this data collected about the number of apps on a
sample of smart phones?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

5. A factory produces 92,000 tubes of toothpaste each day. The quality manager
claims that fewer than 750 defective tubes are produced each day. In a random
sample of 420 tubes of toothpaste, 3 are defective. Is the quality manager’s claim
correct? Explain your answer.

Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Know

75

NAME _________________________________________________________________________________

Lesson 6.1 Sampling and Drawing Inferences

When a population has a large number of data points, a sample can be taken to
help summarize information and make inferences about the entire population. A
random sample has individuals who are chosen by chance, and each member of
the population has an equal chance of being included. In a biased sample, some
members of the population are less likely to be chosen. Samples that are random
are better predictors of trends for the bigger population.

Rosewood Middle School has 714 students. Susan surveys a random sample of
34 students and finds that 9 of them play a sport outside of school. How many
students at the school are likely to play a sport outside of school?

9  71s4; 34s  (9) (714)  6426
34

34s  643246; s  189
34

189 students are likely to play sports outside of the school.

Answer the questions. Show your work.

A high-tech company makes 3,500 widgets a day. The quality department chooses
a random sample of 50 widgets and finds that 3 are defective. How many high tech
widgets per day are likely to be defective?

Grace hears that the average gas price has risen to $2.89 during the gas shortage.
She checks gas prices at stations near her school, and finds that the average is
$3.20. Why are the averages different?

Spectrum Critical Thinking for Math Lesson 6.1
Grade 7 Sampling and Drawing Inferences

76

NAME _________________________________________________________________________________

Lesson 6.2 Comparing Similar Data Sets

What can you infer from the two histograms?

In class 1, no students were shorter than 34 inches or taller than 46 inches.
In class 2, the range of heights is 25 inches, but the range in class 1 is just 12
inches. The median for both classes is 42 inches. 50% of the students in class 1
are between 42 and 46 inches.

12- 8- 6
10 5

10-

8- 6- 4 4
6
Frequency Frequency 4-
6- 4 2- 1
4-

2-

0- 0 0 0-
30 35 40 45 50 55
30 34 38 42 46 50

Heights of Class #1 Heights of Class #2

5- 8- 7
4- 4 4
6-
33
Frequency 3- Frequency 4- 11
2
2-
2- 1
1- 0- 0
0- 0 0
150 200 250 300 350 400 450
150 200 250 300 350 400 450

Weights of Chickens: Soybean Diet (dkg) Weights of Chickens: Sunflower Diet (dkg)

In which range will the median occur for each diet?

What percentage of the chickens are between 300 dkg and 350 dkg for each type
of feed? Round your answers to the nearest percent.

Spectrum Critical Thinking for Math Lesson 6.2
Grade 7 Comparing Similar Data Sets

77

NAME _________________________________________________________________________________

Lesson 6.2 Comparing Similar Data Sets

Class 1: Number of People in 8, 2, 5, 5, 3, 1, 6, 2, 4, 4
Household 3, 5, 4, 4, 5, 4, 3, 2, 4, 4
Class 2: Number of People in
Household

Find the mean, median, and mode of each set of data. How do the data sets
compare?

Class 1 Class 2

Mean: 4; Median: 4 Mean: 3.8; Median: 4

Mode: 2, 4, 5; Range: 7 Mode: 4; Range: 3

This data is spread out fairly evenly This data is more compact and closer to

between 1 and 8. There are a variety the center. There is less variety in sizes.

of household sizes in this class. Four is the most common size.

Find the mean, median, and mode of each set of data. How do the data sets
compare?

Class 1: Teacher Donations to 2 3 7 8 10 11 12 14 15 20 17 20
Charity Fund 14 12 11 12 20 20 20 20
Class 2: Student Donations to 1 2 5 7 8 9 1 11 14 15 17 19 19
Charity Fund 17 11 8 2 2 11

Spectrum Critical Thinking for Math Lesson 6.2
Grade 7 Comparing Similar Data Sets

78

NAME _________________________________________________________________________________

Lesson 6.2 Comparing Similar Data Sets

Box-and-whisker plots can help you interpret the distribution of data. Each section
of a box and whisker plot contains 25% of the data points.

