numerical expressions patterns & relationships · 5. the giant tortoise can move at speeds of...

Topic 8 Numerical expressions

patterns & Relationships

_________ Name

___________

Test Date

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Daily Common Core Review

8-1Name

1. Celia has listened to 5 __ 14 of the songs on her new CD. Which equation could Celia use to find s, the fraction of the songs she has left to listen to?

A 5 ___ 14

1 s 5 9 ___ 14

B 5 ___ 14

1 s 5 14 ___ 14

C 9 ___ 14

1 s 5 14 ___ 14

D 14 ___ 14

1 s 5 5 ___ 14

2. The table below shows how far each skater in a 1-mile race has traveled.

SkatingRace

Skater Ray Scott Theo Walt

Distance(miles) 2 __

5 3 ___

10 1 __

2 1 ___

10

Ifnone of the skaters pass each other, which skater will win the race?

A Ray

B Scott

C Theo

D Walt

3. A dolphin swims 26 miles per hour. How far could a dolphin swim in 9 hours?

A 125 miles

B 184 miles

C 234 miles

D 269 miles

4. Jack needs to finish 5 _ 8 of a school project by Monday. So far he has done 3 _ 8 of the project.

a. Draw a rectangle to show Jack’s project. Shade the fraction that Jack has finished so far.

b. What fraction of the project does Jack have left to do by Monday? Write your answer in simplest form.

5. The giant tortoise can move at speeds of up to 0.17 miles per hour. The top speed for a greyhound is 39.35 miles per hour. How much greater is the greyhound’s speed than the tortoise’s?

6. A school needs 75 light bulbs. Bulbs come in packages of 8. How many packages should the school order?

D 8•1


Practice

8-1Name

Variables and ExpressionsFor questions 1 through 4, write a numerical expression that represents the word phrase.

1. 180 apples divided into 12 baskets

2. 5 more than 3 times 8

3. three times the sum of 13 and 16

4. nine fewer than 72 divided by 3

For 5 through 7, translate each expression into words.

5. 3 20

6. 8 3 (12 2 6)

7. 40 2 2

8. Write two different word phrases for the expression (52 8) ______ 30 .

9. Do 5 3 and 3 5 represent the same expression? Explain.

10. Dan is 12 inches taller than Jay. Jay is 54 inches tall. Which expression shows Dan’s height?

A 54 12 B 54 2 12 C 12 3 54 D 54 ___ 12

11. Explain what the expression (5,903 1,250) 3 6 means.

P 8•1


Quick Check

8-1Name

1. Which shows the phrase “15 divided by 5” as a numerical expression?

A 15 1 5

B 5 3 15

C 5 4 15

D 15 4 5

2. A teacher needs 6 craft sticks for each student in a class for a project. Which numerical expression shows the total craft sticks needed if there are 36 students?

A 6 3 36

B 6 4 36

C 36 1 6

D 6 2 36

3. WritingtoExplain There are 6 people in Susanna’s family (an even number). They went to an amusement park for the day.

a. Admission was $15.00 per person. Write an expression to show the cost of admission for all of them.

b. They split into two equal groups while they were at the park. Write an expression to show the number of people in each group.

c. All of them but three went on the rollercoaster. Write an expression to show the number of people that went on the rollercoaster.

d. They saw another family that had 4 more people in their family than Susanna’s family. Write an expression to show the number of people in that family.

Q 8•1


8-2


Name

1. �Giancarlo�and�his�friends�are�getting�ready�for�a�costume�party.�They�buy�6�packages�of�fake�noses�and�8�packages�of�masks.�Each�package�contains�2�sets.�How�many�total�items�did�they�buy?�

A 28

B 22

C 20

D 16

2. �Which�answer�shows�how�you�can�use�the�Distributive�Property�to�find�the�product�of�16�3�34?

A (10�3�30)�1�(6�3�4)

B (10�3�34)�1�(16�3�4)

C (16�3�3)�1�(16�3�4)

D (16�3�30)�1�(16�3�4)

3. �Leslie�bikes�10�miles�each�day�for�100�days.�How�many�miles�did�she�bike�in�all?�Write�your�answer�in�standard�form�and�in�exponential�notation.

4. �Use�mental�math�to�find�the�product.

