ISBN

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1 GRADE 8

2 ISBN Copright The Continental Press, Inc. No part of this publication ma be reproduced in an form or b an means, electronic, mechanical, photocoping, recording, or otherwise, without the prior written permission of the publisher. All rights reserved. Printed in the United States of America.

3 Table of Contents Introduction...5 Unit Eponents and Radicals EE. Lesson Eponents EE. 8.EE.3 8.EE. 8.EE. Lesson Laws of Eponents... Lesson 3 Scientific Notation...6 Lesson Operations with Scientific Notation... Lesson 5 Radicals... Review Eponents and Radicals...8 Unit Real Numbers NS. Lesson Rational Numbers NS., 8.NS. Lesson Irrational Numbers...36 Review Real Numbers... Unit 3 Linear Relationships EE.5, 8.EE.6 Lesson Proportional Relationships... 8.EE.6 8.EE.7.a 8.EE.7.a, b Lesson Graphing Linear Relationships...8 Lesson 3 Solving Linear Equations...5 Lesson More Solving Equations...56 Review Linear Relationships...6 Unit Sstems of Linear Equations EE.8.a, b Lesson Solving Sstems of Equations Graphicall EE.8.b 8.EE.8.b 8.EE.8.c Lesson Solving Sstems of Equations b Elimination...68 Lesson 3 Solving Sstems of Equations b Substitution...7 Lesson Problem Solving with Sstems of Equations...76 Review Sstems of Linear Equations...8 Unit 5 Functions F., 8.F. Lesson Function Tables F., 8.F. 8.F., 8.F., 8.F.3, 8.F.5 8.F.3, 8.F. Lesson Equations of Functions...88 Lesson 3 Graphs of Functions...9 Lesson Linear Functions...96 Review Functions...

4 Unit 6 Geometr, Part G..a, b, c; 8.G.3 Lesson Translations... 8.G..a, b, c; 8.G.3 Lesson Reflections G..a, b, c; 8.G.3 8.G.3 8.G., 8.G. 8.G.5 Lesson 3 Rotations... Lesson Dilations...6 Lesson 5 Congruence and Similarit... Lesson 6 Angle Relationships... Review Geometr, Part...8 Unit 7 Geometr, Part G.6 8.G.6, 8.G.7 Lesson Proving the Pthagorean Theorem...3 Lesson Appling the Pthagorean Theorem G.8 Lesson 3 Finding Distance Between Points... 8.G.9 Lesson Volume of Solid Figures... Review Geometr, Part...8 Unit 8 Statistics and Probabilit SP. 8.SP., 8.SP.3 8.SP. Lesson Scatter Plots...5 Lesson Lines of Best Fit...56 Lesson 3 Two-Wa Tables...6 Review Statistics and Probabilit... 6 Practice Test...67 Glossar...83

5 Welcome to Finish Line Mathematics for the Common Core State Standards About This Book Finish Line Mathematics for the Common Core State Standards will help ou prepare for math tests. Each ear in math class, ou learn new skills and ideas. This book focuses on the math skills and ideas that are the most important for each grade. It is important to master the concepts ou learn each ear because mathematical ideas and skills build on each other. The things ou learn this ear will help ou understand and master the skills ou will learn net ear. This book has units of related lessons. Each lesson concentrates on one main math idea. The lesson reviews things ou have learned in math class. It provides eplanations and eamples. Along the side of each lesson page are reminders to help ou recall what ou learned in earlier grades. After the lesson come three pages of practice problems. The problems are the same kinds ou find on most math tests. The first page has multiple-choice, or selected-response, problems. Each item has four answers to choose from, and ou must select the best answer. At the top of the page is a sample problem with a bo beneath it that eplains how to find the answer. Then there are a number of problems for ou to do on our own. Constructed-response, or short-answer, items are on the net page. You must answer these items using our own words. Usuall, ou will need to show our work or write an eplanation of our answer in these items. This tpe of problem helps ou demonstrate that ou know how to do operations and carr out procedures. The also show that ou understand the skill. Again, the first item is a sample and its answer is eplained. You will complete the rest of the items b ourself. The last page has one or two etended-response problems. These items are like the short writing items, but the have more parts and are often a little harder. The first part ma ask ou to solve a problem and show our work. The second ma ask ou to eplain how ou found our answer or wh it is correct. This item has a hint to point ou in the right direction. At the end of each unit is a review section. The problems in it cover all the different skills and ideas in the lessons of that unit. The review contains multiplechoice, constructed-response, and etended-response items. A practice test and a glossar appear at the end of the book. The practice test gives ou a chance to tr out what ou ve learned. You will need to use all the skills ou have reviewed and practiced in the book on the practice test. The glossar lists important words and terms along with their definitions to help ou remember them. Introduction 5

6 The Goals of Learning Math Math is everwhere in the world around ou. You use math more than ou probabl realize to help ou understand and make sense of that world. But what does it mean to be good at math? To be good at math, ou need to practice certain habits. And ou need the right attitude. You make sense of problems and do not give up in solving them. You make sure ou understand the problem before ou tr to solve it. You form a plan and then carr out that plan to find an answer. Along the wa, ou ask ourself if what ou are doing makes sense. And if ou do not figure out the answer on the first tr, ou tr another wa. You think about numbers using smbols. You can think about a real-life situation as numbers and operations. You draw conclusions about situations and support them with proof. You use what ou know about numbers and operations to provide reasons for our conclusions and predictions. When ou read or hear someone else s eplanation, ou think about it and decide if it makes sense. You ask questions that help ou better understand the ideas. You model with mathematics. You represent real-life problems with a drawing or diagram, a graph, or an equation. You decide if our model makes sense. You use the right tools at the right time. You know how to use rulers, protractors, calculators, and other tools. More importantl, ou know when to use them. You are careful and accurate in our work. You calculate correctl and label answers. You use the correct smbols and definitions. You choose eactl the right words for our eplanations and descriptions. You look for structure in math. You see how different parts of math are related or connected. You can use an idea ou alread know to help ou understand a new idea. You make connections between things ou have alread learned and new ideas. You look for the patterns in math. When ou see the patterns, ou can find shortcuts to use that still lead ou to the correct answer. You are able to decide if our shortcut worked or not. These habits help ou master new mathematical ideas so that ou can remember and use them. All of these habits will make math easier to understand and to do. And that will make it a great tool to use in our everda life! 6 Introduction

