Common Core Coach. Mathematics. Copyright Protected DO NOT COPY. Dr. Jerry Kaplan Senior Mathematics Consultant. First Edition

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1 First Edition Common Core Coach Mathematics 6 Dr. Jerry Kaplan Senior Mathematics Consultant Common Core Coach, Mathematics, First Edition, Grade 6 T7NA ISBN-: Contributing Writer: Ann Petroni McMullen Cover Design: QA /Bill Smith Cover Illustration: Stephanie Dalton Triumph Learning 6 Madison Avenue, 7th Floor, New York, NY Triumph Learning, LLC. Buckle Down and Coach are imprints of Triumph Learning. All rights reserved. No part of this publication may be reproduced in whole or in part, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission from the publisher. Printed in the United States of America The National Governors Association Center for Best Practices and Council of Chief State School Officers are the sole owners and developers of the Common Core State Standards, Copyright 00. All rights reserved.

2 Contents Domain Ratios and Proportional Relationships..... Lesson Understanding Ratios Lesson Understanding Unit Rates Lesson Using Tables of Equivalent Ratios Lesson Problem Solving: Unit Rates Lesson 5 Using Percent Lesson 6 Using Ratios to Convert Measurement Units Domain Review Domain The Number System Lesson 7 Interpreting and Computing Quotients of Fractions.. 6 Lesson 8 Problem Solving: Dividing with Fractions Lesson 9 Dividing Whole Numbers Lesson 0 Adding and Subtracting Decimals Lesson Multiply and Divide Decimals Lesson Extending Factors and Multiples to GCF and LCM Lesson Locating Positive and Negative Integers on a Line Lesson Understanding Absolute Value Lesson 5 Locating Rational Numbers on a Number Line Lesson 6 Ordering Rational Numbers Lesson 7 Plotting Ordered Pairs on the Coordinate Plane Common Core State Standards 6.RP. 6.RP. 6.RP..a 6.RP..b 6.RP..c 6.RP..d 6.NS. 6.NS. 6.NS. 6.NS. 6.NS. 6.NS. 6.NS.5, 6.NS.6.a, 6.NS.6.c 6.NS.7.c, 6.NS.7.d 6.NS.5, 6.NS.6.c 6.NS.7.a, 6.NS.7.b 6.NS.6.b, 6.NS.6.c Lesson 8 Problem Solving: Using the Coordinate Plane Domain Review Domain Expressions and Equations Lesson 9 Writing and Evaluating Numerical Expressions Lesson 0 Reading and Writing Algebraic Expressions Lesson Evaluating Algebraic Expressions Lesson Generating and Identifying Equivalent Expressions NS.8 6.EE. 6.EE..a, 6.EE..b, 6.EE.6 6.EE..c 6.EE., 6.EE. Problem Solving Fluency Lesson Performance Task

3 Common Core State Standards Lesson Writing and Solving Equations EE.5, 6.EE.6, 6.EE.7 Lesson Lesson 5 Writing and Solving Inequalities Dependent and Independent Variables EE.5, 6.EE.6, 6.EE.8 6.EE.6, 6.EE.9 Lesson 6 Problem Solving: Using Equations EE.9 Domain Review Domain Geometry Lesson 7 Finding the Area of Triangles and Quadrilaterals Lesson 8 Finding the Volume of Rectangular Prisms Lesson 9 Drawing Polygons on the Coordinate Plane Lesson 0 Representing Three-Dimensional Figures Using Nets Lesson Using Nets to Find Surface Area Domain Review Domain 5 Statistics and Probability Lesson Understanding Statistical Variability Lesson Range and Measures of Center Lesson Measures of Variability Lesson 5 Displaying Data Using Dot Plots Lesson 6 Displaying Data Using Box Plots G. 6.G. 6.G. 6.G. 6.G. 6.SP. 6.SP., 6.SP. 6.SP.5.c 6.SP., 6.SP.5.a, 6.SP.5.b, 6.SP.5.c 6.SP., 6SP.5.b, 6.SP.5.c Lesson 7 Lesson 8 Displaying Data Using Histograms Choosing Measures to Fit Distributions Domain 5 Review Glossary Math Tools SP., 6SP.5.a, 6SP.5.b, 6.SP.5.c 6.SP.5a, 6.SP.5.d

4 Grade 5 Grade 6 Grade 7 Grade 5 MD Convert like measurement units within a given measurement system. Grade 5 NBT Understand the place value system. Perform operations with multi-digit whole numbers and with decimals to hundredths. Grade 5 NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Grade 5 G Graph points on the coordinate plane to solve real-world and mathematical problems. Grade 6 RP Understand ratio concepts and use ratio reasoning to solve problems. Grade 7 RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Grade 7 SP Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models.

5 Domain Ratios and Proportional Relationships Lesson Understanding Ratios Lesson Understanding Unit Rates Lesson Using Tables of Equivalent Ratios Lesson Problem Solving: Unit Rates Lesson 5 Using Percent Lesson 6 Using Ratios to Convert Measurement Units Domain Review

6 LESSON Understanding Ratios UNDERSTAND A ratio can compare a part to a part, a part to the whole, or the whole to a part. The word to compares the two terms in a ratio. You can write a ratio in different ways. Write the ratio of squares to all shapes in three different ways. Write the ratio in words. Count the number of squares. There are squares. Count the total number of shapes. There are 8 shapes in all. Write the ratio using the word to. The ratio of squares to all shapes is three to eight. Write the ratio using a symbol. A colon (:) is a symbol used to write a ratio. The ratio of squares to all shapes is :8. Write the ratio using a fraction. Write the first term in the numerator. 8 Write the quantity of all terms in the denominator. The ratio of squares to all shapes is 8. 6 Domain : Ratios and Proportional Relationships

7 Connect For every fish Nate caught fishing from the shore, Brian caught four fish from his boat in the lake. Write a ratio to represent the situation. Then draw a picture to represent the situation. The situation compares two quantities. It compares the number of fish that Nate caught to the number of fish that Brian caught. The situation represents a ratio. Write a ratio for the situation. For every fish Nate caught, Brian caught four fish. This is the same as saying that Nate caught fish for every fish Brian caught. Draw a picture to represent the situation. Nate caught fish for every fish Brian caught. Draw fish for Nate and fish for Brian. Nate Brian The ratio of the number of fish Nate caught to the number of fish Brian caught is to, :, or. MODEL For every two white roses in a bouquet, the florist uses three red roses. Draw a picture to represent the situation. Lesson : Understanding Ratios 7

8 Practice Use the figures below to write each ratio three ways.. green to white figures. circles to stars. green figures to total figures. green stars to white stars HINT Identify what the ratio is comparing. 5. white circles to white figures Use the following address to write each ratio three ways. 65 Riverside Road 6. green circles to green stars 7. numbers to letters 8. vowels to consonants 9. consonants to numbers 0. letters to complete address. complete address to numbers. vowels to letters REMEMBER The order is important in a ratio. Make a drawing to show each ratio. Then write the ratio three ways.. The ratio of books to mystery books on a shelf is nine to two.. The ratio of adults to children at the park is three to ten. 5. The ratio of paws to tails on a tiger is four to one. 8 Domain : Ratios and Proportional Relationships

9 Write a real-world context for each ratio. 6. five to three 7. to 7 8. : 9. 6 Solve. 0. Out of 8 students in the class, wore sneakers to school. What is the ratio of students who wore sneakers to those who did not wear sneakers?. Of the 8 students at a dance camp, 6 are boys. What is the ratio of girls to students at the camp?. ANALYZE The ratio of games the Falcons won to the games they lost is 7:5. What is the ratio of games won to the total number of games played? Explain.. WRITE MATH Write three different ratios for the following situation. Explain what the ratios compare. Kendra has cats and dogs. Lesson : Understanding Ratios 9

10 LESSON Understanding Unit Rates UNDERSTAND A rate is a ratio that compares two quantities with different units of measure. A unit rate is a rate in which the second measurement or amount is unit. Three full vans, all with the same number of seats, hold a total of 6 people. The unit rate is the number of people per van. What is the unit rate? To find the unit rate, find the number of people needed to fill one van. Use drawings to find the unit rate. Draw boxes to represent the vans. Draw bars in each box to represent the people. Draw one bar in each box to represent one person per van. Draw another bar in each box. Then continue to draw one bar in each box until you have drawn 6 bars. Count the number of bars in each box to find the unit rate. Each box has bars. people The unit rate is van, or people per van. 0 Domain : Ratios and Proportional Relationships

11 Connect Three full vans hold 6 people. The number of people per van is the unit rate. What is the number of people per van? Write a ratio to compare the number of people to the number of vans. number of people : number of vans 6: Write the ratio in fraction form. people vans 6 5 Write the fraction as division. The fraction bar means division. Divide the numerator by the denominator. 6 Write the unit rate. 6 5 people The unit rate is van, or people per van. Divide. 6 5 DISCUSS How could you find the unit rate if full vans hold 6 people? Lesson : Understanding Unit Rates

12 EXAMPLE A A recipe calls for eggs to cups of milk. The unit rate is the number of eggs per cup of milk. What is the number of eggs per cup of milk? How many eggs are needed if cups of milk are used? Write a ratio to compare the number of eggs to the number of cups of milk. Write the ratio in fraction form. eggs cups of milk Write the fraction as division. 5 Divide. 5 6 Use the unit rate to find the number of eggs needed for cups of milk. Divide the numerator by the denominator. Write the unit rate eggs The unit rate is cup milk or 6 eggs per cup of milk. Multiply to find the rate for cups of milk. 6 eggs cup milk 5 eggs cups milk eggs are needed if cups of milk are used. TRY A recipe has a ratio of cups of milk to eggs. How many eggs are needed if cups of milk are used? Domain : Ratios and Proportional Relationships

13 EXAMPLE B A recipe for one loaf of bread calls for cup of white flour for each cup of whole wheat flour. Rose wants to make loaves of bread. How much whole wheat flour will she need? How much white flour will she need? Identify the unit rate. The unit rate is cup of white flour per cup of whole wheat flour, to, or :. Multiply to find the amount of whole wheat flour Rose will need. The recipe calls for cup of whole wheat flour. 5 Rose will need cups of whole wheat flour to make loaves of bread. Think about the information you need. The recipe for one loaf of bread calls for cup of whole wheat flour. Rose wants to make loaves of bread. So, she must multiply each ingredient by to find how much flour she will need. Multiply to find the amount of white flour Rose will need. The recipe calls for cup of white flour Rose will need cups of white flour to make loaves of bread. MODEL In a recipe for punch, cup of pineapple juice is used for each cup of orange juice. Draw a diagram to model how many cups of pineapple juice are used with cups of orange juice. Lesson : Understanding Unit Rates

14 Practice Write a ratio for each rate. Then find the unit rate.. 0 miles in 5 hours. 96 words in minutes HINT Use the second term of the ratio to find the unit rate.. 08 people in buses. 6 points in 8 games 5. $.70 for goldfish 6. $ for 7 comic books Find each unit rate grams of protein in servings REMEMBER The second term of a unit rate is always. 8.,068 miles in hours 9. 0 students to 5 teachers 0. $.60 for eggs. 7 strikeouts in games. 90 feet in 5 seconds. students to parents. 60 miles per 0 gallons Domain : Ratios and Proportional Relationships

15 Find each unit rate. Then write which is the better buy. 5. $0.90 for pens or $.75 for 5 pens 6. $.5 for tennis balls or $6. for 8 tennis balls 7. $.9 for apples or $.0 for 0 apples 8. $ for 6 pounds of cereal or $5 for 9 pounds of cereal Choose the best answer. 9. A chef uses cup of cheese per cups of milk in a casserole. How many cups of cheese are needed if the chef uses 6 cups of milk? A. cups B. cups C. 5 cups D. 6 cups Solve.. The coach pays $7 for 8 hamburgers. What is the cost per hamburger? 0. Javier scored 0 goals in 5 soccer games. Using the unit rate, how many goals will he score in 0 games? A. 5 goals B. 0 goals C. 5 goals D. 60 goals. Ayesha s puppy gained 6 pounds in 9 weeks. Using the unit rate, how many pounds did the puppy gain in 5 weeks?. ANALYSIS Compare and contrast rates and ratios. Lesson : Understanding Unit Rates 5

16 LESSON Using Tables of Equivalent Ratios A table can be used to show the relationship between two quantities. You can use equivalent ratios to find a missing value in a table. EXAMPLE A The table shows the relationship between the number of green beads and the number of blue beads Mindy uses when she makes bracelets. Green Beads Blue Beads ? How many blue beads does Mindy use when she uses 0 green beads? Write the first two ratios as fractions. First ratio: green beads blue beads 5 green beads 5 Second ratio: blue beads 5 0 Look for a pattern to see how the first ratio can be changed to the second, third, and fourth ratios The ratios are equivalent. Extend the pattern to find the fifth ratio. Multiply each term in the first ratio by green beads blue beads Mindy uses 5 blue beads when she uses 0 green beads. DISCUSS Explain how to find the number of green beads Mindy uses when she uses 5 blue beads. 6 Domain : Ratios and Proportional Relationships

17 EXAMPLE B The table shows the relationship between the number of dog collars Travis can make and the number of hours he takes to make the collars. Number of Hours 9 5 Number of Dog Collars 8??? Use ratios to complete the table. Decide what ratio the table shows. The table shows the ratio of hours to hours dog collars, or dog collars. Use the equivalent ratios to complete the table. Number of Hours 9 5 Number of Dog Collars Write equivalent ratios for the first ratio and the other ratios in the table. Compare the numerators of the equivalent ratios to decide how the terms are changing in each ratio ? Think: 5 9 So ? Think: So ? Think: 8 5 So CHECK How many dog collars can Travis make in hours? Lesson : Using Tables of Equivalent Ratios 7

18 EXAMPLE C The table shows the relationship between the total number of games and the total number of weeks the games were played. Number of Games Number of Weeks???? Complete the table. Then list the ratios as ordered pairs. Decide what ratio the table shows and which ratio is known. Use the known ratio to find the first ratio in the table. The table shows the ratio of games to weeks, or games weeks. From the table, 8 games were played in week. The ratio is 8. Use the first ratio to find the remaining ratios. 5? Think: 5, so multiply the denominator by. 5 6? Think: 5 6, so multiply the denominator by. 5 0? Think: 5 5 0, so multiply the denominator by 5. As one term in a ratio changes due to multiplication or division, the other unit changes in the same way. 8 5? Think: 8 5, so divide the denominator by TRY Use the equivalent ratios to complete the table. Number of Games Number of Weeks 5 Write the ordered pair for the number of weeks needed to play 6 games. 5 List the ratios as ordered pairs. An ordered pair is shown as (x, y). The first coordinate, or the x-coordinate, is the number of games. The second coordinate, or the y-coordinate, is the number of weeks. The ordered pairs are (, ), (, ), (6, ), (8, ), and (0, 5). 8 Domain : Ratios and Proportional Relationships

19 EXAMPLE D Use the ordered pairs from Example C. Plot the ordered pairs on a coordinate plane. Decide what each axis represents on the coordinate plane. DISCUSS The horizontal axis is called the x-axis. It shows the location of the x-coordinate. In these ordered pairs, the x-coordinate represents the number of games. The vertical axis is called the y-axis. It shows the location of the y-coordinate. In these ordered pairs, the y-coordinate is the number of weeks. Label the axes. Plot the first ordered pair. The first ordered pair is (, ). Start at the origin. Move units to the right. Then move up unit. Plot the point. Number of Weeks y Number of Games Explain how you plotted the remaining ordered pairs. x List the ordered pairs. (, ), (, ), (6, ), (8, ), (0, 5) Plot the remaining ordered pairs. Number of Weeks y Number of Games x Lesson : Using Tables of Equivalent Ratios 9

20 Practice For questions, use equivalent ratios to complete each table.. Number of Hours Number of Miles ??.. Number of Teachers 6 Number of Students???? Number of Words Number of Minutes??? 6? HINT As one unit in a ratio changes, the other unit changes in the same way. For questions and 5, use equivalent ratios to complete each table. List the ratios in each table as ordered pairs. Then plot the ordered pairs on the coordinate plane.. x y 5. Number of Days 5 Inches of Snow y REMEMBER To plot an ordered pair (x, y ), start at the origin. Move horizontally x units and then vertically y units. y x Inches of Snow Days x 0 Domain : Ratios and Proportional Relationships

21 For questions 6 and 7, use the information given to complete the tables. 6. A recipe uses cups of cooked rice and cups of milk. Create a table of equivalent ratios that shows the relationship between the number of cups of rice and the number of cups of milk used for the recipe. Cups of Rice Cups of Milk 7. For every 5 sit-ups Ryan does, he does 0 push-ups. Create a table of equivalent ratios that shows the relationship between the number of sit-ups and the number of push-ups Ryan does. Number of Sit-ups Number of Push-ups Complete the table and graph. 8. Every days, students in a fitness class run miles. Use equivalent ratios to complete the table. Plot the corresponding ordered pairs on the coordinate plane. Number of Days 6 9 Number of Miles Miles y 5 6 Days x Solve. 9. REASON Which of the following ratios is not equivalent to the ratio 5 6? Explain. 0,, 5 8, STRUCTURE Explain how you completed the table of equivalent ratios in question 7. Lesson : Using Tables of Equivalent Ratios

