4 Mathematics. New TEKS Edition STAAR. Instruction. coming September Texas. Grade 4 Sampler includes:

Transcription

1 Mathematics Texas STAAR Instruction New TEKS Edition coming September 0 Grade Sampler includes: Table of Contents, Student Sample Lessons, Teacher Sample Lessons STAAR is a federally registered trademark owned by the Texas Education Agency, and is used pursuant to license.

2 Student Book Sample Lessons Includes two sample lessons: Lesson 0: Compare Fractions Lesson : Understand Fraction Addition and Subtraction

3 Table of Contents Unit : Number and Operations, Part STAAR Reporting Categories and TEKS Lesson Understand Place Value ()(A), ()(B) Lesson Compare and Order Whole Numbers ()(B), ()(C) Lesson Round Whole Numbers ()(D) Lesson Add and Subtract Whole Numbers ()(A) Lesson Multiply by One-Digit Numbers ()(B), ()(D) Lesson Multiply by Two-Digit Numbers ()(C), ()(D) ()(H) Lesson Divide Whole Numbers ()(E), ()(F) ()(H) STAAR Practice Unit : Number and Operations, Part STAAR Reporting Categories and Lesson Understand Fractions ()(A), ()(B) Lesson 9 Understand Equivalent Fractions ()(C) Lesson 0 Compare Fractions ()(D) Lesson Understand Fraction Addition and Subtraction ()(E) Lesson Add and Subtract Fractions 9 ()(E) Lesson Add and Subtract Mixed Numbers 0 ()(E), ()(F) STAAR Practice = STAAR Readiness Standard = STAAR Supporting Standard iii

4 Table of Contents Unit : Number and Operations, Part STAAR Reporting Categories and Lesson Understand Decimals Lesson Relate Decimals and Fractions Lesson Compare and Order Decimals Lesson Add and Subtract Decimals STAAR Practice TEKS ()(B), ()(E) ()(G), ()(H) ()(G) ()(F), ()(E) ()(A) Unit : Algebraic Reasoning, Number and Operations, Part STAAR Reporting Categories and Lesson Estimation and Problem Solving Lesson 9 Model Multi-Step Problems Lesson 0 Number Patterns Lesson Perimeter and Area STAAR Practice ()(G) ()(H), ()(A) ()(B) ()(D) = STAAR Readiness Standard = STAAR Supporting Standard iv

5 Table of Contents Unit : Geometry and Measurement STAAR Reporting Category Lesson Points, Lines, Rays, Angles Lesson Classify Two-Dimensional Figures Lesson Symmetry Lesson Measure and Draw Angles Lesson Add and Subtract With Angles Lesson Convert Measurements Lesson Time and Money Lesson 9 Length, Liquid Volume, and Mass STAAR Practice TEKS ()(A), ()(C) ()(D), ()(A) ()(C) ()(B) ()(C), ()(D) ()(E) ()(A), ()(B) ()(C) ()(C) Unit : Data Analysis and Personal Financial Literacy STAAR Reporting Category Lesson 0 Represent Data Lesson Use Data to Solve Problems Lesson Fixed and Variable Expenses Lesson Profit Lesson Financial Institutions STAAR Practice (9)(A) (9)(B) (0)(A) (0)(B) (0)(E) = STAAR Readiness Standard = STAAR Supporting Standard v

6 Lesson 0 Part : Introduction Compare Fractions Develop Skills and Strategies TEKS..D In the past, you learned to compare fractions using models. Take a look at this problem. Adriana ate of a granola bar and June ate of a same-size granola bar. Which 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. L0: Compare Fractions

7 Part : Introduction Lesson 0 Find Out More Deciding who ate more of the granola bar means comparing the fractions and.., is greater than. is less than. 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. and , 0, so, 0 Reflect Explain how you can tell which fraction is greater, or 0. L0: Compare Fractions 9

8 Part : Modeled Instruction Lesson 0 Read the problem below. Then explore different ways to understand it. A grasshopper weighs about 00 Which weighs more? of an ounce. A beetle weighs of an ounce. 0 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. Compare 0 and 00. First, look at the denominator, 0. Can you multiply 0 by any number to get 00? Yes, Find a fraction equivalent to that has a denominator of 00: Compare the numerators of 0 00 and 00 : 0. So, L0: Compare Fractions

