6th grade math standard test ebook (mc graw hill)

ConsultantThomas L. Harrell

Mathematics Department ChairpersonG.W. Carver Senior High School

Montgomery, AL

GLENCOE MATHEMATICS

Grade 6

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United Statesof America. Except as permitted under the United States Copyright Act, no part of this book maybe reproduced in any form, electronic or mechanical, including photocopy, recording, or anyinformation storage or retrieval system, without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-870354-9 Mastering the California Content Standards Workbook, Grade 6

1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08 07 06 05

Test-Taking Tips

• Go to bed early the night before the test. You will think more clearly aftera good night’s rest.

• Read each problem carefully and think about ways to solve the problembefore you try to answer the question.

• Relax. Most people get nervous when taking a test. It’s natural. Just doyour best.

• Answer questions you are sure about first. If you do not know the answerto a question, skip it and go back to that question later.

• Think positively. Some problems may seem hard to you, but you may beable to figure out what to do if you read each question carefully.

• If no figure is provided, draw one. If one is furnished, mark it up to helpyou solve the problem.

• When you have finished each problem, reread it to make sure your answeris reasonable.

• Become familiar with a variety of formulas and when they should be used.

• Make sure that the number of the question on the answer sheet matches thenumber of the question on which you are working in your test booklet.

© Glencoe/McGraw-Hill Mastering the California ContentStandards Workbook, Grade 6iii

ContentsOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivStudent Recording Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vCalifornia Mathematics Content Standards, Grade 6 . . . . . . . . . . . . . . . vi

Test PracticeDiagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Grade 6 Standards PracticeNumber Sense. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Algebra and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Measurement and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Statistics, Data Analysis, and Probability . . . . . . . . . . . . . . . . . . . . . . . 61Mathematical Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Test PracticeSample Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

© Glencoe/McGraw-Hill Mastering the California ContentStandards Workbook, Grade 6iv

Overview

The material in this booklet is designed to help you meet the CaliforniaMathematics Content Standards for Grade 6.

It contains:

• a Student Recording Chart,

• the California Mathematics Content Standards for Grade 6,

• a Diagnostic Test,

• practice for each Content Standard, and

• a Sample Test.

How to Use This Workbook

Diagnostic Test This test will help you identify any weaknesses you mayhave as you prepare to meet the California Mathematics Content Standardsfor Grade 6. Once you’ve taken the test and it has been graded, complete theStudent Recording Chart that is found on page v. Mark an × in the square foreach question that you answered incorrectly.

Practice If you missed one or two of the questions for a particular contentstandard, you could probably use some extra practice with that standard. TheStudent Recording Chart lists practice pages for each standard. Complete theappropriate practice pages. If you are unsure about how to do some of theproblems, you may want to refer to your mathematics book.

Sample Test After you have completed the practice pages, take the SampleTest on pages 115 to 128.

© Glencoe/McGraw-Hill Mastering the California ContentStandards Workbook, Grade 6

v

Name:

Student Recording Chart

Directions Mark an × by each question from the Diagnostic Test that you answered incorrectly. Ifthere are one or two ×s marked for a standard, write Yes in the Need Practice? box. Then completethe practice pages for that standard.

Strand Number Sense

Standard 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4

Test Questions 1 � 46 � 2 � 47 � 49 � 50 � 19 � 11 � 53 � 10 � 12 � 55 �48 � 51 � 52 � 54 � 13 � 18 � 56 �

Need Practice?

Practice Pages 15-16 17-18 19-20 21-22 23-24 25-26 27-28 29-30

Strand Algebra and Functions

Standard 1.1 1.2 1.3 1.4 2.1

Test Questions 3 � 57 � 58 � 21 � 22 � 59 � 61 � 60 � 14 �

Need Practice?

Practice Pages 31-32 33-34 35-36 37-38 39-40

Strand Algebra and Functions

Standard 2.2 2.3 3.1 3.2

Test Questions 4 � 62 � 5 � 6 � 63 � 64 � 23 � 24 � 25 �

Need Practice?

Practice Pages 41-42 43-44 45-46 47-48

Strand Measurement and Geometry

Standard 1.1 1.2 1.3 2.1 2.2 2.3

Test Questions 15 � 65 � 66 � 67 � 26 � 7 � 27 � 28 � 8 � 29 � 30 �

Need Practice?

Practice Pages 49-50 51-52 53-54 55-56 57-58 59-60

Strand Statistics, Data Analysis, and Probability

Standard 1.1 1.2 2.1 2.2 2.4

Test Questions 68 � 31 � 32 � 33 � 34 � 35 � 36 �

Need Practice?

Practice Pages 61-62 63-64 69-70 71-72 75-76

Strand Statistics, Data Analysis, and Probability

Standard 2.5 3.1 3.2 3.3 3.4

Test Questions 37 � 9 � 69 � 16 � 38 �

Need Practice?

Practice Pages 77-78 79-80 81-82 83-84 85-86

Strand Mathematical Reasoning

Standard 1.1 1.2 1.3 2.3 2.6 2.7 3.1 3.2

Test Questions 39 � 40 � 41 � 42 � 43 � 70 � 44 � 45 �

Need Practice?

Practice Pages 89-90 91-92 93-94 99-100 105-106 107-108 109-110 111-112


vi

California Mathematics Content Standards for Grade 6Number Standard

NUMBER SENSE (NS)

1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Studentssolve problems involving fractions, ratios, proportions, and percentages:

1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on anumber line.

1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ��

ba

� a to b, a:b�.

1.3 Use proportions to solve problems (e.g., determine the value of N if �47

� � �2N1�, find the length of a side

of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems,understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned,and tips.

2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division:

2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explainwhy a particular operation was used for a given situation.

2.2 Explain the meaning of multiplication and division of positive fractions and perform the

calculations �e.g., �58

� � �1156� � �

58

� � �1165� � �

23

��.

2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concretesituations, that use positive and negative integers and combinations of these operations.

2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them tosolve problems with fractions (e.g., to find a common denominator to add two fractions or to find thereduced form for a fraction).

ALGEBRA AND FUNCTIONS (AF)

1.0 Students write verbal expressions and sentences as algebraic expressions and equations; theyevaluate algebraic expressions, solve simple linear equations, and graph and interpret their results:

1.1 Write and solve one-step linear equations in one variable.

1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables.

1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties toevaluate expressions; and justify each step in the process.

1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator.

2.0 Students analyze and use tables, graphs, and rules to solve problems involving rates andproportions:

2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).

2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.

2.3 Solve problems involving rates, average speed, distance, and time.

3.0 Students investigate geometric patterns and describe them algebraically:

3.1 Use variables in expressions describing geometric quantities (e.g., P � 2w � 2�, A � �12

� bh, C � �d—the

formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle,respectively).

3.2 Express in symbolic form simple relationships arising from geometry.

MEASUREMENT AND GEOMETRY (MG)

1.0 Students deepen their understanding of the measurement of plane and solid shapes and use thisunderstanding to solve problems:

� Key Standard * � standard assessed on the CAHSEE


vii

Number Standard

1.1 Understand the concept of a constant such as �; know the formulas for the circumference and area of acircle.

1.2 Know common estimates of � �3.14; �272�� and use these values to estimate and calculate the

circumference and the area of circles; compare with actual measurements.

1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base � height);compare these formulas and explain the similarity between them and the formula for the volume of arectangular solid.

2.0 Students identify and describe the properties of two-dimensional figures:

2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of theseterms.

2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle tosolve problems involving an unknown angle.

2.3 Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).

STATISTICS, DATA ANALYSIS, AND PROBABILITY (PS)

1.0 Students compute and analyze statistical measurements for data sets:

1.1* Compute the range, mean, median, and mode of data sets.

1.2 Understand how additional data added to data sets may affect these computations of measures of central tendency.

1.3 Understand how the inclusion or exclusion of outliers affects measures of central tendency.

1.4 Know why a specific measure of central tendency (mean, median, mode) provides the most usefulinformation in a given context.

2.0 Students use data samples of a population and describe the characteristics and limitations of thesamples:

2.1 Compare different samples of a population with the data from the entire population and identify a situationin which it makes sense to use a sample.

2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, randomsampling) and which method makes a sample more representative for a population.

2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced theconclusions reached.

2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased.

2.5* Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims.

3.0 Students determine theoretical and experimental probabilities and use these to make predictionsabout events:

3.1* Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, treediagrams) and express the theoretical probability of each outcome.

3.2 Use data to estimate the probability of future events (e.g., batting averages or number of accidents permile driven).

3.3* Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event,1 � P is the probability of an event not occurring.

3.4 Understand that the probability of either of two disjoint events occurring is the sum of the two individualprobabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities.

3.5* Understand the difference between independent and dependent events.

MATHEMATICAL REASONING (MR)

1.0 Students make decisions about how to approach problems:



viii

Number Standard

1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information,identifying missing information, sequencing and prioritizing information, and observing patterns.

1.2 Formulate and justify mathematical conjectures based on a general description of the mathematicalquestion or problem posed.

1.3 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex problems.

2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic andalgebraic techniques.

2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, andmodels, to explain mathematical reasoning.

2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms andclear language; support solutions with evidence in both verbal and symbolic work.

2.6 Indicate the relative advantages of exact and approximate solutions to problems and give answers to aspecified degree of accuracy.

2.7 Make precise calculations and check the validity of the results from the context of the problem.

3.0 Students move beyond a particular problem by generalizing to other situations:

3.1 Evaluate the reasonableness of the solution in the context of the original situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation bysolving similar problems.

3.3 Develop generalizations of the results obtained and the strategies sued and apply them in new problemsituations.


Read each question and choose the best answer. Then write theletter for the answer you have chosen in the blank at the rightof each question.

1 The table at the right gives the lowest temperature recorded in a recent year in each of four U.S. cities. Which list shows the temperatures in order fromleast to greatest? NS 1.1A �2, �8, 22, 26B �8, �2, 22, 26C 26, 22, �8, �2D 26, 22, �2, �8

2 In the figure at the right, �JKL is similar to �RQS. What is the length of

—QS? NS 1.3

F 1.6 in.G 2.5 in.H 6.4 in.J 10 in.

3 If 2b � 7 � 5, what is the value of b? AF 1.1A 6B 2C 1D �1

4 Eduardo is making cookies with a recipe that calls for 4 cups of flour.The recipe he is using makes 96 cookies. How many cookies does heget for each cup of flour used? AF 2.2F 22G 24H 48J 92

5 A 32-ounce jar of jelly costs $1.28. At this rate, what is the least that a24-ounce jar of jelly will cost? AF 2.3A $0.86B $0.96C $1.00D $1.70

5 in. 8 in.4 in.J

K

L R

Q

S

CityTemperature

(�F)

San Antonio, TX 22Albany, NY �8Reno, NV �2Fresno, CA 26

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1© Glencoe/McGraw-Hill Mastering the California Content

Standards Workbook, Grade 6

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Diagnostic Test


6 Madison is buying a jar of peanut butter and wants to buy the jar withthe lowest cost per ounce. Which jar should she buy? AF 2.3F $2 for 12 ouncesG $2.50 for 16 ouncesH $4 for 32 ouncesJ $5.50 for 64 ounces

7 If two angles are supplementary and one angle has a measure of 70°,what is the measure of the other angle? MG 2.2A 10°B 20°C 110°D 130°

8 What kind of triangle is shown at the right? MG 2.3F Equilateral triangleG Acute triangleH Obtuse triangleJ Right triangle

9 In how many ways can you line up 4 people from left to right for aphotograph? PS 3.1A 4B 10C 24D 256

10 300,000 � 4,000 � 800 � 76 �F 34,876 NS 2.3G 304,876H 340,876J 348,760

A

CB 35° 55°

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11 What is the first step in calculating the value of � ? NS 2.2

A Find the reciprocal of .

B Find the reciprocal of .

C Find 8 � 4.

D Find 11 � 9.

12 The record high temperature for California is 134°F and the record lowtemperature is �45°F. What is the difference between the record highand low temperatures? NS 2.3F 179°FG 99°FH 91°FJ 89°F

13 At 9:00 A.M., the temperature in Long Beach, California, was 63°F. Thetemperature changes measured during the day were �5°F, �2°F, �1°F,and �4°F. What was the final temperature reading? NS 2.3A 61°FB 65°FC 67°FD 75°F

14 The town in the United States with the lowest elevation is Calipatria,California, which is 184 feet below sea level. How many inches belowsea level is Calipatria? AF 2.1F 552 in.G 1,533 in.H 2,208 in.J 2,108 in.

15 What is the circumference of a circle that has a radius of 4 centimeters?A 2� cm MG 1.1B 4� cmC 8� cmD 16� cm

8�11

4�9

4�9

8�11

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16 Jason has 15 baseball cards that he wants to share with his friend. If 3 of the cards are of players on the Los Angeles Dodgers baseball teamand his friend picks a card at random, what is the probability that thefriend will receive a Los Angeles Dodger card? PS 3.3F 3%G 15%H 20%J 30%

17 Clarissa has a piece of fabric yard wide. She cuts off a strip yard

wide. How much fabric remains? NS 2.1

A yd B yd

C yd D yd

18 Find the product of 4,073 and 38. NS 2.3F 178.714G 44,803H 154,774J 338,059

19 Leroy spent hour talking on the phone on Monday. He talked on the

phone for hour on Wednesday and for 1 hours on Friday. How much

time had Leroy talked on the phone altogether? NS 2.1

A 1 h

B 1 h

C 2 h

D 2 h

20 Eric wants to buy a new mountain bike. If the bike he wants costs $510,how much would he have to save each week in order to have enoughmoney in 15 weeks? AF 1.1F $34 G $36H $38 J $42

3�4

1�2

3�4

1�2

1�4

3�4

1�2

3�4

1�2

3�8

4�0

1�8

5�8

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21 The Taylor family ordered fast food for lunch. They ordered 3 hamburgers each costing h dollars, 2 corn dogs each costing c dollars,and 5 orders of French fries each costing f dollars. Choose the correctexpression to show how much the Taylor family spent on lunch. AF 1.2A 3 � h � 2 � c � 5 � fB 3h � 2c � 5fC h � c � fD 9hcf

22 Arlene bought 2 cans of paint each costing p dollars, 3 rolls of wallpapereach costing w dollars, and 2 brushes each costing b dollars at thehardware store. Write an expression that shows this situation, andevaluate to find the amount Arlene spent if each can of paint cost $15.75,each roll of wallpaper cost $23.35, and each brush cost $4.82. AF 1.2F p � w � b; $43.92G 2p � w � b; $59.67H 2(p � w � b); $87.84J 2p � 3w � 2b; $111.19

23 What is the perimeter of the rectangle shown in the figure at the right? AF 3.1A x � 12B 12xC 24 � 2xD 12 � x

24 The formula for the volume of a rectangular prism is V � �wh, where � is the length of the prism, w is the width of the prism, and h is theheight of the prism. How can you solve this equation for h? AF 3.2F Divide V by �w.G Divide V by w.H Multiply V by w.J Multiply V by �w.

25 For the triangle shown at the right, which statement must be true about the length of side c? AF 3.2A c must be less than 9.B c must be greater than 13.C c must be greater than 5.D c must be less than 4.

c

49

12

x

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26 In the figure at the right, angle R is a right angle. What is true about angles S and T? MG 2.1F They are complementary.G They must also be right angles.H They are supplementary.J They are not measurable.

27 If two angles are complementary and the measure of one angle is 50°,what is the measure of the other angle? MG 2.2A 30°B 40°C 50°D 130°

28 If two angles of a triangle measure 25° and 55°, what is the measure ofthe third angle? MG 2.2F 10°G 20°H 100°J 110°

29 Which of the following is not a possible kind of triangle? MG 2.3A Scalene right triangleB Scalene isosceles triangleC Scalene acute triangleD Scalene obtuse triangle

30 Which of the following is an obtuse triangle? MG 2.3F G

H J

70�

40�

70�

60�

30�

60�

60�

60�

30�

40� 110�

R

TS

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Diagnostic Test


31 If another data value of 4 is added to the data set below, which valuewill change? PS 1.2

0, 1, 2, 4, 4, 5, 7, 8A MeanB MedianC ModeD Range

32 If a data value of 8 is added to the data set below, which measure willchange? PS 1.2

0, 1, 3, 6, 7, 8, 10, 11, 13, 14, 15F MeanG MedianH ModeJ Range

33 In a survey, 33% of a random sample of the seventh-grade students at aschool said they planned to take Algebra in the eighth grade. If therewere 258 students in the seventh grade at the time of the survey, what isa reasonable estimate of the number of students who will be takingAlgebra in the eighth grade the following year? PS 2.1A 300B 133C 85D 75

34 Radio stations conduct surveys to see what type of music people want tohear. What type of sample of the population is the most appropriate?F Random sample PS 2.2G Cluster sampleH Convenience sampleJ Stratified random sample

35 If the number of cars passing through an intersection between 10:00 A.M.and 11:00 A.M. is used to estimate the number of cars that pass throughthat same intersection in one day, this is considered to be an example ofwhat type of sample? PS 2.2A Random sample B Cluster sampleC Convenience sample D Stratified random sample

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36 Which sample below would provide a random sample of students in amiddle school? PS 2.4F Select a group of students eating in one section of the cafeteria.G Assign each student a number, write the numbers on small squares of

paper, put the squares in a bowl, shake the bowl, and select severalnumbers from the bowl.

H Select all students whose last names start with the letter M.J Select all students from one homeroom class.

37 William surveyed 50 people outside of the library and asked them howoften they use the library. From this, he concluded that people in theentire city use the library more that twice in one week. Why is hisconclusion invalid? PS 2.5A The question was not clear.B The sample population is not representative of the entire city

population.C Everyone in the city must be surveyed first to draw a conclusion.D His survey was not long enough.

38 If you select one card from a standard deck of 52 playing cards, what isthe probability of selecting an ace or a king? PS 3.4

F

G

H

J

39 What are the next three numbers in the sequence below? MR 1.1

1, 2, 4, 7, 11, 16, …A 22, 27, 33B 22, 24, 27C 22, 31, 39D 22, 29, 37

40 If you list 4 consecutive whole numbers, what is the greatest possiblenumber of multiples of 3 in your list? MR 1.2F 1 G 2H 3 J 4

1�4

2�13

1�8

1�13

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41 What is the total number of triangles in the figure at the right?A 4 MR 1.3B 6C 7D 8

42 The graph shows the averagetemperature for each month in San Diego, California. What is the best estimate for thetemperature in June? MR 2.3F 57°FG 65°FH 67°FJ 70°F

43 In the 2004 Presidential election, a total of 5,427,055 people inCalifornia voted for John Kerry. What is this value rounded to thenearest thousand? MR 2.6A 5,427,100B 5,427,000C 5,400,000D 5,000,000

44 It takes about 515 wooden 2-by-4’s to frame an average house. What is a reasonable estimate for the number of 2-by-4’s needed to frame 9 average houses? MR 3.1F about 2,000G about 3,000H about 4,000J about 5,000

45 Which is the best first step to use in solving the equation 5x � 10?A Add 5 to both sides of the equation. MR 3.2B Subtract 5 from both sides of the equation.C Multiply both sides of the equation by 5.D Divide both sides of the equation by 5.

