Understand Equivalent Fractions

Focus on Math Concepts Lesson (Student Book pages 0 ) Understand Equivalent Fractions LESSON OBJECTIVES Understand the value of a fraction. Understand how a fraction model represents a fraction. Understand how two fractions are equivalent. Understand how different models can represent the same value. PREREQUISITE SKILLS Represent fractions with denominators,,, 6, or 8 on number lines and using visual models. Identify, create, and explain equivalent fractions. Express whole numbers as fractions. VOCABULARY There is no new vocabulary. Review the following key terms. denominator: the bottom number in a fraction; it tells the total number of equal parts in a whole divide: to separate an amount into equal groups and find the number in each group or the number of groups equivalent fractions: two or more fractions that name the same part of a whole numerator: the top number in a fraction; it tells the number of parts in a whole that are being described THE LEARNING PROGRESSION In Grade, students used number lines to locate unit fractions, and used fraction bars, fraction strips, and area models to recognize and generate equivalent fractions and to compare fractions. In Grade, students extend these understandings to compare fractions with different numerators and different denominators. In this lesson, students focus on extending their understanding of equivalent fractions, using visual models (as was done previously) and by generating equivalent fractions with denominators such as, 0,, and 00. Students use models to build foundational understanding of the effect of multiplying or dividing the numerator and denominator by the same number to generate an equivalent fraction. In later grades, students will be able to generate fractions with any denominator. Students will use knowledge of equivalent fractions to add, subtract, and compare fractions with unlike denominators. Understanding equivalent fractions provides the basis for developing understanding of ratios and proportional thinking. Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com.NF.. MAFS Focus.NF.. Explain why a fraction is equivalent to a fraction (n a) by using visual fraction models, with attention to how the number a b (n b) and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. STANDARDS FOR MATHEMATICAL PRACTICE: SMP,,, 7, 8 (See page A9 for full text.) L: Understand Equivalent Fractions

Part : Introduction Lesson Students explore the meaning of equivalent fractions. They look at visual models. Then they think about ways to divide a fraction model into more parts and the equivalent fractions that are generated as a result. Lesson Part : Introduction Understand Equivalent Fractions What s really going on when fractions are equivalent? Focus on Math Concepts MAFS.NF.. Introduce the Question at the top of the page. Discuss the idea that, if two fractions are equivalent, the model shows the same amount shaded. Read Think with students. Have students underline the part of the sentence that explains how to write an equivalent fraction. Focus attention on the idea of times as many parts and times as many parts shaded. If students underlined divide the same rectangle, ask them to consider what happens when they divide or split the rectangle into equal parts. The result is times as many parts, which implies multiplication. ELL Support Explain that the word equivalent is related to the word equal. Equivalent fractions have equal value but they show their value in a different way. Have students name the equivalent fractions on the page. 0 Equivalent fractions name the same part of a whole. Think about how you could explain to a third grader why and are equivalent. You could shade area models to show and. Both models are the same size. Both show the same amount shaded, so and are equivalent fractions. Think Equivalent fractions show a fraction a different way. Fractions can be written many different ways by changing the number of equal parts in the whole. Start with a rectangle divided into equal parts. Shade one part to show. Divide the same rectangle into equal parts. There are times as many parts and times as many parts shaded. Now out of equal parts are shaded. But, your rectangle still shows shaded. Divide the original rectangle into 8 equal parts. There are times as many parts and times as many parts shaded. Now out of 8 equal parts are shaded. Again, your rectangle shows shaded. So,,, and 8 L: Understand Equivalent Fractions are all equivalent fractions, since they name the same part of a whole. 8 Underline the part that explains how to write a fraction a different way. Hands-On Activity Fold paper to model equivalent fractions. Materials: piece of paper, pre-folded into thirds Give each student a piece of paper pre-folded into thirds, accordion style. Have them open the paper and shade of the page. Have students re-fold the paper and then fold it perpendicular to the original folds. When they open the paper again, ask, Now what part of the paper is shaded? 6 Have students re-fold the paper and fold it one more time, in either direction. Ask, Now what part of the paper is shaded? Mathematical Discourse How would you show someone that and are equivalent? Make two models that are the same size. Divide one into tenths and shade tenths. Divide the other into halves and shade half. Both models have same amount shaded. Look at the model for. What happens to the fraction when you divide the rectangle into equal parts? There are now equal parts, so the denominator is. This is twice as many as before. The numerator is, because twice as many parts are shaded. L: Understand Equivalent Fractions