Active time data is collected from a group of high school students and a group of

elementary students.

High School Students

Elementary Students

50 52 54 56 58 60 62 64 66 68 70 72 74 76 78

The double box and whisker plot shows that the high school students are overall
less active with a median of 59 minutes a day. The middle 50% of students
sampled are active between 54 and 63 minutes a day. The elementary students
are more active. The median is 64 minutes a day. The middle 50% exercise
between 56 and 72 minutes.

This double box-and-whisker plot displays the test scores of students who studied
alone and the scores of students who studied with a study group. Use it to compare
the data sets.

w/o study group
with study group

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Spectrum Critical Thinking for Math Lesson 6.2
Grade 7 Comparing Similar Data Sets

79

NAME _________________________________________________________________________________

Lesson 6.3 Data in the Real World

Tameka is planning a party for her brother. She invites 180 of the people in her
brother’s class. When she sent out invitations, she listed the wrong phone number for
RSVP, so she will not be getting any responses. She is trying to figure out how many
people are planning to come to the party.

a. Tameka decides to ask the 20 students who live in her neighborhood. 12 of
them say that they will be able to come to the party. What is the population in
this event? Is this a random sample? Could this sample be biased? Are these
results too low or too high? Explain.

b. T ameka decides to look at the graduation program and call every 10th person
on the list of graduates to see if they plan to come. She calls 18 people and 8
of them say that they will be able to come. How many people can she expect to
come to the party?

c. I s this a random sample? Could this sample be biased? Compare these results
to the results from the first sample.

Spectrum Critical Thinking for Math Lesson 6.3
Grade 7 Data in the Real World

80

NAME _________________________________________________________________________________

Check What You Learned

Statistics

1. Will wants to survey a sample of students at his school to find out how many
play musical instruments. He surveys students coming out of band class. Is Will’s
sample biased or random? Why?

2. The graph shows a sample of heights of sixth graders and eighth graders.
Compare the data. What can you infer?

66 CHAPTER 6 POSTTEST

55

44

33

22

11

0 0
50-59 60-69 50-59 60-69 70-79

Heights of 6th Graders Heights of 8th Graders

3. A factory produces 74,000 sets of headphones each day. The quality manager
claims that fewer than 600 defective tubes are produced each day. In a random
sample of 310 sets of headphones, 3 are defective. Is the quality manager’s claim
correct? Explain your answer.

Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Learned

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NAME _________________________________________________________________________________

Check What You Learned

Statistics

4. The graph shows a sample of heights of plants that were grown with no fertilizer
and plants that were grown with fertilizer. What can you infer from the box and

whisker plots?

(no fertilizer) (w/fertilizer)


CHAPTER 6 POSTTEST 1 2 3 4 5 6 7 8 9 10 11 12

5. The graph shows a sample group of girls and a sample group of boys, and the
number of books they read during the school year. If there are 200 boys and 200
girls at the school, how many girls and boys read 10 books?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
# of books read during school year (boys)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
# of books read during school year (girls)

Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Learned

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NAME _________________________________________________________________________________ CHAPTER 7 PRETEST

Check What You Know

Probability

1. Of the 50 U.S. states, 13 were the original colonies. If you select 1 state
randomly, how likely is it to be one of the original colonies?

2. David takes 20 shots and scores 6 goals at soccer practice. What is the
experimental probability that he will miss his next shot?

3. Evan hits 6 out of 14 pitches during practice. What does an experimental

probability of 4 describe?
7

4. At Luvski Ski Resort, there are two chair lifts to the top of the mountain. There are
five ski trails to the bottom of the mountain. What is the probability of riding on
Chair 1 and skiing on Trail 3?

Spectrum Critical Thinking for Math Chapter 7
Grade 7 Check What You Know

83

CHAPTER 7 PRETEST NAME _________________________________________________________________________________

Check What You Know

Probability
5. The school picnic is a two-day weekend event. It has been scheduled for May.

The area routinely gets 16 rainy days in May. What is the probability that the
weekend will be dry? Round your answer to the nearest percent.

6. In basketball, Alan makes 1 out of every 4 free throws he tries. What is the
probability that Alan will make his next 3 free throws? Round your answer to the
nearest tenth of a percent.