80

60

40

20

0 J F M A M J J A S O N DMonth

Monthly Average Temperaturesfor San Diego, California

Temperature(�F)

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Diagnostic Test


46 On an extra-large sausage and pepperoni pizza, there are 52 pieces ofpepperoni and 39 pieces of sausage. What is the ratio of pepperonipieces to sausage pieces? NS 1.2F 4 : 5G 4 : 3H 5 : 4J 3 : 4

47 If 10 nautical miles are equivalent to 11.5 statute miles, how manynautical miles are in 230 statute miles? NS 1.3A 265B 231C 229D 200

48 If �ABC is similar to �DEF, find

the length of B�C�. NS 1.3F 4G 5H 8J 10

49 The United States has about 2,000 miles of Pacific coastline, excludingAlaska. If California’s coastline makes up 42% of these miles, abouthow long is California’s coastline? NS 1.4A 21 miB 84 miC 210 miD 840 mi

50 Kelly earns 2% commission on the sales she makes working at a shoestore. If her sales for the day total $925, how much commission didKelly earn for the day? NS 1.4F $2.00G $18.50H $185.00J $462.50

A C D F

EB

6

10 5

3 4

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Diagnostic Test


51 Rex bought a shirt that cost $24. If the sales tax rate is 8%, what wasthe total he paid for the shirt? NS 1.4A $1.92B $24.08C $24.80D $25.92

52 In 2004, $1 in U.S. currency bought 1 Australian dollars. How many

Australian dollars would $4 in U.S. currency buy in 2004? NS 2.1

F 4

G 4

H 5

J 6

53 What is an equivalent way to find � 182? NS 2.2

A Divide 182 by 2.B Multiply 182 by 2.C Add 182 and 2.D Subtract 2 from 182.

54 Which expression is equivalent to � ? NS 2.2

F �

G �

H �

J �

55 What is the least common denominator of and ? NS 2.4

A 20B 27C 30D 60

4�15

7�12

4�3

8�5

3�4

8�5

4�3

5�8

3�4

5�8

3�4

5�8

1�2

1�3

1�3

2�3

1�3

1�3

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Diagnostic Test


56 Jennifer has 54 stickers and 78 pieces of candy. She wants to divide allof the stickers and all of the candy evenly among her friends. What isthe greatest number of friends she can give an equal number of stickersand candy to without having any left over? NS 2.4F 2G 3H 6J 12

57 What value of m will make the equation 12.4 � 32 � m true? AF 1.1A 9.2B 15.6C 19.6D 44.4

58 Martin is 68 inches tall. He is 4 inches shorter than his brother. Whichequation can be used to find the height of Martin’s brother? AF 1.1F b � 4 � 68G 68 � b � 4H 68 � b � 4J 68 � b � 4

59 Which of the following shows an expression that can be obtained from 3 • (4m • 9n) by using the Commutative Property of Multiplication?A 3 • (9n • 4m) AF 1.3B 3 • 4m � 3 • 9nC (3 • 4m) • 9nD 3 � 4m • 9n

60 What is the value of (3 � 2) � (4 � (�1))? AF 1.4F �1G 0H 2J 8

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Diagnostic Test


61 What is the value of (3 � 2)4 � 2(28 � 22)? AF 1.3A �372B 5C 126D 611

62 Angela typed 24 words in 30 seconds. What is her rate in words perminute (wpm)? AF 2.2F 12 wpmG 48 wpmH 54 wpmJ 60 wpm

63 Andrew biked 39 miles in 3.25 hours. What was his average speed?A 12 mi/h AF 2.3B 14 mi/hC 16 mi/hD 18 mi/h

64 A 200-square-foot yard requires 10 pounds of fertilizer. At this rate, howmuch fertilizer will a 500-square-foot yard need? AF 2.3F 20 lbG 22 lbH 25 lbJ 30 lb

65 If C represents the circumference of a circle, r represents the radius ofthe circle, and d represents the diameter of the circle, which ratio belowrepresents the value of �? MG 1.1

A

B

C

D r�C

d�C

C�r

C�d

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62

63

64

65



Go on

Diagnostic Test


66 Which statement is true about the value of �? MG 1.2F � � 3.14G � � 3.14

H � �

J � � 1.73

67 A cylindrical roll of paper towels has a diameter of 6 inches anda height of 11 inches. What is the approximate surface area of the plastic that covers the outside of the roll? (Use � � 3.14.)A 207.24 in2 MG 1.3B 235.5 in2

C 263.76 in2

D 320.28 in2

68 The table at the right gives the number of hours Mika watched television on the weekdays of one week. What is the mode for the times that Mika spent watching television? PS 1.1F 0.5 hG 1 hH 1.5 hJ 2 h

69 A soft drink company is having a giveaway contest in which one can ineach 12-pack purchased is a winner. If two cans have been used from a12-pack and were not winners, what is the probability that the third canis the winner? PS 3.2

A

B

C

D

70 Melissa bought 12 pens that cost $0.89 each. What is the total cost ofthe pens? MR 2.7F $10.68 plus tax G $1.68 plus taxH $1.60 plus tax J $1.55 plus tax

3�10

1�4

1�10

1�12

DayTime

(hours)

Mon. 3.5Tue. 1Wed. 2Thu. 0.5Fri. 0.5

22�7

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67

68

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Diagnostic Test

STOP

Name: Date:



Standards PracticeNumber Sense 1.1

Examples

NS 1.1Compare and order positive and negative fractions, decimals, and mixednumbers and place them on a number line.

1 Which of the following shows the decimals 0.5, 0.7, 1.0, and �0.2 in orderfrom least to greatest?A 0.5, 0.7, 1.0, �0.2B 1.0, 0.7, 0.5, �0.2C �0.2, 0.5, 0.7, 1.0D �0.2, 1.0, 0.7, 0.5

Draw a number line and graph each decimal on the number line.

On a number line, numbers on the left are less than numbers on the right. Sothe decimals graphed increase in value from left to right. From least togreatest, the decimals are �0.2, 0.5, 0.7, 1.0. C

2 Ken finished the bicycle race with a time of 32.3075 minutes, Hark finishedwith a time of 32.3705 minutes, and Joaquin finished with a time of 32.3507seconds. In what order did they finish the race?F Ken, Hark, JoaquinG Ken, Joaquin, HarkH Hark, Joaquin, KenJ Joaquin, Hark, Ken

Since the digits are the same in the tens, ones, and tenths place, look at thedigits in the hundredths place. For Ken’s time, the digit in the hundredthsplace is 0. The digit is 7 in Hark’s time, and 5 in Joaquin’s time. Since theperson with the least time finished first, the order is 32.3075, 32.3507, and32.3705, or Ken, Joaquin, Hark. G

3 The junior high band uniform hats come in three sizes, 6 , 7 and 6 , but

they are labeled small, medium, and large. Which shows the sizes of the hatsin increasing order of size?

A 7 , 6 , 6 B 7 , 6 , 6

C 6 , 6 , 7 D 6 , 6 , 7

Rewrite the mixed numbers as the improper fractions, , , and . Then,

rewrite the fractions using their LCD, 8. So, � � �584�, and � � .

When you compare these fractions, � � . So, 6 � 6 � 7 . D1�4

7�8

3�4

58�8

55�8

54�8

58�8

2�2

29�4

2�2

27�4

55�8

29�4

27�4

1�4

7�8

3�4

1�4

3�4

7�8

7�8

3�4

1�4

3�4

7�8

1�4

7�8

1�4

3�4

–0.2 0 0.2 0.4 0.6 1.0

–0.2

0.8

0.5 0.7 1.0


1 Which point on the number line has a

value of �2 ?

A WB XC YD Z

2 Tony’s batting average is 0.251, Gwen’s is 0.384, Manuel’s is 0.305, andJuana’s is 0.342. Which shows their batting averages in decreasing order?F 0.251, 0.305, 0.342, 0.384 G 0.342, 0.384, 0.305, 0.251H 0.384, 0.342, 0.305, 0.251 J 0.251, 0.342, 0.384, 0.305

3 On a recent visit to a pecan orchard, Richard picked up 6 pounds of

pecans, Delia picked up 6 pounds, Amaya picked up 6 , and Noel

picked up 6 pounds. Who picked up the least amount of pecans?

A Richard B DeliaC Amaya D Noel

4 The table at the right gives the low temperatures for four Montana cities one winter morning. Which city had the warmest low temperature?F BillingsG Great FallsH HelenaJ Missoula

5 Felecia is changing the tire on her bicycle. She needs to find a wrench to

take off the wheel nut. She tried a -inch wrench and it was too big for

the nut. Then she tried a -inch wrench and it was too small for the nut.

Which of the following wrenches should Felecia try to remove the nut?

A in. B in.

C in. D in.13�16

3�8

7�16

5�8

1�2

3�4

City Temperature (°F)Billings �12

Great Falls �18Helena �14

Missoula �17

7�16

3�4

4�5

2�3

1�3 –3 –2 –1 0 1 2 3

W X Y Z

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Interpret and use ratios in different contexts (e.g., batting averages, milesper hour) to show the relative sizes of two quantities, using appropriate notations ( , a to b, a : b).

1 In Ms. Ling’s math class, 7 out of every 8 students turn in their homework ontime. What is the ratio of students who turn in their homework on time tostudents in Ms. Ling’s math class?

A

B 8 : 7C 7(8)

D

A ratio is a comparison of two numbers by division. In this case, the twonumbers to compare are 7 and 8. This ratio could be written as 7 to 8, 7:8,

or . D

2 In the city pet show, 21 out of the 28 birds shown were parakeets. Which ofthe following shows this ratio?F 1 : 8G 3 : 4H 4 : 3J 8 : 1

The ratio of parakeets to the total number of birds is 21 : 28. This ratio can be

written as the fraction . However, is not in simplest form.

�

� The ratio in simplest form is or 3 : 4. G

3 Last Saturday, Aimee rode her bike 12 miles in 2 hours for an average speedof 6 miles per hour. Which expression is used to compute her average speed?A 2 � 12B 2 � 12

C

D 2(12)

Aimee’s speed is a rate, which is a ratio of two measurements with differentunits. The units in this rate are miles and hours. To find Aimee’s speed, dividethe number of miles she traveled by the number of hours it took her to travelthat distance.

� 6 mi/h C12 miles�2 hours

12�2

3�4

21�28

3�4

3 � 7�4 � 7

21�28

21�28

21�28

7�8

7�8

8�7

a�b

Name: Date:




Examples

NS 1.2


1 In the school talent show, 16 out of the 24 participants were girls. Whichof the following shows this ratio?A 3 to 2B 3 to 4C 2 to 3D 4 to 3

2 In one night, a pizza delivery driver delivered 18 pepperoni pizzas, 15 cheese pizzas, and 12 sausage pizzas. What is the ratio of pizzas withmeat to pizzas without meat delivered by the driver?F 2 : 1G 1 : 2H 2 : 3J 3 : 2

3 In an average week, Josefina spends 630 minutes reading books and 210 minutes watching television. What is this ratio in simplest form?

A B

C D

4 During her softball season, Cheryl Jo made 22 hits in 84 times at bat, fora batting average of 0.262. Which expression is used to compute herbatting average?F 22 � 10G 22 � 84H 84 � 22

J

5 In five hours, the Goldberg family traveled 325 miles toward theirvacation destination before they had to stop to refuel their car. Which ofthe following expressions would you use to find their average speed?

A B

C D 325 � 5325�

5

1�65

5�325

22�84

6�1

4�1

3�1

1�3

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Use proportions to solve problems (e.g., determine the value of N if � ,

find the length of a side of a polygon similar to a known polygon). Usecross-multiplication as a method for solving such problems, understandingit as multiplication of both sides of an equation by a multiplicative inverse.

1 Which equation could be the first step in solving the proportion � ?A 18 � 9 � r � 3 B 18 � 3 � r � 9C 9 � 3 � r � 18 D 3r � 162

A proportion is an equation that shows that two ratios are equivalent. In aproportion, the cross products are equal. To solve a proportion, find the crossproducts and then solve for the missing term. The first step in solving the

proportion � is to find the cross products, 18 � 3 � r � 9. B

2 In a recent survey at Mendez Middle School, 42 out of 60 students surveyedsaid that basketball was their favorite sport. If there are 850 students at theschool, how many would be expected to say that basketball was their favoritesport?F 60 G 102H 595 J 748

Assume that the ratio of students surveyed for whom basketball is their favoritesport to the number of students surveyed is equal to the number of students forwhom basketball is their favorite sport in the entire school to the total number

of students in the school, or � . The variable b represents the number

of students in the entire school for whom basketball is their favorite sport.Solve for b by finding the cross products.

�

60b � 42 � 850b � 595 H

3 In the figures at the right, �ABC is similar to �DEF. What is the length of E�F�?A 12 units B 10 unitsC 6 units D 5 units

When geometric figures are similar, their corresponding side lengths areproportional. Use the known side lengths to set up and solve a proportion forthe unknown side length.

�

12x � 60 Find cross products.x � 5 units Solve for x. D

10�x

12�6

12 10

A C

B

6

D F

E

b�850

42�60

b�850

42�60

18�9

r�3

18�9

r�3

N�21

4�7

Name: Date:




Examples

NS 1.3


1 At Barton Junior High, there is one teacher for every 24 students. If thereare 768 students at the school, how many teachers are there?A 32 teachersB 24 teachersC 18 teachersD 12 teachers

2 Gregg has a chocolate chip cookie recipe that makes 64 cookies and callsfor 4 cups of flour. If he only wants to make 48 cookies, how many cupsof flour would he need?F 1 cG 2 cH 5 cJ 3 c

3 Rectangle EFGH is similar to rectangle WXYZ. What is the length of side Y�Z�?A 28 unitsB 21 unitsC 12 unitsD 10 units

4 An architect built a scale model of the new Jonestown Public Library. Thelength of the model building was 36 inches and the width was 20 inches.If the actual library is going to be 126 feet long, how wide will it be?F 226 ftG 70 ftH 48 ftJ 20 ft

5 To make orange juice from concentrate, you mix water with the concentratein the ratio of 3:2. If you want to make orange juice from 6 quarts ofconcentrate, how many quarts of water do you need?A 36 B 18C 12 D 9

3E H

F G

28

7

W Z

X Y

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2

3

4

5




Calculate given percentages of quantities and solve problems involvingdiscounts at sales, interest earned, and tips.

1 At Mr. Ho’s surf shop, all surfboards are on sale for 20% off. How muchmoney will Biff save if he wants to buy a surfboard that costs $540 beforethe discount?A $54B $108C $432D $648

To find out how much Biff will save, find the amount of the discount, or20% of $540.540(0.20) � 108Biff will save $108. B

2 Mr. Ho is required to charge sales tax on every item he sells in his shop. If

the sales tax is 8 %, how much will Sharon’s $411 surfboard cost with tax?

F $34.94G $419.50H $440.50J $445.94

Find the amount of the tax, t, which is 8 % of $411.To find t multiply $411 by 0.085.t � $34.94. Then, add $34.94 to $411 to find the total cost, $445.94. J

3 After Biff considered buying the surfboard, he decided to wait 6 months andput the $432 he was going to spend on the surfboard into a savings account.If the bank pays an annual interest rate of 6% on Biff’s savings account, howmuch interest would he earn in 6 months?A $12.96B $25.92C $36.72D $51.84

To find simple interest, use the formula I � prt where I is the interest, p isthe principal, or the amount of money invested or borrowed, r is the annualinterest rate, and t is the time in years. The principal is $432, the rate is 6%,

and the time, 6 months or year.

I � prt

� ($432)(0.06)� �� $12.96

Biff earned $12.96 interest on his $432 in 6 months. A

1�2

1�2

1�2

1�2

Name: Date:




Examples

NS 1.4


1 Jeremy is buying a jacket that costs $65. The jacket is on sale for 30%off. What is the cost of the jacket before sales tax is added?A $19.50B $30C $35D $45.50

2 Kelli and her friends ate salads for lunch. The total bill was $22. If theywant to leave a 15% tip for the waiter, how much tip should they leave onthe table?F $2.20G $3.30H $4.40J $5.50

3 Two years ago, Tyrone received a gift of $200 from his grandmother forhis birthday. He put the money in a savings account that earns 7% interest.Tyrone now wants to use the money to buy a new bicycle. How muchmoney does he have?A $228B $114C $28D $14

4 Toni is selling candy bars as part of her school band fund-raiser. Theband gets 60% of the money from the sales. The rest goes to pay for thecandy bars. If Toni sells candy bars worth $75, how much money doesthe band get from her sales?F $80 G $60H $45 J $30

5 Devon borrowed $900 from his older brother to buy a new computer. Heagreed to pay back his brother $1,044 in two years. What interest rate isDevon’s brother charging him?A 4.4% B 8%C 10% D 16%

Name: Date:

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2

3

4

5




Solve problems involving addition, subtraction, multiplication, and divisionof positive fractions and explain why a particular operation was used for agiven situation.

1 Charlie mows lawns after school and on weekends to earn extra money.

Charlie’s mower holds gallon of fuel and his string trimmer holds

gallon of fuel. How much fuel must Charlie have to fill both his mower

and string trimmer?

A gal B gal

C gal D gal

To add and , you must rename with a common denominator.

� � � The LCD is 20.

�

Since is in simplest form, the answer is gallon. D

2 If the local hike-and-bike trail is of a mile long, how far did Susanna run if

she ran around the trail 2 times?

F 2 mi G 1 mi

H 1 mi J mi

First write the mixed number 2 as the improper fraction . Then multiply.

� 2 � �

�

� The GCF of 10 and 6 is 2.

� 1

Susanna ran 1 miles. G2�3

2�3

5�3

10�6

5�2

2�3

1�2

2�3

5�2

1�2

3�4

1�2

2�3

1�2

2�3

19�20

19�20

19�20

4�20

15�20

1�5

3�4

1�5

3�4

19�20

7�8

3�4

1�2

1�5

3�4

Name: Date:




Examples

NS 2.1


1 If Monica is 62 inches tall and her mother is 59 inches tall, how much

taller is Monica than her mother?

A 3 B 2

C 2 D 1

2 Shay’s little sister was 1 inches too short to ride the largest roller coaster

at the amusement park. If she is 46 inches tall, how tall must a rider be

in order to ride the roller coaster?

F 48 in. G 48 in.

H 47 in. J 47 in.

3 A radio station plays music about of the time. Using this ratio, about

how many hours of music would you expect the station to play in

6 hours time?

A 8 h B 7 h

C 5 h D 4 h

4 Find � .

F G

H J 6

5 On a recent field trip, of the students in 6th grade at Darwin Middle

School were able to attend an art museum. If 72 students went on thefield trip, what is the total number of 6th graders at the school?