7 Unit Eponents and Radicals Lesson Eponents reviews what an integer eponent is and how to evaluate numbers with these eponents. Lesson Laws of Eponents reviews the rules that show how to multipl and divide numbers with eponents. Lesson 3 Scientific Notation reviews how to epress ver small and ver large quantities using eponential powers of. Lesson Operations with Scientific Notation reviews how to use all four operations to solve problems involving scientific notation. Lesson 5 Radicals reviews what a square root and a cube root are and how to find the values of such numbers. UNIT Eponents and Radicals 7

8 Lesson Eponents 8.EE. Read 5 3 as 5 to the 3rd power. It means the number 5 is a factor 3 times. Numbers with negative eponents can also represent repeated division. 5 When a number is multiplied b itself several times, it is written in eponential form. The number being multiplied is called the base. The eponent shows the number of times it multiplies itself. 5 3 Eponent Base 5 multiplies itself 3 times. Some numbers have negative eponents. These numbers can be rewritten with positive eponents in the denominator of a fraction. 5 5 ( ) ( ) 5 is the same as multipling b itself times. Multiplication is used to find the value of a number in eponential form. What is the value of 7? Alwas use parentheses around a base that is negative. ( )? ( ) ( ) 5 6 A number, n, with an eponent of equals n ( 3) 5 3 ( 5 ) 5 5 A number, n, with an eponent of equals. 8 5 ( 3) 5 ( 5 ) 5 In 7, the base is 7. The eponent is. Multipl the base b itself times. What is the value of 6 3? , In 6 3, the base is 6. The eponent is 3. Rewrite the number as a fraction with a positive eponent ( 3 6 ) 3 Now the base is and the eponent is 3. 6 Multipl the base b itself 3 times UNIT Eponents and Radicals

9 Read each problem. Circle the letter of the best answer. SAMPLE What is the value of 5? A 5 B C 5 D 5 The correct answer is A. The base is positive 5. The eponent is. The negative in front of the base means the answer will be negative. So, 5 5 (5 5) 5 5. Which epression is equivalent to 5? A 5 C B 5 D What is the value of ( 7)? A 7 C 7 B D 3 Which epression is equivalent to the one shown below? ( 3) ( 3) ( 3) ( 3) ( 3) A 3 5 C 3 5 B ( 3) 5 D ( 3) 5 What is the value of 5? A C 5 B D 3 5 What is the value of 3? A, C 3 B 3 D, 6 What is the value of ( 3)? A 9 C 9 B 6 D 6 7 The volume of a cube with a side length of s is given b the formula V 5 s 3. What is the volume, in cubic inches, of a cube with a side length of 8 inches? A C 5 B 9 D 6,56 8 Which epression has the greatest value? A 8 6 C 5 B ( 3) D 5 9 Which equation is true? A ( 5) 5 5 B C 5 D 6 5 ( 6) ( 6) ( 6) ( 6) UNIT Eponents and Radicals 9

10 Read each problem. Write our answer. SAMPLE Talor wrote the equation below. ( ) n 5 What must be the value of n? Answer An base number that has for an eponent will equal. Since the equation is equal to, the eponent n must be. What is the value of ( 3 5 )? Answer Are the epressions 5 and 5 equivalent? Eplain how ou know. Write an epression using multiplication and an epression using division that are equivalent to 7 3. Multiplication Division 3 Are the epressions ( 3) 3 and 3 3 equivalent? Eplain how ou know. UNIT Eponents and Radicals

11 Read each problem. Write our answer to each part. Ariel wrote these epressions Part A Write these epressions in order from least to greatest. Answer Part B Eplain how ou know our answer is correct 5 Kevin wrote the epression 9. Part A What is the value of this epression? Answer How can ou change a negative eponent to a positive eponent? Part B Is the epression ( 9) equivalent to 9? Eplain how ou know. UNIT Eponents and Radicals

12 Lesson Laws of Eponents 8.EE. To combine eponential epressions with the same base, combine the eponents onl. The base stas the same. Laws of eponents can be used to combine eponential epressions with the same base. To multipl eponential epressions, add the eponents. n a n b 5 n (a b) 3 5 ( 3) 5 7 To divide eponential epressions, subtract the eponents. n a n b 5 n (a b) (3 ( )) To change a negative eponent to a positive eponent, rewrite the number as a fraction with a positive eponent To raise an eponential epression to a power, multipl the eponents. (n a ) b 5 n (a b) (3 ) 5 3 ( ) What is the value of ( 3 5 )? First, add the eponents to multipl the epressions in the numerator of the fraction. ( 3 5 ) 5 ( ( 3 5) ) 5 ( ) Net, subtract the eponents to divide the epressions inside parentheses. ( ) 5 (( ) ) 5 ( ) Then multipl the eponents to raise the epression to the power. ( ) 5 ( ) 5 Follow the rules for adding, subtracting, and multipling integers when combining eponents. Finall, rewrite the epression using a positive eponent and evaluate , so ( 3 5 ) 5 6 UNIT Eponents and Radicals