9 Part : Guided Instruction Lesson 0 Connect It Now you will solve the problem from the previous page by finding a common numerator. What is an equivalent fraction for that has a numerator of? 00 One model is divided into 00 equal parts and the other is divided into 0 equal parts. Which has smaller parts? Shade pieces of each model. Which model has a greater area shaded? Which fraction is greater, 00 or 0? Look at the denominators of 00 and. When two fractions have the same 0 numerator and different denominators, how do you know which one is greater? Explain. Try It Use what you just learned to solve these problems. Mel s tomato plant is of a foot tall. Her pepper plant is of a foot tall. Compare the heights of the plants using a symbol. 9 Compare the fractions and using a symbol. 0 L0: Compare Fractions

10 Part : Modeled Instruction Lesson 0 Read the problem below. Then explore different ways to use benchmarks to compare fractions. Jasmine s swimming lesson lasts for of an hour. It takes her of an hour to do 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 and are compared to 0 and. 0 The number line shows that is closer to 0 than is, and that is closer to than is. This means that.. Solve It You can use a benchmark fraction to solve the problem. Another way to compare fractions is by using the fraction as a benchmark. Look at the number line. It shows that is less than and is greater than. So,, and.. Jasmine will spend more time at her swimming lesson than on homework. L0: Compare Fractions

11 Part : Guided Instruction Lesson 0 Connect It Now you will solve a similar problem using as a benchmark. Think about these two fractions: and 0 0 Which fraction is greater than? Which fraction is less than? Which fraction is greater? Explain why. Fill in the blank with the correct symbol to show the comparison. Explain how you can use benchmarks to compare fractions. 0 Try It Use what you just learned to solve these problems. Fill in the blank. Explain how you found your answer. 0 Nathan walked 0 0 distance? Explain. of a mile. Sarah walked 9 of a mile. Who walked a greater 0 L0: Compare Fractions

12 Part : Guided Practice Lesson 0 Study the model below. Then solve problems 9. It is important that both measurements use the same unit! Student Model Becker catches a fish that is of a yard long. To keep the fish, it has to be longer than of a yard. Can Becker keep his fish? Look at how you could show your work using a number line. Pair/Share How else could you solve this problem? Solution: Since is less than, Becker can t keep his fish. Which strategy for comparing do you think works best with these fractions? Myron and Jane are working on the same set of homework problems. Myron has finished of the problems and Jane has 9 finished of the problems. Who has finished more of the homework? Show your work. Pair/Share How did you and your partner choose what strategy to use to solve the problem? Solution: L0: Compare Fractions

13 Part : Guided Practice Lesson 0 Compare the fractions 0 and using the benchmark fraction. Show your work. You already know about how big is! Solution: 9 Janelle walked of a mile. Pedro walked of a mile. Which 0 statement shows how to find the greater fraction? Circle the letter of the correct answer. A B C D and, 0 and. 0 and, 0, and 0. Tina chose B as the correct answer. How did she get that answer? Pair/Share Draw a model to check your answer. There are several ways to compare fractions! Pair/Share How can you find the answer using a benchmark fraction? L0: Compare Fractions

14 Part : TEKS Practice Lesson 0 Solve the problems. Grant needs } cup of raisins and } cup of almonds to make trail mix. Which statement can be used to find out if there are more raisins or almonds in the mix? A B C D } }} and } }} 9 } } and } } } } 9 and } }} } } 9 and } } Tell whether each sentence is True or False. a., True False b. 0. True False c.. True False d. True False e True False Fill in the blank with one of the symbols shown to compare and. 0,. }} 0 } L0: Compare Fractions

15 Part : TEKS Practice Lesson 0 Sam s music teacher told him to practice his trombone for }} of an hour. He spent 0 } of an hour practicing. Did he practice long enough? Show your work. Answer Sam 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 0 equal glasses. Eleanor poured all of her lemonade into equal glasses. Olivia sold glasses of lemonade and Eleanor sold glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions using a symbol. Show your work. Answer sold a greater fraction of her lemonade. Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold 9 }} of her 0 boxes of fruit and Sierra has sold } of her boxes. Which girl has sold a greater fraction 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. L0: Compare Fractions

16 Lesson Part : Introduction Understand Fraction Addition and Subtraction Focus on Math Concepts TEKS..E What s really going on when we add numbers? Adding means joining or putting things together. Think about how you could explain adding to a first grader. You could start at, count on more, and see where you end up: Or, you could put a segment with a length of and a segment with a length of next to each other on a number line to show When you add, 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 and a segment with a length of next to each other to show. Underline the sentence that explains what adding fractions means. 0 0 When you add, you are putting one-fourths together. L: Understand Fraction Addition and Subtraction