6073�20

5. �Use�the�Distributive�Property�to�fill�in�the�blanks.

411385(400�3� )1( 38)

5� �1�88�

5

D 8•2


Practice

8-2Name

Order of OperationsUse the order of operations to evaluate each expression.

1. 4 3 4 1 3 5 2. 3 1 6 3 2 4 3 5

3. 24 2 (8 4 2) 1 6 5 4. (15 2 11) 3 (25 4 5) 5

5. 26 2 4 3 5 1 2 5 6. 15 3 (7 2 7) 1 (5 3 2) 5

7. (8 4 4) 3 (7 3 0) 5 8. 5 3 (6 23) 1 10 4 (8 2 3) 5

9. Which is a true statement, 5 3 4 1 1 5 25 or 3 1 7 3 2 5 17? Explain your answer.

Insert parentheses to make each statement true.

10. 25 4 5 2 4 5 25

11. 7 3 4 2 4 4 2 5 26

12. 3 1 5 3 2 2 10 5 6

13. Insert parentheses in the expression 6 1 10 3 2 so that:

a. the expression equals 32.

b. the expression equals (12 1 1) 3 2.

14. Solve (25 2 7) 3 2 4 4 1 2.

A 18 B 11 C 6 D 5

15. Write two order-of-operation problems. Then trade with a classmate and solve the problems.

P 8•2


Quick Check

8-2NameName

1. What is the value of the expression? Use the correct order of operations.

43 5 1 9 2 2 3 (6 1 1)

A 15

B 69

C 119

D 378

2. What is the first operation that you will perform when you evaluate this expression?

121 (18 2 a) 3 3 4 24

A addition

B subtraction

C multiplication

D division

3. How would you insert parentheses to make the equation true?

23 4 1 6 2 2 1 8 4 4 5 18

A 2 3 4 1 (6 2 2) 1 8 4 4 5 18

B (2 3 4 1 6) 2 2 1 8 4 4 5 18

C 2 3 4 1 6 2 (2 1 8 4 4) 5 18

D 2 3 (4 1 6 2 2) 1 8 4 4 5 18

4. How can this expression be simplified? Use the correct order of operations.

31 2 3 a 1 b 3 4 4 2 2 4

A 2a 1 2b 2 1

B 2a 1 2b 2 2

C 2a 2 2b 1 3

D 5a 1 2b 2 4

5. WritingtoExplain Evaluate the expression below step by step, identifying the operations that you perform and the order in which you perform them.

181 5 1 (5 2 3) 3 6 4 3 2 2

Q 8•2



8-3Name

1. �For�the�following�expression,�which�operation�is�performed�first?�6�3�5�1�6�3�2�2�(8�–�4)

A Addition

B Division

C Multiplication

D Subtraction

2. �Ted�bought�a�CD�for�$14.95�and�a�ball�cap�for�$16.95.�Then�he�purchased�a�book�for�$9.95.�How�much�did�he�spend�in�all?

A $41.85

B $41.95

C $42.85

D $42.95

3. �What�is�the�estimated�sum�of�3,829�1�4,346�1�6,354,�rounded�to�the�nearest�hundred?

A 12,400

B 14,500

C 15,200

D 16,000

4. �What�is�the�sum�of�35.91�1�72.5?

5. �Evaluate�this�expression.13�1�26�2�[(2.8�3�5)�4�7]

A 27

B 30

C 35

D 37

6. Evaluate:� � [(0.3�3�8)�1�(1.5�3�3)]�1�36

� �

� 7.� Evaluate:40�4�[9.6�2�(8�3�0.2)]

� �

D 8•3


Practice

8-3Name

Evaluating Expressions 1. 252(3.13615.3) 2. 162[(4.233.5)29.5]

3. 92[(1224)30.6] 4. [(0.238)1(2.533)]125

5. 424[8.62(830.2)] 6. 2714.23840.2

7. 6.81[(0.537)1(3.133)] 8. 252[(36232.4)1(840.5)]14.5

9. 91[(4.223.3)1(6.440.8)]33 10. 41291(832.3)2151(2.134)

11.Keishaboughtanewpairofskisfor$450.Sheput$120downandgotastudentdiscountof$45.Hermothergaveher1_2ofthebalanceforherbirthday.WhichoftheseexpressionscouldbeusedtofindtheamountKeishastillowesontheskis?

