Revised 2008 GRADE. Mathematics. A Student and Family Guide. Revised Based on TEKS Refinements

GRADE Revised 2008 Mathematics A Student and Family Guide Revised Based on TEKS Refinements

Texas Assessment STUDY GUIDE Texas Assessment of Knowledge and Skills Grade 5 Mathematics A Student and Family Guide Copyright 2008, Texas Education Agency. All rights reserved. Reproduction of all or portions of this work is prohibited without express written permission from the Texas Education Agency.

Cover photo credits: Top left Gabe Palmer/CORBIS; Top right Jim Craigmyle/CORBIS; Bottom right Royalty-Free/CORBIS; Bottom left Jim Craigmyle/CORBIS.

A Letter from the Deputy Associate Commissioner for Student Assessment Dear Student and Parent: The Texas Assessment of Knowledge and Skills (TAKS) is a comprehensive testing program for public school students in grades 3 11. TAKS, including TAKS (Accommodated) and Linguistically Accommodated Testing (LAT), is designed to measure to what extent a student has learned, understood, and is able to apply the important concepts and skills expected at each tested grade level. In addition, the test can provide valuable feedback to students, parents, and schools about student progress from grade to grade. Students are tested in mathematics in grades 3 11; reading in grades 3 9; writing in grades 4 and 7; English language arts in grades 10 and 11; science in grades 5, 8, 10, and 11; and social studies in grades 8, 10, and 11. Every TAKS test is directly linked to the Texas Essential Knowledge and Skills (TEKS) curriculum. The TEKS is the state-mandated curriculum for Texas public school students. Essential knowledge and skills taught at each grade build upon the material learned in previous grades. By developing the academic skills specified in the TEKS, students can build a strong foundation for future success. The Texas Education Agency has developed this study guide to help students strengthen the TEKS-based skills that are taught in class and tested on TAKS. The guide is designed for students to use on their own or for students and families to work through together. Concepts are presented in a variety of ways that will help students review the information and skills they need to be successful on TAKS. Every guide includes explanations, practice questions, detailed answer keys, and student activities. At the end of this study guide is an evaluation form for you to complete and mail back when you have finished the guide. Your comments will help us improve future versions of this guide. There are a number of resources available for students and families who would like more information about the TAKS testing program. Information booklets are available for every TAKS subject and grade. Brochures are also available that explain the Student Success Initiative promotion requirements and the graduation requirements for high school students. To obtain copies of these resources or to learn more about the testing program, please contact your school or visit the Texas Education Agency website at www.tea.state.tx.us/student.assessment. Texas is proud of the progress our students have made as they strive to reach their academic goals. We hope the study guides will help foster student learning, growth, and success in all of the TAKS subject areas. Sincerely, Gloria Zyskowski Deputy Associate Commissioner for Student Assessment Texas Education Agency 3

Contents Mathematics Introduction...................................... 5 Mathematics Chart................................. 9 Objective 1: Numbers, Operations, and Quantitative Reasoning............................ 11 Practice Questions.................................. 36 Objective 2: Patterns, Relationships, and Algebraic Reasoning.............................. 40 Practice Questions.................................. 49 Objective 3: Geometry and Spatial Reasoning.......... 53 Practice Questions.................................. 67 Objective 4: Concepts and Uses of Measurement........ 70 Practice Questions.................................. 93 Objective 5: Probability and Statistics................ 96 Practice Questions................................. 114 Objective 6: Mathematical Processes and Tools........ 120 Practice Questions................................. 135 Mathematics Answer Key.......................... 139 4

MATHEMATICS INTRODUCTION What Is This Book? This is a study guide to help your child strengthen the skills tested on the Grade 5 Texas Assessment of Knowledge and Skills (TAKS). TAKS is a state-developed test administered with no time limit. It is designed to provide an accurate measure of learning in Texas schools. By acquiring all the skills taught in fifth grade, your child will be better prepared to succeed on the Grade 5 TAKS and during the next school year. What Are Objectives? Objectives are goals for the knowledge and skills that a student should achieve. The specific goals for instruction in Texas schools were provided by the Texas Essential Knowledge and Skills (TEKS). The objectives for TAKS were developed based on the TEKS. Each objective is organized into review sections and a practice section. The review sections present examples and explanations of the mathematics skills for each objective. The practice sections feature mathematics problems that are similar to the ones used on the TAKS test. How Can I Use This Book with My Child? First look at your child s Confidential Student Report. This is the report the school gave you that shows your child s TAKS scores. This report will tell you which TAKS subject-area test(s) your child passed and which one(s) he or she did not pass. Use your child s report to determine which skills need improvement. Once you know which skills need to be improved, you can guide your child through the instructions and examples that support those skills. You may also choose to have your child work through all the sections. How Is This Book Organized? This study guide is divided into the six objectives tested on TAKS. A statement at the beginning of each objective lists the mathematics skills your child needs to acquire. The study guide covers a large amount of material, which your child should not complete all at once. It may be best to help your child work through one objective at a time. 5

How Can I Help My Child Work on the Study Guide? When possible, review each section of the guide before working with your child. This will give you a chance to plan how long the study session should be. Sit with your child and work through the study guide with him or her. Pace your child through the questions in the study guide. Work in short sessions. If your child becomes frustrated, stop and start again later. There are several words in this study guide that are important for your child to understand. These words are boldfaced in the text and are defined when they are introduced. Help your child locate the boldfaced words and discuss the definitions. What Are the Helpful Features of This Study Guide? Examples are contained inside shaded boxes. Each objective has Try It problems based on the examples in the review sections. A Grade 5 Mathematics Chart is included on pages 9 10 and also as a tear-out page in the back of the book. This chart includes useful mathematics information. The tear-out Mathematics Chart in the back of the book also provides both a metric and a customary ruler to help solve problems requiring measurement of length. Look for the following features in the margin: Ms. Mathematics provides important instructional information for a topic. Detective Data offers a question that will help remind the student of the appropriate approach to a problem. Do you see that... points to a significant sentence in the instruction. 6

How Should the Try It Problems Be Used? Try It problems are found throughout the review sections of the mathematics study guide. These problems provide an opportunity for a student to practice skills that have just been covered in the instruction. Each Try It problem features lines for student responses. The answers to the Try It problems are found immediately following each problem. While your child is completing a Try It problem, have him or her cover up the answer portion with a sheet of paper. Then have your child check the answer. What Kinds of Practice Questions Are in the Study Guide? The mathematics study guide contains questions similar to those found on the Grade 5 TAKS test. There are two types of questions in the mathematics study guide. Multiple-Choice Questions: Most of the practice questions are multiple choice with four answer choices. These questions present a mathematics problem using numbers, symbols, words, a table, a diagram, or a combination of these. Read each problem carefully. If there is a table or diagram, study it. Your child should read each answer choice carefully before choosing the best answer. Griddable Questions: Some practice questions use a four-column answer grid like the one used on the Grade 5 TAKS test. How Do You Use an Answer Grid? The answer grid contains four columns, the last of which is a fixed decimal point. The answers to all the griddable questions will be whole numbers. Suppose the answer to a problem is 108. First write the number in the blank spaces. Be sure to use the correct place value. For example, 1 is in the hundreds place, 0 is in the tens place, and 8 is in the ones place. Then fill in the correct bubble under each digit. Notice that if there is a zero in the answer, you need to fill in the bubble for the zero. The grid below shows 108 correctly entered. 1 0 1 2 3 4 5 6 7 8 9 0 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Where Can Correct Answers to the Practice Questions Be Found? The answers to the practice questions are in the answer key at the back of the book (pages 139 147). The answer key explains the correct answer, and it also includes some explanations for incorrect answers. After your child answers the practice questions, check the answers. Each question includes a reference to the page number in the answer key. Even if your child chose the correct answer, it is a good idea to read the answer explanation because it may help your child better understand why the answer is correct. 7

Texas Assessment of Knowledge and Skills Grade 5 Mathematics Chart LENGTH Metric Customary 1 kilometer = 1000 meters 1 mile = 1760 yards 1 meter = 100 centimeters 1 mile = 5280 feet 1 centimeter = 10 millimeters 1 yard = 3 feet 1 foot = 12 inches CAPACITY AND VOLUME Metric Customary 1 liter = 1000 milliliters 1 gallon = 4 quarts 1 gallon = 128 fluid ounces 1 quart = 2 pints 1 pint = 2 cups 1 cup = 8 fluid ounces MASS AND WEIGHT Metric Customary 1 kilogram = 1000 grams 1 ton = 2000 pounds 1 gram = 1000 milligrams 1 pound = 16 ounces TIME 1 year = 365 days 1 year = 12 months 1 year = 52 weeks 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Metric and customary rulers can be found on the tear-out Mathematics Chart in the back of this book. 9

Grade 5 Mathematics Chart Perimeter square P = 4 s rectangle P = (2 l) + (2 w) Area square A = s s rectangle A = l w Volume cube V = s s s rectangular prism V = l w h 10

Objective 1 The student will demonstrate an understanding of numbers, operations, and quantitative reasoning. For this objective you should be able to use place value to read, write, compare, and order whole numbers and decimals; use fractions in problem-solving situations; add, subtract, multiply, and divide to solve problems; and estimate to determine reasonable results. How Do You Read Large Numbers? When you read a whole number, start with the digits on the left. Use the commas to help you read the number. Read the number 965,702,318,406. Look at the number in a place-value chart. Hundred Ten Billions Hundred Ten Hundred Ten Millions Thousands Hundreds Tens Ones Billions Billions Millions Millions Thousands Thousands 9 6 5 7 0 2 3 1 8 4 0 6 A comma is used to separate each group of three digits. Look at the number below: 2,167,986,450 Read the three-digit number to the left of the first comma. Then say what they represent. nine hundred sixty-five billion Read the three-digit number to the left of the second comma. Then say what they represent. seven hundred two million Read the three-digit number to the left of the third comma. Then say what they represent. three hundred eighteen thousand Read the last three-digit number. four hundred six Read the number 965,702,318,406 as nine hundred sixty-five billion, seven hundred two million, three hundred eighteen thousand, four hundred six. 11

Objective 1 Try It Read the number 2,309,758,011. Write the number in the place-value chart below. Billions Hundred Ten Hundred Ten Millions Millions Millions Thousands Thousands Thousands Hundreds Tens Ones To read this number, say billion, million, thousand,. Read the number 2,309,758,011 as two billion, three hundred nine million, seven hundred fifty-eight thousand, eleven. How Do You Compare and Order Whole Numbers? Look at the place values of the digits to help you compare and order whole numbers. Look at these two numbers. 4,362,124,631 4,397,125,729 To determine which number is greater, look at them in a place-value chart. Then compare the digits in each place value. Billions Hundred Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Millions Thousands Thousands 4 3 6 2 1 2 4 6 3 1 4 3 9 7 1 2 5 7 2 9 You can use these symbols when you compare numbers. Symbol Meaning is equal to > is greater than < is less than Look at the digits in the billions place first. Both numbers have the digit 4 in the billions place. Look at the digits in the hundred millions place next. Both numbers have the digit 3 in the hundred millions place. Look at the digits in the ten millions place next. Since 9 6, the number with the digit 9 in the ten millions place is greater. 4,397,125,729 4,362,124,631 12