7. Gregg has 12 cards. Half are black, and half are red. He picks 2 cards out of the
deck. What is the probability that both cards are red?

8. Lucy places 5 cards face down on the table and mixes them up. The cards are
numbered 1 through 6. What is the likelihood that her friend Harry will draw an
even-numbered card?

Spectrum Critical Thinking for Math Chapter 7
Grade 7 Check What You Know

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NAME _________________________________________________________________________________

Lesson 7.1 Understanding Probability

The probability of an event measures the likelihood that the event will occur.

Impossible Unlikely Equally Likely/UnlikeLliykely Certain

0 1 1 3 1
4 2 4

0% 0.25 0.50 0.75 100%

25% 50% 75%

The complement of an event is the set of all outcomes not included in the event.

Answer the questions.
What is the sum of the probabilities of an event and its complement?

Students in Ms. Baldwin’s class are picking numbers out of a hat. The hat has 8
pieces of paper. Four pieces of the paper are black, and the other pieces are white.
Where does the probability of picking a white piece of paper out of the hat fall on
the number line above?

Where would the probability of picking a white piece of paper fall on the number
line if there were 6 pieces of white paper and 2 pieces of black paper in the hat?

Spectrum Critical Thinking for Math Lesson 7.1
Grade 7 Understanding Probability

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NAME _________________________________________________________________________________

Lesson 7.2 Frequency Tables

The experimental probability of an event is found by comparing the number of
times the event occurs to the total number of trials. A frequency table is used to
keep track of the trials.

Marvin has a bag of marbles. He removes a marble, records the color, and then
puts the marble back in the bag. The frequency table shows how many times he
picked each color.

Color Frequency Find the experimental probability
Purple 12 for each color.
Pink 10
Orange 15 P (purple)  12  24%
White 13 50
P (pink)  10 
P (orange) 50 20%
15  30%
 50

P (white)  13  26%
50

Students at Prince Middle School were asked about their weekly allowance. Use the
frequency table to calculate the experimental probability for each amount. Show your
work.

Allowance # of students
$15.00 9
$20.00 11
$25.00 12
$30.00 8

Spectrum Critical Thinking for Math Lesson 7.2
Grade 7 Frequency Tables

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NAME _________________________________________________________________________________

Lesson 7.2 Frequency Tables

Answer the questions.

A spinner with 4 equal sections was spun 78 times. Use the frequency table to
calculate the experimental probability of spinning each number. Show your work.
Round your answer to the nearest tenth of a percent.

Number on Spinner Frequency
1 21
2 22
3 18
4 17

What is the probability of not spinning a 3?

A coin was flipped 60 times. The experimental probability of each outcome is shown
in the table below.

Coin Lands On Frequency P (heads)  27  45%
Heads 27 60
Tails 33
P (tails)  33  55%
60

Is this the probability that you expected? Compare the results to your expectations.

Spectrum Critical Thinking for Math Lesson 7.2
Grade 7 Frequency Tables

87

NAME _________________________________________________________________________________

Lesson 7.3 Calculating Probability

Theoretical probability is the probability of an event occurring based on all the
possible outcomes. Theoretical probability can be calculated this way:

P (event)  number of ways the event can occur
total number of possible outcomes

A spinner has 3 equally sized sections labeled A, B, and C. What is the
probability that your spinner landed on section A?

There are 3 possible outcomes, with one of them being A. P (A)  1
3

A bag of marbles contains 5 green marbles, 8 red marbles, and 9 yellow marbles.
Ella chooses one marble at random from the bag. What is the probability that she
picks a green marble? Round your answer to the nearest tenth of a percent.

What is the probability that she does not pick a red marble? Round your answer to
the nearest tenth of a percent.

Spectrum Critical Thinking for Math Lesson 7.3
Grade 7 Calculating Probability

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NAME _________________________________________________________________________________

Lesson 7.4 Probability Models

When all outcomes of an experiment are equally likely, the event has uniform
probability. This probability can be used to predict outcomes.

Vick rolls a number cube. What is the probability that he rolls a prime number? If
he rolls the number cube 30 times, how many times is he expected to roll a prime
number?