17 Part : Introduction Lesson Think Subtracting means separating or taking away. On a number line, you can start with a segment of length and take away a segment of length to show. Look at the whole numbers. Now look at the numerators of the fractions. I think I see a connection When you subtract, you are taking away ones. You can show subtracting fractions on a number line. Start with a segment of length and take away a segment of length to show. 0 0 When you subtract, you are taking away one-fourths. 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 Use your own words to describe what you just learned about adding and subtracting fractions. L: Understand Fraction Addition and Subtraction 9

18 Part : Guided Instruction Lesson Explore It Counting on and using a number line are two ways to think about adding fractions. Count by fourths to fill in the blanks:,,,,,,,, Now label the number line. 0 0 Count by fifths to fill in the blanks:,,,, Now label the number line. 0 Use the number lines above to answer numbers and. What is more than? What is more than? Now try these two problems. Label the number line below and use it to show. Label the number line below and use it to show. 90 L: Understand Fraction Addition and Subtraction

19 Part : Guided Instruction Lesson Talk About It Solve the problems below as a group. Look at your answers to problems and. 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. 0 Label the number line below and use it to show. Try It Another Way Work with your group to use the area models to show adding or subtracting fractions. Show. Show 0 0. L: Understand Fraction Addition and Subtraction 9

20 Part : Guided Practice Lesson Connect It Talk through these problems as a class, then write your answers below. Compare: Draw two different models to show. 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:. What did Rob do wrong? 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. Use the number line and area model below to show L: Understand Fraction Addition and Subtraction

21 Part : Performance Task Lesson Put It Together Use what you have learned to complete this task. Jen has 0 A large dog eats 0 of a kilogram of dog food. Luis has of a kilogram of dog food. 0 of a kilogram in one meal. 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. L: Understand Fraction Addition and Subtraction 9

22 Teacher Resource Book Sample Lessons Includes two sample lessons: Lesson 0: Compare Fractions Lesson : Understand Fraction Addition and Subtraction

23 Develop Skills and Strategies Lesson 0 (Student Book pages ) Compare Fractions Lesson objectives Use symbols (.,,, ) to compare fractions with the same denominator and different numerators. Recognize that fractions with different denominators and the same numerators represent different values. Use benchmark fractions to compare fractions. Recognize that you can only compare two fractions when both refer to the same whole. Prerequisite skills Represent fractions with denominators,,,, or using a number line or visual models. Identify, create, and explain equivalent fractions. Express whole numbers as fractions. Compare fractions whose numerators or denominators are the same. vocabulary There is no new vocabulary. Review the following key terms. compare: to decide if one number is greater than, less than, or equal to another number 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 the Learning Progression In Grade, students used models to compare two fractions with the same numerator or the same denominator by reasoning about their size. In Grade, they extend their understanding of fractions to compare two fractions with different numerators and different denominators. Emphasis is placed on understanding that a comparison only makes sense if the two fractions have the same size wholes. Students use models (e.g., fraction bars, area models, etc.) to compare fractions by creating common numerators or denominators. Students also learn to use benchmark fractions e.g., to compare fractions. They record comparisons using the.,,, and symbols. Students work focuses on visual models and benchmark fractions, rather than an algorithm. Students will not formally address fraction comparison in later grades, but they will later apply their understanding of fraction comparison when they learn to compare decimals and when they compare fractional quantities in their lives. teks Focus..D Compare two fractions with different numerators and different denominators and represent the comparison using the symbols.,, or,. Readiness Standard MatheMaticaL Process standards (MPs):..A,..C,..D,..E,..F (See page A9 for full text. Also see MPS Tips in the lesson.) 0 L0: Compare Fractions

