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Texas 4 Mathematics STAAR ® Instruction New T comin EKS g Sep Edit i temb o er 20 n Gra de 4 S 1 4 ample r inclu des: Table of Con tents, Stude nt Sam ple Le ssons , Teac her Sa mple Lesso ns STAAR is a federally registered trademark owned by the Texas Education Agency, and is used pursuant to license. Student Book Sample Lessons Includes two sample lessons: Lesson 10: Compare Fractions Lesson 11: Understand Fraction Addition and Subtraction Table of Contents Unit 1: Number and Operations, Part 1 STAAR Reporting Categories 1 and 2 TEKS Lesson 1 Understand Place Value 2 4(2)(A)♦, 4(2)(B)✶ Lesson 2 Compare and Order Whole Numbers 8 4(2)(B)✶, 4(2)(C)♦ Lesson 3 Round Whole Numbers 16 4(2)(D)♦ Lesson 4 Add and Subtract Whole Numbers 24 4(4)(A)✶ Lesson 5 Multiply by One-Digit Numbers 34 4(4)(B)♦, 4(4)(D)♦ Lesson 6 Multiply by Two-Digit Numbers 46 4(4)(C)✶, 4(4)(D)✶ 4(4)(H)✶ Lesson 7 Divide Whole Numbers 54 4(4)(E)♦, 4(4)(F)♦ 4(4)(H)✶ STAAR Practice 64 Unit 2: Number and Operations, Part 2 STAAR Reporting Categories 1 and 2 Lesson 8 Understand Fractions 66 4(3)(A)♦, 4(3)(B)♦ Lesson 9 Understand Equivalent Fractions 72 4(3)(C)♦ Lesson 10 Compare Fractions 78 4(3)(D)✶ Lesson 11 Understand Fraction Addition and Subtraction 88 4(3)(E)✶ Lesson 12 Add and Subtract Fractions 94 4(3)(E)✶ Lesson 13 Add and Subtract Mixed Numbers 106 STAAR Practice 118 4(3)(E)✶, 4(3)(F)♦ ✶ = STAAR Readiness Standard ♦ = STAAR Supporting Standard ©Curriculum Associates, LLC Copying is not permitted. iii Table of Contents Unit 3: Number and Operations, Part 3 STAAR Reporting Categories 1 and 2 TEKS Lesson 14 Understand Decimals . . . . . . . . . . . . . . . . . . . . . . 1204(2)(B)✶, 4(2)(E)♦ Lesson 15 Relate Decimals and Fractions . . . . . . . . . . . . . . . . .1264(2)(G)✶, 4(2)(H)♦ 4(3)(G)♦ Lesson 16 Compare and Order Decimals . . . . . . . . . . . . . . . . .1364(2)(F)♦, 4(2)(E)♦ Lesson 17 Add and Subtract Decimals . . . . . . . . . . . . . . . . . . 1484(4)(A)✶ STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Unit 4: A lgebraic Reasoning, Number and Operations, Part 4 STAAR Reporting Categories 2 and 3 Lesson 18 Estimation and Problem Solving . . . . . . . . . . . . . . . 1584(4)(G)♦ Lesson 19 Model Multi-Step Problems . . . . . . . . . . . . . . . . . . 1684(4)(H)✶, 4(5)(A)✶ Lesson 20 Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 1764(5)(B)✶ Lesson 21 Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . .1864(5)(D)✶ STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 ✶ = STAAR Readiness Standard ♦ = STAAR Supporting Standard iv ©Curriculum Associates, LLC Copying is not permitted. Table of Contents Unit 5: Geometry and Measurement STAAR Reporting Category 3 TEKS Lesson 22 Points, Lines, Rays, Angles . . . . . . . . . . . . . . . . . . . 196 4(6)(A)♦, 4(6)(C)♦ Lesson 23 Classify Two-Dimensional Figures . . . . . . . . . . . . . . 208 4(6)(D)✶, 4(6)(A)♦ 4(6)(C)♦ Lesson 24 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4(6)(B)♦ Lesson 25 Measure and Draw Angles . . . . . . . . . . . . . . . . . . . 230 4(7)(C)✶, 4(7)(D)♦ Lesson 26 Add and Subtract With Angles . . . . . . . . . . . . . . . . 240 4(7)(E)♦ Lesson 27 Convert Measurements . . . . . . . . . . . . . . . . . . . . 250 4(8)(A)♦, 4(8)(B)♦ Lesson 28 Time and Money . . . . . . . . . . . . . . . . . . . . . . . . . 260 4(8)(C)✶ Lesson 29 Length, Liquid Volume, and Mass . . . . . . . . . . . . . . 270 4(8)(C)✶ STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Unit 6: Data Analysis and Personal Financial Literacy STAAR Reporting Category 4 Lesson 30 Represent Data . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4(9)(A)✶ Lesson 31 Use Data to Solve Problems . . . . . . . . . . . . . . . . . . 298 4(9)(B)♦ Lesson 32 Fixed and Variable Expenses . . . . . . . . . . . . . . . . . 306 4(10)(A)♦ Lesson 33 Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 4(10)(B)♦ Lesson 34 Financial Institutions . . . . . . . . . . . . . . . . . . . . . . 318 4(10)(E)♦ STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 ✶ = STAAR Readiness Standard ♦ = STAAR Supporting Standard ©Curriculum Associates, LLC Copying is not permitted. v Develop Skills and Strategies Lesson 10 Part 1: Introduction TEKS 4.3.D Compare Fractions In the past, you learned to compare fractions using models. Take a look at this problem. Adriana ate 2 of a granola bar and June ate 2 of a same-size granola bar. Which 4 ·· 5 ·· girl ate more granola bar? Adriana June Explore It Use the math you already know to solve the problem. How many equal pieces of granola bar did Adriana eat? How many equal pieces of granola bar did June eat? Since both girls ate the same number of pieces, what can you look at to find out who ate more? What does the size of the denominator tell you about the size of the pieces of granola bar? Who ate more? Explain why. 78 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 10 Find Out More Deciding who ate more of the granola bar means comparing the fractions 2 and 2 . 4 ·· 2 4 ·· . 2 2 5 ·· 2 is greater than 2 . 5 ·· 4 ·· 5 ·· 5 ·· , 2 4 ·· 2 is less than 2 . 4 ·· 5 ·· What if June’s granola bar was larger than Adriana’s? Would the comparison make sense? To compare fractions, you must use the same-size whole. You can also use equivalent fractions to compare fractions. Look for numbers that you can multiply by the denominators so that the fractions end up with the same denominators. 2 3 4 5 8 and 2 3 5 5 10 4 ·· 20 4 ·· 5 ·· 20 ·· ·· 8 , 10 , so 2 , 2 20 ·· 20 5 ·· 4 ·· ·· 5 ·· Reflect 1 Explain how you can tell which fraction is greater, 2 or 3 . 5 ·· 10 ·· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 79 Part 2: Modeled Instruction Lesson 10 Read the problem below. Then explore different ways to understand it. A grasshopper weighs about 2 of an ounce. A beetle weighs 8 of an ounce. 100 ··· Which weighs more? 10 ·· Picture It You can use models to help compare fractions. The following model shows the weights of the grasshopper and beetle. Grasshopper Beetle Solve It You can use a common denominator to help you solve the problem. It is hard to compare two fractions with different numerators and different denominators. You can write an equivalent fraction for one or both of the fractions so they have a common denominator. Fractions with the same denominator are divided into the same number of equal parts. If fractions have the same denominator, you can just compare the numerators. 8 and 2 . Compare 10 100 ·· ··· First, look at the denominator, 10. Can you multiply 10 by any number to get 100? Yes, 10 3 10 5 100. Find a fraction equivalent to 8 that has a denominator of 100: 8 3 10 5 80 10 10 ··· 100 ·· ·· 10 ·· Compare the numerators of 80 and 2 : 100 ··· 100 ··· 80 . 2 So, 80 . 2 . 100 ··· 80 100 ··· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 10 Connect It Now you will solve the problem from the previous page by finding a common numerator. 2 What is an equivalent fraction for 2 that has a numerator of 8? 100 ··· 3 One model is divided into 400 equal parts and the other is divided into 10 equal parts. Which has smaller parts? 4 Shade 8 pieces of each model. 5 Which model has a greater area shaded? 6 Which fraction is greater, 8 or 8 ? 400 ·· 10 ··· 7 Look at the denominators of 8 and 8 . When two fractions have the same 400 10 ··· ·· numerator and different denominators, how do you know which one is greater? Explain. Try It Use what you just learned to solve these problems. 8 Mel’s tomato plant is 8 of a foot tall. Her pepper plant is 3 of a foot tall. Compare 12 4 ·· ·· the heights of the plants using a symbol. 9 Compare the fractions 3 and 5 using a symbol. 5 10 ·· ·· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 81 Part 3: Modeled Instruction Lesson 10 Read the problem below. Then explore different ways to use benchmarks to compare fractions. Jasmine’s swimming lesson lasts for 2 of an hour. It takes her 1 of an hour to do 3 ·· 6 ·· her homework. Will Jasmine spend more time on her homework or at her swimming lesson? Model It You can use a number line to help you compare fractions. The number line shows where the fractions 2 and 1 are compared to 0 and 1. 3 ·· 1 6 6 ·· 2 3 1 2 0 1 The number line shows that 1 is closer to 0 than 2 is, and that 2 is closer to 1 than 1 is. This means that 2 . 1 . 3 ·· 6 ·· 6 ·· 3 ·· 3 ·· 6 ·· Solve It You can use a benchmark fraction to solve the problem. Another way to compare fractions is by using the fraction 1 as a benchmark. 2 ·· Look at the number line. It shows that 1 is less than 1 and 2 is greater than 1 . So, 1 , 2 and 2 . 1 . 6 ·· 3 3 ·· 6 ·· ·· 6 ·· 2 ·· 3 ·· 2 ·· Jasmine will spend more time at her swimming lesson than on homework. 82 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 3: Guided Instruction Lesson 10 Connect It Now you will solve a similar problem using 1 as a benchmark. Think about these two fractions: 11 and 7 10 ·· 8 ·· 10 Which fraction is greater than 1? 