A 108 B 90

C 60 D 57 3�5

2�5

4�5

3�2

2�3

3�32

1�4

3�8

7�8

3�4

1�4

2�3

1�2

3�4

1�3

3�4

1�6

1�2

2�3

2�3

5�12

7�12

1�3

3�4

1�3

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Explain the meaning of multiplication and division of positive fractions and

perform the calculations �e.g., � � � � �.1 Find the product � .

A

B

C

D

To multiply fractions, multiply the numerators and then multiply thedenominators.

� �

�

� Divide the numerator and denominator by their GCF, 2. C

2 Find the quotient � .

F

G

H

J

� � � To divide by a fraction, multiply by its reciprocal.

� F

3 Which statement is equivalent to � ?

A �

B �

C �

D �

Dividing by a fraction is the same as multiplying by its reciprocal. In this case,

is the reciprocal of , so � is the same as � . D4�5

11�13

5�4

11�13

5�4

4�5

5�4

11�13

4�5

11�13

4�5

13�11

4�5

13�11

4�5

11�13

14�15

7�3

2�5

3�7

2�5

6�35

2�3

3�4

14�15

3�7

2�5

1�2

2�4

3 � 2�4 � 3

2�3

3�4

6�7

1�2

3�8

3�12

2�3

3�4

2�3

16�15

5�8

5�16

5�8

Name: Date:




Examples

NS 2.2


1 Find the product � .

A

B

C

D


F

G

H

J

3 Which statement is equivalent to � ?

A �

B �

C 5 �

D 5 �

4 Which statement describes how to multiply two fractions?F Find the least common denominator and then multiply numerators.G Multiply the first fraction by the reciprocal of the second fraction.H Divide numerators and divide denominators.J Multiply numerators and multiply denominators.


A B

C 3 D 24

3�16

3�64

1�8

3�8

7�5

5�7

7�5

1�5

5�7

1�5

5�7

1�5

1�3

10�27

4�5

5�6

2�3

5�9

1�5

5�24

5�12

20�6

1�4

5�6

Name: Date:

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2

3

4

5




Solve addition, subtraction, multiplication, and division problems, includingthose arising in concrete situations, that use positive and negative integersand combinations of these operations.

1 The lowest point in Death Valley has an elevation of �282 feet. Towne Pass,on one of the highways out of Death Valley, has an elevation of 4,956 feet.What is the difference in elevation between Death Valley and Towne Pass?A 5,238 ftB 4,674 ftC 4,392 ftD 4,110 ft

To find a difference, you subtract. The subtraction expression is 4,956 � (�282). Subtraction is the same as adding the opposite. Theopposite of �282 is 282, so 4,956 � (�282) is the same as 4,956 � 282.Therefore, the difference is 4,956 � 282, or 5,238 ft. A

2 Three friends are driving from Los Angeles to San Francisco. The distancebetween the two cities is 384 miles. If the friends want to split the drivingequally, how far would each of them drive?F 64 miG 96 miH 128 miJ 192 mi

The three friends want to divide the driving task into equal parts. So to findout how far each friend will drive, divide the total distance, 384 miles, by 3.Since 384 � 3 � 128, each friend will drive 128 miles. H

3 Today is Mrs. Alguno’s birthday. She told her students that she is exactly threetimes their age. If the students are all 13 years old, how old is Mrs. Alguno?A 36B 39C 42D 45

Multiply the students’ age, 13, by 3. So, 13 � 3 � 39. Mrs. Alguno is 39 yearsold today. B

Name: Date:




Examples

NS 2.3


1 In 1997, about 1.2 million people lived in San Diego. If Los Angeles hadabout 3 times as many people as San Diego, about how many peoplelived in Los Angeles in 1997?A 3.6 millionB 3 millionC 0.4 millionD 0.3 million

2 The highest point in the Continental United States is Mt. Whitney,California, at 14,494 feet. The highest point in the entire United States is Mt. McKinley in Alaska, at 20,320 feet. How much higher isMt. McKinley than Mt. Whitney?F 34,814 ftG 6,037 ftH 5,826 ftJ �5,286 ft

3 The record high temperature for California is 134°F. The record lowtemperature is �45°F. What is the difference between the record high andrecord low temperatures in California?A 89°FB 91°FC 109°FD 179°F

4 The average width of the state of California is about 250 miles. Theaverage length of the state is about 750 miles. This makes Californiaabout how many times longer than it is wide?F 5 timesG 4 timesH 3 timesJ 2 times

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Determine the least common multiple and greatest common divisor of wholenumbers; use them to solve problems with fractions. (e.g., find a commondenominator to add two fractions or to find the reduced form for a fraction).

1 What is the sum of and ?

A

B

C

D

Find the least common denominator, which is the least common multiple of 3 and 4. The LCD is 12. Rewrite each fraction using the LCD and add therewritten fractions.

� � �

� D

2 Write the fraction in its simplest form.

F

G

H

J

To simplify a fraction, find the greatest common factor (GCF) of the numeratorand denominator. Then divide both the numerator and denominator by the GCF.

� The GCF of 8 and 20 is 4.

� G

3 Find the difference � .

A B

C D

� � � Rewrite using 24 as the denominator.

� Subtract.

� Simplify. B1�8

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5�6

20�24

23�24

5�6

23�24

19�24

1�4

1�8

1�12

5�6

23�24

2�5

8 � 4�20 � 4

8�20

1�3

16�40

2�5

4�10

8�20

11�12

3�12

8�12

1�4

2�3

11�12

3�4

7�12

1�2

1�4

2�3

Name: Date:




Examples

NS 2.4


1 Write in its simplest form.

A

B

C

D

2 What is the least common denominator of and ?

F 20G 40H 60J 80

3 Find � .

A

B

C

D

4 Find � .

F

G

H

J 1

5 Find � .

A B

C D 29�30

25�32

17�24

3�20

7�12

11�15

7�9

13�18

6�27

9�27

2�3

6�7

19�28

5�7

5�28

1�4

3�7

7�10

3�8

1�4

1�2

9�16

3�4

18�32

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Write and solve one-step linear equations in one variable.

1 Chandra is going to launch her model rockets this weekend in the local park.She has three model rockets and she wants to launch each one of them fivetimes. Each launch requires one rocket motor. Which is an equation that can beused to find m, the number of rocket motors Chandra needs for her launches?A m � 3 � 5B 3m � 5C 5m � 3

D m �

When you use verbal phrases to write algebraic equations, words or phrasesoften suggest the operation of the equation. The variable of the equation is m,the number of rocket motors needed. The key to this equation is that she isgoing to launch three rockets, five times each. The word times suggestsmultiplication. So the correct equation is m � 3 � 5. A

2 Which value of d makes the equation d � 9 true?

F 15G 12H 10J 8

To find the value of d that makes the equation true, solve for d. Multiply both

sides of the equation by to isolate d.

d � � 9 �

d � 9 � or 12 G

3 Gina bought a six-pack of bottled water for $3. Write and solve an equationfor b, the price of an individual bottle of water.A b � $2.00B b � $1.50C b � $1.00D b � $0.50

Since there are 6 bottles of water in a six-pack, 6 times the cost of one bottle,b, is $3. Written as an equation, this sentence is 6b � $3. Now solve for b.

6b � $3

b � or $0.50 D$3�6

4�3

4�3

4�3

3�4

4�3

3�4

3�5

Name: Date:



Standards PracticeAlgebra and Functions 1.1

Examples

AF 1.1


1 Which value of c makes the equation 14c � 84 true?A 5B 6C 7D 8

2 Julio is in charge of buying miniature California flags for his fellow bandmembers to place in their hats for the Tournament of Roses parade. If hebuys 1 flag for each of the 120 band members and spends $150, which isan equation that could be used to find f, the cost of one flag?F 120f � 150G 150f � 120H 120 � f � 150J 150 � f � 120

3 Sung wants to buy a pair of $90 in-line skates that are marked down by20%. Which is an equation that can be used to find the number d suchthat 20% of $90 is d?

A � d

B �

C �

D 90d � 20

4 Yolanda and Jessica are hiking in Sequoia National Park. If they hike for

9 hours and take breaks every 1 hours, how many breaks will they take?

F 9G 8H 6J 4

1�2

20�100

d�90

20�90

d�100

90�20

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4




Write and evaluate an algebraic expression for a given situation, using up tothree variables.

1 At a baseball game, Erika and her friends bought four T-shirts and six caps.The T-shirts cost t dollars and the caps cost c dollars. Which expressiondescribes this situation?A 24tcB 6c � 4tC 4t � 6cD 4t � 6c

Look for key words in this problem to help translate it into an expression. T-shirts cost t and caps cost c. They bought four T-shirts, which is representedby 4t. They bought six caps, which is represented by 6c. The and means add.So, 4t and 6c is represented by 4t � 6c. D

2 Chris is buying five ride tickets and two food tickets at a carnival. Theexpression 5r � 2f can be used to find his total cost if r is the cost of oneride ticket and f is the cost of one food ticket. What is the total of Chris’spurchase if one ride ticket costs $3 and one food ticket costs $2?F $19G $18H $16J $12

Evaluate an algebraic expression by replacing the variables with numbers andthen finding the value of the numerical expression. In this case, the value of r is 3 and the value of f is 2. So, replace r with 3 and f with 2 in theexpression 5r � 2f.5(3) � 2(2) � 15 � 4

� 19 F

3 What is the value of a4 � b2 � c if a � 2, b � 6, and c � 9?A 11B 35C 43D 52

Replace the variables with the given values. Then evaluate the numericalexpression.24 � 62 � 9 � 16 � 36 � 9

� 43 C

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Examples

AF 1.2


1 Rafiq bought seven paperback books and two hardback books from anonline used bookstore. If p represents the cost of a paperback book, h represents the cost of a hardback book, and the shipping charge is $10,which expression describes the total cost?A hp � 10B h � p � 10C h � p � 10D 2h � 7p � 10

2 The Rios family is flying to Sacramento to visit relatives. The expression3a � 4c can be used to find the total cost of their trip if a is the cost ofan adult’s airfare and c is the cost of a child’s airfare. What is the totalcost of the trip if adult airfare is $109 and child airfare is $94?F $703G $376H $327J $203

3 What is the value of 4x � 2y � if x � 3, y � 7, and z � 8?

A 27B 3C �1D �2

4 What is the value of 2q3 � 7r2 � s0 if q � 3, r � 4, and s � 5?F 165G 190H 273J 454

8�z

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Apply algebraic order of operations and the commutative, associative, anddistributive properties to evaluate expressions, and justify each step in theprocess.

1 What is the value of 3(2 � 52) � 5(23 � 3)?A 14B �14C �55D �124

Following the order of operations solves this problem. Do all operationswithin the grouping symbols first, then powers, then multiply and dividefrom left to right, then add and subtract from left to right.

3(2 � 52) � 5(23 � 3) � 3(2 � 25) � 5(8 � 3) 52 � 25; 23 � 8� 3(�23) � 5(11) 2 � 25 � �23; 8 � 3 � 11� �69 � 55 3(�23) � �69; 5(11) � 55� �14 B

2 Which expression is equivalent to 3ab � (2c � d)?F (d � 3ab) � 2cG 3(ab � 2c � d)H (3ab � 2c) � dJ (3ab � 2c) � d

Solve this problem by using both the commutative and associative properties.The commutative property says that a � b � c � c � b � a. The associativeproperty says that (a � b) � c � a � (b � c). The only expression that isequivalent using these properties is answer choice F.

3 The expression 3y(x � 8) is equivalent to which of the following?A 3y � x � 8B 3x � 24C 3y � 24D 3xy � 24y

This problem is an example of the distributive property, which states that a(b � c) � ab � ac. Multiply both terms within the parentheses by 3y to getthe answer, 3xy � 24y. D

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Examples

AF 1.3


1 Which expression do you obtain if you use the commutative property ofaddition with the expression 12x2 � 8y2 � 5z?A x2y2z(12 � 8 � 5)B 12 � 8 � 5(x2y2z)C (12x2 � 8y2) � 5zD 5z � 8y2 � 12x2

2 Using the commutative property of multiplication, the expression 4b(2a �c) is equivalent to which of the following expressions?F 8ab � 4bcG (8ab) � cH (2a � c)4bJ 4b(�c � 2a)

3 Which expression is equivalent to (�9)(2g � 4)?A 2g � 9 � 4B �18g � 36C 18g � 36D �7g � 5

4 What is the value of 3(7 � 23) � 13 � 2(42 � 10)?F 52G 44H 37J 25

5 Evaluate b(a � 3) � 2b � 5 if a � 6 and b � 8.A 61B 40C 36D 17

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Solve problems manually by using correct order of operations or by using ascientific calculator.

1 In the expression 34(2 � 53), which should be found first?A 34

B 2 � 52

C 53

D 34 � 2

Following the order of operations, do operations within grouping symbolsfirst. Since 2 � 53 is inside the grouping symbols, do 53 first because powersare done before all other operations. C

2 What is the value of 12 � (40 � 4) � 6?F 8G 18H 36J 48

According to the order of operations, do operations within grouping symbolsfirst, then multiplication or division from left to right, followed by addition andsubtraction from left to right. With a scientific calculator that has parentheseskeys, enter the expression exactly as it is written (including the parentheses) toget the correct value.

12 � (40 � 4) � 6 � 12 � (36) � 6� 12 � 6� 18 G

3 Find the value of 25 � [(�7)(3) � 62 � 2].A 35B 29C 24D �7

According to the order of operations, the expression is evaluated in thefollowing steps. With a scientific calculator that has parentheses keys, enterthe expression exactly as it is written (including the parentheses) to get thecorrect value.

25 � [(�7)(3) � 62 � 2]� 32 � [(�7)(3) � 36 � 2] Evaluate powers first.� 32 � [�21 � 18] Multiply and divide from left to right.� 32 � (�3) Complete the operations inside of the

grouping symbols.� 32 � (�3) � 35 Add and subtract from left to right. A

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Examples

AF 1.4


1 To find the value of 73 � 2 � [(�5)(21) � 42], which operation will youuse last?A additionB subtractionC multiplicationD division

2 What is the value of (4 � 8) � (12 � (�7))?F 31G 7H �7J �12

3 Find the value of 6(3 � 8) � 3 � 9.A 31B 27C 14D 5.5

4 What is the value of the expression (5 � 23) � 17 � 24?F 23G 20H 17J 12

5 Find the value of 43 � [(�2)(11 � 15)] � (72 � 24)(2).A 178B 94C 58D 9

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Convert one unit of measure to another (e.g. from feet to miles, fromcentimeters to inches).

1 Dale is the manager of the track team. For each track meet, he fills a 16-quart cooler with water. How many gallons of water does the cooler hold?A 64 galB 16 galC 4 galD 0.5 gal

When converting from a smaller unit to a larger unit within the same system,divide. Since there are 4 quarts in 1 gallon, divide the number of quarts in thecooler by 4. Since 16 � 4 � 4, the cooler holds 4 gallons. C

2 The marine animals at Sea World in San Diego eat an average of 4,500pounds of food per day. How many ounces of food do the animals eat?F 281 ozG 36,000 ozH 72,000 ozJ 144,000 oz

When converting from a larger unit to a smaller unit within the same system,multiply. Since there are 16 ounces in 1 pound, multiply the number ofpounds of food by 16.4,500 � 16 � 72,000The marine animals at Sea World eat 72,000 ounces of food per day. H

3 The main span of the Golden Gate Bridge is 4,200 feet long. About howmany meters long is this span? 1 ft � 0.305 mA 13,771 mB 4,505 mC 3,895 mD 1,281 m

To convert between standard units and metric units, use a conversion factor.In this case, use the conversion factor 1 ft � 0.305 m.4,200 � 0.305 � 1,281 mThe main span of the Golden Gate Bridge is 1,281 meters long. D

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Examples

AF 2.1


1 By 2003, 10% of the vehicles sold in California must run on electricmotors. If an electric car can travel 125 miles before it must be recharged,how many feet can it travel before being recharged?A 5,155 ftB 5,405 ftC 125,000 ftD 660,000 ft

2 Bridalveil Fall in Yosemite National Park is approximately 620 feet high.About how many yards high is Bridalveil Fall?F 1,947 ydG 1,860 ydH 310 ydJ 207 yd

3 Barbara has a Spanish test in 72 hours. How many days from now isBarbara’s Spanish test?A 3 daysB 4 daysC 5 daysD 6 days

4 The distance from Monterey to San Luis Obispo is about 140 miles. Howmany kilometers is this? 1 mi � 1.6 kmF 88 kmG 224 kmH 280 kmJ 326 km

5 Ming bought 5 gallons of ice cream that she wants to serve one cup at atime. How many servings does she have?A 60 servingsB 80 servingsC 100 servingsD 120 servings

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Demonstrate an understanding that rate is a measure of one quantity perunit value of another quantity.

1 The speed limit on some interstate highways in California is 70 miles per hour,

which can be written as the fraction . This quantity is an example ofwhich of the following?A rateB timeC proportionD distance

Notice that the ratio has a denominator of one unit. In this case, the

unit is 1 hour. A ratio that compares a measure of one quantity to a unit valueof another quantity is a rate. A

2 Raul bought 4 pounds of grapes for $6. What was the cost per pound?F $1.00G $1.50H $2.00J $2.50

Write a ratio with $6 as the numerator and 4 pounds as the denominator.Then find an equivalent ratio with a denominator of 1 pound.

� �

Raul’s grapes cost $1.50 per pound. G

3 On her keyboarding test, Rochelle typed 175 words in 5 minutes. What is theaverage number of words per minute that Rochelle can type?A 5B 30C 35D 50

Use the information given to write a ratio with 175 in the numerator and 5 inthe denominator. Then find an equivalent ratio with a denominator of 1 minute.

� �

Rochelle averages a rate of 35 words per minute. C

35 words�

1 min175 � 5�

5 � 5175 words��

5 min

$1.50�1 pound

$6 � 4�4 � 4

$6�4 pounds

70 miles�

1 hour

70 miles�

1 hour

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Examples

AF 2.2


1 For the pep squad fund-raiser, Jack sold 154 candy bars in 7 days. Whatis the average number of candy bars he sold per day?A 22B 18C 12D 7

2 Rosa and her two friends paid a total of $123 for admission toDisneyland. What was the cost per person?F $62G $41H $27J $19

3 Serge is helping his grandmother make pastries for a family reunion. Therecipe calls for 4 cups of flour and makes 48 pastries. How many pastriesdoes he get for each cup of flour used?A 4B 6C 12D 18

4 Alberto bought a 10-day fishing license for $10.25 and caught 5 rainbowtrout. What was the cost of the license per fish?F $51.25G $20.50H $10.25J $2.05

5 Delta bought a quart of orange juice for $1.60. What is the price of theorange juice per ounce?A $0.80B $0.50C $0.16D $0.05

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Solve problems involving rates, average speed, distance, and time.