Part : Introduction Lesson Students explore equivalent fractions based on thirds. This gives them an opportunity to see that there are many ways for the number of parts to differ even when fractions are the same size. Read Think with students. Direct their attention to the model of. Point out that this is a different fraction from the one on the previous page, where the model started with. Ask students what they notice about the parts in and the parts in. [There are more parts in but the entire shaded area is the same in each model.] Emphasize that multiplying by is not the same as multiplying by. Multiplying by would change the value. Have students read and reply to the Reflect directive. Part : Introduction Think Every fraction has many equivalent fractions. You can start with any fraction and change the way the whole is divided to get an equivalent fraction. This model is divided into equal parts. The shaded section shows the fraction. has times as many parts shaded and 6 times as many equal parts. has times as many equal parts and times as many parts shaded as. All three models have the same shaded area. So,, 6, and are equivalent. You can also multiply the numerator and denominator of by the same number. times as many equal parts and times as many parts Think of times shaded: 6. as many as times as many equal parts and times as many parts shaded:. Reflect Explain how you can find equivalent fractions. Possible answer: You can divide a model into different numbers of equal parts. If the same area of the same shape is shaded, the fractions are equivalent. You can also multiply the numerator and denominator by the same number. 6 Lesson L: Understand Equivalent Fractions Visual Model Explore equivalent fractions on a number line. Materials: number lines, markers Draw five number lines from 0 to, one above the other. On the top number line, mark and label thirds. On the next number line, mark and label fourths. Mark and label sixths, eighths, and twelfths on the next three number lines, respectively. Have a volunteer use one color marker to circle all the fractions that are equivalent to., 6, 8, 6 Have a volunteer use another color to circle all the fractions that are equivalent to. 6, Have a volunteer use another color to circle the fractions that are equivalent to. 8, Ask students how they think number line models are like and not like area models. Mathematical Discourse What would happen if we divided the model into sixths using vertical lines? Would the models still show equivalent fractions? Yes. There would still be shaded parts out of 6, and the shaded part would still be the same size. What questions can you ask to tell if two fraction models are equivalent? Are the wholes the same size? Is the same amount shaded in each model? Did the numerator and denominator change in the same way? L: Understand Equivalent Fractions

Part : Guided Instruction Lesson Students explore the effects of dividing fraction models into more or fewer parts. They analyze changes to the size of the parts and the number of parts. 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. As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Hands-On Activity to help redirect the thinking of students who struggle. Look for evidence of understanding that doubling the number of parts means that the size of the parts is decreased by a factor of. Students apply their understanding to answer problems 6 and 7. 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: If students struggle with question 7, they may not understand the need to reduce the numerator by a factor of 0 when the denominator is reduced by a factor of 0. Help the student identify 0 equal parts in the model and count the shaded parts. [] Point out that each of the 0 equal parts has 0 of the original parts in it, for a total of 00 original parts. Ask the student how many original parts are shaded [0] and what happened to make this new equivalent fraction. [0 times fewer parts, each composed of 0 smaller parts were marked. Out of those 0 larger parts, are shaded.] Part : Guided Instruction Lesson Explore It Dividing models is one way to think about equivalent fractions. The model shows. How many equal parts make up the whole? Draw more lines on the circle to make 8 equal parts. Compare the equal parts to the 8 equal parts. How many times as many parts are there now? times Now how many parts are shaded? Why are there two times as many parts shaded as there were in the model? The circle is now divided into two times as many parts. Use the model above to answer problems and. If of the original parts were shaded, how many of the 8 parts would be shaded? 6 If all 8 parts were shaded, how many of the original parts would be shaded? Now try these two problems. 6 Draw a model to show and then divide it into a different number of parts to find an equivalent fraction. Possible answer: L: Understand Equivalent Fractions Hands-On Activity 7 This model shows 0. If the model 0 had only 0 equal parts, how many would be shaded? Provide concrete experience with equivalent fractions. Materials: Square pieces of paper, ruler, pencil Give students a square piece of paper. Have them imagine it is their favorite sandwich. Students use rulers to draw cut in the sandwich, making equal pieces. Say that people want to share the sandwich. Have students explain how to make equal-sized pieces. Then have them draw the cut. Say that 8 people want to share the sandwich. Ask, What will happen to the pieces? [There will be twice as many and each piece will be twice as small.] Students draw two cuts to make eighths. Suggest that only two people are going to share the sandwich. Ask, How many eighths does each person get? [ eighths] L: Understand Equivalent Fractions