11 Which fraction is less than 1? 12 Which fraction is greater? Explain why. 13 Fill in the blank with the correct symbol to show the comparison. 11 10 ·· 7 8 ·· 14 Explain how you can use benchmarks to compare fractions. Try It Use what you just learned to solve these problems. 15 Fill in the blank. Explain how you found your answer. 16 Nathan walked 5 10 ·· 3 4 ·· 10 of a mile. Sarah walked 19 of a mile. Who walked a greater 20 ·· 10 ·· distance? Explain. L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 83 Part 4: Guided Practice Lesson 10 Study the model below. Then solve problems 17–19. Student Model It is important that both measurements use the same unit! Becker catches a fish that is 3 of a yard long. To keep the fish, 12 ·· 1 it has to be longer than of a yard. Can Becker keep his fish? 3 ·· Look at how you could show your work using a number line. 1 2 3 4 5 6 7 8 9 10 11 0 12 12 12 12 12 12 12 12 12 12 12 1 1 3 Pair/Share How else could you solve this problem? Which strategy for comparing do you think works best with these fractions? Solution: 2 3 Since 3 is less than 1 , Becker can’t keep his fish. 12 ··· 3 ·· 17 Myron and Jane are working on the same set of homework problems. Myron has finished 7 of the problems and Jane has 9 ·· 2 finished of the problems. Who has finished more of the 3 ·· homework? Show your work. Pair/Share How did you and your partner choose what strategy to use to solve the problem? 84 Solution: L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 4: Guided Practice 18 Compare the fractions 3 and 7 using the benchmark fraction 1 . 12 2 ·· ·· 10 ·· Show your work. Lesson 10 You already know about how big 1 is! 2 ·· Pair/Share Draw a model to check your answer. Solution: 19 Janelle walked 3 of a mile. Pedro walked 6 of a mile. Which 10 ·· 6 ·· statement shows how to find the greater fraction? Circle the letter There are several ways to compare fractions! of the correct answer. A 3 5 6 and 6 , 6 6 ·· B C D 12 ·· 12 ·· 10 ·· 3 5 6 and 6 . 6 12 12 ·· 10 ·· ·· 6 5 3 and 3 , 3 10 ·· 5 5 ·· 6 ·· ·· 3 , 1 and 6 . 1 6 ·· 2 10 ·· 2 ·· ·· 6 ·· Tina chose B as the correct answer. How did she get that answer? Pair/Share How can you find the answer using a benchmark fraction? L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 85 Part 5: TEKS Practice Lesson 10 Solve the problems. 1 2 2 cup of raisins and 3 cup of almonds to make trail mix. Which statement Grant needs } } 4 3 can be used to find out if there are more raisins or almonds in the mix? 8 2 4 3 4 2 6 3 6 2 6 3 6 9 B 5 } and } 5 } } 4 5 3 6 C 5 } and } 5 }} } 4 3 9 12 D 5 } and } 5 } } 4 7 3 9 Tell whether each sentence is True or False. b. c. d. e. 2, 6 15 ··· True False 7 .7 8 ·· True False 1.3 8 ·· True False 254 6 ·· True False 30 5 3 50 ··· True False 5 ·· 10 ··· 2 ·· 4 ·· 500 ···· Fill in the blank with one of the symbols shown to compare 5 and 5 . 10 ··· , 5 }} 10 86 3 5 }} and } 5 }} } 4 3 12 12 a. 3 2 A 5 8 ·· . 5 } 8 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 5: TEKS Practice 4 Lesson 10 5 of an hour. He spent Sam’s music teacher told him to practice his trombone for }} 10 2 of an hour practicing. Did he practice long enough? } 6 Show your work. Answer Sam 5 practice long enough. Olivia and Eleanor each made the same amount of lemonade to sell at a lemonade stand. Olivia poured all of her lemonade into 10 equal glasses. Eleanor poured all of her lemonade into 5 equal glasses. Olivia sold 7 glasses of lemonade and Eleanor sold 2 glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions using a symbol. Show your work. sold a greater fraction of her lemonade. Answer 6 9 of her Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold }} 10 5 of her boxes. Which girl has sold a greater fraction boxes of fruit and Sierra has sold } 8 of her boxes of fruit? Draw a model to show your answer. Show your work. Answer has sold a greater fraction of her boxes of fruit. L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 87 Focus on Math Concepts Lesson 11 Part 1: Introduction TEKS 4.3.E Understand Fraction Addition and Subtraction What’s really going on when we add numbers? Adding means joining or putting things together. Think about how you could explain adding 2 1 3 to a first grader. You could start at 2, count on 3 more, and see where you end up: 2 . . . 3 . . . 4 . . . 5. Or, you could put a segment with a length of 2 and a segment with a length of 3 next to each other on a number line to show 2 1 3. 1 0 1 1 1 2 1 1 3 4 5 6 7 8 9 10 When you add 2 1 3, you are putting ones together. Think Adding fractions means joining or putting together parts of the same whole. You can put a segment with a length of 2 and a segment 4 ·· 3 with a length of next to each other to show 2 1 3 . 4 4 ·· 4 ·· ·· 1 4 0 4 1 4 1 4 1 4 2 4 1 4 3 4 0 Underline the sentence that explains what adding fractions means. 1 4 4 4 1 5 4 6 4 7 4 8 4 2 When you add 2 1 3 , you are putting one-fourths together. 4 ·· 88 4 ·· L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 11 Think Subtracting means separating or taking away. Look at the whole numbers. Now look at the numerators of the fractions. I think I see a connection. On a number line, you can start with a segment of length 5 and take away a segment of length 2 to show 5 2 2. 1 1 0 1 1 2 1 1 1 1 3 4 5 6 7 8 9 10 When you subtract 5 2 2, you are taking away ones. You can show subtracting fractions on a number line. Start with a segment of length 5 and take away a segment of length 2 to show 5 2 2 . 4 ·· 4 ·· 1 4 0 4 1 4 1 4 1 4 1 4 1 4 2 4 3 4 4 ·· 1 4 1 4 4 4 0 4 ·· 1 5 4 6 4 7 4 8 4 2 When you subtract 5 2 2 , you are taking away one-fourths. 4 ·· 4 ·· Now you’ll have a chance to think more about how adding or subtracting fractions is like adding or subtracting whole numbers. You may find that using number lines or area models can help you explain your thinking. Reflect 1 Use your own words to describe what you just learned about adding and subtracting fractions. L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. 89 Part 2: Guided Instruction Lesson 11 Explore It Counting on and using a number line are two ways to think about adding fractions. 2 Count by fourths to fill in the blanks: 1 , 2 , 4 ·· 4 ·· , 4, 5, 4 ·· 4 ·· , , , Now label the number line. 0 1 4 2 4 4 4 5 4 10 4 3 Count by fifths to fill in the blanks: 1 , 2 , 5 ·· 5 ·· , , Now label the number line. 0 1 5 2 5 6 5 Use the number lines above to answer numbers 4 and 5. 4 What is 1 more than 6 ? 4 4 ·· ·· 5 What is 1 more than 3 ? 5 5 ·· ·· Now try these two problems. 6 Label the number line below and use it to show 2 1 1 . 4 ·· 4 ·· 7 Label the number line below and use it to show 3 1 1 . 4 ·· 4 ·· 90 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 11 Talk About It Solve the problems below as a group. 8 Look at your answers to problems 2 and 3. How is counting by fractions the same as counting with whole numbers? How is it different? 9 Label the number line below and use it to show 7 2 2 . 8 ·· 8 ·· 5 1 10 Label the number line below and use it to show 2 . 6 ·· 6 ·· Try It Another Way Work with your group to use the area models to show adding or subtracting fractions. 1 2 11 Show 1 . 8 ·· 8 ·· L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. 6 2 2. 12 Show 10 ·· 10 ·· 91 Part 3: Guided Practice Lesson 11 Connect It Talk through these problems as a class, then write your answers below. 2 1 13 Compare: Draw two different models to show 2 . 3 ·· 3 ·· 14 Explain: Rob had a large pizza and a small pizza. He cut each pizza into fourths. He took one fourth from each pizza and used the following problem to show their sum: ··14 1 ··14 5 ··24 . What did Rob do wrong? 15 Demonstrate: Think about how you would add three whole numbers. You add two of the numbers first, and then add the third to that sum. You add three fractions the same way. 1 1 3 1 4. Use the number line and area model below to show ·· 10 ·· 10 ·· 10 92 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 4: Performance Task Lesson 11 Put It Together 16 Use what you have learned to complete this task. Jen has 4 of a kilogram of dog food. Luis has 3 of a kilogram of dog food. 10 ·· A large dog eats 2 of a kilogram in one meal. 10 ·· 10 ·· A Write two different questions about this problem that involve adding or subtracting fractions. i ii B Choose one of your questions to answer. Circle the question you chose. Show how to find the answer using a number line and an area model. L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. 