Objective 1 Try It Use a place-value chart to order these numbers from greatest to least. 6,490,781,004 63,758,902,449 6,837,200,312 67,554,309,377 Hundred Ten Hundred Ten Hundred Ten Billions Millions Thousands Hundreds Tens Ones Billions Billions Millions Millions Thousands Thousands 6 4 9 0 7 8 1 0 0 4 Write the numbers in the place-value chart. The first one has been done for you. Look at the digits in the ten billions place. The numbers and both have a in the ten billions place. They are the two greatest numbers. For these two numbers, compare the digits in the billions place. Since, the number. The greatest number is. The next-greatest number is. Look at the remaining two numbers. They both have a in the billions place. So compare the digits in the hundred millions place. Since, the number. The least number is. The numbers in order from greatest to least are The numbers 63,758,902,449 and 67,554,309,377 both have a 6 in the ten billions place. Since 7 3, the number 67,554,309,377 63,758,902,449. The greatest number is 67,554,309,377. The next-greatest number is 63,758,902,449. They both have a 6 in the billions place. Since 8 4, the number 6,837,200,312 6,490,781,004. The least number is 6,490,781,004. The numbers in order from greatest to least are 67,554,309,377 63,758,902,449 6,837,200,312 6,490,781,004 13

Objective 1 What Are Decimals? Decimals are another way to write fractions with denominators such as 10, 100, and 1,000. Decimals and fractions both name part of a whole. A decimal names part of a whole that has been divided into 10, 100, 1,000, or more parts. 3 The fraction is written as the decimal 0.3. 1 0 7 The fraction is written as the decimal 0.07. 1 00 9 The fraction is written as the decimal 0.009. 1,0 00 Look at the decimal below: Decimal point 1.47 When a number with a decimal is written in words, the -ths ending tells you that those digits belong on the right side of the decimal point. The decimal point separates the whole part of the number from the fractional part of the number. There is a 1 to the left of the decimal point, so there is one whole. There is a 47 to the right of the decimal point. This means 47 out of 100 parts. The decimal point means and. The number 1.47 is read: one and forty-seven hundredths. Looking at decimals in a place-value chart can help you read and understand them. Read the decimal 12.34 in the place-value chart. Tens Ones. Tenths Hundredths 1 2. 3 4 Read the number to the left of the decimal point, twelve. Say the word and to represent the decimal point. Read the number to the right of the decimal point, thirty-four. Then say the place-value name of the last digit on the right, hundredths. Read the number 12.34 as twelve and thirty-four hundredths. 14

Objective 1 Read the number 5.826 in the place-value chart. Ones. Tenths Hundredths Thousandths 5. 8 2 6 Read the number to the left of the decimal point, five. Say the word and to represent the decimal point. Read the number to the right of the decimal point, eight hundred twenty-six. Then say the place-value name of the last digit on the right, thousandths. Read the number 5.826 as five and eight hundred twenty-six thousandths. Here are 10 blocks put together to model thousandths. Each block is divided into 100 small squares. There are a total of 1,000 small squares. The number 1,000 is the denominator of the fraction. The numerator of the fraction is the number of small squares that are shaded. The model shows 64 small squares shaded. 64 64 The model represents the fraction. The fraction is read as 1,000 1,000 sixty-four thousandths. The equivalent decimal should be written to the thousandths place. 64 0.064 1,000 The decimal 0.064 is read as sixty-four thousandths. The decimal 64 0.064 represents the fraction. 1,000 15

Objective 1 How Do You Compare and Order Decimals? Look at the values of the digits to help you compare and order decimals. Compare these decimals. 2.9 2.925 Placing zeros at the end of a decimal numeral does not change its value. 5.3 5.30 12.84 12.840 576.03 576.030 To determine which decimal is greater, look at the numbers in a place-value chart. You can write zeros at the end of 2.9 until it has the same number of digits to the right of the decimal point as 2.925. The number 2.9 2.900. Ones. Tenths Hundredths Thousandths 2. 9 0 0 2. 9 2 5 Look at the ones place. Both numbers have a 2 in the ones place. Look at the tenths place. Both numbers have a 9 in the tenths place. Look at the hundredths place. Since 2 0, then 2.925 2.900. Try It Use the place-value chart to order these decimals from greatest to least. 1.19 3.417 3.6 1.1 Should you place zeros at the end of any of these numbers? Ones. Tenths Hundredths Thousandths.... Write the numbers in the place-value chart. First look at the digits in the ones place. The numbers and both have a 3 in the ones place. They are the two greatest numbers. For these two numbers, compare the digits in the tenths place. Since, the number. The greatest number is ; it should be written first. The next-greatest number is ; it should be written next. 16

Look at the remaining two numbers. Compare the digits. Both Objective 1 numbers have a in the ones place and the tenths place. Look at the hundredths place. Since, the number. The least number is ; it should be written last. Now, list the numbers in order from greatest to least. The numbers 3.417 and 3.600 both have a 3 in the ones place. Since 6 4, the number 3.600 3.417. The greatest number is 3.6. The next-greatest number is 3.417. Both numbers have a 1 in the ones place and the tenths place. Since 9 0, the number 1.190 1.100. The least number is 1.1; it should be written last. Listed from greatest to least, the numbers are: 3.6 3.417 1.19 1.1. How Do You Add or Subtract Decimals? To add or subtract decimals, vertically line up the decimal points of the numbers. If necessary, put zeros after the last digits so that each number has the same number of digits to the right of the decimal point. This will not change the value of the number. Then add or subtract as usual. Remember to regroup when necessary. Be sure to include the decimal point in the answer. Find the sum of 3.24, 6.7, and 0.29. Write the numbers in a column. Line up the decimal points. 1 1 3.24 6.70 0.29 10.23 Put a zero at the end of 6.7 so that it has the same number of digits. Add the hundredths. Add the tenths. Add the ones. Bring the decimal point straight down in the answer. The sum is 10.23. 17

Objective 1 In subtraction problems, remember to check your answer by adding. For example, 1 1 2.85 +2.15 5.00 Hannah buys a magazine for $2.85. If she pays for it with a five-dollar bill, how much change should she receive? Use subtraction to find how much change Hannah should receive. Write the numbers in a column and line up their decimal points. A five-dollar bill can be written as $5.00. 9 41010 $5.00 2.85 $2.15 Subtract the hundredths. You need to regroup, but there are no tenths. Regroup 5 ones as 4 ones and 10 tenths. Regroup 10 tenths as 9 tenths and 10 hundredths. Now subtract the hundredths. Subtract the tenths. Subtract the ones. Be sure to place the decimal point in the answer. Hannah should receive $2.15 in change. Try It Travis ran 1.5 kilometers on Monday, 2.25 kilometers on Wednesday, and 0.9 kilometer on Friday. How many kilometers did he run in all? Use the operation of to solve this problem. Use the space below to find the answer. Do you need to place zeros at the end of the decimal numbers? Will you need to regroup?.... The sum of the numbers is. Travis ran a total of kilometers. Use the operation of addition to solve this problem. 1.50 2.25 0.90 4.65. The sum of the numbers is 4.65. Travis ran a total of 4.65 kilometers. 18

Objective 1 When Do You Use Multiplication and Division to Solve Problems? Use multiplication to combine groups of equal size. Use division to separate a whole into groups of equal size. A baker uses 325 pounds of flour each week to make pies and cakes. How many pounds of flour does the baker use in 26 weeks? Multiply to find the total number of pounds of flour. 13 Multiply the ones. 325 26 1950 Multiply the tens. 1 325 26 1950 6500 The zero is a place holder. Finally, add the products. 325 26 1950 6500 325 26 8,450 8450 The baker uses 8,450 pounds of flour. The principal needs 200 tacos for lunch on field day. The tacos come 12 in a package. What is the least number of packages of tacos the principal will need for the lunch? Divide to find how many packages of tacos the principal will need. 1 12 20 0 12 8 Divide: 20 12 Multiply: 1 12 Subtract: 20 12 16 12 20 0 12 80 72 8 Bring down the zero Divide: 80 12 Multiply: 6 12 Subtract: 80 72 200 12 16 R8 There is a remainder of 8. If the principal buys 16 packages of tacos, she will not have enough. She will need 8 more tacos to have 200. The principal will have to buy one more package to have enough tacos. She will need to buy 17 packages. Do you see that... 19

Objective 1 Try It There are 178 baseball players who want to play in the summer league. There can be no more than 15 players on each team. What is the least number of teams that can be formed for the summer league? Use the operation of to find the number of teams that can be formed. 1 7 8 178 = with a remainder of. Since there is a remainder of 13, more team can be formed. What does the remainder tell you to do? 1 =. The least number of teams that can be formed for the summer league is. Use the operation of division to find the number of teams that can be formed. 11 15 1 7 8 15 28 15 13 178 15 11 with a remainder of 13. Since there is a remainder of 13, 1 more team can be formed. 11 1 12. The least number of teams that can be formed for the summer league is 12. What Are Factors? Numbers multiplied together are called factors. For example, 6 and 5 are factors of 30 because 6 5 30. Some other factors and their products are shown below. Factor Factor Product 3 7 21 8 6 48 9 7 63 5 12 60 20

Objective 1 To find all the factors of a number, write down all the factor pairs for that number. For example, the factor pairs for 15 are 1 15 and 3 5. So the factors, listed in order, are 1, 3, 5, and 15. What Are Common Factors? Factors shared by two or more numbers are called common factors of those numbers. For example, 5 is a common factor of 10 and 25 because 5 is a factor of 10 and also a factor of 25. Find all the common factors of 30, 45, and 50. List the factors of each number. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 45: 1, 3, 5, 9, 15, 45 Factors of 50: 1, 2, 5, 10, 25, 50 Circle the factors that the three numbers have in common. The common factors of 30, 45, and 50 are 1 and 5. Try It Find all the common factors of 27 and 36. The factors of 27 are:,,, and. The factors of 36 are:,,,,,,,, and. The common factors of 27 and 36 are:,, and. The factors of 27 are: 1, 3, 9, and 27. The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors of 27 and 36 are: 1, 3, and 9. 21

