Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber Photo Credits Cover Joe McBride/CORBIS; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 11 12) The McGraw-Hill Companies; v (5 10 13 14) File Photo; viii Dynamic Graphics Group/ Creatas/Alamy; viii Jeremy Woodhouse/Getty Images; ix Glen Allison/Getty Images; 2–3 Kenneth Eward/Photo Researchers,Inc.; Photodisc/Getty Images; 10 Jeffrey L Rotman/Peter Arnold,Inc.; 13 Photodisc/Getty Images; 16 (b) Kevin Sanchez/Cole Group/Getty Images; 16 (t) The McGraw-Hill Companies, Inc.; 18 CORBIS; 24 Image Source/SuperStock; 29 Manchan/Getty Images; 32 Jeff Maloney/Getty Images; 38 G.K & Vikki Hart/Getty Images Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878212 MHID: 0-07-878212-0 Printed in the United States of America 10 055/027 16 15 14 13 12 11 10 09 08 07 California Math Triumphs Volume 5B California Math Triumphs Volume Place Value and Basic Number Skills 1A Chapter Counting 1A Chapter Place Value 1A Chapter Addition and Subtraction 1B Chapter Multiplication 1B Chapter Division 1B Chapter Integers Volume Fractions and Decimals 2A Chapter Parts of a Whole 2A Chapter Equivalence of Fractions 2B Chapter Operations with Fractions 2B Chapter Positive and Negative Fractions and Decimals Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Volume Ratios, Rates, and Percents 3A Chapter Ratios and Rates 3A Chapter Percents, Fractions, and Decimals 3B Chapter Using Percents 3B Chapter Rates and Proportional Reasoning Volume The Core Processes of Mathematics 4A Chapter Operations and Equality 4A Chapter Math Fundamentals 4B Chapter Math Expressions 4B Chapter Linear Equations 4B Chapter Inequalities Volume Functions and Equations 5A Chapter Patterns and Relationships 5A Chapter Graphing 5B Chapter Proportional Relationships 5B Chapter The Relationship Between Graphs and Functions Volume Measurement 6A Chapter How Measurements Are Made 6A Chapter Length and Area in the Real World 6B Chapter Exact Measures in Geometry 6B Chapter Angles and Circles iii Authors and Consultants AUTHORS Frances Basich Whitney Kathleen M Brown Dixie Dawson Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California Math Curriculum Staff Developer Washington Middle School Long Beach, California Math Curriculum Leader Long Beach Unified Long Beach, California Philip Gonsalves Robyn Silbey Kathy Vielhaber Mathematics Coordinator Alameda County Office of Education Hayward, California Math Specialist Montgomery County Public Schools Gaithersburg, Maryland Mathematics Consultant St Louis, Missouri Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California Dinah Zike Educational Consultant, Dinah-Might Activities, Inc San Antonio, Texas CONSULTANTS Assessment Donna M Kopenski, Ed.D Math Coordinator K–5 City Heights Educational Collaborative San Diego, California iv Instructional Planning and Support ELL Support and Vocabulary Beatrice Luchin ReLeah Cossett Lent Mathematics Consultant League City, Texas Author/Educational Consultant Alford, Florida Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc CONTRIBUTING AUTHORS California Advisory Board CALIFORNIA ADVISORY BOARD Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan Linda Anderson Cheryl L Avalos Bonnie Awes Kathleen M Brown 4th/5th Grade Teacher Oliveira Elementary School, Fremont, California Mathematics Consultant Retired Teacher Hacienda Heights, California Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California Math Curriculum Staff Developer Washington Middle School Long Beach, California Carol Cronk Audrey M Day Jill Fetters Grant A Fraser, Ph.D Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California Classroom Teacher Rosa Parks Elementary School San Diego, California Math Teacher Tevis Jr High School Bakersfield, California Professor of Mathematics California State University, Los Angeles Los Angeles, California Eric Kimmel Donna M Kopenski, Ed.D Michael A Pease Chuck Podhorsky, Ph.D Mathematics Department Chair Frontier High School Bakersfield, California Math Coordinator K–5 City Heights Educational Collaborative San Diego, California