22 LESSON Problem Solving: Unit Rates READ Felicia s Ride Felicia takes hours to ride her bike 56 miles. She rides at a constant speed during the -hour ride. What is her unit rate in miles per hour? PLAN Write an equation to represent the problem. The number of miles per hour Felicia rides is her unit rate. Let m 5 the number of miles Felicia rides per hour. The number of hours Felicia rides multiplied by her unit rate is equal to the distance she rides. m 5 56 SOLVE Step : Choose an operation to solve the equation. Use the opposite operation to isolate the variable. The opposite of multiplication is. Step : Solve the equation. Divide both sides by. m 5 56 m 5 56 Step : Simplify. On the left side of the equation, 5, so the s cancel, leaving m or m. On the right side of the equation, compute 56. CHECK m 5 56 m 5 Substitute the value found for m in the original equation. m Felicia s unit rate is miles per hour. Domain : Ratios and Proportional Relationships

23 read There are instructors per 8 students at a tennis camp. At that rate, how many instructors are needed for 0 students? plan First find the unit rate of students to instructors. Then use the unit rate to find the number of instructors needed for 0 students. solve Step : Find the unit rate. Write the ratio of students to instructors: students instructors 5 Divide to find the unit rate: 8 5 The unit rate is students per instructor. Step : Write an equation using the unit rate. Let n 5 the number of instructors needed for 0 students. The number of students divided by the number of students per instructor is equal to the number of instructors needed. 0 5 n Step : Simplify the equation. check n 5 Tennis Camp Use a tape diagram to check the relationship between the number of instructors and the number of students. instructors 5 students 6 The tape diagram supports the relationship. It shows that the number of students is always 6 times the number of instructors. There is instructor for 6 students, instructors for students, instructors for 8 students, instructors for students, and instructors for 0 students. So 8 instructors are needed for 0 students. 0 Lesson : Problem Solving: Unit Rates

24 read The Farmer s Market At the farmer s market, 8 apples cost $5.0. If each apple costs the same amount, what is the price per apple? plan Write an equation to represent the problem. The price per apple is the unit cost. The number of apples times the cost per apple is equal to the total cost. Let p 5 the price per apple. 8 p 5 $5.0 Find the unit cost. solve Find the unit cost. Write a ratio of cost to number of apples: Simplify to find the unit cost: So the unit cost is. check Substitute the unit rate for p in the original equation. 8 p The price per apple is. Domain : Ratios and Proportional Relationships

25 read An order of 5 paintbrushes costs $.50. If each paintbrush costs the same amount, what is the unit cost? What is the cost for an order of paintbrushes? plan First, find the unit cost of paintbrush. Let c 5 the cost of one paintbrush. $ _ c Then use the unit cost to find the total cost of paintbrushes. solve Find the unit cost, c c 5 Find the total cost of paintbrushes. check 5 Use a double number line to check the relationship between the number of paintbrushes and the total cost. Paintbrushes ordered Art Supply Order 0 5 Total cost, $ The double number line supports the relationship. It shows that the total cost is always $0.70 times the number of paintbrushes ordered. The unit cost of a paintbrush is. An order of paintbrushes costs. Lesson : Problem Solving: Unit Rates 5

26 Practice Use the -step problem-solving process to solve each problem.. READ It takes Karl 6 hours to drive 7 miles. If he drives at a constant speed during the 6 hours, what is his unit rate? At this rate, how far will Karl drive in 8 hours? PLAN SOLVE CHECK. Harper bought chairs for $. What is the unit price? How much will 6 chairs cost? 6 Domain : Ratios and Proportional Relationships

27 . A painter uses tubes of black paint for every 6 tubes of white paint to paint a mural. At that rate, how many tubes of black paint are used if 5 tubes of white paint are used?. At Bob s Binders, a set of notebooks costs $.0. At Pam s Paper Place, a set of notebooks costs $5.0. What is the unit price of the notebooks at each store? Which is the better buy? 5. Ms. Compra bought 5 markers for $.75. At this price, how much will markers cost? How much will 0 markers cost? Lesson : Problem Solving: Unit Rates 7

28 5 LESSON Using Percent EXAMPLE A At the book fair, 0% of the books sold were science related. For every 00 books sold, how many were science related? Use a model to show 0%. A percent is a special ratio. It is a rate per 00. A 0 by 0 grid has 00 equal parts. 0% means 0 out of 00 equal parts. Shade 0 parts to show 0%. Use the ratio to find the number of science-related books sold per 00. Write an equivalent ratio to show the percent. 0% is equivalent to 0 per 00 or % % of out of 00 books sold were science related. For every 00 books sold, 0 were science related. MODEL Explain how to use the model to show how many books sold per 00 were not science related. What percent of the books sold were not science related? 8 Domain : Ratios and Proportional Relationships

29 EXAMPLE B In a jar of 500 beads, 0% are red. How many of the beads are red? Understand what you have to find. Find 0% of 500 to find the number of red beads. Write an expression for 0% of % of 500 means 0% times % of % Write the percent as a ratio. 0% Simplify the expression. 0% Substitute the ratio for the percent Write 500 as a fraction. 5 5, Multiply the numerators. Then multiply the denominators Simplify. Use a tape diagram to illustrate the percents. 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% DISCUSS red beads total number of beads There are 50 red beads in the jar. Explain how to use the tape diagram to find 80% of 500. Lesson 5: Using Percent 9

30 EXAMPLE C is 70% of what number? Make a tape diagram to model the problem. If you know the part and the percent, you can find the whole. 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% is part of the whole. It is 70% of the whole. Use the tape diagram to find the size of each part. There are 7 equal parts up to. Think: What number times 7 is equal to? 7 5, so each part increases by. Each part is 0% on the tape diagram, so 0% of the whole is. Complete the tape diagram. 0% of the whole is. Count by s to complete the tape diagram. 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% CHECK What is 70% of 0? Explain. Use the tape diagram to find the whole. 00% is the whole. 0 is 00% of the whole. is 70% of 0. 0 Domain : Ratios and Proportional Relationships

31 read Problem Solving At Shaun s Skate World, 8 of the skateboards in stock are on sale. If 0% of the skateboards in stock are on sale, how many skateboards are in stock? plan 8 is 0% of the skateboards in stock. Use a tape diagram to model the problem. Complete the tape diagram to find the number of skateboards in stock. solve Make the tape diagram. 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% 8 There are equal parts up to 8. Think: What number times is equal to 8? 5 8, so each part increases by. Each part is 0% on the tape diagram, so 0% of the whole is. Count by to complete the tape diagram. 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% What number is 00% of the tape diagram? check Look back at the completed tape diagram. What is 0% of the total? Does the number match the quantity in the problem? There are skateboards in stock at Shaun s Skate World. Lesson 5: Using Percent

32 Practice Write a percent for each situation.. 0 out of 00 people. 7 out of 0 stores HINT Write an equivalent ratio to show the rate per votes for Derek per 00 voters Find the percent of each number.. 0% of 0 REMEMBER You can use a tape diagram to model the percent. 5. 0% of % of % of % of % of % of % of % of 00 Make a tape diagram to find each percent.. 5 is 0% of what number?. 75 is 50% of what number? 5. is 0% of what number? 6. 8 is 80% of what number? 7. 0 is 60% of what number? is 0% of what number? Domain : Ratios and Proportional Relationships

33 Choose the best answer. 9. What is 5% of 00? A. 5 B. 0 C. 0 D is 60% of what number? A. B. C. 0 D. 7 Solve.. A play ran for 00 performances. The theater was full for 85% of the performances. For how many performances was the theater not full?. In a batch of lightbulbs, 0% are tinted. If 5 lightbulbs are tinted, how many light bulbs are in the batch? 5. CRITIQUE Taylor says that 0% of 50 is the same amount as 50% of 0. Is Taylor correct? Explain.. In a survey of 800 students, 70% said they liked pop music. How many students surveyed like pop music?. Keith spent 60% of his birthday money at the mall. If he spent $ at the mall, how much money did Keith receive for his birthday? 6. WRITE MATH Explain how to use a tape diagram to find 90% of 80. Lesson 5: Using Percent

34 6 LESSON Using Ratios to Convert Measurement Units Objects can be measured using either the customary system or the metric system. We can measure length, mass, and capacity. We can also convert the measurements within a system. EXAMPLE A Which is longer, 5 feet or 58 inches? Write the ratio for feet to inches. You can use ratios to convert measurement units. In a ratio, as one unit changes, the other unit changes in the same way. Use the measurement equivalence: foot 5 inches. feet inches 5 Continue writing equivalent ratios until the number of inches is 58 or greater. ft 5? in. Multiply both terms of the ratio by So ft 5 6 in. ft 5? in. Multiply both terms of the ratio by So ft 5 8 in. 5 ft 5? in. Multiply both terms of the ratio by So 5 ft 5 60 in. Write an equivalent ratio to show the number of inches in feet. Multiply both terms of the ratio by. 5 5 So ft 5 in. DISCUSS Compare. 5 ft 5 60 in. 60 in.. 58 in. So 5 ft. 58 in. 5 feet is longer than 58 inches. Explain how to use equivalent ratios to find the number of feet in 6 yards. Use Math Tool: Tables of Measurement Units. Domain : Ratios and Proportional Relationships

35 EXAMPLE B How many quarts are in gallons? Write the ratio of quarts to gallons. Use the measurement equivalence: quarts 5 gallon. quarts gallons 5 Use a tape diagram to model the ratio. Extend the tape diagram to find the equivalence. Use the tape diagram to show gallons. Draw quarts for every gallon. Quarts Gallons Use the tape diagram to record the number of quarts in gallons. There are quarts in gallons. The tape diagram shows that quarts 5 gallon. Quarts Gallons TRY Use a tape diagram to find how many pints are in quarts. Lesson 6: Using Ratios to Convert Measurement Units 5

36 EXAMPLE C Which is longer, 7 centimeters or 7 meters? Write the ratio of centimeters to meters. Use the measurement equivalence: 00 centimeters 5 meter. centimeters meters 5 00 Compare 7 cm and 7 m. 7 m cm 7 cm, 700 cm So 7 cm, 7 m 7 meters is longer than 7 centimeters. Write an equivalent ratio to find the number of centimeters in 7 meters. Multiply both terms of the ratio by So 700 cm 5 7 m TRY How many centimeters are in 50 millimeters? Use Math Tool: Tables of Measurement Units. 6 Domain : Ratios and Proportional Relationships

37 Relevant Ratios Use Math Tool: Tables of Measurement Units for this activity. Work with a partner to complete the tables.. Choose six different objects in your classroom. For example, choose the door in your classroom. For the time table, choose an event such as the length of a song. Record the names of the objects and the event in the first empty column in the tables.. Estimate the measures of the objects and the event you chose. Record the values in the Estimate column. Do not use the same unit of measure for all the objects.. Select units to which you will convert your estimated measures. Record these units in the Conversion Units column. For example, if your estimate was in feet and you are going to convert feet to inches, record inches in this column.. Determine the ratios you will use to convert your estimated measures. Record these ratios in the Conversion Ratio column. 5. Write the estimates in the converted units in the Conversion column. Length Capacity Weight Length Capacity Mass Customary Units Object Estimate Conversion Units Metric Units Object Estimate Conversion Units Conversion Ratio Conversion Ratio Conversion Conversion Time Event Estimate Conversion Units Conversion Ratio Conversion Lesson 6: Using Ratios to Convert Measurement Units 7

38 Use Math Tool: Tables of Measurement Units. Use ratios to convert each measurement. Practice. ft 5 yd. 96 oz 5 lb. h 5 min HINT Check the Math Tool: Tables of Measurement Units to make accurate conversions.. 0 mm 5 cm 5. 6 L 5 ml 6. 9,000 g 5 kg 7. 6 gal 5 qt 8. ft 5 in km 5 cm 0.. kg 5 g. 6 pt 5 c. 0 h 5 d Compare. Write.,,, or s min. 5 yd 75 in mm 80 cm REMEMBER You can change yards to feet and then feet to inches. 6. lb oz 7. 6 km 6,00 cm 8. g 50 mg 9. qt 6 pt 0. 5 L kl. 0 ft 6 yd. 70 cm 7 m. gal 5 pt. d 8 h Solve. 5. Which is a longer piece of ribbon, one that is 65 millimeters long or one that is 55 centimeters long? How do you know? 6. Which pitcher has a greater capacity, one that holds gallons or one that holds quarts? How do you know? 8 Domain : Ratios and Proportional Relationships

39 7. Which melon has a greater mass, one that is kilograms or one that is,00 grams? How do you know? 8. Which class lasts longer, one that is hours long or one that is 75 minutes long? How do you know? Choose the best answer. 9. Which is the longest? A. km B. 5 m C.,500 cm D.,000 mm Solve.. Rylie drinks cups of water from a -quart bottle of water. How much water is left in the bottle?. REASON How can you use the metric prefixes milli -, centi -, and kilo - to help you decide if a metric conversion is reasonable? 0. Which has the least capacity? A. gal B. qt C. 6 pt D. 5 c. A track is 00 meters long. How many times must Julius run around the track to run kilometers?. CONSTRUCT How many cups equal gallons? Explain how you did the conversion. Lesson 6: Using Ratios to Convert Measurement Units 9

40 DOMAIN Review Use the following information for questions. Jeremy s password is the combination of letters and numbers shown below. Use Jeremy s password to write each ratio three ways. 7 R Q A 5 8 E. letters to numbers. odd to even numbers. vowels to letters Find each unit rate.. 70 kilometers per 5 liters 5.,76 yards in 8 seconds 6. $5.85 for 9 rolls 7. hits in games markers in boxes 9. 8 words in 6 minutes For questions 0 and, use equivalent ratios to complete each table. 0. Number of Hours Number of Meters 8. Number of Pages Number of Books 8 Write a percent for each situation.. 75 out of 00 students. out of 0 movies. 0 votes for Erin per 00 voters 0

41 Find the percent of each number. 5. 0% of % of % of % of 600 Use a tape diagram to find each percent. 9. is 0% of what number? is 70% of what number?. 5 is 0% of what number?. 8 is 60% of what number? Use ratios to convert each measurement. Use Math Tool: Tables of Measurement Units.. 6 km 5 m. d 5 h 5. 5 ft 5 yd 6. 6 gal 5 qt cm 5 mm 8. 7,000 ml 5 L 9. 0 c 5 qt 0. 5 ft 5 in.. oz 5 lb. Every days, Kareem swims 0 laps. Use ratios to complete the table. Plot the corresponding ordered pairs on the coordinate plane. y Number of Days 6 8 Number of Laps Laps Days x

42 Choose the best answer.. Kimberly uses cup of walnuts for every cups of raisins when making trail mix. How many cups of walnuts will she need if she uses 8 cups of raisins? A. c B. c C. 5 c D. 6 c 5. In a rock collection, 60% of the rocks are quartz. If the collection contains 5 rocks, how many are quartz? A. 8 B. C. 5 D. 9 Solve.. A ceramics teacher orders 5 blocks of gray clay for every blocks of red clay. How many blocks of gray clay will the teacher order if she orders 5 blocks of red clay? A. 7 B. 0 C. 5 D The ribbon on a spool is 0 yards long. How many 6-inch pieces of ribbon can be cut from the spool? A. 0 B. 0 C. 50 D Jesse ran nearly three laps around the track for every lap his grandfather walked. Write the ratio of Jesse s laps around the track to his grandfather s laps. 8. A recipe has a ratio of cups of sugar to 6 cups of flour. How many cups of flour are used for each cup of sugar in the recipe? 9. Uma paid $ for 6 pillows. What is the unit price? How much would 9 pillows cost? 0. It takes Levar hours to drive 5 miles. What is Levar s unit rate? At this rate, how far will Levar drive in 7 hours?