Part : Guided Instruction Lesson Students use their understanding of equivalent fraction models to explore multiplying and dividing to generate equivalent fractions. They use relationships between the size and number of parts to recognize the need to multiply or divide both the numerator and denominator by the same number. Organize students into 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. Make sure that students understand why they must multiply or divide both the numerator and denominator by the same number. In problems and, students calculate equivalent fractions and represent the results in a visual model. This provides a visual check for their thinking. Direct the group s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group s solutions to problems and. SMP Tip: Use this page to deepen students thinking about the structure of fractions (SMP 7). Have them explain what the denominator signifies (the number of parts the whole has been divided into) and what the numerator signifies (the number of parts out of the whole that you actually have). Draw out their ideas about the structure of the equivalent fractions they create by multiplying and dividing. Students should recognize that when the total number of parts (denominator) increases (multiplication) or decreases (division), the parts you actually have (numerator) increase or decrease in the same relationship. Part : Guided Instruction Lesson Talk About It Solve the problems below as a group. 8 Write the equivalent fractions from problems and. Multiply both the numerator and denominator of by the same number to get. 8 What number did you use? Why does this make sense? Multiply by. There are twice as many equal parts and twice as many parts shaded in as in 8. What happens if you divide both the numerator and the denominator in by? 8 You get. 9 To find an equivalent fraction to, Beth divided by to get in the denominator. 6 8 What should Beth do to find the numerator? What are the equivalent fractions? Divide by. 6 8 and are equivalent. 0 Fill in the missing numbers to find an equivalent fraction to 6. 6 0 Try It Another Way Work with your group to model equivalent fractions. Shade the model to show. Then show 0 equal parts and write an equivalent fraction. 0 Shade the model to show. Then show equal parts and write an equivalent fraction. L: Understand Equivalent Fractions Mathematical Discourse and 8 When you create equivalent fractions with multiplication or division, what patterns do you see? The number of parts in the whole increases (multiplication) or decreases (division) and the number of parts you have increases or decreases by the same factor. Also, the size of the parts decreases (multiplication) or increases (division) by the same factor. How can you tell that 6 and 0 are equivalent fractions, without drawing a diagram? Because if you multiply the numerator () by, you get 0. If you multiply the denominator (6) by, you get. 6 L: Understand Equivalent Fractions