A 450212014542 C 4502(120245)42

B [4502(120245)42] D [4502(120145)]42

12. (733.4)2[(2.835)2(4.332)]116.Givetheorderofoperationsastudentsolvingthisproblemwouldusetoevaluatetheexpression.Solve.

P 8•3


Quick Check

8-3Name

3. Whichexpressionhasavalueof66?

A (1643.2)3210.23(8164)

B [(1643.2)3(210.2)]38164

C 164[(3.232)1(0.238)]164

D [164(3.232)10.23(8164)]

4. WritingtoExplain Selasimplifiedtheexpression253[63 (2528)112]45andfoundtheanswertobe770.Rudisimplifiedthesameexpressionandfoundtheanswertobe570.Whichgirliscorrect?Explain.

Rudi: Sela: 253[63(2528)112]45 253[63(2528)112]45 253[63(17)112]45 253[15028112]45 253[102112]45 253[142112]45 253[114]45 25315445 25311445 25315445 2,85045 3,85045 570 770

1. Evaluatetheexpression[(1232.4)2(8.440.6)]12725.1.

A 9.84

B 14.16

C 36.7

D 49.3

2. Evaluatetheexpression42294[(2.437)1(0.632)].

A 41.5

B 38.875

C 3.8875

D 3.6667

Q 8•3


8-4


Name

1. �Ms.�Chang�drew�a�quadrilateral�on�the�board.�Which�word�best�describes�the�figure�Ms.�Chang�drew?

A Rectangle

B Square

C Rhombus

D Trapezoid

2. �What�digit�is�in�the�ten�thousands�place�in�the�number�34,278,915?

A 1

B 3

C 5

D 7

3. �While�doing�his�math�homework,�Ramon�used�4�×�6�=�24�to�solve�a�division�fact.�What�division�fact�could�Ramon�have�been�solving?

A 24�÷�4�=�6

B 12�÷�4�=�3

C 24�÷�3�=�8

D 18�÷�6�=�3�

4. �Annie�tutors�8�students�each�week.�She�charges�each�student�$72�a�week.�How�much�money,�in�dollars,�does�Annie�earn�each�week?

5. �What�fraction�is�modeled�by�the�shaded�circles�below?�Explain�how�you�wrote�the�fraction.

D 8•4


Practice

8-4Name

Addition and Subtraction ExpressionsFind a rule and write the missing number for each table.

1.

r 19 24 32 37

7 12 20

3.

s 10 15 25 30

5 10 25

5.

w 3 6 9 12

6 12 15

2.

a 6 9 12 15

40 46 49

4.

b 16 19 22 26

35 38 42

6.

n 51 42 33 24

40 31 13

7. Evaluate the expression 15 − n when n = 9.

8. Which expression stands for “32 more than a number d”?

A 32 × d

B 32 − d

C 32 + d

D 32 ÷ d

9. Explain what the variable represents in an addition or subtraction expression.

P 8•4


Quick Check

8-4Name

1.What is a rule for this table?

x 17 25 32 41? 11 19 26 35

A x − 6

B x + 6

C x − 7

D x + 7

2.What is a rule for this table?

p 7 15 19 23? 29 37 41 45

A p + 19

B p − 22

C p + 22

D p − 19

3.How many boys will participate in the track meet if there are 21 girls participating in the meet?

Numberofgirlsinthetrackmeet

12 15 18 21

Numberofboysinthetrackmeet

8 11 14

A 17

B 16

C 15

D 14

4. WritingtoExplain Juliet’s Book Shop is having a sale. What expression can Juliet write to figure out sale prices? Explain how you found the expression. Then, use the expression to find the sale prices when the regular prices are $30, $35, and $40.

Regularprice(p) $10 $15 $20 $25 $30 $35 $40

Saleprice $8 $13 $18 $23

Q 8•4



8-5Name

1. �In�standard�form,�what�is�four�hundred�two�thousand,�nine�hundred�eighty-one?

A 42,981

B 402,981

C 420,981

D 429,081

2. �The�parents�of�fourth-grade�students�at�Mulcahey�School�were�invited�to�attend�an�open�house.�The�parents�were�put�into�6�equal�groups.�Each�group�had�31�parents.�How�many�parents�attended�the�open�house?