43 Oatmeal Serving: cup (0 grams) Nutrition Information Total carbohydrate Dietary fiber g Protein 5g Calories 50 From fat 5 Recipe Water c Oats _ c C op y D rig O ht N O Pr ot T C ec O PY ted This label shows some nutrition information and a recipe for one serving of oatmeal. 7 g. How many cups of oats are needed for 8 servings of oatmeal? How do you know?. How many cups of oats are needed for 5 servings of oatmeal? How do you know?. What is the ratio of total carbohydrate to protein in one serving of oatmeal? Does the ratio change with additional servings of oatmeal? Explain.. In oatmeal, 50% of the dietary fiber is soluble fiber. How many grams of the dietary fiber in one serving of oatmeal is soluble fiber? Explain. 5. What percent of one serving of oatmeal is made up of dietary fiber? Explain. Domain Review CC_MTH_G6_SE_D_Final.indd /06/ :0 PM

44 Grade 5 Grade 6 Grade 7 Grade 5 NBT Understand the place value system. Perform operations with multi-digit whole numbers and with decimals to hundredths. Grade 5 NF Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Grade 6 NS Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to the system of rational numbers. Grade 7 RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Grade 7 NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Grade 7 EE Use properties of operations to generate equivalent expressions. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Grade 5 G Graph points on the coordinate plane to solve real-world and mathematical problems. Grade 7 G Draw, construct, and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

45 Domain The Number System Lesson 7 Interpreting and Computing Quotients of Fractions Lesson 8 Problem Solving: Dividing with Fractions Lesson 9 Dividing Whole Numbers Lesson 0 Adding and Subtracting Decimals Lesson Multiply and Divide Decimals Lesson Extending Factors and Multiples to GCF and LCM Lesson Locating Positive and Negative Integers on a Number Line Lesson Understanding Absolute Value Lesson 5 Locating Rational Numbers on a Number Line Lesson 6 Ordering Rational Numbers Lesson 7 Plotting Ordered Pairs on the Coordinate Plane Lesson 8 Problem Solving: Using the Coordinate Plane.. 0 Domain Review

46 LESSON 7 Interpreting and Computing Quotients of Fractions UNDERSTAND Use models to help divide fractions. Divide 8 means find how many groups of 8 are in. Use models to show. Place 8 models under the models for. Separate the 8 models into groups of There are groups of 8 in Domain : The Number System

47 Connect Divide. 8 To divide fractions, first write the reciprocal of the divisor. The divisor is 8. Its reciprocal is 8. Multiply the fractions. Multiply the numerators. Then multiply the denominators Write the division problem as a multiplication problem. Multiply the dividend by the reciprocal of the divisor Write the product in simplest form CHECK Use multiplication to check your answer. 8 5 Lesson 7: Interpreting and Computing Quotients of Fractions 7

48 EXAMPLE Divide Write the division problem as a multiplication problem. Multiply the dividend by the reciprocal of the divisor Multiply the fractions. Write the product in simplest form Use models to justify the answer There are groups of 5 in TRY Divide Domain : The Number System

49 read Problem Solving A board is yard long. Jenna cuts the board into pieces that are 6 yard long. How many pieces are there? plan Find how many groups of 6 are in. Divide. 6 solve To divide, multiply by the reciprocal of the divisor Write the product in simplest form. check Use models to check. 6 6 Explain how the models justify the quotient. 6 6 Jenna cut the board into pieces. Lesson 7: Interpreting and Computing Quotients of Fractions 9

50 Practice Draw models to help you to find the quotient HINT Find how many groups of 0 are in _ Write the related multiplication problem for each division problem REMEMBER Multiply by the reciprocal of the divisor Divide. Write the quotient in simplest form Domain : The Number System

51 Choose the best answer. 9. Divide. A. B. C. 6 5 D A. B. C D. Solve.. How many 6 -pound hamburgers can be made from pound of hamburger meat?. A piece of string is meter long. How many pieces of string that are 0 meter long can Wyatt cut? 5. CONCLUDE Why is of 8 different than 8?. A relay race is 5 mile long. Each runner runs 0 mile. How many runners are in the relay?. Julia has quart of orange juice. She will fill each glass with 6 quart of juice. How many glasses can she fill? 6. DECiDE Which is greater: 9 or 9? Explain why. Lesson 7: Interpreting and Computing Quotients of Fractions 5

52 8 LESSON Problem Solving: Dividing with Fractions READ A race is miles long. There are water stations every mile along the route. How many water stations are along the route? PLAN Write an equation to represent the problem. Let n 5 the number of water stations. 5 n SOLVE Write the division problem as a Multiply the dividend by the 5 Write the whole number as an improper fraction. Then multiply. Write the product in simplest form CHECK Water Stations Explain how the models below justify the quotient. problem. of the divisor. There are water stations along the route. 5 Domain : The Number System

53 Frame It read A rectangular frame that is foot long has an area of 5 8 square foot. How wide is the frame? plan The area of a rectangle is length multiplied by width. So the width is equal to the area divided by the. Write an equation to represent the problem. Let w 5 the width of the frame. w solve To solve the equation, multiply the dividend by the Multiply. Write the product in simplest form. w 5 check 5 5 of the divisor. Use the relationship between multiplication and division to check the quotient. Because multiplication and division are opposite operations, the product of the quotient multiplied by the divisor should equal the dividend. If 5 8 5, then Check the product. 5 5 So, 5 8 5, because The frame is foot wide. Lesson 8: Problem Solving: Dividing with Fractions 5

54 Practice Use the -step problem-solving process to solve each problem.. READ Setsuko has a rectangular piece of fabric that is 5 6 yard wide. The area of the fabric is 5 6 square yard. How long is Setsuko s fabric? PLAN solve CHECK. How much trail mix will each person get if 6 people share pound of trail mix equally? 5 Domain : The Number System

55 . A potter has 9 pounds of clay. He uses 0 pound of clay for each bowl he makes. How many bowls can the potter make?. How long is a rectangular field with a width of 5 mile and an area of 50 square mile? 5. How many cup servings of rice are in 8 cup of rice? Explain why your answer is reasonable. Lesson 8: Problem Solving: Dividing with Fractions 55

56 9 LESSON Dividing Whole Numbers EXAMPLE A Divide. 9,75 6 CHECK The first digit of the quotient will be in the thousands place. Divide the 9 thousands by 6. That is thousands in each group There are 5 thousands left. Bring down the. Divide the tens by 6. That is 5 tens in each group There are tens left. Check your answer using multiplication. Remember to add the remainder. Bring down the 7. Divide the 57 hundreds by 6. That is 9 hundreds in each group There are hundreds left. Bring down the 5. Divide the 5 ones by 6. That is 7 ones in each group R The remainder is ones. Write it next to the quotient. 9,75 6 5,957 R 56 Domain : The Number System

57 EXAMPLE B Divide. 7,6 Set up the problem. There are not enough ten thousands to divide by, 7 6 so the first digit is in the thousands place. Divide the 7 thousands by. CHECK That is 5 thousands in each group Bring down the 6. Divide the 6 tens by. That is 9 tens in each group Use multiplication to check your answer. 5 Bring down the. There are not enough hundreds to divide, so write 0 in the quotient in the hundreds place Bring down the. Divide the 0 ones by. That is 6 ones in each group ,6 5 5,096 Lesson 9: Dividing Whole Numbers 57

58 EXAMPLE C It cost $58,6 to make 8 antennas. How much did it cost to make each antenna? Divide. 58,6 8 5 n Decide where to place the first digit in the quotient The first digit will be in the ten thousands place. DISCUSS Divide Subtract. Bring down the 8 thousands. Divide the 78 thousands Subtract. Bring down the hundreds. Divide the 6 hundreds Subtract. Bring down the tens. Divide the 9 tens Subtract. Bring down the 6 ones. Divide the 6 ones Subtract. Estimate to check that the answer is reasonable. Use compatible numbers to estimate the quotient. 58,6 is about 00,000 8 is close to 0. A reasonable estimate of the quotient is 00, , ,000,5 is close to 5,000, so the answer is reasonable. It cost $,5 to make each antenna. Why might you prefer to use an estimate to check that a quotient is reasonable instead of using multiplication to check the answer? 58 Domain : The Number System

59 Problem Solving Division Sentence Search Each dividend can be divided by one of the divisors to create one of the quotients. Match the divisors and the quotient with the correct dividend to write a true division sentence. Dividends Divisors Quotients, ,978 5, 5 6, , , ,6 86,6 59,97 9,068 Lesson 9: 9: Dividing Whole Numbers 59

60 Practice Estimate each quotient.. 59, , ,9 6 HINT Compatible numbers are numbers that are easy to divide. Use 6 6 to estimate. Divide. Check your answer.. 5 6,80 REMEMBER Divide, multiply, and subtract in each step , , , , , , , ,0 Find each missing number. Dividend Divisor Quotient Remainder., , , , , Domain : The Number System

61 8. Choose a 5-digit number and a -digit number. Both must be greater than. Write a real-world problem about dividing the 5-digit number by the -digit number. Then solve the problem. 9. Write a real-world problem about dividing $5,775 by $5. Then solve the problem. Solve. 0. There were,800 visitors to a special exhibit at the museum. The exhibit lasted 0 days. What was the daily average number of visitors to the special exhibit?. A tour group spent $,96 on airline tickets. There are 8 people in the group. How much did each person spend on an airline ticket?. Kristy bought a car for $6,. She is paying for the car with 6 equal monthly payments. How much is each payment?. The concert hall has 5,50 seats. There are 78 rows of seats in the hall. Each row has the same number of seats. How many seats are in each row?. DEDUCE How could the fact help you estimate the quotient of 6,57 6? 5. ASSESS Sometimes a real-world problem involving division has a remainder. To complete the solution, you have to interpret the remainder. Explain why. Lesson 9: Dividing Whole Numbers 6

62 0 LESSON Adding and Subtracting Decimals EXAMPLE A Find the sum Line up the decimal points in the addends Add from right to left Use zeros as placeholders Place the decimal point in the sum DISCUSS Why do you line up the decimal points when you add decimal numbers? 6 Domain : The Number System

63 EXAMPLE B Add y Line up the decimal points Add. TRY Place the decimal point in the sum Add w Estimate to check that the sum is reasonable. Round each addend to the nearest hundred rounds to rounds to 00. The estimated sum is about 00 00, or 500. The exact sum, 7.7, is close to 500. So the sum is reasonable. y Lesson 0: Adding and Subtracting Decimals 6

64 EXAMPLE C Find the difference Set up the problem to subtract vertically. Line up the decimal points Subtract from right to left Estimate to check that the sum is reasonable. Round each addend to the nearest hundred rounds to rounds to 00. The difference is about 00 00, or 00. The exact difference, 57., rounds to 00. So the difference is reasonable Use zeros as placeholders Place the decimal point in the difference TRY Subtract d 6 Domain : The Number System

65 read Problem Solving At the tournament, Team A scored points. Team B scored points. How many more points did Team A score than Team B? plan Write an equation to represent the problem. Let p 5 the difference between the two team s points p solve Line up the decimal points. Then subtract check Use addition to check the answer. Line up the decimal points in the addends. Does the total equal Team A s points? Team A scored more points than Team B. Lesson 0: Adding and Subtracting Decimals 65

66 Add or subtract. Check your answer. Practice HINT To subtract in each place, add a zero to 5. as a placeholder REMEMBER Align the decimal point to add or subtract in the same place Find the missing number Domain : The Number System

67 Choose the best answer.. Allie rode her bike. kilometers from home to the park. She rode for 7.85 kilometers around the park. Then she rode back home along her original route. How many kilometers did Allie ride in all? A. 8.8 km B. 8.7 km C..5 km D. 6.5 km 5. Mark s batting average is 0.. Last year, it was By how much has Mark s batting average increased? A. 0. B C D Write a real-world problem that can be solved by adding two decimals. Then solve the problem. Solve. 7. A diver scores 6.89, 7.95, 6.5, and 8.05 in four dives. What is the diver s total score? 9. CONCLUDE When you compute with decimals, you should always check that your answer is reasonable. Why? 8. Kanye has to drive 9 miles to visit his grandparents. He drives 85.6 miles and then stops to get gas. How many miles does Kanye have left to drive? 0. JUSTIFY Why do you always line up the decimal points when you add or subtract decimal numbers? Lesson 0: Adding and Subtracting Decimals 67

68 LESSON Multiply and Divide Decimals EXAMPLE A Find the product Multiply as with whole numbers You can also estimate the product to place the decimal point. Count the decimal places in the factors. Add. Put the total number of decimal places in the product.. 8 decimal places 5. 7 decimal places decimal places The product is about 0 5, or is closer to 50 than or TRY Find each product Domain : The Number System

69 EXAMPLE B A bag of pears weighs 6.75 pounds. The pears cost $.9 per pound. What is the total cost of the pears? Multiply $.9 5 c Multiply as with whole numbers. 5 6 $ Check that the product is reasonable. The product is about 7, or. is close to 6.5, so the product is reasonable. Place the decimal point in the product. $. 9 decimal places decimal places 6. 5 decimal places Find the cost to the nearest cent. DISCUSS Why do you round money amounts to the nearest hundredth? Round the product to the hundredths place. 6.5 The digit to the right of the hundredths place is less than 5. Round down. To the nearest cent, 6.5 rounds to 6.. The total cost of the pears is $6.. Lesson : Multiply and Divide Decimals 69

70 EXAMPLE C Divide CHECK Make the divisor a whole number by multiplying the divisor and the dividend by Place a zero in the tenths place in the dividend and continue to divide Bring down the 0 tenths. Divide the 90 tenths Divide as with whole numbers Subtract. Bring down the 5 ones. Divide the 95 ones Subtract. Use multiplication to check your answer Place the decimal point in the quotient. Place the decimal point above the decimal point in the dividend Domain : The Number System

71 read Problem Solving Nima drove kilometers in. hours. What was her average speed? plan Write an equation to represent the problem. Let k 5 the number of kilometers Nima drove each hour k solve Make the divisor a whole number by multiplying the divisor and the dividend by Place the decimal point in the quotient. Divide as with whole numbers. Divide until there is no remainder check. The quotient, k, is. Nima drove at an average speed of kilometers per hour. Lesson : Multiply and Divide Decimals 7

72 Practice Write the number of decimal places that will be in the product HINT Add the number of decimal places in the factors. Rewrite each problem so that the divisor is a whole number REMEMBER Multiply the divisor and the dividend by the same multiple of ten. Find the product. Check that your answer is reasonable Find the quotient. Check your answer Domain : The Number System

73 0. When you divide a decimal by a number greater than, how does the quotient compare with the dividend? When you divide a decimal by a number less than, how does the quotient compare with the dividend? Give examples to support your answer.. Write a real-world problem that can be solved by multiplying two decimals. Then solve the problem.. Write a real-world problem that can be solved by dividing two decimals. Then solve the problem. Solve.. Andie bought.85 meters of cord. The cord cost $0.98 per meter. What was the total cost of the cord? 5. Sameer worked for 6. hours each day for 5 days. He earned $8.75 per hour. How much did Sameer earn in all? 7. PROVE Explain how to find the product of How do you know that the product is reasonable?. Naomi drove 70. miles on 8. gallons of gas. How many miles per gallon did her car get? 6. Ming bought 9.8 gallons of gas for $7.7. How much did she pay per gallon? 8. SUMMARIZE Explain how to divide a decimal by a decimal. Lesson : Multiply and Divide Decimals 7

74 LESSON Extending Factors and Multiples to GCF and LCM UNDERSTAND You can use area models or grids to find the greatest common factor (GCF) of two numbers. Find the GCF of 8 and 0. Think of 8 and 0 as the dimensions of a rectangle. To find the GCF using models, find the dimensions of the largest square that can tile the entire rectangle with no gaps or overlaps. The largest square that fits inside the rectangle is a square against one side with length 8 units long. Two 8 by 8 squares fit inside the rectangle. However, they do not fill the rectangle The largest square that fits inside the remaining rectangle is a by square. Two by squares fit and completely fill the original rectangle Since the original rectangle is now filled, is the GCF. Notice that the entire 8 by 0 rectangle can be filled with the by squares. 8 The GCF of 8 and 0 is. 0 7 Domain : The Number System

75 Connect Find the GCF of 8 and 0. List all the factors of each number. Factors of 8:,,, 8 Factors of 0:,,, 5, 0, 0 Find the common factors of 8 and 0. Find the GCF. Find the greatest factor that appears in both lists. Factors of 8:,,, 8 Factors of 0:,,, 5, 0, 0 is the greatest factor that appears in both lists. The GCF of 8 and 0 is. Underline the common factors. These are the factors in both lists. Factors of 8:,,, 8 Factors of 0:,,, 5, 0, 0 MODEL Use an area model to find the GCF of 6 and Lesson : Extending Factors and Multiples to GCF and LCM 75

76 EXAMPLE A The GCF of two numbers is. The two numbers are between 5 and 6. The greater number has two more factors than the lesser number. What are the numbers? Decide which of the numbers between 5 and 6 have as a factor. TRY Is a factor of: 6? Yes? No 7? No? Yes 8? No? No 9? Yes? No 0? No 5? Yes Which pairs of numbers have as the GCF? 6 and 9? Yes 6 and? No 6 and 5? No 9 and? Yes 9 and 5? Yes and 5? Yes Find the GCF of and 8. List the factors of 6, 9,, and 5. 6:,,,, 6, 9,, 8, 6 9:,,, 9 :,,, 6, 7,,, 5:,, 5, 9, 5, 5 The greater number has two more factors than the lesser number. Does 9 have more factors than 6? No Does have more factors than 6? No Does 5 have more factors than 6? No Does have more factors than 9? No Does 5 have more factors than 9? Yes Does 5 have more factors than? No The numbers are 9 and Domain : The Number System

77 EXAMPLE B Use the distributive property to express the sum below. 5 Find the GCF of 5 and. List the factors of 5 and. Factors of 5:,,, 6, 9, 8, 7, 5 Factors of :,,,, 6, The GCF of 5 and is 6. Rewrite the expression as a sum with the factors. 5 5 (6 9) (6 ) Write each addend with 6 as a factor Use the distributive property. Use the GCF as the factor that is distributed to each term in the sum. (6 9) (6 ) 5 6(9 ) 5 5 6(9 ) CHECK How do you know that the equation 5 5 6(9 ) is true? Lesson : Extending Factors and Multiples to GCF and LCM 77