13 Read each problem. Circle the letter of the best answer. SAMPLE What is the value of ( ) 6 ( ) 3? A 6 B 6 C 6 D 6 The correct answer is A. Two eponential epressions with the same base,, are being divided. The law of eponents sas ou can subtract the eponents: ( ) (6 3) 5 ( ) 3. Then evaluate: ( )( )( ) 5 6. Which epression is equivalent to ? A 8 8 C 6 8 B 8 5 D 6 5 What is the value of ( 3 )? A 6 C 6 B D 3 Simplif the epression below. ( 6) 8 ( 6) A 6 C ( 6) B 6 6 D ( 6) 6 What is the value of 3 3 7? A 8 C 9 B 9 D 7 5 What is the value of this epression? 6 3 A 6 C B D 6 6 Simplif the epression below. ( 6 ) A C 8 3 B D 6 7 What is the value of this epression? A C 5 B D 5 UNIT Eponents and Radicals 3

14 Read each problem. Write our answer. SAMPLE The two epressions below have the same value. 6 What is the value of n? n 5 Answer First simplif each epression: 6 5 (6 ) 5 n 5 5 (n 5) 5 The eponents n 5 and are the same, so n 5 5, and n 5. 8 Simplif ( 3 5 ). Write our answer using a positive eponent. Answer 9 What is the value of 3? Show our work. Answer Courtne thinks the value of (3 ) 3 3 is. Is she correct? Eplain how 3 ou know. UNIT Eponents and Radicals

15 Read each problem. Write our answer to each part. Michael wrote this epression. 6 3 Part A What is the value of the epression? Answer Part B Write an eponential epression using division that has the same value as the one Michael wrote. Answer Look at this epression. 3 ( 3 ) Part A What is the value of this epression? Show our work. What is the value of an number raised to the power? Answer Part B Tro found the value of this epression b first subtracting the eponents of the numbers inside the parentheses. Marni found the value of the epression b first multipling each term of the fraction b the eponent. If all their math work is correct, will Tro and Marni get the same value? Eplain how ou know. UNIT Eponents and Radicals 5

16 Lesson 3 Scientific Notation 8.EE.3 A number in scientific notation should alwas have onl one digit to the left of the decimal point. The decimal point for an integer is to the right of its ones place The zeros after the last digit in a decimal can be dropped without changing the value of the number A number in scientific notation is ver large if the power of is positive. A number in scientific notation is ver small if the power of is negative. Scientific notation can be used to write ver large or ver small numbers. In scientific notation, a number greater than or equal to and less than is multiplied b a power of. You can move the decimal point in numbers to change them from standard form to scientific notation and from scientific notation to standard form. Write the number 5,8, in scientific notation. Place a decimal point to the right of the first digit to make a number greater than or equal to and less than. 5,8, Multipl this number b a power of. The eponent is equal to the number of places the decimal point moved. The decimal point moved 7 places, so the power is 7. 5,8, Write the number in standard form. Move the decimal point the same number of places as the number in the eponent. Then drop the power of. The negative eponent means the number is ver small and the decimal point moves to the left You can estimate amounts using scientific notation. A penn is about meter thick. A roll of coins contains 5 pennies. Estimate the thickness of the roll of coins. In scientific notation, The eponent,, increases the original eponent b, from 3 to. Round the decimal part.5 to.5. Then multipl b 5 to get 75. The thickness of the roll of coins is about meter. Numbers in scientific notation can have onl one digit to the left of the decimal point, so A roll of coins is about meter thick. 6 UNIT Eponents and Radicals

17 Read each problem. Circle the letter of the best answer. SAMPLE The Great Lakes have a water area of about. 3 5 square kilometers. One of them, Lake Huron, has a water area of about 6. 3 square kilometers. What approimate fraction of the total water area of the Great Lakes does Lake Huron represent? A 6 B 5 C D 3 The correct answer is C. First change each number from scientific notation to standard form: , and ,. Round, to a compatible number:,. Then divide this into 6,: 6,, Lake Huron represents of the total water area. Which number is written in correct scientific notation? A 5, C 5. 3 B.5 3 D About. 3 6 people live in Washington, D.C. How is this number written in standard form? A. C, B. D,, 3 How is 8,3,, written in scientific notation? A C B.83 3 D An ant can lift a total of kilogram. What is this number written in standard form? A.33 C 3,3 B.33 D 33, 5 The land area of Connecticut is about square miles. The land area of New Meico is about. 3 5 square miles. About how man times greater is the land area of New Meico than the land area of Connecticut? A C 5 B D 6 How is.9863 written in scientific notation? A C B D The diameter of Venus is approimatel. 3 kilometers wide. The diameter of Jupiter is about times wider than this. Which is the best estimate for the width of the diameter of Jupiter? A. 3 C. 3 6 B. 3 5 D. 3 8 UNIT Eponents and Radicals 7

18 Read each problem. Write our answer. SAMPLE There are one billion nanoseconds in one second. Write the number of nanoseconds in one second using scientific notation. Answer First, write one billion as a number in standard form. One billion is,,,. Then write this number in scientific notation. The decimal point goes directl after the. Since the decimal point is moved 9 places, the number is A compan made $3,5, in sales last ear. Write this number in scientific notation. Answer 9 Write the number in standard form. Answer The mass of a proton is about, times greater than the mass of an electron. The mass of an electron is about kilogram. What is the approimate mass, in kilograms, of a proton? Show our work. Answer A skscraper is. 3 inches tall. A bug is. 3 inch tall. How man times greater is the height of the skscraper than the height of the bug? Show our work. Answer 8 UNIT Eponents and Radicals