Objective 1 How Are Decimals Related to Fractions? Decimals and fractions both name part of a whole or part of a group. A decimal shows part of a whole that has been divided into 10, 100, 1,000, or more parts. A fraction shows part of a whole that has been divided into any number of parts. The denominator tells the number of equal parts into which the whole has been divided. The numerator tells how many of the equal parts have been selected. Models can represent both fractions and decimals. Look at this model. The model is divided into 100 equal parts, and 39 of the parts are shaded. You can represent the shaded part of the model in two ways. As a fraction: Use 100 as the denominator. Use 39 as the numerator. 39 The fraction is. 1 00 As a decimal: Write the decimal to the hundredths place. Use 39 as the number of hundredths. The decimal is 0.39. 39 Both the fraction and the decimal 0.39 are read as 1 00 thirty-nine hundredths. 39 The fraction and the decimal 0.39 are equivalent. 1 00 22

Objective 1 Try It Each large square in this model represents 1 whole. What fraction does the model represent? What decimal does the model represent? The model shows large squares completely shaded. In the third large square, out of equal parts are shaded. Write the number as a fraction: as a decimal: In words, you read this number as. The model shows two large squares completely shaded. In the third large square, 71 out of 100 equal parts are shaded. Written as a fraction, the 71 number is 2, and as a decimal, it is 2.71. The number is read as 1 00 two and seventy-one hundredths. 23

Objective 1 How Do You Find Equivalent Fractions? Equivalent fractions are fractions that have the same value but are written differently. Equivalent fractions name the same part of a whole in different ways. Look at the circles below: 1 2 2 4 4 8 Any fraction with the same number in both its numerator and its denominator is equal to 1. 3 1 3 2 1 2 The fractions 1 2, 4 2, and 8 4 all name the same part of a whole. To find an equivalent fraction, multiply or divide the numerator and the denominator by the same number. This is the same as multiplying or dividing by a fraction that is equal to 1. To find an equivalent fraction with a larger denominator, multiply by a fraction equal to 1. Here are some fractions that are equivalent to 4 6. Original Fraction Equivalent Fraction Equal to 1 Multiply Fraction 4 6 2 2 4 2 8 6 2 12 8 12 4 6 3 3 4 3 6 3 12 18 12 18 4 6 10 10 4 10 40 6 10 60 40 60 To find an equivalent fraction with a smaller denominator, divide by a fraction equal to 1. Look at these examples. Original Fraction Equivalent Fraction Equal to 1 Divide Fraction 4 6 2 2 4 2 2 6 2 3 2 3 9 15 3 3 9 3 3 15 3 5 3 5 12 28 4 4 12 4 3 28 4 7 3 7 24

Objective 1 6 What are two equivalent fractions for? 12 To find an equivalent fraction with a denominator larger than 2 3 4 12, multiply by a fraction equal to 1. You could use,,, or 2 3 4 any other fraction equal to 1. 12 24 6 2 12 2 6 is a fraction equivalent to. 1 2 To find an equivalent fraction with a denominator smaller than 3 12, divide by a fraction equal to 1. You could use because 3 3 is a factor of both 6 and 12. 6 3 12 3 2 6 is a fraction equivalent to. 4 12 6 12 2 The fractions,, and are equivalent fractions. 12 24 4 12 24 2 4 Try It 5 Find two equivalent fractions for, one with a larger denominator 15 and one with a smaller denominator. To find an equivalent fraction with a larger denominator, 4 by a fraction equal to. For example, use. 4 5 4 15 4 5 is a fraction equivalent to. 15 To find an equivalent fraction with a smaller denominator, by a fraction equal to. For example, use because 5 is a factor of both 5 and 15. 5 5 15 5 5 5 25

Objective 1 5 is a fraction equivalent to. 15 5 The fractions,, and are fractions. 15 To find an equivalent fraction with a larger denominator, multiply by a fraction equal to 1. 5 4 20 15 4 60 20 60 5 is a fraction equivalent to. 15 To find an equivalent fraction with a smaller denominator, divide by a fraction equal to 1. 5 5 15 5 1 5 is a fraction equivalent to. 3 15 5 20 1 The fractions,, and are equivalent fractions. 15 60 3 1 3 How Do You Compare Fractions? To compare fractions that have the same denominator, compare their numerators. Compare the fractions 3 6 and 6 5. Do you see that... Both fractions have the denominator 6. Compare the numerators to find how many parts are in each fraction. Since 3 5, then 3 6 6 5. It is also true that 5 3. So you can also say that 5 6 6 3. 26

Objective 1 To compare fractions that have different denominators: First find a common denominator for the two fractions. A common denominator is one that is the same in two or more fractions. To find a common denominator for two fractions, list the multiples of each denominator, then find one multiple that is in both lists. Rewrite each fraction as an equivalent fraction with the common denominator. Compare the numerators of the two rewritten fractions. A multiple of a number is the product of that number and any other factor. Compare the fractions 3 4 and 6 5. Which fraction is greater? The denominators 4 and 6 are different. Find a common denominator by listing the multiples of 4 and 6. A number found in both lists of multiples is a common multiple. It is also a common denominator for the two fractions. Multiples of 4: 4, 8, 12, 16,... Multiples of 6: 6, 12, 18, 24,... A common multiple of 4 and 6 is 12. A common denominator is 12. Rewrite 3 4 and 6 5 to have 12 as their denominators. 3 4? 12 5 6? 12 3 22 4 22 5 22 6 22? 12? 12 3 3 9 3 4 3 1 2 4 9 12 5 2 1 0 5 6 2 12 6 1 0 12 Now both fractions have a denominator of 12. Compare the 10 9 numerators. Since 10 9, then. 12 12 10 5 9 3 5 3 5 Since and, then. The fraction is greater 12 6 12 4 6 4 6 3 than the fraction. 4 27

Objective 1 Sometimes it is necessary to list more than the first four multiples. Try It Bart and Pablo had small pizzas for lunch. Bart ate 5 6 of his pizza. Pablo ate 7 8 of his pizza. Who ate more of his pizza? Find a common denominator for 5 6 and 8 7. List the first four multiples of the two denominators. Multiples of 6:,,, Multiples of 8:,,, A common multiple of 6 and 8 is. A common denominator is. Find fractions equivalent to 5 6 and 8 7 with as their common denominator. 5 6? 24 5 6 24 7 8? 24 7 8 24 5 7 Since, the fraction 6 8 is greater. (circle one) ate more of his pizza. The first four multiples of 6 are: 6, 12,18, and 24. The first four multiples of 8 are: 8, 16, 24, and 32. A common multiple of 6 and 8 is 24. A common denominator is 24. Find fractions equivalent to 5 6 and 8 7 with 24 as their common denominator. 5 4 20 6 4 24 7 3 21 8 3 24 Since 21 20, the fraction 7 8 is greater. Pablo ate more of his pizza. 28

Objective 1 What Are Mixed Numbers and Improper Fractions? Mixed numbers and improper fractions are two ways to name fractions greater than 1. A mixed number includes a whole number and a 1 fraction. For example, 4 is a mixed number. The whole number is 4, 3 1 and the fraction is. 3 1 Numerator Whole number 4 3 Denominator An improper fraction has a numerator that is greater than or equal to 9 its denominator. The fraction is an improper fraction because the 4 numerator 9 is greater than the denominator 4. 9 Numerator 4 Denominator A mixed number and an improper fraction can both name the same amount. 1 2 = 4 9 4 1 The mixed number 4 can be written as an improper fraction. 3 Multiply the denominator 3 by the whole number 4. 3 4 12 Add the numerator to this product. 12 1 13 13 becomes the numerator of the improper fraction. Keep the original denominator of 3. 13 3 1 13 The mixed number 4 is equivalent to the improper fraction. 3 3 Do you see that... 29

Objective 1 22 The improper fraction can be written as a mixed number. 7 Divide the numerator 22 by the denominator 7. 3 R1 7 22 21 1 22 7 3 with a remainder of 1 The 3 in the quotient becomes the whole number in the mixed number. The remainder can be written in fraction form using the remainder as the numerator and keeping the original denominator. 1 3 7 22 1 The improper fraction is equivalent to the mixed number 3. 7 7 Try It Write the improper fraction 14 5 as a mixed number. the numerator by the denominator. 1 6 6 The quotient is a whole number in the mixed number. The remainder 4 becomes the of the fraction. The is 5. The improper fraction 14 5 is equivalent to the mixed number. Divide the numerator 14 by the denominator 5. The quotient 2 is the whole number in the mixed number. The remainder 4 becomes the numerator of 14 the fraction. The denominator is 5. The improper fraction is equivalent to 4 5 the mixed number 2. 5 30

Objective 1 How Do You Add or Subtract Fractions with Common Denominators? To add fractions with common denominators, add the numerators. Then write the sum over the common denominator. 1 6 6 2 1 2 3 6 6 1 6 + = 2 6 3 6 To subtract fractions with common denominators, subtract the numerators. Then write the difference over the common denominator. 7 4 7 4 3 1 2 1 2 12 1 2 7 12 = 4 12 3 12 Rasheed bought 1 8 pound of green onions and 8 5 pound of carrots. 1 8 lb 5 8 lb How many pounds of vegetables did Rasheed buy altogether? Find the sum of the two fractions. Since the denominators are the same, add the numerators. 1 5 6 8 8 8 6 Rasheed bought pound of vegetables. 8 31

Objective 1 Try It A cook measures lemon juice and orange juice in two measuring cups. How much more lemon juice than orange juice does she have? 1 cup 3 4 cup 1 2 cup 1 4 cup 1 cup 3 4 cup 1 2 cup 1 4 cup Lemon juice Orange juice The drawing shows cup of lemon juice. The drawing shows cup of orange juice. The number sentence shows how much more lemon juice than orange juice the cook has. The drawing shows 3 4 cup of lemon juice and 4 1 cup of orange juice. The number sentence 3 4 4 1 4 2 shows how much more lemon juice than orange juice the cook has. When Should You Estimate an Answer? When you do not need an exact answer to a problem, you can estimate to find an answer that is close to the exact answer. For example, some problems ask about how many or approximately how much. Use estimation when solving such problems. One way to estimate is to round numbers in a problem before working it out. You can round decimals in the same way you round whole numbers. A number line or a set of rounding rules can help you. For example, to round a decimal to the nearest tenth, look at the digit in the hundredths place. If the digit in the hundredths place is 0, 1, 2, 3, or 4, leave the digit in the tenths place the same. If the digit in the hundredths place is 5, 6, 7, 8, or 9, round the digit in the tenths place to the next-higher value. 32