78 EXAMPLE C Find the least common multiple (LCM) of 6 and 9. List some multiples of each number. Multiples of 6: 6,, 8,, 0, 6,, 8, 5 Multiples of 9: 9, 8, 7, 6, 5, 5 Find the common multiples. Find the LCM. Find the least multiple that appears in both lists. Multiples of 6: 6,, 8,, 0, 6,, 8, 5 Multiples of 9: 9, 8, 7, 6, 5, 5 8 is the least multiple that appears in both lists. The LCM of 6 and 9 is 8. Underline the common multiples. These are the multiples found in both lists. Multiples of 6: 6,, 8,, 0, 6,, 8, 5 Multiples of 9: 9, 8, 7, 6, 5, 5 DISCUSS How can you choose when to stop while making a list of multiples to find the LCM? 78 Domain : The Number System

79 S o l v e t h e R i d d l e s The GCF of two numbers less than or equal to is. Their LCM is 0. What are the numbers? The LCM of two numbers less than or equal to is 6. The GCF of the numbers is. What are the numbers? The GCF of two numbers is. The numbers are between 0 and 0. What are the numbers? The LCM of two numbers less than or equal to is 0 more than. What are the numbers? 5 The GCF of two numbers less than 00 is. The difference between the numbers is 6. The greater number is a multiple of 0. What are the numbers? 6 Write your own GCF riddle. Exchange your riddle with classmates and solve. 7 Write your own LCM riddle. Exchange your riddle with classmates and solve. Lesson : Extending Factors and Multiples to GCF and LCM 79

80 Practice Use area models or grids to find the GCF of each pair of numbers.., 0. 6,. 9, 5 HINT The lesser number can always form at least one square. Find the GCF of each pair of numbers.. 5, , 6. 7, , 8., , 60 0., 0. 5, 5. 6, 8 Find the LCM of each pair of numbers.., REMEMBER The first multiple of a number is the number itself.. 5, 7 5., , 8 7., 5 8., , 6 0., 9. 8,., 0. 9,., 80 Domain : The Number System

81 Use the distributive property to express each sum with the GCF factored out Solve.. Consider the numbers between 0 and 0. Which number has the greatest number of factors?. What is the GCF of, 0, and 6?. What is the LCM of,, and 8?. Find the GCF and the LCM of 9 and. 5. Tristan has 5 apples and 0 pears that he is putting into gift baskets. Each basket will have the same number of apples and pears. What is the greatest number of baskets Tristan can make with no fruit left over? Explain. 6. At Poultry Paradise, turkey burgers are sold in packages of 8. Whole-grain buns are sold in packages of 6. What is the least number of turkey burgers and buns Carly can buy to have an equal number of each? Explain. 7. DEMONSTRATE What is the least common factor of any pair of numbers? Explain. 8. ANALYZE Why are you able to find a greatest common factor but not a greatest common multiple? Lesson : Extending Factors and Multiples to GCF and LCM 8

82 LESSON Locating Positive and Negative Integers on a Number Line Negative numbers are less than zero and are located to the left of 0 on a horizontal number line. Positive numbers are greater than zero and are located to the right of 0. EXAMPLE A Locate 9 and its opposite on a number line. DISCUSS The number line shows positive and negative integers. The integer 9 can be written as 9 or 9. The opposite of a number lies the same distance from 0 on a number line but in the other direction. The opposite of 9 is 9. To plot a point at 9, count 9 units to the right of 0 and draw a point. To plot a point at 9, count 9 units to the left of 0 and draw a point The number line above shows the location of 9 and its opposite, 9. EXAMPLE B Find the opposite of the opposite of 6. Find the opposite of 6. The opposite of 6 is the integer with the opposite sign that lies the same distance from 0. The opposite of a negative number is a positive number. So the opposite of 6 is 6. Find the opposite of the opposite of 6. The opposite of 6 is 6. To find the opposite of the opposite, find the opposite of 6. The opposite of 6 is 6. The opposite of the opposite of 6 is 6. What number is its own opposite? Explain. 8 Domain : The Number System

83 EXAMPLE C In a football game, the team with possession of the ball gained 5 yards. What is the opposite of gaining 5 yards? Decide what 0 means in the situation. In this situation, 0 is the place on the field where the team with possession starts. From that point, the team gained 5 yards. Write the opposite of 5 as an integer. The opposite of 5 is 5. TRY Interpret the meaning of 5 yards. The number 5 yards means losing 5 yards from 0, the place on the field where the team with possession starts. The opposite of gaining 5 yards is 5 yards or losing 5 yards. EXAMPLE D The low temperature in a -hour period was 7 F. What is the opposite of 7 F? Decide what 0 means in the situation. In this situation, 0 is the point on the temperature scale that separates the positive temperatures from the negative temperatures. Positive temperatures are above 0, and negative temperatures are below 0. What is the opposite of F? Interpret the meaning of 7 F in this situation. In this situation, 7 F means 7 F below 0. Find the opposite of 7 F. The opposite of 7 F is 7 F. In this situation, 7 F means 7 degrees above 0. The opposite of 7 F is 7 F or 7 F above 0. Lesson : Locating Positive and Negative Integers on a Number Line 8

84 Practice Locate each integer and its opposite on a number line. What is each integer s location in relation to 0? Use Math Tool: Blank Number Lines HINT You can use intervals that are multiples when drawing a number line. Find the opposite of the opposite of each number What is the opposite of the opposite of any integer? How do you know? Write an integer to represent each situation. REMEMBER Find the opposite. Then find the opposite of the opposite.. 0 feet below sea level. C below 0. gaining 6 points. a bank deposit of $0 5. a loss of yards in a football game 6. a debit of $5 8 Domain : The Number System

85 Write an integer to represent each situation. Explain the meaning of 0 in the situation. Then describe the opposite situation, and write an integer to represent it. 7. Going up 6 flights of stairs 8. A $5 withdrawal from an account 9. Losing 8 pounds 0. A credit of $0. A $00 loss in an investment. 8 meters above sea level Solve.. WRITE MATH Describe a situation that can be represented by the integer 0 and its opposite.. ARGUE Why is it important to understand the meaning of 0 when using integers to represent real-world situations? Lesson : Locating Positive and Negative Integers on a Number Line 85

86 LESSON Understanding Absolute Value EXAMPLE A Evaluate 8. Understand the notation. Write: 8 Read: the absolute value of 8 Locate 8 on a number line Use the definition of absolute value to evaluate 8. The absolute value of a number is its distance from is 8 units from units from 0 DISCUSS Why is the absolute value of a number always greater than or equal to 0? 86 Domain : The Number System

87 EXAMPLE B A submarine is cruising at 65 feet. How many feet below the surface is the submarine? Use absolute value to represent the situation. The problem asks for the distance of the submarine from the surface. Find 65. Interpret the meaning of the absolute value. The absolute value represents the distance of the submarine from the surface. Since a distance is always greater than or equal to 0, the location of the submarine is a positive distance from the surface. The submarine is located 65 feet below the surface. TRY Evaluate 65. The absolute value of a number is its distance from is 65 units to the left of 0 on a number line. So A weather balloon records data at,000 feet above the ocean. How many feet above sea level is the weather balloon? Lesson : Understanding Absolute Value 87

88 EXAMPLE C Audrey has an account balance less than $. Is her debt greater than or less than $? Locate on a number line. 0 Evaluate. 5 0 Use the number line to show an amount less than. 0 An account balance less than $ means that Audrey s debt is greater than $. Audrey s debt is greater than $. MODEL Thatcher has an account balance of more than $0. Is his debt greater than or less than $0? Explain. 88 Domain : The Number System

89 read Problem Solving Yoshiro returned a shirt to the store. He received a credit of $5 to spend at the store. If he spends $0, will the credit cover the purchase? plan Use integers to represent the credit and the amount spent. Use absolute value to compare the credit amount to the amount Yoshiro spends. solve What is the credit amount as an integer? What is the amount Yoshiro spends as an integer? What is the absolute value of the credit amount? What is the absolute value of the amount Yoshiro spends? Compare the absolute value of the credit amount to the absolute value of the amount Yoshiro spends. Does the $5 credit cover the $0 purchase? check How do you know that your answer is reasonable? The $5 credit cover the $0 purchase. Lesson : Understanding Absolute Value 89

90 Evaluate the absolute value of each integer. Practice HINT The absolute value of an integer is its distance from 0 on a number line What values can n have to make each equation true? 9. n 5 REMEMBER Distance can be measured to the left of 0 or to the right of 0 on a number line. 0. n 5. n 5. n 5 0. n n 5 89 Use absolute value to represent each situation. Then solve. 5. The lake reaches a depth of 85 meters. How many meters below the surface is the lake bottom? 7. A liquid freezes at a temperature of 6 F. How many degrees below 0 does the liquid freeze? 6. Mr. Murray had a return of $0 on a stock investment. What was Mr. Murray s loss on the stock? 8. An airplane is flying at an elevation of 5,000 feet. How many feet above sea level is the airplane flying? 90 Domain : The Number System

91 Solve. 9. Izabella has an account balance of more than $6. Is her debt greater than or less than $6? Explain. 0. Sterling has an account balance less than $0. Is his debt greater than or less than $0? Explain.. Phoebe has an account balance more than $5. How does her debt compare to a debt of $0? Explain.. Write a problem with a real-world context using CONCLUDE Which is the greater debt: $ or $0? Use absolute value to explain.. Hector has a credit of $5 to spend at a bicycle shop. If he spends $8, will the credit cover the purchase? Explain.. Write a problem with a real-world context using DEFINE Explain what the absolute value of an integer is. Lesson : Understanding Absolute Value 9

92 5 LESSON Locating Rational Numbers on a Number Line EXAMPLE A Locate on the number line. 0 Decide if the point lies to the left of 0 or to the right of 0. Negative rational numbers lie to the left of 0 on a horizontal number line. The number is negative, so it lies to the left of 0. Count units to the left of 0 to locate the point. The number line is divided into fourths. Each interval to the left of 0 represents. Start at 0. Count fourths from 0 to. Then count more fourths from to. Plot a point at. Label the point. 0 MODEL The number line above shows on the number line. Locate on the number line. 9 Domain : The Number System

93 EXAMPLE B A stock fell.6 points. Plot the amount the stock fell on a number line. Write a rational number to represent the situation. Let 0 represent the starting value of the stock. Since the stock fell, its value is less than the starting value. The rational number.6 represents the amount the stock fell. Draw a number line..6 is between and. Draw a horizontal number line divided into tenths from to. Label the integers. Locate.6 on the number line. Start at and count 0.6 units to the left. Plot a point at.6. Label the point..6 DISCUSS The number line above shows.6, the amount the stock fell. How is locating.6 on a number line different than locating.6 on a number line? How is it similar? Lesson 5: Locating Rational Numbers on a Number Line 9

94 EXAMPLE C Jerry owed Jordan $.50. Jerry paid back Jordan $.50. How much does Jerry still owe Jordan? Use rational numbers to represent how much Jerry owed Jordan and how much Jerry paid him back. Jerry owed Jordan $.50:.5 Jerry paid Jordan back $.50:.5 Locate.5 on a number line..5 is between and 5. From, count 0.5 units to the left. Plot and label a point Use the same number line to determine how much Jerry owes Jordan. Jerry paid back $.50. Jerry paid back the amount he originally owed Jordan. From.5, move right.5 units on the number line So Jerry owes Jordan $0. From the number line, Jerry owes Jordan $0. TRY If you lose $.5 in the morning and you find $.5 in the afternoon, how much money did you lose in all? Explain. 9 Domain : The Number System

95 EXAMPLE D From the ground, Maribel climbed 65 feet up a cliff face. Then she rappelled 65 feet down the cliff face. Where was Maribel after she rappelled down the cliff face? Use rational numbers to represent how far Maribel climbed up the cliff face and how far she rappelled down. Up: 65 Down: 65 Use the same number line to show how far Maribel rappelled down the cliff face. 65 ft 0 65 ft Use a number line to show how far Maribel climbed up the cliff face. On the number line, 0 represents the ground. Locate 65 on a number line. 65 ft 0 Maribel rappelled down the same distance she went up the cliff face So she is back at the place where she started. After she rappelled down the cliff face, Maribel was back on the ground. TRY At the entrance to a cave, Josh climbs down 0 feet into a chamber. Then he climbs back up 0 feet from the chamber. Where is Josh after he climbs back up? Lesson 5: Locating Rational Numbers on a Number Line 95

96 Practice Name each point on the number line below using both a fraction and a decimal. A B C D E A. B. C. D 5. E HINT Start at 0 and count units from right to left to name a point less than 0. Plot and label each point on the number line below Juliet deposited $8.75 in a checking account. Draw a horizontal number line to show the deposit. What does 0 mean on your number line? Juliet writes a check for $8.75. How does this change the amount from when she deposited $8.75 into the account? Why? 96 Domain : The Number System

97 Solve.. From sea level, a helicopter rises to an elevation of 5.8 meters. Then it descends 5.8 meters. What is the elevation of the helicopter after it descends? Explain.. Mr. Sampson lost 5 pounds. Then he gained 5 pounds. How did Mr. Sampson s weight change from the time he lost the weight to the time he gained the weight? Explain.. In the morning a stock rises.85 points. By closing time the stock falls.85 points. Use rational numbers to record the stock changes in the morning and by closing time. How much did the stock change from the morning to closing? Explain. 6. Landon earns $6.50. He spends $6.50 for a concert ticket. How much money does Landon have left? Use a sum of rational numbers to explain. 5. From the surface, a diver descends to a depth of 5 6 feet. Then the diver rises 5 6 feet. Draw a number line to show the situation. Where is the diver when she rises to the surface? Explain. 7. Write a negative rational number greater than 50. Write the opposite of your rational number. Then add the number and its opposite. What is the sum? 8. CONTRAST How is locating a rational number on a number line similar to locating an integer? How is it different? 9. JUSTIFY Why is the sum of a rational number and its opposite always equal to 0? Lesson 5: Locating Rational Numbers on a Number Line 97

98 6 LESSON Ordering Rational Numbers EXAMPLE A Compare and 5. Locate each integer on a number line MODEL Compare the positions of the integers on the number line. When oriented from left to right on the number line, 5 lies to the left of. In other words, lies to the right of 5. Use a number line to compare and. Use symbols to compare the integers. On a number line, the lesser number lies to the left of the greater number. Since 5 lies to the left of, 5,. On a number line, the greater number lies to the right of the lesser number. Since lies to the right of 5,. 5. So 5, or Domain : The Number System

99 EXAMPLE B The freezing point of nitrogen is 6. K. The freezing point of salt water is 70.6 K. Which has a lower freezing point? Compare 6. and Locate each rational number on a number line TRY Use symbols to compare the rational numbers. Use the number line to compare the numbers. 6. lies to the left of So 6., lies to the right of 6.. So At midnight, the temperature at the ski resort was 9.5 C. At sunrise, the temperature was.8 C. Use symbols to compare the temperatures. When was it warmer? Decide which has a lower freezing point. Since 6., 70.6, nitrogen has a lower freezing point than salt water. Nitrogen has a lower freezing point than salt water. Lesson 6: Ordering Rational Numbers 99

100 EXAMPLE C Order,.6, and. from least to greatest. Locate each rational number on a number line Compare the positions of the integers on the number line.. lies to the left of. lies to the left of.6. So.,,.6 shows the numbers ordered from least to greatest. Notice also that.6 lies to the right of. lies to the right of.. So.6... shows the numbers ordered from greatest to least. From least to greatest, the numbers are.,, and.6. MODEL Use a number line and comparison symbols to order 8,., and from greatest to least. 00 Domain : The Number System

101 read Problem Solving A scientist takes temperature readings at three locations on a mountain at 8 a.m. The temperature at the base is 0. C. The temperature at mid-mountain is C. The temperature at the summit is.8 C. Order the temperatures from greatest to least. Which location has the warmest temperature at 8 a.m.? plan Use a number line to order the temperatures from to. solve Locate each temperature on a number line. Which is the warmest temperature? How do you know? Which is the coolest temperature? How do you know? Use comparison symbols to write the temperatures from greatest to least. check How can you use your number line to support your answer? From greatest to least, the temperatures are,, and. The warmest temperature at 8 a.m. was at. Lesson 6: Ordering Rational Numbers 0

102 Practice Compare. Write. or, HINT Compare the signs of the numbers Order from least to greatest...75, 0,.,.5, , 0.58, 6. 0.,., Order from greatest to least. 7. 6, 6., , 5, REMEMBER You can use a number line to help order rational numbers. 9. 5, 0., ,, 6. On a number line, is located to the right or to the left of? Write two comparisons to support your answer.. On a number line, is 6 located to the right or to the left of 9? Write two comparisons to support your answer. 0 Domain : The Number System

103 Write a real-world example for each inequality statement.. 0.., ,.5 Solve. 7. The freezing point of nitrogen is C. The freezing point of mercury is 8.87 C. Which has a lower freezing point? Explain. 9. At opening bell, Stocks A, B, and C had the same value. By the closing bell, Stock A s value was., Stock B s value was 0.9, and Stock C s value was. Which stock gained the most? Which stock lost the most? Explain. 8. The melting point of nitrogen is 95.8 C. The melting point of solid oxygen is 8. C. Which has a higher melting point? Explain. 0. Hannah ran her first lap around the track in 50.5 seconds. She ran her second lap in 5. seconds. She ran her third lap in 9 seconds. Which lap was Hannah s slowest lap? Which lap was Hannah s fastest lap? Explain.. DEMONSTRATE Explain how to use a number line to order any three rational numbers from least to greatest. Give an example.. WRITE MATH Explain why every positive rational number is greater than every negative rational number. Lesson 6: Ordering Rational Numbers 0