1 Nadja flew with her family from Austin, Texas, to San Jose, California. The

trip was about 1,470 miles and took about 3 hours. What was the averagespeed of the plane?A 490 mi/hB 420 mi/hC 355 mi/hD 276 mi/h

Divide the number of miles Nadja traveled by the number of hours the triptook to find the average speed.

1470 � 3 � 420

The average speed of the plane was 420 mi/h. B

2 Elbert rides his bike to school every morning. He travels at an average speedof 12 mi/h and the trip takes him 15 minutes. How far does Elbert live fromhis school?F 48 miG 24 miH 8 miJ 3 mi

To find the number of miles Elbert rides his bike, multiply his rate by theamount of time he travels. Since his rate is in miles per hour, convert the timehe travels from minutes to hours before multiplying.

rt � d

� �� h� � d mi 15 min = h

3 mi � dElbert rides his bike 3 miles to school. J

3 On a recent trip to the Sierra Nevada Mountains, Dolores rode a bus thattraveled 60 mi/h for 150 miles. How long did the trip take?

A 4 h B 3 h

C 2 h D 1 h

Since the formula is rt � d, you can divide the number of miles traveled bythe rate to find the time of the trip.

150 � 60 � 2

The trip took 2 hours. C1�2

1�2

3�4

1�2

1�4

1�2

1�4

1�4

12 mi�

1 h

1�2

1�2

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Examples

AF 2.3


1 Dmitri is writing a 15-page research paper for his English class at a rate

of 1 pages per hour. How long will it take him to finish the paper?

A 30 hB 18 hC 15 hD 12 h

2 On a recent trip to Mt. Shasta, Blanche’s father drove an average of 68 miles

per hour for of an hour before they stopped to eat. How far did they

travel before they stopped?F 51 miG 60 miH 91 miJ 136 mi

3 Conception rode the train from San Jose to Sacramento. The trip is

150 miles and took 3 hours. What was the average speed of the train?

A 55 mi/hB 40 mi/hC 35 mi/hD 25 mi/h

4 The first plane to fly faster than the speed of sound was flown in Californiaby Chuck Yeager in 1947. The plane flew at a speed of about 662 milesper hour. If Mr. Yeager flew at that speed for 15 minutes, how far wouldhe travel?F 9,330 miG 6,620 miH 165.5 miJ 99 mi

3�4

3�4

1�4

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Use variables in expressions describing geometric quantities (e.g.,

P � 2w � 2�, A � bh, C � �d—the formulas for the perimeter of a

rectangle, the area of a triangle, and the circumference of a circle,respectively).

1 Which expression shows the perimeter of the rectangle shown at the right?A 2abB abC 2a � 2bD a � b

The formula for the perimeter of a rectangle is P � 2w � 2�, where w is thewidth and � is the length. In the given rectangle, the width is a and the lengthis b, so the perimeter is 2a � 2b. C

2 The equation for the area of the triangle at the right is

30 � bh. Which quantity is represented by b in the

equation?F 5.3G 7H 9J 12

The formula for the area (A) of a triangle is A � bh, where b is the base

and h is the height. In the given triangle, the base is 12, so 12 is representedby b in the equation. J

3 Margarita is racing her go-cart around a circular track that has a diameter of50 yards. What is the length of the track? (Use 3.14 for .)A 157 ydB 150 ydC 94 ydD 75 yd

The length of the track is the same as the circumference of the track. Theformula for the circumference C of a circle is C � d, where d is thediameter of the circle. The problem states to use the approximation 3.14 for, so don’t use the key on a calculator for . If no approximation for isgiven, then use the key on a calculator.

C � dC � (3.14)(50) Substitute 3.14 for and 50 for d.C � 157The track is 157 yards long. A

1�2

1�2

12

7 95.3

a

b

1�2

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Examples

AF 3.1

best answer. Then write the letter for the answer you havechosen in the blank at the right of each question.

1 What is the perimeter of the rectangle shown atthe right?A 7 � sB 7sC 2(7s)D 14 � 2s

2 A compact disc has a diameter of 12 cm. What is its circumference? (Use 3.14 for �.)F 376.8 cmG 48.43 cmH 37.68 cmJ 36 cm

3 The area of the triangle at the right is represented by which of the following expressions?

A wy

B xz

C 2xzD 2yx

4 If the circumference of the circle at the right is 45 cm, which equation could be used to find s?

F s �

G s � 45 � 3.14

H s �

J s � 45�

5 Find the perimeter of the rectangle at the right.A 8cB 10cC 2c � 4cD 4c � c

4c

c

3.14�

45

45�3.14

s

1�2

1�2

x

w yz

7

s

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Express in symbolic form simple relationships arising from geometry.

1 The formula A � �w, where � is length and w is width is used to find thearea of a rectangle. If you know the area and width, how can you find thelength?A divide the area by the widthB divide the area by the lengthC multiply the area by the widthD multiply the area by the length

Since area and width are known, solve the equation A � �w for �, theunknown length.

A � �w

� �

So, to find the length, divide the area by the width. A

2 If the formula for the area of a parallelogram is A � bh, where b is the lengthof the base and h is the height, which of the following expressions representsthe area of a parallelogram where the length of the base is 5 times the height?F 2(5h)G 5h � hH 5h2

J 5bh

Using the formula for the area of a parallelogram, A � bh, substitute thegiven values for the length of the base and height.

A � bh

A � (5h)(h)

A � 5h2 H

3 The diameter of a circle, d, is equal to twice the radius, r. Which equationrepresents this situation?

A d �

B d � 2rC d � 2 � rD d � r � 2

The problem states that the diameter of a circle is twice the radius. Twice is the same as 2 times, so the radius must be multiplied by 2. Therefore, d � 2r. B

r�2

A�w

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Examples

AF 3.2


1 Daniel knows that the area of his bedroom is 156 square feet. He wants toknow the length and the width. He measured the width, 12 feet, with atape measure. Why does he not need to measure to find the length?A He can multiply the area and the width to find the length.B He can subtract the width from the area to find the length.C He can divide the area by the width to find the length.D He can add the area and the width to find the length.

2 If the base of a triangle is half of its height, which equation is true?F b � 2hG h � 2b

H h � b

J b � � h

3 Cora lives in a house that is in the shape of a hexagon. What is theperimeter of her house if all sides have length s?A 5sB 6sC 7sD 8s

4 Two rectangles A and B are similar. If rectangle A has a width of w and alength of � and rectangle B has a width of 5w, what is the length ofrectangle B?F 5�G 5w

H w

J 5 � w

1�5

1�2

1�2

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Understand the concept of a constant number such as �; know theformulas for the circumference and area of a circle.

1 Which of the following expressions best represents for a circle ofcircumference C and diameter d?A 3.14

B

C

D 3.14159

The Greek letter is used to represent the irrational number that is thequotient of the circumference of a circle divided by the diameter. It is thesame for every circle. The other answer choices are all commonapproximations of . C

2 What is the circumference of a circle that has a radius of 4 cm?F 4 cmG 8 cmH 16 cmJ 24 cm

The formula for the circumference of a circle is C � d, where d is thediameter, or C � 2r, where r is the radius. Substitute 4 for r in the secondversion of the formula and solve for C.

C � 2rC � 2(4) or 8 G

3 What is the area of a circle that has a diameter of 12 inches?A 24 in2

B 36 in2

C 48 in2

D 144 in2

The formula for the area A of a circle is A � r2 where r is the radius. The

radius of a circle is the diameter. In this case the radius is 6 because

� �(12) � 6. So, substitute 6 for r in the formula and solve for A.

A � r2

A � (6)2 or 36 B

1�2

1�2

C�d

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Standards PracticeMeasurement and Geometry 1.1

Examples

MG 1.1


1 Which of the following expressions represents the circumference of acircle with a diameter of 17 meters?A 289 mB 34 mC 17 mD 53.38 m

2 A circular swimming pool has an area of 144 square feet. What is itsdiameter?F 72 ftG 24 ftH 12 ftJ 6 ft

3 Sheila wears a silver ring that has a diameter of 19 millimeters. If she cutthe ring and flattened it out until it was straight, how long would thesilver strip be?A 19 � mmB 19 mmC 38 mmD 361 mm

4 Russ has a circular throw rug with a radius of 2 meters. How much floorspace does his rug cover?F 4 m2

G 2 m2

H m2

J m2

5 Mercedes has a camping tent with a round floor that has an area of 9 square feet. What is the radius of the floor of her tent?A 18 ftB 9 ftC 4.5 ftD 3 ft

1�4

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Know common estimates of � �3.14; � and use these values to estimate

and calculate the circumference and the area of circles; compare withactual measurements.

1 Enrique likes to play disc golf. He has a disc that has a diameter of 28 cm.What is the circumference of his disc?A 56 cmB 84 cmC 88 cmD 96 cm

Both 3.14 and are approximations for that can be used to find the

circumference of a circle. In this case, works best because you cansimplify it before you multiply.

C � d

C � �

C � 88 Simplify. Use � � and d � 28.

The circumference of his disc is 88 cm. C

2 A basketball goal is a circle with a diameter of 18 inches. What is the area ofa basketball goal?F 28.26 in2

G 56.52 in2

H 254.34 in2

J 1,017.36 in2

Solve the formula for the area of the circle. Use 3.14 as an approximation for because the answer choices are given as decimal values.

A � r2

A � (3.14)(9)2

A � (3.14)(81) or 254.34 Use � 3.14 and r � 9.A basketball goal has an area of 254.34 in2. H

3 Cheryl drives a pickup truck. The tires on her truck are 31 inches in diameter.How far will her truck move if the tires turn around one complete time?A 97.34 in. B 48.67 in.C 31 in. D 15.5 in.

Find the circumference.C � dC � (3.14)(31) Use � 3.14 and d � 31.C � 97.34The truck moves 97.34 inches. A

22�7

28�1

22�7

22�7

22�7

22�7

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Examples

MG 1.2


1 A bowling ball has a diameter of 8.6 inches. How far does a bowling ballroll each time it rotates?A 54 in.B 27 in.C 17.2 in.D 8.6 in.

2 What is the area of the circle shown at the right?F 414 mm2

G 441 mm2

H 1385 mm2

J 5,538 mm2

3 Joaquin is wrapping a gift that is in a cylindrical box. The box has aradius of 8 inches. How many inches of ribbon will he need to wrapcompletely around the cylinder?A 50.24 in.B 100.48 in.C 200.96 in.D 803.84 in.

4 Jessica’s little sister has an inflatable pool that has a play area of 28.26 square feet. What is the diameter of the pool?F 2 ftG 3 ftH 6 ftJ 9 ft

5 Mark marked the wheel of his bicycle and then measured the distancethat the bicycle traveled as the wheel turned around exactly one time. Thedistance was about 81.5 inches. Which of the following is the bestapproximation for the diameter of the wheel?A 20 in.B 21 in.C 25 in.D 26 in.

42 mm

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Know and use the formulas for the volume of triangular prisms and cylinders(area of base � height); compare these formulas and explain the similaritybetween them and the formula for the volume of a rectangular solid.

1 A cereal box has a base that is a rectangle measuring 10 inches by 2 inches,with a height of 12 inches. What is the volume of this box?A 20 in3

B 24 in3

C 120 in3

D 240 in3

This box is a rectangular prism because its base is a rectangle. The volume Vof a rectangular prism is V � �wh, where � is the length of the base, w is thewidth of the base, and h is the height.

V � �whV � (10)(2)(12) or 240The box has a volume of 240 in3. D

2 Find the volume of the triangular prism shown at the right. The height of the prism is 5 centimeters.The base of the prism is a triangle that has a base length of 15 centimeters and a height of 20 centimeters.F 1,500 cm3

G 750 cm3

H 300 cm3

J 150 cm3

The volume of a triangular prism V is V � Bh, where B is the area of thebase of the prism, and h is the height. Since the base of a triangular prism is a

triangle, you already know how to find its area (A � bh). Area of base

B � bh, or (15)(20), which equals 150. V � Bh, or (150)(5), which

equals 750. The volume of the triangular pyramid is 750 cm3. G

3 A cylindrical oatmeal container has a circular base with a radius of 6 centimeters and is 20 centimeters high. What is the volume of thecontainer? (Use 3.14 for .)A 2,260.8 cm3 B 376.8 cm3

C 113.04 cm3 D 18.84 cm3

The volume of a cylinder, like a prism, is equal to the area of the base timesthe height, or V � r2h. V � (3.14)(6)2(20) or 2,260.8. So, the volume of thecontainer is 2,260.8 cm3. A

1�2

1�2

1�2

20 cm

5 cm

15 cm

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Examples

MG 1.3


1 A cylindrical mailing tube has a circular base with a diameter of 10centimeters and is 50 centimeters high. What is the volume of the tube?(Use 3.14 for .)A 785 cm3

B 1,570 cm3

C 3,925 cm3

D 15,700 cm3

2 Alyssa’s grandfather is a farmer in the San Joaquin Valley. He has acylindrical grain silo that is 80 feet tall, and has a base diameter of 16 feet. What is the volume of the silo?F 2,009.6 ft3

G 4,019.2 ft3

H 16,076.8 ft3

J 64,307.2 ft3

3 An architect wants to build a building on this oddly shaped lot in the middle of a Y-intersection. If the building is 50 feet tall, what would be its volume?A 450,000 ft3

B 225,000 ft3

C 9,000 ft3

D 4,500 ft3

4 A rectangular prism with volume V has a base with area A. A triangularprism with the same height as the rectangular prism has a base with area

. What is the volume of the triangular prism?

F V 2

G V 3

H A3

J V�2

A�2

120 ft75 ft

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Identify angles as vertical, adjacent, complementary, or supplementary andprovide descriptions of these terms.

Use the figure shown at the right to answer Questions 1–2.

1 Which of the following pairs of angles are adjacent angles?A �5 and �6B �2 and �4C �1 and �5D �4 and �6

Adjacent angles have the same vertex and have a common side, but have nointerior points in common. Of the pairs of angles listed, only �5 and �6 areadjacent angles. A

2 Which pair of angles are vertical angles?F �2 and �7G �3 and �6H �2 and �8J �3 and �4

Vertical angles are non-adjacent angles formed by intersecting lines. Twointersecting lines form two pairs of vertical angles. In the figure above, �2and �8 are vertical angles, as well as �1 and �7, �3 and �5, and �4 and�6. The measures of vertical angles are equal. H

3 If two angles are supplementary, what is the sum of their measures?A 90°B 180°C 270°D 360°

The sum of the measures of supplementary angles is 180°. In the figureabove, �1 and �8 are supplementary. B

1 8 �726354

m

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Examples

MG 2.1


Use the figure shown at the right to answer Questions 1–2.

1 What is the measure of �4 if m�1 � x°?A x°B 90° � x°C 180° � x°D 180° � x°

2 What is the measure of �3 if m�1 � x°?F x°G 90° � x°H 180° � x°J 180° � x°

3 Two angles, �A and �B, are adjacent angles. Which of the following isalways true of these angles?A If �A has a measure of 50°, then �B has a measure of 130°.B If �A has a measure of 50°, then �B has a measure of 40°.C �A and �B share a common side.D �A and �B have equal measures.

4 If �X has a measure of 47° and �Y has a measure of 53° which of thefollowing statements is true?F The angles are supplementary.G The angles are complementary.H The angles are vertical.J The angles are neither supplementary, complementary, nor vertical.

5 Two angles each measure 45°. Which of the following is never true?A The angles are complementary.B The angles are adjacent.C The angles are vertical angles.D The angles are supplementary.

124

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Use the properties of complementary and supplementary angles and the sumof the angles of a triangle to solve problems involving an unknown angle.

1 If �C and �D are complementary, and �C has a measure of 23°, what is themeasure of �D?A 237°B 157°C 113°D 67°

Complementary angles have measures whose sum is 90°, so you know thatm�C � m�D � 90°. Since m�C is 23°, substitute and solve for m�D.

23° � m�D � 90°m�D � 90° � 23° or 67° D

2 If �G and �H are supplementary, and �G has a measure of 87°, what is themeasure of �H?F 3°G 93°H 103°J 267°

Supplementary angles have measures whose sum is 180°, so you know thatm�G � m�H � 180°. Since m�G is 87°, substitute and solve for m�H.

87° � m�H � 180°m�H � 180° � 87° or 93° G

3 In the triangle shown at the right, the measure of �Y is 25°. What is the measure of �X?A 65°B 90°C 115°D 155°

The sum of the measures of the angles of a triangle is 180°. This is a righttriangle as indicated by the square marking on �Z. The measure of �Y isgiven also. Subtract the two known angle measures from 180° to find themeasure of �X.

m�X � 180 � 90° � 25°m�X � 65° A

Z

Y

X

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Examples

MG 2.2


1 In the figure shown at the right, the measure of �2 is 75°. What is the measure of �1?A 15°B 105°C 165°D 255°

2 Two angles, �A and �B are complementary. The measure of �B is twicethe measure of �A. What is the measure of �A?F 30°G 45°H 60°J 75°

3 In �EFG, �E and �F have the same measure. Which of the followingcannot be true?A m�G � 60°B m�G � 90°C m�E � m�F � m�G � 180°D m�E 90°

4 In the figure shown at the right, �S and �T are supplementary. What is the measure of �T?F 238°G 148°H 122°J 32°

58�

S

T

Q

R

1 2

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Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isoscelestriangle).

1 The Transamerica Pyramid in San Francisco is a 48-story building. Eachoutside face of the building is a triangle with two sides of equal length. Whatis true about these triangles?A They are equilateral triangles.B They are obtuse triangles.C They are isosceles triangles.D They are right triangles.

An isosceles triangle has at least two congruent sides. C

2 Which of the following terms does not match the figure shown at the right?F SquareG ParallelogramH QuadrilateralJ Trapezoid

The figure shown is a square, which has four congruent angles and fourcongruent sides. It is a parallelogram because its opposite sides are paralleland congruent. It is a quadrilateral because it has four sides. It is not atrapezoid because it has more than one pair of opposite sides that are parallel. J

3 Which of the following is an equilateral triangle?A

B

C

D

An equilateral triangle has three congruent sides. A

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Examples

MG 2.3


1 What is the best description of the triangle shown at the right?A IsoscelesB ScaleneC RightD Equilateral

2 Which of the following descriptions does not describe the figure shown at the right?F TrapezoidG QuadrilateralH ParallelogramJ Rhombus

3 Which of the following is an isosceles right triangle?A

B

C

D

4 What is the best description of the figure shown at the right?F TriangleG Acute triangleH Equilateral triangleJ Isosceles triangle

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Compute the range, mean, median, and mode of data sets.

Use the following table of predicted high temperatures for Big Surfor the week of June 19th to answer Questions 1–3.