Objective 1 Jill and Simon are walking along a trail that is shaped like a triangle. The first side of the trail is 2.76 kilometers long. The second side of the trail is 2.39 kilometers long. The third side of the trail is 0.63 kilometer long. About how long is this trail? Jill and Simon use different methods to round the numbers. Jill s method: Jill rounds the length of each side of the trail to the nearest tenth of a kilometer. 2.76 rounded to the nearest tenth is 2.8. 2.39 rounded to the nearest tenth is 2.4. 0.63 rounded to the nearest tenth is 0.6. Jill adds the rounded numbers to find the approximate length of the trail. 2.8 2.4 0.6 5.8 Jill says the trail is about 5.8 kilometers long. Simon s method: Simon rounds the length of each side of the trail to the nearest kilometer. This is the same as rounding to the nearest whole number or ones place. 2.76 rounded to the nearest whole number is 3. 2.39 rounded to the nearest whole number is 2. 0.63 rounded to the nearest whole number is 1. Simon adds the rounded numbers to find the approximate length of the trail. 3 2 1 6 Simon says the trail is about 6 kilometers long. Both methods show correct rounding. Even though Jill and Simon rounded the numbers to different places, both estimates are reasonable. Do you see that... 33

Objective 1 Another way to estimate is by using compatible numbers. Compatible numbers are numbers that are easy to add, subtract, multiply, or divide. Using compatible numbers makes the computation easier. Walter is estimating the total number of marbles that he and five of his friends have. The number of marbles each person has is as follows: 125 73 145 90 120 50 To estimate the sum, you can use compatible numbers. For this example, group together numbers that equal about 200. 125 73 is close to 200. 50 145 is close to 200. 90 120 is close to 200. 200 200 200 600. Walter estimates that he and his friends have about 600 marbles altogether. When using compatible numbers to estimate a product or quotient, change the numbers to other numbers that form a basic fact. This can help you solve the problem in your head. In the fall, 86 students played soccer. If there were 9 teams, about how many players were on each team? To find about how many, estimate the answer. Use compatible numbers, or numbers that are easy to divide, to make the problem easier. Use basic facts to help you. Find a number close to 86 that you can divide by 9 in your head. 86 is close to 90, and 90 divides easily by 9 Divide to find about how many players were on each team. 90 9 = 10 There were about 10 players on each team. 34

Objective 1 Try It Suzanne borrowed $358 from her mother to buy a computer. In order to pay back the entire $358, Suzanne gave her mother the same amount of money each month for 1 year. About how much money did Suzanne give her mother each month? Use to find about how much money Suzanne will give her mother each month. There are months in a year. Find a number close to 358 that divides easily by 12. Use the basic fact 36 12 = 3 to help. $358 is close to. 360 12 Suzanne gave her mother about each month. Use division to find about how much money Suzanne will give her mother each month. There are 12 months in a year. $358 is close to $360. 360 12 30. Suzanne gave her mother about $30 each month. Now practice what you ve learned. 35

Objective 1 Question 1 Which numeral has the digit 8 in the millions place? A 1,807,629 B 82,531,044 C 28,162,751 D 8,629,794,312 Question 2 How is the number three hundred billion, seven hundred eighty thousand, four hundred thirty-nine written as a numeral? A 3,780,000,439 B 300,000,780,439 C 3,780,439 D 300,780,439 Answer Key: page 139 Answer Key: page 139 Question 3 Dylan wants to put the numbers below in order from greatest to least. 231,392,159 5,301,229,038 204,515,724 3,033,431,602 Which list shows the numbers in order from greatest to least? A 204,515,724 231,392,159 3,033,431,602 5,301,229,038 B 5,301,229,038 231,392,159 204,515,724 3,033,431,602 C 5,301,229,038 3,033,431,602 231,392,159 204,515,724 D 3,033,431,602 231,392,159 5,301,229,038 204,515,724 Answer Key: page 139 36

Objective 1 Question 4 Which list shows the numbers in order from least to greatest? A 0.11 0.16 0.33 0.9 B 0.9 0.33 0.16 0.11 C 0.9 0.16 0.11 0.33 D 0.11 0.33 0.16 0.9 Question 5 Which number is less than 7.424? A 7.52 B 7.43 C 7.425 D 7.42 Question 6 Answer Key: page 139 Answer Key: page 139 How is the numeral 1.035 written in words? A One thousand thirty-five B One and thirty-five hundredths C One and thirty-five thousandths D One thousand thirty-five hundredths Question 7 Jackie is trying to write the improper fraction 7 in different ways. Which of the following is 3 7 equivalent to? 3 4 A 1 6 1 B 2 3 1 C 3 2 3 D 1 4 Answer Key: page 140 Question 8 3 Which fraction is NOT equivalent to? 9 1 A 3 15 B 45 6 C 18 5 D 12 Answer Key: page 140 Question 9 Betty is ordering a cake for a party. She can cut it into eighths or twelfths. Which of the following 1 1 correctly compares and? 8 12 1 1 A 8 12 1 1 B 8 12 1 1 C 8 12 D Not here Answer Key: page 140 Answer Key: page 140 37

Objective 1 Question 10 The model below shows 150 1,000 shaded. Which decimal represents the shaded part of this model? A 0.50 B 0.050 C 0.015 D 0.150 Answer Key: page 140 Question 11 Bob bought a CD for $11.95. He gave the cashier $20.00. How much change did Bob receive? A $7.95 B $8.05 C $8.15 D $8.25 Question 12 Karen ordered 15 cases of cookies to sell for her soccer team. Each case had 288 cookies in it. How many cookies did she order in all? A 1,728 B 4,320 C 4,280 D Not here Answer Key: page 140 Answer Key: page 141 38

Objective 1 Question 13 A group of 12 friends spent a total of $156 at the carnival. If each friend spent the same amount of money, how much did each friend spend? Record your answer and fill in the bubbles. Be sure to use the correct place value. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Question 16 3 Tristan has 4 yards of rope. He cuts off a piece 1 4 of rope 2 yards long. Which number sentence 4 can be used to find the length of rope, in yards, Tristan will have left after he cuts off a piece 1 2 yards long? 4 3 1 A 4 2 4 4 3 1 1 B 4 2 2 4 4 4 3 1 C 4 2 4 4 3 1 1 D 4 2 2 4 4 4 9 9 9 Answer Key: page 141 Answer Key: page 141 Question 14 Mr. Cohen is buying notepads for a school trip. There are 54 students going on the trip. Each student needs 1 notepad. Notepads are sold in packages of 5. How many packages of notepads does Mr. Cohen need to buy? Question 17 Monica s family went on a trip. The distances they drove each day are shown below. 59.6 miles, 79.5 miles, 89.8 miles, 70.3 miles About how far did Monica s family drive altogether on their trip? A 10 A 400 miles B 7 B 280 miles C 11 C 320 miles D 9 D 300 miles Answer Key: page 141 Answer Key: page 141 Question 15 Which list shows all the common factors of 12, 24, and 36? A 1, 2, 3, 4, 6, 12 B 2, 6, 12 C 1, 2, 3 D 1, 3, 4, 6, 8 Answer Key: page 141 Question 18 Tickets for a play cost $3.25 each. If 417 people bought tickets for the play, which is closest to the total amount of money made in ticket sales? A $1,200 B $1,000 C $800 D $400 Answer Key: page 141 39

Objective 2 The student will demonstrate an understanding of patterns, relationships, and algebraic reasoning. For this objective you should be able to make generalizations based on observed patterns and relationships; and describe relationships mathematically. How Can You Describe the Relationship Between Sets of Data? Sometimes you might need to know how sets of numbers are related. Looking at the data in a list, table, or chart can help you see a relationship. The school library received boxes of new books. The table below shows the number of books in each box. New Books Number of Number of Boxes Books 2 36 3 54 4 72 5 90 Look at the relationship between the numbers. There are 36 books in 2 boxes. 2 36 There are 54 books in 3 boxes. 3 54 There are 72 books in 4 boxes. 4 72 There are 90 books in 5 boxes. 5 90 The relationship shown in the table is: The number of boxes times 18 equals the number of books. 40

Objective 2 Rowan is saving money each week. The table below shows the amount of money he had saved after different numbers of weeks. Rowan s Savings Week Amount of Money 4 $100 5 $125 6 $150 7 $175 What is the relationship between the amount of money Rowan saved and the number of weeks he saved? Look at the pairs of numbers in the table: 4 and $100 5 and $125 6 and $150 7 and $175 How is the number of weeks related to the amount of money saved? 4 100 4 25 = 100 5 125 5 25 = 125 6 150 6 25 = 150 7 175 7 25 = 175 The amount of money saved is 25 times the number of weeks. 41

Objective 2 Try It The table below shows the total number of sheets of paper in different numbers of notebooks. Paper in Notebooks Number of Notebooks Number of Sheets of Paper 2 150 3 225 4 300 5 375 Describe the relationship between the number of notebooks and the number of sheets of paper in the notebooks. Look at the pairs of numbers: 2 and 150 3 and 225 4 and 300 5 and 375 How is the number of sheets of paper related to the number of notebooks? 2 150 3 225 4 300 5 375 The number of sheets of paper is times the number of notebooks. 2 75 150 3 75 225 4 75 300 5 75 375 The number of sheets of paper is 75 times the number of notebooks. 42

Objective 2 What Are Prime Numbers and Composite Numbers? A prime number is a number that has only 1 and itself as factors. For example, 17 is a prime number. The only factors of 17 are 1 and 17. The number 1 is neither prime nor composite. A composite number has 1, itself, and at least one other number as its factors. For example, 21 is a composite number. In addition to 1 and 21, 21 also has 3 and 7 as factors. The factors of 21 are 1, 3, 7, and 21. Determine whether each number below is prime or composite. List the factors of each number. If the only factors of a number are 1 and itself, it is a prime number. If the number has other factors, it is a composite number. Number Factors Prime or Composite? 2 1, 2 Prime (The only factors are 1 and 2.) 3 1, 3 Prime (The only factors are 1 and 3.) 4 1, 2, 4 Composite 5 1, 5 Prime (The only factors are 1 and 5.) 6 1, 2, 3, 6 Composite 7 1, 7 Prime (The only factors are 1 and 7.) 8 1, 2, 4, 8 Composite 9 1, 3, 9 Composite 10 1, 2, 5, 10 Composite 11 1, 11 Prime (The only factors are 1 and 11.) 12 1, 2, 3, 4, 6, 12 Composite 13 1, 13 Prime (The only factors are 1 and 13.) 14 1, 2, 7, 14 Composite 15 1, 3, 5, 15 Composite The prime numbers from this list are 2, 3, 5, 7, 11, and 13. The composite numbers from this list are 4, 6, 8, 9, 10, 12, 14, and 15. You do not need to know all the factors of a number to decide whether it is a composite number. You need to know only that it has at least one factor in addition to 1 and itself. Do you see that... 43

Objective 2 Try It Is the number 28 a prime number or a composite number? Think about all the factors of 28. 28 has 1 and itself as factors because 1. 28 is an even number, so another factor is. 2 28. Another factor pair of 28 is and 7. 7 28. Since the number 28 has factors besides 1 and 28, it is a number. 28 has 1 and itself as factors because 1 28 28. 28 is an even number, so another factor is 2. 2 14 28. Another factor pair of 28 is 4 and 7. 4 7 28. Since the number 28 has factors besides 1 and 28, it is a composite number. 44