104 7 LESSON Plotting Ordered Pairs on the Coordinate Plane EXAMPLE A Plot (, ) on a coordinate plane. Which quadrant is the point located in? Identify the quadrants on a coordinate plane. Two perpendicular number lines intersect to form the coordinate plane. The horizontal number line is the x-axis. The vertical number line is the y-axis. The axes intersect at the origin. The axes divide the coordinate plane into four quadrants. II y III IV 5 Locate the quadrant. I x Plot (, ) on a coordinate plane. Points on a coordinate plane can be located by ordered pairs. The first number of an ordered pair tells how far to move left or right from the origin. The second number tells how far to move up or down. To plot (, ), start at the origin, (0, 0). The first number is, so move units to the left of 0. The second number is positive, so move units up. Plot and label the point. (, ) y x You can use the signs of the numbers in an ordered pair to find the quadrant. Quadrant I: (, ) Quadrant II: (, ) Quadrant III: (, ) Quadrant IV: (, ) The first number in (, ) is negative and the second number is positive, so the point lies in Quadrant II. The point (, ) is shown on the coordinate plane. It lies in Quadrant II. TRY Plot (, ) on the coordinate plane. Which quadrant is the point located in? 0 Domain : The Number System

105 EXAMPLE B Plot (, 5) and (, 5) on a coordinate plane. How are the ordered pairs different? In which quadrant is each point located? Plot each point on a coordinate plane. To plot (, 5), start at the origin. Move units to the right and 5 units down. To plot (, 5), start at the origin. Move units to the left and 5 units up. Label each point. (, 5) y (, 5) Find the quadrants where the points are located. Because the signs of the coordinates are different, the ordered pairs lie in different quadrants. (, 5) is in Quadrant IV. (, 5) is in Quadrant II. The points are shown on the coordinate plane. The coordinates of the ordered pairs differ by their signs. (, 5) is in Quadrant IV, and (, 5) is in Quadrant II. x TRY Compare the numbers in the ordered pairs. The first numbers in the ordered pairs have opposite signs. The second numbers in the ordered pairs also have opposite signs. Plot (, 5) and (, 5) on the coordinate plane. Which quadrant is each point located in? Lesson 7: Plotting Ordered Pairs on the Coordinate Plane 05

106 EXAMPLE C From Zoe s house, the hospital is blocks to the east and blocks south. Her school is blocks to the west and block south. The park is blocks to the west and blocks north. The library is blocks to the east and block north. Plot the locations of the hospital, school, park, and library on a coordinate grid in relation to Zoe s house. Write an ordered pair for each location. Zoe s house lies at the origin. Each unit represents block. East lies to the right of the origin, and west lies to the left of the origin. North lies above the origin, and south lies below the origin. Hospital: (, ) School:, Park:, Library:, Plot each point on a coordinate grid. Plot a point at the origin for Zoe s house. Then plot the points to show the locations of each place. Label what each point represents. Park School 5 5 y Zoe s House Hospital Library x MODEL A grocery store is unit west and units south of Zoe s house. Plot and label a point to show the grocery store on the coordinate grid. The coordinate grid above shows the location of the hospital, school, park, and library in relation to Zoe s house. 06 Domain : The Number System

107 EXAMPLE D Reflect rectangle ABCD across the y-axis. y A B D C 5 x Decide how the coordinates of an ordered pair change when the point is reflected across the y-axis. When a point is reflected across the y-axis, the point lies on the opposite side of the y-axis. This means that the first number in the ordered pair changes its sign and the second number stays the same. Plot and label the reflection of each point. Connect the points to form a rectangle. Write the ordered pairs of each reflected point. Each point is reflected across the y-axis. Change the sign of the first number in the original point to write the ordered pair of the reflection. Use a prime 9 to represent the reflection. Ordered Pairs Point Reflection A(5, ) A9 (5, ) B(, ) B9 (, ) C(, 5) C9 (, 5) D(5, 5) D9 (5, 5) y A D B B 5 C The reflection of rectangle ABCD is rectangle A9 B9 C9 D9. It is shown on the above grid. C A D x DISCUSS How do the coordinates of an ordered pair change when the point is reflected across the x-axis? Explain. Lesson 7: Plotting Ordered Pairs on the Coordinate Plane 07

108 Practice Plot and label each point on the coordinate grid. Identify the quadrant where the point is located. y x. C (, ). D (, ). E (.5, ).5 is between and. Identify the quadrant where each ordered pair is located. HINT. F (5, ) 5. G, 6. (, ) 7. (, ) 8. (0, 5) REMEMBER You can use a coordinate grid to find the quadrant. 9. (6, 9) 0.,. (0.5, 0.5). (5, 5). (8.75, ). (9, 0) 5. (5, 00) 6. (, 6) 7. (0.75,.5) 08 Domain : The Number System

109 8. From the park, Harun s house is blocks to the west and blocks south. Eli s house is blocks to the west and blocks north. Lily s house is blocks to the east and block north. Elena s house is 5 blocks to the east and blocks south. Plot the locations of the Harun s, Eli s, Lily s, and Elena s houses on the coordinate grid in relation to the park. y 9. Reflect trapezoid CDEF across the y-axis. C D F 5 E y x x 0. Reflect parallelogram JKLM across the x-axis y J K M L x Solve.. INTERPRET When the first number in an ordered pair is 0, where is the point located? When the second number in an ordered pair is 0, where is the point located? How do you know? Lesson 7: Plotting Ordered Pairs on the Coordinate Plane 09

110 8 LESSON Problem Solving: Using the Coordinate Plane read Library Walk On a coordinate plane, a school is located at (, ). The library is located at (, ). If each unit on the plane represents one block, how far does Kyle walk to go to the library after school? plan Plot the points and on a coordinate plane. Use absolute value to find the distance between the points. solve Plot each point on a coordinate plane. What is the first number in each ordered pair? Because the first numbers of the ordered pairs are the same, you can add the distances of both points from the x-axis to find the distance between the points. To find the distance of each point from the x-axis, find the absolute value of the second number in the ordered pair. 5 5 So the distance from (, ) to the x-axis is The distance from (, ) to the x-axis is units. units. Add the absolute values to find the distance between the points. y x 5 How many blocks away from the school is the library? check Count the vertical units between the points. How many vertical units from (, ) to (, )? Kyle walks blocks to go to the library after school. 0 Domain : The Number System

111 read From Luke to Leah Each unit on the coordinate plane below represents one mile. The ordered pair for Luke s house is (, ). Leah s house is west of Luke s house. Luke and Leah live 9 miles apart. What ordered pair represents Leah s house? plan Plot the point for Leah s house units west of Luke s house. solve Plot a point at for Luke s house. Because Leah s house is west of Luke s house, count units to the Plot a point for Leah s house at. check to reach Leah s house. Find the distance between the points. Are the second coordinates in the ordered pairs the same? Because the second coordinates in the ordered pairs are the same, add the distances of both points from the y-axis to find the distance between the points. To find the distance of each point from the y-axis, find the absolute value of the first coordinates in the ordered pairs. Find each absolute value. Add the absolute values. 5 What is the distance between the points? Is Leah s house 9 miles from Luke s house? The ordered pair for Leah s house is y x Lesson 8: Problem Solving: Using the Coordinate Plane

112 Practice Use the -step problem-solving process to solve each problem.. READ On a coordinate plane map, the soccer stadium is located at (, 8). The bus stop is located at (6, 8). If each unit represents one block, how far will Renata have to walk from the bus stop to the stadium? PLAN solve CHECK. Each unit on a coordinate plane represents one kilometer. One end of a street starts at (0, 5). The street ends at (0, 5). How long is the street? Domain : The Number System

113 . An artist draws a square on a coordinate plane. The coordinates of two corners on the square are (, ) and (, ). The sides of the square are units long. What are the coordinates of the other two corners on the square?. Each unit on a coordinate plane represents one mile. The mall is at (6, 5). Colt s office is 8 miles due south of the mall. What are the coordinates of Colt s office on the grid? 5. On a coordinate plane, Akira s house is located at (, 9). Her dance studio is located at (, 7). If each unit represents one block, how far is it from Akira s house to her dance studio? Lesson 8: Problem Solving: Using the Coordinate Plane

114 DOMAIN Review Divide. Write the quotient in simplest form Add or subtract. Check your answer Find the product. Check that your answer is reasonable Find the quotient. Check your answer. 5. 8, , ,

115 Find the GCF of each pair of numbers.., 0. 8, 6. 5, Find the LCM of each pair of numbers.. 5, 8 5., 6., 0 Use the distributive property to express each sum with the GCF factored out Write an integer to represent each situation. Explain the meaning of 0 in the situation. Then describe the opposite situation, and write an integer to represent it feet below sea level 0. 5 F above 0. a gain of 8 points Find the opposite of the opposite of each number On a number line, is located to the right or to the left of? Write two comparisons to support your answer. 5

116 7. On a number line, is located to the right or to the left of? Write two comparisons to support your answer. Write the quadrant where each ordered pair is located. 8. (, 6) 9. (9, ) 0. (5, ). (, 8) Write the location of each point on the number line below using both a fraction and a decimal. A B C D 0. A. B. C 5. D Solve. 6. How wide is a rectangular table with a length of yard and an area of square yard? 7. Harry has an account balance of more than $8. Is his debt greater than or less than $8? 8. The surface temperature on Mars can range from 7. C to 07 C. Which temperature is lower? 9. Each unit on a coordinate plane represents one mile. One end of a road starts at (, ). The road ends at (6, ). How long is the road? 6

117 Decode Ordered the Pairs Step : Use Math Tool: Coordinate Planes. Plot points in different quadrants. Step : Code the ordered pairs. For each point, write clues for each coordinate in the ordered pair. C op y D rig O ht N O Pr ot T C ec O PY ted Clues may include operations, factors, multiples, opposites, and absolute values. For example: The first coordinate in this ordered pair is the opposite of the GCF of 8 and. The second coordinate is the quotient of The ordered pair is (, 5). Step : Exchange clue sets with a classmate, and decode the ordered pairs. Plot the points on another coordinate plane and compare solutions. Domain Review CC_MTH_G6_SE_D_Final.indd 7 7 /06/ : PM

118 Grade 5 Grade 6 Grade 7 Grade 5 OA Write and interpret numerical expressions. Analyze patterns and relationships. Grade 5 NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Grade 6 EE Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Grade 7 RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Grade 7 EE Use properties of operations to generate equivalent expressions. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Grade 7 G Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 8

119 Domain Expressions and Equations Lesson 9 Writing and Evaluating Numerical Expressions Lesson 0 Reading and Writing Algebraic Expressions.... Lesson Evaluating Algebraic Expressions Lesson Generating and Identifying Equivalent Expressions Lesson Writing and Solving Equations Lesson Writing and Solving Inequalities Lesson 5 Dependent and Independent Variables Lesson 6 Problem Solving: Using Equations Domain Review

120 9 LESSON Writing and Evaluating Numerical Expressions EXAMPLE A Write a numerical expression for using an exponent. Identify the base in the expression. The base or repeated factor is Identify the exponent. 8 Write the expression using an exponent. Using an exponent, can be written as 8. EXAMPLE B Expand the expression. 9 Identify the base and the exponent in the expression There are three 8s. So the exponent is. TRY In 9, the base is 9 and the exponent is. Use an exponent to write an expression for. Expand the expression. The base, 9, is repeated times The expression 9 can be written as Domain : Expressions and Equations

121 EXAMPLE C Use the order of operations to evaluate the expression. 6 ( ) Do the operations within parentheses first. 6 ( ) Multiply and divide from left to right The value of the expression 6 ( ) is 750. EXAMPLE D Evaluate the expression. 7 6 Evaluate the exponent. Evaluate the exponent Multiply and divide from left to right TRY 9 6 Evaluate the expression Add and subtract from left to right Lesson 9: Writing and Evaluating Numerical Expressions

122 Write each expression using an exponent. Practice HINT Count the number of times the factor is multiplied. Expand each expression.. 6 REMEMBER The exponent tells how many times to repeat the factor. Evaluate each expression Evaluate each expression ( ) ( ) Domain : Expressions and Equations

123 Choose the best answer.. Which shows the expression using an exponent? A. 5 B. 5 C. 5 D. 5. Which shows how to expand 7? A. 7 B. 7 C. D What is the value of the expression? 00 9 A. 6 B. 0 C. 7 A. 56 B. 7 C. 8 D. 6 Solve.. Which expression has a value of? A. B. ( ) C. 6 ( ) D. 6 ( ) D What is the value of the expression? 6. What is the value of the expression? 6 6 A. B. 6 C. D EXAMINE Is the value of the same as the value of 5? Explain. 8. ANALYZE Explain how to use the order of operations to evaluate the expression. Lesson 9: Writing and Evaluating Numerical Expressions

124 0 LESSON Reading and Writing Algebraic Expressions EXAMPLE A Write an algebraic expression to represent 8 less than four times a number. Choose a variable for the number. Write four times the number as one of the terms. Let n 5 the number four times n n or n The coefficient of n is. The coefficient tells you to multiply the number by. Write the expression. 8 less than four times a number n 8 Identify the operation. The phrase less than indicates subtraction. 8 less than means to subtract 8 from n. An algebraic expression to represent 8 less than four times number is n 8. TRY Write an algebraic expression to represent 6 more than twice a number. Domain : Expressions and Equations

125 EXAMPLE B Write a verbal expression for the algebraic expression x. Understand the first operation used in the expression. x means a number, x, divided by. Understand the second operation used in the expression. Use the operations to write a verbal expression. more than a number divided by A verbal expression to represent x is more than a number divided by or a number divided by plus. means add. TRY Write a verbal expression for the algebraic expression m 5. Lesson 0: Reading and Writing Algebraic Expressions 5

126 EXAMPLE C Write an algebraic expression for the verbal expression times the sum of and m. This expression has two terms. Write an algebraic expression for the sum of and m. Identify the operation. To find the sum, you need to add. m or m Write an algebraic expression for times m. Use parentheses to group the addition expression. ( m) An algebraic expression for times the sum of and m is ( m) or ( m). DISCUSS The second factor in (8 m) is the difference of two terms. What are the two terms? 6 Domain : Expressions and Equations

127 read Problem Solving Holly read twice as many books as Amber read this month. Amber read more books this month than last month. Write an algebraic expression for the number of books Holly read this month. plan Step : Write an expression for the number of books Amber read this month. Let b 5 number of books Amber read last month Step : Write an expression for the number of books Holly read this month. solve Step : This month, Amber read b books. Step : Holly read twice as many books as Amber this month. check This month, Holly read (b ) books. Does the expression match the verbal description in the problem? An algebraic expression for the number of books Holly read this month is. Holly read books this month. Lesson 0: Reading and Writing Algebraic Expressions 7

128 Practice Write an algebraic expression for each verbal expression. Let x 5 the number.. 5 more than a number. 0 decreased by a number HINT Decreased means to subtract.. the quotient of and a number. 8 times a number 5. subtract 9 from a number 6. multiply a number by 7 7. divide a number by 0 8. the sum of 5 and a number Write a verbal expression for each algebraic expression. 9. s 0. y REMEMBER The coefficient is a factor of the product.. 6 k. p 5. 8 m. x Write an algebraic expression for each verbal expression. Let n 5 the number. 5. the product of and the sum of 6 and a number 6. 5 more than the quotient of a number and 7. the sum of and the product of 8 and a number 8 Domain : Expressions and Equations

129 Choose the best answer. 8. Which algebraic expression represents the verbal expression subtract from x? A. x B. x C. x D. x 9. Which algebraic expression represents the verbal expression the sum of 8 and the quotient of n and? A. 8 (n ) B. 8 (n ) C. 8(n ) D. 8(n ) 0. Which verbal expression represents the algebraic expression 0 b? A. the sum of 0 and b B. 0 multiplied by b C. subtract 0 from b D. divide 0 by b Solve.. Jonathan had some DVDs. He gave 6 of them away. Write an algebraic expression for the number of DVDs Jonathan has left.. Which algebraic expression does not show a product? A. y B. 8y C. 5(6 y) D. (y ). A clown made times as many balloon hats as balloon dogs. He made more balloon dogs than balloon swords. Write an algebraic expression for the number of balloon hats the clown made.. CONCLUDE In the algebraic expression 7 y, what is the coefficient of y? Explain. 5. WRITE Describe a situation that can be written as an algebraic expression. Explain what the variable in your expression represents. Lesson 0: Reading and Writing Algebraic Expressions 9

130 LESSON Evaluating Algebraic Expressions EXAMPLE A Evaluate 6 8x for x 5 9. Substitute 9 for x in the expression. TRY 6 8x 6 8(9) Evaluate the expression 6 8x for x 5. Use the order of operations to evaluate the expression. 6 8(9) Divide (9) Multiply. 8(9) Add. 79 When x 5 9, 6 8x is equal to Domain : Expressions and Equations

131 EXAMPLE B Evaluate b c 5 for b 5 and c 5. Substitute for b in the expression. b c 5 c 5 Substitute for c in the expression. c 5 Use the order of operations to evaluate the expression. () 5 Multiply. 5 () 5 Multiply. () 5 5 Subtract. 5 5 Subtract. 7 The value of the expression b c 5 when b 5 and c 5 is 7. () 5 TRY Evaluate the expression b c 5 for b and c 5.5. Lesson : Evaluating Algebraic Expressions