A 25

B 37

C 180

D 186

3. �What�mixed�number�represents�the�shaded�parts�of�the�rectangles?

A 1��3�_�8��

B 1��1�_�8��

C ��5�_�8��

D ��3�_�8��

4. �There�are�14�students�in�Mrs.�McGovern’s�club,�and�only�half�of�the�students�returned�their�permission�slips�on�time.�How�many�students�returned�their�permission�slips�on�time?

5. Complete�the�following�pattern:

96��84��72��60� � �

�Write�a�rule�for�the�pattern,�and�explain�how�you�used�the�rule�to�complete�the�pattern.

D 8•5


Practice

8-5Name

Multiplication and Division ExpressionsFind a rule and write the missing number for each table.

1.

m 6 7 8 9

54 63 81

3.

z 24 18 9 0

8 3 0

5.

e 5 7 9 11

42 54 66

7.

s 3 8 10 16

60 160 320

2.

k 14 21 49 63

2 3 9

4.

q 2 3 4 5

14 21 28

6.

l 64 48 32 24

8 6 4

8.

d 30 25 15 5

5 3 1

9. Evaluate the expression 48 ÷ n when n = 6.

10. Which expression means “3 times a number h”?

A 3 × h B 3 − h C 3 + h D 3 ÷ h

11. How could you change Exercise 5 so that your rule uses the inverse operation?

P 8•5


Quick Check

8-5Name

1. What is a rule for the table?

s 35 40 45 50? 7 8 9 10

A s − 28

B s × 5

C s ÷ 5

D s ÷ 7

2. What is a rule for the table?

g 3 7 8 9? 18 42 48 54

A g + 15

B g ÷ 6

C g × 6

D g − 15

3. Use the table to find the score a student should receive for answering 19 questions correctly.

Numberofquestionsansweredcorrectly 3 16 18 19

Score 15 80 90

A 5

B 85

C 92

D 95

4. WritingtoExplain The Ramirez family’s favorite snack is yogurt. Mrs. Ramirez buys yogurt in different-sized cases.

Numberofcontainersinacase

10 20 35

Howmanydayspassbeforeitisalleaten

2 4 7

How many days would it take for a case of 25 yogurts to be eaten? Explain your reasoning.

Q 8•5


8-6


Name

1. Which answer shows the algebraic expression for the following phrase?

Six times a number plus seven

A 6 + n × 7

B 6 + 7n

C 6 × 7 + n

D 6n + 7

2. Evaluate the following expression.

(44 + 14) – 19

A 38

B 39

C 45

D 49

3. Evaluate the following expression for n = 14.

46 – 2n

A 17

B 18

C 20

D 22

4. Molly is writing a paper for her history class. To complete the paper, Molly must spend 2 hours on research and 1 hour writing each page.

Use the expression to evaluate the number of hours needed for 10 pages.

Use the expression to evaluate the number of hours needed for 19 pages.

5. Evaluate the following expressions for y = 22.

149 – 5y

248 + 3y

6y – 51

D 8•6


Practice

8-6Name

Patterns: Extending TablesIn 1–3, fill in the tables and find the relationship between the corresponding terms.

1. Rule A: Start with 40. Subtract 5. Rule B: Start with 63. Subtract 5.

Start First Term Second Term Third TermRule A 40 35Rule B 63 58

2. Rule C: Start with 64. Divide by 4. Rule D: Start with 128. Divide by 4.

Start First Term Second Term Third TermRule C 64Rule D 128

3. Rule G: Start with 3. Multiply by 2. Rule H: Start with 6. Multiply by 2.

Start First Term Second Term Third TermRule GRule H

In 4–7, find the relationship between the corresponding terms in each rule.

4. Rule A: Start with 47. Subtract 3.

Rule B: Start with 59. Subtract 3.

5. Rule A: Start with 9. Add 4.

Rule B: Start with 22. Add 4.

6. Rule A: Start with 6. Multiply by 2.

Rule B: Start with 18. Multiply by 2.

7. Rule A: Start with 64. Divide by 2.

Rule B: Start with 256. Divide by 2.