Objective 2 Arrays and area models can show prime numbers and composite numbers. 2 1 2 1 2 This model shows 2 1 and 1 2. The only factors of 2 are 1 and 2. Therefore, 2 is a prime number. A factor pair shows 2 factors of a given product. One factor pair of 18 is 3 and 6. 7 1 7 This model shows 7 1 and 1 7. The only factors of 7 are 1 and 7. Therefore, 7 is a prime number. 7 1 8 1 8 This model shows 8 1, 1 8, 4 2, and 2 4. The factors of 8 are 1, 2, 4, and 8. Therefore, 8 is a composite number. 2 4 8 1 4 2 Factor trees can show whether a number is prime or composite. Look at the factor trees for the numbers 16 and 17 below. 16 17 2 x 8 1 x 17 2 x 4 2 x 2 The number 16 has other factors besides 1 and itself. So it is not a prime number. It is a composite number. The number 17 has no other factors besides 1 and itself. So it is a prime number. 45

Objective 2 Try It Use factor pairs, arrays, or factor trees to determine whether these numbers are prime or composite. 18 23 31 27 The prime numbers are and. The composite numbers are and. The number 18 is a composite number. Its factors are 1, 2, 3, 6, 9, and 18. The number 23 is a prime number; it has no other factors besides 1 and 23. The number 31 is also prime; it has no other factors besides 1 and 31. The number 27 is a composite number. Its factors are 1, 3, 9, and 27. The prime numbers are 23 and 31. The composite numbers are 18 and 27. An equation can consist of numbers, mathematical symbols, and variables. Variables are letters such as x or y that represent a number. For example, 4 x 7 is an equation where x is the variable. How Can You Describe Relationships Mathematically? You can describe relationships mathematically by replacing words and sentences with numbers, symbols, and variables. Describing relationships with numbers, symbols, and variables can help you solve problems. Follow these steps to help create number sentences or equations to describe and solve problems. Read the whole problem carefully. Determine which numbers you know and which ones you need to find. Look for the relationship between these numbers. Write a number sentence or equation that represents the relationship. 46

Objective 2 In one middle school 250 students play school sports. Each student plays only one sport. The graph shows the number of students who participate in tennis, volleyball, or soccer. The remaining students play basketball. 120 Sports Participation 100 Number of Students 80 60 40 20 0 Tennis Volleyball Soccer Basketball Sport Read the whole problem. Look for what you know and what you need to find. There are 250 students in all who play sports. The bar graph shows that 40 students play tennis, 80 students play volleyball, and 100 students play soccer. Subtract these numbers from 250 to find the number of students who play basketball. Write an equation that can be used to represent b, the number of students who play basketball. The equation 250 40 80 100 b represents the number of students who play basketball. 47

Objective 2 Patrick bought 2 hats for $4 each and a video game for $45. Write an equation that represents the amount Patrick spent. Read the whole problem. Look at what you know and what you need to find. Patrick bought 2 hats for $4 each. The expression 4 4 represents the total amount spent on hats. He spent $45 on a video game. Use addition to represent the amount he spent in all. The equation 4 4 45 represents the total amount of money Patrick spent. Try It Georgia had a collection of 503 stickers. She bought 50 new stickers, and then her sister gave her 35 more stickers. Write a number sentence that can be used to represent s, the total number of stickers Georgia now has in her collection. Georgia had a total of stickers in her collection. She bought new stickers. Her sister gave her more stickers. The number sentence s can be used to represent the total number of stickers Georgia now has. Georgia had a total of 503 stickers. She bought 50 new stickers. Her sister gave her 35 more stickers. The number sentence s 503 50 35 can be used to represent the total number of stickers Georgia now has. Now practice what you ve learned. 48

Objective 2 Question 19 The table below shows the ages of Jeannette and her brother Mark at different times. Age in Years Jeannette Mark 4 15 8 19 12 23 Which statement is true about the relationship between Jeannette s and Mark s ages? A B C D Mark is 4 years older than Jeannette. Jeannette is 11 years younger than Mark. When Jeannette is 16 years old, Mark will be 24 years old. When Mark is 16 years old, Jeannette will be 6 years old. Answer Key: page 141 Question 20 Look at the pattern of numbers below. 1, 7, 8, 15, 23, 38, 61, 99 Which statement best describes the rule for determining the last 6 numbers shown in this pattern? A B C D Each number is 6 more than the number before it. Each number is the sum of the 2 numbers before it. Each number is 7 more than the number before it. Each number is the product of the 2 numbers before it. Answer Key: page 142 49

Objective 2 Question 21 The table shows the cost of different numbers of zoo tickets. Which equation can be used to find t, the cost of 8 tickets? A t 2 8 B t 4 8 C t 1 8 D t 2 8 Number of Tickets Zoo Tickets Cost of Tickets 2 $4 4 $8 6 $12 10 $20 Question 22 Some factor pairs for the composite number 24 are shown below. 1 24 2 12 3 8 What factor pair is missing from this list? A 4 5 B 4 6 C 4 7 D 4 9 Question 23 Which of these is a prime number? Answer Key: page 142 A 27 B 39 C 33 D 29 Answer Key: page 142 Answer Key: page 142 50

Objective 2 Question 24 Leah used tiles to show all the arrays for the numbers 3, 4, 5, and 6. 3 4 1 3 1 4 3 1 4 1 2 2 5 6 1 5 1 6 2 3 5 1 3 2 6 1 Which of these numbers are prime? A 3 and 5 B 3 and 6 C 4 and 6 D 4 and 5 Answer Key: page 142 51

Objective 2 Question 25 The table below shows the weight, in ounces, of different numbers of peanuts. Peanuts Weight (ounces) Total Number of Peanuts 3 5 10 20 120 200 400 800 What is the relationship between the total number of peanuts and the weight of the peanuts in ounces? A The number of ounces is 117 less than the total number of peanuts. B The total number of peanuts is 400 times the number of ounces. C The total number of peanuts is 40 times the number of ounces. D The number of ounces is 780 less than the total number of peanuts. Answer Key: page 142 Question 26 Last night Gianna did her homework. She spent 20 minutes on math, 15 minutes on reading, and 30 minutes on science. Which equation can be used to find m, the total number of minutes Gianna spent on her homework last night? A B C D 20 15 30 m 20 15 30 m 20 15 30 m 20 15 30 m Answer Key: page 142 52

Objective 3 The student will demonstrate an understanding of geometry and spatial reasoning. For this objective you should be able to identify essential attributes of two-dimensional and three-dimensional geometric figures; describe the results of transformations; and locate and name points on a coordinate grid. What Are Parallel Lines? Parallel lines are lines that are the same distance apart at all points. Parallel lines never intersect. Several examples of parallel lines are shown below. R T S V A C B D J K L M Line AB can also be named line BA. You can also look at a figure and identify line segments that are parallel. Line segments in figures are parallel if they are the same distance apart at all points. Which two line segments in the figure below appear to be parallel? W X Z Y Line segment WX and line segment ZY appear to be the same distance apart at all points. Line segments WX and ZY appear to be parallel. 53

Objective 3 What Are Perpendicular Lines? Perpendicular lines are lines that intersect to form right angles. Angle GXF is a right angle. Line EF is perpendicular to line GH. G A small is placed at the vertex of an angle to show that it is a right angle. E X F H X T A X B S C You can also look at a figure and identify line segments that are perpendicular. Which line segments in the rectangle below are perpendicular? A B D C The rectangle shows a right angle at each of the vertices: A, B, C, and D. The two line segments that meet at each vertex form a right angle and are perpendicular to each other. Line segments AB and BC are perpendicular. Line segments BC and CD are perpendicular. Line segments CD and AD are perpendicular. Line segments AD and AB are perpendicular. 54

Objective 3 Try It Look at square KLMN below. K L N M Which line segments are parallel? Opposite sides of a square are parallel. Parallel lines never or. Line segment and line segment are parallel. Line segment and line segment are parallel. Which line segments are perpendicular? There is a angle at each vertex of a square. Perpendicular lines intersect at right angles. Line segment and line segment are perpendicular. Line segment and line segment are perpendicular. Line segment and line segment are perpendicular. Line segment and line segment are perpendicular. Parallel lines never intersect or cross. Line segment KL and line segment NM are parallel. Line segment KN and line segment LM are parallel. There is a right angle at each vertex of a square. Line segment KL and line segment KN are perpendicular. Line segment KL and line segment LM are perpendicular. Line segment NM and line segment LM are perpendicular. Line segment NM and line segment KN are perpendicular. 55

Objective 3 Do you see that... How Can You Describe a Two-Dimensional Figure? You can describe a two-dimensional figure (or plane figure) by counting the number of sides, vertices, and angles the figure has. Polygons are closed two-dimensional figures with straight sides. A circle is a closed two-dimensional figure, but it is not a polygon because it has no sides. You should be able to recognize and describe the following polygons. Figure Polygons Congruent Description Sides Noncongruent Sides Triangle 3 sides 3 vertices 3 angles All sides equal in length A rectangle is a quadrilateral with opposite sides congruent and 4 right angles. A square is a rectangle with all sides congruent. Quadrilateral 4 sides 4 vertices 4 angles All sides equal in length Pentagon 5 sides 5 vertices 5 angles All sides equal in length Hexagon 6 sides 6 vertices 6 angles All sides equal in length Octagon 8 sides 8 vertices 8 angles All sides equal in length 56

Objective 3 Try It Look at the figures below. Count the number of sides, vertices, and angles. Then name the figure. A B C D Figure A has sides, vertices, and angles. Opposite sides appear to be. Figure A is a. Figure B has sides, vertices, and angles. Figure B is a. Figure C has sides, vertices, and angles. All sides appear to be. Figure C is a. Figure D has sides, vertices, and angles. Figure D is a. Figure A has 4 sides, 4 vertices, and 4 angles. Opposite sides appear to be congruent. Figure A is a rectangle. Figure B has 5 sides, 5 vertices, and 5 angles. Figure B is a pentagon. Figure C has 4 sides, 4 vertices, and 4 angles. All sides appear to be congruent. Figure C is a square. Figure D has 6 sides, 6 vertices, and 6 angles. Figure D is a hexagon. 57