132 EXAMPLE C What is the area of the triangular scarf? 5 cm 6 cm Write an expression to represent the area. bh In the expression, b represents the base and h represents the height of the triangle. Substitute 7 cm for b and 6 cm for h in the expression. (7 cm)(6 cm) 7 cm From the diagram, identify the values for the base and the height. b 5 7 cm h 5 6 cm Evaluate the numerical expression for the area. DISCUSS What is the area of a triangle with a height of 6 inches and a base of inches? (7 cm)(6 cm) 97 cm 86 cm The area of the scarf is 86 cm. Domain : Expressions and Equations

133 Problem Solving read Lauren s bedroom is shaped like a rectangle. She wants to carpet the entire floor. A diagram of the floor is shown below. ft 8 ft How many square feet of carpeting does she need? plan You need to find the area of the rectangle. The area is represented by the expression lw, where l 5 length and w 5 width. From the diagram, identify the values for the length and the width. l 5 8 ft w 5 ft Substitute the measures of the length and width in the expression. solve Substitute 8 ft for l and lw 8 ft ft ft for w in the expression. 8 ft ft 5 ft The area of Lauren s bedroom floor is ft. check ft 8 ft 5 ft Lauren needs square feet of carpeting for her bedroom floor. Lesson : Evaluating Algebraic Expressions

134 Practice Evaluate the algebraic expression for the given value of the variable.. d for d n for n 5 6. x 6 for x 5 7 HINT Evaluate the exponent first.. 8k 9 for k y for y c 9 for c a 7 6 for a w.5 for w Evaluate each algebraic expression for x 5.5 and y x 0y. 6y x REMEMBER Use the order of operations to evaluate after substitution.. 9 7x y. y x x y 5. y 0 5 x b for b 5.7 Evaluate the algebraic expression for the given values of the variables. 6. 9a 6b for a 5 8 and b x 5y for x 5.9 and y w 9x for w 5 and x p q for p and q 5 Domain : Expressions and Equations

135 Choose the best answer. 0. What is the value of the expression x y when x 5 and y 5? A. 9 B. 0 C. 8 D. Solve.. What is the area of a square with a side length of centimeters? Use the expression s, where s is the side length of the square. A. square centimeters B. 8 square centimeters C. square centimeters D., square centimeters. This expression,.8c, shows how to convert from degrees Celsius to degrees Fahrenheit, where C represents the temperature in degrees Celsius. What is the temperature in degrees Fahrenheit when the temperature is 5 C?. The expression 5 9 (F ) shows how to convert from degrees Fahrenheit to degrees Celsius, where F represents the temperature in degrees Fahrenheit. What is the temperature in degrees Celsius when the temperature is F?. EVALUATE The perimeter of a rectangle is represented by the algebraic expression l w, where l represents the length and w represents the width of the rectangle. Find the perimeter of a rectangle that has a length of feet and a width of 6 feet. 5. SUMMARIZE Explain how to evaluate an algebraic expression with two variables. Lesson : Evaluating Algebraic Expressions 5

136 LESSON Generating and Identifying Equivalent Expressions EXAMPLE A Write an expression equivalent to y y y y y. TRY Identify the variable in the expression. The variable is y. Add the terms and write an equivalent expression. y y y y y 5y The expression y y y y y is equivalent to the expression 5y. EXAMPLE B Write an expression equivalent to 7( x). Use the distributive property. 7( x) (7 ) (7 x) Write an expression equivalent to the expression below. m m m Identify the number of terms in the expression. There are 5 terms in the expression. The coefficient of each term, y, is. Identify the operation. The expression involves addition. Use the order of operations to simplify the expression. (7 ) (7 x) 8 7x The expression 7( x) is equivalent to the expression 8 7x. 6 Domain : Expressions and Equations

137 EXAMPLE C Write an expression equivalent to 0a b. Identify the terms. The first term is 0a. The second term is b. Find the greatest common factor (GCF) of the terms. The GCF of 0a and b is 8. DISCUSS Rewrite the expression, and then factor out the 8. 0a b (8 5a) (8 b) 8(5a b) An expression equivalent to 0a b is 8(5a b). EXAMPLE D Are these equivalent expressions? Use the distributive property to simplify (x 8). (x 8) ( x) ( 8) x (x 8) and x 8 Are the expressions (y 5) and y 5 equivalent? Explain. Write each term using the GCF as a factor. 0a 5 8 5a b 5 8 b Add to simplify the second expression. x 8 x Compare the expressions. The expressions x and x are the same. The expressions (x 8) and x 8 6 are equivalent expressions. Lesson : Generating and Identifying Equivalent Expressions 7

138 Practice Write an equivalent expression for each expression.. x x. c c c c. p p HINT Identify the operation.. y y y y 5. n n n n n 6. a a a a a Use the distributive property to write an equivalent expression for each expression. 7. (x ) 8. 6( a) 9. (c d) 0. 7(5x ). 5(m n). 8 6y 5 8 Use the GCF to write an equivalent expression for each expression.. y. 7 x 5. 0c 5 6. r s 7. c d 8. x 0y 8 Domain : Expressions and Equations

139 Are the two expressions equivalent? Write yes or no. 9. 7(x y) and x y 0. c 9d and (c d). 5(6a b) and 0b 0a. y y 5w and (y w) Choose the best answer.. Which expression is equivalent to 6( x)? A. 0 x B. 6x C. 0 6x D. x Solve.. Which expression is equivalent to (7b c)? A. b 6c B. b c C. 0b 5c D. 0b c 5. IDENTIFY Write two equivalent expressions you could use to show the perimeter of this triangle. t t t 6. ANALYZE One bookshelf is.a b inches long. Another shelf is 5 times as long. Write an expression you could use to show the length of the longer shelf. Lesson : Generating and Identifying Equivalent Expressions 9

140 LESSON Writing and Solving Equations EXAMPLE A Is 8 a solution of the equation x 5 7? Understand what it means for a value to be a solution of an equation. A value is a solution of an equation if substituting the value for the variable in the equation makes the equation true. Evaluate Substitute 8 for x in the equation. x Write? over the equal sign. Compare the expressions on both sides of the equation. DISCUSS Explain why 5 is not a solution of the equation x 5 7. In a true equation, the expressions on both sides of the equal sign have the same value. Since and 7 5 7, the equation is a true equation. Yes, 8 is a solution of the equation x Domain : Expressions and Equations

141 EXAMPLE B Are any of the following numbers a solution of the equation 5 5 y 6? 9,,, 5 Substitute 9 for y in the equation. TRY 5 5 y Substitute 9 for y. 5 0 Simplify the right side of the equation. 5 9 is not a solution. Substitute for y in the equation. 5 5 y Substitute for y Simplify the right side of the equation. 5 5 is not a solution. Are any of the following numbers a solution of the equation 5 5 y 6? 9,, Substitute for y in the equation. 5 5 y Substitute for y Simplify the right side of the equation. 5 5 is not a solution. Substitute 5 for y in the equation. 5 5 y Substitute 5 for y Simplify the right side of the equation The equation is true. 5 is a solution. The solution to the equation is 5. Lesson : Writing and Solving Equations

142 EXAMPLE C Logan has 8 model cars. His grandparents give him some more model cars. Now he has model cars. How many model cars did Logan s grandparents give him? Choose a variable. Write an equation. Let c 5 the number of model cars Logan s grandparents gave him. 8 c 5 To solve an addition equation, use the inverse operation. Addition and subtraction are inverse operations because they undo each other. You can find the value of c by subtracting 8 from both sides. This will get c by itself on one side and the value of c on the other side. 8 c c 5 8 Subtract 8 from both sides of the equation. c 5 Check the equation. Substitute for c in the equation. 8 c The equation is true. The solution is. Logan s grandparents gave him model cars. Simplify both sides of the equation. MODEL Write an equation for the following problem. Deanne worked 5 hours this week. That was hours less than she worked last week. How many hours did she work last week? Domain : Expressions and Equations

143 READ Problem Solving Keke did 7 sit-ups at the end of the month. This was 6 times as many sit-ups as she was able to do at the beginning of the month. How many sit-ups was Keke able to do at the beginning of the month? PLAN Choose a variable. Write an equation that expresses the problem situation. Use the inverse operation to solve the equation. SOLVE Let s 5 6s 5 What operation is the inverse of multiplication? Divide both sides of the equation by CHECK Substitute 6s 5 5 6s s 5 for s in the equation. to solve. Simplify. 5 The equation is true. The solution is. Keke was able to do sit-ups at the beginning of the month. Lesson : Writing and Solving Equations

144 Practice Is the given value a solution of the equation? Write yes or no.. 6. k 5 6.; k y 5 5; y 0 8 HINT Does the value make a true equation?. x ; x 0.. w 5 5 5; w Write which number is a solution of the equation. 5. 8x 5 5. Try:.,., b 5 6 Try: 5 6,, 6 7. y 5 7 Try: 6, 6, 6 REMEMBER Test each value in the equation. 8. n Try: 6.9, 7.5, 7.65 Use inverse operations to solve. Check your answer k b 9. 6y 5 8. m x s x 5 6. a n Domain : Expressions and Equations

145 Solve. Check your answer w 9. m y x 8. s c 5 8 Choose the best answer.. Which of the following is a solution of the equation? Solve d A. d 5 6 B. d 5 0 C. d 5 56 D. d Tanya is 6 inches tall. She is 7 inches taller than May. Write and solve an equation to find May s height. 8. Write a real-world problem that can be solved by the equation y Then solve your problem. 5. Which of the following is a solution of the equation? n A. n 5 B. n 5 C. n 5 68 D. n Oliver earns $9 per hour. Write and solve an equation to find how many hours he must work to earn $5. 9. Write a real-world problem that can be solved by the equation 5x 5 5. Then solve your problem. 0. ANALYZE How does using an inverse operation help you solve an equation?. EXPLAIN When solving an equation, why do you apply the inverse of the given operation to both sides? Lesson : Writing and Solving Equations 5

146 LESSON Writing and Solving Inequalities EXAMPLE A Emma needs to score at least 8 on her next social studies test to earn an A in the class. Write an inequality to represent the situation. Choose a variable. Choose the inequality symbol. TRY Let s 5 the score Emma needs to get on her test. Write the inequality. s 8 Check that the inequality expresses the situation. The solution set is all numbers greater than or equal to 8. Inequality Symbol Verbal Clues, Less than, fewer than # Less than or equal to, at most. Greater than, more than Greater than or equal to, at least Emma needs at least 8. Use the inequality symbol. This means that Emma must score 8 or any score higher than 8 on the test. She can score 8, 8, 85, and so on, up to and including 00, to earn an A in the class. The inequality s 8 expresses the situation. s 8 Todd caught a fish that weighed less than pounds. Write an inequality to represent the situation. 6 Domain : Expressions and Equations

147 EXAMPLE B Are any of the following numbers in the solution set of the inequality x 78 # 5? 8, 7, Substitute 8 for x in the inequality. x 78 # ? 5 Substitute 8 for x. 06 # 5 Simplify the left side of the inequality. The inequality is true. 8 is a solution. Substitute for x in the inequality. x 78 # 5 78? 5 Substitute for x. 9 # 5 Simplify the left side of the inequality. The inequality is not true since is not a solution. 8 and 7 are in the solution set of the inequality x 78 # 5. Substitute 7 for x in the inequality. x 78 # ? 5 Substitute 7 for x. 5 # 5 Simplify the left side of the inequality. The inequality is true since is a solution. DISCUSS How does the result of substituting 7 for x in the inequality above help you find other solutions? What are three other solutions? Lesson : Writing and Solving Inequalities 7

148 EXAMPLE C Solve the inequality. Graph the solution. n. Use the inverse operation to solve the inequality. n. n. Divide both sides of the inequality by. n. Simplify both sides of the inequality. Graph the solution on a number line. Draw a number line. An open circle around a number means that the value is not in the solution set. A closed circle around a number means that the value is in the solution set. Since n., is not a solution. Draw an open circle around. Then shade to the right of. Check the solution set. The solution set is all numbers greater than. So numbers such as.8,., 5, 6, and so on are all in the solution set of the inequality. Substitute.8 for n in the inequality. n. (.8).? The solution of the inequality is n.. The graph of the solution is shown above. DISCUSS How would the graph of the solution to n be different than the graph of the solution to n.? Explain. 8 Domain : Expressions and Equations

149 read Problem Solving Write and solve an inequality that describes the problem below. Then graph the solution. Yuan walked more miles than Dennis walked. Yuan walked at most 5 miles. How many miles did Dennis walk? plan Choose a variable. Write an inequality. Solve the inequality. Then graph the solution on a number line. solve Let m 5 m is the number of miles Yuan walked. Yuan walked at most 5 miles. This means the number of miles he walked is less than or equal to. Write the inequality: m Solve the inequality. What operation is the inverse of addition? Subtract m m Graph the solution on the number line. from both sides of the inequality to solve Why does the number line for the solution set start at 0? check Decide if the solution set makes sense. Dennis walked at most. This means he walked miles or less. Yuan walked at most 5 miles, and Dennis walked fewer miles than Yuan. Dennis walked at most miles. Lesson : Writing and Solving Inequalities 9

150 Practice Write an inequality for each situation. Use x for the variable.. Noah has fewer than 8 tennis balls. HINT Use verbal clues to choose the inequality symbol.. Mariposa read at least 5 pages of the book.. Jaxon rode no more than 8 miles.. Kira spent more than $85. Decide whether the given value is a solution of the inequality. Write yes or no. 5. y 8, 8. Try: y 0 7 b. 0 Try: b n. 05 Try: n w Try: w 0 7 Write an inequality for each graph. Use x for the variable x 9 Try: x s, 7 Try: s REMEMBER An open circle means that the point is not in the solution set Domain : Expressions and Equations

151 Solve each inequality. Graph the solution. Use Math Tool: Blank Number Lines. 5. w m y x, s c, 8. b 6.. Solve. h 5 5. Kareem spent $8 less than Marco spent at the mall. Kareem spent more than $5. How much did Marco spend at the mall?. Jamie has times as many pencils as pens. She has at least 8 pencils. How many pens does Jamie have? 5. DEDUCE In which direction on the number line do you draw the solution graph of an inequality that has a. or a symbol? In which direction do you draw the solution graph of an inequality that has a, or a symbol? 6. JUSTIFY How far to the left can you extend the graphs when graphing the solution sets for questions and? Explain. Lesson : Writing and Solving Inequalities 5

152 5 LESSON Dependent and Independent Variables An equation such as y 5 x shows how the variables x and y are related. The value of y depends on the value of x, so x is the independent variable and y is the dependent variable. The equation shows that whatever the value of x is, the value of y is less than that value. You can use a table of values to show this relationship. Each line of the table can also be expressed as an ordered pair (x, y). So the ordered pairs for this table are (5, ), (, 0), and (5, ). x y TRY EXAMPLE A The table at the right shows the relationship between the weight in pounds on Earth, E, and the weight on Mars, M. The equation M 5 0.8E also models the relationship. Which is the independent variable? Which is the dependent variable? Use words to express the equation. Equation: M 5 0.8E Verbal expression of the equation: The weight on Mars equals 0.8 times the weight on Earth. The table shows the relationship between the weight in pounds on the moon, m, and the weight on Earth, e. The equation e 5 6.0m also models the relationship. Which is the independent variable? Which is the dependent variable? Why? m e E M Use the equation to fill in the table. For each weight on Earth, multiply the weight by 0.8 to find the weight on Mars. So to fill in the values in the table, select the weight on Earth, then use the equation to find the weight on Mars. Decide which is the independent variable and which is the dependent variable. You are choosing an Earth weight and applying the rule to find a Mars weight. For the equation M 5 0.8E, E is the independent variable; whatever value you choose for E determines the value of M, the dependent variable. 5 Domain : Expressions and Equations

153 EXAMPLE B The table shows the relationship between the number of pizzas, n, and the total cost in dollars of the pizzas, c. Write an equation to model the relationship. Which is the independent variable? Which is the dependent variable? Look for a relationship between the variables in the table. The first row in the table shows that for each pizza bought, the cost is $. Decide which is the independent variable and which is the dependent variable. n c Use the relationship to write an equation. The total cost in dollars, c, is times the number of pizzas, n. So c 5 n models the relationship. The total cost depends on the number of pizzas. The independent variable is the number of pizzas, n. The dependent variable is the total cost, c. The equation c 5 n models the relationship. The independent variable is n, and the dependent variable is c. DISCUSS How can you verify that the equation works for every value in the table? Lesson 5: Dependent and Independent Variables 5