Instructional Math Coach Aspire Public Schools Oakland, California Math Director City Heights Educational Collaborative San Diego, California Arthur K Wayman, Ph.D Frances Basich Whitney Mario Borrayo Melissa Bray Professor Emeritus California State University, Long Beach Long Beach, California Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA Teacher Rosa Parks Elementary San Diego, California K–8 Math Resource Teacher Modesto City Schools Modesto, California v California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction Melody McGuire Math Teacher California College Preparatory Academy Oakland, California 6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California Eppie Leamy Chung Monica S Patterson Teacher Modesto City Schools Modesto, California Educator Aspire Public Schools Modesto, California Judy Descoteaux Rechelle Pearlman Mathematics Teacher Thornton Junior High School Fremont, California 4th Grade Teacher Wanda Hirsch Elementary School Tracy, California Paul J Fogarty Armida Picon Mathematics Lead Aspire Public Schools Modesto, California 5th Grade Teacher Mineral King School Visalia, California Lisa Majarian Anthony J Solina Classroom Teacher Cottonwood Creek Elementary Visalia, California Lead Educator Aspire Public Schools Stockton, California vi Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Bobbi Anne Barnowsky Volume 5A Functions and Equations Chapter Patterns and Relationships 1-1 Sort and Classify .4 KAF1.1, 1SDAP1.1 1-2 Patterns .13 1SDAP2.1, 2SDAP2.1 Progress Check .20 1-3 Number Relationships 21 2SDAP2.1, 3AF2.2, 1-4 Solve Equations .27 3AF2.1, 4AF1.5 Progress Check .33 Assessment Study Guide .34 Chapter Test .38 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Standards Practice 40 Dana Meadows near Yosemite National Park Chapters and are contained in Volume 5A Chapters and are contained in Volume 5B Standards Addressed in This Chapter KAF1.1 Identify, sort, and classify objects by attribute and identify objects that not belong to a particular group (e.g., all these balls are green, those are red) 1SDAP1.1 Sort objects and data by common attributes and describe the categories 1SDAP2.1 Describe, extend, and explain ways to get to a next element in simple repeating patterns (e.g., rhythmic, numeric, color, and shape) 2SDAP2.1 Recognize, describe, and extend patterns and determine a next term in linear patterns (e.g., 4, 8, 12 , the number of ears on one horse, two horses, four horses) 3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit) 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4) 4AF1.5 Understand that an equation such as y = 3x + is a prescription for determining a second number when a first number is given vii Contents Chapter Graphing Standards Addressed in This Chapter 2-1 Bar Graphs and Picture Graphs 44 1SDAP1.2, 2SDAP1.1, 2SDAP1.2 2-2 Line Plots 53 3SDAP1.3, 2SDAP1.1, 2SDAP1.2 Progress Check .60 2-3 Ordered Pairs 61 5SDAP1.5, 4MG2.0, 5SDAP1.4 2-4 Coordinate Grids 67 4MG2.1, 5SDAP1.4, 5SDAP1.5 Progress Check .75 Assessment Study Guide .76 Chapter Test .80 Standards Practice 82 2SDAP1.1 Record numerical data in systematic ways, keeping track of what has been counted 2SDAP1.2 Represent the same data set in more than one way (e.g., bar graphs and charts with tallies) 3SDAP1.3 Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot) 4MG2.0 Students use twodimensional coordinate grids to represent points and graph lines and simple figures 4MG2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line) 5SDAP1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph 5SDAP1.5 Know how to write ordered pairs correctly; for example, (x, y) viii Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Golden Gate Bridge, San Francisco 1SDAP1.2 Represent and compare data (e.g., largest, smallest, most often, least often) by using pictures, bar graphs, tally charts, and picture graphs Contents Chapter Proportional Relationships 3-1 Linear Patterns 3AF2.1, 3AF2.2 3-2 Ratios and Rates 11 3AF2.1, 6AF2.1 Progress Check .18 3-3 Proportional Reasoning 19 3AF2.1, 6NS1.3 Assessment Study Guide .26 Chapter Test .28 