24 Part : Introduction Lesson 0 At A GlAnce 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. 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. Work through Explore It as a class. Have students describe the problem and state what needs to be done mathematically (compare the fractions). Have students explain the model. (See Mathematical Discourse, below.) 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 classmates or with classmate? Why? Students should apply this reasoning to explain that Adriana s pieces were bigger than June s. MpS tip: Some students may be tempted to use the model alone as justification for saying that Adriana ate more. Encourage them to use the model mathematically (..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.). Lesson 0 Part : Introduction Compare Fractions L0: Compare Fractions Mathematical Discourse Develop Skills and Strategies In the past, you learned to compare fractions using models. Take a look at this problem. Adriana ate of a granola bar and June ate of a same-size granola bar. Which 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? 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 pieces. Who ate more? Explain why. Both girls ate the same number of pieces. Adriana s pieces are larger than June s pieces. So, Adriana ate more. TEKS..D What does the model tell you? The two rectangles show equal sized wholes. One has equal parts; the other has equal parts. Two parts are shaded in each. It looks like more space is shaded in the rectangle with 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. L0: Compare Fractions

25 Part : Introduction Lesson 0 At A GlAnce Students use symbols to compare fractions. Then they reflect on the importance of comparing fractions from same-size wholes. Step By Step Read Find Out More as a class. Explain that you can show either fraction first in the comparison;. is the same as,. Make sure students understand what. and, mean and how 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. Part : Introduction Find Out More Deciding who ate more of the granola bar means comparing the fractions and.. is greater than., is less than. 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. 0 and 0 0 0, 0 0, so, Reflect Explain how you can tell which fraction is greater, or 0. Possible answer: Multiply the numerator and denominator by to find an equivalent fraction in tenths: 0. Since 0. 0,. 0. Lesson 0 Hands-On Activity Investigate the importance of comparing fractions from same-size wholes. L0: Compare Fractions 9 Materials: 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 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. 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 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. L0: Compare Fractions

26 At A GlAnce Part : Modeled Instruction Students use a model to study a problem involving fraction comparison. They solve the problem using common denominators. SteP By SteP Read the problem at the top of the page as a class. Have students identify this as fraction comparison. 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. Read Solve It as a class. Have students identify a common denominator. [hundredths] MPS tip: Students must apply knowledge of equivalent fractions to solve this problem. Encourage them to analyze mathematical relationships to connect mathematical ideas (..F) as they search for a viable common denominator. Ask, How could you rewrite as tenths? [You can t 00 because you would have to divide by 0.] How could you rewrite as hundredths? [Multiply both 0 numbers by 0 to get 0 hundredths.] Visual Model compare the fractions using a number line. After completing the page, present another model. Draw two number lines from 0 to, one above the other. On one number line, mark and label tenths. On the other number line, mark and label every 0 hundredths: 0, 0, etc. Mark 0 hundredths between each tenth. Guide students to recognize that the number lines are the same length. Have a volunteer mark on the hundredths 00 number line. Have another volunteer mark 0. Students compare the two values. 00, 0 or Part : Modeled Instruction Read the problem below. Then explore different ways to understand it. A grasshopper weighs about 00 of an ounce. A beetle weighs of an ounce. 0 Which weighs more? Picture It You can use models to help compare fractions. The following model shows the weights of the grasshopper and beetle. Grasshopper Beetle L0: Compare Fractions Mathematical Discourse What do you notice about the two models? Lesson 0 Lesson 0 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. Compare 0 and 00. First, look at the denominator, 0. Can you multiply 0 by any number to get 00? Yes, Find a fraction equivalent to that has a denominator of 00: Compare the numerators of 0 00 and 00 : 0. So, One model has 0 parts and the other has 00 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 00. The beetle s weight is given in tenths, so that model is divided into 0. 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. L0: Compare Fractions

27 At A GlAnce Part : Guided Instruction Students revisit the problem on page 0. 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 0. Work through problems 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 may challenge students because it can be confusing that the greater number actually means smaller parts. Encourage students to create and use representations (..E) as they work on this problem. Ask, What does a numerator of 00 mean? If students struggle, draw a different example on the board. Use a simpler model showing compared to. Elicit that the fraction with the 0 smaller denominator always has bigger parts. Part : Guided Instruction Connect It L0: Compare Fractions Lesson 0 Now you will solve the problem from the previous page by finding a common numerator. What is an equivalent fraction for that has a numerator of? One model is divided into 00 equal parts and the other is divided into 0 equal parts. Which has smaller parts? the one divided into 00 equal parts Shade pieces of each model. Which model has a greater area shaded? the one divided into 0 pieces Which fraction is greater, 00 or 0? 0 Look at the denominators of 00 and. When two fractions have the same 0 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 SolutIonS Solution:, or. ; Students may use common numerators. is a common numerator ( ). and. ths are smaller than nds, so,. ERROR ALERT: Students who wrote Lesson 0 Try It Use what you just learned to solve these problems. Mel s tomato plant is of a foot tall. Her pepper plant is of a foot tall. Compare the heights of the plants using a symbol., or. 9 Compare the fractions and using a symbol or 0,. may have reasoned that since. and., the fraction must be greater. Have students draw fraction bar models to illustrate that.. Then help them find common numerators and solve. 9 Solution:. 0 or, ; Students may recognize 0 that and., so L0: Compare Fractions

