NAME _________________________________________________________________________________
Final Test Chapters 1–7
Answer the questions. Show your work.
5. The circumference of a circular garden is 42 meters. A gardener is digging a
straight line along the diameter of the garden at a rate of 10 meters per hour.
How many hours will it take the gardener to dig across the garden? Use 3.14 for
p. Round your answer to the nearest hundredth.
6. Thomas spins a spinner 25 times. The results are shown in the table. Based on the
results of the experiment and your best guess, how does the size of the section
containing #5 compare to the size of the section containing #6?
Number Frequency
1 2
2 4
3 1
4 8
5 2
6 8
7. 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 1–7 CHAPTERS 1–7 FINAL TEST
Grade 7 Final Test
99
NAME _________________________________________________________________________________
Final Test Chapters 1–7
Answer the questions. Show your work.
8. A college football stadium holds 25,000 fans. In a random sample of 30 fans, 26
were wearing the colors of the home team. Predict the number of fans who are
wearing the colors of the home team.
9. If it takes Joe 15 hours to make 3 cornhole boards, how long will it take him to
make 11 cornhole boards?
10. J ack bought 4 turkey sandwiches and 2 bags of apple slices for $22.60. If the
apple slices cost $0.75 per bag, how much did each sandwich cost?
CHAPTERS 1–7 FINAL TEST Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test
100
NAME _________________________________________________________________________________
Final Test Chapters 1–7
Answer the questions. Show your work.
11. Identify the mistake that was made in simplifying the expression. Then, correctly
simplify the expression.
5 (a 3) (6a 12) 7a
5a 1 2 6a 12 7a
(5a 6a 7a) (2 12)
4a 10
12. On a road map, the distance between two cities is 12.6 centimeters. What is the
actual distance if the scale on the map is 2 cm:50 mi. How long would it take a
driver traveling 70 miles per hour to go from one city to the next city?
13. Several puppies from 2 different breeds were weighed. The puppies’ weights in CHAPTERS 1–7 FINAL TEST
pounds are shown in the table. What can you infer from the data?
Breed A
Breed B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test
101
Answer Key
Page 5 Page 6
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 1 PRETEST CHAPTER 1 PRETEST Check What You Know
Adding and Subtracting Rational Numbers Adding and Subtracting Rational Numbers
Compare the values using , , or . Solve the problems. Show your work.
<1. <2. u214.375u ______ u13.75u
227 1 _____ 27 1 11. Lucy has scored 40 points in her trivia game so far. She answers 2 more
3 4 questions correctly and scores 20 points. Then, she answers a question
incorrectly and loses 25 points. What is her final score?
Write the additive inverse of each number.
40 + 20 + (–25) = 35
3. 242.6 4__2__._6 4. 13.25 2__1__3_.25
Lucy's final score is 35.
Solve the problems. 6. 9 3 2 5 1 6 1
4 3 6 12. A hiker takes a trail that increases his altitude by 26.9 feet. He switches to
5. 13.45 0.025 2 5.7 another trail that will decrease his altitude by 35.6 feet. What is his overall
9 9 + 6 2 change in altitude?
13.45 12 12
+ 0.025 26.9 + (–35.6) = –8.7
– 5.7 – 5 4 10 7
12 12 The hiker's altitude decreased by 8.7 feet.
7.775
13. The coldest record temperature in Belton is 29 degrees. The highest
temperature on record is 102 degrees. What is the difference between the two
Use a number line to complete the following problem–s.21 temperatures? 102 – (–9) = 111
)
7. 2 3 (2 1
4 2
– 1 41 25
– 1 1 – 30 +3 5 The difference is 111 degrees.
4 4 5
8. 21.75 3
1.25 25 0 14. Shane lost 5 1 pounds. Shonda lost 3 1 pounds. How much more weight did
1.25 2 4
–1.75 Shane lose? –5 2 – (–3 1 ) =
4 4 =
Solve the problems. Show your work. Shane lost 1 2
4 4
9. 24.35 21.48 10. 1 453215(203 1 ) 2 more pounds. –5 + 3 1 –2 1
2 4 4
–4.35 – 1.48 –5.83 1180 + 13 3
4 10 5 11 0
Spectrum Critical Thinking for Math
Grade 7 Chapter Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Know
Check What You Know
6
5
Page 7 Page 8
Lesson 1.1 NAME ________________________________________________________________________________ Lesson 1.2 NAME ________________________________________________________________________________
Absolute Value and Rational Numbers Additive Inverse
Absolute value is the distance between a number and zero on a number line. Opposite numbers have the same absolute value . Two numbers that can be added
Numbers that are opposites will have the same absolute value. together to equal zero are called additive inverses .
2 2 5 2 2 5 2 Emelia’s mother gives her $4 to go to the store . At the store, Emelia spends $4 on
3 3 3 3 a bag of oranges . How much money does Emelia have left?
21 2 2 2 1 0 1 2 1 Spent $4
3 3 3 3 Given $4
Answer the questions. 25 0 1 2 3 4 5
If u X uu Y u and both X and Y are negative numbers, describe the location of point X Combine the amount of money that Emelia was given ($4) with the amount she
spent (2$4) to find how much she has left . 4 (24) = 0 . Emelia has $0 left .
in relation to the location of point Y on a number line.
Point X has to be located to the right of point Y on
a number line. Because the absolute value of X is Answer the questions using the additive inverse . Show your work .
less than the absolute value of Y, the distance of X Hermine is making a skirt . She bought 5 1 yards of fabric . She used 3 1 yards to
4 2
from 0 is less than the distance of Y from 0. make the skirt . How many yards can she give away if she wants to end up with no
Compare u28 1 u and u 7 2 u. Explain your thinking. fabric? 1 3
3 3 1 4 4
3 2 + ? = 5 She needs to give away 1 yards .
|–8 1 | > |7 2 | because |–8 1 | is further from
3 3 3 5 1 – 3 1 = 21 – 7 = 21 – 14 = 7 =1 3
2 4 2 4 2 4 4 4 4
zero on the number line than |7 3 |. Juan earned $28 mowing a lawn and $15 walking a dog . The next day, he bought
a t-shirt for $20 and a jacket for $22 . Was the sum of what he earned the additive
Write a statement comparing u E u and u F u. Explain your thinking. inverse of what he spent?
|E|<|F| No . Juan earned 28 + 15 for a total of + $43 .
E0 F He spent 20 + 22, for a total of –$42 . He did not
spend the additive inverse because 43 + (–42) ≠ 0 .
Point E is 2 units from 0 and point F is 3 units
from 0.
Spectrum Critical Thinking for Math Lesson 1.1 Spectrum Critical Thinking for Math Lesson 1 .2
Grade 7 Absolute Value & Rational Numbers Grade 7 Additive Inverse
7 8
102
Answer Key
Page 9 Page 10
Lesson 1.3 NAME ________________________________________________________________________________ Lesson 1.4 NAME ________________________________________________________________________________
Adding & Subtracting Rational Numbers Adding Positive and Negative Numbers
When you add or subtract fractions, the denominators must be the same . When The sum of 2 positive numbers is a positive 2 2 1 23 5 25 21355
you add or subtract decimals, the place values must be aligned . 23 13
number farther to the right of the
2345
Corey bought 4 3 ounces of cashews for $2 .30 and 3 1 ounces of pistachios for first addend on the number line.
5 10 25 24 23 22 21 0 1
$1 .24 . What is the total amount of nuts that he bought? What was the total The sum of 2 negative numbers is a negative
number farther to the left of the first addend.
amount of money he spent?
4 3 3110 4 6 3 1 2 .30 The sum of a positive and a negative 221351
5 10 10 1 .24 number will be positive if the positive 13
number has a greater absolute value.
7170 ounces of nuts $ 3 .54 total spent 24 23 22 21 0 1 2 3 4
Solve the problem . Show your work . The sum of a positive and a negative 2 3 1 2 5 21
number will be negative if the negative 12
Royal’s new smartphone has 32 gigabytes (GB) of memory . She has the following number has a greater absolute value.
25 24 23 22 21 0 1 2 3
apps and files on her phone . Does she have enough space for the operating system
upgrade that uses 9 .1GB? 32 .00 25 .327
Royal's Phone –2 .64 –2 .1 Write always, sometimes, or never below each statement. Give an example to
show your answer.
Program Size (GB) 29 .360 23 .227
Operating System 2 .64 –1 .203 –5 .12 When adding two numbers, the sum is greater than each of the addends.
Games 1 .203 28 .157 18 .107
0 .08 –0 .08 –4 .592 Sometimes. Ex. 5 + (–9) = –4 ; –4 is not greater than
Calculator App 2 .75 5. (examples will vary)
Downloaded Files
When adding two negative numbers the sum is greater than each of the addends.
Other Apps 2 .1 28 .077 13 .515
Never. Ex. –5 + (–9) = –14; the sum of 2 negative
Videos 5 .12 –2 .75 numbers is more negative. (examples will vary)
Photos 4 .592 25 .327
When adding two numbers with opposite signs, the absolute value determines the
32 .000 Royal will have enough sign of the sum.
– 13 .515 space to upgrade the
operating system . Always. Ex. –5 + 9 = 4 ; the sum is positive because
18 .485 the absolute value of 9 is greater. (examples will vary)
Lesson 1 .3
Spectrum Critical Thinking for Math Adding & Subtracting Rational Numbers Spectrum Critical Thinking for Math Lesson 1.4
Grade 7 Grade 7 Adding Positive and Negative Numbers
9
10
Page 11 Page 12
Lesson 1.4 NAME ________________________________________________________________________________ Lesson 1.5 NAME ________________________________________________________________________________
Adding Positive and Negative Numbers Subtracting Positive and Negative Numbers
Distance and direction also help to determine the sum of positive and negative Subtracting a number is the same as adding the additive inverse of the second
fractions and decimals on a number line. number and applying the rules of integer addition.
Answer the questions. 4 2 7 5 4 1 (27) 5 23
Is an estimate of 3 for the sum of 12 3 and 15 2 reasonable? Explain your answer. 4 2 (27) 5 4 1 7 5 11
4 3
Write always, sometimes, or never below each statement. Give an example to
This is not accurate. |12 3 |<|–15 2 |, so the sign of show your answer.
4 3
the sum is negative. A better estimate would be –3. When subtracting two numbers, the difference is always less than the two numbers.
Is an estimate of 4 accurate for the sum of 24.24 and 27.8 accurate? Explain Sometimes. Ex. –4 – (–1) = –3 ; –3 is not less than
your answer. –4. (examples may vary)
This is accurate because |24.24|<|–27.8|, so the sign Subtracting a negative number is the same as adding the absolute value of that
number.
of the sum is negative. +1 1
2 Always. Ex. 6 – (–7) = 6 + 7 = 13; this is the
Use a number line to answer the questions. same as adding the absolute value of –7.
(example may vary)
53 1 1
2 (1 2 ) 5 Answer the question. Show your work.
5 0 3 1 5 Jamal wrote the following on his paper: 10 2 (210) 5 10 210 5 0. What was his
2 mistake? What is the correct answer?
– 0.5 Jamal did not add the additive inverse. Jamal
should have rewritten the problem as
–32.5 1 (0.5 ) 5 0 5 10 + 10 = 20.
5 –3 –2.5
–4 3
4
–5 1 3
4 1 (4 4 ) 5
5 – 1 0 5
4
Spectrum Critical Thinking for Math Lesson 1.4 Spectrum Critical Thinking for Math Lesson 1.5
Grade 7 Adding Positive and Negative Numbers Grade 7 Subtracting Positive and Negative Numbers
11 12
103
Answer Key
Page 13 Page 14
Lesson 1.5 NAME ________________________________________________________________________________ Lesson 1.6 NAME ________________________________________________________________________________
Subtracting Positive and Negative Numbers Adding with Mathematical Properties
The rules that apply to integers also apply when subtracting positive and negative Mathematical properties can be used to add rational numbers quickly .
fractions and decimals. 21 11 13 (25) (27)
4 2 (27.5) 5 4 1 7.5 5 11.5 Commutative Property: (a b b a) 21 (25) (27) 13 11
4 2 (27 1 ) 5 4 1 7 1 5 11 1 Associative Property: {21 (25) (27) 13} 11
2 2 2 (a b) c a (b c) (213 13) 11
Answer the questions.
Is an estimate of 8 as the difference between 4.7 and 23.3 reasonable? Explain. Identity Property of Addition: a 0 a 0 11 11
3.3 4.7 Simplify the expressions using mathematical properties .
25 24 23 22 21 0 1 2 3 4 5 23 .25 4 .2 3 .2 (22 .1) 0 .05
Yes, this is a reasonable estimate. There are exactly 8 units (–3 .25 + 3 .2 + 0 .05) + {4 .2 + (–2 .1)}
0 + (4 .2 – 2 .1)
between these two numbers on the number line. 2 .1
On a number line, what is the difference between 261.5 and 223.4?
–61.5 +23.4 –38.1
270 260 250 240 230 220 210 0 10 2 1 (23 1 ) 1 2 2 2
2 3 2 3 3
1 1 1 2 2
|–61.5| – |–23.4| = –38.1 (2 2 + 2 )+(–3 3 + 3 +2 3 )
Evaluate: 23 3 2 63 1 5 187 – 505 – 318 = –39 6 3+(–2 2 +2 2 )
8 8 8 8 8 8 3 3
= –39 3 3+0
4
4 18 7 (29) 3 (29) 3
Evaluate: 217.56 2 13.43 5 (4 + 7 + 3) + {(18 + (–9) + (–9)}
–17.56 + (–13.43) = – 30.99 (4 + 10) + {18 + (–18)}
14 + 0
Spectrum Critical Thinking for Math Lesson 1.5 Spectrum Critical Thinking for Math 14 Lesson 1 .6
Grade 7 Subtracting Positive and Negative Numbers Grade 7 Adding with Mathematical Properties
13 14
Page 15 Page 16
Lesson 1.7 NAME ________________________________________________________________________________ Lesson 1.7 NAME ________________________________________________________________________________
Rational Numbers in the Real World Rational Numbers in the Real World
Solve the problems . Show your work . Solve the problems . Show your work .