Objective 3 A three-dimensional figure has length, width, and height. How Can You Describe a Three-Dimensional Figure? You can describe a three-dimensional figure (or solid figure) by counting the number of vertices, edges, and faces the figure has. Edge Vertex Face A face is a flat surface in the shape of a two-dimensional figure. An edge is a line segment where two faces meet. A vertex is a point where three or more edges meet. The plural of vertex is vertices. Here are some three-dimensional figures you should be able to recognize and describe. Three-Dimensional Figures Figure Example Description Triangular prism 2 triangular faces 3 rectangular faces 9 edges 6 vertices Rectangular prism 6 rectangular faces 12 edges 8 vertices Cube 6 square faces 12 edges 8 vertices Square pyramid 1 square face 4 triangular faces 8 edges 5 vertices Triangular pyramid 4 triangular faces 6 edges 4 vertices 58

Objective 3 Try It Ernest gave his father a candy bar in a box shaped like the one below. chocolate candy bar The box has edges. The box has vertices. The box has rectangular faces. The box has triangular faces. How many more edges than vertices does this box have? edges vertices This box has more edges than vertices. How many faces does this box have? rectangular faces triangular faces This box has faces. The box has 9 edges. The box has 6 vertices. The box has 3 rectangular faces. The box has 2 triangular faces. 9 edges 6 vertices 3. This box has 3 more edges than vertices. 3 rectangular faces 2 triangular faces 5. This box has 5 faces. 59

Objective 3 Some three-dimensional figures have a curved surface. Some of these figures also have one or two flat circular surfaces called bases. Curved surface Bases Here are some three-dimensional figures with curved surfaces that you should be able to recognize and describe. Three-Dimensional Figures Figure Example Description Cylinder 1 curved surface 2 circular bases Cone 1 curved surface 1 circular base Sphere 1 curved surface 60

Objective 3 What Are Transformations? Transformations are ways of moving a figure in a plane. Three kinds of transformations are translations, reflections, and rotations. The result of a translation, reflection, or rotation is a figure that is congruent to the original figure. A translation is a sliding movement. A figure can be translated up, down, left, right, or diagonally by sliding it. Do you see that... A reflection is a mirror image across a line. A rotation is a turning movement around a point. In a rotation, the figure moves in a circular path. Rotates around this point 61

Objective 3 What transformation is represented in the diagram? The figure has been moved down. This is a translation. Try It Name the transformation in each diagram. A B C The figures in diagram A show a mirror image. This transformation is a. The figures in diagram B show a sliding movement diagonally. This transformation is a. The figures in diagram C show a turning movement around a point. This transformation is a. The figures in diagram A show a mirror image. This transformation is a reflection. The figures in diagram B show a sliding movement diagonally. This transformation is a translation. The figures in diagram C show a turning movement around a point. This transformation is a rotation. 62

Objective 3 What Is a Coordinate Plane? A coordinate plane is a grid used to locate points. A point is located by using an ordered pair of numbers. The two numbers that form the ordered pair are called the point s coordinates. Every coordinate plane has a special point called the origin. The coordinates of the origin are (0, 0). The horizontal line is called the x-axis. The first number of an ordered pair tells how many units the point is to the right of the origin. The first number goes right. The vertical line is called the y-axis. The second number of an ordered pair tells how many units the point is above the origin. The second number goes up. Look at the coordinate grid below. The coordinates of point P are (2, 5). y 6 5 P (2, 5) Origin (0, 0) 4 3 2 1 0 1 2 3 4 5 6 x The first number of point P is 2. It is 2 units to the right of the origin. The second number of point P is 5. It is 5 units above the origin. 63

Objective 3 Look at the coordinate grid. y 6 5 4 C B 3 2 A 1 0 1 2 3 4 5 6 x What are the coordinates of points A, B, and C? Starting from the origin, point A is located 4 units to the right and 2 units up. The ordered pair (4, 2) shows the location of point A. Starting from the origin, point B is located 2 units to the right and 4 units up. The ordered pair (2, 4) shows the location of point B. Starting from the origin, point C is located 1 unit to the right and 4 units up. The ordered pair (1, 4) shows the location of point C. 64

Objective 3 Try It Look at the coordinate grid. 6 y 5 4 R 3 2 T 1 0 S 1 2 3 4 5 6 x What are the coordinates of points R, S, and T? Point R is units to the right and units up. The What happens when a point is on the x-axis or y-axis? ordered pair (, ) shows the location of point R. Point S is units to the right and units up. The ordered pair (, ) shows the location of point S. Point T is units to the right and units up. The ordered pair (, ) shows the location of point T. Point R is 6 units to the right and 4 units up. The ordered pair (6, 4) shows the location of point R. Point S is 3 units to the right and 0 units up. The ordered pair (3, 0) shows the location of point S. Point T is 0 units to the right and 2 units up. The ordered pair (0, 2) shows the location of point T. 65

Objective 3 Try It The grid below shows a map of a neighborhood. 9 8 y 7 6 Library 5 4 3 2 1 0 BOOKSTORE Store School 1 2 3 4 5 6 7 8 9 x The school is units to the right of the origin and unit above the origin. The school is located at (, ). The store is units to the right of the origin and units above the origin. The store is located at (, ). The library is units to the right of the origin and units above the origin. The library is located at (, ). The school is 4 units to the right of the origin and 1 unit above the origin. The school is located at (4, 1). The store is 6 units to the right of the origin and 3 units above the origin. The store is located at (6, 3). The library is 5 units to the right of the origin and 8 units above the origin. The library is located at (5, 8). Now practice what you ve learned. 66

Objective 3 Question 27 Which figure appears to have exactly one pair of parallel sides? Question 29 Which of the following figures has exactly two bases? A B A B C D Cylinder Cone Sphere Not here Answer Key: page 143 C Question 30 Dennis has a bank shaped like the figure below. D Answer Key: page 142 Question 28 The figure below is a rectangle. W X Which statement is true about this figure? A This figure has 8 more edges than vertices. B This figure has 3 more vertices than faces. C This figure has 5 more vertices than faces. D This figure has 3 more edges than vertices. Z Y Which statement about the figure is true? A B C D Line segment WX is parallel to line segment XY. Line segment YZ is parallel to line segment XY. Line segment WX is perpendicular to line segment XY. Line segment WX is perpendicular to line segment YZ. Answer Key: page 142 Answer Key: page 143 67

Objective 3 Question 31 Which of these shows only a rotation? A C B D Answer Key: page 143 Question 32 The pictures below show 3 pairs of figures that each represent a single transformation. 1 2 3 Which of the following correctly lists the transformations in order from 1 to 3? A Reflection, rotation, reflection B Translation, reflection, rotation C Rotation, reflection, translation D Translation, rotation, reflection Answer Key: page 143 68

Objective 3 Question 33 Look at the coordinate grid below. y 6 5 4 3 2 1 n 0 1 2 3 4 5 6 x Which ordered pair appears to be on line n? A (3, 3) B (3, 4) C (4, 3) D (4, 4) Answer Key: page 143 Question 34 The grid below represents a map of a playground. y 7 6 5 4 Sandbox Swing 3 2 1 Tree Slide 0 1 2 3 4 5 6 7 x Which ordered pair best represents the location of the sandbox? A (4, 3) B (1, 2) C (2, 4) D (5, 6) Answer Key: page 143 69

Objective 4 The student will demonstrate an understanding of the concepts and uses of measurement. For this objective you should be able to apply measurement concepts involving conversions, length, perimeter, area, volume, time, and temperature to solve problems. In the United States most people use the customary system of measurement. Scientists and people in most other countries use the metric system. When Do You Use Measurement? You can use measurement for many things. You can measure your leg to find out how long it is. You can place a bag of peanuts on a scale to find out how much it weighs. If you look at a clock to tell the time or at a thermometer to read the temperature, you are using measurement. How Do You Convert Units of Measurement? In some problems you will need to convert one unit of measurement to another, such as pounds to ounces or centimeters to meters. To convert a smaller unit to a larger unit, divide by the number of smaller units there are in one of the larger units. To convert a larger unit to a smaller unit, multiply by the number of smaller units there are in one of the larger units. Large units Small units Use the information from the Mathematics Chart to help convert units of measure. Texas Assessment of Knowledge and Skills Grade 5 Mathematics Chart LENGTH Metric Customary 1 kilometer = 1000 meters 1 mile = 1760 yards 1 meter = 100 centimeters 1 mile = 5280 feet 1 centimeter = 10 millimeters 1 yard = 3 feet 1 foot = 12 inches CAPACITY AND VOLUME Metric Customary 1 liter = 1000 milliliters 1 gallon = 4 quarts 1 gallon = 128 fluid ounces 1 quart = 2 pints 1 pint = 2 cups 1 cup = 8 fluid ounces MASS AND WEIGHT Metric Customary 1 kilogram = 1000 grams 1 ton = 2000 pounds 1 gram = 1000 milligrams 1 pound = 16 ounces 70

Objective 4 On a baseball field the distance from home plate to first base is 90 feet. How many yards is it from home plate to first base? Use the Mathematics Chart to compare feet and yards. 1 yard 3 feet? yards 90 feet A foot is a smaller unit than a yard. To convert a smaller unit to a larger unit, divide. To convert 90 feet to yards, divide 90 feet by the number of feet in one yard. 90 feet 3 feet per yard 30 yards The distance from home plate to first base is 30 yards. Carlo has 6 gallons of water in a large container. He is using a one-quart bucket to empty the large container. How many times will he have to fill the bucket to empty the large container? Use the Mathematics Chart to compare gallons and quarts. 1 gallon 4 quarts 6 gallons? quarts A gallon is a larger unit than a quart. To convert a larger unit to a smaller unit, multiply. To convert 6 gallons to quarts, multiply 6 gallons by the number of quarts in one gallon. 6 gallons 4 quarts per gallon 24 quarts Carlo will have to fill the one-quart bucket 24 times to empty the large container. 71

Objective 4 Pasta Mario has 2 kilograms of pasta in a total of 4 boxes. Each box is the same size. How many grams of pasta are in each box? Use the Mathematics Chart to compare kilograms and grams. 1 kilogram 1,000 grams 2 kilograms? grams A kilogram is a larger unit than a gram. To convert a larger unit to a smaller unit, multiply. To convert 2 kilograms to grams, multiply 2 kilograms by the number of grams in one kilogram. 2 kilograms 1,000 grams per kilogram 2,000 grams Mario has 2,000 grams of pasta in a total of 4 boxes. Now find how many grams of pasta are in each box. Divide 2,000 grams by 4 boxes to find the number of grams in each box. 2,000 grams 4 boxes 500 grams per box Each box of pasta contains 500 grams. Try It A cooler contains 20 liters of water. How many milliliters of water does the cooler contain? 1 liter milliliters A is a larger unit than a. To convert liters to milliliters, by the number of milliliters in one liter. liters 1,000 milliliters per liter milliliters The cooler contains milliliters of water. 1 liter 1,000 milliliters A liter is a larger unit than a milliliter. To convert liters to milliliters, multiply by the number of milliliters in one liter. 20 liters 1,000 milliliters per liter 20,000 milliliters The cooler contains 20,000 milliliters of water. 72