28 At A GlAnce Part : Modeled Instruction Students explore a problem involving fraction comparison. A number line and a benchmark fraction help students understand the quantities being compared. SteP By SteP Read the problem at the top of the page as a class. Discuss the meaning of the problem. 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. Read Solve It as a class. Have a volunteer describe the meaning of. [It is midway between 0 and, so it is easy to find.] Work through the comparisons and guide students to the solution. Remind students to relate the mathematical answer to the problem context. ell Support 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. Materials: Number line from 0 to, cards with a variety of fractions written on them (denominators of,,,,, 0,, 00) Have students label on the number line. Give students cards with a variety of fractions. Students take a card and place it on the number line between 0 and or between and. Ask students to explain each fraction s placement. Discuss any fractions they are not sure about. Part : Modeled Instruction Read the problem below. Then explore different ways to use benchmarks to compare fractions. L0: Compare Fractions Mathematical Discourse How do you know that is less than? Sixths are smaller than halves. Lesson 0 Lesson 0 Jasmine s swimming lesson lasts for of an hour. It takes her of an hour to do 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 and are compared to 0 and. 0 The number line shows that is closer to 0 than is, and that is closer to than is. This means that.. Solve It You can use a benchmark fraction to solve the problem. Another way to compare fractions is by using the fraction as a benchmark. Look at the number line. It shows that is less than and is greater than. So,, and.. Jasmine will spend more time at her swimming lesson than on homework. How does a number line help you compare fractions? It helps you see which fractions are closer to 0 and which are closer to. You can mark off different sized pieces for easy comparison. How does a benchmark fraction, such as, help you compare fractions? You know how large is, so it is a useful reference point. It is often easier to compare a fraction to than to another fraction. By comparing two fractions to, you can often see how they compare to each other. L0: Compare Fractions

29 At A GlAnce Part : Guided Instruction Students compare two fractions using as a benchmark. Then they solve problems using benchmark fractions or other methods. SteP By SteP Introduce the idea that is not the only fraction that students can use as a benchmark fraction. Walk through problems 0 with students. They should see that 0 is the same as, and 0 0 is more than 0. Also, is the same as, and is less 0 than. Have students share and explain their answers to problem. Encourage students to ask each other questions to clarify the reasoning. MPS tip: Help students use benchmark fractions strategically (..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 0, is a useful benchmark because both fractions in the problem are near on a number line. Part : Guided Instruction Connect It Now you will solve a similar problem using as a benchmark. Think about these two fractions: 0 and 0 Which fraction is greater than? 0 Which fraction is less than? Fill in the blank with the correct symbol to show the comparison. 0 Explain how you can use benchmarks to compare fractions. L0: Compare Fractions Lesson 0 Which fraction is greater? Explain why. Since 0 is greater than and is less than, must be greater than 0. You can compare both fractions to the same number to see which one is greater than, less than, or equal to that benchmark. The fraction that is greater than the benchmark is greater than the one that is less than or equal to the benchmark. Try It Use what you just learned to solve these problems. Fill in the blank. Explain how you found your answer., 0 I used as a benchmark. 0 and.. So,. 0. Nathan walked 0 0 of a mile. Sarah walked 9 of a mile. Who walked a greater 0 distance? Explain. 0 out of 0 is equal to. 9 out of 0 is 9, which is less than. Nathan 0 walked a greater distance than Sarah. Lesson 0 try It SolutIonS Solution:, ; Students may use as 0 a benchmark to determine that and. 0. Therefore, 0,. Solution: Nathan; Students may realize that 0 is the same as and 9 is less than. 0 0 Therefore, 0 is a greater distance. 0 ERROR ALERT: Students who chose 9 may have 0 reasoned that 9 parts is more than 0 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).. L0: Compare Fractions