93 Teacher Resource Book Sample Lessons Includes two sample lessons: Lesson 10: Compare Fractions Lesson 11: Understand Fraction Addition and Subtraction Develop Skills and Strategies Lesson 10 (Student Book pages 78–87) Compare Fractions Lesson objectives the Learning Progression • Use symbols (., ,, 5) to compare fractions with the same denominator and different numerators. In Grade 3, students used models to compare two • Recognize that fractions with different denominators and the same numerators represent different values. denominator by reasoning about their size. In Grade 4, • Use benchmark fractions to compare fractions. compare two fractions with different numerators and • Recognize that you can only compare two fractions when both refer to the same whole. different denominators. Emphasis is placed on Prerequisite skiLLs the two fractions have the same size wholes. Students • Represent fractions with denominators 2, 3, 4, 6, or 8 using a number line or visual models. compare fractions by creating common numerators or • Identify, create, and explain equivalent fractions. fractions with the same numerator or the same they extend their understanding of fractions to understanding that a comparison only makes sense if use models (e.g., fraction bars, area models, etc.) to denominators. Students also learn to use benchmark • Express whole numbers as fractions. fractions 1 e.g., 1 2 to compare fractions. They record • Compare fractions whose numerators or denominators are the same. work focuses on visual models and benchmark vocabuLary formally address fraction comparison in later grades, There is no new vocabulary. Review the following key terms. but they will later apply their understanding of fraction compare: to decide if one number is greater than, less than, or equal to another number when they compare fractional quantities in their lives. 2 ·· comparisons using the ., ,, and 5 symbols. Students’ fractions, rather than an algorithm. Students will not comparison when they learn to compare decimals and greater than (.): a comparison of two numbers that says one has greater value than the other less than (,): a comparison of two numbers that says one has less value than the other teks Focus 4.3.D Compare two fractions with different numerators and different denominators and represent the comparison using the symbols ., 5, or ,. Readiness Standard MatheMaticaL Process standards (MPs): 4.1.A, 4.1.C, 4.1.D, 4.1.E, 4.1.F (See page A9 for full text. Also see MPS Tips in the lesson.) 140 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 10 At A GlAnce Develop Skills and Strategies Students explore a fraction comparison problem involving fractions that have the same numerator but different denominators. They use an approach they already know. They explain that the fractions have the same number of pieces but the fraction with the greater denominator has smaller pieces. Lesson 10 Part 1: Introduction TEKS 4.3.D Compare Fractions In the past, you learned to compare fractions using models. Take a look at this problem. Adriana ate 2 of a granola bar and June ate 2 of a same-size granola bar. Which 4 ·· 5 ·· girl ate more granola bar? Adriana Step By Step • Tell students that this page shows a way to compare fractions using a visual model. • Have students read the problem at the top of the page. Explore It Use the math you already know to solve the problem. • Work through Explore It as a class. How many equal pieces of granola bar did June eat? the model alone as justification for saying that Adriana ate more. Encourage them to use the model mathematically (4.1.E) by reasoning about the number of pieces and sizes of the pieces. Discuss how the number of pieces in each square relates to the denominators and how the number of shaded pieces relates to the numerators. ell Support Have students identify the comparison word in this problem [more]. Connect this word to the mathematical term greater than. Have students think of other words that might be used to compare quantities (e.g., bigger, larger, longer, taller, most, etc.). Do the same with less than (e.g., less, fewer, smaller, shorter, least, etc.). L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 2 Since both girls ate the same number of pieces, what can you look at to find out who ate more? the size of the pieces What does the size of the denominator tell you about the size of the pieces of granola bar? The greater denominator means there are more and smaller • Have students explain the model. (See Mathematical Discourse, below.) MpS tip: Some students may be tempted to use 2 How many equal pieces of granola bar did Adriana eat? • Have students describe the problem and state what needs to be done mathematically (compare the fractions). • Ask student pairs or groups to explain their answers for the last two bullets. Look for understanding that a larger denominator means the whole is broken into more pieces, which means each piece is smaller. Ask, Would you rather share your favorite treat with 3 classmates or with 1 classmate? Why? Students should apply this reasoning to explain that Adriana’s 2 pieces were bigger than June’s. June pieces. Who ate more? Explain why. Both girls ate the same number of pieces. Adriana’s 2 pieces are larger than June’s 2 pieces. So, Adriana ate more. 78 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse • What does the model tell you? The two rectangles show equal sized wholes. One has 4 equal parts; the other has 5 equal parts. Two parts are shaded in each. It looks like more space is shaded in the rectangle with 4 parts. • Why do we look at the denominators to find out who ate more? When numerators are the same, the number of pieces is the same. So we compare sizes of the pieces. The denominator tells how many pieces. The more pieces, the smaller they are. • How can you use the model to find out who ate more? We see more shaded area in Adriana’s granola bar. June’s granola bar has more pieces, so they are smaller than Adriana’s. Therefore, June ate less than Adriana did. 141 Part 1: Introduction Lesson 10 At A GlAnce Students use symbols to compare fractions. Then they reflect on the importance of comparing fractions from same-size wholes. Part 1: Introduction Lesson 10 Find Out More Deciding who ate more of the granola bar means comparing the fractions 2 and 2 . 4 ·· 5 ·· Step By Step • Read Find Out More as a class. • Explain that you can show either fraction first in the comparison; 2 . 2 is the same as 2 , 2 . Make sure 4 ·· 5 ·· 5 ·· 4 ·· students understand what . and , mean and how 2 . 2 4 5 ·· ·· 2 is greater than 2 . 4 5 ·· ·· 2 , 2 5 4 ·· ·· 2 is less than 2 . 5 4 ·· ·· What if June’s granola bar was larger than Adriana’s? Would the comparison make sense? the direction of the sign shows the comparison. • Discuss the question about the sizes of the granola bars. Have students explain their thinking. • Students answer Reflect on their own. Consider having students share their ideas with a partner. Then discuss as a group. Reinforce the idea that the comparison doesn’t make sense unless the wholes are the same size. Point out that often students will see fractions in a problem with no diagram. To compare fractions, you must use the same-size whole. You can also use equivalent fractions to compare fractions. Look for numbers that you can multiply by the denominators so that the fractions end up with the same denominators. 2 3 4 5 8 and 2 3 5 5 10 4 ·· 20 4 ·· 5 ·· 20 ·· ·· 8 , 10 , so 2 , 2 20 ·· 20 5 ·· 4 ·· ·· 5 ·· Reflect 1 Explain how you can tell which fraction is greater, 2 or 3 . 5 ·· 10 ·· Possible answer: Multiply the numerator and denominator by 2 to find an equivalent fraction in tenths: 4 . Since 4 . 3 , 2 . 3 . 10 ··· Hands-On Activity 10 ··· 10 ·· 5 ··· 10 ··· 79 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Investigate the importance of comparing fractions from same-size wholes. Materials: 8 squares of paper of four different sizes with two of each size, cards, markers • Have students divide different squares into thirds, fourths, fifths, sixths, and tenths. • Have students shade 2–5 parts in each square and write each resulting fraction on a card. • Set out the squares so that two different-sized squares and two same-sized squares are paired. • Students discuss whether or not each pair of fractions can be compared, and why. Fractions of different-sized wholes cannot be compared, even if they have the same numerators or same denominators. Fractions of the same-sized wholes can be compared; students have not yet learned how to compare fractions of same-sized wholes that have different numerators and different denominators, but they will learn to do that in this lesson. 