154 EXAMPLE C The graph shows the number of dollars, d, Sariah earns for each hour, h, she works. List the ordered pairs for the points shown on the graph. Which is the independent variable? Which is the dependent variable? Dollars 0 d Hours h DISCUSS List the ordered pair for each point shown on the graph. The first number in the ordered pair is the number on the horizontal axis, h, that lines up with the point. The second number in the ordered pair is the number on the vertical axis, d, that lines up with the point. The ordered pairs are (, 0), (, 0), (, 60), (, 80), and (5, 00). How can you use the ordered pairs to write an equation that models the relationship shown by the graph? What is the equation? Decide which is the independent variable and which is the dependent variable. In an ordered pair, the first number represents the independent variable. The second number represents the dependent variable. The ordered pairs are in the form (h, d). The independent variable is h, and the dependent variable is d. This makes sense because the number of dollars Sariah earns depends on the number of hours she works. The ordered pairs are (, 0), (, 0), (, 60), (, 80), and (5, 00). The independent variable is h, and the dependent variable is d. 5 Domain : Expressions and Equations

155 read Problem Solving The graph shows Gavin s age, g, and Celia s age, c. Identify the independent variable and the dependent variable. What equation expresses the relationship of Gavin s age and Celia s age? plan List the ordered pairs. Identify the independent variable and the dependent variable. Relate the value of each c variable to the value of the corresponding g variable. Write the equation for the relationship between Gavin s and Celia s ages. solve When you read an ordered pair on the graph, whose age do you read first? Why? The ordered pairs are in the form (, ) and are (, ), (, 5), (, ), (, ), (, ). The independent variable is. The dependent variable is. How is each c value related to the corresponding g value? c is greater than g. The equation for the relationship between Gavin s age and Celia s age is. Celia s Age c Gavin s Age g check Use the graph. How old is Celia when Gavin is 5? Use the equation c 5 g and substitute 5 for g. Does the equation c 5 g match the information on the graph? The independent variable is. The dependent variable is. The equation is. Lesson 5: Dependent and Independent Variables 55

156 Practice. The table shows the relationship between the age of a plant in months, m, and the height of the plant, h. Which is the independent variable? Which is the dependent variable? HINT The height of the plant depends on its age. m h 8 6. The table shows the relationship between the number of quarts, q, and the number of cups, c. Write an equation to model the relationship. Which is the independent variable? Which is the dependent variable? REMEMBER For an equation in the form y 5 x or y 5 x, y is the dependent variable.. The table shows the relationship between Aubrey s age, A, and Malin s age, M. Which is the independent variable? Which is the dependent variable? q c A M Domain : Expressions and Equations

157 . Evan makes wooden puzzle boxes. The graph shows the number of dollars, d, Evan earns for each puzzle box, b, he sells. List the ordered pairs shown on the graph. Which is the independent variable? Which is the dependent variable? Write an equation that expresses the relationship between the number of puzzle boxes sold and the amount Evan earns. d Dollars Earned Number Sold b 5. The graph shows the distance, d, Mr. Benson traveled each hour, h, on a trip. List the ordered pairs shown on the graph. Which is the independent variable? Which is the dependent variable? Write an equation that expresses the relationship between the number of miles traveled and the number of hours of travel time. Distance (in miles) d Time (in hours) h 6. SUMMARIZE Which axis represents the independent variable on a graph? Which axis represents the dependent variable on a graph? Explain. 7. ANALYZE How are an equation, a table of values, a set of ordered pairs, and a graph of the equation related? Does the independent variable change in the representations? Does the dependent variable change in the representations? Lesson 5: Dependent and Independent Variables 57

158 6 LESSON Problem Solving: Using Equations READ Avery wants to rent a mountain bike. It costs $5 per hour to rent the bike. The equation c 5 5h represents the cost, c, of renting the mountain bike for h hours. Analyze the variables. Use the equation to create a table of values. Then graph the values. Use the graph to determine the cost if Avery rents the mountain bike for 7 hours. PLAN Analyze the variables. Decide which is the dependent variable and which is the independent variable. Make a table of values for the equation c 5 5h. Include at least five ordered pairs in the table. SOLVE Mountain Bike Rental The dependent variable is c, and the independent variable is h. Use the equation. Complete the table. h c 5 5h c (h, c) c 5 5h 5 5() (, 5) c 5 5h 5 5() 5 0 (, 0) c 5 5h 5 5() 5 (, ) c 5 5h 5 5( ) 5 (, ) 5 c 5 5h 5 (5) 5 (, ) Plot the ordered pairs on the coordinate plane. Then connect the points with a straight line. CHECK How do you know that your graph is accurate? The graph of c 5 5h is shown on the right. The cost for Avery to rent the mountain bike for 7 hours is c h 58 Domain : Expressions and Equations

159 read Carnival Time A ticket for each ride at the carnival costs $. Write an equation to represent the cost, c, of going to the carnival and riding r rides. Analyze the variables. Graph the equation. Use the graph to determine how many rides you could go on if you have $0. plan Decide which is the dependent variable and which is the independent variable. Make a table of values. Use the table of values to write the equation. Then use the table of values to graph the equation. solve The total cost, c, depends on the number of rides, r. The dependent variable is. The independent variable is. Make a table of values. r c 5 Write an equation to represent this situation. Graph the equation. Then connect the points with a straight line. check How do you know that your graph is accurate? An equation to represent the situation is is shown above. If you have $0, you can go on c The graph of the equation rides. r Lesson 6: Problem Solving: Using Equations 59

160 Practice Use the -step problem-solving process to solve each problem.. READ Each box at a market contains 6 oranges. Make a table of values to show the relationship between the number of boxes, b, and the number of oranges, n, that are at the market. Analyze the variables. Write an equation to represent the relationship. Then graph the relationship. Use Math Tool: Grids to draw the graph. How many boxes of oranges are in the market if there are 8 oranges? PLAN solve CHECK. Diego is years older than May. Write an equation to show the relationship between Diego s age, D, and May s age, M. Make a table of values. Then graph the relationship. Use Math Tool: Grids to draw the graph. How old will Diego be when May is 8 years old? 60 Domain : Expressions and Equations

161 . Akira earns $8 each day she cares for her neighbor s dog. Write an equation to represent the relationship between the amount of money Akira earns, m, and the number of days, d, that she cares for the dog. Then graph the relationship. Use Math Tool: Grids to draw the graph. How much will Akira earn if she cares for the dog for 5 days?. Hugo rides his bike at an average rate of 5 miles per hour. Write an equation to represent the relationship between the distance, d, that Hugo rides and the number of hours, h, that he rides. Then graph the relationship. Use Math Tool: Grids to draw the graph. How far will Hugo ride if he rides his bike for hours? 5. Every time Luke puts a dime into the parking meter, he gets 0 minutes of parking time. The equation p 5 0d represents the time, p, he can park after he puts d dimes into the meter. Analyze the variables. Use the equation to create a table of values. Then graph the values. Use Math Tool: Grids to draw the graph. If he puts 5 dimes into the meter, can Luke park for more than or for less than hour? Lesson 6: Problem Solving: Using Equations 6

162 DOMAIN Review Evaluate each expression ( ) Write an algebraic expression for each verbal expression. Let n 5 the number more than a number 8. 6 decreased by a number 9. the quotient of 5 and a number 0. subtract from a number Write a verbal expression for each algebraic expression.. 6m. w. 6 y. x 8 Evaluate the algebraic expression for the given value of the variable y for y c for c a 8 for a w.9 for w 5. 6

163 Choose the best answer. 9. Which expression is equivalent to d d d d d? A. 5 d B. 5d C. d 5 D. d 5 0. Which expression is equivalent to w w w? A. w B. w C. w D. w Write an equivalent expression for each expression.. s s s. p p p p p. x x. n n n n Use the distributive property to write an equivalent expression for each expression. 5. ( y 5) 6. 5( k) 7. 7(n p) 8. 0(6w x) Write which of the numbers, if any, is a solution of the equation. 9. 5x 5.5 Try: 0.7, 0.8, y 5 5 Try:, 8, 8 Write which numbers, if any, are solutions of the inequality.. n 9, 5 n, 5, 6, 7, 9. 6x. 8 x,,, 5. y 7 # 9 y, 0, 6, 8. r $ 50 r 0, 5, 7, 0 6

164 For questions 5 8, write an inequality for each situation. 5. Kendall has at least 0 sports cards. 6. Ryder hiked no more than 8 miles. 7. Adela saved more than $6. 8. Kazuo lost fewer than 9 points. 9. Lauren is 6 years older than Erik. Write an equation to show the relationship between Lauren s age, L, and Erik s age, E. Make a table of values. Then graph the relationship. Use Math Tool: Grids to draw the graph. How old will Lauren be when Erik is 8 years old? Solve. 0. Mulan earns $ per hour. Write and solve an equation to find how many hours she worked if she earned $.. Dennis spent $5 less on lunch than he spent on dinner. Dennis spent at most $ on dinner. Write and solve an inequality to find how much Dennis spent on lunch.. CREATE Write a word problem that can be solved by the equation 08 r 5.. Rewrite Refer to question and suppose that Dennis spends at least $ on dinner. Modify the equation and your solution to fit the new problem. 6

165 In the balance-scale puzzles shown here, the value of x will not be a number. Instead, it will be one or more shapes. Example: The two scales below are both in balance. What shape or shapes does x represent? x In this puzzle, x diamond are balanced by 6 squares and diamond is balanced by squares. To find the shape(s) represented by x, you can remove the diamond from the left side of the first scale. To keep the first scale in balance, you also need to remove the equivalent of diamond from the right side of the scale. The second scale shows that the equivalent of diamond is squares. When you remove diamond from the left and squares from the right, then x will be by itself on the left side of the first scale. It will be balanced by squares. So in this puzzle, x represents squares. Now try the puzzle below on your own, with a partner, or with a small group. (Beware! This puzzle is not as easy as the one shown above.) The three scales below are all in balance. What shape or shapes does x represent? Explain your solution. Scale : Some Weighty Questions Scale : x Scale : Domain Review 65

166 Grade 6 Grade 7 Grade 8 Grade 5 NBT Perform operations with multi-digit whole numbers and with decimals to hundredths. Grade 5 NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Grade 5 MD Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Grade 5 G Graph points on the coordinate plane to solve real-world and mathematical problems. Classify two-dimensional figures into categories based on their properties. Grade 6 G Solve real-world and mathematical problems involving area, surface area, and volume. Grade 7 G Draw, construct, and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 66

167 Domain Geometry Lesson 7 Finding the Area of Triangles and Quadrilaterals Lesson 8 Finding the Volume of Rectangular Prisms Lesson 9 Drawing Polygons on the Coordinate Plane Lesson 0 Representing Three-Dimensional Figures Using Nets Lesson Using Nets to Find Surface Area Domain Review

168 7 LESSON Finding the Area of Triangles and Quadrilaterals UNDERSTAND Area is the number of square units (units ) in the interior of a figure. Find the area of the quadrilateral. The quadrilateral is a parallelogram. Count the number of whole squares inside. Count the number of partial squares inside. There are 0 whole squares. Add the number of whole and partial squares to find the area. 0 5 The area of the parallelogram is square units. Relate the area of the parallelogram to the area of a rectangle having the same length and width as the base and height of the parallelogram. Draw the height of the parallelogram from one corner of the parallelogram to its base. This forms a right triangle. Notice that you can make a right triangle with the same area by drawing the height outside the triangle and extending the base. Both triangles have a height of units and a base of units. The area of the rectangle is 6, or, square units. The area of the parallelogram is square units. Each partial square is of a whole square. There are halves. This is the same as whole squares. The models show that the area of a parallelogram with base b and height h is the same as the area of a rectangle with base b and height h. The area of the parallelogram is square units Domain : Geometry

169 Connect Find the area of the parallelogram. cm 6 cm Write the formula for the area of a parallelogram. A 5 bh Find the area. A 5 bh A 5 6 cm cm A 5 cm Substitute 6 for b and for h in the formula. Simplify. The area of the parallelogram is cm. Identify the base and the height of the parallelogram. The base is 6 cm. The height forms a right angle with the base. The height is cm. DISCUSS Why do you include cm in the answer for the area of the parallelogram? Lesson 7: Finding the Area of Triangles and Quadrilaterals 69

170 EXAMPLE A Find the area of the right triangle. 6 8 Write the formula for the area of a triangle. A 5 bh Find the area. A 5 bh A A 5 units Substitute 8 for b and 6 for h in the formula. Simplify. Identify the base and the height of the triangle. The base is 8 units. The height is 6 units. Find the area of the triangle by decomposing a rectangle. Draw a horizontal line from the top of the triangle and a vertical line from the end of the base to create a rectangle. DISCUSS Why is it true that the area of the triangle that resulted from decomposing the rectangle is half the area of the rectangle? 6 The area of the rectangle is 6 8, or 8, units. 8 The area of the triangle is half the area of the rectangle: 8 units or units. The area of the triangle is units. 70 Domain : Geometry

171 EXAMPLE B Find the area of the triangle. 7 Identify the base and the height of the triangle. Write the formula for the area of a triangle to find the area. The base is 7, or 0, units. The height forms a right angle with the base. The height is units. Find the area of the triangle by decomposing a rectangle. Create a rectangle by using the base of the triangle as its length and the height of the triangle as its width. The area of the rectangle is 0, or 0, units. The area of the triangle to the left of the height line is half the area of the rectangle with length and width : or 6 units. The area of the triangle to the right of the height line is half the area of the rectangle with length 7 and width : 7 or units. So the area of the decomposed triangle is 6, or 0, units. The area of the triangle is 0 units. A 5 bh A 5 0 A 5 0 unit Substitute 0 for b and for h in the formula. Simplify. 0 7 MODEL Find the area of the triangle by decomposing a rectangle Lesson 7: Finding the Area of Triangles and Quadrilaterals 7

172 EXAMPLE C Find the area of the trapezoid. 5 yd yd 8 yd 6 yd Find familiar figures that make up the trapezoid. The trapezoid is composed of two triangles and a rectangle. Label the left triangle I, the middle rectangle II, and the right triangle III. 5 I II III 8 Use the rectangle area formula to find the area of figure II. A 5 lw A 5 8 yd 5 yd A 5 0 yd 6 Substitute 8 yd for l and 5 yd for w in the formula. Simplify. Use the triangle area formula to find the area of figure I. A 5 bh A 5 yd 5 yd A 5 0 yd Use the triangle area formula to find the area of figure III. A 5 bh A 5 6 yd 5 yd Substitute yd for b and 5 yd for h in the formula. Simplify. Substitute 6 yd for b and 5 yd for h in the formula. A 5 5 yd Simplify. 5 Add to find the total area. A 5 area of figure I area of figure II area of figure III A 5 0 yd 0 yd 5 yd A 5 65 yd Simplify. The area of the trapezoid is 65 yd. TRY Suppose the height of the trapezoid above is changed to yards. What is the area of the new trapezoid? 7 Domain : Geometry

173 read Problem Solving Erik drew the diagram to the right of his irregularly shaped garden to find its total area. What is the area of Erik s garden? plan Find familiar figures that make up the garden. Find the area of each figure. Add the areas to find the total area. ft ft 0 ft solve The top part of the garden diagram is shaped like a parallelogram. The base is ft, and the height is ft. The bottom part of the garden diagram is shaped like a triangle. The base is ft, and the height is ft. Use the formula for the area of a A 5 to find the area of the top part. A 5 Substitute values for b and h. A 5 ft Simplify. Use the formula for the area of a A 5 to find the area of the bottom part. A 5 Substitute values for b and h. A 5 ft Simplify. Find the total area of the garden. 5 check Look back at the garden diagram. Estimate the total area of the garden. Does your answer seem reasonable? Explain. The area of Erik s garden is. Lesson 7: Finding the Area of Triangles and Quadrilaterals 7

174 Practice Find the area of each figure.. parallelogram. rectangle. triangle. square b 5 5 cm l 5 ft b 5 m s 5.6 mi h 5 8 cm w 5 ft h 5 9 m Find the area of each figure in. 6 in m 7 in. 9.. km. km 6 in. 9. m 7. HINT 8 yd. km.8 km 0. A square is a special parallelogram with sides length s. yd REMEMBER The height forms a right angle with the base. 9 ft in. ft 5 ft 8 ft 7 Domain : Geometry

175 Solve.. A rectangular blanket has an area of square feet. The blanket is 6 feet long. How wide is the blanket?. A triangular sail has a base of feet and a height of 5 feet. What is the area of the sail?. The area of a square window is 8 square inches. How long is each side of the window? 5. A square poster board has sides that are 0 inches long. When the triangular flaps at the sides are opened, the poster board takes the shape of a trapezoid. The base of each triangle is inches. What is the area of the trapezoid poster board? 0 in. in.. Andrea s garden is shaped like a parallelogram with a base length of 6.5 meters and a height of.8 meters. She wants to increase the area by doubling the base length and increasing the height by.5 meters. By how much will the changes increase the area of the garden? Explain. 6. A rectangular table is. meters long and.5 meters wide. The table has triangular flaps that can be raised on opposite sides. The base of each flap is. meters. What is the area of the table when both flaps are raised?. m.5 m. m 7. CONSTRUCT A banner is shaped like a parallelogram. The banner has a base length of.5 meters and an area of.55 square meters. Explain how to find the height of the banner. 8. WRITE MATH Explain how the area of a parallelogram is related to the area of a triangle when both figures have the same base and height. Lesson 7: Finding the Area of Triangles and Quadrilaterals 75