19 Read the problem. Write our answer to each part. The weights, in ounces, of different animals are shown in the table below. Animal Weight of Animals Weight (ounces) Elephant Cat.9 3 Mouse Zebra Part A Approimatel how man times heavier is an elephant than a mouse? Answer Part B Eplain how ou found our answer. It ma help to change the numbers in scientific notation to standard form and use compatible numbers. UNIT Eponents and Radicals 9

20 Lesson Oper a t i o 8.EE. n Scientific Notation s w i t h It is easier to adjust the smaller number to have the same power of as the larger number. To change a number in scientific notation to have a larger power of, follow these steps:. Find n, the number the power of increases b.. Move the decimal point to the left n places. Remember to add eponents when multipling powers of and subtract eponents when dividing powers of. The commutative propert sas ou can add or multipl numbers in an order and the result will be the same. a b 5 b a a 3 b 5 b 3 a Numbers in scientific notation can be combined using basic operations. To add or subtract, rewrite the problem so that both numbers have the same power of. Find the sum Rewrite so is to the 8th power: Add the decimal numbers. The power of stas the same (.75.9) So, To multipl or divide, first multipl or divide the decimal numbers. Then multipl or divide the powers of. Find the product. (5.5 3 )( ) Rewrite the problem using the commutative propert. (5.5 3 )( ) 5 ( )( 3 5 ) Multipl the decimal numbers. Then multipl the powers of. ( )( 3 5 ) Write the result in proper scientific notation So, (5.5 3 )( ) UNIT Eponents and Radicals

21 Read each problem. Circle the letter of the best answer. SAMPLE The temperature at the surface of the sun is approimatel. 3 degrees Fahrenheit. The temperature at its center is approimatel degrees Fahrenheit. About how man times greater is the temperature at the center of the sun than at its surface? A B C.7 3 D The correct answer is B. Division is used to determine how man times greater one number is than another. Write a division epression: ( ) (. 3 ). Divide the decimal numbers: Then divide the powers of : So, ( ) (. 3 ) Rob correctl combined and and got What operation did he use? A addition C multiplication B subtraction D division The area of North America is about square miles. The area of South America is about square miles. What is the approimate total area, in square miles, of both North and South America? A C.65 3 B D What is.5 ( )? A C B D Television ratings show people watched show X and people watched show Y. Which statement is true? A.6 3 more people watched X. B more people watched X. C more people watched Y. D more people watched Y. 5 A light-ear is the distance light travels in a ear. One light-ear is about miles. The sun is about light-ears from Earth. About how man miles is the sun from Earth? A C B D A scale drawing of an insect is 7. cm long. The actual length of the insect is cm. How man times smaller is the actual length compared to the scale drawing? A C B 5 D 5 UNIT Eponents and Radicals

22 Read each problem. Write our answer. SAMPLE One megabte is approimatel btes. One gigabte is. 3 3 megabtes. Approimatel how man btes are in one gigabte? Answer Multipl to find the number of btes in one gigabte: ( )(. 3 3 ) 5 (.5 3.)( ) So, one gigabte is about btes. 7 Find the quotient. Write our answer in proper scientific notation Answer 8 The length of a rectangle is. 3 kilometer. The width of the rectangle is kilometer. What is the area, in square kilometers, of this rectangle? Show our work. Answer 9 Katherine found the difference of as Eplain whether or not Katherine is correct. UNIT Eponents and Radicals

23 Read the problem. Write our answer to each part. The approimate densities of some chemical elements are shown in the table below. Densit of Elements Element Densit (kilograms per cubic centimeter) Calcium Gold.93 3 Silver.5 3 Sodium Part A What is the densit, in kilograms per cubic centimeter, of gold and sodium together? Show our work. Answer Part B How man times greater is the densit of silver than of calcium? Eplain how ou know. Which operation would ou use to find how man times greater one number is than another? UNIT Eponents and Radicals 3

24 Lesson 5 Radicals 8.EE. A number multiplied b itself is a squared number The radical smbol X is used to show a square root. The number under the radical is called the radicand. 5 Radicand The first ten perfect squares are A number multiplied b itself twice is a cubed number The first five perfect cubes are A perfect square is the product of a number and itself. A square root is the inverse of a perfect square. The square root of a number is the number that is squared to equal. What is the value of? Think: What number squared equals? Since, or 3, equals, 5. A perfect cube is the product of the same three numbers. The cube root of a number is the number that is cubed to equal. The smbol 3 represents the cubed root of. What is 3 33? Think: What number cubed equals 33? Since 7 3, or equals 33, The square root and the cube root of a number are not alwas whole numbers. Between what two consecutive whole numbers is 5? Find the closest perfect squares greater and less than and is between 9 and 6, so 5 is between 7 and 8. Since 5 is closer to 9 than 6, 5 is closer to 7 than 8. Between what two consecutive whole numbers is 3? Find the closest perfect cubes greater and less than and is between 6 and 5, so 3 is between and 5. UNIT Eponents and Radicals

25 Read each problem. Circle the letter of the best answer. SAMPLE Which value is closest to 3? A 5.8 B 6.7 C. D. The correct answer is A. Find cubes of numbers that are just under and just over. Since and , 3 is between 5 and 6. Since is closer to 6 than it is to 5, 3 is closer to 6 than it is to 5. A reasonable estimate for 3 is 5.8. Which of these numbers is a perfect square? 5 What is the perimeter of the triangle below? A C 6 B D 8 36 ft ft The area of a square picture is 6 square inches. What is the length of the picture? A inches C 8 inches B 6 inches D inches 3 Which of the following numbers is not a perfect cube? A 7 C 5 B 5 D 9 Between what two consecutive whole numbers is 75? A 5 and 6 C 7 and 8 B 6 and 7 D 8 and 9 6 ft A ft C ft B ft D ft 6 Which value is closest to 3? A 3.5 C 5. B. D Which radical is closest in value to 7.5? A B 5 55 C 65 D 75 8 Preeti wrote the equation below. n Which number is closest in value to n? A.3 C 8. B 6. D 9. UNIT Eponents and Radicals 5

