AUTHORS Carter • Cuevas • Day • Malloy • Kersaint • Luchin • McClain Molix-Bailey • Price • Reynosa • Silbey • Vielhaber • Willard
Bothell, WA • Chicago, IL • Columbus, OH • New York, NY
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Copyright © 2015 McGraw-Hill Education All rights reserved. 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 the prior written consent of McGraw-Hill Education, including, but not limited to, network storage or transmission, or broadcast for distance learning. McGraw-Hill is committed to providing instructional materials in Science, Technology, Engineering, and Mathematics (STEM) that give all students a solid foundation, one that prepares them for college and careers in the 21st century. Send all inquiries to: McGraw-Hill Education 8787 Orion Place Columbus, OH 43240 ISBN: 978-0-02-135911-0 (Volume 2) MHID: 0-02-135911-3 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 RMN 15 14 13 12 Common Core State Standards© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Understanding by Design® is a registered trademark of the Association for Supervision and Curriculum Development (“ASCD”).
CONTENTS IN BRIEF Units organized by CCSS domain California Math is organized into units based on groups of related standards called domains. This year, you will study and understand the five domains shown below.
UNIT 1 Domain 6.RP
Ratios and Proportional Relationships Chapter 1 Ratios and Rates Chapter 2 Fractions, Decimals, and Percents
Corbis/P Corb is/Punch unchStoc Stockk Corbis/PunchStock
UNIT 2 Domain 6.NS
The Number System Chapter 3 Compute with Multi-Digit Numbers Chapter 4 Multiply and Divide Fractions Chapter 5 Integers and the Coordinate Plane
LatinStock Collection/Alamy
UNIT 3 Domain 6 6.EE
Expressions and Equations Chapter 6 Expressions Chapter 7 Equations Chapter 8 Functions and Inequalities
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UNIT 4 Domain 6.G
Geometry Chapter 9 Area Chapter 10 Volume and Surface Area
Mitch Reardon/Stone/Getty RReardo Reeard eardon/S ardon/S n/SSt Stone/ one/Gett Gettyy Images Images
UNIT 5 Domain 6.SP
Statistics and Probability Chapter 11 Statistical Measures Chapter 12 Statistical Displays
L. Mou M outon/Photo Mouton/PhotoAlto
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UNIT 1
Ratios and Proportional Relationships
IEW UNIT PROJECT PREV page 2
Chapter 1
Ratios and Rates What Tools Do You Need? What Do You Already Know? Are You Ready?
4 5 6 7 Lesson 1 Factors and Multiples 15 Inquiry Lab: Ratios 19 Lesson 2 Ratios 27 Inquiry Lab: Unit Rates 31 Lesson 3 Rates
Essential Question HOW do you use equivalent rates in the real world?
39 Lesson 4 Ratio Tables 47 Lesson 5 Graph Ratio Tables 55
Problem-Solving Investigation: The Four-Step Plan
p. 31
Mid-Chapter Check
58
67 Inquiry Lab: Ratio and Rate Problems 71 Lesson 7 Ratio and Rate Problems 79
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81 83 84
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Chapter Review Performance Task Reflect
21st Century Career in Chemistry
(t)©FLPA/Alamy; (b)Brand X Pictures/PunchStock
59 Lesson 6 Equivalent Ratios
Ratios and Proportional nal R Relationships
Chapter 2
Fractions, Decimals, and Percents What Tools Do You Need? What Do You Already Know? Are You Ready?
86 87 88 89 Lesson 1 Decimals and Fractions 97 Inquiry Lab: Model Percents 101 Lesson 2 Percents and Fractions 109 Lesson 3 Percents and Decimals
Essential Question WHEN is it better to use a fraction, a decimal or a percent?
117 Lesson 4 Percents Greater than 100% and Percents Less than 1%
125
Problem-Solving Investigation: Solve a Simpler Problem
Mid-Chapter Check
128
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(t)StockTrek/Getty Images; (c)Inti St Clair/Digital Vision/Getty Images; (b)Corbis/PunchStock
129 Lesson 5 Compare and Order Fractions, Decimals, and Percents
137 Lesson 6 Estimate with Percents 145 Inquiry Lab: Percent of a Number 147 Lesson 7 Percent of a Number 155 Lesson 8 Solve Percent Problems 163
21st Century Career in Movies
Chapter Review Performance Task Reflect
UNIT PROJECT
p. 155
165 167 168
169
People Everywhere vii
UNIT 2 The Number System
IEW UNIT PROJECT PREV page 172
Chapter 3
Compute with Multi-Digit Numbers What Tools Do You Need? What Do You Already Know? Are You Ready?