30 Part : Guided Practice Lesson 0 Part : Guided Practice Lesson 0 Part : Guided Practice Lesson 0 It is important that both measurements use the same unit! Study the model below. Then solve problems 9. Student Model Becker catches a fish that is of a yard long. To keep the fish, it has to be longer than of a yard. Can Becker keep his fish? Compare the fractions 0 and using the benchmark fraction. Show your work. Possible answer: 0, and. so 0, You already know about how big is! Pair/Share How else could you solve this problem? Look at how you could show your work using a number line Since is less than, Becker can t keep his fish. Solution: Solution: 0, or. 0 9 Janelle walked of a mile. Pedro walked of a mile. Which 0 statement shows how to find the greater fraction? Circle the letter of the correct answer. Pair/Share Draw a model to check your answer. There are several ways to compare fractions! Which strategy for comparing do you think works best with these fractions? Pair/Share How did you and your partner choose what strategy to use to solve the problem? Myron and Jane are working on the same set of homework problems. Myron has finished of the problems and Jane has 9 finished of the problems. Who has finished more of the homework? Show your work. Possible answer: 9 Since 9. 9, 9. Solution: Myron has finished more of the homework. A and, 0 B and. 0 C 0 and, D, and 0. Tina chose B as the correct answer. How did she get that answer? Possible answer: Tina found an equivalent fraction, but compared them incorrectly. She thought that is greater than because is greater than 0. 0 Pair/Share How can you find the answer using a benchmark fraction? L0: Compare Fractions L0: Compare Fractions At A GlAnce 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. SteP By SteP Ask students to solve the problems individually. Circulate and provide support. 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. SolutionS Ex Since,, Becker can t keep his fish. The number line shows twelfths and thirds. Solution: Myron finished more of the homework; Multiply the numerator and denominator of by so both fractions have a denominator of 9. (DOK ) Solution: 0, or. ; Compare the fractions 0 to. Use common denominators or common numerators. 0,,.. (DOK ) 9 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. (DOK ) L0: Compare Fractions

31 Part : TEKS Practice Lesson 0 Part : TEKS Practice Lesson 0 Part : TEKS Practice Lesson 0 Solve the problems. Grant needs } cup of raisins and } cup of almonds to make trail mix. Which statement can be used to find out if there are more raisins or almonds in the mix? A B C D } }} and } 9 }} } } and } } } } 9 and } }} } } 9 and } } Tell whether each sentence is True or False. a. b. c. d. e., True True True True True False False False False False Fill in the blank with one of the symbols shown to compare and 0. }} 0,., } Sam s music teacher told him to practice his trombone for }} of an hour. He spent 0 } of an hour practicing. Did he practice long enough? Show your work. Answer Sam 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 0 equal glasses. Eleanor poured all of her lemonade into equal glasses. Olivia sold glasses of lemonade and Eleanor sold glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions using a symbol. Possible answer: Olivia sold of her lemonade and Eleanor 0 Show your work. sold of hers. 0 ; 0., so Olivia sold more. 0 Answer Possible answer: }, } and }} 0 }, so }, }} 0 Olivia sold a greater fraction of her lemonade. Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold }} 9 of her 0 boxes of fruit and Sierra has sold } of her boxes. Which girl has sold a greater fraction of her boxes of fruit? Draw a model to show your answer. Show your work. did not 9 has a bigger area shaded, so it is greater than 0. Answer Rachel has sold a greater fraction of her boxes of fruit. 9 0 L0: Compare Fractions L0: Compare Fractions AT A GlAncE Students compare fractions to solve word problems that might appear on a mathematics test. SoluTionS Solution: A; Find a common denominator:. Multiply numerator and denominator of by and numerator and denominator of by. Then numerators can be compared. (DOK ) Solution: a. False; b. False; c. True; d. False; e. True (DOK ) Solution:,; of 0 equal parts is a smaller amount than of equal parts. (DOK ) Solution: did not; Compare the fractions using the benchmark fraction. and 0,. So,,. (DOK ) 0 Solution: Olivia; Compare the fractions and 0. Students may find a common denominator and write as tenths. (DOK ) Solution: Rachel; See possible student work above. Compare the shaded parts of the bar models. (DOK ) L0: Compare Fractions