A number cube has 3 prime numbers (2, 3, 5). There are 6 possible outcomes.

P (prime number)  3  50%
6

0.5  30  15

There is a 50% chance of rolling a prime number. If Vick rolls the number cube

30 times, it is expected that he will roll a prime number 15 times

Answer the questions. Show your work.
A spinner has 20 equal sections, numbered 1 through 20.
a. What is the probability that the spinner will land on a multiple of 3?

b. I f the spinner is spun 42 times, how many times can it be expected to spin a
multiple of 3?

c. What is the probability that it will not spin a multiple of 4?

Spectrum Critical Thinking for Math Lesson 7.4
Grade 7 Probability Models

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NAME _________________________________________________________________________________

Lesson 7.5 Other Probability Models

When a probability event has unequal odds, the outcomes are not equally likely to
occur.

A spinner has 4 equal sections. 2 of the sections are yellow, one of the sections

is purple, and the other section is green. What is the probability that the spinner

lands on yellow?

P (yellow)  2  50%
4

What is the probability of not spinning purple?

P (not purple)  1 1  3  75%
4 4

Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.

A grocery store randomly selects an item to be on sale each day

Item # of Days on Sale

Ice Cream 4

Oranges 5

Chicken 3

Chips 5

Eggs 4



a. What is the probability that the item on sale will be ice cream or chips?

b. What is the probability that oranges or chicken will not be on sale?

Spectrum Critical Thinking for Math Lesson 7.5
Grade 7 Other Probability Models

90

NAME _________________________________________________________________________________

Lesson 7.5 Theoretical vs. Experimental Probability

Theoretical probability is what is expected to happen based on likely outcomes.
Experimental probability is what actually happens.

Suppose you toss a coin 25 times, and it lands tails up 11 times. Compare the
experimental probability and the theoretical probability.

Theoretical probability: 1  50%
2
11
Experimental probability: 25  44%

The experimental probability is less than the theoretical probability. It is impossible
to meet the experimental probability because there are an odd number of coin
tosses.

Thomas spins a spinner 40 times. The results are shown in the table. Based on the
results of the experiment, use your best guess to draw the spinner.

Number Frequency
1 9
2 11
3 12
4 8

Spectrum Critical Thinking for Math Lesson 7.6
Grade 7 Theoretical vs. Experimental Probability

91

NAME _________________________________________________________________________________

Lesson 7.7 Understanding Compound Events

When two or more things are happening at one time in an experiment, it is a
compound event. The probability of each event is multiplied.

What is the probability of rolling a 2 and then a 6 when rolling a number cube

twice? 1
6
P (2) 

P (6)  1
6
P (2, then 6)  1  1  1
6 6 36

Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.

A standard spinner is arranged so that the numbers 1 to 15 share equal space.

a. What is the probability of getting a 9 on two consecutive spins?

b. What is the probability of not getting a 9 on two consecutive spins?

What is the probability of rolling a 2 on a standard number cube and then
getting heads on a coin toss?

What is the probability of not rolling a 6 on a number cube and then getting
heads on a coin toss?

Spectrum Critical Thinking for Math Lesson 7.7
Grade 7 Understanding Compound Events

92

NAME _________________________________________________________________________________

Lesson 7.7 Understanding Compound Events

The Fundamental Counting Principle says that when there are m ways to do one
thing, and n ways to do another, then the product of m and n is the possible
number of outcomes for both events. A tree diagram can help you visualize this.

An ice cream shop offers vanilla, strawberry, and chocolate ice cream. A
customer can choose a regular cone, a sugar cone, or a cup. What is the
probability of getting strawberry ice cream on a sugar cone?

There are 3 flavors and 3 serving options. Reg

3 3 3  9, so there are nine possible outcomes. V Sugar
Cup
There is one possible combination of Reg
strawberry ice cream and sugar cone.
S Sugar
P (strawberry sugar cone)  1  11% Cup
9
Reg

C Sugar
Cup

Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.

A salad bar has croutons, raisins, sunflower seeds, and cranberries available
as toppings. Teresa wants 2 different toppings on her salad. How many possible
2-topping combinations can Teresa choose? What is the probability of having
croutons and sunflower seeds on her salad?