142 Real-World connection compare fractions in everyday recipes. Materials: recipes, measuring cups and measuring spoons, sand or rice, bowls Provide students with a variety of recipes that have fractions of cups, tablespoons, and teaspoons in their ingredients. Have students read through the recipes and write down the fractions they see and arrange them by unit. Students write all the fractions of cups in one group, the fractions of tablespoons in another group, and the fractions of teaspoons in a third group. Have them use visual models and ., ,, and 5 to compare the fractions in each group. If time allows, show students the measuring cups and spoons and have them measure the fractional quantities using sand or rice to compare amounts. L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 2: Modeled Instruction Lesson 10 At A GlAnce Part 2: Modeled Instruction Students use a model to study a problem involving fraction comparison. They solve the problem using common denominators. Lesson 10 Read the problem below. Then explore different ways to understand it. A grasshopper weighs about 2 of an ounce. A beetle weighs 8 of an ounce. 100 ··· Which weighs more? SteP By SteP 10 ·· Picture It • Read the problem at the top of the page as a class. You can use models to help compare fractions. The following model shows the weights of the grasshopper and beetle. • Have students identify this as fraction comparison. Grasshopper Beetle • Ask students to describe the models in Picture It and explain how to use these to solve the problem. • Point out that the fractions have different denominators. Show this on the models. Solve It You can use a common denominator to help you solve the problem. It is hard to compare two fractions with different numerators and different denominators. You can write an equivalent fraction for one or both of the fractions so they have a common denominator. Fractions with the same denominator are divided into the same number of equal parts. If fractions have the same denominator, you can just compare the numerators. • Read Solve It as a class. Have students identify a common denominator. [hundredths] 8 and 2 . Compare 10 100 ·· MPS tip: Students must apply knowledge of ··· First, look at the denominator, 10. Can you multiply 10 by any number to get 100? Yes, 10 3 10 5 100. Find a fraction equivalent to 8 that has a denominator of 100: equivalent fractions to solve this problem. 8 3 10 5 80 10 ··· 100 ·· 10 ·· 10 ·· Encourage them to analyze mathematical Compare the numerators of 80 and 2 : 100 ··· as they search for a viable common denominator. Ask, How could you rewrite 2 as tenths? [You can’t 100 ··· 80 . 2 relationships to connect mathematical ideas (4.1.F) So, 80 . 2 . 100 ··· 80 100 ··· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 100 ··· because you would have to divide 2 by 10.] How could you rewrite 8 as hundredths? [Multiply both 10 ·· numbers by 10 to get 80 hundredths.] Mathematical Discourse Visual Model compare the fractions using a number line. • After completing the page, present another model. • Draw two number lines from 0 to 1, one above the other. On one number line, mark and label tenths. On the other number line, mark and label every 10 hundredths: 10, 20, etc. Mark 10 hundredths between each tenth. • Guide students to recognize that the number lines are the same length. • Have a volunteer mark 2 on the hundredths 100 ··· number line. Have another volunteer mark 8 . 10 ·· • Students compare the two values. 2 , 8 or 8 . 2 . 3 ··· 100 10 10 100 4 ·· ··· ··· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. • What do you notice about the two models? One model has 10 parts and the other has 100 parts. The models are the same size. Different numbers of parts are shaded. • Why are the models divided into different numbers of parts? The grasshopper’s weight is given in hundredths, so that model is divided into 100. The beetle’s weight is given in tenths, so that model is divided into 10. • How can you compare these fractions? Make both fractions have a common denominator so that both wholes have the same number of parts. Using the common denominator, write an equivalent fraction for the beetle’s weight. 143 Part 2: Guided Instruction Lesson 10 At A GlAnce Students revisit the problem on page 80. They use common numerators, along with a visual model, to compare the fractions and solve the problem. SteP By SteP • Tell students that Connect It refers to the problem on page 80. • Work through problems 2–7 with students. Students should understand that, when the numerators are the same, the same number of pieces are shaded in each fraction. Then they need to look at the total number and size of the pieces in the whole to determine which fraction has the greater portion shaded. • Students work with a partner to solve the Try It problems. Circulate and support students’ work. • Have students explain how they found the solutions and discuss their explanations with the class. MPS tip: Connect It problem 7 may challenge Part 2: Guided Instruction Lesson 10 Connect It Now you will solve the problem from the previous page by finding a common numerator. 2 What is an equivalent fraction for 2 that has a numerator of 8? 100 ··· 8 400 ···· 3 One model is divided into 400 equal parts and the other is divided into 10 equal parts. Which has smaller parts? the one divided into 400 equal parts 4 Shade 8 pieces of each model. 5 Which model has a greater area shaded? 6 Which fraction is greater, 8 or 8 ? 400 ·· 10 ··· the one divided into 10 pieces 8 10 ··· 7 Look at the denominators of 8 and 8 . When two fractions have the same 400 10 ··· ·· numerator and different denominators, how do you know which one is greater? Explain. The fraction with the smaller denominator has bigger parts, so it is greater. The numerators show the same number of parts, but the parts are different sizes. Try It Use what you just learned to solve these problems. 8 Mel’s tomato plant is 8 of a foot tall. Her pepper plant is 3 of a foot tall. Compare 12 4 ·· ·· the heights of the plants using a symbol. 8 , 3 or 3 . 8 4 4 ··· 12 ·· ·· 12 ··· 9 Compare the fractions 3 and 5 using a symbol. 3 5 10 ·· ·· . 5 or 5 , 3 5 ··· 10 10 ·· 5 ·· ··· students because it can be confusing that the greater number actually means smaller parts. 81 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Encourage students to create and use representations (4.1.E) as they work on this problem. Ask, What does a numerator of 400 mean? If students struggle, draw a different example on the board. Use a simpler model showing 2 5 ·· compared to 2 . Elicit that the fraction with the 10 ·· smaller denominator always has bigger parts. tRy It SolutIonS 8 Solution: 8 , 3 or 3 . 8 ; Students may use 12 ·· 4 ·· 4 ·· 12 ·· common numerators. 24 is a common numerator (3 3 8). 24 5 8 and 24 5 3 . 36ths are smaller than 36 ·· 12 ·· 32nds, so 8 , 3 . 12 ·· 32 ·· 4 ·· 4 ·· ERROR ALERT: Students who wrote 8 . 3 may 12 ·· 4 ·· have reasoned that since 12 . 4 and 8 . 3, the fraction 8 must be greater. Have students draw 12 ·· fraction bar models to illustrate that 3 . 8 . Then 4 ·· 12 ·· help them find common numerators and solve. 9 Solution: 3 . 5 or 5 , 3 ; Students may recognize 5 10 10 5 ·· ·· ·· ·· 5 1 3 1 that 5 and . , so 3 . 5 . 10 2 5 2 5 10 ·· ·· ·· ·· ·· ·· 144 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 3: Modeled Instruction Lesson 10 At A GlAnce Part 3: Modeled Instruction Students explore a problem involving fraction comparison. A number line and a benchmark fraction help students understand the quantities being compared. Lesson 10 Read the problem below. Then explore different ways to use benchmarks to compare fractions. Jasmine’s swimming lesson lasts for 2 of an hour. It takes her 1 of an hour to do 3 ·· 6 ·· her homework. Will Jasmine spend more time on her homework or at her swimming lesson? SteP By SteP Model It • Read the problem at the top of the page as a class. You can use a number line to help you compare fractions. The number line shows where the fractions 2 and 1 are compared to 0 and 1. 3 ·· • Discuss the meaning of the problem. 1 6 • Read Model It as a class. Have students describe the features of the number line and explain where the fractions from the problem are located. 6 ·· 2 3 1 2 0 1 The number line shows that 1 is closer to 0 than 2 is, and that 2 is closer to 1 than 1 is. 6 3 3 6 ·· ·· ·· ·· This means that 2 . 1 . 3 ·· 6 ·· • Read Solve It as a class. Have a volunteer describe the Solve It meaning of 1 . [It is midway between 0 and 1, so it is You can use a benchmark fraction to solve the problem. easy to find.] Look at the number line. It shows that 1 is less than 1 and 2 is greater than 1 . Another way to compare fractions is by using the fraction 1 as a benchmark. 2 ·· 2 ·· 6 ·· So, 1 , 2 and 2 . 1 . 6 ·· • Work through the comparisons and guide students to the solution. 3 ·· 3 ·· 2 ·· 3 ·· 2 ·· 6 ·· Jasmine will spend more time at her swimming lesson than on homework. • Remind students to relate the mathematical answer to the problem context. ell Support 82 Review the term benchmark fraction. The term benchmark originally came from surveying, or measuring land. Surveyors made a cut into stone to help measure the height of the land the same way every time. These cuts were called benchmarks because they served as a “bench” for the surveyor’s leveling tools. We use benchmark fractions as a reference point for comparing other fractions. Hands-On Activity compare fractions to 1 . 