Lynn begins with a bank balance of $64 . After three checks are written, the account Monica is monitoring the amount of water in a container in her backyard . During
now has a balance of 2$13 . What was the total amount of the checks?
the rainy season, it rains each day . Then, the sun comes out and evaporates some
of the water in the container . The table below tracks the amount of rain added and
64 – (–13) = 64 + 13 = 77 evaporated each week . What is the final height of the water1? 110 8 1190
Height (cm) 10
+ =
The total amount of the checks was $77 . Beginning amount 1 1 1190 + (– 130) = 1160 = 1 3
10 5
Aaron picks peaches from his family’s orchard and sells them at a farmers market . Rain 4 1160 + 5 = 11110 = 2110
Each morning, he picks more peaches and adds them to the unsold peaches from Evaporation 5 10
the previous day . Last week he picked and sold the amounts shown in the table . How
many pounds did he have at the end of the week? 3
10
Rain 1 2110 + (– 120) = 1190
Evaporation 2 1190 + 2140 = 31130 = 4130
4 .1 24 .3 Rain 1
+ 19 .8 – 12 .9 5
Beginning amount Weight (pounds) 23 .9 11 .4 2 2
Picked 4 .1 – 21 .6 + 15 .3 5
Sold 19 .8
Picked 21 .6 2 .3 26 .7 Evaporation 7 4130 – 7 = 3160 = 3 3
Sold 17 .7 + 17 .7 – 23 .9 10 10 5
Picked 18 .1
Sold 22 .4 20 .0 2 .8 The final height of the water is 3 3 cm .
Picked 12 .9 – 18 .1 5
Sold 15 .3
23 .9 1 .9 Mrs . McCoy’s original loan balance was $3,467 . She made her regular payment of
+ 22 .4 $291 as well as an extra payment of $79 . She was charged $23 in interest . Write
and simplify an expression to find Mrs . McCoy’s new balance .
There are –3467 + 291 + 79 – 23 =
2 .8 pounds remaining .
–3176 + 79 – 23 Mrs . McCoy's new
–3097 – 23 balance is $3,120 .
–3120
Spectrum Critical Thinking for Math Lesson 1 .7 Spectrum Critical Thinking for Math Lesson 1 .7
Grade 7 Rational Numbers in the Real World Grade 7 Rational Numbers in the Real World
15 16
104
Answer Key
Page 17 Page 18
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Check What You Learned
Adding and Subtracting Rational Numbers Adding and Subtracting Rational Numbers
Solve the problems. Show your work.
The table shows stock market gains and losses that were recorded over a 5-day
period. Use the table to answer the questions. Show your work.
Day 1 2 3 4 5 5. The goal of Tarik’s card game is to have a score of 0. Find two more cards he
Stock X 1$6.75 2$12.50 1$21 2$9.20 1$4.35 could pick to win if he is holding cards with the following values: 27, 3, 4, 29.
Day 1 2 3 4 5 CHAPTER 1 POSTTEST CHAPTER 1 POSTTEST sum of current cards: –7 + 3 + 4 + (–9) = –9
Stock Y 1$15 2$22.60 1$18.90 2$14.25 1$7.25 The total of the two new cards needs to be 9.
Answer can vary.
1. On which day did Stock X have the biggest change (gain or loss)? Some possible answers: 5,4; –1,10; 3,6
1 3
Day 3, because +21 has the largest absolute 6. Jacob finds a piece of metal that is 2 5 inches thick. He files off 10 inch. He adds
value.
a protective covering to the metal that is 2 inch thick. Then, he polishes the metal
2. On which day did Stock Y have the biggest change? 10
1
Day 2, because –22.6 has the largest absolute and removes an additional 10 inch. What is the final thickness of the metal?
value.
2 1 – 3 = 2120 – 3 = 1190 The metal is 2
3. How much was lost in total on day 4 for Stock X and Stock Y? 5 10 10
inches thick after
–9.20 1190 + 2 = 2110 being polished.
+ –14.25 The total loss was $23.45. 10
–23.45 2 – = 21 1
10 10
4. For Stock Y, what was the difference between day 3 and day 2? an herb garden in her back yard. This year, she planted 1 of her
7. Raquel planted 2
18.9 18.9
– –22.6 + 22.6 The difference was $41.50. garden with chives, 1 with oregano, and 1 with parsley. What portion of the
4 8
41.5
garden is still unplanted?
1 = 4 ; 1 = 2 1 of the garden is
2 8 4 8 8
4 + 2 + 1 = 7 still available.
8 8 8 8
1 – 7 = 8 – 7 = 1
8 8 8 8
Spectrum Critical Thinking for Math Chapter 1 Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Learned Grade 7 Check What You Learned
17 18
Page 19 Page 20
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 2 PRETEST CHAPTER 2 PRETEST Check What You Know
Multiplying and Dividing Rational Numbers Multiplying and Dividing Rational Numbers
Solve the problems. Show your work. 8 . True or false: 25 (217) 25 25 (25) (217)
1. 6 3 1 6 10 = 60 = 20 True . This is true because of the Commutative
3 1 3 3
4.75
×5 Property .
.25 9 . Write 5 as a decimal . .833
3.50 6
20.00
2. 5 4.75 23.75 6qw5 .000
48 0 .83
3. 10 (22) (23) 5 –20 (–3) = 60 20
18
20
18
4. 22 (211) (25) 5 22 (–5) = –110 2
10 . Katie is digging a hole to plant a tree . She digs 1 foot deep each day for 3
5. 10 1 4 5 1 51 ÷ 51 = 51 10 = 510 = 2 days . Write and evaluate a numeric expression to represent this situation .
5 10
5 10 5 51 255 –1 3 = –3
Katie had dug 3 feet after 3 days .
11 . A tree fell in Kyle’s back yard during a storm . Each day, he cuts a 2 1 foot
1 3
3
19 section off of the tree . If the tree was 23 feet tall, how many days will it take
1025qw19475
6. 194.75 4 10.25 5 him to cut up the entire tree?
– 1025
7. 49 4 (27) 5 –7 23 1 ÷ 2 1 =
9225 3 3
70 7
– 9225 3 ÷ – 3 = It will take 10 days to cut the
entire tree .
0 70 3 =
3 7
210
21 = 10
Spectrum Critical Thinking for Math Chapter 2 Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Know Grade 7 Check What You Know
19 20
105
Answer Key
Page 21 Page 22
Lesson 2.1 NAME ________________________________________________________________________________ Lesson 2.1 NAME ________________________________________________________________________________
Multiplying Rational Numbers Multiplying Rational Numbers
The distributive property can be used to multiply a whole number and a rational Malachi has to pack 84 toiletry bags for the homeless shelter. He has packed 1 of
number . 4
the boxes. How many bags are left for him to pack?
Distributive Property: a (b c) a b a c 1 84 5 1 (80 1 4) 5 1 80 1 1 45
4 4 4 4
2 3 2 2(3 2 ) 2 4 .75 2(4 0 .75) 20 1 1 5 21
3 3 Malachi has already packed 21 bags, so he has 63 more bags to pack.
2 3 2 2 6 4 2 4 2 0 .75 =
3 3
6 1 1 7 1 8 1 .5 Solve the problems. Show your work.
3 3
9 .5 Leanne bought 20 pounds of potting soil for her new plants. The soil costs $3.17 per
pound. How much did Leanne spend on potting soil?
Use the distributive property to find the p(5rod+uct .51
4
45 1 ) 20 3.17 = 20 (3 + 0.17)
5 20 3 + 20 0.17
1
4 5+4 5 60 + 3.4
= 63.4
20 + 4 = 20 4 Leanne spent $63.40.
5 5
10 17 .135 Thuy has a cookie recipe that calls for 4 1 cups of flour. He wants to quadruple the
10 (17 + 0 .135) 3
10 17 + 10 0 .135 recipe. How much flour should he use?
170 + 1 .35 = 171 .35 4 (146++3134) =4 4 + 4 1 Thuy should use
+ 1 3
1
7 10 2 7 (10 + 2 ) = 16 3 17 1 cups.
7 7 7 3
7 10 + 2 1
7 17 3
Rochelle walked around the block to exercise. Each lap is 1 3 miles. How far did she
5
70 + 2 = 72 walk if she walked around the block 4 times? 3 3
5 5
8 5 .125 8 (5 + 0 .125) Rochelle walked 4 (1 + ) = (4 1) + (4 )
8 5 + 8 0 .125
6 2 miles. 4 + 12 =4 + 2 2
5 5 5
40 + 1 = 41 2
6 5
Spectrum Critical Thinking for Math Lesson 2 .1 Spectrum Critical Thinking for Math Lesson 2.1
Grade 7 Multiplying Rational Numbers
Grade 7 Multiplying Rational Numbers
21
22
Page 23 Page 24
Lesson 2.2 NAME ________________________________________________________________________________ Lesson 2.2 NAME ________________________________________________________________________________
Proving the Rules for Multiplying Integers Proving the Rules for Multiplying Integers
Multiplication is repeated addition . A number line can be used to model integer When multiplying more than 2 factors, the same rules apply. Multiply 2 factors at
multiplication . a time.
2 3 can be modeled by moving 2 units to the right 3 times . (22) 3 can be
modeled by moving 2 units to the left 3 times . 28 1 (22) 4 5 28 (22) 4 5
16 4 5 64
22 22 22 2 2 2
Find the product. –6 4
27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 23 2 4 5
2 3 6; –2 (3) = –6 = –24
2 (23) can be modeled by graphing the opposite of 2 units to the right 3 times . 23 (22) (24) 5 6 (–4)
2 2 2 = –24
27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 21 (22) (23) (24) 5 2 (–3) (–4)
23 2 6; 2 (–3) = –6 = – 6 (–4)
(22) (23) can be modeled by graphing the opposite of 2 units to the left
3 times .
27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 = 24 6 (–4) (–5) (–2) =
(22) (23) 6 –24 (–5) (–2) =
22 (23) (24) (25) (22) 5 120 (–2) =
Answer the questions based on the models above . –240
Given a numeric expression in the form a b c d e …, how can you predict if
If the product of 2 integers is positive, what must be true about the sign of each the product will be positive or negative before you begin your calculations? Give an
factor? example.
The sign of the factors must either be both positive The number of negative factors affects the sign of the
or both negative . Examples will vary . product. An even number of negative factors will result in
If the product of 2 integers is negative, what must be true about the sign of each a positive answer. An odd number of negative factors will
factor? result in a negative product.
One of the signs of the factors must be negative Ex. 2 3 (–4) (–5) (–5) (–1) = 6 20 5 = 600
while the other one is positive . Examples will vary . Ex. 2 3 (–4) (–5) (–5) = 6 20 (–5) = –600
Spectrum Critical Thinking for Math Lesson 2 .2 Spectrum Critical Thinking for Math Lesson 2.2
Grade 7 Proving the Rules for Multiplying Integers Grade 7 Proving the Rules for Multiplying Integers
23 24
106
Answer Key
Page 25 Page 26
Lesson 2.3 NAME ________________________________________________________________________________ Lesson 2.4 NAME ________________________________________________________________________________
Dividing Rational Numbers Dividing Integers
To divide mixed numbers, rewrite To divide decimals, multiply the Dividing is the opposite of multiplying .
them as improper fractions and then divisor and dividend by a factor of Rewrite 235 4 7 as 7 __ 235 .
multiply the reciprocal of the divisor. 10 that will make the divisor a whole
number. We know that 25 will finish the equation, because a negative factor times a
positive factor is a negative product .
Oscar needs 1h21asin5ch31 eisncohfetswoinfetwfoinre. Cassie has a 4.35-foot piece of
a project. He The rules of integer multiplication also apply to integer division . The quotient
of two integers with the same sign is positive . The quotient of two integers with
How many projects can he make? wood. She needs to cut it into 0.29- different signs is negative .
5 1 4 1 1 5 foot pieces. How many pieces can she 2144 4 24 26
3 2 2144 4 (224) 6
make? 15
16 4 3 5 16 2 5 29qw435
3 2 3 3 Multiply the divisor and
dividend by 100. 2 29
32 5 3 5 145
9 9
He can make 3 5 projects. She can make 15 pieces. 2145 Answer the questions .
9 0
Solve the problems. Show your work. A number with an absolute value of 42 was divided by a number with an absolute
value of 7 . The quotient is -6 . Write two possible numeric equations .
4) (–5) (–2) = Anderson spent $11.76 on some vegetables. How many pounds did he buy if the
4 (–5) (–2) = –42 ÷ 7 = –6
cost was $1.47 per pound? 8 42 ÷ (–7) = –6
120 (–2) = 147qw1176
11.76 4 1.47 Anderson bought A number with an absolute value of 63 was divided by a number with an absolute
–240 multiply by 100 –1176 8 pounds of value of 9 . The quotient is 7 . Write two possible numeric equations .
1176 4 147 0 vegetables. 63 ÷ 9 = 7
–63 ÷ (–9) = 7
Jen buys a piece of fabric that is 6 7 yards long. She wants to make pillow covers
1 8 Complete the equations .
2
that require 1 yards of fabric each. How many pillow cases can she make?