Part : Guided Practice Lesson Students demonstrate their understanding of equivalent fractions by modeling and calculating equivalent fractions. They explain their reasoning and then connect their knowledge of equivalent fractions to a real-life situation. Discuss each Connect It problem as a class using the discussion points outlined below. Compare: Students should use a variety of methods: multiplication, division, logical reasoning, or drawing a model. Ask, What does this fraction represent? [ Six sixths equal whole. Any fraction with the same numerator and denominator is equivalent.] Illustrate: Share an illustration of what happens if they do not multiply the numerator and denominator by the same number. Then have students explain why the fractions are not equivalent. [They don t take up the same area.] Choose: Have students describe the problem and explain the math involved. Determine which measuring cup they can use to make a fraction equivalent to. Students can then suggest ways to find a solution and discuss why the ideas will or will not work. Try students ideas to determine what approach leads to an answer. Part : Guided Practice Lesson Connect It Talk through these problems as a class, then write your answers below. Compare: Use different methods to find two fractions that are equivalent to. Possible answer: Illustrate: Explain why you can multiply both the numerator and denominator by the same number to make an equivalent fraction. Draw a model to show an example. Possible answer: Choose: Think about the cooking problem below. Fia needs of a cup of brown sugar. She only has a -cup measuring cup and a -cup measuring cup. Which should she use, and why? 8 She should use the -cup measuring cup because she can make an 8 equivalent fraction that has a denominator of 8 by multiplying both the numerator and denominator of by. L: Understand Equivalent Fractions 6 6 6 has times as many equal parts and times as many parts shaded as. 6 Three times as many means multiplying the numerator and denominator by. 6 SMP Tip: Problem provides an opportunity for students to construct viable arguments and critique the reasoning of others (SMP ). Encourage students to think out loud and test out their ideas with their peers. Ask students if they agree or disagree with their classmates ideas. If students think that a line of reasoning is incorrect, ask them to demonstrate what the problem is and explain why it does not work, using concepts of equivalent fractions. If they agree, they should be able to help demonstrate why the approach works. If students struggle, suggest that they write next to and also write next 8 to. Ask if they can change thirds to fourths. [No.] Can they change eighths to fourths? [Yes.] L: Understand Equivalent Fractions 7

Part : Performance Task Lesson Students demonstrate their understanding and application of equivalent fraction concepts by drawing a model and calculating equivalent fractions. Direct students to complete the Put It Together task on their own. 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. If students struggle, have them review the meaning of the numerator and the denominator. Ensure that their model starts with 0 equal parts with 6 parts shaded. If time permits, have students share their models and their equivalent fractions and explain how they used multiplication or division to check. SCORING RUBRICS A Points Expectations The initial model shows 0 equal parts with 6 parts shaded. The equivalent fraction models are correct and there are two different equivalent fractions. The initial model shows 0 equal parts with 6 parts shaded. One equivalent fraction model is correct. 0 The initial model may or may not be correct. Neither equivalent fraction model is correct. B Part : Performance Task Lesson Put It Together 6 Use what you have learned to complete this task. A Draw a model to show the fraction 6 and two equivalent fractions. Possible answer: B How can you use multiplication and division to check your equivalent fractions in Part A? Why does this work? Possible answer: Start with 6. Divide the numerator and denominator by to get. Multiply the numerator and denominator in 6 by to get. The numerator is the number of parts shaded and the 0 denominator is the number of parts. If you multiply or divide both by the same number, you are not changing the whole or the area that is shaded. You are only changing the number of equal parts and number of parts shaded. L: Understand Equivalent Fractions 6 0 Points Expectations The multiplication or division calculations are correct and match the equivalent fraction models. One of the multiplication or division calculations is correct and matches one of the equivalent fraction models. Or, both multiplication or division calculations are correct and show fractions that are equivalent to 6, but they do not match the equivalent fraction models. 0 Neither calculation is correct. 0 8 L: Understand Equivalent Fractions

Differentiated Instruction Lesson Intervention Activity Use models to find missing numbers. Materials: fraction bars Distribute fraction bars to pairs of students. Tell students to use fourths to model the fraction. Then, have students align fraction bars below their model to show the same amount. Ask students, How many twelfths did you use? [9] What fraction, with a denominator of, is equivalent to? 9 Write the comparison on the board, 9, and discuss the relationship between the numerators and denominators. What number do you multiply by to get 9? [] What number do you multiply by to get? [] Repeat with 8 fraction bars to find 6 8. On-Level Activity Use fraction circles to model equivalent fractions. Materials: Fraction circles Organize students in small groups and distribute fraction circles. Have students show and write the fraction. Have students use fraction circles to find as many fractions as they can that are equivalent to. Have students write all the equivalent fractions they found. Repeat the activity for. Challenge Activity Identify equivalent fractions. Materials: Cards with fractions written on them so that each fraction has at least equivalent fractions in the deck Organize students in pairs. Give each pair a deck of fraction cards. Students take turns drawing two cards from the pile and stating whether or not the fractions are equivalent. They must prove that the fractions are or are not equivalent by multiplying, dividing, or drawing a visual model. L: Understand Equivalent Fractions 9