26 Read each problem. Write our answer. SAMPLE The volume of a cube with a side length of s is s 3 cubic units. The volume of the cube below is 5 cubic centimeters. s cm What is the area, in square centimeters, of each side of the cube? Answer First find s, the side length of the cube. Since , The area of each square side on the cube is s. So, the area of each side is square centimeters. 9 The base of a square pramid has an area of 65 square inches. What is the length, in inches, of each side of the square base? Answer Mateo thinks the cube root of, is. Is he correct? Eplain how ou know. Between which two consecutive whole numbers is 5? Eplain how ou found our answer. 6 UNIT Eponents and Radicals

27 Read each problem. Write our answer to each part. The floor of a square bedroom has an area of 69 square feet. Part A What is the length, in feet, of each side of the bedroom floor? Answer Part B The floor of a square famil room has an area twice as great as the area of this bedroom floor. Is the length of each side of the famil room floor twice as great as the length of each side of the bedroom floor? Eplain how ou know. Find the area of the famil room floor. Is this area a perfect square? 3 Ale wants to estimate the value of 3. Part A Between which two consecutive whole numbers is 3? Answer Part B Estimate the value of 3 to the nearest tenth. Eplain how ou know our estimate is correct. UNIT Eponents and Radicals 7

28 Review Eponents and Radicals Read each problem. Circle the letter of the best answer. Which epression is equivalent to 6 5? A 65 C B 6 5 D The area of a square dance floor is 96 square feet. What is the length of each side of the dance floor? A feet C 8 feet B 6 feet D feet 3 A bee s wings can beat about times an hour. How is this number written in standard form? A 6,8 C 6,8, B 68, D 6,8, What is the value of (3 ) 3? A 9 C 7 B D 5 Which number is equivalent to.635? A C B D About people in South Africa speak English. English is the primar language for about of these people. The rest speak it as a secondar language. About how man speak English as a secondar language? A. 3 6 C.33 3 B. 3 7 D Which of these equations is true? A B ( 5 ) C 5 D 3 5 ( ) ( ) ( ) 8 The longest bone in the human bod averages about m in length. The shortest bone averages m. About how man times greater is the length of the longest bone than the shortest bone? A C, B 5 D 5, 8 UNIT Eponents and Radicals

29 Read each problem. Write our answer. 9 What is the value of (? 5 ) Answer One ear, India produced more than,5, tons of bananas. How is this number written in scientific notation? Answer Are the epressions ( ) and equivalent? Eplain how ou know. Between which two consecutive whole numbers is 3 3? Answer 3 What is the value of the epression 6 9? Answer What is the value of 95. ( )? Write our answer in scientific notation. Answer 5 Tia thinks the quotient (. 3 9 ) (. 3 3 ) is equal to Eplain whether or not Tia is correct Damian wrote the equation n 3 5. What is the value of n? Answer UNIT Eponents and Radicals 9

30 Read each problem. Write our answer to each part. 7 Mr. Wler wrote the epression Part A What is the value of this epression? Answer Part B Write an eponential epression using division that is equivalent to the one Mr. Wler wrote. Answer 8 The United States government produced about.5 billion pennies one ear. Part A How is.5 billion written in scientific notation? Answer Part B That same ear, about half-dollars were produced. How man more pennies were produced that ear than half-dollars? Eplain how ou know. 3 UNIT Eponents and Radicals

31 Unit Real Numbers Lesson Rational Numbers reviews what a rational number is and what numbers make up the rational numbers. Lesson Irrational Numbers reviews what an irrational number is, what numbers make up the irrational numbers, and how to find approimate values of irrational numbers. UNIT Real Numbers 3

32 Lesson Rational Numbers 8.NS. The real numbers are made up of rational numbers and irrational numbers. Whole numbers include the counting numbers,, 3,, and. Integers include whole numbers and their opposites. A fraction with a numerator that is equal to or greater than the denominator is called an improper fraction. A rational number is an number that can be written as a fraction. All whole numbers, integers, fractions, improper fractions, and mied numbers are rational numbers. Show that each of these numbers is a rational number , so.5 is rational , which is an improper fraction, 3 3 so 6 is rational , so 3 is rational. 5, so is rational. Some decimal numbers are rational numbers. A decimal is rational if it terminates or repeats. A terminating decimal is a decimal number whose digits end. A repeating decimal is a decimal number with digits that repeat in a pattern. Which of these numbers are rational numbers? The bar smbol above digits in a decimal is used to show digits that repeat Ellipses ( ) indicate that a number continues is a terminating decimal, so it is rational.. and are repeating decimals, so the are rational. 3 9 can be written as the improper fraction 9, so it is rational and.39 are neither terminating nor repeating decimals, so the are not rational. 3 UNIT Real Numbers