6 7 ÷ 1 1 7 452 4 ______ 13 31 –65______ 4 (21) 231
8 2 She can make 4 12 ______ 4 5 213
55 ÷ 3 = 55 2 pillow cases. –2238 4______ 19 21______ 4 (23) 27 –254 4 ______ 227
8 2 8 3
110 55 7
24 = 12 = 4 12
Spectrum Critical Thinking for Math Lesson 2.3 Spectrum Critical Thinking for Math Lesson 2 .4
Grade 7 Dividing Rational Numbers Grade 7 Dividing Integers
25 26
Page 27 Page 28
Lesson 2.5 NAME ________________________________________________________________________________ Lesson 2.6 NAME ________________________________________________________________________________
Multiplying and Dividing with Properties Converting Rational Numbers Using Division
Commutative Property: The order in ab5ba Fractions can be converted to decimals using long division. If a decimal in the
which numbers are multiplied does not change answer is repeating, draw a line over the digits that repeat.
the product. (a b) c 5 a (b c)
Associative Property: The grouping of Rewrite 1 as a division problem. .2
factors does not change the product. a (b 1 c) 5 a b 1 a c 5 5qw1.0
a (b 2 c) 5 a b 2 a c
Distributive Property: The multiple of a 1 = 0.2 1.0
sum is the multiple of each addend separately a15a 5 0
added together.
Identity Property: The product of a factor a050 Rewrite 2 as a division problem. .222
and 1 is the factor. 04a50 9 9qw2.000
Properties of Zero: The product of a factor 2 = 0.2 18
and 0 is 0. The quotient of the dividend 0 and 9 20
any divisor is 0. 18
20
Answer the questions. Show your work. .875
Use the given property to evaluate the expression. 7 of Kiara’s homework is done. Write this as a decimal. 8qw7.000
8 –64
Commutative Property: 21.4 5 (210) 5 –1.4 (–10) 5 =
14 5 = 70 60
[(–3) + (– 1 )] ÷ 1 = .875 –56
2 + (– 1) 2 2 40
Distributive Property: 23 1 4 1 5 –40
2 2
(–3) 2 = –6 + (–1) = 7 0
2 Li invited 4 of the 11 people in her math group to the study session. Write the
fraction of students who were not invited as a decimal. .6363
1 11qw7.0000
Associative & Identity Properties: 2 3 (23 4) (22) 5
(– 1 ) (–3) 4 (–2) = 1 4 (–2) – 66
3 = 4 (–2) = –8 0.63 40
– 33
Zero Property: (210 1 10) 4 17 5 0 ÷ 17 = 0 70
– 66
40
– 33
Spectrum Critical Thinking for Math Lesson 2.5 Spectrum Critical Thinking for Math 7 Lesson 2.6
Grade 7 Multiplying and Dividing with Properties Grade 7
Converting Rational Numbers Using Division
27
28
107
Answer Key
Page 29 Page 30
Lesson 2.7 NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Rational Numbers in the Real World Check What You Learned
Multiplying and Dividing Rational Numbers
Negative and positive numbers are used to represent how far something is above Use the distributive property to find the product .
or below a point of reference . The point of reference, such as sea level, zero
balance, or target amount, is considered to be zero . 7
10
1 . 20 5 2 . 19 10 .25
Five and a half feet below sea level 25 7 20 (5 + 7 ) 100 + 140 19 10 + 19 0 .25
8 10 190 + 4 .75
194 .75
$100 .25 balance in the bank 100 .25 10 100 + 14 = 114
20 (5 + 7 )
10
Write and evaluate a numeric expression to represent each situation . CHAPTER 2 POSTTEST
Solve the problem . Show your work .
Marcy is tracking the depth of a baby shark in the ocean . The baby shark swims
3 more feet below sea level each day . At this rate, how deep will the shark be 3 . The Apps 'R' Us store charges $1 .09 per app . Susan downloaded 2 apps on
swimming after 5 days?
Friday and 3 apps on Saturday . How much did she spend?
–3 5 = –15
1 .09 2 + 1 .09 3
The baby shark will be 15 feet below sea level
after 5 days . 1 .09 (2 + 3) Susan spent $5 .45 .
1 .09 5
5 .45
4 . Without multiplying, find the sign of the product . Explain your thinking .
The construction worker adds 1 of a bucket of concrete mix to the sidewalk 6 times to 227 14 (210) (272) 45
4
1 The product will be negative because there is an odd number
8
fill the mold . The concrete was too high, so he removes of a bucket 2 times to level of negative factors . The first two negative factors result in a
it out . What is the overall amount of concrete used to make the sidewalk? positive product . When you multiply, that positive product times
1 6 + (– 1 ) 2 the remaining negative factor, the product becomes negative .
4 8
6 2 5 . Use a number line to find the product: 5 (23) =
4 8
+ (– ) 5 of a bucket was used . –5 (–3) = –15
4
6 + (– 1 ) –5–5 –5
4 4
5
4 220 –15 0 20
Spectrum Critical Thinking for Math Lesson 2 .7 Spectrum Critical Thinking for Math Chapter 2
Grade 7 Rational Numbers in the Real World Grade 7 Check What You Learned
29 30
Page 31 Page 32
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Check What You Learned
Multiplying and Dividing Rational Numbers Multiplying and Dividing Rational Numbers
Solve the problems . Show your work . Solve the problems. Show your work.
6 . The school club collected toys to donate to young children . Club members 9. Jayvon collected 9 of the 12 hidden treasures in his video game. Write as a
wrapped each coat individually . They used 8 rolls of wrapping paper . Each roll decimal. .75
1 12qw9.00
was 47 2 feet long . Each gift box used 9 1 feet of wrapping paper . How many gifts
2
were wrapped? – 84 0.75
47 1 ÷ 9 1 190 = 5 gifts per roll CHAPTER 2 POSTTEST CHAPTER 2 POSTTEST 60
2 2 38
95 ÷ 19 5 7 = 35 gifts wrapped – 60
2 2 0
95 2 10. The high temperature in Alaska was recorded for 5 straight days in the winter.
2 19
The recorded temperatures were 25.5, 22, 1, 0, and 24 degrees. What was
7 . Anna went to the store and bought 3 items that cost $10 .75, $8 .90, and $5 .10 . the average temperature during this period of time? –2.1
5qw10.5
What was the average cost of the items? 3qw248 .7 .255 –5.5 + (–2) + 1 + 0 + (–4) = –10.5
– 24 5 5 – 10
10 .75 + 8 .90 + 5 .10 = 24 .75 05
07 The average temperature was –5
–6 –2.1 degrees.
0
15 11. During the first quarter of a game, a football team made 3 plays that each
–15
resulted in a loss of 3 yards. The team also made 4 plays that each resulted in a
0
8 . How can the commutative and associative properties be used to make this problem gain of 2 yards. What was the team’s net gain or loss at the end of the quarter?
easier to solve?
2 2 4 (26) 2 .25 36 3 (–3) + 4 2
3 –9 + 8
–1
The 4 and −6 can be switched (commutative), then the
2– 23.25ancadn−b6ecmanulbtipelimedulttioplgieedt The team had a total loss of 1 yard at the end
to get 4 . The 4 and the of the 1st quarter.
9 . Then 4 and 9 can be
easily multiplied to get 36 .
Spectrum Critical Thinking for Math Chapter 2 Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Learned Grade 7 Check What You Learned
31 32
108
Answer Key
Page 33 Page 34
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 3 PRETEST CHAPTER 3 PRETEST Check What You Know
Expressions, Equations, and Inequalities Expressions, Equations, and Inequalities
Name the property represented (associative, commutative, or distributive) . 7. What is the difference between solving an equation and an inequality?
1 . 3(x y) 3x 3y _D__i_s_t_r_i_b__u__t_iv__e When solving an inequality, the inequality
2 . 4x 2y 2y 4x _C__o__m__m___u__t_a__tive sign switches if you have to multiply or divide
by a negative number.
3 . (2x y) z 2x (y z) _A__s_s__o__c_i_a__t_i_v_e 8. Solve the inequality: 5x 3 38 5x + 3 ≤ 38
Solve the problems. Show your work.
4 . Sheldon bought 6 pieces of gum for $0 .35 each, 10 pieces of licorice for $0 .15 – 3 – 3
each, and 2 candy bars for $1 .25 each . Write and evaluate an expression for the 555xx ≤≤ 33555 x ≤ 7
total amount Sheldon spent on candy .
t = 6(0 .35) + 10(0 .15) + 2(1 .25) 1
3
t = 2 .10 + 1 .50 + 2 .50 9. Tracy needs less than 12 yards of fabric to make costumes for the school play.
t = 6 .10 7x + 9 = –5 – 4S4he1661 al r e+ a d fy <h a –s1 44261 yards of fabric. How much more fabric can she buy?
1
5 . Solve for x: 7x 9 25 –9 –9 3 Tracy can buy any amount of fabric
1 1
7x = –14 6 less than 8 6 yards.
7x = – 14 1 0 . E r ic b o u gf h<t w 8ant61s a new computer that costs more than $1,275. His
7 7
x = –2
grandmother gives him $475. He makes $40 for each lawn that he mows. How
6 . Naomi put the same amount of money in the bank each week for 9 weeks . She
many lawns will he have to mow? Write and solve an inequality.
took $50 out to go to the fair . She had $143 .50 left in the account . How much
was she putting into the account each week? 9x – 50 = 143 .50 Eric will have to mow 475 + 40l > 1275
more than 20 lawns.
+ 50 + 50 – 475 – 475
She was putting $21 .50 9x = 193 .50 40l > 800
into the account each week . 9x = 193 .50 40l > 84000
9 9 40
l > 20
x = 21 .50
Spectrum Critical Thinking for Math Chapter 3 Spectrum Critical Thinking for Math Chapter 3
Grade 7 Check What You Know
Grade 7 Check What You Know
34
33
Page 35 Page 36
Lesson 3.1 NAME ________________________________________________________________________________ Lesson 3.2 NAME ________________________________________________________________________________
Properties and Equivalent Expressions Creating Expressions to Solve Problems
The Commutative, Associative, and Distributive properties can be used to create A rectangle has a length of 5x 1 2 and a width of 3x – 4. What is the perimeter
equivalent expressions . of the rectangle? Use the properties to simplify the expression.
7x 2 5x Original expression P 5 2l 1 2w 5 2(5x 1 2) 1 2(3x 2 4)
7x (2 5x) Associative Property 10x 1 4 1 6x 2 8 Distributive Property
10x 1 (4 1 6x) 2 8 Associative Property
7x (5x 2) Commutative Property 10x 1 (6x 1 4) 2 8 Commutative Property
(10x 1 6x) 1 (4 2 8) Associative Property
(7x 5x) 2 Associative Property x (10 1 6) 1 (4 2 8) Distributive Property
x (7 5) 2 Distributive Property 16x 1 (24) 516x 2 4
The perimeter of the rectangle is 16x 24.
12x 2 Commutative Property
Solve the problems.
Use the Commutative, Associative, and Distributive Properties to simplify the A jewelry store is having a sale. All necklaces are 25% off. Using number properties,
expressions . write two equivalent expressions that can be used to calculate the sales price of any
17x 6 13x 23 17x + (6 + 13x) – 3 necklace at the store. n – 0.25n
17 x + (13x + 6) – 3
Associative Property: (17x + 13x) + (6 – 3) n(1– 0.25) Distributive Property
x(17 + 13) + (6 – 3)
Commutative Property: 30x + 3
Associative Property: 0.75n Commutative Property
Distributive Property: Joan pays her daughter $10 a week plus $5 per chore she completes. She pays
Commutative Property: her younger son $7 a week plus $3 for each chore he completes. Using the number
1 (x 212) 1 (x 8) (4411413414xxxxx+–+++31(–(12P3+12roxp+xe)21rti+–e21xs a3(+n–x)d)3E4++q+u4iv4a4le)nt properties, write two equivalent expressions. Assume that each child does the same
4 2
Distributive Property: number of chores. (10 + 5c) + (7 + 3c)
(10 + 5c) + (3c + 7)
Associative Property: Commutative Property 10 + (5c + 3c) + 7
Associative Property
Commutative Property:
Associative Property: Distributive Property 10 + (5 + 3) c + 7
Distributive Property: Commutative Property 10 + 7 + 8c
17 + 8c
Commutative Property: Spectrum Critical Thinking for Math
Spectrum Critical Thinking for Math Lesson 3 .1 Lesson 3.2
Expressions
Grade 7 Grade 7 Creating Expressions to Solve Problems
35 36
109
Answer Key
Page 37 Page 38
Lesson 3.3 NAME ________________________________________________________________________________ Lesson 3.4 NAME ________________________________________________________________________________
Using Variables and Expressions Numeric and Algebraic Solutions
A problem can be solved by writing an expression that is equal to the unknown Chelsea is driving across the country. The trip is 2,035 miles. She takes 3 days to
variable .
drive. The first day, she drove 615 miles. The second day she drove 1 1 times as
3
far. How far did she drive the 3rd day?
Darryl bought 2 pairs of pants, 3 shirts, and 1 pair of shoes . How much did he Solve working backward: Solve with equation:
spend if the pants cost $31 .25 each, the shirts cost $17 .50 each, and the shoes
were $50 .75? Day 1: 2035 615 1420 miles remaining 615 1 1 (615) x 2035
3
s amount spent
s 2(31 .25) 3(17 .50) 50 .75 Day 2: 1 1 (615) 820 615 820 x 2035
s 62 .50 52 .50 50 .75 165 .75 3
1435 x 2035
Darryl spent $165 .75 . 1420 820 600 miles
1435 1435
Chelsea drove 600 miles the 3rd day.
x 600
Solve each problem working backward. Then, solve with an equation.
Write and simplify expressions to solve the following problems . A chef adds 2 more cups of cheese to the original amount in a recipe. She doubles
Bruno and Mark were shopping for school supplies . Mark bought 3 packs of pencils, the total amount to 6 cups. What was the original amount given in the recipe?