2 ·· Materials: Number line from 0 to 1, cards with a variety of fractions written on them (denominators of 3, 4, 5, 6, 8, 10, 12, 100) • Have students label 1 on the number line. 2 ·· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse • How do you know that 1 is less than 1 ? 6 2 ·· ·· Sixths are smaller than halves. • How does a number line help you compare fractions? It helps you see which fractions are closer to 0 and which are closer to 1. You can mark off different sized pieces for easy comparison. • How does a benchmark fraction, such as 1 , help you 2 ·· compare fractions? • Give students cards with a variety of fractions. You know how large 1 is, so it is a useful • Students take a card and place it on the number reference point. It is often easier to compare a line between 0 and 1 or between 1 and 1. 2 ·· 2 ·· • Ask students to explain each fraction’s placement. Discuss any fractions they are not sure about. L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 2 ·· fraction to 1 than to another fraction. By 2 ·· comparing two fractions to 1 , you can often see 2 ·· how they compare to each other. 145 Part 3: Guided Instruction Lesson 10 At A GlAnce Part 3: Guided Instruction Students compare two fractions using 1 as a benchmark. Then they solve problems using benchmark fractions or other methods. Lesson 10 Connect It Now you will solve a similar problem using 1 as a benchmark. Think about these two fractions: 11 and 7 10 ·· SteP By SteP 11 Which fraction is less than 1? • Introduce the idea that 1 is not the only fraction that 2 ·· 10 ··· 10 ·· 10 ··· 8 ·· 11 10 ·· 7 . 8 ·· 14 Explain how you can use benchmarks to compare fractions. 10 ·· You can compare both fractions to the same number to see which one is more than 10 . Also, 8 is the same as 1, and 7 is less greater than, less than, or equal to that benchmark. The fraction that is 8 ·· greater than the benchmark is greater than the one that is less than or equal to the benchmark. • Have students share and explain their answers to problem 14. Encourage students to ask each other questions to clarify the reasoning. Try It Use what you just learned to solve these problems. 15 Fill in the blank. Explain how you found your answer. 5 10 ·· I used 1 as a benchmark. 5 5 1 and 3 . 1 . So, 3 . 5 . 2 ·· 10 ··· 16 Nathan walked MPS tip: Help students use benchmark fractions strategically (4.1.C). When students discuss their comparison, guide them to consider what benchmark fraction will be most helpful in solving a given problem. For example, in problems 10–13, 1 is a useful benchmark because both fractions in the problem are near 1 on a number line. 8 ·· 13 Fill in the blank with the correct symbol to show the comparison. They should see that 10 is the same as 1, and 11 is 8 ·· 7 8 ·· Since 11 is greater than 1 and 7 is less than 1, 11 must be greater than 7 . • Walk through problems 10–13 with students. 10 ·· 10 ··· 12 Which fraction is greater? Explain why. students can use as a benchmark fraction. than 8 . 8 ·· 8 ·· 11 10 Which fraction is greater than 1? 2 ·· 4 ·· 2 ·· 4 ·· , 3 4 ·· 10 ··· 10 of a mile. Sarah walked 19 of a mile. Who walked a greater 20 ·· 10 ·· distance? Explain. 10 out of 10 is equal to 1. 19 out of 20 is 19 , which is less than 1. Nathan 20 ··· walked a greater distance than Sarah. 83 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. tRy It SolutIonS 15 Solution: 5 , 3 ; Students may use 1 as 10 ·· 4 ·· 2 ·· a benchmark to determine that 5 5 1 and 3 . 1 . Therefore, 5 , 3 . 10 ·· 10 ·· 2 ·· 4 ·· 2 ·· 4 ·· 16 Solution: Nathan; Students may realize that 10 is the same as 1 and 19 is less than 1. 20 ·· Therefore, 10 is a greater distance. 10 ·· ERROR ALERT: Students who chose 19 may have 20 ·· 10 ·· reasoned that 19 parts is more than 10 parts. Remind students that they also need to take into account the total number of parts that each mile is divided into (the denominators of the fractions). 146 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Part 4: Guided Practice Lesson 10 Part 4: Guided Practice Lesson 10 Study the model below. Then solve problems 17–19. Student Model It is important that both measurements use the same unit! Part 4: Guided Practice 18 Compare the fractions Lesson 10 3 and 7 using the benchmark fraction 1 . 12 2 ·· ·· 10 ·· Show your work. You already know about how big 1 is! 2 ·· Becker catches a fish that is 3 of a yard long. To keep the fish, 12 ·· Possible answer: 3 , 1 and 7 . 1 so 3 , 7 10 ·· 2 12 ·· 2 10 ··· 12 ··· ··· ··· it has to be longer than 1 of a yard. Can Becker keep his fish? 3 ·· Look at how you could show your work using a number line. Pair/Share 1 2 3 4 5 6 7 8 9 10 11 0 12 12 12 12 12 12 12 12 12 12 12 1 1 3 Pair/Share How else could you solve this problem? Which strategy for comparing do you think works best with these fractions? Solution: 2 3 19 Janelle walked Since 3 is less than 1 , Becker can’t keep his fish. 12 ··· 3 ·· problems. Myron has finished 7 of the problems and Jane has 9 ·· finished 2 of the problems. Who has finished more of the 3 ·· homework? Show your work. 9 ·· Pair/Share How did you and your partner choose what strategy to use to solve the problem? 84 9 ·· 9 ·· There are several ways to compare fractions! A 3 5 6 and 6 , 6 6 ·· B C D 12 12 ·· 10 ·· ·· 3 5 6 and 6 . 6 12 12 ·· 10 ·· ·· 6 5 3 and 3 , 3 10 ·· 5 5 ·· 6 ·· ·· 3 , 1 and 6 . 1 6 ·· 2 10 ·· 2 ·· ·· 6 ·· Tina chose B as the correct answer. How did she get that answer? Possible answer: Tina found an equivalent fraction, but Possible answer: 2 5 6 Since 7 . 6 , 7 . 2 3 of a mile. Pedro walked 6 of a mile. Which 10 ·· 6 ·· statement shows how to find the greater fraction? Circle the letter of the correct answer. 17 Myron and Jane are working on the same set of homework 3 ·· Draw a model to check your answer. 3 , 7 or 7 . 3 10 ··· 12 12 ··· 10 ··· Solution: ··· 9 ·· compared them incorrectly. She thought that 6 is greater 3 ·· 12 ··· than 6 because 12 is greater than 10. 10 ··· Pair/Share How can you find the answer using a benchmark fraction? Solution: Myron has finished more of the homework. L10: Compare Fractions 85 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. ©Curriculum Associates, LLC Copying is not permitted. At A GlAnce SolutionS Students study an example that uses a number line and equivalent fractions to compare two fractions. Then they solve word problems involving fraction comparison using a variety of methods. Ex Since 3 , 1 , Becker can’t keep his fish. 12 ·· 3 ·· The number line shows twelfths and thirds. 17 Solution: Myron finished more of the homework; Multiply the numerator and denominator of 2 by SteP By SteP 3 so both fractions have a denominator of 9. • Ask students to solve the problems individually. Circulate and provide support. (DOK 2) • Watch for students who struggle with the reasoning required for using benchmark fractions or finding common numerators. • When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. 18 Solution: 3 , 7 or 7 . 3 ; Compare the fractions 10 ·· 12 ·· 12 ·· 10 ·· to 1 . Use common denominators or common 2 ·· numerators. 3 , 1 , 7 . 1 . (DOK 2) 10 ·· 2 ·· 12 ·· 2 ·· 19 Solution: A; Use common numerators and then look at the denominators to compare. Explain to students why the other two answer choices are not correct: C is not correct because fifths are greater than sixths. D is not correct because 3 5 1 . (DOK 3) 6 ·· L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. 3 ·· 2 ·· 147 Part 5: TEKS Practice Lesson 10 Part 5: TEKS Practice Lesson 10 Solve the problems. 1 2 4 2 cup of raisins and 3 cup of almonds to make trail mix. Which statement Grant needs } } 4 3 can be used to find out if there are more raisins or almonds in the mix? 2 8 3 2 4 3 4 2 6 3 6 5 }} and } 5 }} } 4 3 12 12 B 5 } and } 5 } } 4 5 3 6 C 5 } and } 5 }} } 4 3 9 12 D 2 5 6 and 3 5 6 } } } } 4 7 3 9 Show your work. 2 , 1 and 5 5 1 , so 2 , 5 Possible answer: } }} } 6 } 6 }} 2 10 } 2 10 Answer Sam 5 Tell whether each sentence is True or False. 2, 6 15 ··· b. 7 .7 10 8 ··· ·· c. 1.3 2 8 ·· ·· d. 254 4 6 ·· ·· e. True 5 ·· True 30 5 3 50 ··· 500 ···· 3 False 3 False 3 True False True 3 False 3 True False Lesson 10 5 of an hour. He spent Sam’s music teacher told him to practice his trombone for }} 10 2 of an hour practicing. Did he practice long enough? } 6 9 A a. 3 Part 5: TEKS Practice practice long enough. Olivia and Eleanor each made the same amount of lemonade to sell at a lemonade stand. Olivia poured all of her lemonade into 10 equal glasses. Eleanor poured all of her lemonade into 5 equal glasses. Olivia sold 7 glasses of lemonade and Eleanor sold 2 glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions using a symbol. Possible answer: Olivia sold 7 of her lemonade and Eleanor 10 ··· Show your work. sold 2 of hers. 2 5 4 ; 7 . 