174 175 176 177 Lesson 1 Add and Subtract Decimals 185 Lesson 2 Estimate Products
Essential Question HOW can estimating be helpful?
193 Lesson 3 Multiply Decimals by Whole Numbers
201 Lesson 4 Multiply Decimals by Decimals 209 Inquiry Lab: Multiply by Powers of 10 211
Problem-Solving Investigation: Look for a Pattern (t)Kim Karpeles/Axiom Photographic Agency/Getty Images; (b)Rolf Bruderer/Blend Images/Getty Images
Mid-Chapter Check
214 215 Lesson 5 Divide Multi-Digit Numbers 223 Lesson 6 Estimate Quotients 231 Lesson 7 Divide Decimals by Whole Numbers
239 Lesson 8 Divide Decimals by Decimals
247
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Chapter Review 249 Performance Task 251 Reflect 252
21st Century Career in Design
The Number System
Chapter 4
Multiply and Divide Fractions What Tools Do You Need? 254 What Do You Already Know? 255 Are You Ready? 256 257 Lesson 1 Estimate Products of Fractions 265 Lesson 2 Multiply Fractions and Whole Numbers 273 Lesson 3 Multiply Fractions 281 Lesson 4 Multiply Mixed Numbers 289 Lesson 5 Convert Measurement Units 297
Essential Question WHAT does it mean to multiply and divide fractions?
Problem-Solving Investigation: Draw a Diagram
Mid-Chapter Check 300 301
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(t)©Royalty-Free/Corbis; (b)Fuse/Getty Images
305
Inquiry Lab: Divide Whole Numbers by Fractions Lesson 6 Divide Whole Numbers by Fractions
313 Inquiry Lab: Divide Fractions 317 Lesson 7 Divide Fractions 325 Lesson 8 Divide Mixed Numbers 333 Chapter Review 335 Performance Task 337 Reflect 338
21st Century Career in Culinary Arts
p. 257
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The Number System
Chapter 5
Integers and the Coordinate Plane What Tools Do You Need? 340 What Do You Already Know? 341 Are You Ready? 342 343 Inquiry Lab: Integers 345 Lesson 1 Integers and Graphing 353 Inquiry Lab: Absolute Value 355 Lesson 2 Absolute Value Essential Question
363 Lesson 3 Compare and Order Integers 371
Problem-Solving Investigation: Work Backward
Mid-Chapter Check
HOW are integers and absolute value used in real-world situations?
374
387 Lesson 5 Compare and Order Rational Numbers 395 Lesson 6 The Coordinate Plane 403 Lesson 7 Graph on the Coordinate Plane 411 Inquiry Lab: Find Distance on the Coordinate Plane
415 p. 387
Chapter Review 417 Performance Task 419 Reflect 420
421
Get Out the Map! x
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UNIT PROJECT
21st Century Career in Art
(t)Henrik Sorensen/The Image Bank/Getty Images; (c)Brand X Pictures/PunchStock; (b)LatinStock Collection/Alamy
Inquiry Lab: Number Lines 375 379 Lesson 4 Terminating and Repeating Decimals
UNIT 3 Expressions and Equations
IEW UNIT PROJECT PREV page 424
Chapter 6
Expressions What Tools Do You Need? 426 What Do You Already Know? 427 Are You Ready? 428 429 Inquiry Lab: Structure of Expressions 433 Lesson 1 Powers and Exponents Essential Question 441 Lesson 2 Numerical Expressions
HOW is it helpful to write numbers in different ways?
449 Lesson 3 Algebra: Variables and Expressions
457 Inquiry Lab: Write Expressions 461 Lesson 4 Algebra: Write Expressions 469
Problem-Solving Investigation:
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(t)Steve Mason/Photodisc/Getty Images; (b)Jupiterimages/Comstock Images/Getty Images
Act It Out
Mid-Chapter Check
472 473 Lesson 5 Algebra: Properties 481 Inquiry Lab: The Distributive Property 485 Lesson 6 The Distributive Property 493 Inquiry Lab: Equivalent Expressions 495 Lesson 7 Equivalent Expressions 503
Chapter Review 505 Performance Task 507 Reflect 508
21st Century Career in Engineering
p. 495
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Expressions and Equations
Chapter 7
Equations What Tools Do You Need? 510 What Do You Already Know? 511 Are You Ready? 512 513 Lesson 1 Equations 521 525
533 535 543
Inquiry Lab: Solve and Write Addition Equations Lesson 2 Solve and Write Addition Equations
Essential Question HOW do you determine if two numbers or expressions are equal?