Standards Practice 30 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Wind turbines, Altamont Chapters and are contained in Volume 5A Chapters and are contained in Volume 5B Standards Addressed in This Chapter 3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit) 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4) 6NS1.3 Use proportions to solve problems (e.g., determine the value of N N if = _, find the length of a side of a 21 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 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches ix Contents Chapter The Relationship Between Graphs and Functions 4-1 Introduction to Functions 34 Standards Addressed in This Chapter 5AF1.5 4-2 Graph Linear and Nonlinear Equations 41 7AF3.0, 7AF3.1 Progress Check 50 4-3 Direct Variation .51 7AF3.3, 7AF3.4 4-4 Slope 59 7AF3.3, 7AF3.4 Progress Check 66 Assessment Study Guide 67 Chapter Test 72 Standards Practice 74 x 7AF3.0 Students graph and interpret linear and some nonlinear functions 7AF3.1 Graph functions of the y = nx2 and y = nx3 and use in solving problems 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle) Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc Redwood National Park 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid The ratio is constant The constant of variation is _3 or means that it cost The constant of variation $ for pony ride BOOKS The graph shows the ratio of the cost, y, to the number of books Diego buys, x Use the graph to find the constant of variation Interpret the direct variation Point A B C Number of Books, x Cost, y 12 _4 _8 12 _ y = x y = x y = x $PTU 11 _3 = = = _4 _4 = = y The ratio x is constant The constant of variation is The constant of variation book _4 means that it cost _4 or $4 for $ Z # " Y /VNCFSPG#PPLT Copyright © by The McGraw-Hill Companies, Inc Do the ordered pairs in the table represent a linear or nonlinear function? Explain 12 Point x y A B 10 C 15 E 10 25 linear This function is constant 13 D 20 because the ratios are Point A B C D E x 18 21 y 16 14 This function is not constant nonlinear because the ratios are GO ON Lesson 4-3 Direct Variation 57 Vocabulary Check Write the vocabulary word that completes each sentence 14 ratio A(n) is a relationship between two quantities in which the first measures a certain number of units and the second measures another number of units Constant of variation 15 is the rate of change in a direct variation 16 10 Writing in Math Interpret the ratio _ where y represents the number of ten-dollar bills, and x represents the number of hundred-dollar bills _ The ratio 10 means that there are 10 ten-dollar bills in 1 one-hundred-dollar bill Spiral Review Write a function and make a function table 17 MONEY Ferdinand pays $15 every month for a daily newspaper subscription How much money does Ferdinand spend each year to receive a newspaper every day? (Lesson 4-1, p 34) y= 15x Cost ($), y 45 90 12 135 180 Ferdinand will spend $ 180 each year to receive a newspaper every day Find each unit price 18 (Lesson 3-2, p 11) Ricardo bought 15 erasers for $1.50 19 The unit price is $ 0.10 for eraser 20 LaShonda bought juice boxes for $4.50 Julie bought mugs for $60 The unit price is $ 12 21 The unit price is $ 0.75 for juice box Dah-Chou bought picture frames for $120 The unit price is $ 15 picture frame 58 Chapter The Relationship Between Graphs and Functions for mug for Copyright © by The McGraw-Hill Companies, Inc Number of Months, x Lesson 4-4 Slope KEY Concept The slope of a line illustrates the ratio of the number of units of rise to the number of units of run for a linear function You can find the slope of a line from the graph of that line or by using the slope formula change in y Δy Slope = m = _ = _ Δx change in x = "MPXFSDBTFN SFQSFTFOUTTMPQF y2 - y1 m = x2 - x1 , where x2 ≠ x1 Copyright © by The McGraw-Hill Companies, Inc Look at the graph at the right Choose two points on the line, such as (-3, -2) and (0, 0) 5IFCMVF BSSPXTTIPX UIFSJTF VOJUT 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of the graph 7AF3.4 Plot the values of quantities whose ratios