4 , so Olivia sold more. 5 5 ··· 10 ··· 10 ··· 10 ·· ·· Answer 6 did not Olivia sold a greater fraction of her lemonade. 9 of her Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold }} 10 5 of her boxes. Which girl has sold a greater fraction boxes of fruit and Sierra has sold } 8 of her boxes of fruit? Draw a model to show your answer. Fill in the blank with one of the symbols shown to compare 5 and 5 . 10 ··· , 5 }} 10 , 5 8 ·· Show your work. . 9 10 5 } 8 5 8 9 has a bigger area shaded, so it is greater than 5 . 10 8 ··· ·· Rachel Answer has sold a greater fraction of her boxes of fruit. 86 L10: Compare Fractions 87 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. AT A GlAncE Students compare fractions to solve word problems that might appear on a mathematics test. SoluTionS 1 Solution: A; Find a common denominator: 12. Multiply numerator and denominator of 2 by 4 and 3 ·· numerator and denominator of 3 by 3. Then 4 ·· numerators can be compared. (DOK 2) ©Curriculum Associates, LLC Copying is not permitted. 4 Solution: did not; Compare the fractions using the benchmark fraction 1 . 5 5 1 and 2 , 1 . So, 2 , 5 . (DOK 1) 6 ·· 2 ·· 10 ·· 2 ·· 6 ·· 2 ·· 10 ·· 5 Solution: Olivia; Compare the fractions 7 and 2 . 10 ·· 5 ·· Students may find a common denominator and write 2 as tenths. (DOK 1) 5 ·· 6 Solution: Rachel; See possible student work above. Compare the shaded parts of the bar models. (DOK 1) 2 Solution: a. False; b. False; c. True; d. False; e. True (DOK 1) 3 Solution: ,; 5 of 10 equal parts is a smaller amount than 5 of 8 equal parts. (DOK 1) 148 L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Differentiated Instruction Lesson 10 Assessment and Remediation • Ask students to compare 7 and 4 and to show a visual model and explain their work. 10 5 ·· ·· • For students who are struggling, use the chart below to guide remediation. • After providing remediation, check students’ understanding. Ask students to compare 3 and 5 . 8 12 ·· ·· • If a student is still having difficulty, use STAAR Ready Instruction, Level 3, Lesson 18. If the error is . . . Students may . . . To remediate . . . 7 is greater than not understand the reason for comparing fractions with the same number of pieces (numerators). Have students draw same-size bar models of 7 and 4 . Point out that the 10 5 ·· ·· size of the pieces is not the same, so you can’t compare numerators. Ask students to come up with a common numerator (such as 28) so that the number of pieces is the same and write equivalent fractions for 7 and 4 10 5 ·· ·· so they can compare. not understand the reason for comparing fractions with same-size pieces (denominators). Have students draw same-size bar models of 7 and 4 . Ask if the number 10 5 ·· ·· of shaded pieces is the same (no) and explain that therefore you can’t compare based on the size of the pieces. Ask students to come up with a common denominator (such as 10) so that the size of the pieces is the same and write equivalent fractions for 7 and 4 so they can compare. not understand when it is appropriate to use a benchmark fraction. Explain that when both fractions are greater than (or less than) the benchmark fraction, you don’t have enough information to compare. Have students make a number line from 0 to 1. Help them mark and label tenths and fifths. Have students locate 7 and 4 on the number line to 10 5 ·· ·· make the comparison. 10 ·· 4 because 7 . 4 5 ·· 7 is greater than 10 ·· 4 because 10 . 5 5 ·· 7 5 4 because 5 ·· 7 . 1 and 4 . 1 10 ·· 2 5 ·· 2 ·· ·· 10 ·· 10 ·· 5 ·· Hands-On Activity Challenge Activity Draw models to compare fractions. Compare three or more fractions. Materials: 1-cm grid paper, scissors, markers or pencils Materials: Fractions written on cards Have students work with a partner. Provide students with 1-centimeter grid paper. Instruct each student in the pair to draw and cut out two 3-by-4 arrays. Have them use the arrays to show halves, thirds, fourths, sixths and twelfths. Have each student color part of each of their models and write a fraction to show the shaded part. Have students compare the fractions using ., ,, or 5. Have students repeat by drawing and cutting out two 2-by-8 arrays. They should use the array to model and compare halves, fourths, eighths, and sixteenths. Ask students if they can use these models to compare eighths and twelfths. [No, because the models are different sizes.] Have students turn over one of their 2-by-8 arrays and draw 12 equal parts. Ask students if they could use this array and another of their 2-by-8 arrays to compare eighths and twelfths [yes], and have them do so. L10: Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. Give a pair or small group of students a pile of cards with fractions written on them. Students set out three, four, or even five fractions and place them in order from least to greatest. The strategy is to choose one fraction and then compare it to another fraction. Then choose a third fraction and compare it to each of the already ordered fractions. Then choose a fourth fraction and compare to each, and so on. To make the comparisons, students may draw number lines or visual models or compare to benchmark fractions using what they know about equivalent fractions. 149 Focus on Math Concepts Lesson 11 (Student Book pages 88–93) Understand Fraction Addition and Subtraction Lesson objectives the Learning Progression • Understand addition as joining parts. One goal of the Texas Essential Knowledge and Skills for Mathematics is to develop a deeper understanding of fractions by using a progression of concepts from simple to complex. This lesson prepares students for the conceptual shift involved in progressing from adding and subtracting whole numbers to adding and subtracting fractions. Students are guided to think of operations with fractions as very much like operations with whole numbers. • Understand subtraction as separating parts. • Extend their understanding of addition and subtraction of whole numbers to addition and subtraction of fractions. • Use fraction models to add and subtract fractions with like denominators. Prerequisite skiLLs Students see that you can count with unit fractions In order to be proficient with the concepts in this lesson, students should: just as you count with whole numbers. And because • Know addition and subtraction basic facts. do arithmetic with them. If you walked 2 of a • Understand the meaning of fractions. mile (2 fifths) yesterday and 4 of a mile (4 fifths) today, • Identify numerators and denominators. • Write whole numbers as fractions. vocabuLary There is no new vocabulary. Review the following key terms. numerator: the top number in a fraction; it tells the number of equal parts that are being described you can count with unit fractions, you can also 5 ·· 5 ·· altogether you walked 6 of a mile (6 fifths; because 5 ·· 2 things plus 4 more of those things is 6 of those things). Students use the meaning of fractions and the meanings of addition and subtraction that were built in earlier grades to understand why the procedures for adding and subtracting fractions make sense. denominator: the bottom number in a fraction; it tells the total number of equal parts in the whole teks Focus 4.3.E Represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations. Readiness Standard MatheMaticaL Process stanDarDs (MPs): 4.1.A, 4.1.C, 4.1.D, 4.1.E, 4.1.F, 4.1.G (See page A9 for full text. Also eee MPS Tips in the lesson.) 150 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 11 At A GlAnce Focus on Math Concepts Students explore the idea that adding fractions is not essentially different from adding whole numbers. Lesson 11 4.3.E What’s really going on when we add numbers? expression 2 1 3. 4 ·· Adding means joining or putting things together. Think about how you could explain adding 2 1 3 to a first grader. You could start at 2, count on 3 more, and see where you end up: 2 . . . 3 . . . 4 . . . 5. Step By Step Or, you could put a segment with a length of 2 and a segment with a length of 3 next to each other on a number line to show 2 1 3. • Introduce the Question at the top of the page. 1 • Help students relate the number line diagram to the sum 2 1 3. 0 1 3 4 5 6 7 8 9 10 You can put a segment with a length of 24 and a segment ·· Underline the sentence that explains what adding fractions means. with a length of 3 next to each other to show 2 1 3 . 4 ·· 1 4 made up of 5 one-fourths. 0 4 • If students need additional support with locating 1 4 1 4 4 ·· 1 4 2 4 1 4 3 4 0 4 ·· 1 4 4 4 1 5 4 6 4 7 4 8 4 2 When you add 2 1 3 , you are putting one-fourths together. fractions on a number line, have them build a 4 ·· 88 4 ·· L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse To extend students’ understanding of decomposing fractions, follow these steps: • Draw and label a number line on the board from 0 to 2 like the one on the page showing fourths. • Ask students to think of two different fractions that you could put together that would give you the same sum as adding 2 and 3 . 