Develop Skills and Strategies Lesson 6 (Student Book pages ) Add and Subtract Fractions LESSON OBJECTIVES Add fractions with like denominators. Subtract fractions with like denominators. Use fraction models, number lines, and equations to represent word problems. PREREQUISITE SKILLS In order to be proficient with the concepts/skills in this lesson, students should: Understand addition as joining parts. Understand subtraction as separating parts. Know addition and subtraction basic facts. Understand the meaning of fractions. Identify numerators and denominators. Write whole numbers as fractions. Compose and decompose fractions. THE LEARNING PROGRESSION In keeping with the Common Core goal of developing a deeper understanding of fractions, this lesson extends students understanding of fraction addition and subtraction. Students use visual models and equations to represent and solve word problems involving the addition and subtraction of fractions referring to the same whole and having like denominators. Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com.NF..a.NF..d 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 MAFS Focus.NF.. Understand a fraction with a. as a sum of fractions a b b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. STANDARDS FOR MATHEMATICAL PRACTICE: SMP,,,, 6, 7, 8 (See page A9 for full text.) 8 L6: Add and Subtract Fractions

Part : Introduction Lesson 6 Students read a word problem and answer a series of questions designed to explore the connection between adding and subtracting fractions and adding and subtracting whole numbers. Tell students that this page models building the solution to a problem one step at a time and writing to explain the solution. Have students read the problem at the top of the page. Work through Explore It as a class. Ask students to explain how they figured out the answers for how many cards Lynn and Paco received altogether, and for how many cards Todd received. Guide students to understand that they needed to join and take away the numbers of cards to answer the questions. Be sure to point out that equals the total number of cards,. Remind students that the whole is represented by the set, or pack, of cards. Ask student pairs or groups to explain their answers for the remaining questions. Encourage students to explain the connection between adding and subtracting fractions and whole numbers. [When adding or subtracting whole numbers, you join or separate whole numbers. And, when adding or subtracting fractions, you join or separate parts of a set or whole.] Lesson 6 Part : Introduction Add and Subtract Fractions In Lesson, you learned that adding fractions is a lot like adding whole numbers. Take a look at this problem. L6: Add and Subtract Fractions Mathematical Discourse Develop Skills and Strategies Lynn, Paco, and Todd split a pack of baseball cards. Lynn got cards, Paco got cards, and Todd got the rest of the cards. What fraction of the pack did Todd get? Explore It Use the math you already know to solve the problem. How many cards did Lynn and Paco get altogether? 7 How many cards did Todd get? There are cards in the pack. What fraction represents the whole pack of cards? If Lynn got cards out of, that means she got of the pack. If Paco got cards out of, what fraction of the pack did he get? 7 What fraction of the pack did Lynn and Paco get altogether? Explain how you could find the fraction of the pack that Todd got. Possible answer: Todd got cards. There are cards in the pack. If the numerator tells the number of cards Todd got, and the denominator tells the number of cards in the pack, then Todd got of the pack. MAFS.NF..a.NF..d What does the denominator of a fraction tell you? Listen for responses that include the phrase equal parts of a whole or equal parts of a set. What does the numerator of a fraction tell you? Students responses should indicate an understanding that the numerator tells you the number of equal parts you are talking about. L6: Add and Subtract Fractions 9