1 What is the range of predicted high temperatures for Big Sur for the week?A 8°B 5°C 4°D 2°

The range of a set of data is the difference between the greatest number andleast number in the set. In this case, the highest predicted temperature is 81°and the lowest is 73°. Since 81 � 73 � 8, the range is 8°. A

2 Find the mean and median of the predicted high temperatures.F mean � 79°, median � 78°G mean � 78°, median � 81°H mean � 78°, median � 79°J mean � 74°, median � 81°

The mean is the same as the average of the temperatures.

mean �

� 78°The median is the middle number in a set of data when the data are arrangedin numerical order: 73°, 77°, 79°, 80°, 81°.The median is 79°. H

3 Find the mode of the predicted high temperatures.A 81°B 77°C 73°D no mode

The mode of a set of data is the number that appears most often. In this case,all of the numbers appear only once. Therefore, there is no mode for this setof data. D

77° � 80° � 81° � 79° � 73°��

5

Day Mon. Tue. Wed. Thu. Fri.Temperature 77° 80° 81° 79° 73°

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Standards PracticeStatistics, Data Analysis, and Probability 1.1

Examples

SDAP 1.1


Every Saturday, Rachael rides the BART subway from her homein San Francisco to her grandmother’s house in Berkeley. Thefollowing table shows the amount of time the trip took eachweek for five weeks. The time includes walking time to and fromthe stations. Use this information to answer Questions 1–4.

1 What is the range of times for Rachael’s trips?A 10 minB 8 minC 6 minD 5 min

2 What is the mean amount of time for Rachael’s trips?F 50 minG 49 minH 48 minJ 46 min

3 What is the median amount of time in the data set?A 53 minB 51 minC 49 minD 43 min

4 Which of the following is the mode for the data set?F 53 minG 51 minH 50 minJ 49 min

Week 1 2 3 4 5Time (min) 51 49 53 43 49

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Understand how additional data added to data sets may affect thecomputations of measures of central tendency.

1 Darnell has the following scores on his math tests for the grading period: 78,80, 86, and 94. If he scores 83 on his next test, which of the following valueswill change?A MeanB MedianC ModeD Range

The median of his current tests is 83, so adding a score of 83 will not changethe median. There is no mode, nor would there be after adding the new testscore, so the mode does not change. Since the 83 is neither the high nor thelow score, the range does not change. The mean is the only value that willchange after the new test score is added. A

2 After her last exam, Melinda’s median exam score changed from an 89 to a91. She was happy because she needed at least a 90 average to get an A inthe class. Why might Melinda still not have an A?F Average is the same as mode, not median.G Average is the same as range, not median.H Average is the same as mean, not median.J Average is a value that is different from mean, median, and mode.

If Melinda needs a 90 average to have an A, then her mean score needs to be 90. Even though her median score is 91, her mean could still be lowerthan 90. H

3 Joshua collects model cars. The mean value of the cars in his collection is$35 and the median price is $50. Which of the following will not affect themean price of his cars?A Adding a car that costs $10B Adding a car that costs $35C Adding a car that costs $50D Adding a car that costs $100

Adding a car that costs the same as the mean price will not affect the mean. B

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Examples

SDAP 1.2


1 A taxi driver in Los Angeles has had 5 fares this morning. He drove his 5 customers 12, 30, 14, 22, and 17 miles each. If he picks up another fare and drives the customer 19 miles, which of the following values willchange?A MeanB MedianC ModeD Range

2 Because she lives on the beach in La Jolla, Mrs. Alvarez always hasgrandchildren visiting her during summer. If she makes a set of data ofthe numbers of grandchildren that she has visiting each week, the data sethas a small range. But for July 4th, the number of grandchildren visitingher will greatly change the range of her data. What can you say about thenumber of grandchildren visiting her during the 4th?F The number is probably about the same as the other weeks.G There is not enough information to say how the data will be affected.H The number is probably much greater than the other weeks.J Mrs. Alvarez will not have enough room for all of the grandchildren.

3 The table gives the heights of 5 buildings in San Francisco. If the height of the Transamerica Pyramid,which is 853 feet tall, wereadded to the table, which value would change the most?A MeanB MedianC ModeD Range

Building Height (ft)345 California Building 615101 California Street 600California Center 600Pacific Gas and Electric Bldg. 598Embarcadero Center, No. 4 570

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Understand how the inclusion or exclusion of outliers affects measures ofcentral tendency.

Use the data in the following table to answer Questions 1–3.

1 Which of the measures would be most representative of the data?A MeanB MedianC ModeD Range

Because Movie 1 earned so much more than the others, it is called an outlier.Outliers can also be much less than the other members of a set of data.Outliers affect mean and range the most. In this case, the median, 239, ismost representative of the data. B

2 Which would be most affected if the number 1 movie were removed from the table?F MeanG MedianH ModeJ Minimum

Movie 1 is an outlier, and outliers affect the mean and range the most. Sincerange is not an answer choice, then it would affect the mean the most. F

3 If the number 6 movie earned about $163 million and was included in thetable, would it be an outlier?A Yes, because it is $8 million less than the number 5 movie.B Yes, because it is less than the median of the data set.C No, because it is equivalent to the median of the data set.D No, because it is only $8 million less than the number 5 movie.

Movie 1 is an outlier, and it earned $149 million more than Movie 2. Thenumber 6 movie only earned $8 million less than Movie 5, so it would not beconsidered an outlier. D

Box Office Earnings for Five Top MoviesMovie Rank Earnings (Rounded to the nearest million dollars)

1 4312 2823 2394 2055 171

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Examples

SDAP 1.3


Use the data in the following table to answer Questions 1–3.

1 In the data set above, which of the following best describes thepopulation of Los Angeles?A MedianB RangeC ModeD Outlier

2 Based on the data above, which best represents the population of the fourlargest cities in California?F Mean populationG Median populationH Population rangeJ Population mode

3 Which statement is true about the mean of the data?A Because of Los Angeles, it does not represent the data well.B It is not affected by the Los Angeles data.C The mean is less affected by Los Angeles than the median or mode.D The mean is 1,040,945.

4 The population of the 5th largest city in California, Long Beach, was461,522 in 2000. If Long Beach were added to the table, which would itaffect the most?F MeanG MedianH ModeJ Range

Populations of the Top FourCalifornia Cities in 2000

City PopulationLos Angeles 3,694,820San Diego 1,223,400San Jose 894,943

San Francisco 776,733

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Know why a specific measure of central tendency (mean, median, mode)provides the most useful information in a given context.

1 Rochelle surveyed every student in the 6th grade to find out his or herfavorite meal in the school cafeteria. Which measure would she use to findthe most popular meal from this data?A MeanB MedianC ModeD Range

Since the data collected were not numerical, Rochelle cannot find the mean,median, or range. She can, however, find the mode, which is the mostpopular meal. C

2 Students who scored in the top half of their class are exempt from Mr.White’s math final exam. However, Mr. White did not release a list of namesof students who are exempt. He only released the mean, median, mode, andrange of scores of his class. Along with their own scores, which of thefollowing should the students use to find out if they scored in the top half?F MeanG MedianH ModeJ Range

Students should compare their own scores to the median, since the median isthe middle number when a set of data is arranged in numerical order. G

3 Alicia’s family is thinking about moving to Palm Springs, but they areconcerned about the summer heat. To find out more, they decided to look upclimate data for Palm Springs. All they could find were daily hightemperatures. Which of the following additional pieces of information wouldhelp them most?A The daily mean temperatures for AugustB The daily median temperatures for AugustC The modes for the daily temperatures in AugustD The daily range of temperatures in August

Since they already know the daily high temperatures for August, the dailyrange of temperatures would be the most helpful. Knowing the hightemperature and the range, they can figure out how cool it gets in themorning, as well as how hot it is in the afternoon. D

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Examples

SDAP 1.4


1 Mrs. Kim told her math class that anyone who scored below the classaverage on her exam would have to take the exam again. Which of thefollowing do students need to know to find out whether they have to takethe exam again?A The mean exam scoreB The median exam scoreC The exam score modeD The range of exam scores

2 Alberto wanted to know if more people in his community drive cars,trucks, or SUVs, so he recorded the number and type of vehicles thatdrove through a busy intersection during rush hour. Which of thefollowing measures will tell him which type of vehicle is most popular?F MeanG MedianH ModeJ Range

3 For a class photo, the taller half of the students were asked to stand in theback row, and the shorter half were asked to sit in the front row. Which ofthe following measures will help students decide where to stand for thephoto?A MeanB MedianC ModeD Range

4 Alberto got a summer job with his brother’s landscape company. He wastold that he would mow an average of 8 lawns per day. If Alberto wants toknow what are the fewest and the most lawns he might have to mow,which of the following will help?F MeanG MedianH ModeJ Range

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Compare different samples from a population with the data from the entirepopulation and identify when it makes sense to use a sample.

1 Recently, about 11% of the population of California lived in the city of LosAngeles. If this percent has remained about the same, about how many of300 randomly selected Californians would you expect to live in Los Angeles?A 11B 22C 33D 44

Use a proportion and solve for the unknown value.

�

100n � 3,300 Cross multiply.n � 33 Divide each side by 100.

Out of 300 randomly surveyed Californians, 33 would live in Los Angeles. C

2 Out of 60 students who were surveyed, 9 said that skateboarding was theirfavorite sport. If the 450 students in the whole school were surveyed, abouthow many would be likely to say that skateboarding is their favorite sport?F 45G 60H 68J 397

Use a proportion and solve for the unknown value. 68 students would sayskateboarding was their favorite sport. H

3 To find out what kinds of hobbies students in her school have, Juliette askedthe 8 students in her drama club. Which of the following best describesJuliette’s sample choice?A It is a good sample because it is small.B It is a poor sample because it contains all girls.C It is a good sample because it is not random.D It is a poor sample because it is small and not very random.

Juliette’s sample is poor because it is small and not very random. She needsto survey more students, and to survey students from more than just thedrama club. D

n�300

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Examples

SDAP 2.1


1 Rafael needs to find out what kinds of music the DJ should play for theupcoming school dance. Which of the following would make the bestsurvey?A His fellow basketball team members.B The students in his homeroom.C Every 10th student who enters the cafeteria at lunch.D The students in the school band.

2 If 120 out of 200 residents of a small town who were surveyed said thatthey did not want a discount store to be built in their town, about howmany of the town’s 1,900 residents would be likely to have the samefeelings?F 1,800G 1,140H 320J 200

3 Mrs. Ghish wants to find out where the 6th graders at Yolo MiddleSchool want to go for their class field trip. Which of the following wouldmake the best sample?A Every 10th student who walks into the school building in the morning.B Every 5th student who enters the school cafeteria.C Ten students chosen at random from the school roster.D One out of every 5 sixth graders she finds in the hallway between

classes.

4 Recently, roughly 2% of all Californians lived in San Francisco. Supposethis rate has remained constant. If you survey 250 Californians atrandom, about how many would live in San Francisco?F 50G 25H 5J 2.5

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Identify different ways of selecting a sample (e.g., convenience sampling,responses to a survey, random sampling) and which makes a sampling morerepresentative for a population.

1 Suppose a textbook company sends a letter to the parents of 200 students inCalifornia asking whether the students would like to answer questions to helpthem design a new math textbook. Which of the following best describes thismethod of selecting a sample.A RandomB ConvenienceC SurveyD Cluster

This is a survey. The parents of 200 students were asked whether theirchildren could answer questions. Those who say yes will become the samplefor the population. C

2 Suppose that the number of people who visit Yosemite National Park on June 1is used to estimate the total number of people who will visit the park duringthe summer. This would be an example of which of the following methods ofselecting a sample?F RandomG ConvenienceH SurveyJ Cluster

Since this survey takes place on only one day, it is designed mostly forconvenience. A more random method of selecting a sample would look at thenumber of visitors on more than one day, or on parts of several days. G

3 During the 2000 Census, the Census Bureau sent a longer form, with morequestions, to about 1 out of every 6 households, chosen indiscriminately. Thelonger form is an example of what type of sample selection method?A RandomB ConvenienceC SurveyD Cluster

Since the longer form is sent to households indiscriminately, this is anexample of random sampling. A

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Examples

SDAP 2.2


1 The school newspaper asked every eighth student in the hallway howmuch homework they had on an average night. Which of the followingbest describes this method of selecting a sample?A RandomB ConvenienceC SurveyD Cluster

2 A cable television network sends letters to 1,000 Californians askingthem if they would like to take part in focus groups to help the networkdecide what kinds of new shows to air. This is an example of which of thefollowing methods of selecting a sample?F RandomG ConvenienceH SurveyJ Cluster

3 Suppose Caltrans wants to know how many miles most Californians driveper week. To find out, they decide to ask everyone who comes into theDMV on the first day of each month. Which method are they using toselect their sample?A RandomB ConvenienceC SurveyD Cluster

4 Mr. Jefferson, the school principal, wants to find out how many studentsplan on attending the basketball tournament this weekend by taking arandom sample. Which should he do?F Ask the students who come into the office.G Ask the students who play basketball.H Go to four classrooms and ask the students in them.J Ask every 10th student who enters the school in the morning.

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Analyze data displays and explain why the way in which the question wasasked might have influenced the results obtained and why the way in whichthe results were displayed might have influenced the conclusions reached.

1 Which of the following survey questions would not influence the responsegiven?A Which television programs do you like to watch in the evening?B Do you prefer to read books or magazines in the evening?C Do you like to spend time with your family or by yourself in the

evenings?D What do you do when you are at home in the evening?

Question D is the only one that does not influence the response givenbecause it does not suggest a specific activity. Question A suggests watchingtelevision, Question B suggests reading, and Question C suggests spendingtime with family or alone. D

The graphs at the right show sales for Gino’s Italian Restaurant for one year. Use the graphs to answer Questions 2 and 3.

2 Which is a correct statement about the graphs?F The range of sales is greater for Graph A

than for Graph B.G Graph A gives the impression that sales

dropped only slightly.H Graph A and Graph B show the exact

same sales data.J Graph B gives the impression that sales

dropped greatly.

Look closely at the two graphs and you will see that the only difference between them is the vertical scale. Both graphs show the exact same sales data. H

3 Why would Graph A be considered misleading?A The range on the vertical scale is much smaller than that of Graph B.B The months should be along the vertical axis.C The points are not connected in a straight line.D The points are too close together.

Because the range on the vertical axis of Graph A is much smaller than thatof Graph B, it makes the drop in sales look much more drastic. However,both graphs show the same data. A

Jan. Apr. Jul. Oct.

Graph A

Sales($1,000s)

35

33

31

Jan. Apr. Jul. Oct.

Graph B

Sales($1,000s)

4035302520151050

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Examples

SDAP 2.3


The graphs show the number of poundsof paper that students at Chavez MiddleSchool recycled in eight weeks. Use thegraphs to answer Questions 1 and 2.

1 Which statement best describes the graphsabove?A Graph A could be considered misleading.B Both of the graphs have the same scale.C Graph B makes the increase in paper

recycling look less impressive.D Both graphs show the same recycling data.

2 Why could Graph B be considered misleading?F The vertical scale does not begin at zero.G The horizontal scale does not begin at zero.H It shows incorrect data.J It includes data that was not shown in Graph A.

3 Which of the following questions would likely influence the responsegiven?A What activities do you enjoy during summer vacation?B Do you prefer indoor or outdoor activities during the summer?C Describe your usual summer vacation activities.D What activities do you associate with summer vacation?

4 Which of the following questions would most likely not influence theresponse given?F Which do you enjoy more, math or English?G What is your favorite school subject?H Most students prefer science to foreign language; which do you

prefer?J If you had to choose between taking journalism or art, which would

you pick?

1 2 3 4Week

5 6 7 8

Graph A

Pounds

1,000800600400200

0

1 2 3 4Week

5 6 7 8

Graph B

Pounds

900800700600500400300

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Identify data that represent sampling errors and explain why the sample(and the display) may be biased.

1 Lucinda asked her parents what their favorite soft drink was. Both of themsaid that iced tea was their favorite soft drink. From this, Lucinda concludedthat the most popular kind of soft drink for adults is iced tea. Why is herconclusion not correct?A Lucinda only gave her parents two types of soft drinks to choose from.B We do not know how old Lucinda’s parents are.C Lucinda did not use a large enough sample.D Lucinda should have asked two adults who were not family members.

The population that Lucinda is asking about is all adults. She only asked twoadults about their favorite soft drink. For a population this large, 2 is not alarge enough sample. C

2 Fidel lives in Fresno. He wants to find out which professional sports team isthe favorite of people in Fresno. Which would be the best way for Fidel toselect a sample of people to find out which team they like?F Ask 100 people at a junior high football game.G Ask 100 of his fellow sixth graders as they enter the school building.H Stand on a busy street corner in Fresno and ask 100 people of all ages.J Pass out a questionnaire at school and ask people to mail it back.

If he asks people at a football game, they may be biased toward football, andFidel’s question does not specify a sport. If he asks students in his school, heis limiting his sample to one age group, and the population is made up ofmany age groups. The best way for Fidel to select a sample would be to standon a busy street corner in Fresno and ask 100 people of all ages. H

3 Brook asked students in his school the survey question, “Cats and dogs arethe most popular pets; what is your favorite pet?” Why is this survey biased?A She only asked students.B She mentioned cats before dogs.C She asked a specific question.D She suggested answers.

Brook is asking students about their favorite pets, but she implied that theanswer should be cats or dogs. Some students may have favorite pets that arenot cats or dogs. D

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Examples

SDAP 2.4


1 A newspaper wants to find out what is the favorite holiday ofCalifornians. Which would be the best way to select a sample for thissurvey?A Survey 100 people of all ages on a street corner in Los Angeles.B Survey 1,000 middle school students in all parts of California.C Ask people who read the paper to e-mail their responses.D Survey 1,000 people of all ages from all parts of the state.

2 Rodrigo asked people who were stopped in their cars at a busyintersection in Los Angeles during rush hour what was their usual way oftraveling to work. Why is his survey biased?F He limited the choices.G He asked only people who were driving cars.H He asked only people who were riding public transportation.J He asked only people at intersections.

3 Delta asked students in her middle school what their favorite type oftelevision show was. Almost all of them said that the situation comedywas their favorite. From this, Delta concluded that the most popularshows on television must be situation comedies. What is wrong with herconclusion?A Delta did not use a large enough sample.B The question she asked was biased.C She asked too many people.D Delta asked too many different age groups.

4 Gilbert asked 100 people at a California beach whether surfing orsunbathing was their favorite beach activity. Why is this survey biased?F Gilbert only asked people at the beach.G He should have asked people of different age groups.H One hundred people is too large a sample for the question.J Gilbert limited the question.

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Identify claims based on statistical data and, in simple cases, evaluate thevalidity of the claims.

1 An airline that flies out of Oakland and not San Francisco claims that in asurvey of 500 people, 95% prefer to fly out of Oakland rather than SanFrancisco. Which information about the survey would be most likely to makepeople think the airline’s claim is misleading?A The survey only included people who were flying on business.B The survey was of people who were passengers on the airline.C The survey was taken only during one month of the year.D Only 25% of the people surveyed lived in the Bay Area.