are always the same Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities VOCABULARY slope the ratio of the change in the y-value to the corresponding change in the x-value in a linear function Z • Start at point (-3, -2) • Move up parallel to the y-axis (rise) units • Move to the right parallel to the x-axis (run) units • You are at another point on the line, (0, 0) Y 5IFSFE BSSPXTTIPX UIFSVO VOJUT • From the point (0, 0) move up (rise) units • Move to the right (run) units You are at another point on the line, (3, 2) +2 +2 You can continue to move up units and right units because the slope of the line is constant y -2 x -3 +3 Another way to find the slope is to use the points (0, 0) and (3, 2) in the formula y2 - y1 _ - m = _ x2 - x1 = - = +3 GO ON Lesson 4-4 Slope 59 Example YOUR TURN! Graph the function y = 2x + and determine its slope Graph the function y = x - and determine its slope _ Complete a function table for the equation +1 +1 +1 +1 x -2 -1 y -1 +2 +2 +2 +2 Complete a function table for the equation +2 Z +2 -2 y -9 -8 -7 -6 -5 -1 -1 -1 -1 Graph the ordered pairs on the graph Z Y Y From the points on the graph, determine rise the slope run Find the change in ( ) To move from (-2, -1) to (-1, 1), you move up units and right unit The change in x-values and the change in y-values are constant from one point to another So, the rise is units, and the run is unit rise Slope = run = or Z Y @@ Z Y From the points on the graph, determine rise the slope run Find the change in ( ) x-values and the change in y-values To move from the point ( -2 , -9 ) to the point ( , -8 ), you will move up unit(s) and right unit(s) The change in x-values and the change in y-values are constant from one point to another So, the rise is units rise _ Slope = run = Chapter The Relationship Between Graphs and Functions unit, and the run is Copyright © by The McGraw-Hill Companies, Inc x-values and the change in y-values 60 +2 x Graph the ordered pairs on the graph +2 Example YOUR TURN! Determine the slope of the graph Determine the slope of the graph @@ Z Y $ Y # Complete a function table for the graph Point x y A 2 B _ _ _ " Point A B C x -5 -2 -5 y y2 - y1 5-8 m= x -x = =0 (-2) 2 3 The slope of the line is - Y Complete a function table for the graph C -2 Substitute the x and y values in the slope formula to find the slope of the line Copyright © by The McGraw-Hill Companies, Inc Z @@ Z Y Z Substitute the x and y values in the slope formula to find the slope of the line -5 - ( -2 ) _ y2 - y1 m= x -x = = - _3 The slope of the line is - Who is Correct? _ _ run Slope = rise = Landon _ _ rise Slope = run = _3 @@ Z Y What is the slope of the line on the graph? Dale - Jena _ _ rise Slope = run -2 = Z Y Circle correct answer(s) Cross out incorrect answer(s) GO ON Lesson 4-4 Slope 61 Guided Practice Use the graph to answer each question $ (4, -2) What is the location of Point A? @@ Z Y Y # (0, -1) What is the location of Point B? Z " Step by Step Practice Determine the slope of the function y = 3x - Step Complete a function table for the equation x y -9 -6 -3 Step Plot the points on the graph Step From the points on the graph, determine the slope rise run Find the change in x-values and the change in y-values ( ) To move from ( , -9 ) to ( you move up unit(s) and right , -6 ), Z Z Y Y unit(s) So, the rise is units , and the run is unit rise _ 3 Slope = run = or Graph each function and determine its slope Z y = - x - _ The slope is -3 @@ Z Y Y 62 Chapter The Relationship Between Graphs and Functions y = 6x + The slope is Z Z Y Y Copyright © by The McGraw-Hill Companies, Inc The change in x-values and the change in y-values are constant from one point to another Step by Step Problem-Solving Practice Problem-Solving Strategies Draw a diagram Look for a pattern Guess and check ✓ Write an equation Work backward Solve PRICES Gasoline costs $2 per gallon Graph an equation to represent the cost of purchasing x gallons of gas Understand Read the problem Write what you know $2 a gallon Gasoline costs Plan Pick a strategy One strategy is to write an equation Let x represent the