Part : Introduction Lesson 6 Students use fraction models to review adding and subtracting fractions. Read Find Out More as a class. Point out that when you have a set of objects, the denominator represents the total number of objects in the set. Since there are baseball cards in the pack, that means there are parts in the set. The number of cards that each person has represents the numerator of the fraction. Remind students that when you have a whole object that is divided into equal parts, the denominator shows the total number of parts. Note that the whole pizza was divided into 8 equal slices, so the denominator is 8. If there are 7 slices remaining, then the numerator of the fraction is 7. If more slices are taken away, then there are slices left, and the numerator of the fraction is. Have students read and reply to the Reflect directive. Part : Introduction L6: Add and Subtract Fractions Lesson 6 Find Out More We often use fractions in real life. Sometimes they refer to parts of a set of objects, like the baseball card problem. In that problem, the whole is the pack, and cards means there are parts of the whole. Each person got baseball cards from the same pack, so each fraction refers to the same whole. When you add or subtract baseball cards, the whole will stay the same because the cards are all from the same pack of. Fractions in real life can also refer to equal parts of a whole object. Lynn, Paco, and Todd might share a pizza cut into 8 slices. The whole is the pizza, and 8 slices means there are 8 equal parts of the whole. Even if a person takes away slice or slices from the pizza, the whole will stay the same. Reflect Describe another example of a set of objects or a whole object divided into fractions. Possible answer: You can think of a full egg carton as a set of objects. Each egg is of the set. Hands-On Activity Use models to add fractions. Materials: Drawing paper and notebook paper Distribute drawing paper and a piece of notebook paper to each student. Tell students to use scissors to cut out equal-sized cards. Explain to students that the cards represent one pack of cards, or one whole set, and that there are parts in the set. Tell students to hold up cards. Have students write the name of the fraction represented by the cards on their paper. Review the meaning of the fraction. [ cards out of ] Then, repeat with 7 cards. Tell students to add (join) the fractions and write the sum on their paper. Have a volunteer explain how they determined their answer. If time permits, repeat for additional fraction pairs. Real-World Connection Encourage students to think about everyday places or situations where people might need to add or subtract like fractions. Have volunteers share their ideas. Examples: cooking, construction site, distances on a map 60 L6: Add and Subtract Fractions

Part : Modeled Instruction Lesson 6 Students use models and number lines to review adding fractions. Part : Modeled Instruction Read the problem below. Then explore different ways to understand it. Lesson 6 Read the problem at the top of the page as a class. Read Picture It. Have a volunteer name the denominator of the fraction in the problem. [0] Point out that each pot is of the total number of pots. Guide students to recognize that since Josie painted of the pots and Margo painted, the picture is shaded to represent the number of pots each girl painted, for Josie and for Margo. Have students count aloud to find the sum. Direct students to look at the number line in Model It. Emphasize that the number line is divided into tenths to represent the total number of pots. You may wish to draw the number line on the board and have a volunteer demonstrate the jumps to the right to add tenths to. SMP Tip: Help students make sense of the problem and generalize that the same properties that apply to whole numbers apply to fractions. (SMP ) Concept Extension Illustrate the commutative property of addition. Ask, What if I drew the starting point at instead of? Could I still solve the problem? To emphasize the point, draw a number line on the board with a point at. Then, have students explain how to count on from to find the answer. Encourage a volunteer to come to the board and demonstrate how to find the sum. Josie and Margo made 0 clay pots in art class. Josie painted of the pots. Margo painted of the pots. What fraction of the clay pots did they paint? Picture It You can use models to help understand the problem. The following model shows the pots. Each pot is of the total number of pots. Josie painted pots, and Margo painted pots. They painted a total of 7 pots. J J J J J J M M J J J M M M M M M tenths tenths 7 tenths Model It You can also use a number line to help understand the problem. The following number line is divided into tenths, with a point at. 6 7 8 9 0 0 0 0 0 0 0 0 0 0 Start at and count tenths to the right to add. 6 7 8 9 0 0 0 0 0 0 0 0 0 0 L6: Add and Subtract Fractions Mathematical Discourse How could you use fractions to label 0 and on the number line? Students may suggest that you can write both as a number out of 0, so 0 and 0. What is another way you could solve the problem? Responses may mention using fraction strips. You could line up three strips and four strips in a single row. Then, you could count how many tenths you have altogether. L6: Add and Subtract Fractions 6