4 packs of paper, and 2 notebooks . Bruno bought 2 packs of pencils, 3 packs Total: 6 cups 2(x + 2) = 6
2x + 4 = 6
of paper, and 1 notebook . Packs of pencils cost $3 .20, paper costs $0 .75, and athdeddinogub2lec:up26s:=
–4 –4
notebooks cost $4 .50 . How much did the supplies cost altogether? Before 3 2x = 2
Before
c = cost of supplies
c = 3 .20(3 + 2) + 0 .75(4 + 3) + 4 .50(2 + 1) 2x 2
2 2
c = 3 .20(5) + 0 .75(7) + 4 .50(3) 3 – 2 = 1 cup = x=1
c = 16 + 5 .25 + 13 .5 Five less than 3 times a number is 25. What is the number?
c = 34 .75 The total cost was $34 .75 .
How would the amount that Bruno and Mark spent change if the pencils were 25%
3n – 5 = 25
off and the paper was 1 off? 1
3 3
5 .25(1– 5 less than what number is 25: +5 +5
c = 16(1 – 0 .25) + ) + 13 .50 25 + 5 = 30
c = 16(0 .75) + 5 .25( 2 ) + 13 .50 30 3n =30
3 3 3n 30
c = 12 + 3 .50 + 13 .50 3 times what number is 30: = 10 3 = 3
c = $29 savings = 34 .75 – 29 .00 They will save $5 .75 . n = 10
Spectrum Critical Thinking for Math Lesson 3 .3 Spectrum Critical Thinking for Math Lesson 3.4
Grade 7 Using Variables and Expressions Grade 7 Numeric and Algebraic Solutions
37 38
Page 39 Page 40
Lesson 3.5 NAME ________________________________________________________________________________ Lesson 3.6 NAME ________________________________________________________________________________
Equations in the Real World Using Variables to Express Inequalities
Jerri is teaching Jesse how to play a new video game. They play a round against Inequalities have more than one number as a part of the solution. Inequalities can
each other. Jerri’s score is 100 less than 3 times Jessie’s score. Their scores add up be solved the same way that equations are solved. If you multiply or divide by a
to 1400. Write and solve an equation to find out each of their scores. negative number to solve, the inequality sign needs to be reversed.
x 5 Jesse’s score; 3x 100 5 Jerri’s score 4x 5 1500 Solve the inequality: 3(x 5) 1 77. Is 10 a part of the solution?
x 3x 100 5 1400 4 4
x 5 375
4x 100 5 1400
100 5 100 3(x 5) 1 77 10 is not a part of the solution. The
4x 51,500 inequality sign had to change because we
Jesse scored 375. Jerri scored 3x 15 1 77 had to divide by a negative number.
3(375) 100 5 1,025.
3x 14 77
14 14
Write and solve an equation for each problem. 3x 63
3x 63
3 3
Ruth paid $50.25 for a dress. The original price was $67. What was the discount on x 21
the dress? Solve each inequality. Show your work.
67(1 – r) = 50.25 –67r = –16.75 Is 20 a part of the solution? 3p – 9 – 5p > 23
67 – 67r = 50.25 –67 – 67 3(p 3) 5p 23 – 2p – 9 > 23
– 67 – 67 r = 0.25
Yes, –20 is a part +9 +9
– 67r = – 16.75 The dress was marked down of the solution. –2p > 32
0.25 100 = 25%. Is 100 a part of the solution? –2p > 32
1.25(x 16) 140.75 –2 –2
Sayeed sold magazine subscriptions for the school fundraiser and raised $21.25 in p < –16
donations. Robyn sold three-quarters of the number of magazine subscriptions that No, 100 is not
Sayeed sold and raised $15.50 in donations. Together, they raised $127.75. How a part of the 1.25x + 20 ≤ 140.75
solution. – 20 – 20
much did each student make in magazine subscriptions?
1.25x ≤ 120.75
1.75m + 36.75 = 127.75 m = Sayeed's magazine sales 1.25x ≤ 120.75
0.75m = Robyn's magazine 1.25 1.25
– 36.75 – 36.75 sales x ≤ 96.6
m = 52; Sayeed raised $52 and
1.75m =91 Robyn raised 0.75 52 = $39
1.75m = 91 in magazine subscriptions.
1.75 1.75
Spectrum Critical Thinking for Math Lesson 3.5 Spectrum Critical Thinking for Math Lesson 3.6
Grade 7 Equations in the Real World Grade 7 Using Variables to Express Inequalities
39 40
110
Answer Key
Page 41 Page 42
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Lesson 3.7 Inequalities in the Real World Check What You Learned
Expressions, Equations, and Inequalities
Niki has saved $132 . She earns $12 an hour babysitting . She wants to buy a 1 . Write an equivalent expression using the Commutative, Associative, and
tablet that costs no more than $264 . How many hours does she have to babysit Distributive properties .
to earn enough money? 8x 2y 2 4 3y 2 3
132 12h 264 Niki will have to work no more than 11
hours to earn enough money .
2132 2132 8x + (2y – 4) + 3y – 3 Associative Property
8x + (–4 + 2y) + 3y – 3 Commutative Property
12h 132 8x – 4 + (2y + 3y) –3 Associative Property
8x – 4 + (2 + 3) y – 3 Distributive Property
12h 132 8x – 4 – 3 + 5y Commutative Property
12 12 8x + 5y – 7 Commutative Property
h 11 CHAPTER 3 POSTTEST
Write and solve an inequality for the scenario . 2 . Write two equivalent expressions to represent the perimeter of a triangle that has
2 sides with length 3x 2 1 and 1 side with length 2x 2 .
A laser tag arena offers two payment plans for laser tag games . Plan A charges $6
per game plus a one-time membership fee of $35 . Plan B offers unlimited games 2(3x – 1) + 2x + 2
for a year for a one-time membership fee of $149 . What is the minimum number of 6x – 2 + 2x + 2
games you would have to play in order for the unlimited plan to be the best deal? 8x
g = number of games 3 . Tia has 4 more than 1 the number of pairs of earrings that Ebony has . Together,
6g + 35 ≥ 149 2
– 35 – 35 they have 25 pairs of earrings . How many pairs of earrings does each girl have?
6g ≥ 114
6g ≥ 114 e = Ebony's earrings; 1 e + 4 = Tia's earrings
66 e 2
g ≥ 19 + 1 e + 4 = 25
2 e = 14
3
You would need to play a minimum of 19 games 2 e + 4 = 25 Ebony has 14 pairs of
in order for the unlimited plan to be the best – 4 –4
deal .
3 e = 21 earrings and Tia has
2 1
2 3 2 2 (14) + 4 = 11 pairs of
3 2 3
e = 21 earrings .
Spectrum Critical Thinking for Math Lesson 3 .7 Spectrum Critical Thinking for Math Chapter 3
Grade 7 Inequalities in the Real World
Grade 7 Check What You Learned
41
42
Page 43 Page 44
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned CHAPTER 4 PRETEST Check What You Know
Expressions, Equations, and Inequalities Ratios and Proportional Relationships
4 . Solve for x: 26x 1 .5 8 .7 1 . Cheyenne can type 1 of a page of her essay in 1 of an hour . How many pages
2 2
–6x + 1 .5 = 8 .7
can she type in 1 hour?
– 1 .5 – 1 .5 Cheyenne can 1 ÷ 1 = 1 2 = 2 =1
type 1 page in 2 2 2 1 2
–6x = 7 .2 an hour .
–6x = 7 .2 1 12 =
–6 –6 2 1
x = –1 .2 CHAPTER 3 POSTTEST
1 1 =6
5 . Is 0a part of the solution? 2 3 (x 2 27) 2 17 1 2
– 1 (x – 27) – 17 ≤ 1 2 . Graph the values in the table to see if they represent a proportional relationship .
3
1
– 3 x + 9 – 17 ≤1 x ≥ –27 12
0 is a part of the solution . 11
– 1 x – 8 ≤ 1 x248 10
3 y 3 6 12
+8+8 9
1 The values represent 8
– 3 x ≤9 a proportional 7
– 1 relationship . 6
–3 3 x ≤ –3 9 5
4
6 . Rhonda is buying a video game system that costs $325 . She also wants to buy 3
2
1
an equal number of strategy games and action games . Strategy games cost $20 0 1 2 3 4 5 6 7 8 9 10 11 12
each, and action games cost $35 each . How many games can she buy if she 3 . Use the table to find the constant of proportionality .
spends no more than $435?
325 + 20g + 35g ≤ 435 x 40 80 120 k= 30 = 3
y 30 60 90 40 4
325 + 55g ≤ 435
– 325 – 325
55g ≤ 110 Rhonda can buy at most
g≤2 2 of each type of game .
Spectrum Critical Thinking for Math Chapter 3 Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Learned Grade 7 Check What You Know
43 44 111
Answer Key
Page 45 Page 46
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 4 PRETEST Lesson 4.1 Comparing Unit Rates
Ratios and Proportional Relationships
4 . Wayne takes 5 steps every time that Jade takes 7 steps . What is the constant of A rate is a special ratio of two values with different units . When one of the values
is 1, it is a unit rate . The two values can be divided to calculate the unit rate .
proportionality? Use it to write an equation .
k= 5 Carson can read 5 1 pages of his history textbook in 1 of an hour . How many
7 2 6
pages can he read in 1 hour? 1 1
2 6
5 5 4
7
w= j 11 6
2 1
66 33
2
5 . Given the graph, what is the constant of proportionality?
10 Carson can read 33 pages in 1 hour .
9
8 Solve the problems . Show your work .
1 7
2
k= 6 8,4 Penny is comparing two recipes . One recipe calls for 1 stick of butter for 3 cups of
4 4
5 4,2 1 2
4 milk . The other recipe calls for 2 stick of butter for 1 3 cups of milk . Which recipe has
3 2,1 more butter per 1 cup of milk?
2
1 ÷ 3 b13uttsetricpk eorf 1 1 ÷ 5 b13u0ttestricpkeor f1
1 4 4 cup of milk . 2 3 cup of milk .
1
0 1 2 3 4 5 6 7 8 9 10 1 4 1 2 3 3
4 3 3 5 10
6 . Write an equation using the constant of proportionality from #5 . = =
y= 1 x 1 3 the first recipe has more butter per cup of milk .
2 3 10
Fran ran 4 1 miles in 2 of an hour . Fred ran 6 1 miles in 3 of an hour . Who ran the
2 5 2 5
fastest?
9 ÷ 2 Fran ran 13 ÷ 3 Fred ran
2 5 1 2 5 5
7 . Use the equation in #6 to predict the value of y when x 50 11 4 miles 10 6 miles
y= 1 (50) 9 5 per hour . 13 5 per hour
2 2 2 2 3
45 1 65 5
4 = 11 4 6 =10 6
y = 25 Fran ran the fastest .
Spectrum Critical Thinking for Math Chapter 4 Spectrum Critical Thinking for Math Lesson 4 .1
Grade 7 Check What You Know Grade 7 Comparing Unit Rates
45 46
Page 47 Page 48
Lesson 4.2 NAME ________________________________________________________________________________ Lesson 4.2 NAME ________________________________________________________________________________
Testing Proportional Relationships Testing Proportional Relationships
When two quantities have a proportional relationship, this means the ratio of one You can also test proportionality by cross-multiplying . If a relationship is
quantity to the other quantity is constant. When graphed on a coordinate plane, proportional, the cross products will be equal . Use cross products to check the
the proportional relationship will form a straight line through the origin. proportionality .
Does this represent a proportional relationship? 13 Is 4 .5 , 6 .75 proportional?
12 2 3
11
Time (minutes) 2 46 Pages read 10 4 .5 3 2 6 .75?
Pages read 4 8 12 13 .5 13 .5?
9
8 Yes, the relationship is proportional .
7
6 Cross-multiply to determine if each relationship is proportional .
5
The graph is forms a straight line that 4 Time (hours) 2 3 .5 5
goes through the origin. It is proportional. 3 Time 116 238 340
2
1
0 1 2 3 4 5 6 7 8 9 10
Time (minutes)
Graph these relationships to determine if they are proportional.
Number of Pounds 2 3 6 Georgia uses 4 pencils in 2 weeks, 7 2 = 3 .5
pencils in 3 weeks, and 8 pencils in 116 238
4 weeks.
Cost 2.50 3.75 7.50 2(238) = 3 .5(116)
10 10 476 = 406
9 6,7.50 9 4,8 Not proportional
8 8
costs 7 pencils 7 3,7
6
2,23546.,530.75 Mike is trying to choose a data plan for his phone . 3GB costs $27, 4GB costs $36,
5 2,4 and 7GB costs $63 .
4
3 3 3 4 4 7
27 36 36 63
2 2 = =
1 1
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 3(36) = 4(27) 4(63) = 7(36)
pounds weeks
The graph does not form a
The graph forms a straight 108 = 108 252 = 252
line that goes through the straight line that goes through Proportional
origin. It is proportional. the origin. It is not proportional.
Spectrum Critical Thinking for Math Lesson 4.2 Spectrum Critical Thinking for Math Lesson 4 .2
Testing Proportional Relationships Grade 7 Testing Proportional Relationships
Grade 7
48
47
112
Answer Key
Page 49 Page 50
Lesson 4.3 NAME ________________________________________________________________________________ Lesson 4.4 NAME ________________________________________________________________________________
Constants of Proportionality Using Equations to Represent Proportions
A unit rate can also be called a constant of proportionality . The constant of The constant of proportionality can be used to write an equation to represent the
proportionality, k, is the ratio of the output variable to the input variable . relationship.