Inquiry Lab: Solve and Write Subtraction Equations Lesson 3 Solve and Write Subtraction Equations Problem-Solving Investigation: Guess, Check, and Revise
Mid-Chapter Check 546 547
559 561
Chapter Review Performance Task Reflect
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571 573 574
Inquiry Lab: Solve and Write Division Equations Lesson 5 Solve and Write Division Equations 21st Century Career in Music
p. 535
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551
Inquiry Lab: Solve and Write Multiplication Equations Lesson 4 Solve and Write Multiplication Equations
Expressions and Equations
Chapter 8
Functions and Inequalities What Tools Do You Need? 576 What Do You Already Know? 577 Are You Ready? 578 579 Lesson 1 Function Tables 587 Lesson 2 Function Rules 595 Lesson 3 Functions and Equations
Essential Question HOW are symbols, such as <, >, and =, useful?
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(t)Island Effects/Vetta/Getty Images; (c)Andersen Ross/Photodisc/Getty Images; (b)Antonio M. Rosario/Photographer’s Choice/Getty Images
603 Lesson 4 Multiple Representations of Functions
611
Problem-Solving Investigation: Make a Table
Mid-Chapter Check
614 Inquiry Lab: Inequalities 615 617 Lesson 5 Inequalities 625 Lesson 6 Write and Graph Inequalities 633 Inquiry Lab: Solve One-Step Inequalities 635 Lesson 7 Solve One-Step Inequalities 643
p. 595
21st Century Career in Atmospheric Science
Chapter Review 645 Performance Task 647 Reflect 648
UNIT PROJECT
649
It’s Out of This World xiii
UNIT 4 Geometry
IEW UNIT PROJECT PREV page 652
Chapter 9
Area What Tools Do You Need? 654 What Do You Already Know? 655 Are You Ready? 656 657 Inquiry Lab: Area of Parallelograms 661 Lesson 1 Area of Parallelograms 669 Inquiry Lab: Area of Triangles 673 Lesson 2 Area of Triangles Essential Question 681 Inquiry Lab: Area of Trapezoids 685 Lesson 3 Area of Trapezoids 693
HOW does measurement help you solve problems in everyday life?
Problem-Solving Investigation: Draw a Diagram
Mid-Chapter Check 696 (t)Paul Sutherland/National Geographic/Getty Images; (b)Sean Justice/Taxi/Getty Images
697 Lesson 4 Changes in Dimensions 705 Lesson 5 Polygons on the Coordinate Plane
713 Inquiry Lab: Area of Irregular Figures 717 Lesson 6 Area of Composite Figures 725
21st Century Career in Community Planning
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727 729 730
p. 661
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Chapter Review Performance Task Reflect
Geometry
Chapter 10
Volume and Surface Area What Tools Do You Need? 732 What Do You Already Know? 733 Are You Ready? 734 735 Inquiry Lab: Volume of Rectangular Prisms 739 Lesson 1 Volume of Rectangular Prisms 747 Lesson 2 Volume of Triangular Prisms 755
Problem-Solving Investigation:
Essential Question HOW is shape important when measuring a figure?
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(t)Philipp Nemenz/The Image Bank/Getty Images; (c)Joshua Ets-Hokin/Photodisc/Getty Images; (b)Mitch Reardon/Stone/Getty Images
Make a Model
Mid-Chapter Check
758 759 763
Inquiry Lab: Surface Area of Rectangular Prisms Lesson 3 Surface Area of Rectangular Prisms
771 Inquiry Lab: Nets of Triangular Prisms 773 Lesson 4 Surface Area of Triangular Prisms 781 Inquiry Lab: Nets of Pyramids 783 Lesson 5 Surface Area of Pyramids 791
Design
p. 783
Chapter Review Performance Task Reflect
UNIT PROJECT
21st Century Career in Interior
793 795 796 797
A New Zoo xv
UNIT 5 Statistics and Probability
IEW UNIT PROJECT PREV page 800
Chapter 11
Statistical Measures What Tools Do You Need? 802 What Do You Already Know? 803 Are You Ready? 804 805 Inquiry Lab: Statistical Questions 809 Lesson 1 Mean 817 Lesson 2 Median and Mode 825
Problem-Solving Investigation: Use Logical Reasoning
Essential Question HOW are the mean, median, and mode helpful in describing data?