176 8 LESSON Finding the Volume of Rectangular Prisms UNDERSTAND Volume is the number of cubic units needed to fill the space in a three-dimensional figure. Volume is measured in cubic units (units ). Find the volume of the rectangular prism. in. in. in. Count the number of cubes in the top layer of the prism. There are 0 cubes in the top layer. Find the total number of cubes that fill the prism. There are layers of cubes that fill the prism. Each layer has 0 cubes. There are 0, or 0, unit cubes that fill the prism. Find the volume of one unit cube. Each unit cube is inch by inch by inch. The volume of one unit cube is inch inch inch, or 8 cubic inch. Find the volume of the rectangular prism in cubic inches. Multiply the number of cubic inches in one unit cube by the number of unit cubes that fill the prism The volume of the rectangular prism is cubic inches. 76 Domain : Geometry

177 Connect Find the volume of the rectangular prism. in. in. in. Write the formula for the volume of a rectangular prism. V 5 lwh, where l 5 length, w 5 width, and h 5 height Find the volume. Identify the length, width, and the height of the rectangular prism. The length is inches. The width is inch. The height is inches. V 5 lwh V 5 in. in. in. Substitute for l, for w, and for h in the formula. 5 5 in. _ in. in. Write improper fractions for the mixed numbers. 5 5 in. Multiply. 5 in. Simplify. The volume of the rectangular prism is in. TRY What is the volume of a rectangular prism that is yard long, yard wide, and yards tall? Lesson 8: Finding the Volume of Rectangular Prisms 77

178 EXAMPLE A Find the volume of the rectangular prism that has a length (l ) of 5 feet, a width (w) of feet, and a height (h) of 8 feet. Write the formula for the volume of a rectangular prism. V 5 lwh Find the volume. TRY V 5 lwh 5 5 ft ft 8 ft. Substitute the values for l, w, and h in the formula. 5 6 ft 5 ft 6 ft Write improper fractions for the mixed numbers ft Multiply ft Simplify ft The volume of the rectangular prism is ft. EXAMPLE B What is the height of a box that has the shape of a rectangular prism, if the volume is 05 cubic inches and the area of the base is 0 square inches? Use the formula for the volume of a rectangular prism. The area of the base is known, so substitute B for lw in the formula. V 5 Bh What is the height of a rectangular prism that has a volume of 0 cubic yards if its base is yards long and yards wide? Find the height. V 5 Bh Substitute known values for V and B h h 0 Divide both sides by 0. 5 h So h 5 inches. The height of the box is inches. 78 Domain : Geometry

179 read Problem Solving A gift box shown at the right is packed with small cubic inch blocks. The blocks are packed tightly with no spaces between. How many blocks are in the gift box? What is the volume of the box in cubic inches? plan Find how many inch lengths are equivalent to the length, width, and height. Multiply the products to find the total number of blocks. Use the formula for the volume of a. solve The length is inches, so the box is, or blocks long. The width is inches, so the box is, or blocks wide. The height is inches, so the box is, or blocks high. So the total number of blocks Use the formula to find the volume. V 5 V 5 Substitute values for l, w, and h. V 5 Change to improper fractions. V 5 in. Multiply. V 5 in. Simplify. 9 in. in. in. check The volume of each block is inch inch inch, or in. Is the total number of blocks times the volume of each block equal to the volume of the box in cubic inches? The number of blocks in the box is. The volume of the box is in. Lesson 8: Finding the Volume of Rectangular Prisms 79

180 Practice Find the volume of each figure.. ft ft Find the volume ft of each unit cube. HINT. yd yd yd Find the volume of each rectangular prism... cm. cm 6 cm. l 5 cm, w 5 8 cm, h 5.6 cm 5. B 5 6 m, h 5 58 m REMEMBER You can use the area of the base to find the volume of a prism. 6. l 5.5 m, w 5 8 m, h 5.5 m 7. l 5 9 in., w 5 in., h 5 5 in. 8. B 5 yd, h 5 yd 9. l 5 ft, w 5 ft, h 5 ft 0. l 5 5 in., w in., h 5 in.. B 5 6 ft, h 5 ft 80 Domain : Geometry

181 Find the missing dimension of each rectangular prism.. l 5 cm, w 5?, h 5 cm, V 5 9 cm. B 5 8 in., h 5?, V in.. l 5?, w 5 ft, h 5 ft, V 5 ft 5. l 5 in., w 5 in., h 5?, V in. 6. B 5?, h 5 8 yd, V 5 76 yd Solve. 7. A cereal box is 6 inches long, inches wide, and 0 inches tall. What is the volume of the cereal box? 9. The floor of a rectangular storage bin has an area of 7 square feet. The volume of the bin is 70 cubic feet. How tall is the storage bin?. ANALYZE Explain why area is measured in square units and volume is measured in cubic units. 8. A display case is shaped like a cube. Each side of the display case is 8 inches long. What is the volume of the display case? 0. An aquarium has a volume of 9,55 cm. The aquarium is.5 centimeters wide and 0 centimeters tall. How long is the aquarium?. Demonstrate Draw a rectangular prism. Select lengths for the length, width, and height. Label the dimensions. Explain how to find the volume of the prism using the two formulas for volume. Lesson 8: Finding the Volume of Rectangular Prisms 8

182 9 LESSON Drawing Polygons on the Coordinate Plane EXAMPLE A Plot the given points, and connect them in order to form a polygon. A (, ); B (, ); C (, ); D (, ); E (0, 5) Then use coordinates to find the number of units from one point to another for two points that lie on the same horizontal line. TRY Plot each point A D 5 Look at line segment BC. B y 5 The coordinates for point B are (, ), and the coordinates for point C are (, ). Since line segment BC is horizontal, both points have the same y-coordinate. Use the x-coordinates to find the number of units from either point to the other. E How many units are there from point P at (5, ) to point Q at (8, )? Use coordinates. Then check by plotting the points and counting units. You can use Math Tool: Coordinate Grid to plot the points. C x Connect the points in the order they were given. Then connect the first and last points. Draw 5 line segments to form a polygon. B y 5 C A E D 5 Find the absolute value of the difference of the x-coordinates. 5 5 Points B and C lie on the same horizontal line. The number of units from point B to point C is. x 8 Domain : Geometry

183 EXAMPLE B Plot the given points, and connect them in order to form a polygon. F (, ); G (, ); H (, 5); J (5, ); K (5, ) Then use coordinates to find the number of units from one point to another for two points that lie on the same vertical line. Plot each point. TRY K J Look at line segment JK. H The coordinates for point J are (5, ), and the coordinates for point K are (5, ). Since line segment JK is vertical, both points have the same x-coordinate. 5 y Use the y-coordinates to find the number of units from either point to the other. How many units are there from point R at (5, 7) to point S at (5, 9)? Use coordinates. Then check by plotting the points and counting units. You can use Math Tool: Coordinate Grid to plot the points. F G x Connect the points in the order they were given. Then connect the first and last points. Draw 5 line segments to form a polygon. K J H Find the absolute value of the difference of the y-coordinates y Points J and K lie on the same vertical line. The number of units from point J to point K is 7. F G x Lesson 9: Drawing Polygons on the Coordinate Plane 8

184 Practice Use the information given below for questions. The center of Tiny Town could be laid out on a coordinate plane with each unit representing one block. The bank would be located at (, ), the stadium at (, ), the grocery store at (, ), and the movie theater at (, ).. Which two places in Tiny Town lie on the same horizontal line? How do you know? HINT Look for ordered pairs with the same y-coordinate.. How many blocks is it from the bank to the grocery store? REMEMBER Find the absolute value of the difference of the different coordinates. Use the information given below for questions 8.. Plot the given locations in Tiny Town on the coordinate plane. 5 y The coordinates of the vertices of trapezoid JKLM are J(, ), K(, ), L (, ), and M(, ).. Which sides of the trapezoid are horizontal line segments? 5. How many units long is the vertical line? x 6. How many units long is side KL? 7. How many units long is side MJ? 8. Using a grid from Math Tool: Coordinate Planes, plot the points to form trapezoid JKLM. 8 Domain : Geometry

185 Use the information given below for questions 9 5. The stations for different points of interest along a scenic drive in a state park could be laid out on a coordinate plane with each unit representing one mile. The following shows the coordinates of each station. Station A: (5, ) Station B: (, ) Station C: (, 5) Station D: (5, 5) Station E: (5, ) Station F: (, ) Station G: (, ) Station H: (, ) Station I: (, ) Station J: (5, ) Find the distance between the stations. Indicate whether the line along which the stations lie is horizontal or vertical. 9. Station A to Station B 0. Station B to Station C. Station E to Station F. Station F to Station G. Station I to Station J. Station J back to Station A 5. Using a grid from Math Tool: Coordinate Planes, plot and label the stations in order. Draw the polygon formed when the stations are connected. Solve. 6. EXPLAIN Refer back to the information given for questions 8. Can you use the methods presented in this lesson to find the length of side JK? Explain why or why not. 7. DESCRIBE Describe the procedure you would follow to find the length of each side of the rectangle formed when points (, 5), (, 5), (, ), and (, ) are connected. Lesson 9: Drawing Polygons on the Coordinate Plane 85

186 0 LESSON Representing Three-Dimensional Figures Using Nets A three-dimensional figure that has two congruent, parallel, triangular bases and three rectangular sides is a triangular prism. A net is a flat pattern that can be folded to form a three-dimensional figure. EXAMPLE A The figure on the right is the net of a three-dimensional figure. If the net were folded along the dashed lines, what three-dimensional figure would it form? Identify the shapes that make up the net. Each two-dimensional figure on the net represents one face of the threedimensional figure. The net is made from rectangles and triangles. So faces of the three-dimensional figure are rectangles and faces are triangles. Create the figure. The net can be folded along the dashed lines. Think of folding the rectangular faces up along the longer dashed lines first. The top and bottom rectangles will meet along the two long, solid lines. Then the triangles on the ends can be folded to close up the figure. The three-dimensional figure formed is shown below. DISCUSS If the triangles were on opposite sides of the middle rectangle, would the net still be able to be folded into a triangular prism? What if the triangles were on opposite sides of the top rectangle? Trace, modify, and cut out to experiment with different nets. Is there only one correct net for a threedimensional figure, or could there be more than one? The three-dimensional figure has rectangular faces and triangular faces. The triangular faces are parallel. The three-dimensional figure represented by the net is a triangular prism. 86 Domain : Geometry

187 A pyramid has a single base and has triangular faces that all meet at the same point. A pyramid is named by the shape of its base. EXAMPLE B Draw the net of the rectangular pyramid shown below. Identify the shape of each face. CHECK The base, or bottom face, of the pyramid is a rectangle. Each side face of the pyramid is a triangle. There are triangular faces. How can you check that your net represents the three-dimensional figure? Draw the net. First draw a rectangle to represent the base. Used dashed lines to represent the base. Then draw triangles along each side of the rectangle to represent the triangular faces. If the triangles were folded up along the dashed lines and slanted so they all met at one point, a rectangular pyramid would be formed. The net above is a net of the rectangular pyramid. Lesson 0: Representing Three-Dimensional Figures Using Nets 87

188 Practice What three-dimensional figure does each net represent?... HINT Draw a net for each three-dimensional figure.. 5. REMEMBER Use dashed lines to represent the folds. A pyramid is named by the shape of its base Domain : Geometry

189 Solve. 7. Viola folded the net shown to create a three-dimensional figure. What figure did she make? 8. Mason drew the nets shown to represent two different three-dimensional figures. How are the figures similar? How are they different? What are the figures? 9. A tent is shaped like the three-dimensional figure shown. What three-dimensional figure is the tent? Draw a net of the tent. 0. A cereal box is shaped like the three-dimensional figure shown. What three-dimensional figure is the box? Draw a net of the cereal box.. DESCRIBE How do you use a net to determine the three-dimensional figure it represents?. WRITE MATH Explain how to draw the net of a three-dimensional figure. Lesson 0: Representing Three-Dimensional Figures Using Nets 89

190 LESSON Using Nets to Find Surface Area EXAMPLE What is the surface area of the triangular prism? 5 cm cm 6 cm 5 cm Draw a net of the triangular prism. Draw the net. Label the lengths of the sides. 5 cm 5 cm 6 cm cm 5 cm Find the surface area of the triangular prism. To find the surface area, add the areas of the all the faces of the triangular prism. SA 5 75 cm 90 cm 75 cm cm cm SA 5 6 cm The surface area of the triangular prism is 6 cm. Use the net to find the area of each face of the triangular prism. D A B C 5 cm 5 cm E 6 cm cm 5 cm Find the area of the rectangular faces. Face A has length 5 cm and width 5 cm. Area of face A: A 5 lw 5 5 cm 5 cm 5 75 cm Face B has length 5 cm and width 6 cm. Area of face B: A 5 lw 5 5 cm 6 cm 5 90 cm DISCUSS Face C has length 5 cm and width 5 cm. Face C has the same area as face A. Find the area of the triangular faces. Face D has base 6 cm and height cm. Area of face D: A 5 bh 5 6 cm cm 5 cm Face E has the same area as face D. Why do you need only the areas of faces A, B, and D to find the surface area of the triangular prism? 90 Domain : Geometry

191 READ Problem Solving Raymond has a cereal box that he is planning to paint for a project. The dimensions of the box are shown to the right. What is the surface area that he will paint? PLAN Draw a net of the cereal box. The box is shaped like a. Label the sides of the net that correspond to the cereal box. Use the net to find the of each face. Then add the areas to find the. SOLVE Draw the net, and label the sides. Each face of the box is a. Use the formula area of each face. Find the area of each of the six faces of the box. Face : Face : Face : Face : Face 5: Face 6: Add to find the total surface area. to find the in. in. 0 in. 0 in. 0 in. in. in. in. in. 5 CHECK Does your answer seem reasonable? Explain. Raymond has to paint a surface area of in.. Lesson : Using Nets to Find Surface Area 9

192 Practice Draw a net to find the surface area of each three-dimensional figure.. 6 ft. in. 6 ft HINT 6 ft Each face is a square. 8 in. 9 in cm cm 6 cm REMEMBER Only two of the three rectangular faces are congruent. 0 cm. m m 6. 5 m 0 in. 6 in. 9 in. cm cm cm 9 Domain : Geometry

193 Solve. 7. The dimensions of Femi s tent are shown to the right. Draw a net and use it to find the surface area of her tent. 6 ft 8. Elaine s room is in the shape of a rectangular prism 5 feet long, feet wide, and 0 feet tall. Elaine paints the four walls and the ceiling but not the floor. How much surface area does Elaine paint? 9. A Plexiglas display case is shaped like the triangular prism to the right. What is the surface area of the prism? 0. DEMONSTRATE The net for a rectangular prism is shown to the right. Use the net to find the surface area of the rectangular prism. Show your work. 9 cm cm 0 ft 8 ft in. 5 in. 7 cm cm ft in. 5 in. 7 cm 9 cm cm 7 cm cm. Classify What are the similarities and differences for a rectangular pyramid and a triangular prism? How do the nets of these figures help you identify them? 7 cm Lesson : Using Nets to Find Surface Area 9

194 DOMAIN Review Find the area of each figure.. 5 ft. 9 m 6 ft m. cm 5 cm 8 cm Find the volume of each rectangular prism.. 6 yd 0 yd 5. B 5 78 ft, h 5 ft 6. l 5. cm, w 5.8 cm, h 5 6 cm 7. l 5 8 in., w 5 9 in., h 5 5 in. 8. B 5 yd, h 5 7 yd 6 yd 6 yd 9. l 5 6 ft, w 5 5 ft, h 5 ft 0. l 5 8 in., w 5 8 in., h 5 5 in. Use the following information for questions. The coordinates of the vertices of pentagon CDEFG are C(5, 5), D(, 5), E(, ), F(, ) and G(5, ).. How long is side CD?. How long is side DE?. How long is side CG?. Use a grid from Math Tool: Coordinate Planes to plot the points. Draw the polygon formed when the points are connected. 9

195 Draw a net for each three-dimensional figure Draw a net to find the surface area of each three-dimensional figure. 7. ft 6 ft 8 ft 8. cm cm 0 cm 0 cm 95

196 Choose the best answer. 9. A rectangular card has an area of 0 square inches. The card is 6 inches wide. How long is the card? A. 8 in. B. 8 in. C. 0 in. D in. 0. A block in the shape of a cube has sides that are inches long. What is the volume of the block? A. 8 8 in. B. 8 in. C. 5 in. D in. Solve.. The top of a shoe box has an area of 88 square inches. The volume of the shoe box is 96 cubic inches. How tall is the shoe box?. A paperweight shaped like a cube has sides that are 8 centimeters long. Draw a net to find the surface area of the paper weight.. EXPLAIN Refer back to question. To find the length of line segment DE, which coordinates did you use and which did you ignore? Why?. DEMONSTRATE How can you decompose this figure into rectangles and use them to find the area of the polygon?

197 . Choose a rectangular prism in your classroom. Examples include books and boxes of different sizes. Object chosen:. Measure the length, width, and height of the prism. l 5 w 5 h 5. Draw the rectangular prism. Label the sides of your drawing using your measurements.. Find the volume of the rectangular prism. 5. Draw a net of the rectangular prism. Label the side lengths on the net. 6. Find the surface area of the rectangular prism. 7. Make a copy of your net, and exchange it with a classmate. Find the surface area of your partner s rectangular prism. Draw your partner s rectangular prism, and label its dimensions. Find the volume of your partner s rectangular prism. Compare answers with your partner. Domain Review 97