33 Read each problem. Circle the letter of the best answer. SAMPLE Which of the following is not a rational number? A B 3 C 3 5 D The correct answer is D. A rational number is an number that can be written as a fraction. This includes all fractions, mied numbers, improper fractions, integers, and terminating or repeating decimals. The decimal number in choice A terminates, a mied number is in choice B, and an improper fraction is in choice C. All of these are rational numbers. The decimal in choice D does not terminate or repeat, so it cannot be written as a fraction and is not rational. Which tpe of number is not rational? Which of these numbers is not rational? A B C D negative integer improper fraction repeating decimal non-terminating decimal A C B 93 D Julius thinks the number 6. 5 is rational. Is he correct? Which decimal is rational? A 6.33 B C 8.36 D A B C D Yes, because the decimal repeats. Yes, because the decimal terminates. No, because the number is not a fraction. No, because the number is not an integer. 3 Which statement is true of the numbers 39 7 and.3 8? A B C D 39 Onl is rational. 7 Onl.3 8 is rational. Both numbers are rational. Both numbers are not rational. 6 Which equation shows wh.33 is a rational number? A B C D UNIT Real Numbers 33

34 Read each problem. Write our answer. SAMPLE Tler wrote these numbers Which of these numbers are rational? Answer A rational number is an number that can be written as a fraction. Mied numbers like 6, integers like, terminating decimals like.89, and repeating decimals like can all be written as fractions. So, all are rational. 7 Write a decimal number that is not a rational number. Answer 8 Show that.357 is a rational number. 9 Gregor thinks 6.9 is not a rational number because it is not a fraction. Is Gregor correct? Eplain how ou know. Use a calculator to change to a decimal. What kind of decimal is it? 7 Answer 3 UNIT Real Numbers

35 Read each problem. Write our answer to each part. Look at this set of numbers Part A Which of these numbers are not rational? Answer Which decimals repeat? Which decimals terminate? Part B Eplain how ou know our answer is correct Cassandra changed 3 to a decimal. Part A Write the decimal number that is equivalent to 3. Answer Part B Which tpe of decimal best describes 3 terminating, not terminating, repeating, or not repeating? Eplain how ou know. UNIT Real Numbers 35

36 Lesson I rational Numbers 8.NS., 8.NS. Rational numbers and irrational numbers make up the real numbers. A non-terminating decimal is a decimal number whose digits do not end. A non-repeating decimal is a decimal number whose digits do not repeat. The square root of a number that is not a perfect square is an irrational number. The square root of a fraction can be written as the square root of the numerator divided b the square root of the denominator Common approimations for p are 3. and 7. An irrational number is a real number that cannot be written in fraction form. Irrational numbers are decimals that are non-terminating and non-repeating. Most square roots are irrational numbers. The number pi (p) is also irrational. Pi equals the ratio of the circumference of a circle to its diameter. The value of p is alwas Eplain whether these numbers are rational or irrational is a terminating decimal, so it is rational is a non-terminating and non-repeating decimal, so it is irrational. 6 can be written as the square root of the quotient of two 8 perfect squares, 6 8 5, so it is rational. 9 is not the square root of a perfect square, so it is irrational. Rational numbers approimate the value of irrational numbers. Draw a dot to show the approimate location of on this number line. 3 Find the roots of two perfect squares that lies between. 5 and and 5, so the value of is between and. Since is a little closer to than it is to, a good approimation for is.. 36 UNIT Real Numbers 3

37 Read each problem. Circle the letter of the best answer. SAMPLE Which number is closest in value to 3? A 3. B 5.6 C 9.9 D 8.7 The correct answer is B. Find two consecutive perfect squares that surround 3 and. Use these to approimate the values of 3 and. Since 3 is between and, 3 is between and : 3 <.7. Since is between 9 and 6, is between 9 and 6 : < 3.3. Multipl the approimate values: Which of these numbers is irrational? A 9 5 B C D 9 5 Which statement best eplains wh is an irrational number? A B C D It is negative. It is not a fraction. It is a not a whole number. It is a non-terminating decimal. 3 Which of these numbers lies between and.5 on a number line? A B 3 C 7 D 8 Which statement is true? A p 5 7 C, Which number line shows the approimate location of 8? A B C D Which statement best describes the value of p? A It is less than. B It is between and.5. C It is between.5 and 3. D It is greater than B p, 3. D. 3.5 UNIT Real Numbers 37

38 Read each problem. Write our answer. SAMPLE Write these numbers in order from least to greatest p Answer Find rational approimations for 9 and p. The number 9 is almost directl between and 5, so 9 is approimatel 9.5. Since p is approimatel 3., p is approimatel (3.) The numbers, in order from least to greatest, are , 9, p. 7 Write a number between 7.6 and 7.5 that is irrational. Answer 8 Which of these numbers are irrational? Answer 9 Place a dot on the number line below to show the approimate location of 5. Is the number rational or irrational? Eplain how ou know. 38 UNIT Real Numbers

39 Read the problem. Write our answer to each part. Jessica wrote the numbers in the table below to represent the side lengths of four different squares. SIDE LENGTHS OF SQUARES Square A B C Side Length (in.) p 5 D 6 6 Part A Which of these squares have side lengths that are irrational numbers? Answer Part B List these side lengths in order from least to greatest. Eplain how ou know our answer is correct. Find decimal equivalents or approimations for all side lengths. Then compare the decimals. UNIT Real Numbers 39

40 Review Real Numbers Read each problem. Circle the letter of the best answer. Which of the following is a rational number? A p C Which number is closest in value to X on the number line below? X B 8 D 9.33 What tpe of decimal number is 7. 7? A repeating C non-repeating A B 6 8 C D 3 B terminating D non-terminating 3 Which of these numbers is not rational? A B C D Which statement best describes the number 8.88? A B It is rational because it is a repeating decimal. It is rational because it is a terminating decimal. Which of these numbers comes between 9 7 and 7? A C C D It is irrational because it is a repeating decimal. It is irrational because it is a nonterminating decimal. B D 5 Which decimal cannot be written in fraction form? A. 987 C.3535 B.573 D Which equation shows wh. 6 is a rational number? A B C D UNIT Real Numbers