Days of Car Rental 3 5 6 A recipe calls for 1 cup of sugar for every 1 1 cups of flour. Write an equation
Cost of Car Rental 96 .75 161 .25 193 .50 2 3
to calculate how much flour to use for each cup of sugar. How many cups of flour
output will be used if 2 1 cups of sugar are used?
input 2
96 .75 161 .25 193 .25
k ; k 3 32 .25; 5 32 .25; 5 32 .25 flour 1 1
sugar 3
k cups of 8 2 2
1 3 3
The constant of proportionality is 32 .25 . The cost of renting a car is $32 .25 per 2
day .
f (2 2 )s f (2 2 ) (2 1 ) ( 8 ) ( 5 ) 40 6 2 cups of flour
3 3 2 3 2 6 3
Use the information to calculate the constant of proportionality for each table . What Write an equation using the constant of proportionality to represent each relationship
does the constant of proportionality mean in the context of the data given? described.
Gallons of Gas 3 8 15 It takes 1 1 gallon of gas to mow 1 acre of grass. How much gas does it take to
Price 6 .54 17 .44 32 .70 2 2
1
mow 2 2 acres of grass? 1 1
2
k = 6 .54 = 2 .18; 17 .44 = 2 .18; 32 .70 = 2 .18 k= gas gallons = 1 =3
3 8 15 acres of grass
The constant of proportionality is 2 .18 . Milk costs g = 3a 2
$2 .18 per gallon . g = 3 (2 1 ) = 7 1 gallons
2 2
1 1
Time (hours) 1 .75 0 .25 2 .5 There are 880 feet in 6 of a mile. How many feet are in 1 3 miles?
21 3 30
Distance Biked feet 880
(miles) k= mile = 1 = 5280
k = 21 = 12; 3 = 12; 30 = 12 f = 5280m 6
1 .75 0 .25 2 .5
1
The constant of proportionality is 12 . The person f = 5280 (1 3 ) = 7,040 feet
was biking 12 miles per hour .
Spectrum Critical Thinking for Math Lesson 4 .3 Spectrum Critical Thinking for Math Lesson 4.4
Grade 7 Constants of Proportionality Grade 7 Using Equations to Represent Proportions
49 50
Page 51 Page 52
Lesson 4.5 NAME ________________________________________________________________________________ Lesson 4.6 NAME ________________________________________________________________________________
Proportions on the Coordinate Plane Proportions in the Real World
A constant of proportionality can be found using the graph of a proportional Answer the questions. Show your work.
relationship . Identify an ordered pair (x,y) on the line . The constant of
proportionality is k y/x . Juice is sold at the grocery store in several different sizes. The prices are shown in
the table.
Given the graph, calculate the constant of proportionality . Value ($) 0 .90 Size (fl. oz.) Price ($) Unit Price ($)
What does the constant of proportionality represent 0 .70 16 1.89
on the graph? 0 .50 32 3.49 0.12
64 7.59 0.11
k 0 .50 0 .10 0 .30 128 9.99 0.12
5 0.08
0 .10
The constant of proportionality is 0 .10 . 0 12345678 a. Complete the table with the unit prices of each size. Round to the nearest
It represents the value of each dime: $0 .10 . # of dimes
hundredth. 1.89 = 0.12 7.59 = 0.12
16 64
3.49 9.99
Calculate the constant of proportionality for each graph . What does the constant of 32 = 0.11 128 = 0.08
proportionality represent? 12 b. If you wanted 64 fluid ounces of juice, which would be the best way to
11 purchase it?
2 .25 10
1 .75 It would be best to buy 2 32-oz. bottles for
1 .25 9 3.49 2 = $6.98 instead of 1 64-oz bottle for $7.59.
0 .75 8
0 .25 7 Amount of Bill 19.00 35.00 72.00
6
0 1234567 5
4
3
2
1
1 2 3 4 5 6 7 8 9101112 Tip 3.42 6.30 12.96
How much would 7 tokens be worth? How many blue marbles are there if Calculate the constant of proportionality shown in the table above. What does this
k= value = 150 there are 3 red marbles? constant mean within the context of the data given? How much would the tip be if the
game tokens 3
k= blue = 12 = 4 bill was $95.00?
red 9 3
= 0 .50 k = 3.42 = 0.18
Each game token is worth 19.00
$0 .50 . There are 4 blue marbles for
7 tokens would be worth The constant of proportionality is 0.18. This means that the
0 .50 7 = $3 .50 . every 3 red marbles . There
4 customers paid an 18% tip.
would be 3 (3) = 4 blue
marbles if there are 3 red tip = 0.18 95.00 = $17.10
Spectrum Critical Thinking for Math marbles . Lesson 4 .5 Spectrum Critical Thinking for Math Lesson 4.6
Grade 7 Grade 7 Proportions in the Real World
Proportions on the Coordinate Plane 52
51
113
Answer Key
Page 53 Page 54
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Lesson 4.6 Proportions in the Real World Check What You Learned
Answer the questions. Show your work. Ratios and Proportional Relationships
One half of a can of paint covers 150 square feet of a wall. 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?
a. Create a table to represent this relationship. 2 ÷ 1 Pool A is 3 ÷ 3 Pool B is
3 5 5 filling 1
Cans of paint 1 1 2 2 = 4 filling 2 2 gallon per
Area covered 2 300 600 3 4 gallons p3er 3 5 minute .
8 1 5 3
150 3
area covered
(square feet) minute .
CHAPTER 4 POSTTEST
b. Create a graph to represent this relationship 2 2 15 = 1
3 15
100
900 Pool A is being filled faster .
800
700 2 . Graph the values in the table to see if they represent a proportional relationship .
600
500 Time 1 1 1 2
400 2 2
300 Amount Painted (square feet)
200
100 120
110
0 1 2 3 4 5 6 7 8 9 10 100
cans of paint 90
80
c. What is the constant of proportionality? What does it represent? 70 28 84 112
60
k= 150 = 300 Each can of paint can cover 50
1 300 square feet of the wall. 40
30
2 20 Yes, it is proportional
10 because it forms a
d. Write an equation to represent the relationship between cans of paint used and straight line that passes
1 2 3 4 5 6 7 8 9 10 through the origin .
how much of the wall is covered. How much of the wall can be covered by 2 1
2
cans of paint?
w = 300p 1
2
w = 300(2 ) = 750 square feet
750 square feet of wall can be
covered.
Spectrum Critical Thinking for Math Lesson 4.6 Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Learned
Grade 7 Proportions in the Real World
54
53
Page 55 Page 56
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Mid-Test Chapters 1–4
Ratios and Proportional Relationships 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
3 . Use the table to find the constant of proportionality . debits?
Miles Walked 1 1 3 3 1 57 – (–14) =
Calories Burned 4 4 2 57 + 14 = 71
The total amount of the debits was $71 .
25 175 350
pk r=op2o5rt÷ion41alit=y 100 . The constant of CHAPTER 4 POSTTEST 2 . Astrid played a board game in math class . She ended up with these cards:
is 100 .
3 .9 24 .2 23 .9 4 .1 6 .8 28 .5 210 .8
4 . Write an equation using the constant of proportionality in #3 on the previous a . The score is the sum of the card values . What was her score? Show your work .
page . How many calories would you expect to burn if you walk 3 1 miles? 3 .9 + (–3 .9) + 4 .1 + (–4 .2) + 6 .8 + (–8 .5) + (–10 .8)
5 0 + (–0 .1) + 6 .8 + (–8 .5) + (–10 .8)
6 .7 + (–8 .5) + (–10 .8)
c = 100m c = 100( 16 ) You would –1 .8 + (–10 .8)
c = 320 5 expect to burn –12 .6
c = 100(3 1 ) 320 calories .
5 b . In order to win the game, the absolute value of a score must be less than 12 .
pay ($)160
5 . Use the graph to create a table 140 Is it possible for Astrid to win? Explain why or why not .
of values . Find the constant of CHAPTERS 1–4 MID-TEST120
|–12 .6| = 12 .6
proportionality and write an 100 12 .6 > 12
80 Astrid cannot win . The absolute value of her score is
greater than 12 .
equation . What does it mean 60
within the context of the graph? 40 3 . A number j is positive and another number k is negative . Based on this
20 information, can you determine whether j 2 k is positive or negative? Explain .
How much would the pay be 10
after 25 hours of work? 0 2 6 10 14 18
hours worked (h)
Hours worked 6 10 14
60
Pay (dollars) 100 140
k = 60 = $10 p =10h j – k will always be positive . Since k is a negative number,
6 its additive inverse will be added to j, resulting in a
p = 10(25)
positive number .
The pay is $10 per hour . p = $250
Spectrum Critical Thinking for Math Chapter 4 Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Check What You Learned Grade 7 Mid-Test
55 56
114
Answer Key
Page 57 Page 58
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Mid-Test Chapters 1–4 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 7. Determine whether the expression 2(x 3) is equal to (4x 1 ) (8x 5 1 ).
4 8 2 2
pound, how many cans can it hold? If you add more support so the shelf can hold Identify the properties you used in your solution steps.
1 1
5 1 pounds, how many cans can the shelf hold now? Show your work. 4x + 2 – 8x + 5 2 Distributive Property
4
3 3 1 3
3 4 ÷ 8 = 5 4 ÷ 8 = 4x – 8x + 1 + 5 1 Commutative Property
2 1 + 2 Associative Property
15 8 = 21 8 = (4x – 8x) + ( 2 5 1 )
4 3 4 3 2
120 168
12 = 10 12 = 14 –4x + 6 = –2(2x – 3) Distributive Property
The shelf can hold 10 cans. The shelf can now hold 14 –2(2x – 3) = –2(x – 3)
cans.
8. A plumber charges $110 for a service call and $65 for each hour of work after
5. Mary Ellen says that the expression 5 7 8(4 2) simplifies to a
the first hour. Let h represent the hours the electrician works on a service call.
negative number because if you multiply three negative numbers, the final answer Write an expression to represent the cost. How many hours did it take the plumber
will be negative. Is she correct? Show why or why not. to complete a job if the total cost is $240? 195 = 65h
–5 + 7 + 8(–4 –2) = The answer is positive. Although c = cost; h = hours
–5 + 7 + 8(8) = the expression has 3 negative c = 110 + 65(h –1) 195 = 65h
–5 + 7 + 64 = numbers, it is not purely a 65 65
2 + 64 = multiplication problem. This CHAPTERS 1–4 MID-TEST CHAPTERS 1–4 MID-TEST 240 = 110 + 65 (h – 1) h=3
problem also has addition. You 240 =110 + 65h - 65
66 must use the order or operations 240 = 65h + 45 The job took 3 hours.
to simplify this expression. –45 –45
6. A number is multiplied by 3 , divided by 1 , and then divided by 7 . The 9. Candace has $65.25. She spent $31.50 on a new throw rug for her bedroom.
4 2 8 She wants to buy some matching throw pillows. How many pillows can she buy if
the pillows cost $11.25 each?
resulting number is 96. Work backward to get the original number by performing
opposite operations.
96 (– 7 ) = –84 31.50 + 11.25p ≤ 65.25
8
1 –31.50 –31.50
–84 = –42
–42 ÷ 2 11.25p ≤ 33.75
(– 3 ) = –42 (– 4 ) = 56
4 3 11.25p ≤ 33.75 Candace can buy no
11.25 11.25 more than 3 pillows.
The original number is 56. p≤3
Chapters 1–4
Spectrum Critical Thinking for Math Chapters 1–4 Spectrum Critical Thinking for Math Mid-Test
Grade 7 Mid-Test
Grade 7
57
58
Page 59 Page 60
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Mid-Test Chapters 1–4 CHAPTER 5 PRETEST Check What You Know
10 . A museum is keeping track of the number of visitors per day . Geometry
15,000 1. Find the length of the missing side for the pair of similar triangles.
14,000
13,000 12 ft. 14 ft. 12 = 14
12,000 18 ft. 18 x
11,000
10,000 18 ft. 12x = 252
9,000 27 ft. 12x = 252
8,000 12 12
7,000 x = 21ft.
6,000
5,000 1 2 3 4 5 6 7 8 9 1011121314 2. Can these lengths form a triangle?
4,000
3,000 Side 1: 9 cm
2,000
1,000 Side 2: 5 cm
0 Side 3: 11 cm
a . Create a table that represent the proportional relationship . 9 + 5 > 11
9 + 11 > 5
# of days 3 6 9 5 + 11 > 9
# of visitors 1000 2000 3000 These sides can form a triangle.
b . What is the constant of proportionality to the nearest hundredth? What does it CHAPTERS 1–4 MID-TEST 3. Name the shape that is created by each cross section.
represent on the graph?
k = 100 = 333 .333
3
The museum has about 333 visitors per day .
c . Write an equation to represent the relationship .
v = 333 .33d
d . Predict how many people will have visited the museum after 33 days . rectangle triangle
v = 333 .33(33) Spectrum Critical Thinking for Math Chapter 5
v = 10999 .89 Grade 7 Check What You Know
They can expect about 11,000 visitors after 33 days . 60
115
Spectrum Critical Thinking for Math Chapters 1–4
Grade 7 Mid-Test
59
Answer Key
Page 61 Page 62
NAME ________________________________________________________________________________ CHAPTER 5 PRETEST Lesson 5.1 NAME ________________________________________________________________________________
Check What You Know Scale Drawings
Geometry 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
4. Find the circumference and area of the circle. Use 3.14 for . Round answers to proportional relationship between the original object and the scale drawing.
the nearest thousandth, if necessary.
4.3 m A _______ square feet Benita makes a scale drawing so she can rearrange her room. Her actual room is
C _______ feet 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
C = d A = r2 4.3 )2 should she draw it on her diagram?
C = 3.14 4.3 A = 3.14 ( 2
C = 13.502 m 14.515 m2 scale factor inches 6 in. 0.5 in.
feet 12 ft. 1 ft.
The scale factor can also be written as 0.5 in.:1 ft. 7 in.
6 in.
5. If AGB is 120 degrees, what is the measure of HGE?
A B HGE _1__2__0__ degrees 4 ft. 0.5 in. 2 inches
H 1 ft.
C The measure of HGE is also
G E 120 degrees because they are 8 ft. 0.5 in. 4 inches
1 ft.