32 Differentiated Instruction Lesson 0 Assessment and Remediation Ask students to compare and and to show a visual model and explain their work. 0 For students who are struggling, use the chart below to guide remediation. After providing remediation, check students understanding. Ask students to compare and. If a student is still having difficulty, use STAAR Ready Instruction, Level, Lesson. If the error is... Students may... To remediate... is greater than not understand the Have students draw same-size bar models of and. Point out that the 0 0 because. reason for comparing size of the pieces is not the same, so you can t compare numerators. Ask fractions with the students to come up with a common numerator (such as ) so that the same number of number of pieces is the same and write equivalent fractions for and pieces (numerators). 0 so they can compare. is greater than 0 because 0. because 0. and 0. Hands-On Activity not understand the reason for comparing fractions with same-size pieces (denominators). not understand when it is appropriate to use a benchmark fraction. Draw models to compare fractions. Materials: -cm grid paper, scissors, markers or pencils Have students work with a partner. Provide students with -centimeter grid paper. Instruct each student in the pair to draw and cut out two -by- 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. Have students repeat by drawing and cutting out two -by- 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 -by- arrays and draw equal parts. Ask students if they could use this array and another of their -by- arrays to compare eighths and twelfths [yes], and have them do so. Have students draw same-size bar models of and. Ask if the number 0 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 0) so that the size of the pieces is the same and write equivalent fractions for and so they can compare. 0 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. Help them mark and label tenths and fifths. Have students locate and on the number line to 0 make the comparison. Challenge Activity Compare three or more fractions. Materials: Fractions written on cards 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. L0: Compare Fractions 9

33 Focus on Math Concepts Lesson (Student Book pages 9) Understand Fraction Addition and Subtraction Lesson objectives Understand addition as joining parts. 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 In order to be proficient with the concepts in this lesson, students should: Know addition and subtraction basic facts. Understand the meaning of fractions. 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 denominator: the bottom number in a fraction; it tells the total number of equal parts in the whole the Learning Progression 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. Students see that you can count with unit fractions just as you count with whole numbers. And because you can count with unit fractions, you can also do arithmetic with them. If you walked of a mile ( fifths) yesterday and of a mile ( fifths) today, altogether you walked of a mile ( fifths; because things plus more of those things is 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. teks Focus..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):..A,..C,..D,..E,..F,..G (See page A9 for full text. Also eee MPS Tips in the lesson.) 0 L: Understand Fraction Addition and Subtraction

34 Part : Introduction Lesson At A GlAnce Students explore the idea that adding fractions is not essentially different from adding whole numbers. A number line diagram gives meaning to the expression. Step By Step Introduce the Question at the top of the page. Help students relate the number line diagram to the sum. Read Think with students. Reinforce the idea that fractions are numbers. Guide students to recognize that just as the number is made up of ones, the number is made up of one-fourths. If students need additional support with locating fractions on a number line, have them build a number line by putting fraction strips end-to-end, creating a concrete model to show. Lesson Part : Introduction Understand Fraction Addition and Subtraction What s really going on when we add numbers? L: Understand Fraction Addition and Subtraction Focus on Math Concepts Adding means joining or putting things together. Think about how you could explain adding to a first grader. You could start at, count on more, and see where you end up: Or, you could put a segment with a length of and a segment with a length of next to each other on a number line to show When you add, 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 and a segment with a length of next to each other to show. 0 0 When you add, you are putting one-fourths together. TEKS..E Underline the sentence that explains what adding fractions means. concept extension To extend students understanding of decomposing fractions, follow these steps: Draw and label a number line on the board from 0 to 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 and. Have a volunteer go to the board to show the two fractions on the number line. and in either order Mathematical Discourse 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. L: Understand Fraction Addition and Subtraction

35 Part : Introduction Lesson 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. Step By Step Read Think with students. Discuss how the number line represents the problem. Show how to subtract on the number line. (start at and count back ) Ask a volunteer to explain how to use the number line to find. Provide fraction strips for students who need more support. Have students read and reply to the Reflect directive. Visual Model Tell students that you will use a number line to show. Draw a number line from 0 to 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 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. (..E) Part : Introduction Think Subtracting means separating or taking away. On a number line, you can start with a segment of length and take away a segment of length to show When you subtract, you are taking away ones. You can show subtracting fractions on a number line. Start with a segment of length and take away a segment of length to show. 0 L: Understand Fraction Addition and Subtraction Mathematical Discourse How would you explain subtracting in your own words? Listen for phrases such as take apart or take away. Lesson 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 from. 0 When you subtract, you are taking away one-fourths. 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 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. Look at the whole numbers. Now look at the numerators of the fractions. I think I see a connection. 9 L: Understand Fraction Addition and Subtraction