41 Read each problem. Write our answer. 9 What two whole numbers does the value of p lie between? Answer Which of the numbers below are rational? Answer Which of these numbers has the greatest value? 7 p Answer Show that.7 is a rational number. 3 Write a rational number that lies between and 3. Answer Write a number between 9 and that is irrational. Answer 5 Which is greater, p or? Eplain how ou know. 7 UNIT Real Numbers

42 Read each problem. Write our answer to each part. 6 Reill changed 6 to a decimal. Part A Which tpe of decimal best describes 6 terminating, not terminating, repeating, or not repeating? Eplain how ou know. Part B Can a fraction be written as a non-terminating decimal? Eplain how ou know. 7 Dao writes the irrational number. Part A Place a dot on the number line below to show the approimate location of. 3 5 Part B Eplain how ou found our answer. UNIT Real Numbers

43 Unit 3 Linear Relationships Lesson Proportional Relationships reviews what proportional relationships are and compares them in equation and graphical forms. Lesson Graphing Linear Relationships reviews how to graph linear equations through the origin on a coordinate plane and through an point on the -ais of a coordinate plane. Lesson 3 Solving Linear Equations reviews how to solve linear equations with one solution, no solution, and infinitel man solutions. Lesson More Solving Equations reviews how to solve multiple-step linear equations with rational number coefficients. UNIT 3 Linear Relationships 3

44 Lesson Proportional Relationships 8.EE.5, 8.EE.6 A constant is a value that does not change. In 5 3, the constant is 3. The slope of a line shows how the change in one variable relates to the change in the other variable. change in Slope 5 change in For points (, ) and (, ), slope 5. Slope is a constant since it is the same throughout a proportional relationship. A unit rate is the ratio, or rate, for one unit of a given quantit. The unit rate $3 lb shows a cost of $3 per pound. A proportional relationship eists between two quantities when one is a constant multiple of the other. In 5 m, a proportional relationship eists between the quantities and since the constant, m, multiplies. In a proportional relationship, the constant is called the slope. Graphs and equations in the form 5 m can be used to show proportional relationships. Compare the equation 5 with the proportional relationship graphed here. Which has a greater slope, the equation or the line in the graph? The slope of the equation 5 is the constant,. The slope of the line in the graph is the ratio that compares the change in cost to the change in weight. 3 When the cost is $3, the weight is lb: slope 5 6 When the cost is $6, the weight is lb: slope When the cost is $9, the weight is 3 lb: slope The slope of the line is 3 or 3. Since 3., the line in the graph has a greater slope. In the graphed eample above, ou can see the slope is the same at each point. The slope simplifies to the same unit rate. Cost ($) 8 6 DELI SALAD COSTS 3 5 Weight (lb) UNIT 3 Linear Relationships

45 Read each problem. Circle the letter of the best answer. SAMPLE Similar triangles GHJ and KHL are shown on the coordinate plane. Which statement must be true of the slope of GH? 8 6 K G A It is the same as the slope of GJ. B C It is the same as the slope of KH. It is twice the slope of GJ. D It is twice the slope of KH. H L J 6 8 The correct answer is B. Slope is the ratio of the change in -values to the change in -values. GH has endpoints (, 8) and (, ). The slope of GH is The slope of GJ is The slope of KH is The slopes of GH and KH are both 3. Lace walks at a speed of.5 miles per hour. Which equation models this unit rate? A 5.5 C 5.5 B 5.5 D 5.5 Triangle MNP is similar to triangle QRS. M N Which sides have the same slope? A Q P MP and QS C QR and QS S R 3 The average speed, in miles per hour, a jet flies is graphed on the coordinate plane below. Distance (mi),,8,6,,, 8 6 JET SPEED Time (hr) Which equation models this relationship? A 5 C 5, B 5 5 D 5, B MP and NP D QR and NP UNIT 3 Linear Relationships 5

46 Read each problem. Write our answer. SAMPLE The equation 5 5 models the rate Tasha charges for tutoring hours. This graph models the rate Lizzie charges for tutoring. How do the rates Tasha and Lizzie charge for tutoring compare? Answer Amount Charged ($) LIZZIE S TUTORING RATES 6 8 Hours The rate charged is the same as the slope of each model. The slope of the equation is 5, so Tasha charges $5 per hour. The slope of the line is divided b for an point on the graph, so the slope 5 5. Lizzie charges $ per hour of tutoring, which is 5 $5 less per hour than Tasha. Based on the graph above, write the equation that models Lizzie s charge for tutoring. Answer 5 Triangles CDE and CFG are similar, as shown here. Eplain wh the slope of CE is the same as the slope of CG. 6 A factor makes 6 brushes in hours. Does the equation 5 5 model the number of brushes made each hour? Eplain how ou know. 8 6 F D G E C UNIT 3 Linear Relationships

47 Read each problem. Write our answer to each part. 7 Gavin bought cubic ards of topsoil for $75. Part A What is the unit rate, in cost per cubic ard, for the topsoil? Answer What operation do ou use to find the cost per ard? Part B Write an equation in the form 5 m to show this relationship. Eplain how ou know our equation is correct. 8 Triangle DEF is shown here. Triangle DGH is similar to triangle DEF. The coordinates of verte H are (, 5) 8 D F Part A What are the coordinates of verte G? 6 E Answer Part B Eplain how ou know. 6 8 UNIT 3 Linear Relationships 7