D
The bed should be 2 inches by 4 inches on the diagram.
F
vertical angles.
6. What is the volume of a rectangular prism with a length of 10mm, a width of Solve the problem. Show your work.
8mm, and a height of 5mm?
V = 10 8 5 = 400 mm3 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 2 cm 5 cmm; 2 5
5 m ? 5 w
= =
5 cm 2w = 25; w = 12.5 m
7. What is the combined area of a rectangle with a length of 13.2 in. and a width of 2 cm 20 mcm; A = lw; A = 50(12.5) = 625 m2
4.1 in., and a triangle with a base of 13.2 in. and a height of 6.5 in.? 5m ? P = 2l + 2w; P = 2(50) + 2(12.5)
A = 13.2(4.1) = 54.12 in.2 = 2 = 20 P = 125 m
5 l
A= 21are(a13=.25)4(6.1.52) = 42.9 in.2
Total + 42.9 = 97.02 in.2 2l = 100; l = 50 m
Spectrum Critical Thinking for Math Chapter 5 Spectrum Critical Thinking for Math Lesson 5.1
Grade 7 Check What You Know Grade 7 Scale Drawings
61 62
Page 63 Page 64
Lesson 5.2 NAME ________________________________________________________________________________ Lesson 5.3 NAME ________________________________________________________________________________
Forming Triangles Cross Sections of 3-Dimensional Figures
The sum of the lengths of two sides of a triangle must be greater than the length of A cross section is the intersection of a 3-dimensional figure and a plane . Here
the third side. are some examples:
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?
463
634
346
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 Intersect each 3-D figure with the given 2-D shape .
triangle with the straws. Will he be successful? If not, which straw should he replace?
What is the minimum length of the replacement?
3+4<9 Don will not be successful. He could A rectangle A square
3+9>4 replace the 3 cm straw with a straw
4 + 9 >3 longer than 5 cm (x + 4 > 9) so that
he could make the triangle.
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 A quadrilateral A triangle
is an integer.
x+2+x+1>–3 x+2+x–3>x+1 x+1+x–3>x+2
2x + 3 > x – 3 2x – 1 > x + 1 2x – 2 > x + 2
x>–6 x>2 x>4
The minimum value of x is 5.
Spectrum Critical Thinking for Math Lesson 5.2 Spectrum Critical Thinking for Math Lesson 5 .3
Grade 7 Forming Triangles Grade 7 Cross Sections of 3-Dimensional Figures
63 64
116
Answer Key
Page 65 Page 66
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Lesson 5.4 Circles: Circumference Lesson 5.5 Circles: Area
The perimeter of a circle is called the circumference . The area of a circle is A pr2, where p 3.14.
C 2r (r radius) or C d (d diameter), 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 plate has a diameter of 10 inches . What is the circumference of the plate?
A 314 square feet
C 3 .14(10) 31 .4 inches Answer the questions. Show your work. Use 3.14 for p.
Answer the questions . Show your work . Use 3 .14 for . 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.
In college basketball rules, the ball can have a maximum circumference of 30 inches . A = 3.14( 32 )2
What is the maximum diameter of a basketball to the nearest hundredth? 2
30 ≤ 3 .14d A = 3.14(256)
30 ≤ 3 .14d A = 803.8 square feet
3 .14 3 .14
A frying pan has a diameter of 11 inches. What is the area to the nearest square
d ≤ 9 .55 inches
inch of the smallest cover that will fit on top of the frying pan?
The diameter has to be 9 .55 inches or less . A = 3.14( 11 )2
2
A = 3.14(30.25)
A round stained-glass window has a circumference of 195 inches . What is the radius
of the window to the nearest inch? A = 95 square inches
2(3 .14)r = 195 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
6 .28r = 195 driving? Round your answer to the nearest tenth.
6 .28r = 195 A = 3.14(25)2
6 .28 6 .28
r = 31 inches A = 3.14(625)
A = 1,962.5 square miles
The radius of the window is 31 inches .
Spectrum Critical Thinking for Math Lesson 5 .4 Spectrum Critical Thinking for Math Lesson 5.5
Grade 7 Circles: Circumference Grade 7 Circles: Area
65 66
Page 67 Page 68
Lesson 5.6 NAME ________________________________________________________________________________ Lesson 5.7 NAME ________________________________________________________________________________
Angle Relationships Area of Composite Shapes
When two lines intersect, they form angles that have special relationships. Shapes that are composed of other shapes are called composite shapes The area
• Vertical angles have the same measure. of a composite shape is equal to the sum of each shape it is made of.
• Supplementary angle are two angles with the sum of 180 degrees
• Complimentary angles are two angles with the sum of 90 degrees. Find the area of the composite shape:
828 p 3m
What is the value of p? 11.4 m
The angles are supplementary, so the sum is 180.
Area = area of rectangle + area of semicircle
p 82 180 area of rectangle lw (11.4)(3) 34.2 m2
82 82
Area of semicircle 1 pr2 1 (3.14) ( 3 )2 3.53 m2
p 98 degrees 2 2 2
34.2 3.53 37.73 m2
If 4 is a right angle and 5 measures 40 degrees, find the measures of the Find the area of the composite shapes. Show your work. Use 3.14 for p. Round
answers to the nearest hundredth.
remaining angles.
GH 5 = 2 (vertical angles) 5 ft. 1 bh = ( 1 )(3)(5) = 7.5 ft.2
2 = 40° 3 ft. 22
3
L 4I 2 1 + 2 = 180 (supplementary) Area of semicircle =
1 + 40 = 180 1 pr2 = 1 (3.14) ( 5 )2 = 9.81 ft.2
51 J 22 2
K Sum of each area = 7.5 + 9.81 = 17.31 ft.2
1 = 140° 3.1 cm bh = ( 1 )(4.2)(7.4) = 15.54 cm2
7.4 cm 2
lw = (7.4)(3.1) = 22.94 cm2
2 + 3 = 90 (complementary)
40 + 3 = 90 1 pr2 = 1 (3.14) (7.4)2 = 21.49 cm2
3 = 50° 2 2 (3.14)2(32.1)2 = 3.77
4.2 cm 1/2 pr2 = 1 cm2
2
15.54 + 22.94 + 21.49 + 3.77 = 63.74 cm2
Spectrum Critical Thinking for Math Lesson 5.6 Spectrum Critical Thinking for Math Lesson 5.7
Grade 7 Angle Relationships Grade 7 Area of Composite Shapes
67 68
117
Answer Key
Page 69 Page 70
Lesson 5.8 NAME ________________________________________________________________________________ Lesson 5.9 NAME ________________________________________________________________________________
Volume of Rectangular Prisms Volume of Triangular Prisms
Volume is the amount of space an object occupies. The volume of a rectangular A triangular prism is a prism whose base is a triangle. The volume of a triangular
prism can be calculated using the formula V bh, where b area of the base prism is the product of the area of the base and the height of the prism.
and h height. The area of the base is b lw, where l length and w width.
Volume bh, where b 1 bh
The volume of a rectangular prism is 210 cm3. If it has a length of 5 cm and a 2
width of 6 cm, what is the height?
The triangular base has a height of 3 cm and a
V 210 cm3
210 (5)(6)h base of 8 cm. The height of the prism is 12 cm.
210 30h b 1 (8)(3) 1 (24) 12 cm2
h 70 cm 2 2
V (12)(12) 144 cm3
3 cm 12 cm
8 cm
Find the volume of each figure. Show your work. V = bh
Answer the questions. Show your work. b= 1 bh
2
Penny is using colored sand to fill a jar that is shaped like a rectangular prism. The 1
bag of sand contains 150 cubic inches. The base of the prism is 6.5 inches by 7.4 b= 2 (6)(4) = 12 cm2
inches. The height of the box is 2.2 inches. Will all the sand fit in the jar?
6 cm V = (12)(12) = 144 cm3
V = 6.5 7.4 2.2 = 105.82 in.3
12 cm
150 – 105.82 = 44.18 cubic inches of the sand
will not fit into the jar. 4 cm Volume of rectangular prism = lwh
(4)(3)(12.5) = 150 cm3
A rectangular prism has a volume of 966 ft3. The prism’s height is 4 feet, and its Volume of triangular prism = bh
length is 14 feet. What is its width?
b= 1 (3)(4) = 6 = (6)(12.5) = 75 cm3
966 = (4)(14)w 4 cm 2
966 = 56w Total area = 150 + 75 = 225 cm3
966 = 56w 4 cm
56 56
12.5 cm
17.25 ft. = w
3 cm
Spectrum Critical Thinking for Math Lesson 5.8 Spectrum Critical Thinking for Math Lesson 5.9
Grade 7 Volume of Rectangular Prisms Grade 7 Volume of Triangular Prisms
70
69
Page 71 Page 72
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Check What You Learned
Geometry Geometry
1. At the photo lab, a customer brings in a photograph that is 4 inches wide by 6 4. A mini pancake has a circumference of 3 centimeters. A regular pancake has a
inches high. The customer wants the photograph enlarged to 20 inches wide by
25 inches high. Can this be done? Explain your reasoning. circumference of 6 centimeters. Is the area of the regular pancake twice the area
of the mini pancake? Use 3.14 for .
4 = 20 No, this cannot be done C = 2r C = 2r No, the area of the
6 25 because the cross-products
are not equal. 3 = 2r 6 = 2r larger pancake is
100 ≠ 120 CHAPTER 5 POSTTEST CHAPTER 5 POSTTEST 3 32=rc222mr 6 = 2r 4 times the area of
2 2 2 the mini pancake.
r=
2. If a triangle XYZ has two sides with lengths of 5 cm and 8 cm, what is the r = 3 cm
maximum and minimum length of the third side? Explain your answer. Assume A=
that the length of the third side is an integer. A = r2
5+8>x 5+x>8 x+8>5 The maximum A = ( 3 )2 7.065 cm2 A = (3)2
13 > x –5 –5 –8 –8 length is 12 cm, A = 9 2 = A = 9 = 28.26 cm2
x>3 and the minimum 4
x > –3 length is 4 cm.
5. If 4 is a right angle and 5 40°, find the measure of the remaining angles.
m 3. What are two possible shapes that can be formed by a cross section of this C 3 + 4 + 5 = 180
cm3 shape? Describe the angle of the cross section. 3 + 90 + 40 = 180
= bh A 54 3 = 50°
5) = 75 cm3 1B 2 3 D 5 + 1 = 180
5 cm3 40 + 1 = 180
E 1 = 140°
F 1 + 2 = 180
140 + 2 = 180
A cross section that is parallel to the base (the 2 = 40°
base is the triangle) will form a triangle.
A cross section that is perpendicular to the base
will form a rectangle.
Spectrum Critical Thinking for Math Chapter 5 Spectrum Critical Thinking for Math Chapter 5
Grade 7 Check What You Learned Grade 7 Check What You Learned
71 72
118
Answer Key
Page 73 Page 74
NAME ________________________________________________________________________________ CHAPTER 6 PRETEST NAME ________________________________________________________________________________
Check What You Learned CHAPTER 5 POSTTEST Check What You Know
Geometry Statistics
6. Find the volume of the figure. 1. Are the following samples biased or random? Explain your answer.
V = sum of volume of each section a. Wendy wants to find out the favorite sports of the students at her school. She
2 mm (2 mm)(10 mm)(2 mm) = 40 mm3 asks 25 students who were at the basketball team tryouts.
(9 mm)(2 mm)(5 mm)= 90 mm3 This is a biased sample. The answers are
more likely to be “basketball” since she is
10 mm 40 mm3 + 90 mm3 = 130 mm3 asking people who are at basketball tryouts.
5 mm 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 mm
2 mm 9 mm This is a random sample. Technology is used
to randomly select students.
7. Bill wants to fill this triangular prism 2 full of water. How much water does he
3 2. The graph represents a sample of football players’ heights. If there were 100
players, how many players could be expected to be 70 inches tall?
need? V = bh
b= 1 bh Football Players’ Heights (in.) 3 x
2 10 100
1 m2 3 =
20 m b= 2 (12)(16) = 96 Number of Players
2
16 m V = 96 m2 (20 m) = 1920 m3 3 10 = 30
1 10 10 100
12 m
0
If you only want to fill this up 2 of the 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 x = 30 players
way, you will need: 3
Heights (in.)
2
3 (1920 m3) = 1280 m3
Spectrum Critical Thinking for Math Chapter 5 Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Learned Grade 7 Check What You Know
73 74
Page 75 Page 76
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 6 PRETEST Lesson 6.1 Sampling and Drawing Inferences
Statistics
3 . A sample of people were asked how far they drive to work . What percentage 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
of people drive 6 miles to work? Round your answer to the nearest tenth of 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
percent . 5 = x members of the population are less likely to be chosen. Samples that are random
13 100 are better predictors of trends for the bigger population.
1 2 3 45 6 7 8 9 10 13x = 500 Rosewood Middle School has 714 students. Susan surveys a random sample of
Distance to Work from Home 34 students and finds that 9 of them play a sport outside of school. How many
13x = 500 students at the school are likely to play a sport outside of school?
13 13
x = 38 .5% 9 71s4; 34s (9) (714) 6426
4 . What can you infer from this data collected about the number of apps on a 34
sample of smart phones? 34s 643246; s 189
34
189 students are likely to play sports outside of the school.
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 Answer the questions. Show your work.
The median number of apps is 10 . The A high-tech company makes 3,500 widgets a day. The quality department chooses
minimum is 1 and the largest number is 25 . a random sample of 50 widgets and finds that 3 are defective. How many high tech
The middle 50% have between 5 and 18 apps . w5i3d0gets=pe3r 5dxa0y0are likely to be defective? 50x = 10500
50 50
5 . A factory produces 92,000 tubes of toothpaste each day . The quality manager 50x = 3(3500)
claims that fewer than 750 defective tubes are produced each day . In a random 50x = 10500 x = 210
sample of 420 tubes of toothpaste, 3 are defective . Is the quality manager’s claim Grace hears that the average gas price has risen to $2.89 during the gas shortage.
correct? Explain your answer . She checks gas prices at stations near her school, and finds that the average is
x = 3 The manager’s claim is $3.20. Why are the averages different?