4 ·· 2 1 Adding fractions means joining or putting together parts of the same whole. number 5 is made up of 5 ones, the number 5 is 4 ·· concept extension 1 1 Think • Guide students to recognize that just as the number line by putting 1 fraction strips end-to-end, 4 ·· creating a concrete model to show 2 1 3 . 4 4 ·· ·· 1 When you add 2 1 3, you are putting ones together. • Read Think with students. Reinforce the idea that fractions are numbers. 4 ·· • Have a volunteer go to the board to show the two fractions on the number line. 3 1 and 4 in 4 ·· either order 4 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC TEKS Understand Fraction Addition and Subtraction A number line diagram gives meaning to the 4 ·· Part 1: Introduction Copying is not permitted. 4 ·· • How would you explain adding in your own words? Responses should include phrases such as “join” or “put together.” • How is adding fractions like adding whole numbers? Students may mention that, in both cases, you are putting things together. • Can you think of another way to explain adding fractions? Students may suggest that you can count on with fractions just like you count on with whole numbers. 151 Part 1: Introduction Lesson 11 At A GlAnce Students explore the idea that subtracting fractions is not essentially different from subtracting whole numbers. A number line diagram gives meaning to the expression 5 2 2 . 4 ·· Part 1: Introduction Think Subtracting means separating or taking away. • Read Think with students. • Discuss how the number line represents the problem 5 2 2. Show how to subtract on the number line. (start at 5 and count back 2) • Ask a volunteer to explain how to use the number line to find 5 2 2 . Provide 1 fraction strips for 4 4 4 ·· ·· ·· students who need more support. • Have students read and reply to the Reflect directive. Look at the whole numbers. Now look at the numerators of the fractions. I think I see a connection. On a number line, you can start with a segment of length 5 and take away a segment of length 2 to show 5 2 2. 4 ·· Step By Step Lesson 11 1 1 0 1 1 2 1 1 1 1 3 4 5 6 7 8 9 10 When you subtract 5 2 2, you are taking away ones. You can show subtracting fractions on a number line. Start with a segment of length 5 and take away a segment of length 2 to show 5 2 2 . 4 ·· 4 ·· 1 4 0 4 1 4 1 4 1 4 1 4 1 4 2 4 3 4 4 4 0 4 ·· 4 ·· 1 4 1 4 1 5 4 6 4 7 4 8 4 2 When you subtract 5 2 2 , you are taking away one-fourths. 4 ·· 4 ·· Now you’ll have a chance to think more about how adding or subtracting fractions is like adding or subtracting whole numbers. You may find that using number lines or area models can help you explain your thinking. Reflect Visual Model • Tell students that you will use a number line to show 5 2 3 . 8 ·· 8 ·· 1 Use your own words to describe what you just learned about adding and subtracting fractions. Possible answer: I learned that adding and subtracting fractions is just like adding and subtracting whole numbers. When the denominators are the same, you can just add or subtract the numerators. 89 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. • Draw a number line from 0 to 1 on the board. • Ask students for ideas on how to divide the line so that you can use it to help you solve the problem. • Have students explain why dividing the line into eighths makes sense. • Label 0 and 1 on the line and have students provide labels for the other marks as you move your finger along the line. • Ask a volunteer to show how to find the answer to the problem using the number line. MpS tip: In the Visual Model activity, students are asked to create and use a representation and explain why dividing the line into eighths makes sense. (4.1.E) 152 Mathematical Discourse • How would you explain subtracting in your own words? Listen for phrases such as “take apart” or “take away.” • How is subtracting fractions like subtracting whole numbers? Students may note that subtracting means taking away. It doesn’t matter what kinds of numbers you’re subtracting. • Do you see a connection between the whole numbers and the numerators of the fractions on this page? Students may mention that the whole numbers and the numerators of the fractions are the same numbers, and to answer both problems you subtract 2 from 5. L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 11 At A GlAnce Part 2: Guided Instruction Students use number lines to answer questions, reinforcing the understanding that fractions are numbers. Explore It Counting on and using a number line are two ways to think about adding fractions. 2 Count by fourths to fill in the blanks: 1 , 2 , 4 ·· 4 ·· Step By Step 3 6 7 8 9 , 4 , 5 , ·· 4 , ·· 4 , ·· 4 , ·· 4 4 ·· 4 ·· 4 ·· Now label the number line. • Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class. 0 5 ·· • Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. STUDENT MISCONCEPTION ALERT: Some students may think that a fraction is always less than 1. If this misconception persists, use fraction strips to demonstrate fractions less than, equal to, and greater than 1. Then, encourage students to use the fraction strips to show and name other fractions greater than 1. L11: Understand Fraction Addition and Subtraction Copying is not permitted. 2 4 3 4 ·· 4 4 5 4 6 4 ·· 7 4 ·· 8 4 ·· 9 4 ·· 10 4 4 5 5 , ·· 5 , ·· Now label the number line. 0 1 5 2 5 3 5 ·· 4 5 ·· 5 5 ·· 6 5 Use the number lines above to answer numbers 4 and 5. 7 4 What is 1 more than 6 ? 4 4 ·· ·· 4 ·· 5 What is 1 more than 3 ? 5 5 ·· ·· 5 ·· 4 Now try these two problems. 6 Label the number line below and use it to show 2 1 1 . 4 ·· 4 ·· • If students need more support, suggest that they count out loud to help them fill in the missing numbers in problems 2 and 3. • To help students answer problem 4, have them put their finger on 6 on the number line, then count on 4 ·· by 1 . Similarly, to answer problem 5, have them put 4 ·· their finger on 3 on the number line and count on 5 ·· by 1 . 1 4 3 3 Count by fifths to fill in the blanks: 1 , 2 , ·· 5 5 ·· 5 ·· • As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking. ©Curriculum Associates, LLC Lesson 11 0 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 7 Label the number line below and use it to show 3 1 1 . 4 ·· 4 ·· 0 4 90 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse • In which direction on the number line do you move when adding? Explain. Responses might include the fact that adding means joining so you will be getting segments that are longer or an answer farther to the right than the number you started with. • For problem 5, will the answer change if you find 3 more than 1 ? Explain. 5 ·· 5 ·· Listen for responses that demonstrate an understanding that you can add two numbers in any order and get the same sum. 153 Part 2: Guided Instruction Lesson 11 At A GlAnce Students use number lines to show subtracting fractions. Then they use models to show adding and subtracting fractions. Part 2: Guided Instruction Lesson 11 Talk About It Solve the problems below as a group. 8 Look at your answers to problems 2 and 3. How is counting by fractions the same Step By Step • Organize students in pairs or groups. You may choose to work through the first Talk About It problem together as a class. • Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking. as counting with whole numbers? Possible answer: When you count with whole numbers, you count by ones. When you count with fractions, the numerator counts by ones as long as the denominators are the same. How is it different? Possible answer: When you count by fractions, you are counting by parts. 9 Label the number line below and use it to show 7 2 2 . 8 ·· 8 ·· 0 • When sharing ideas about problems 9 and 10, be sure to emphasize that when labeling the number line, numerators count on by ones, but the denominator remains the same. 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 10 8 11 8 12 8 11 6 12 6 5 1 10 Label the number line below and use it to show 2 . 6 ·· 6 ·· MpS tip: During this time, you may choose to ask a particular group to prepare to share their thinking or solution. Encourage students to justify their thinking using precise mathematical language. (4.1.G) 1 8 0 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 10 6 Try It Another Way Work with your group to use the area models to show adding or subtracting fractions. 1 2 11 Show 1 . 8 ·· 8 ·· 6 2 2. 12 Show 10 ·· 10 ·· 91 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. • Direct the group’s attention to Try It Another Way. Have a volunteer from each group come to the board to draw the group’s solutions to problems 11 and 12. Hands-On Activity Use fraction strips to subtract fractions. Materials: strips of paper, markers, scissors • Model how to fold the strip of paper in half, in half again, and in half a third time. Tell students to unfold the strips and use a marker to show the 8 equal sections. • Direct students to cut out each section. Ask them to name the fraction that represents each section. 