48 Lesson Graphing Linear Relationships 8.EE.6 A coordinate plane has a horizontal ais, called the -ais, and a vertical ais, called the -ais. To plot point (, ) on a coordinate plane, first locate point along the -ais. From there, move units up or down and plot the point. The point (, b) represents the -intercept. The slope of a line is the ratio of the change in the -value over the change in the -value. Slope is sometimes referred to as rise over run. Slope 5 change in change in 5 rise run For points (, ) and (, ), the slope formula is To find the slope from a graph, ou can count the rise and the run from an point on the line to the net point. 8 UNIT 3 Linear Relationships A graph of a linear relationship with the equation 5 m goes through the origin, or center of a plane. The constant m represents the slope, or steepness, of the line. Draw the graph of the equation 5 on a coordinate plane. 3 Make a table. Pick values for. Find the values of that make the equation true. Then plot the points and connect them A linear relationship with the equation 5 m b goes through the point b on the -ais. The point b is called the -intercept. Write the equation of the line shown on this coordinate plane. The line touches the -ais at 3. So the -intercept is b 5 3 or (, 3). Use it and another point in the slope formula. (, 3) and (, ) ( slope 5 m 5 3) 5 5 In 5 m b, m 5 and b 5 3. So the equation of the line is

49 Read each problem. Circle the letter of the best answer. SAMPLE What is the slope, m, and the -intercept, b, of this line? A m 5 and b 5 3 B m 5 3 and b C m 5 and b D m 5 3 and b 5 5 The correct answer is D. The -intercept, b, is the point where the line touches the -ais. Since the line goes through the origin, the -intercept is. From the origin, count the rise over the run to the net point of the graph. This is the slope. From the origin, rise unit rise up and run 3 units left to the net point: m 5 run Which equation has a slope of 6 and a -intercept of 5? A C 5 (5 6) B D 5 (6 5) Which statement is true of m, the slope of this line? Ton graphed the line 5 on this coordinate plane. Line A 6 Line B 6 Line D 6 6 Line C Which line did Ton graph? A line A C line C B line B D line D A m, C, m, B, m, D m. UNIT 3 Linear Relationships 9

50 Read each problem. Write our answer. SAMPLE What is the slope of this line? Answer Locate two points on the line. Then use the slope formula: slope 5. For points (, 3) and (, 3), slope 5 3 ( 3) 5. The slope is What is the equation of the line shown on this coordinate plane? Answer 5 Hakim graphs a line that slants downward from left to right. What must be true of the slope of this line? Answer 6 Paige graphs the equation 5 5. What are the slope and the -intercept of this equation? Slope -intercept 5 UNIT 3 Linear Relationships

51 Read each problem. Write our answer to each part. 7 Eric wants to graph the line 5. Part A Make a table of values to find five points on the graph of this line Be careful when subtracting numbers with negative signs. Part B Graph this line on the coordinate plane above. 8 Willow has the equations 5 and 5 3. Part A Graph and label the equations on the coordinate plane below Part B How are these lines alike? How are the different? Eplain. UNIT 3 Linear Relationships 5

52 Lesson 3 Solving Linear Equations 8.EE.7.a Addition and subtraction are inverse operations. Multiplication and division are inverse operations. To keep an equation balanced, alwas perform the same operation to both sides. n n p p You can check that an answer is correct b substituting the value of the variable back into the original equation. It should make the equation true for 5 3() true An equation is a number sentence that shows two epressions are equal. An equation can contain numbers and variables. A variable is a smbol or letter that represents an unknown number. A solution to an equation is the value of the variable that makes the equation true. To solve an equation for a variable, use inverse operations. Inverse operations are like opposite operations. The undo each other. Solve the equation In this equation, 9 multiplies. Divide both side of the equation b 9 to undo the multiplication and solve for. 9 ( 9) 5 3 ( 9) 5 3 Some equations require two steps to solve. For these, first undo the addition or subtraction. Then undo the multiplication or division. Solve the equation The two operations in this equation are multiplication and addition. First, subtract 9 from both sides of the equation to undo the addition Net, divide both sides b to undo the multiplication. 5 5 Check: ( ) Since is a true statement, 5 is the solution. 5 UNIT 3 Linear Relationships

53 Read each problem. Circle the letter of the best answer. SAMPLE What is the solution to 5 3 3? A 5 B 3 C 3 D 5 The correct answer is B. This equation uses multiplication and subtraction. First add 3 to both sides to undo the subtraction: , so Then divide both sides b 3 to undo the multiplication: , so 3 5. What is the first operation ou should use to solve the equation 5 5? A addition C multiplication B subtraction D division What is the solution to the equation d 5 6? A d 5 C d 5 8 B d 5 8 D d 5 3 Which steps should be taken to solve the equation 6 5 5? A B C D first add, then multipl first multipl, then add first divide, then subtract first subtract, then divide What is the solution to the equation 8r 6 5? A r 5 C r 5 5 What value of z makes this equation true? z 5 5 A z 5 8 C z 5 3 B z 5 3 D z What is the solution to this equation? m 3 5 A m 5 C m 5 8 B m 5 7 D m What value of q makes the equation true? 3q 5 7 A 5 C 3 B 3 D What value of k makes this equation true? k 7 5 B r 5 D r 5 A C B D UNIT 3 Linear Relationships 53

54 Read each problem. Write our answer. SAMPLE Madison thinks the solution to the equation z 7 5 is 3. Is she correct? Eplain. Answer No, Madison is not correct. To check if 3 is a solution, substitute that number for z into the equation: ( 3) 7 5. Then simplif and see if the result is true: Since 3?, the number 3 is not the solution. 9 What value of n makes the equation n 3 5 true? Show our work. Answer Write a two-step equation that has 5 as its solution. Answer Is 7 the solution to the equation w 7 5? Eplain how ou know. 5 UNIT 3 Linear Relationships