92000 420 correct . There are likely to
The averages are different because Grace’s sample was
420x = 276,000 be about 657 defective tubes biased. She did not randomly select the gas stations in the
x = 657 .14 of toothpaste a day . city. She only took data from the gas stations that were in
her area.
Spectrum Critical Thinking for Math Chapter 6 Spectrum Critical Thinking for Math Lesson 6.1
Grade 7 Check What You Know Grade 7 Sampling and Drawing Inferences
75 76
119
Answer Key
Page 77 Page 78
Lesson 6.2 NAME ________________________________________________________________________________ Lesson 6.2 NAME ________________________________________________________________________________
Comparing Similar Data Sets Comparing Similar Data Sets
What can you infer from the two histograms? 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
In class 1, no students were shorter than 34 inches or taller than 46 inches . Class 2: Number of People in
In class 2, the range of heights is 25 inches, but the range in class 1 is just 12 Household
inches . The median for both classes is 42 inches . 50% of the students in class 1
are between 42 and 46 inches . Find the mean, median, and mode of each set of data. How do the data sets
compare?
12- 8- 6
10 5
10-
8- 6- 4 4 Class 1 Class 2
Frequency 6- 6 Frequency 4- Mean: 4; Median: 4 Mean: 3.8; Median: 4
4 2- 1
4-
Mode: 2, 4, 5; Range: 7 Mode: 4; Range: 3
2-
0- 0 0 0- This data is spread out fairly evenly This data is more compact and closer to
30 35 40 45 50 55
30 34 38 42 46 50 between 1 and 8. There are a variety the center. There is less variety in sizes.
Heights of Class #1 Heights of Class #2 of household sizes in this class. Four is the most common size.
5- 8- 7 Find the mean, median, and mode of each set of data. How do the data sets
4- 4 4 compare?
6-
33
Frequency 3- Frequency 4- 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
2- 2 11 Class 2: Student Donations to 1 2 5 7 8 9 1 11 14 15 17 19 19
2- 1 Charity Fund 17 11 8 2 2 11
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) mean = 2 + 3 + 7 + 8 + 10 + 2(11) + 3(12) + 2(14) + 15 + 17 + 6(20) = 13.4
20
In which range will the median occur for each diet?
median = 13 mode = 20
Soybeans: There are 14 values . Sunflowers: There are 12
mean = 2(1) + 3(2) + 5 + 7 + 2(8) + 9 + 3(11) + 14 + 15 + 2(17) + 2(19) = 9.55
The median value will be values . The median value will 19
between 200 and 250 . be between 300 and 350 . median = 9 mode = 2, 11
What percentage of the chickens are between 300 dkg and 350 dkg for each type The teachers donated more to charity. They had a
higher mean and median with a mode of $20.
of feed? Round your answers to the nearest percent .
soybean: 3 = .21 = 21% sunflower: 7 = 0 .58 = 58%
14 12
Spectrum Critical Thinking for Math Lesson 6 .2 Spectrum Critical Thinking for Math Lesson 6.2
Grade 7 Comparing Similar Data Sets Grade 7 Comparing Similar Data Sets
77 78
Page 79 Page 80
Lesson 6.2 NAME ________________________________________________________________________________ Lesson 6.3 NAME ________________________________________________________________________________
Comparing Similar Data Sets Data in the Real World
Box-and-whisker plots can help you interpret the distribution of data. Each section Tameka is planning a party for her brother. She invites 180 of the people in her
of a box and whisker plot contains 25% of the data points. 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
Active time data is collected from a group of high school students and a group of people are planning to come to the party.
elementary students.
a. Tameka decides to ask the 20 students who live in her neighborhood. 12 of
High School Students
Elementary Students them say that they will be able to come to the party. What is the population in
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 this event? Is this a random sample? Could this sample be biased? Are these
The double box and whisker plot shows that the high school students are overall results too low or too high? Explain.
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 The population is the 180 students who were invited.
are more active. The median is 64 minutes a day. The middle 50% exercise This is not a random sample. She only chose people
between 56 and 72 minutes. who lived in her neighborhood. This could be biased
because these students may be more likely to come to
the party because they live closer to her brother. These
results are likely too high.
This double box-and-whisker plot displays the test scores of students who studied b. Tameka decides to look at the graduation program and call every 10th person
alone and the scores of students who studied with a study group. Use it to compare
the data sets. on the list of graduates to see if they plan to come. She calls 18 people and 8
w/o study group of them say that they will be able to come. How many people can she expect to
with study group
come to the party?
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
8 = x She can expect 80 people to
The students who studied alone had a median of 75 for 18 180 come to the party.
their test grade. The grades ranged from 55 to 95. The 18x = 1440 x = 80
middle 50% made between a 62 and an 85. The students
who studied in a group had a median of 90 with grades c. Is this a random sample? Could this sample be biased? Compare these results
that ranged from 65 to 95. The middle 50% made between to the results from the first sample.
an 82 and 92. I can infer that the students that studied in a
group did better. This is a random sample. This is more likely not to be
biased. This is a lower number of expected guests than
the first sample. This is more realistic since it was a
random sample.
Spectrum Critical Thinking for Math Lesson 6.2 Spectrum Critical Thinking for Math Lesson 6.3
Grade 7 Comparing Similar Data Sets Grade 7 Data in the Real World
79 80
120
Answer Key
Page 81 Page 82
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Check What You Learned
Statistics Statistics
1 . Will wants to survey a sample of students at his school to find out how many 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
play musical instruments . He surveys students coming out of band class . Is Will’s whisker plots?
(no fertilizer)
sample biased or random? Why?
(w/fertilizer)
This is a biased sample . Everyone coming out of band class plays
an instrument . 1 2 3 4 5 6 7 8 9 10 11 12
2 . The graph shows a sample of heights of sixth graders and eighth graders . The plants that were grown with no fertilizer were shorter
and more consistent in height. The median height was about
Compare the data . What can you infer? 4.4 inches, with the middle 50% being between 4 and about
5.4 inches tall. The plants grown with fertilizer are taller. The
66 CHAPTER 6 POSTTEST CHAPTER 6 POSTTEST median was 9 inches, with the middle 50% being between
about 8.1 and about 10.4 inches.
55
5. The graph shows a sample group of girls and a sample group of boys, and the
44 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?
33
22
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
11
# of books read during school year (boys)
0 0
50-59 60-69 50-59 60-69 70-79
Heights of 6th Graders Heights of 8th Graders
The median height for 6th graders is between 50 and 59 inches .
There are no students taller than 69 inches . The median height for
8th grade students is between 60 and 69 inches . About 43% of
the 8th graders are taller than 69 inches .
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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
correct? Explain your answer .
x = 3 310x = 222000 # of books read during school year (girls)
74000 310
Boys: 2 = x 14x = 400 Girls: 1 = x 14x = 200
x = 716 .1 14 200 14 200
The manager’s claim is incorrect . There are likely to be about 716 x = 28.6; about 29 boys x = 14.3; about 14 girls
defective sets of headphones a day .
Spectrum Critical Thinking for Math Chapter 6 Spectrum Critical Thinking for Math Chapter 6
Grade 7 Check What You Learned Grade 7 Check What You Learned
81 82
Page 83 Page 84
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Know CHAPTER 7 PRETEST CHAPTER 7 PRETEST Check What You Know
Probability Probability
1. Of the 50 U.S. states, 13 were the original colonies. If you select 1 state 5 . The school picnic is a two-day weekend event . It has been scheduled for May .
randomly, how likely is it to be one of the original colonies? 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 .
13
50 = 26% 15 15 = 225 = 23%
31 31 961
2. David takes 20 shots and scores 6 goals at soccer practice. What is the
experimental probability that he will miss his next shot?
6 14 6 . In basketball, Alan makes 1 out of every 4 free throws he tries . What is the
20 20
P(miss) = 1 – = = 70% probability that Alan will make his next 3 free throws? Round your answer to the
nearest tenth of a percent .
3. Evan hits 6 out of 14 pitches during practice. What does an experimental P(three free throws) = 1 1 1 = 1 = 1 .6%
4 4 4 64
probability of 4 describe?
7
1 .6% chance
P(misses pitch) It is the probability that he misses
a pitch. 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?
4. At Luvski Ski Resort, there are two chair lifts to the top of the mountain. There are P(red, red) = 1 1 = 1 = 25%
five ski trails to the bottom of the mountain. What is the probability of riding on 2 2 4
Chair 1 and skiing on Trail 3?
T1 T1 25% chance
T2 T2
Chair 1 T3 Chair 2 T3 8 . Lucy places 5 cards face down on the table and mixes them up . The cards are
T4 T4 numbered 1 through 6 . What is the likelihood that her friend Harry will draw an
T5 T5 even-numbered card?
Chapter 7 P(even) = 3 = 50%
Check What You Know 6
P(chair 1, trail 3) = 1 = 10% 83 50% chance
10
Spectrum Critical Thinking for Math Spectrum Critical Thinking for Math Chapter 7
Grade 7 Grade 7 Check What You Know
84
121
Answer Key
Page 85 Page 86
Lesson 7.1 NAME ________________________________________________________________________________ Lesson 7.2 NAME ________________________________________________________________________________
Understanding Probability Frequency Tables
The probability of an event measures the likelihood that the event will occur. 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
Impossible Unlikely Equally Likely/UnlikeLliykely Certain keep track of the trials.
0 1 1 3 1 Marvin has a bag of marbles. He removes a marble, records the color, and then
4 2 4 puts the marble back in the bag. The frequency table shows how many times he
picked each color.
0% 0.25 0.50 0.75 100% Find the experimental probability
for each color.
25% 50% 75% Color Frequency
Purple 12
The complement of an event is the set of all outcomes not included in the event. Pink 10 12
Orange 15 50
White 13 P (purple) 24%
Answer the questions. P (pink) 10 20%
P (orange) 50 30%
15
50
What is the sum of the probabilities of an event and its complement? P (white) 13 26%
50
The sum is 1. The event has a 100% chance of
either occurring or not occurring. Students at Prince Middle School were asked about their weekly allowance. Use the
Students in Ms. Baldwin’s class are picking numbers out of a hat. The hat has 8 frequency table to calculate the experimental probability for each amount. Show your
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 work.
the number line above?
Allowance # of students P($15) = 9 = 22.5%
It is as likely as not likely to pick a white piece of $15.00 9 40
paper out of the hat. $20.00 11
$25.00 12 P($20) = 11 = 27.5%
Where would the probability of picking a white piece of paper fall on the number 40
line if there were 6 pieces of white paper and 2 pieces of black paper in the hat?
$30.00 8 P($25) = 12 = 30%
The probability of picking a white piece of paper 40
would be likely.
P($30) = 8 = 20%
40
Spectrum Critical Thinking for Math Lesson 7.1 Spectrum Critical Thinking for Math Lesson 7.2
Grade 7 Understanding Probability Grade 7 Frequency Tables
85 86
Page 87 Page 88
Lesson 7.2 NAME ________________________________________________________________________________ Lesson 7.3 NAME ________________________________________________________________________________
Frequency Tables Calculating Probability
Answer the questions. Theoretical probability is the probability of an event occurring based on all the
possible outcomes. Theoretical probability can be calculated this way:
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. P (event) number of ways the event can occur
total number of possible outcomes
Round your answer to the nearest tenth of a percent.
P(1) = 21 = 26.9%
Number on Spinner Frequency 78 A spinner has 3 equally sized sections labeled A, B, and C. What is the
probability that your spinner landed on section A?
1 21
2 22 P(2) = 22 = 28.2% There are 3 possible outcomes, with one of them being A. P (A) 1
3 18 78 3
4 17
P(3) = 18 = 23.1%
78 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
P(4) = 17 = 21.8% picks a green marble? Round your answer to the nearest tenth of a percent.
78
What is the probability of not spinning a 3? P(green) = 5 = 22.7%
22
P(not 3) = 1 – 0.231 = 0.769 = 76.9%
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% What is the probability that she does not pick a red marble? Round your answer to
60
the nearest tenth of a percent.
Is this the probability that you expected? Compare the results to your expectations. P(not red) = 1 – 8 = 1 – .364 = .636 = 63.6%
22
Since there were 2 possible outcomes, a 50% was
expected for each side of the coin. This did not
happen because this is experimental probability.
Spectrum Critical Thinking for Math Lesson 7.2 Spectrum Critical Thinking for Math Lesson 7.3
Grade 7 Frequency Tables Grade 7 Calculating Probability
87 88
122
Answer Key
Page 89 Page 90
Lesson 7.4 NAME ________________________________________________________________________________ Lesson 7.5 NAME ________________________________________________________________________________
Probability Models Other Probability Models
When all outcomes of an experiment are equally likely, the event has uniform When a probability event has unequal odds, the outcomes are not equally likely to
probability . This probability can be used to predict outcomes . occur.
Vick rolls a number cube . What is the probability that he rolls a prime number? If A spinner has 4 equal sections. 2 of the sections are yellow, one of the sections
he rolls the number cube 30 times, how many times is he expected to roll a prime
number? is purple, and the other section is green. What is the probability that the spinner
lands on yellow?
P (yellow) 2 50%
4
A number cube has 3 prime numbers (2, 3, 5) . There are 6 possible outcomes .
P (prime number) 3 50% What is the probability of not spinning purple?
6
0 .5 30 15
There is a 50% chance of rolling a prime number . If Vick rolls the number cube P (not purple) 1 1 3 75%
4 4
30 times, it is expected that he will roll a prime number 15 times
Answer the questions . Show your work . Answer the questions. Show your work. Round your answers to the nearest tenth of a
percent.