number of gallons and y represent the cost y = 2x Use the equation to make a function table +1 +1 +1 Number of Gallons, x Cost, y $PTU JO Solve +2 +2 +2 Z Z Y Y /VNCFSPG(BMMPOT Copyright © by The McGraw-Hill Companies, Inc Graph the ordered pairs in the table on the graph To find the slope, move up right unit(s) unit(s) and to the The change in x-values and the change in y-values constant are from one point to another rise _ or slope = run = Check Use the formula for slope to check your answer y2 - y1 - (0) m = = = x2 - x1 1-0 _2 GO ON Lesson 4-4 Slope 63 MONEY Jay earns $12 per hour Graph an equation to represent the total amount Jay earns if he works x hours Include at least three points on the graph Check off each step ✔ Understand ✔ Plan ✔ Solve ✔ Check %PMMBST&BSOFE Write an equation Let x represent the hours and y represent the dollars y = 12x 12 The slope is Z Y Y /VNCFSPG)PVST Draw two lines on the graph to the right One line The second line should should have a slope of have a slope of - What you notice about direction of the two lines? Z @@ Z Y _ _ Skills, Concepts, and Problem Solving Graph each function and determine its slope y = - 6x + 10 Z Y The slope is - 64 y = 4x - Chapter The Relationship Between Graphs and Functions Z Y The slope is Y @@ Z Y Copyright © by The McGraw-Hill Companies, Inc The line with the positive slope of moves up as you travel from left to right along the x-axis The line with the negative slope of - moves down as you travel from left to right along the x-axis Z SWIMMING Ines can swim laps in 20 minutes Graph an equation to represent the number of minutes it takes Ines to swim x laps .JOVUFT 11 Write an equation Let x represent the laps that Ines can swim and y represent the minutes y = 7x The slope is 12 FITNESS It costs $40 a month to belong to Rosemill Athletic Club Graph an equation to represent the total cost of belonging to the club for x months .POUIT Write an equation Let x represent the cost and y represent the months y = 40x The slope is 40 Z Z Y Y -BQT Z Z Y Y $PTU Vocabulary Check Write the vocabulary word that completes each sentence Slope 13 14 is the ratio of the rise over the run Writing in Math Explain how to find the slope for the equation y = -2x + To find the slope, first determine different points on the line It helps to the change in x-values and the change in y-values to determine the slope the change in y-values The slope is the rise or run the change in x-values _ Spiral Review 15 MONEY The graph shows the ratio of the cost, y, to the number of tickets sold, x Use the graph to find the constant of variation of the cost to the number of tickets sold Then interpret the constant of variation (Lesson 4-3, p 49) Tickets Sold, x Cost, y 10 20 30 40 50 10 _ The ratio is is or constant 10 The constant of variation of $PTU Copyright © by The McGraw-Hill Companies, Inc make a table Next, plot the points on a coordinate graph Finally, find , so the constant of variation 10 _ means it cost $ 10 for Z Z Y Y 5JDLFUT4PME ticket Lesson 4-4 Slope 65 Chapter Progress Check (Lessons 4-3 and 4-4) Interpret each ratio 7AF3.4 The ratio of the number of ears, y, to the number of people, x, is _ The ratio means that there are ears for each person The ratio of the number of legs, y, to the number of centipedes, x, 100 is _ The ratio 100 means that there are 100 legs on each centipede # Determine the slope of the graph 7AF3.3 What is the “rise” of the line? -2 What is the slope of the line? What is the “run” of the line? Z Z ĕ ... 10 09 08 07 California Math Triumphs Volume 5B California Math Triumphs Volume Place Value and Basic Number Skills 1A Chapter Counting 1A Chapter Place Value 1A Chapter Addition and Subtraction... feedback and suggestions for improving the effectiveness of the mathematics instruction Melody McGuire Math Teacher California College Preparatory Academy Oakland, California 6th and 7th Grade Math. .. School Bakersfield, California Math Coordinator K–5 City Heights Educational Collaborative San Diego, California Instructional Math Coach Aspire Public Schools Oakland, California Math Director City