Mid-Chapter Check 828 829 Lesson 3 Measures of Variation (t)Mike Powell/Allsport Concepts/Getty Images; (b)Dave & Les Jacobs/Blend Images/Getty Images
837 Lesson 4 Mean Absolute Deviation 845 Lesson 5 Appropriate Measures 853
21st Century Career in Marine Biology
Chapter Review 855 Performance Task 857 Reflect 858
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p. 809
Statistics and Probability
Chapter 12
Statistical Displays What Tools Do You Need? 860 What Do You Already Know? 861 Are You Ready? 862 863 Lesson 1 Line Plots 871 Lesson 2 Histograms 879 Lesson 3 Box Plots 887
Problem-Solving Investigation: Use a Graph
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(t)Doug Armand/Stone/Getty Images; (c)Clarissa Leahy/Photodisc/Getty Images; (b)L. Mouton/Photo Alto
Mid-Chapter Check 890 Essential Question
891 Lesson 4 Shape of Data Distributions 899 Inquiry Lab: Collect Data 901 Lesson 5 Interpret Line Graphs
WHY is it important to carefully evaluate graphs?
909 Lesson 6 Select an Appropriate Display 917 Inquiry Lab: Use Appropriate Units and Tools 919
21st Century Career in Environmental Science
p. 891
Chapter Review 921 Performance Task 923 Reflect 924
UNIT PROJECT Let’s Exercise
925
Glossary Selected Answers Index Work Mats Foldables
GL1 SA1 IN1 WM1 FL1
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Common Core State Standards for MATHEMATICS, Grade 6 California Math, Course 1, focuses on four critical areas: (1) using concepts of ratio and rate to solve problems; (2) understanding division of fractions; (3) using expressions and equations; and (4) understanding of statistical reasoning.
Content Standards Domain 6.RP
Ratios and Proportional Relationships R • Understand ratio concepts and use ratio reasoning to solve problems.
Domain 6.NS
The Number System T • Apply and extend previous understandings of multiplication and division to divide fractions by fractions. • Compute fluently with multi-digit numbers and find common factors and multiples. • Apply and extend previous understandings of numbers to the system of rational numbers.
Domain 6.EE
Expressions and Equations E • Apply and extend previous understandings of arithmetic to algebraic expressions. • Reason about and solve one-variable equations and inequalities. • Represent and analyze quantitative relationships between dependent and independent variables.
Domain 6.G
Geometry G • Solve real-world and mathematical problems involving area, surface area, and volume.
Domain 6.SP
Statistics and Probability S • Develop understanding of statistical variability. • Summarize and describe distributions.
Mathematical Practices
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Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
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1 2 3 4 5 6 7 8
Common Core State Standards for Mathematics
Track Your Common Core Progress These pages list the key ideas that you should be able to understand by the end of the year. You will rate how much you know about each one. Don’t worry if you have no clue before you learn about them. Watch how your knowledge grows as the year progresses!
I have no clue.
I’ve heard of it.
I know it! Before
After
6.RP Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
6.NS The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
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6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
For more about the Common Core State Standards go to commoncoresolutions.com.
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Common Core State Standards for Mathematics
Before
6.NS The Number System
After
continued
Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. d. Distinguish comparisons of absolute value from statements about order.
6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.EE Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
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Common Core State Standards for Mathematics
Before
6.EE Expressions and Equations
After
continued
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
6.EE.3 Apply the properties of operations to generate equivalent expressions. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).
Reason about and solve one-variable equations or inequalities. 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.9 Use variables to represent two quantities in a real-world problem that
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change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
For more about the Common Core State Standards go to commoncoresolutions.com.
xxi
Common Core State Standards for Mathematics
Before
After
6.G Geometry Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.SP Statistics and Probability Develop understanding of statistical variability. 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Summarize and describe distributions. 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5 Summarize numerical data sets in relation to their context, such as by:
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a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