Spectrum Critical Thinking for Math Lesson 7.7
Grade 7 Understanding Compound Events

93

NAME _________________________________________________________________________________

Lesson 7.8 Probability in the Real World

Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.

A retail store is having a contest. The randomly selected prize will be a can opener,
a gift card, or a set of towels. The store cashier will spin a spinner with the numbers
5–8 to see whether every 5th, 6th, 7th, or 8th customer will win a prize.

a. Create a tree diagram to show all the possible outcomes in this situation.

b. What is the probability that every 5th person will win a can opener or a gift
card?

c. What is the probability that every 6th or 7th person will win a set of towels?

d. What is the probability that a customer will not win a can opener?

Spectrum Critical Thinking for Math Lesson 7.8
Grade 7 Probability in the Real World

94

NAME _________________________________________________________________________________

Check What You Learned

Probability
Answer the questions. Show your work.

1. An auto company conducted a survey with a random sample of 500 people to
find out which type of vehicle they preferred to drive. The results are shown below.

Favorite Vehicle Number of People

Compact 75 CHAPTER 71 POSTTEST

Sedan 45

SUV 95

Pickup 90

Station Wagon 95

Minivan 100

a. What is the probability that a randomly selected survey participant prefers to
drive an SUV? Write it as a decimal.

b. I f 1,500 people were surveyed, how many would you expect to prefer to drive
an SUV? Explain your answer.

2. Mr. Rose randomly selects names to see who will give the first book report. There
are 10 boys and 14 girls in his class. What is the probability that he will select a
girl’s name?

Spectrum Critical Thinking for Math Chapter 7
Grade 7 Check What You Learned

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CHAPTER 7 POSTTEST NAME _________________________________________________________________________________

Check What You Learned

Probability
Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.
3. Of the original 56 signers of the Declaration of Independence, 4 represented

North Carolina. If you selected 1 signer randomly, how likely is it that he
represented North Carolina?

4. Every seventh-grade student is eating in the cafeteria. Juwarne is a seventh-grade
student. How likely is it that she is in the cafeteria?

5. Kobe makes 15 of 20 free throws at basketball practice. What is the experimental
probability that he will miss his next free throw?

6. At the barbershop, there are 2 chairs for customers to wait in. There is a rack with
5 magazines for customers to read while they wait. How many possible choices of
chairs and magazines do the barbershop customers have?

Spectrum Critical Thinking for Math Chapter 7
Grade 7 Check What You Learned

96

NAME _________________________________________________________________________________

Final Test Chapters 1–7

Answer the questions. Show your work.

1. These temperature changes in a vat of liquid were noted by a scientist performing
a chemical experiment. What was the net temperature change from the first
Monday to the second Monday?

Monday 4.6 °C
Tuesday 10.2 °C
Wednesday 20.3 °C
Thursday 23.5 °C
Friday
Saturday 4.2 °C
Monday 14.4 °C
26.9 °C

2. Serena took care of Jason’s large fish tank while he was on vacation. The tank lost
water through evaporation, and Serena added more water as shown in the table.
In total, how much water will be gained or lost by the time Jason returns from
vacation?

Day Water Lost Water Added
(in quarts) (in quarts)
Mon.
Tue. 3 5
Wed. 4 8

1 7 CHAPTERS 1–7 FINAL TEST
2 8

5 1
8 2

Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test

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NAME _________________________________________________________________________________

Final Test Chapters 1–7

Answer the questions. Show your work.

3. The chart shows the high and low temperature in Anchorage for a week.

Temperature in Anchorage (°F)

Sun Mon Tues Wed Thurs Fri Sat

High 3° 5° 6° 7° 2° 15° 1°
Low
8° 12° 21° 17° 15° 25° 18°

a. Find the average of the high temperatures. Round your answer to the nearest
tenth of a degree.

b. Find the average of the low temperatures. Round your answer to the nearest
tenth of a degree.

4. The terms 8x, 5z, 15y, z, 2x and another term are added to form an expression.
When simplified, this expression equals 2 (3z  5x). Identify the missing term
and write the expression.

CHAPTERS 1–7 FINAL TEST Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test

98