3 ··18 4 Have them label each section. • Write 7 2 5 on the board. Have students use their 8 8 ·· ·· strips to show that the difference is 2 . 8 ·· 154 Mathematical Discourse • What is another name for 8 ? 12 ? Explain your 8 ·· 6 ·· thinking. Students should recognize that eight 1 pieces 8 ·· make up 1 whole and that twelve 1 pieces make 6 ·· up 2 wholes. • Can you think of another way to show finding a difference on a number line? Students may mention adding up to subtract. For example, to find 7 2 2 , you might start 8 ·· 8 ·· at 2 and think, “What do I need to add to 8 ·· get to 7 ?” 8 ·· L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Part 3: Guided Practice Lesson 11 At A GlAnce Part 3: Guided Practice Students demonstrate their understanding of adding and subtracting fractions as they talk through three problems. Lesson 11 Connect It Talk through these problems as a class, then write your answers below. 2 1 13 Compare: Draw two different models to show 2 . 3 ·· 3 ·· Step By Step Possible answers: • Discuss each Connect It problem as a class using the discussion points outlined below. 1 3 0 2 3 3 3 14 Explain: Rob had a large pizza and a small pizza. He cut each pizza into fourths. He took one fourth from each Compare: pizza and used the following problem to show their sum: ··14 1 ··14 5 ··24 . • You may choose to have students work in pairs to encourage sharing ideas. Each partner draws a different model. What did Rob do wrong? Possible answer: Rob’s addition is correct, but he cannot add one fourth of the large pizza and one fourth of the small pizza in this way because the wholes are not the same. • For a quick and easy assessment, have students draw their models on small whiteboards or paper and hold them up. Choose several pairs to explain their models to the class. • Use the following to lead the class discussion: Explain how you knew the number of parts to draw in the whole. How did you show subtraction in your model? How are the models the same? How are they different? 15 Demonstrate: Think about how you would add three whole numbers. You add two of the numbers first, and then add the third to that sum. You add three fractions the same way. 1 1 3 1 4. Use the number line and area model below to show ·· 10 ·· 10 ·· 10 0 92 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Demonstrate: Explain: • The second problem focuses on the importance of the whole and the fact that you cannot add or subtract fractions unless they refer to the same whole. • Read the problem together as a class. Ask students to continue to work in pairs to discuss and write their responses about what Rob did wrong. • Begin the discussion by asking questions, such as: What fraction describes a slice of the larger pizza? 3 ··14 4 What fraction describes a slice of the smaller pizza? 3 1 4 4 ·· Are both 1 s the same size? [no] Why not? [the whole 4 ·· • This discussion gives students an opportunity to think about problems that involve adding three fractions. MpS tip: Ask students to show how to use a number line as a tool to model the sum of three whole numbers. (4.1.C) • Discuss how you can add three (or more) fractions in the same way as adding whole numbers as long as you are talking about the same type of fractions. Have students explain how they used the models to show the sum. • Remind students to start at 0 when labeling the number line. pizzas are not the same size] Why doesn’t it make sense to add these two fractions? [the wholes are not the same] L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. 155 Part 4: Performance Task Lesson 11 At A GlAnce Part 4: Performance Task Students write two questions that can be answered using some or all of the given information about the problem situation. Then they answer one of the questions. Lesson 11 Put It Together 16 Use what you have learned to complete this task. Jen has 4 of a kilogram of dog food. Luis has 3 of a kilogram of dog food. 10 10 ·· ·· A large dog eats 2 of a kilogram in one meal. 10 ·· Step By Step A Write two different questions about this problem that involve adding or subtracting fractions. • Direct students to complete the Put It Together task on their own. i Possible answer: How much dog food do Jen and Luis have altogether? ii Possible answer: How much more dog food does Jen have than Luis? • Explain to students that the questions they write do not have to use all of the given information. B Choose one of your questions to answer. Circle the question you chose. Show how to find the answer using a number line and an area model. Possible answers: • As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed. 0 • If time permits, have students share one of their questions with a partner and show how to find the answer to their partner’s question using a visual model. 1 10 2 10 4 10 ··· 3 10 1 4 10 5 10 6 10 7 10 8 10 9 10 10 10 3 10 ··· ScorinG ruBricS L11: Understand Fraction Addition and Subtraction See student facsimile page for possible student answers. A points expectations 2 156 The response demonstrates the student’s mathematical understanding of adding and subtracting fractions. Both questions can be answered using the information given in the problem. 1 An effort was made to accomplish the task. The response demonstrates some evidence of verbal and mathematical reasoning, but the student’s questions may contain some misunderstandings. 0 There is no response or the response shows little or no understanding of the task. ©Curriculum Associates, LLC B Copying is not permitted. 93 points expectations 2 Both a number line and an area model are correctly drawn and labeled to show the solution to the problem. 1 Only one model is correctly drawn and labeled or the models drawn may contain minor errors. Evidence in the response demonstrates that with feedback, the student can revise the work to accomplish the task. 0 There are no models drawn or the models show no evidence of providing visual support for solving the problem. L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. Differentiated Instruction Lesson 11 Intervention Activity On-Level Activity Use fraction strips to model adding and subtracting fractions. Find a sum greater than one. Guide students through the following steps in finding the sum of two fractions whose sum is greater than one. You may have students work in groups and present their results to the class. Materials: fraction strips Write an addition expression on the board, such as 2 1 3 . Have students lay 1 fraction strips end-to-end 8 8 ·· ·· 1. Have students draw a picture of two pizzas of equal size, both cut into 8 equal pieces. Tell students that two slices of one pizza and five slices of the other pizza were eaten. Ask students to write a fraction to represent the leftover slices. 8 ·· to show the sum. Ask them to tell you how many 1 s there are in all. Continue with similar problems. 8 ·· Include expressions whose sums are greater than one, such as 3 1 2 . 4 ·· 4 ·· 2. Ask students to imagine replacing missing slices of one pizza with remaining slices from the other pizza. Have students draw another pizza to represent this situation and write a fraction to represent the leftover slices. Write a subtraction expression on the board, such as 5 2 2 . Have students lay 1 fraction strips end-to6 ·· 6 ·· 6 ·· end to show 5. Then have them “take away” 2 . Ask 6 ·· 6 ·· them to tell you how many 1 s are left. Continue with 3. Ask students to write a number sentence that 6 ·· similar problems. Be sure to provide expressions that represents the total amount of pizza left. include fractions greater than one, such as 6 2 3 . 5 ·· 3 ··68 1 ··38 5 ··98 4 5 ·· 4. Ask students to write a fraction equivalent to 9 that makes it easier to see how many whole 8 ·· pizzas are left over. 3 11 4 8 ·· Challenge Activity Write a question for the answer given. Write the following problem on the board: The answer is 7 . What could the question be? 8 ·· Encourage students to think about both addition and subtraction. Provide number lines, area models, or fraction strips for support as necessary. Note the methods students use. Do they just guess, work out their problem, check to see if it’s correct, and then adjust their responses if necessary? Do they use a visual model or do they work symbolically? If time permits, give students (or pairs or groups) practice with similar problems. You might ask them to write two questions for each answer you supply, one using addition and one using subtraction. L11: Understand Fraction Addition and Subtraction ©Curriculum Associates, LLC Copying is not permitted. 157