These passengers could not have flown out of San Francisco because theairline does not fly out of that airport. Therefore, the passengers are mostlikely going to say that they prefer to fly out of Oakland because they like theairline, not because they like one airport or the other. B

2 Monique asked all the students in her middle school’s science classes toname their favorite type of novel. The results showed that most preferredscience fiction. Based on this, Monique recommended to the school librarianthat the library should buy more science fiction books. Why is herconclusion invalid?F She did not count the number of people she surveyed.G Her sample is not large enough.H The sample is not random.J She should have included students in the math classes as well.

Monique only asked students in the science classes. A better survey wouldhave asked students at random in all classes, not just science classes.Therefore, her sample is not random. H

3 A maker of aspirin claims that 2 out of 3 doctors surveyed prefer their brandof aspirin to other pain relievers. Why might this data be misleading?A The sample is too large.B The sample should contain people other than doctors.C The sample contains different kinds of pain relievers.D The number of doctors in the sample is not known.

The number of doctors in the sample is not known. Therefore, the surveycould have a sample that is too small, which makes the claim possiblymisleading. D

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Examples

SDAP 2.5


1 An insurance company claims that 80% of Californians have earthquakeinsurance. Which information about the survey would be most likely tomake people think the company’s claim is misleading?A The survey only included 100 people in two California counties.B The survey included both homeowners and renters.C The survey included both city and rural residents.D The survey took place in a year when there were no major

earthquakes.

2 A researcher believes that higher speed limits on California highwayscause more accidents. Which of the following factors should she notconsider when she gathers data?F Accidents on highways where the speed limit has been increased.G Accidents on highways where the speed limit has not changed.H The number of cars traveling on highways with and without increased

speed limits.J Whether the accidents happen during the day or night.

3 Jerome surveyed 50 students in the morning before school. He askedwhether the school day should start later in the day. From his survey, heconcluded that school should not start later in the day. Why is hisconclusion invalid?A The question was not clear.B The question was limited.C He only surveyed students who arrive at school early.D His survey did not include enough students.

4 Allison’s family is moving to Los Angeles. They read that Los Angelesonly receives about 12 inches of rain per year. From this information,they concluded that the climate of Los Angeles must be like that of adesert. What is wrong with their conclusion?F They only looked at the average temperature.G They only looked at rainfall data for one month.H They should look at the amount of rainfall over several years.J They did not consider other climate factors such as temperature.

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Represent all possible outcomes for compound events in an organized way(e.g. tables, grids, tree diagrams) and express the theoretical probability ofeach outcome.

1 The tree diagram shows the possibilities for choosing between three types of cola (A, B, or C) in two different sizes (small or large) at the concession stand of the city middle school track meet. How many different cola choices are there?A 2B 3C 6D 9

Notice in the tree diagram that each of the three types of cola can be servedin either a small size or a large size, for a total of 6 choices. C

This diagram shows the possible outcomes from rolling two six-sided number cubes. Use this information to answerQuestions 2 and 3 below.

2 How many ways are there to roll atleast one “2”?F 1G 2H 6J 11

Look at the diagram for outcomes that have at least one “2.” There are 11 outcomes where at least one of the number cubes showed a “2.” J

3 What is the theoretical probability of rolling at least one “2” when twonumber cubes are rolled?

A B

C D

Theoretical probability is the ratio of favorable outcomes to the total possibleoutcomes. In this case, favorable outcomes are those in which a 2 is rolled.There are 11 favorable outcomes and 36 total outcomes. The theoretical

probability is . A11�36

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4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

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Examples

SDAP 3.1


Use this tree diagram to answer Questions 1 and 2.

1 The tree diagram shows the differentcombinations of shirt, pants, and shoesthat Ashok can wear. How manycombinations does he have?A 2B 3C 4D 12

2 What is the theoretical probability that Ashok will choose to wear loafers,no matter which pants or shirt he wears?

F

G

H

J

3 Yolanda has a penny, nickel, dime, and quarter in her pocket. What is thetheoretical probability that she will pull out a coin, flip it, and get heads?

A

B

C

D

4 What is the theoretical probability that Yolanda will pull out all fourcoins, flip them all, and get heads on all four?

F G

H J 1�16

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Use data to estimate the probability of future events (e.g., batting averagesor number of accidents per mile driven).

1 A football player threw 20 passes during his last game, 7 of which werecomplete. If he keeps completing passes at the same rate, how many passeswill he complete if he throws 100 passes?A 5B 7C 35D 100

Use a proportion to solve this problem.

�

20n � 700 Cross multiply.n � 35 Divide each side by 20.

In 100 passes, he will complete 35. C

2 Of the last 100 visitors who entered Joshua Tree National Park, 85 were justvisiting for the day while 15 were going to camp overnight in the park. Outof the next 20 visitors who enter the park, how many will likely be there tocamp overnight?F 15G 7H 3J 1

Based on the given information, the probability of a visitor camping in the

park overnight is , or . Therefore, of the next 20 visitors who enter the

park, 3 will likely camp overnight. H

3 Angelina is fly-fishing in the Sierra Nevada. She hiked to 15 differentlocations on a stream and caught 3 trout. What is the probability that she willcatch a trout at the next location she fishes?

A

B

C

D

From the information given, the probability of Angelina catching a trout at

any given location is , which simplifies to . D1�5

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Examples

SDAP 3.2


1 If a basketball player for the UCLA Bruins takes 60 free throws andmakes 27, how many free throws will she make in 160 attempts?A 81B 72C 54D 9

2 Leona has made an A on 15 of the 18 math tests that she has taken thisyear. What is the probability that she will make an A on her next test,which is the final exam?

F

G

H

J

3 Elton went to the beach 9 out of the past 12 Saturdays. In the next 8Saturdays, how many times will Elton likely go to the beach?A 2 timesB 4 timesC 6 timesD 8 times

4 Mr. Brandies estimated that he spends 1 out of every 4 minutes of hisdrive to work stuck in traffic jams. If it takes Mr. Brandies 50 minutes toget to work, about how much time does he spend in traffic jams?F about 30 minG about 20 minH about 18 minJ about 13 min

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Represent probabilities as ratios, proportions, decimals between 0 and 1, andpercentages between 0 and 100, and verify that the probabilities computedare reasonable; know that if P is the probability of an event, 1 � P is theprobability of an event not occurring.

1 Speedy Airlines claims that 7 out of 8 of its flights out of LAX are on time.What is the probability that the next Speedy Airlines flight from LAX will beon time?A 90%B 87.5%C 50%D 12.5%

The probability of a flight being on time can be represented by the ratio which is 87.5% when written as a percent. B

2 A Stanford University baseball player has a batting average of 0.225. If hisaverage stays constant, how many hits can he expect to get in his next 200 atbat?F 12.5G 22.5H 25J 45

Multiply his average by the number of at-bats to find the number of hits he willget. In his next 200 at-bats, the player can expect 200 � 0.225 or 45 hits. J

3 Approximately 11% of the population of California lives in the city of LosAngeles. If you meet 100 Californians at random, how many on average willnot be from Los Angeles?A 11B 22C 66D 89

If the probability of an event occurring is P, then the probability of the eventnot occurring is 1 � P. So, the probability of meeting a Californian who isnot from Los Angeles is 1 � 11%, or 89%. Then, out of 100 people, 89% is89 people. So, out of 100 Californians, 89 would not be from Los Angeles. D

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Examples

SDAP 3.3


1 Suppose you roll two number cubes. What is the probability that the sumof the numbers on the two cubes will be 7?

A 33 %

B 16 %

C 13 %

D 11 %

2 A softball player has a batting average of 0.362. What is the probabilitythat she will not make a hit at her next at-bat?F 0.638G 0.623H 0.362J 0.263

3 If a meteorologist says that a given area has a 20% chance of rain, what isthe probability that it will not rain?

A

B

C

D

4 A ski resort at Lake Tahoe is completely full 5 out of 7 nights a weekduring ski season. What is the probability that a skier will get a roomduring ski season if she shows up without a reservation?F 71.4%G 57.1%H 32.8%J 28.6%

5 What is the probability of an event that always occurs?A 0.100 B 100C 1 D 0.01

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Understand that the probability of either of two disjoint events occurring isthe sum of the two individual probabilities and that the probability of oneevent following another, in independent trials, is the product of the twoprobabilities.

1 Danielle has a bag that contains 25 marbles. Seven of the marbles are greenand three are blue. What is the probability that Danielle would reach into thebag and randomly choose either a green or blue marble?

A

B

C

D

Since there are seven green marbles, the probability of picking a green marble

is . Since there are three blue marbles, the probability of picking a blue

marble is . However, if Danielle can pick either a green or blue marble, she

has 7 � 3 or 10 choices out of 25. Therefore, the probability of picking

either a green or blue marble is � , which is or . C

2 Robert and his sister are playing a board game. According to the rules of thegame, any player who rolls a double (the same number on both numbercubes) gets to repeat their turn. What is the probability of rolling a double?

F

G

H

J

There are 6 numbers on the cubes, so there are 6 ways a player can roll adouble. G

3 What is the probability of tossing heads on a coin and a 4 on a number cube?

A B

C D

Both events are independent. Multiply the probability of each event to get the

probability of both events occurring. Therefore, � � . C1�12

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Examples

SDAP 3.4


1 Coach Brown’s PE class has 5 sixth graders, 30 seventh graders and 3 eighth graders. If she picks a student at random to run an errand, whatis the probability that the student will be either a sixth or eighth grader?

A

B

C

D

2 A cookie jar has 36 cookies in it. If 12 of the cookies are chocolate chip,and 6 are sugar cookies, what is the probability of reaching into the jarand pulling out either a chocolate chip or sugar cookie?

F

G

H

J

3 Darren is watching cars at two intersections in downtown Los Angeles. At each intersection, equal numbers of cars go straight, turn left, or turnright. What is the probability that a particular car will go straight throughboth intersections?

A B

C D

4 Alejandra is going to fly from the Los Angeles area to the San FranciscoBay area. There are 4 airports she could leave from in the Los Angelesarea, and 3 that she could fly to in the Bay area. What is the probabilitythat she will fly out of Burbank in the LA area and arrive at Oakland inthe Bay area?

F G

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Understand the difference between independent and dependent events.

You are in a pool of 5 contestants for a Hollywood game show. Foreach round of the game, a contestant is picked from the pool toplay. Once a contestant plays a round, that contestant is removedfrom the pool. Use this situation to answer Questions 1–3.

1 Suppose that after each round, a new contestant is brought from the audienceto keep 5 contestants in the contestant pool. What is the probability that youwill be picked to play in each round?

A

B

C

D

Because the contestant who is picked to play in each round is replaced, thefirst round does not affect the probability in the following rounds. These are

independent events. The probability of being picked in each round is . A

2 Suppose that the contestants in the contestant pool are not replaced after eachround. What would your probability be of being picked in the second roundand then in the third round?

F , G ,

H , J ,

These are dependent events because one person was removed from the poolin the first round. So, there are four contestants left for the second round, and

the probability of being picked is , and then for the third round. H

3 What is the probability of being picked in the fourth round if the winningcontestants are not replaced after each round?

A B

C D

This is a dependent event because the number of contestants in the pooldecreases after each round. In the fourth round only two contestants are left,

so your probability is . D1�2

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Examples

SDAP 3.5


1 There are 14 girls and 13 boys in Amelia’s math class. The teacher ispicking two girls and two boys to represent the class in a math contest. Ifthe teacher has already picked one girl, what is the probability that Ameliawill be picked?

A

B

C

D

2 Ashok needs to roll a 12 on two number cubes to win a board game. If herolled a 6 with his first number cube, what is the probability that he willroll a 6 with the second one, so he will have 12?

F

G

H

J

3 Lee has a bag of jellybeans. The bag contains 21 licorice, 14 root beer,and 12 vanilla beans. If she eats 3 licorice beans, what is the probabilitythat she will reach into the bag and randomly pull out a vanilla bean?

A B

C D

4 Paco and Tommy flipped a coin to see who would get to ride first onPaco’s new skateboard. When Paco lost the toss, he suggested that they gofor 2 out of 3 flips. Why is Paco still not likely to ride the skateboard first?F Each coin flip is a dependent event, which does not change the

probability.G 2 out of 3 is the same probability as 1 out of 2.H Each successive coin flip has a higher probability.J Paco is less likely to win 2 coin tosses than Tommy is to win 1 coin toss.

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Analyze problems by identifying relationships, distinguishing relevant fromirrelevant information, identifying missing information, sequencing andprioritizing information, and observing patterns.

1 Zachary and his family are going on a camping trip to Kings CanyonNational Park. The trip is 150 miles, and they estimate it will take 3 hours forthe drive and 45 minutes for rest stops. How long will the trip take?A 45 minB 50 minC 3 hD 3 h 45 min

The number of miles that Zachary’s family will travel to their destination isirrelevant information because the problem asks for the time of the trip.Simply add the driving time to the rest stop time to find the total, 3 hours and45 minutes. D

2 The afternoon high temperature today in El Centro was 28 degrees warmerthan the morning low temperature. What do you need to know to find today’smorning low temperature?F The average high temperatureG The average low temperatureH Today’s afternoon high temperatureJ Today’s humidity level

The temperature range was 28 degrees, so the morning low temperature was28 degrees less than the afternoon high temperature. You need to know theafternoon high temperature to solve this problem. H

3 Mrs. Moreno picked Julie and Hannah to go to the board to solve the firstquiz problem. She then picked Miguel and Scott to solve the second problem.Angela and Linda were picked to solve the third problem. Which of thefollowing describes the pattern Mrs. Moreno is using to pick students?A girl, girl, boy, boy, girl, girl, …B girl, boy, girl, boy, …C boy, boy, girl, girl, boy, boy, …D girl, boy, girl, girl, boy, boy, …

Mrs. Moreno picked two girls, then two boys, then two girls. A

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Standards PracticeMathematical Reasoning 1.1

Examples

MR 1.1


1 Which number comes next in the pattern?

40, 30, 21, 13, 6, 0, …

A �4B �5C �6D �8

2 Rodrigo is paying his mother $75 every month for 6 months for his newbike. He has had the bike for 4 months now. How much did the bike cost?F $750G $600H $450J $300

3 Nicoleta had 4 math exams during the grading period. She made an 88 onher first and third exams, and a 93 on her fourth exam. She has a 100average on her homework. What additional information do you need toknow in order to find her average exam grade?A Her in-class work average.B The number of weeks in the grading period.C The number of homework assignments she had.D Her score on the second exam.

4 Anton and Trudy went to a USC Trojans football game with $25 each.The game lasted about 4 hours and USC won with a score of 17–12.When they left, Anton and Trudy had a total of $12.75 between them.How much money did they spend at the game?F $37.25G $24.50H $12.25J $5.00

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Formulate and justify mathematical conjectures based upon a generaldescription of the mathematical question or problem posed.

1 Mr. Malcolm has 23 students in his 1st period math class. Which of thefollowing statements about his class is true?A He can easily divide the class into smaller study groups.B He will have even rows and columns of students in the desks.C He will not be able to divide the class into groups containing the same

number of students.D He can use all the desks in his classroom to make six even rows of desks.

Since 23 is a prime number, it is only divisible by itself and 1. Therefore, Mr. Malcolm will not be able to divide the class into groups containing thesame number of students. C

2 Suppose you pick any even number between 100 and 500, and then multiplythat number by 5. Which of the following could be the product?F 1,525G 1,610H 1,642J 1,835

The digit in the ones place of the product must be a zero because any evennumber multiplied by 5 is a multiple of 10. For example, 2 � 5 � 10, 4 � 5 � 20, etc. Therefore, only 1,610 could be the product. G

3 If a and b are whole numbers and a is a factor of b, then which of thefollowing is not possibly a correct statement?A a � bB a � bC a bD a ≠ b

A factor is a number that divides into a whole number with a remainder ofzero. Since it must divide into the number, it cannot be greater than thenumber, so the answer is a b. C

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Examples

MR 1.2


1 What is the greatest number of even numbers that you can have in a listof 13 consecutive whole numbers?A 5B 6C 7D 8

2 Mrs. Nguyen holds 20 note cards with 20 consecutive numbers written onthem, one number on each card. She tells you to pick a group ofconsecutive cards so you are certain that one of the cards has a numberthat is divisible by 5. How many cards should you pick?F 5G 4H 3J 2

3 If the distance between Redding and Chico is a multiple of 2, which ofthe following distances could be the correct one?A 77B 75C 73D 72

4 Suppose an odd whole number is subtracted from an even whole number.Which of the following cannot possibly be the difference?F 524G 525H 563J 571

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Determine when and how to break a problem into simpler parts.

1 Paulo wants to buy a snowboard that is marked down $40. If the sales tax is

8 %, which is the first step for finding the price he will pay for the snowboard?

A Find 8 % of $40.

B Divide $40 by 8 %.

C Subtract $40 from the original price.

D Add 8 % to the original price.

Paulo must pay sales tax on the final price of the snowboard. Since there is a$40 discount, he must subtract $40 from the original price to find the finalprice. C.

2 For which kind of situation would it be most useful to break a problem intosimpler parts?F Find the area of a circle of radius 3 feet.G Find the speed of a car that averages 5 miles in 10 minutes.H Find the difference in temperature between 23°F and �16°F.J Find how many handshakes will occur when 8 students each shake hands

with each other.

It is most useful to break a problem into simpler parts when the numbers in theproblem are very large or when the problem is based on a pattern or sequence.In the situations described in choices F, G, and H, you need to calculate aspecific value. In J, you could start with 2 students, then 3 students, then 4 students to see if there is a pattern. Once you find a pattern with fewerstudents, then you can solve the problem with 8 students. J

3 A window washing company has been hired to wash the windows on a glassbuilding. Each of the four sides of the building is 64 feet high and 100 feetlong. The manager wants to divide the work evenly among 4 crews. Howmany square feet of windows will each crew have to wash?A 25,600 ft2

B 6,400 ft2

C 256 ft2

D 64 ft2

Since the building has 4 sides of the same dimensions, assume that each ofthe four crews will wash one side of the building. Therefore, there is noreason to find the total area of the whole building since you would multiplyone side by 4 and then divide by 4. So, simply find the area of one side ofthe building which is 64 � 100, or 6,400 ft2. B

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Examples

MR 1.3


1 Marzieh’s sixth grade science class went on a field trip to theExploratorium in San Francisco. The trip cost $3 per student foradmission to the Exploratorium, and $4 for lunch. Which is not areasonable first step for finding the total cost of the trip?A Multiply the number of students by 3.B Multiply the number of students by 4.C Subtract $3 from $4.D Add $3 and $4.