Objective 4 Try It Paul is making signs for the school dance. He has a roll of paper 5 feet 3 inches long. If Paul divides the paper into 3 equal pieces to make the signs, how long will each sign be in feet and inches? First convert the measurement 5 feet 3 inches to inches. 1 foot inches 5 feet 12 inches per foot inches inches inches inches To find the length of each sign, divide by 3. 3 equal pieces Each sign will be inches long. Since inches is greater than 1 foot, convert this measurement to feet and inches. 12 1 R Each sign will be 1 9 long. 1 foot 12 inches. 5 feet 12 inches per foot 60 inches. 60 inches 3 inches 63 inches. 63 3 equal pieces 21. Each sign will be 21 inches long. Since 21 inches is greater than 1 foot, convert this measurement to feet and inches. 21 12 1 R9. Each sign will be 1 foot 9 inches long. 73

Objective 4 How Do You Measure Length? To find the length of an object, use a ruler. There are two rulers on the Mathematics Chart in the back of this book. Use the ruler below to measure this screw to the nearest 1 2 inch. 0 1 2 3 4 Inches The 0-inch mark on the ruler is placed at the left end of the screw. The right end of the screw aligns with the mark on the ruler that is about halfway between 3 inches and 4 inches. 1 The length of the screw is about 3 inches. 2 Use the ruler on the Mathematics Chart to measure the line segment next to the paper clip to the nearest millimeter. Place the 0-centimeter mark on the ruler at one end of the paper clip. Read the mark on the ruler that aligns with the other end of the paper clip. The paper clip is between 4 and 5 centimeters long. This is the same as between 40 and 50 millimeters long. The end of the paper clip aligns with the mark on the ruler that is 9 marks past the 4. The paper clip is about 49 millimeters long. 74

Objective 4 Try It Mai rode a bus to school. After school she rode a bus to the library, and then she rode another bus home. Use the inch ruler on the Mathematics Chart and the map below to find the total distance Mai rode. Mai s house Library Scale 1 inch 2 miles School What is the total distance Mai rode in miles? The distance from Mai s house to the school on the map is inches. The distance from the school to the library on the map is inch. The distance from the library to Mai s house on the map is inches. The map s scale shows that inch represents miles. The total distance equals inches miles per inch. Mai rode a total of miles. The distance from Mai s house to the school on the map is 2 inches. The distance from the school to the library on the map is 1 inch. The distance from the library to Mai s house on the map is 2 inches. The total distance is 2 1 2 5 inches. The map s scale shows that 1 inch represents 2 miles. The total distance equals 5 inches 2 miles per inch. Mai rode a total of 10 miles. 75

Objective 4 How Do You Find the Perimeter of a Figure Using a Formula? Perimeter is the distance around a figure. To find the perimeter of a figure, add the lengths of all its sides. Some figures have certain attributes that let you find the perimeter in a different way. Look at the square below. The side length is shown in centimeters. What is the perimeter of the square in centimeters? 8 cm Remember what you know about a square. All sides of a square are congruent. So each side of this square is 8 cm long. Find the perimeter of the square. You could add the lengths of all the sides: 8 8 8 8 32 Since all four sides are the same length, you could multiply the length of one side by the number of sides: 4 sides 8 cm 32 cm Both strategies show that the perimeter of the square is 32 cm. 76

Objective 4 A rectangular garden has a length of 12 yards and a width of 8 yards. What is the perimeter of the garden? 8 yd 12 yd Remember what you know about a rectangle and use it to find the perimeter of the rectangular garden. Opposite sides of a rectangle are congruent. The length of the rectangular garden is 12 yards. There are 2 sides that are 12 yards long. 2 12 yards 24 yards. The width of the rectangular garden is 8 yards. There are 2 sides that are 8 yards wide. 2 8 yards 16 yards. Add 24 and 16 to find the perimeter: 24 yards 16 yards 40 yards The perimeter of the rectangular garden is 40 yards. 77

Objective 4 For squares and rectangles, you can use a formula to find the perimeter. A formula is an equation that uses numbers, mathematical symbols, and letters to help solve a problem. The formulas for finding the perimeter of a square and a rectangle can be found on the back of the Mathematics Chart. Grade 5 Mathematics Chart Perimeter square P = 4 s rectangle P = (2 l) + (2 w) Area square A = s s rectangle A = l w Look at the rectangle below. 5 feet 2 feet The formula for the perimeter of a rectangle is P (2 l) (2 w). Read this as Perimeter equals two times the length plus two times the width. Find the perimeter using a formula. The formula for the perimeter of a rectangle is P (2 l) (2 w). The letter P stands for the perimeter, the letter l stands for the length, and the letter w stands for the width. Substitute the values for the length and width of the rectangle into the formula. The length (l) is 5 ft, and the width (w) is 2 ft. Then solve for the perimeter. P (2 l) (2 w) P (2 5) (2 2) Multiply 2 5 and 2 2 P 10 4 Add 10 and 4 P 14 The perimeter of the rectangle is 14 feet. 78

Objective 4 A square mirror has a side length of 16 inches. Find the perimeter of the square mirror using a formula. The formula for the perimeter of a square is P 4 s. The letter P stands for the perimeter, and the letter s stands for the side length. Substitute the value for the side length of the square into the formula. Each side (s) is 16 inches. Then solve for the perimeter. P 4 s P 4 16 P 64 The perimeter of the square mirror is 64 inches. The formula for the perimeter of a square is P 4 s. Read this as Perimeter equals four times the length of the side. 79

Objective 4 Try It The picture frame below is 24 centimeters long and 16 centimeters wide. What is the perimeter of the picture frame? 24 cm 16 cm Use the formula to find the perimeter of the rectangle. P ( ) ( ) P (2 ) (2 ) P P The perimeter of the picture frame is centimeters. P (2 l) (2 w) P (2 24) (2 16) P 48 32 P 80 The perimeter of the picture frame is 80 centimeters. 80

Objective 4 How Do You Find the Area of a Figure Using a Formula? Area is the measure of how many square units a figure covers. To find the area of a figure, you could count the number of square units the figure covers. Some figures have certain attributes that let you find the area in other ways. In the rectangle below, there are 5 columns and 3 rows. Area is measured in square units, such as square feet or square meters. The number of columns and the number of rows represent the dimensions of the rectangle. You could multiply the number of columns by the number of rows to find the total number of square units. This is the same as multiplying the length by the width. In this rectangle, the length is 5 units and the width is 3 units. If you multiply the length by the width, the area is 5 3 15 square units. For squares and rectangles, you can use a formula to find the area. A formula is an equation that uses numbers, mathematical symbols, and letters to help solve a problem. The formulas for finding the area of a square and a rectangle can be found on the back of the Mathematics Chart. rectangle P = (2 l) + (2 w) Area square A = s s rectangle A = l w Volume cube V = s s s rectangular prism V = l w h 81

Objective 4 Bonita drew a rectangle 7 centimeters long and 3 centimeters wide. 3 cm 7 cm The formula for the area of a rectangle is A l w. Read this as Area equals length times width. Use a formula to find the area of the rectangle. The formula for the area of a rectangle is A l w. The letter A stands for the area, the letter l stands for the length, and the letter w stands for the width. Substitute the values for the length and width of the rectangle into the formula. The length (l) is 7 cm, and the width (w) is 3 cm. Then solve for the area. A l w A 7 3 A 21 The area of Bonita s rectangle is 21 square centimeters. 82

Objective 4 A square is a special kind of rectangle. All the sides of a square are the same length. Each side can be represented by the letter s. Find the area of a square with a side that is 6 inches long. 6 in. Use the formula for the area of a square. The formula for the area of a square is A s s. The letter A stands for the area, and the letter s stands for the side length. Substitute the value for the side length into the formula. Each side (s) is 6 inches. Then solve for the area. A s s A 6 6 A 36 The area of the square is 36 square inches. The formula for the area of a square is A s s. Read this as Area equals side times side. 83

Objective 4 Try It Carolyn measured the front cover of her photo album and found that it was 9 inches wide and 12 inches long. What is the area of the front cover? 9 inches 12 inches The length of the front cover is inches. The width of the front cover is inches. Use the formula A to find the area of a rectangle. A A The area of the front cover is square inches. The length of the front cover is 12 inches. The width of the front cover is 9 inches. A l w A 12 9 A 108 The area of the front cover is 108 square inches. 84

Objective 4 Try It Ty is using square tiles for an art project. The side of each tile measures 5 centimeters. What is the area of each tile? 5 cm Use the formula A to find the area of a square. A A The area of each tile is square centimeters. A s s A 5 5 A 25 The area of each tile is 25 square centimeters. How Do You Find the Volume of a Figure Using a Formula? The volume of an object is the number of cubic units it takes to fill the object. One way to find the volume of an object is to count the number of cubes it takes to fill it. The cube below has edges that each measure 1 unit long. It has a volume of 1 cubic unit. 1 cubic unit Volume is measured in cubic units, such as cubic inches or cubic centimeters. 1 unit 1 unit 1 unit 85

Objective 4 The rectangular prism shown below is made of 1-unit cubes. What is its volume? The prism has two layers. Find the number of cubic units in each layer. Top layer Bottom layer The top layer has a width of 3 units and a length of 4 units. The top layer is made up of 3 4 12 cubic units. The bottom layer also has 12 cubic units, even though you cannot see them all in the original figure. Since there are 2 layers, multiply the number of cubic units in one layer by 2. 12 2 24 The volume of the rectangular prism is 24 cubic units. A rectangular prism has three dimensions: length, width, and height. In a model of a rectangular prism, the number of columns, the number of rows, and the number of layers represent the dimensions of the prism. Multiplying the number of columns by the number of rows and then multiplying by the number of layers is the same as multiplying the length by the width by the height. For cubes and rectangular prisms, you can use a formula to find the volume. A formula is an equation that uses numbers, mathematical symbols, and letters to help solve a problem. The formulas for finding the volume of a cube and a rectangular prism can be found on the back of the Mathematics Chart. rectangle A = l w Volume cube V = s s s rectangular prism V = l w h 86

Objective 4 Look again at this rectangular prism. 2 3 4 Use the formula to find the volume of the rectangular prism. The formula for the volume of a rectangular prism is V l w h. The letter V stands for the volume, the letter l stands for the length, the letter w stands for the width, and the letter h stands for the height. Substitute the values for the length, width, and height of the rectangular prism into the formula. The length (l) is 4 units, the width (w) is 3 units, and the height (h) is 2 units. Then solve for volume. V l w h V 4 3 2 Multiply 4 3 V 12 2 Multiply 12 2 V 24 The volume of the rectangular prism is 24 cubic units. The formula for the volume of a rectangular prism is V l w h. Read this as Volume equals length times width times height. A cube is a special kind of prism. All the dimensions of a cube are the same. Each side length can be represented by the letter s. Lance uses a box shaped like a cube for storing his toys. Each side of the box is 20 inches. What is the volume of Lance s toy box? Use a formula to find the volume of the cube. The formula for the volume of a cube is V s s s. The letter V stands for the volume, and the letter s stands for the side length. Substitute the value for the side length into the formula. Each side (s) is 20 inches. Then solve for the volume. V s s s V 20 20 20 Multiply 20 20 V 400 20 Multiply 400 20 V 8,000 The volume of Lance s toy box is 8,000 cubic inches. The formula for the volume of a cube is V s s s. Read this as Volume equals side times side times side. 87