P 8•6


Quick Check

8-6Name

1. WhatistheruleusedforRuleB?

A Startat0.Add2.

B Startat2.Add6.

C Startat0.Add6.

D Startat0.Multiplyby6.

2. WhatistherelationshipbetweenthecorrespondingtermsinRulesAandB?

A RuleBisalways4greaterthanRuleA.

B RuleAisalways3timesasmuchasRuleB.

C RuleAisalways4greaterthanRuleB.

D RuleBisalways3timesasmuchasRuleA.

For1–2,usethetable.

Start FirstTerm SecondTerm ThirdTermRuleA 0 2 4 6RuleB 0 6 12 18

3. WritingtoExplain Sammyiscountingthenumberofwingsonthebeesinhisuncle’sbeekeepinghive.Sammymakesaruletohelphim:Startwith0.Add2foreachbee.Whatrulecouldheuseforcountingthelegs?Rememberthatbeeshave6 legs.Howarethecorrespondingtermsinthesesequencesrelated?

Q 8•6


8-7


Name

1. Which answer shows the operation that should be done first in the following expression?

15 (16 4) (5 2)

A addition

B subtraction

C multiplication

D division

2. Using the order of operations, which answer shows the correct value of the following expression?

22 (8 5) (12 3)

A 10

B 34

C 40

D 100

3. Which answer is correct? (45 16) 12

A 41

B 27

C 19

D 17

4. Pam’s suitcase can weigh up to 50 pounds. She is planning to pack 5 school books, each of which weighs 2 pounds. She will also pack 3 notebooks weighing 1 pound each, and 6 boxes of golf balls weighing 2 pounds each. Her clothing weighs a total of 16 pounds. Write and evaluate an expression to show how many pounds Pam has to pack.

WillPam’s suitcase be under the 50-pound weight limit?

5. Insert parentheses to make each statement true.

47 36 12 44

18 2 16 0

50 2 5 30

14 7 2 1

D 8•7


Practice

8-7Name

Problem Solving: Use Reasoning 1. Christinacollectsstamps.Shehas47stampsinall.Shehas

20stampsfromEurope.ThenumberofAfricanstampsis2timesthenumberofAsianstamps.Howmanystamps fromeachofthesethreecontinentsdoesshehave?Use reasoningtowritethisasanumericalexpression.

2. Writeaproblemthatcanbesolvedbyactingitoutandusingreasoning.

3. Apublicpoolopenedforthesummer.Atotalof246peoplecameswimmingoverthefirst3daysitwasopen.Onthefirstday,79cametoswim.Onthesecondday,104peopleswam.Howmanypeopleswamonthethirdday?

4. Marissaearned$480inthesummer.Ifsheearned$40aweek,howmanyweeksdidshework?

A 48 B 12 C 10 D 9

5. Howcouldyouusecubestohelpyouwriteanexpressionandmodeltheproblem?

P 8•7


Quick Check

8-7Name

1. Mike has 15 coins in his pocket. One third of the coins are dimes and the rest are nickels. How much money does he have? You can use reasoning to solve the problem.

A $0.50

B $0.75

C $1.00

D $1.50

2. Linda has 50 pages in her scrapbook. She used 8 pages for things about her family. Two-thirds of the remaining pages are about her friends, and the rest are blank. How many blank pages remain? You can use reasoning to solve the problem.

A 12

B 14

C 21

D 48

3. A car dealership has 64 vehicles on its lot. Of the total, 1 _ 4 are pickup trucks and there are twice as many cars as there are SUVs. How many of each type of vehicle are on the lot. You can use reasoning to solve the problem.

A 16 pickup trucks, 32 cars, 16 SUVs

B 24 pickup trucks, 32 cars, 16 SUVs

C 16 pickup trucks, 16 cars, 32 SUVs

D 22 pickup trucks, 28 cars, 14 SUVs

4. Writing toExplain There are 42 houses on Main Street. Three less than half of the houses are made of brick. Of the rest of the houses, 3 _ 4 are painted white and the rest painted other colors. Explain how you could use objects to act out this problem. Then use reasoning to solve the problem and fill in the table.

House NumberTotal 42BrickWhite

Other colors

Q 8•7

numerical expressions patterns & relationships · 5. the giant tortoise can move at speeds of...

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