2 Stoney is buying himself a set of golf clubs. The $199 clubs are on salefor 25% off. The sales tax is 8%. Which is the best second step forfinding how much he will pay for the clubs?F Find 25% of $199.G Find 8% of $199H Find 8% of $149.25J Find 25% of $214.52

3 Jessica wants to find the surface area of a crate. How can she break thisproblem into simpler parts?A Find the area of the front, side, and top surfaces. Then multiply the

sum of the three areas by 2.B Find the volume and divide by 3.C Find the area of the largest side and multiply by 6.D Find the area of the base and multiply by the height.

4 In which situation would it be most useful to break a problem intosimpler parts?F Round 42,198 to the nearest hundred.G Find the product of 492 � 10.H Find the area of a square with a side that measures 2 feet.J Find the number of ways Louise can arrange 5 ornaments on a shelf.

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Use estimation to verify the reasonableness of calculated results.

1 Elijah and his father are driving from their home in Barstow to visit hisgrandmother in Las Vegas. The trip is about 152 miles and his father willaverage about 62 miles per hour. Which of the following expressions wouldprovide the most reasonable estimate for the time the trip will take?A 150 � 60B 160 � 60C 150 � 65D 155 � 65

To estimate, round both the dividend and the divisor to the nearest ten. So,152 rounds down to 150 and 62 rounds down to 60. Therefore, 150 � 60 isthe most reasonable estimate. A

2 Sonya found the product of 24,879 and 0.046 several times using acalculator. Use an estimate to choose the correct product.F 114,434G 11,443.4H 1,144.34J 1.14434

Use the compatible numbers 25,000 and 0.04 to get the product of 1,000. Theanswer closest in value to the estimate is 1,144.34. H

3 Sergio bought 9.7 gallons of gasoline costing $1.699 per gallon. Use anestimate to choose the correct total cost.A $15.41B $15.99C $16.48D $18.04

Round 9.7 to 10 and $1.699 to $1.70. Then multiply 10 � $1.70 is $17. Theanswer closest in value to the estimate is $16.48. C

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Examples

MR 2.1


1 When they are standing in formation on the football field, the JeffersonMiddle School band has 8 members on each of 12 yard lines. Which ofthe following expressions would be best to use to obtain a reasonableestimate for the number of students in the band?A 10 � 12B 8 � 12C 10 � 10D 9 � 9

2 Chase found the quotient of 74,879 and 257 several times using acalculator. Use an estimate to choose the correct quotient.F 29.14G 291.36H 2,913.58J 29,135.79

3 The prices for soft drinks at a convenience store range from $0.59 to$1.29. What is the best estimate for the total cost of four soft drinks fromthe store?A $2.40B $4.00C $4.80D $5.20

4 Lori bought 189.7 shares of stock costing $19.50 per share. Use anestimate to choose the correct total cost.F $36.99G $369.92H $3,699.15J $36,991.50

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Apply strategies and results from simpler problems to more complexproblems.

1 Find 111,111,1112.A 12,345,677,654,321B 12,345,678,987,654,321C 123,456,789,987,654,321D 1,234,567,890,987,654,321

Solve simpler problems and look for a pattern. 112 � 121, 1112 � 12,321,and 1,1112 � 1,234,321. Continuing the pattern, 111,111,1112 is12,345,678,987,654,321. B

2 Find the sum of all the counting numbers from 1 to 4,000.F 44,040,000G 8,002,000H 4,001,000J 414,141

Look for a pattern in a simpler problem: 1 � 2 � 3 … 8 � 9 � 10.Observe that each pair of numbers in the sequence adds to 11.The sum of the sequence from 1 to 10 is 11 � 5 � 55.Since there are 5,000 numbers in the sequence from 1 to 4,000, there are2,000 pairs of numbers, and each pair has a sum of 4,001.2,000 � 4,001 � 8,002,000 G

3 The four triangles in the parallelogram shown at the right are all the same size. What is thearea of each of these triangles?A 80 cm2

B 40 cm2

C 20 cm2

D 10 cm2

The formula for the area of a parallelogram is A � bh where b is the lengthof the base and h is the height. The problem states that triangles in theparallelogram are all the same size. Because there are four triangles, each one

must have the area of the entire parallelogram. So, find the area of the

parallelogram and then multiply it by . A � 5 � 16 � , or 20 cm2. C1�4

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Examples

MR 2.2


1 Wynona knows that 310 � 59,049. What is the best way she can use thisinformation to find 312 if she does not have a calculator?A Multiply 3 by itself 12 times.B Multiply 59,049 by 3.C Multiply 59,049 by 9.D Find 32 and then add the result to 59,049.

2 If 108 � 100,000,000 and 109 � 1,000,000,000, how many zeros would1059 have if it were written in standard form?F 60G 59H 58J 57

3 Refer to the figure at the right. Suppose the small squares numbered 1, 2, 3, and 4 each have sides 1 centimeter long. You can use the fact that ABCD has sides 4 centimeters long and that the small squares are 1 centimeter on a side to figure out that each of the shaded regions has an area of 6 square centimeters. Suppose you placed 5 squares, each 2 centimeterson a side, corner to corner in a similar way. If you draw a large squarethat contains the diagonal strip of squares and shade above and below thestring of squares, what is the area of each shaded region?A 35 cm2

B 40 cm2

C 45 cm2

D 50 cm2

1

2

3

4

A D

CB

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Estimate unknown quantities graphically and solve for them by usinglogical reasoning, arithmetic, and algebraic techniques.

Use the graph at the right to answer Questions 1 and 2.

1 What is the corresponding y-value for an x-value of �1.75?A �1B �2C �2.5D �3

Estimate the location of �1.75 on the x-axis. Then look down to find wherethe graph intersects with the x-value of �1.75. On this graph, an x-value of�1.75 corresponds with a y-value below �2 but above �3. Answer choice C,�2.5, is the closest value. C

2 Which equation best describes points (x, y) on the graph?

F y � x � 1

G y � 2xH y � 2x � 1J y � 2x � 1

All points on the graph are solutions to its equation. So, pick a point such as(1, 3) and substitute these values for x and y for the possible equations to findout which equation is true for the values. The only equation that is true whenthe values (1, 3) are substituted for x and y is y � 2x � 1. J

3 The results from a one-week cookie sale are shown in the graph at the right. How many boxes of cookies were sold on Thursday?A 13B 9C 8D 5

Look on the horizontal axis for Thursday. Since there are two Ts (for Tuesdayand Thursday), look for the one that is between Wednesday and Friday. Then,look up to see where the graph crosses the grid mark for Thursday. The graphcrosses above 12 but below 14. Answer choice A, 13, most closely matchesthe value on the graph. So 13 boxes were sold on Thursday. A

M W FS T TDay of the Week

Cookie Sales

Numberof Boxes

Sold

S

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Examples

MR 2.3



1 What is the corresponding y-value for an x-value of 2?A 3

B

C 2

D

2 Which equation best describes the points (x, y) that are on the graph?

F y � � x

G y � � x

H y � x

J y � x

The graph shows the average hightemperatures for each month in SanFrancisco. Use thegraph to answerQuestions 3 and 4.

3 Which months have high temperatures above 70 degrees?A May and JuneB June and JulyC July and AugustD June through October

4 Between which two months does the average high temperature change themost?F October and November G February and MarchH April and May J December and January

January

FebruaryMarch April May

JuneJuly

August

September

October

November

December

Average High Temperature80757065605550

Temperature(�F)

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Use a variety of methods, such as words, numbers, symbols, charts, graphs,tables, diagrams, and models, to explain mathematical reasoning.

Use the chart at the right to answer Questions 1 and 2.

1 The chart at the right tells how many SUVs and trucksRaquel saw on a recent trip to Napa. If she saw a total of 98 vehicles (cars, trucks, and SUVs), how many were cars?A 58B 56C 42D 24

Each tic mark indicates one SUV or truck. Count the number of SUVs andtrucks and subtract that number from 98 to find the number of cars thatRaquel saw. 98 � 42 � 56. Raquel saw 56 cars. B

2 Suppose Raquel sees 120 vehicles on her way back home from Napa. Whichof the following is the best estimate for the number of those vehicles that willbe cars?F 68G 75H 90J 105

Since Raquel saw 98 vehicles on her trip to Napa, the number of cars she saw(56) is a little more than 56% of the total number of vehicles. So, find thenumber in the answer choices above that is closest to 56% of 120, or120 � 0.56 � 67.2. F

3 At John Sutter Middle School, 235 of the 614 students participate in extracurricular sports. The graph shows the percent of the students who participate in the four different sports. How many students play basketball?A 35B 42C 64D 94

From the chart, you know that 40% of the students who play sports playbasketball. Simply find 40% of the number of students who play sports, or235 � 0.4 � 94. D

Basketball40%

Track15%

Volleyball18%

Football27%

SUVs and Trucks

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Examples

MR 2.4


The graph shows fruit box sales for each week of the Chowchilla Union High School FFA club fundraiser. Use this graph to answer Questions 1 and 2.

1 In which two weeks do students appear to havesold the same number of fruit boxes?A week 1 and week 2B week 1 and week 3C week 2 and week 3D week 2 and week 4

2 From the graph, which would you say is the best estimate for the averagenumber of boxes sold each week?F 25G 30H 35J 40


3 In a recent year, the top 5 agriculturalproducing states sold $76.6 billion worth of agricultural products. Use the graph to estimate California’s agricultural sales.A $26 billionB $15 billionC $13 billionD $9 billion

4 California’s agricultural sales were about how much greater than Texas’sagricultural sales?F $42 billionG $26 billionH $16 billionJ $10 billion

California34%Nebraska

14%

Illinois13%

Iowa18% Texas

21%

Top Five Agricultural States

50403020100

Fruit Boxes Sold

BoxesSold

1 2Week

3 4

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Express the solution clearly and logically using the appropriatemathematical notation and terms and clear language; support solutionswith evidence in both verbal and symbolic work.

1 Mai Li has found rock climbing shoes that range in price from $47 to $93.Which sentence best describes the price p of a pair of shoes that she isconsidering buying?A p � $47B p � $93C p $47D $47 � p � $93

Since the shoes that Mai Li has found range in price from $47 to $93, theprice of the shoes she’s considering buying could be anywhere from $47 to$93. The only sentence that describes this range is $47 � p � $93. D

2 Which sentence best describes the graph shown at the right?F x 3G x � 3H �7 � x � 3J x �7

The graph starts at 3 and extends to the left end of the number line. Since thepoint where the graph begins at 3 is an open circle, you know that the valuesgraphed are less than but not equal to 3. Since the graph has an arrow on theleft end, you know that the graph continues on to infinity. So, the correctsentence is x � 3. G

3 Which of the given values is greater than 1?A 9.25 � 10�5

B 7.06 � 10�2

C 1.58 � 10�1

D 1.34 � 102

The values are all expressed in scientific notation. The exponent of 10 ineach expression tells which way and how many places to move the decimalpoint when writing the number in standard form. A negative exponent meansthat the decimal point is moved to the left, resulting in a number less than 1.A positive exponent means move the decimal point at the right, resulting in anumber greater than 1. So, 1.34 � 102 � 134, which is greater than 1. D

–2 2 3 4 5–1 60 71 8

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Examples

MR 2.5


1 Jamail is shopping for a fly fishing rod and reel. The lowest priced rodand reel that he has found was $34.99. Which sentence best describes theprice p of the rods and reels Jamail has found?A p � $34.99B p $34.99C p $34.99D p � $34.99


2 Which expression represents theMarch average temperature forFresno?F 5.50 � 101

G 5.50 � 10�1

H 6.90 � 102

J 4.53 � 101

3 Which expression represents the average temperature t on any given dayin Fresno?A t � 82B t 43C 43 � t � 82D 43 t 82

4 The graph shown is not labeled. Which of the following must be the sentence that is represented by the graph?F y 4G y �5H y �3J y � �4

80

60

40

20

0 J F M A M J J A S O N DMonth

Monthly Average Temperaturesfor Fresno, California

Temperature(�F)

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Indicate the relative advantages of exact and approximate solutions toproblems and give answers to a specified degree of accuracy.

1 Which distance would it be most important to know to the nearest meter?A The width of your deskB The height of your school buildingC The distance from San Diego to TijuanaD The distance from Los Angeles to Washington, DC

Look for the distance that would be measured in meters. The height of yourschool building would most likely be measured in meters, so rounding to thenearest meter would make the most sense. The other listed distances wouldnot be measured in meters, so it would not make sense to round them to thenearest meter. B

2 Every weekday, Julio’s father drives from their home in Redwood City to hisjob in San Francisco. The round trip is 53 miles. About how far does he drivein one five-day workweek?F 200 miG 250 miH 300 miJ 350 mi

It is much easier to find 50 � 5 than 53 � 5, so an approximation isappropriate for the solution to this problem. Julio’s father drives about 250 miles each week. G

3 What is the value of to the nearest thousandth?

A 0.187B 0.1875C 0.188D 0.19

To convert a fraction to a decimal, divide the numerator by the denominator.In this case 3 � 16 � 0.1875. Finding the value of 0.1875 to the nearestthousandth means rounding it to the thousandths place. Seven is the numeralin the thousandths place. Since the numeral at the right is 5, round the 7 upto 8. The result is 0.188. C

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Examples

MR 2.6


1 Cherie is measuring her height with a tape measure that is marked incentimeters. How should she record her height?A To the nearest millimeterB To the nearest centimeterC To the nearest inchD To the nearest meter

2 Duane rides his bike a distance of 4,237 meters from his house to schoolevery day. About how far is Duane’s house from his school?F 2 kmG 3 kmH 4 kmJ 5 km

3 Ricardo’s mother bought some stock in a computer company that cost

$28 per share. How much did the stock cost in dollars and cents?

A $28.66B $28.67C $28.68D $28.69

4 In science class, Jewel and her lab partner measured the mass of 35 jellybeans with a triple-beam-balance. The pointer on the balancescale ended up just a little past 40 grams. The scale is marked in wholegrams. What measurement should they record?F 40 gG 40.1 gH 40.25 gJ 40.5 g

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Make precise calculations and check the validity of the results from thecontext of the problem.

1 Ms. Aguilera needs to put trim around her classroom bulletin board. Thewidth of the bulletin board is 85 cm and the height is 130 cm. How muchtrim does she need?A 170 cmB 215 cmC 260 cmD 430 cm

The bulletin board is a rectangle. The perimeter of a rectangle is equal totwice the width plus twice the length. First estimate by rounding the widthand length to 100 cm, which gives you a perimeter estimate of 400 cm. Then,find the actual perimeter as 2(85) � 2(130) � 430. Check to make sure theanswer is reasonable compared to the estimate. D

2 The Sears Tower in Chicago is 1,454 feet high. The First Interstate WorldCenter Building in Los Angeles is 1,018 feet high. How much taller is theSears Tower than the First Interstate World Center Building?F 2,472 ftG 454 ftH 436 ftJ 418 ft

First estimate by rounding the heights to 1,450 and 1,000 feet. This gives adifference of about 450 feet. Then subtract to find the difference.1,454 � 1,018 � 436. Is this reasonable when compared to the estimate? Yes. H

3 Darius wants to hang the United States and California flags on his front porch.

He has a wooden rod that is 7 feet long that he wants to cut in half to hang

each flag on. At what point should he mark the rod so that he can cut itexactly in half?

A 3 ft B 3 ft

C 4 ft D 4 ft

First estimate. Half of 7 feet would be 3 feet. Then divide.

7 � 2 � �

� � or 3 . This is reasonable compared to the estimate, so Darius

should mark the rod to cut it in half at 3 feet. B7�10

7�10

37�10

1�2

37�5

1�2

37�5

2�5

1�2

1�5

7�10

1�2

2�5

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Examples

MR 2.7


1 The area of California is 158,869 square miles. Of that area, 155,973 squaremiles is land and the rest is covered by water. How much of the area inCalifornia is covered by water?A 5,280 mi2

B 4,842 mi2

C 3,508 mi2

D 2,896 mi2

2 Milk products are California’s top selling agricultural products. In 1997,California produced $3.626 billion in milk products. The second rankingagricultural product was grapes. In 1997, California produced $2.819 billionin grapes. How much higher was the value of milk products in 1997?F 6.445 billionG 1.253 billionH 0.807 billionJ 0.512 billion

3 Angela’s little sister has an inflatable swimming pool that is 24 cm high,240 cm long and 120 cm wide. How much water will the pool hold?A 691,200 cm3

B 28,800 cm3

C 5,760 cm3

D 2,880 cm3

4 Jason needs to cover the floor of his room before he paints the walls. Thefloor of his room has an area of 140 square feet. He has drop cloths thateach cover 35 square feet. How many of the drop cloths does he need tocover the floor?F 1G 2H 3J 4

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Evaluate the reasonableness of the solution in the context of the originalsituation.

1 Folsom Middle School has 420 students. On any given day, about 6% of thestudents are absent. Which is the most reasonable answer for the number ofstudents who are absent on a given day?A 21B 25C 28D 32

Round the 6% down to 5% and mentally find 5% of 420 for an easy estimateof 21 students. However, since 6% of 420 is a little more than 5% of 420, 25 is a more reasonable answer than 21. B

2 Delbert hopes to earn between $800 and $1,000 mowing lawns this summer.If he makes $20 for each lawn he mows, what is a reasonable number oflawns he must mow during the summer to reach his goal?F 40 or fewerG 40H between 40 and 50J 50 or more

At $20 per lawn, Delbert would need to mow 40 lawns to make $800, and 50 lawns to make $1,000, so he will need to mow between 40 and 50 lawns. H

3 Sandra has to go to the dentist for a checkup every 4 months. What is themost reasonable estimate for the number of times she’ll have to go to thedentist in a 3-year period?A 6B 9C 12D 18

Since there are 12 months in a year, then Sandra goes to the dentist 3 times ayear. In a 3-year period, Sandra would likely go to the dentist 9 times. B

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Examples

MR 3.1


1 Samuel is helping his mother fertilize their lawn. The lawn has an area of11,000 square feet and each bag of fertilizer covers about 2,000 squarefeet. How many bags will they need in order to cover the entire lawn?A 3B 4C 5D 6

2 Jacque is flying a kite. She has tied three 48-foot rolls of string togetherto fly the kite. If she has let out all of the string, what is a reasonableestimate for the height of the kite?F 120 ftG 150 ftH 175 ftJ 200 ft

3 Michael is having a clambake on the beach at Big Sur and he has invited9 of his friends. He estimates that each person will eat 12 clams, but he isnot sure whether everyone will show up. What is the most reasonableestimate for the number of clams that will be eaten at Michael’sclambake?A 120B 115C 100D 25

4 Delia has a job handing out surveys at a shopping mall. Usually between5% and 7% of the surveys she hands out are completed and returned toher. If she hands out 144 surveys one evening, how many can she expectto be returned?F 7 or fewerG between 7 and 12H exactly 12J more than 12

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Note method of deriving the solution and demonstrate conceptualunderstanding of the derivation by solving similar problems.

1 To solve the equation 4x � 3 � 25, first add 3 to each side and then divideeach side by 4. How can you solve the equation 40 � 6y � 8?A Add 40 to both sides and then divide by 6.B Add 6 to both sides and then divide by 8.C Add 8 to both sides and then divide by 40.D Add 8 to both sides and then divide by 6.