A spinner has 20 equal sections, numbered 1 through 20 . A grocery store randomly selects an item to be on sale each day
a . What is the probability that the spinner will land on a multiple of 3? Item # of Days on Sale
6 Ice Cream 4
P(multiple of 3) = 20 = 30% Oranges 5
3
There are 6 multiples of 3 between 1 and 20 (3, 6, 9, Chicken 5
Chips 4
12, 15, 18) Eggs
b . If the spinner is spun 42 times, how many times can it be expected to spin a
multiple of 3?
0 .3 42 = 12 .6; about 13 times a. What is the probability that the item on sale will be ice cream or chips?
c . What is the probability that it will not spin a multiple of 4? P(ice cream or chips) = 4+5 = 9 = 42.9%
21 21
There are 5 multiples of 4 (4,8,12,16,20) .
P(not multiples of 4) = 1 – 5 b. What is the probability that oranges or chicken will not be on sale?
= 75% 20
5 15 P(no oranges or chicken) = 1 – 5+3 = 1 – 8 = 61.9%
1– 20 = 20 21 21
Spectrum Critical Thinking for Math Lesson 7 .4 Spectrum Critical Thinking for Math Lesson 7.5
Probability Models Grade 7 Other Probability Models
Grade 7
89 90
Page 91 Page 92
Lesson 7.5 NAME ________________________________________________________________________________ Lesson 7.7 NAME ________________________________________________________________________________
Theoretical vs . Experimental Probability Understanding Compound Events
Theoretical probability is what is expected to happen based on likely outcomes . When two or more things are happening at one time in an experiment, it is a
Experimental probability is what actually happens . compound event . The probability of each event is multiplied .
Suppose you toss a coin 25 times, and it lands tails up 11 times . Compare the What is the probability of rolling a 2 and then a 6 when rolling a number cube
experimental probability and the theoretical probability .
twice? 1
6
P (2)
Theoretical probability: 1 50% P (6) 1
2 6
11 1 1 1
Experimental probability: 25 44% P (2, then 6) 6 6 36
The experimental probability is less than the theoretical probability . It is impossible Answer the questions . Show your work . Round your answers to the nearest tenth of a
to meet the experimental probability because there are an odd number of coin percent .
tosses .
Thomas spins a spinner 40 times . The results are shown in the table . Based on the A standard spinner is arranged so that the numbers 1 to 15 share equal space .
results of the experiment, use your best guess to draw the spinner .
a . What is the probability of getting a 9 on two consecutive spins?
Number Frequency P(two consecutive numbers) = 1 1 = 1 = 0 .4%
15 15 225
19
2 11 b . What is the probability of not getting a 9 on two consecutive spins?
3 12 P(not two consecutive numbers) = 1 – 1 = 224 = 99 .6%
225 225
48
P(1) = 9 = .225 .225(360°) = 81° What is the probability of rolling a 2 on a standard number cube and then
40 .275(360°) = 99°
11 .30(360°) = 108° getting heads on a coin toss?
40 .20(360°) = 72°
P(2) = = .275 P(2, heads) = 1 1 = 1 = 8 .3%
6 2 12
P(3) = 12 = .30
40 What is the probability of not rolling a 6 on a number cube and then getting
8 heads on a coin toss?
P(4) = 40 = .20
5 1 5
All of the experimental probabilities are close to 25% . This P(not 6, then heads) = 6 2 = 12 = 41 .7%
indicates that each section of the spinner is 1 of the area .
4
Spectrum Critical Thinking for Math Lesson 7 .6 Spectrum Critical Thinking for Math Lesson 7 .7
Grade 7 Understanding Compound Events
Grade 7 Theoretical vs . Experimental Probability
92
91
123
Answer Key
Page 93 Page 94
Lesson 7.7 NAME ________________________________________________________________________________ Lesson 7.8 NAME ________________________________________________________________________________
Understanding Compound Events Probability in the Real World
The Fundamental Counting Principle says that when there are m ways to do one Answer the questions . Show your work . Round your answers to the nearest tenth of a
thing, and n ways to do another, then the product of m and n is the possible percent .
number of outcomes for both events . A tree diagram can help you visualize this .
A retail store is having a contest . The randomly selected prize will be a can opener,
An ice cream shop offers vanilla, strawberry, and chocolate ice cream . A a gift card, or a set of towels . The store cashier will spin a spinner with the numbers
customer can choose a regular cone, a sugar cone, or a cup . What is the 5–8 to see whether every 5th, 6th, 7th, or 8th customer will win a prize .
probability of getting strawberry ice cream on a sugar cone?
a . Create a tree diagram to show all the possible outcomes in this situation .
There are 3 flavors and 3 serving options . Reg can opener can opener
3 3 3 9, so there are nine possible outcomes . V Sugar 5th gift card 7th gift card
Cup
There is one possible combination of Reg set of towels set of towels
strawberry ice cream and sugar cone .
S Sugar can opener can opener
Cup 6th gift card 8th gift card
P (strawberry sugar cone) 1 11%
9 Reg
C Sugar
Cup
set of towels set of towels
Answer the questions . Show your work . Round your answers to the nearest tenth of a b . What is the probability that every 5th person will win a can opener or a gift
percent . card?
A salad bar has croutons, raisins, sunflower seeds, and cranberries available P(5th, can opener or giftcard) = 2 = 16 .7%
12
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? 1 c . What is the probability that every 6th or 7th person will win a set of towels?
raisins 12
croutons sunflower seeds = 8 .3% 2
P(6th 7th, 12
cranberries or towels) = = 16 .7%
croutons
raisins sunflower seeds d . What is the probability that a customer will not win a can opener?
cranberries croutons
raisins
sunflower seeds P(not can opener) = 8 = 66 .7%
12
croutcornasnberries
cranberries raisins Lesson 7 .7 Spectrum Critical Thinking for Math Lesson 7 .8
sunflower seeds Understanding Compound Events Grade 7 Probability in the Real World
Spectrum Critical Thinking for Math
Grade 7 94
93
Page 95 Page 96
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Check What You Learned Check What You Learned
Probability Probability
Answer the questions . Show your work .
Answer the questions . Show your work . Round your answers to the nearest tenth of a
percent .
1 . An auto company conducted a survey with a random sample of 500 people to 3 . Of the original 56 signers of the Declaration of Independence, 4 represented
find out which type of vehicle they preferred to drive . The results are shown below .
North Carolina . If you selected 1 signer randomly, how likely is it that he
Favorite Vehicle Number of People represented North Carolina? 4
Compact 75 56
Sedan 45 CHAPTER 71 POSTTEST CHAPTER 7 POSTTEST = 7 .1%
SUV 95
Pickup 90 4 . Every seventh-grade student is eating in the cafeteria . Juwarne is a seventh-grade
95
Station Wagon 100 student . How likely is it that she is in the cafeteria?
Minivan
It is certain 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?
a . What is the probability that a randomly selected survey participant prefers to P(miss) = 1 – 15 = 5 = 25%
drive an SUV? Write it as a decimal . 20 20
95 = 0 .19 6 . At the barbershop, there are 2 chairs for customers to wait in . There is a rack with
500 5 magazines for customers to read while they wait . How many possible choices of
chairs and magazines do the barbershop customers have?
b . If 1,500 people were surveyed, how many would you expect to prefer to drive
an SUV? Explain your answer . 19% of 1,500 people is
285 people .
0 .19 1500 = 285 M1 M1
2 . Mr . Rose randomly selects names to see who will give the first book report . There M2 M2
are 10 boys and 14 girls in his class . What is the probability that he will select a 1st M3 2nd M3
girl’s name?
M4 M4
P(girl) = 14 = 58 .3%
24 M5 M5
Spectrum Critical Thinking for Math Chapter 7 There are 10 possibilities . Chapter 7
Grade 7 Check What You Learned Check What You Learned
Spectrum Critical Thinking for Math
95 Grade 7
96
124
Answer Key
Page 97 Page 98
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Final Test Chapters 1–7 Final Test Chapters 1–7
Answer the questions . Show your work . Answer the questions . Show your work .
1 . These temperature changes in a vat of liquid were noted by a scientist performing 3 . The chart shows the high and low temperature in Anchorage for a week .
a chemical experiment . What was the net temperature change from the first
Monday to the second Monday? Temperature in Anchorage (°F)
Monday 4 .6 °C Sun Mon Tues Wed Thurs Fri Sat
Tuesday 210 .2 °C
Wednesday 20 .3 °C High 3° 5° 26° 27° 2° 215° 1°
Thursday 223 .5 °C Low
Friday 28° 212° 221° 217° 215° 225° 218°
Saturday 4 .2 °C
Monday 214 .4 °C a . Find the average of the high temperatures . Round your answer to the nearest
226 .9 °C tenth of a degree .
–26 .9 – 4 .6 = –31 .5°C 3 + 5 + (–6) + (–7) + 2 + (–15) + 1 = –2 .4°F
7
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 . b . Find the average of the low temperatures . Round your answer to the nearest
In total, how much water will be gained or lost by the time Jason returns from tenth of a degree .
vacation?
(–8) + (–12) + (–21) + (–17) + (–15) + (–25) + (–18) = –16 .6°F
Day Water Lost Water Added 7
(in quarts) (in quarts)
Mon . 4 . The terms 8x, 5z, 15y, z, 2x and another term are added to form an expression .
Tue . 3 5 When simplified, this expression equals 2 (3z 5x) . Identify the missing term
Wed . 4 8 and write the expression .
1 7 CHAPTERS 1–7 FINAL TEST 2(3z + 5x) = 6z + 10x
2 8 5z + z + 15y + 8x + 2x = 6z + 15y + 10x
The missing term is –15y .
5 1
8 2
– 3 + 5 – 1 + 7 – 5 + 1 = 1 quarts CHAPTERS 1–7 FINAL TEST
4 8 2 8 8 2 8
Spectrum Critical Thinking for Math Chapters 1–7 Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test Grade 7 Final Test
97 98
Page 99 Page 100
NAME ________________________________________________________________________________ NAME ________________________________________________________________________________
Final Test Chapters 1–7 Final Test Chapters 1–7
Answer the questions . Show your work . Answer the questions . Show your work .
5 . The circumference of a circular garden is 42 meters . A gardener is digging a 8 . A college football stadium holds 25,000 fans . In a random sample of 30 fans, 26
straight line along the diameter of the garden at a rate of 10 meters per hour . were wearing the colors of the home team . Predict the number of fans who are
How many hours will it take the gardener to dig across the garden? Use 3 .14 for wearing the colors of the home team .
. Round your answer to the nearest hundredth . 26 = x 30x = 650000
30 25000 30 30
C = d rt = d
42 = 3 .14d 10t = 13 .38 30x = (26)(25000) x = 21666 .67
d =13 .38 m t = 1 .34 hours 30x = 650000
6 . Thomas spins a spinner 25 times . The results are shown in the table . Based on the Approximately 21,667 fans are predicted to be
results of the experiment and your best guess, how does the size of the section wearing colors of the home team .
containing #5 compare to the size of the section containing #6?
Number Frequency P(5) = 2 = 8% 9 . If it takes Joe 15 hours to make 3 cornhole boards, how long will it take him to
12 25 make 11 cornhole boards?
24 P(6) = 8 = 32% 15 = x 3x = 165
31 25 3 11 3 3
48 Section #6 is 4 times as 3x = 165 x = 55
52 large as Section #5 .
68 It will take 55 hours to make 11 cornhole boards .
7 . A number j is positive and another number k is negative . Based on this 10 . Jack bought 4 turkey sandwiches and 2 bags of apple slices for $22 .60 . If the
information, can you determine whether j k is positive or negative? Explain . apple slices cost $0 .75 per bag, how much did each sandwich cost?
No, it depends on whether the absolute value of j or the 4t + 2(0 .75) = 22 .6 t = 5 .275
absolute value of k is larger . If the absolute value of j is
larger, then the sum is positive . If the absolute value of k CHAPTERS 1–7 FINAL TEST CHAPTERS 1–7 FINAL TEST 4t + 1 .5 = 22 .6 Each sandwich is
is larger, then the sum is negative . about $5 .28 .
– 1 .5 –1 .5
4t = 21 .1
4 4
Spectrum Critical Thinking for Math Chapters 1–7 Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test Grade 7 Final Test
99 100
125
Answer Key
Page 101
NAME ________________________________________________________________________________
Final Test Chapters 1–7
Answer the questions. Show your work.
11. Identify the mistake that was made in simplifying the expression. Then, correctly
simplify the expression. 5(a – 3) + (6a + 12) – 7a =
5 (a 3) (6a 12) 7a 5a – 15 + 6a + 12 – 7a =
5a 2 6a 12 7a
(5a 6a 7a) (2 12) (5a + 6a – 7a) +
4a 10 (–15 + 12) =
4a – 3
The 5 was added to the –3 instead of multiplied
when the 5 was distributed.
12. On a road map, the distance between two cities is 12.6 centimeters. What is the
actual distance if the scale on the map is 2 cm:50 mi. How long would it take a
driver traveling 70 miles per hour to go from one city to the next city?
2 cm = 12.6 2x = 630
50 mi. x 2 2
2x = (50)(12.6) x = 315
2x = 630 The cities are 315 miles apart.
13. Several puppies from 2 different breeds were weighed. The puppies’ weights in
pounds are shown in the table. What can you infer from the data?
Breed A CHAPTERS 1–7 FINAL TEST
Breed B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Breed B is larger than Breed A. Breed B also has
a larger range of weights than Breed A.
Spectrum Critical Thinking for Math Chapters 1–7
Grade 7 Final Test
101
126
NAME _________________________________________________________________________________
Notes
NAME _________________________________________________________________________________
Notes
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