Objective 4 Try It The rectangular prism below is made of 1-centimeter cubes. 3 cm 4 cm 5 cm What is the volume of this prism? The length is cm. The width is cm. The height is cm. Use the formula V to find the volume of this rectangular prism. V V V The prism has a volume of cubic centimeters. The length is 5 cm. The width is 4 cm. The height is 3 cm. V l w h V 5 4 3 V 20 3 V 60 The prism has a volume of 60 cubic centimeters. 88

Objective 4 How Do You Solve Problems Involving Elapsed Time? Time can be measured in several ways. Daily activities are measured in seconds, minutes, and hours. Longer periods of time are measured in days, weeks, months, and years. Some problems require you to calculate the amount of time that has passed between one event and another. Gene got a new watch for his birthday. He used the watch to find how many minutes he walked his dog. Gene and his dog left the house at 10:30 A.M. They returned at 11:13 that morning. For how many minutes did Gene walk his dog? From 10:30 to 11:00 is 30 minutes. 7 6 11 12 1 10 98 2 3 5 4 From 11:00 to 11:13 is 13 minutes. 7 6 11 12 1 10 98 2 3 5 4 30 minutes 13 minutes 43 minutes Gene walked his dog for 43 minutes. Emilio and his friends went to a movie. The movie started at 4:30 and lasted 2 hours 20 minutes. What time was the movie over? Start at 4:30 and add 2 hours: 4:30 2 hours is 6:30. Add 20 minutes to 6:30: 6:30 20 minutes is 6:50. The movie was over at 6:50. 89

Objective 4 Jamie was at the dentist s office from 9:25 until 10:40. How many hours and minutes was Jamie at the dentist? From 9:25 to 10:25 is 1 hour. Count by fives from 10:25 to 10:40. 5 5 5 15 10:25 10:30 10:35 10:40 From 10:25 to 10:40 is 15 minutes. Jamie was at the dentist s office for 1 hour 15 minutes. Try It Sam and his family went to visit his grandmother. They left their house at 8:15 A.M. They arrived at his grandmother s house at 11:05 A.M. How long did it take Sam and his family to get to his grandmother s house? The family started out at 8:15 A.M. 8:15 to 10:15 is hours. 10:15 to 11:05 is minutes. It took hours minutes for Sam and his family to get to his grandmother s house. 8:15 to 10:15 is 2 hours. 10:15 to 11:05 is 50 minutes. It took 2 hours 50 minutes for Sam and his family to get to his grandmother s house. 90

Objective 4 How Do You Solve Problems Involving Changes in Temperature? Temperature is a measure of how hot or cold something is. Temperature is measured with a thermometer. Temperature is measured in degrees Fahrenheit ( F) in the customary system and in degrees Celsius ( C) in the metric system. The thermometers below show some common temperatures measured in the customary system and in the metric system. 220 200 212 F Boiling point of water 100 C 100 180 80 160 140 120 100 80 60 95 F Temperature on a summer day 35 C 60 40 20 What do the marks on the thermometer represent? 40 32 F Freezing point of water 0 C 0 F C The temperature at 8:00 A.M. was 50 F. By 10:00 A.M. the temperature had increased 7 F. From 10:00 A.M. to 2:00 P.M., the temperature increased 3 F per hour. What was the temperature at 2:00 P.M.? First find the temperature at 10:00 A.M. At 8:00 A.M. the temperature was 50 F. By 10:00 A.M. the temperature was 7 F higher than it was at 8:00 A.M. 50 7 57 At 10:00 A.M. the temperature was 57 F. Then find the number of degrees the temperature increased from 10:00 A.M. to 2:00 P.M. The temperature increased 3 F per hour from 10:00 A.M. to 2:00 P.M. 91

Objective 4 From 10:00 A.M. to 2:00 P.M. is 4 hours. Use multiplication to find the number of degrees the temperature increased. 4 hours 3 F per hour 12 F Next find the temperature at 2:00 P.M. At 10:00 A.M. the temperature was 57 F. At 2:00 P.M. the temperature was 12 F higher than it was at 10:00 A.M. 57 12 69 At 2:00 P.M. the temperature was 69 F. Now practice what you ve learned. 92

Objective 4 Question 35 A rectangular prism is shown below. Question 37 A rectangular playground has a length of 110 yards and a width of 70 yards. 3 cm 5 cm What is the volume of the prism? 2 cm 70 yd A 10 cubic centimeters B 25 cubic centimeters C 21 cubic centimeters D 30 cubic centimeters Question 36 Answer Key: page 143 At 8:00 A.M. the temperature is 45 F. If the temperature increases 5 F each hour, what will the temperature be at 2:00 P.M.? What is the perimeter of the playground in yards? Record your answer and fill in the bubbles. Be sure to use the correct place value. 0 1 2 3 0 1 2 3 0 1 2 3 110 yd A 51 F B 50 F 4 5 6 4 5 6 4 5 6 C 75 F D 70 F 7 8 9 7 8 9 7 8 9 Answer Key: page 143 Answer Key: page 143 93

7 4 8 5 9 6 3 Objective 4 Question 38 Harry takes a bus to visit his cousin. The bus leaves at 10:40 A.M. The bus trip takes 3 hours 45 minutes. What time is it when the bus arrives at Harry s destination? A B C D 2:25 P.M. 1:45 P.M. 3:45 P.M. 1:25 P.M. Answer Key: page 144 Question 40 Band practice started at 1:30 P.M. and was over at 3:25 P.M. How long was band practice? A B C D 2 hours 5 minutes 2 hours 55 minutes 1 hour 55 minutes 1 hour 5 minutes Answer Key: page 144 Question 39 A cell phone has the dimensions shown below. 3 cm PQRS GHI * 0 TUV OPER # 1 2 JKL ABC WXYZ TALK BACK END MNO DEF 8 cm 1 cm Question 41 How many meters are equal to 50 kilometers? A 5,000 m B 500 m C 5 m D 50,000 m Which of the following equations can be used to find V, the volume of the cell phone in cubic centimeters? A V 8 3 1 B V 8 3 1 C V 8 3 1 D V 8 3 1 Answer Key: page 144 Answer Key: page 144 94

Objective 4 Question 42 The drawing below represents a rectangular patio floor. Use the ruler on the Mathematics Chart to measure the dimensions of the drawing to the nearest centimeter. If each centimeter on the drawing represents 1 meter, which of the following is closest to the perimeter of the actual patio floor? A 16 meters B 18 meters C 20 meters D 25 meters Answer Key: page 144 Question 43 The area of a rectangular sidewalk is 856 square feet. The width of the sidewalk is 8 feet. What is the length of the sidewalk? A 420 feet B 6,848 feet C 107 feet D 17 feet Question 44 Jesse s father is 6 feet 2 inches tall. How many inches tall is Jesse s father? A 74 inches B 62 inches C 72 inches D 68 inches Answer Key: page 144 Answer Key: page 144 95

Objective 5 The student will demonstrate an understanding of probability and statistics. For this objective you should be able to describe and predict the results of a probability experiment; and solve problems by collecting, organizing, displaying, and interpreting sets of data. What Is Probability? Probability is a way of describing how likely it is that a particular outcome, or result, will occur. A fraction can be used to describe the results of a probability experiment. The numerator of the fraction is the number of favorable outcomes for the experiment. The denominator of the fraction is the number of all possible outcomes for the experiment. number of favorable outcomes number of possible outcomes The spinner below is divided into 4 equal sections. Each section is a different color. If the spinner is spun one time, what is the probability of the spinner landing on the color yellow? Red Blue Yellow Green You can use a fraction to describe the results of this probability experiment. The favorable outcome for this experiment is landing on the color yellow. There is only one section of the spinner labeled yellow. There is only one favorable outcome. The numerator of the fraction is 1. There are 4 sections on the spinner: red, blue, yellow, and green. The number of possible outcomes is 4. The denominator of the fraction is 4. number of favorable outcomes number of possible outcomes 1 4 Do you see that... The probability of the spinner landing on the color yellow is 1 1 out of 4, or. 4 96

Objective 5 Mike has several coins in his pocket: 3 dimes, 2 nickels, 1 quarter, and 2 pennies. Mike reaches into his pocket and takes out one coin at random. What is the probability that Mike will pick a dime? A favorable outcome for this experiment is to pick a dime. There are 3 dimes in his pocket. There are 3 favorable outcomes. There are a total of 3 2 1 2 coins in his pocket. There are 8 coins in all in his pocket. There are 8 possible outcomes. The probability that Mike will pick a dime is the fraction: number of favorable outcomes number of possible outcomes 3 8 A random event occurs when a selection is made without looking. The probability that Mike will pick a dime is 3 out of 8, or 3 8. 97

Objective 5 Try It Joel has a bag with colored cubes in it. There are 5 red cubes, 2 white cubes, and 4 blue cubes. Joel will take a cube out of the bag at random. What is the probability that he will take out a red cube? A favorable outcome for this experiment is to take a cube out of the bag. There are red cubes in the bag. There are favorable outcomes. The expression represents the total number of cubes in the bag. There are cubes in the bag. There are possible outcomes. The probability that Joel will pick a red cube is. A favorable outcome for this experiment is to take a red cube out of the bag. There are 5 red cubes in the bag. There are 5 favorable outcomes. There are a total of 5 2 4 cubes in the bag. There are 11 cubes in the bag. There are 11 possible outcomes. The probability that Joel will pick a red cube is 1 51. 98

Objective 5 How Can You Use Experimental Results to Make Predictions? Sometimes you need to predict the probability of an event based on the results of an experiment. If you know the probability of an event, you can use it to predict the probability of that event occurring in the future. John plays baseball. This season he has been at bat 36 times. He got a hit 6 of those times. Based on this information, how many times can John expect to get a hit in his next 12 times at bat? Find the fraction that describes the number of hits John got out of 36 times at bat. number of favorable outcomes (getting a hit) number of possible outcomes (number of times at bat) 6 3 6 6 The probability that John got a hit is. 36 Use what you know about equivalent fractions to find the number of hits John can expect to get in his next 12 times at bat. 6 36 John can expect to get 2 hits in his next 12 times at bat. 6 3 36 3 12 2 12 Photodisc 99