To solve a two-step equation, “undo” the operations in reverse order. So,undo the subtraction by adding 8 to both sides of the equation. Then undo themultiplication by dividing both sides by 6. D

2 To convert kilometers to meters, multiply the number of kilometers by 1,000.How would you convert meters to kilometers?F Multiply the number of meters by 1,000G Divide the number of meters by 1,000H Subtract 1,000 from the number of metersJ Add 1,000 to the number of meters

Since you multiply by 1,000 to convert kilometers to meters, undo theoperation by dividing by 1,000 to convert meters to kilometers. G

3 Sean knows that approximately 10% of the population of California lives inLos Angeles. To find the population of Los Angeles, he multiplied the totalpopulation of California by 0.1. If about 3% of the population of Californialives in San Francisco, how can Sean find the population of San Francisco?A Multiply the total population of California by 0.03B Multiply the total population of California by 0.3C Multiply the total population of California by 0.1D Divide the total population of California by 0.03

To find the percent of a number, change the percent to a decimal and thenmultiply the decimal by the number. In this case, 3% becomes 0.03. Thenmultiply the total population of California by 0.03 to find 3% of thepopulation of California, which is about the population of San Francisco. A

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Examples

MR 3.2


1 To convert inches to centimeters, multiply the number of inches by 2.54.How would you convert 20 centimeters to inches?A 20 � 2.54B 2.54 � 20C 20 � 2.54D 20 � 25.4

2 To solve the equation z � 8 � 1, subtract 8 from both sides. How canyou solve the equation �5 � b � 6F Add 6 to both sides.G Multiply both sides by 6.H Subtract 6 from both sides.J Divide both sides by 6.

3 Sylvia knows that 20% of 250 is 50. What is 20% of 320?A 72B 64C 50D 32

4 If 1,000 millimeters equals 1 meter then what is 1,000 milliliters equal to?F 1 LG 10 LH 100 LJ 1,000 L

5 Joshua solved the equation 4x � 320 by dividing both sides of the

equation by 4. How would he solve the equation � 70?

A Divide both sides by 5.B Subtract 5 from both sides.C Add 5 to both sides.D Multiply both sides by 5.

y�5

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Develop generalizations of the results obtained and the strategies used andapply them in new problem situations.

1 Jessica knows that the solution to the equation x � 12 � 17 is 5. What is thesolution to the equation 29 � 4 � y?A 33B 25C 17D 12

Jessica solved the problem by subtracting 12 from both sides of the equation.Solve 29 � 4 � y similarly by subtracting 4 from both sides to get 25 � y. B

2 If 10% of 1,200 is 120 and 20% of 1,200 is 240, find 30% of 2,000.F 200G 400H 600J 800

Notice that finding 10% of 1,200 is the same as finding 1,200 � 0.1 and thatfinding 20% of 1,200 is the same as finding 1,200 � 0.2. So, finding 30% ofa number should be the same as multiplying the number by 0.3. In that case,30% of 2,000 is 2,000 � 0.3, or 600. H

3 When Albert simplifies (8 � 12) � 7 he gets 140 and when he simplifies 80 � (11 � 6) he gets 16. What will be the result when he simplifies 4 � (3 � 5) � 7?A 25B 18C 10D 4

Notice that in Albert’s simplifications, he did the operations within theparentheses first because (8 � 12) � 7 is the same as 20 � 7, or 140, and 80 � (11�6) is the same as 80 � 5, or 16. So, when he simplifies theexpression 4 � (3 � 5) � 7, he should do the operation within theparentheses first.

4 � (3 � 5) � 7 � 4 � 8 � 7 Simplify within the parentheses first.� 32 � 7 Multiply.� 25 The answer is 25. A

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Examples

MR 3.3


1 If the solution to the equation w � 11 � 62 is 73, then what is thesolution to the equation 45 � �13 � x?A 73B 58C 32D 28

2 Gerald found that 15% of 1,000 is 150 and 16% of 1,000 is 160. What is17% of 1,500?F 170G 200H 255J 275

3 When Serena simplified the expression 2 � 3 � 5, the result was 17.When she simplified 8 � 27 � 3 � 4 the result was 3. What will be theresult when she simplifies 7 � 22 � 4 � 12?A 300B 77C 53D �19

4 If the solution to the equation 2x � 1 � 5 is 2, then what is the solutionto the equation 3x � 8 � 44?F 12G 9H 6J 2

5 Georgio knows that 3 � 2 � 6, 3 � 3 � 9, 3 � 4 � 12, and 3 � 5 � 15.So, which of the following must be the product of 3 � 1,258,757?A 3,776,271B 3,782,458C 3,759,512D 3,781,864

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1 Which place in the decimals indicates that 23.5395 is greater than23.5164?

A TensB TenthsC HundredthsD Thousandths

2 Several classes from Kinsey Middle School visited a science museum.For every 25 students there were 2 adults. If there were 125 students inthe group, how many adults went with them?

F 8G 9H 10J 15

3 Masao bought a 5-pack of paper for $7. Which equation could be usedto find c, the cost for one pack of paper?

A 5 � c � 7B 7 � c � 5

C � 5

D 5c � 7

4 The 6th grade class at Harrison Middle School sold 125 tickets to theschool dance in 5 days. What is the average number of tickets the 6thgrade class sold per day?

F 5G 21H 25J 30

5 Nicoletta is traveling along Interstate 10 in Los Angeles, California, to goto Houston, Texas. If her average speed is 65 miles per hour, how longwill it take her to travel the 1,560 miles between the two cities?

A 12 hB 17.2 hC 24 hD 42 h

c�7

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Sample Test


6 Katie needs to travel 165 miles and her average speed is 55 miles perhour. How long will it take her to reach her destination?F 115 minG 10 hH 4 hJ 3 h

7 Which statement is always true for two adjacent angles?A Their measures have a sum of 180°.B They are equal in measure.C Their measures have a sum of 90°.D They share a common side.

8 Which of the following is not a possible kind of triangle?F Acute isosceles triangleG Obtuse equilateral triangleH Acute equilateral triangleJ Obtuse isosceles triangle

9 Kichi wants to create a 2-digit secret code. Each of the digits can be anyof the numbers from 0 through 4. How many different codes can Kichimake?A 8B 10C 16D 25

10 300,000 � 40,000 � 800 � 76 �F 34,876G 304,876H 340,876J 348,760

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Sample Test


11 Dividing by is equivalent to multiplying by what number?

A

B 3C 5D 8

12 The record high temperature for San Diego, California, is 111°F and therecord low temperature is 25°F. What is the difference between thesetwo temperatures?F 86°FG 96°FH 106°FJ 136°F

13 Jeremy read the water level changes in the Kings River in Californiafour times during the summer. He made his measurements to the nearestfoot and used a negative sign to indicate a decrease in the water level.His four measurements were 1 foot, �1 foot, �3 feet, and 1 foot. Whatwas the overall increase or decrease in the water level?A 2 ftB 1 ftC �1 ftD �2 ft

14 The Two California Plaza Building in Los Angeles is 750 feet tall. Howmany yards tall is the building?F 25 ydG 220 ydH 250 ydJ 2,250 yd

15 Which is a correct formula for the circumference of a circle that has aradius of r units and a diameter of d units?A C � 2�dB C � �r2

C C � 2�rD C � �d2

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Sample Test


16 A number cube contains 2 odd prime numbers. If the number cube isrolled, what is the probability that the number rolled will be an oddprime number?

F

G

H

J

17 Eight students won $120 in a science competition. How much will eachstudent receive if they share the winnings equally?A $12B $13C $14D $15

18 Multiply 5,063 and 45.F 227,835G 45,567H 27,834J 25,335

19 Find the sum of and .

A

B

C

D

20 A shoe store advertisement stated that the store was selling 2 pairs ofshoes for $57.95. If a family of 6 bought a pair of shoes for eachmember of the family, how much did they spend?F $347.95G $173.85H $165.90J $115.90

5�20

5�15

11�15

11�5

2�15

3�5

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1�3

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Sample Test


21 Zenia is buying end-of-school gifts for her teachers. She decides to givethe female teachers engraved pendants that cost $16 each and give themale teachers engraved pens that cost $18 each. If f represents thenumber of female teachers Zenia has and m represents the number ofmale teachers she has, which expression could be used to determine thetotal cost of her gifts?A 16 � 18 � m � fB 16f � 18mC f � mD 16m � 18f

22 Alma is buying 2 pairs of blue jeans and 3 sweaters for her camping tripto Mt. Shasta. The expression 2b � 3s can be used to find her total costif b is the cost of one pair of blue jeans and s is the cost of one sweater.What is the total cost of her purchases if one pair of blue jeans costs $32and one sweater costs $26?F $58G $80H $90J $142

23 A formula for the area of a triangle is A � bh, where b is the base and

h is the height of the triangle. Which quantity can you be sure stands forthe length of one of the sides of the triangle?

A

B A

C bD h

24 The formula for the area of a rectangle is A � �w, where � is the lengthand w is the width of the rectangle. Which formula can be used to findthe length, �, of the rectangle if you know the area and the width?

F � �

G � �

H � � AwJ � � A � w

A�w

w�A

12

1�2

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Sample Test


25 The area, A, of a triangle is equal to the product of the length of the

base, b, and the height, h. Which equation can you use to find the heightof the triangle if you know the area and the base?

A h �

B h �

C h � AbD h � 2Ab

Use the figure at the right to answer Questions 6 and 7.

26 Without knowing any measurements, what is true about angle 1 and angle 2?F They are complementary angles.G They are supplementary angles.H They are adjacent angles.J They are vertical angles.

27 If angle 2 has a measure of 90°, what is the sum of the measures ofangle 1 and angle 3?A 360°B 180°C 90°D 45°

28 If one of two complementary angles has a measure of 83°, what is themeasure of the other angle?F 180°G 97°H 90°J 7°

29 What kind of triangle is shown at the right?A Equilateral triangleB Acute triangleC Obtuse triangleD Right triangle

A

CB 35° 55°

1 32

2A�b

A�2b

1�2

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Sample Test


30 Which figure is an equilateral triangle?F G

H J

31 The table at the right shows the heights of 5 buildings inLos Angeles. If the height of the Wells Fargo Tower,which is 740 feet tall, wereincluded in the data, whichmeasure would show the least amount of change?A MeanB MedianC ScaleD Range

32 If a data value of 12 is added to the data set below, which measure willchange?

4, 5, 5, 9, 10, 11, 13, 14, 18, 20, 23

F MeanG MedianH ModeJ Range

33 Which measure could be used to find the type of shirt most commonlyworn by students in the 6th grade?A MeanB ModeC MedianD Range

Building Height (feet)

First Interstate World Center 1,018First Interstate Tower 858Two California Plaza 750Gas Company Tower 749333 South Hope Bldg. 743

12

10 11

10

10 10

8

10 11

12

10 10

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34 In a survey, of the students in a large random sample of students at a

school said they liked the lunches that were served at school. If there are425 students in the school, what is a reasonable estimate of the numberof students who like the school lunches?F 85G 340H 400J 531

35 Many television stations survey a portion of the population to see whichshows are being watched the most. What type of sample of thepopulation is the most appropriate?A Random sampleB Cluster sampleC Convenience sampleD Stratified random sample

36 A video store manager wants to add more children’s movies to thestore’s selection. To determine whether this is a good idea, the store staffpoll some of the people who come in the video store on the weekend.This is considered an example of what type of sample?F RandomG SurveyH ConvenienceJ Cluster

37 The graph at the right shows the number of farms in the United States during a period of 57 years. Why could the graph be considered misleading?A The scale for the year is not in

equal intervals.B The scale for the number of farms in not in equal intervals.C The number of farms is decreasing from year to year.D The points are connected on the graph.

1940 1950 1960Year

Farms in the United States

Numberof Farms(millions)

1970 1980 1997

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38 In 1997, 1 of the 15 players with 100 or more at-bats on the AnaheimAngels baseball team finished the season with a batting average of 0.300or higher. In 1998, 3 of the 13 players with 100 or more at-bats on theAnaheim Angels baseball team finished with a batting average of 0.300 orhigher. If a player with 100 or more at-bats is selected at random fromeach of the two teams, what is the probability that they both had abatting average of 0.300 or higher?

F

G

H

J

39 Juan bought 3 apples and 4 pears at 25¢ each, peanuts at $1.49 perpound, and a 10-pound bag of potatoes for $2.49. What else do you needto know in order to find out the total cost of the shopping trip?A The cost of an appleB The number of potatoes purchasedC The number of pounds of peanuts purchasedD The number of pounds of potatoes purchased

40 The table below lists the first six powers of 9. Shading shows the powersthat end in the digit 1. The exponents for these powers are circled. Whichof the following describes the exponents that give a power of 9 that endsin the digit 1?

F Powers of 2G Multiples of 2H Factors of 8J Factors of 12

Exponent (n)

Power (9n)

1

9

2

81

3

729

4

6,561

5

59,049

6

531,441

1�65

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Sample Test


41 Which statement is always true about the sum of a pair of numbers ifeach of the numbers gives a remainder of 3 when divided by 5?A The digit in the ones place of the sum must be 6.B The digit in the ones place of the sum must be 1.C The digit in the ones place of the sum must be 1 or 6.D The digit in the ones place of the sum can be 1, 6, or 7.

42 Using the table at the right, what is the value of 27?F 32G 64H 128J 256

43 In a recent year, the average number of miles per gallon of gasoline forcars driven in California was 18.02 miles per gallon. Which statementdescribes the meaning of this average?A The number of miles driven each day by a car in California was

18.02 miles.B The number of gallons of gasoline used each day by a car in

California was 18.02 gallons.C The number of gallons of gasoline used by cars in California divided

by the number of miles driven was 18.02.D The number of miles driven by cars in California divided by the

number of gallons of gasoline used was 18.02.

44 Logan plans to read 55 books in one year. What is a reasonable estimatefor the amount of time he can take to read each book?F 3 daysG 1 weekH 21 daysJ 5 weeks

45 If the sum of the digits of a whole number is divisible by 3, then thewhole number itself is divisible by 3. Which digit could you use in placeof the underlined digit in the number shown below to get a number thatis divisible by 3?

29,379,134

A 5 B 6C 7 D 8

Power Value21 222 423 824 16

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46 On Tuesday, Eric studied math for 35 minutes and English for 25 minutes.What is the ratio of time spent studying English to the time spentstudying math?F 4 to 5G 5 to 4H 5 to 7J 7 to 5

47 For the triangles shown at the right, �STUis similar to �ZYX. What is the length of

—YZ?

A 6B 8C 18D 24

48 If �ABC is similar to �DEF, find the length of D�E�.F 2G 3H 4J 5

49 What is the total cost of a pair of pants that costs $30 if the state sales

tax is 6 %?

A $1.95B $28.05C $31.95D $36.50

50 The seven national parks that are entirely in California cover almost3,000,000 acres. If the Channel Islands Park covers a little more than8% of the total acreage, about how many acres are contained in theChannel Islands National Park?F 2,400 G 24,000H 240,000 J 2,400,000

51 A $78 portable CD player is on sale for 20% off the regular price. Whatis the sale price of the CD player?A $15.60 B $39.00C $58.00 D $62.40

1�2

A C

D F

Ex

B5

10

6

S U

X Z

YT

12

12

18

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52 According to the Food Guide Pyramid, cup of fruit juice is equivalent

to 1 serving from the fruit group. Using this guide, 4 cups of fruit

juice are equivalent to how many servings from the fruit group?

F 3 G 3

H 5 J 6

53 Which explanation describes how to multiply two fractions?

A Multiply the first fraction by the reciprocal of the second fraction.B Rewrite the fractions with common denominators, then add the

numerators.C Rewrite the fractions with common denominators, then divide the

numerators.D Multiply numerators and multiply denominators.

54 Which expression is equivalent to � ?

F � G �

H � J �

55 By which number should you divide the numerator and denominator to

write in simplest form?

A 4 B 6C 9 D 12

56 Angie gets paid every 6 days. Antoine gets paid every 14 days. If theyboth are paid today, how many more days will it be until the next timethey both are paid on the same day?F 21 days G 30 daysH 42 days J 84 days

57 Which value of b makes �6b � 42 true?A 8 B 7C �7 D �8

36�60

4�3

5�4

3�4

5�4

4�3

4�5

3�4

4�5

3�4

4�5

1�4

3�4

3�8

1�2

3�4

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58 Sarah is keeping a journal and wants to buy a new book to write in. Sheis looking at a book that contains 488 blank pages for writing. If Sarahwrites 8 pages per week, which is an equation that can be used to findw, the number of weeks Sarah can write in the book until it is full?F w � 488 • 8G 8w � 488

H � 488

J w �

59 Which expression is equivalent to (�4)(y � 9)?A 4(y � 9) B (�4 � y) � (�4 � 9)C (�4)y � (�4)(�9) D �4y � 9

60 What is the value of 4 � [23 � (�3)(�1)]?F �9 G �1H 16 J 15

61 To find the value of 62 � 12 � (6 � 3 � 2), which operation will youuse last?

A Addition B SubtractionC Multiplication D Division

62 Alyssa ran the 40-yard dash in 6 seconds. How many feet per seconddid she run?

F 6 ft/s G 20 ft/s

H 34 ft/s J 80 ft/s

63 Jacque traveled 261 miles in 4 hours. What was her average speed?

A 58 mi/h B 60 mi/hC 62 mi/h D 65 mi/h

64 Jermaine needs to ride his bicycle 10 miles to train for a race. If hisaverage speed was 15 miles per hour, how long did it take him to bikethe 10 miles?F 1 h 30 min G 50 minH 45 min J 40 min

1�2

2�3

8�488

w�8

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65 Which formula is true for the area of a circle with radius r and diameter d?A A � �r2

B A � �(2r)2

C A � �dD A � 2�r

66 Which of the following is often used as an approximate value for �?F 3.12G 3.13H 3.14J 3.15

67 Which expression can you use to find the approximate volume of the cylinder shown at the right?

A 10 • 7 • B • (52) • 7

C • (102) • 7 D • (5 • 7)2

68 Austin’s test scores are 83, 84, 90, 89, 79, 86, 84, and 93. What is themedian of Austin’s test scores?F 84 G 85H 86 J 87

69 Of the 9 professional baseball All-Star games played from 1995 to 2003, 6 were won by the American League. Using the data, what is theprobability that the American League wins an All-Star game?

A B

C D 1

70 Keisha is making a square pillow with side lengths of 1 feet. If she

wants to put trim around the edges of the pillow, how much trim willshe need for the entire pillow?

F 6 ft G 5 ft

H 4 ft J 3 ft1�3

2�3

2�3

2�3

2�3

3�5

2�3

1�3

22�7

22�7

22�7

22�7

7 cm

10 cm

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6th grade math standard test ebook (mc graw hill)

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