Part : Guided Instruction Students revisit the problem on page to learn how to add fractions using equations. Then, students solve addition word problems. Read Connect It as a class. Be sure to point out that the questions refer to the problem on page. Review the meanings of numerator (the number of equal parts of a set you have) and denominator (the total number of equal parts the set is divided into). Ask, If Josie and Margo only made 8 pots, what fraction would represent of the pots? 8 Emphasize that adding fractions is like adding whole numbers. Say, When finding the number of pots Josie and Margo painted altogether, you add the numerators of the fractions and write that sum over the denominator. ELL Support Write the word tenths on the board. Circle the letters that spell ten in the word and write the number 0 below it. Repeat using the word eighths. Have students write tenths and eighths on a piece of paper. Next to the words, have them write fractions associated with the words. If time allows, repeat with other fraction words. Part : Guided Instruction Connect It Now you will solve the problem from the previous page using equations. How do you know that each pot is of the total number of pots? Possible answer: The denominator tells the total number of pots. The numerator tells the number of pots that you are talking about. L6: Add and Subtract Fractions Lesson 6 Lesson 6 What do the numerators, and, tell you? Possible answer: tells the number of pots that Josie painted. tells the number of pots that Margo painted. How many clay pots did Josie and Margo paint altogether? Write equations to show what fraction of the clay pots Josie and Margo painted altogether. Use words: tenths tenths 7 tenths Use fractions: TRY IT SOLUTIONS 7 Solution: ; Students may show on a number line divided into thirds and count mark to the right. They also may write the equation. 8 Solution: of a meter; Students may show on a number line divided into fifths and count marks to the right. They also may write the equation. ERROR ALERT: Students who wrote both the numerators and the denominators. 7 6 Explain how you add fractions with the same denominator. Possible answer: Add the numerators and leave the denominator as is. Try It Use what you just learned to solve these problems. Show your work on a separate sheet of paper. 7 Lita and Otis are helping their mom clean the house. Lita cleaned of the rooms. Otis cleaned of the rooms. What fraction of the rooms did Lita and Otis clean altogether? 8 Mark s string is of a meter long. Bob s string is of a meter long. How long are the two strings combined? of a meter 7 0 ( or ) added 6 L6: Add and Subtract Fractions

Part : Modeled Instruction Lesson 6 Students use models and number lines to review subtracting fractions. Part : Modeled Instruction Read the problem below. Then explore different ways to understand it. Lesson 6 Read the problem at the top of the page as a class. Read Picture It. Guide students to recognize that Alberto s water bottle is divided into 6 equal parts. Ask, What do the 6 equal parts represent? (the denominator) What do the shaded parts represent? (the numerator, or how much water is in the bottle) Point out that sixths are being taken away since Alberto drank parts of the water bottle. Ask, What is? [] Say, So, sixth of Alberto s water bottle still has water in it. Tell students to look at the number line in Model It. Point out that the number line is divided into sixths to represent the 6 equal parts of Alberto s water bottle. Have a volunteer count jumps to the left from to 6 subtract sixths. Ask, What number did [volunteer s name] land on? 6 Say, So, both the model and number line show that sixth of Alberto s water bottle still has water in it. Concept Extension Help students see the relationship between the picture and the number line. Draw the number line on the board. Then, draw the -full water bottle turned on its side above the 6 number line, making sure each part of the water bottle is lined up with its tick mark on the number line. Point out that on the number line lines up with 6 the top of the water bottle. Then, cross out (or erase) one part of the water bottle at a time, moving from right to left along the number line. After parts are crossed out (or erased) to show the water Alberto drank, point out to students that the remaining water is lined up with the -mark on the number line. 6 6 Alberto s water bottle had of a liter in it. He drank of a liter. What fraction 6 6 of the bottle still has water in it? Picture It You can use models to help understand the problem. The following model shows the water bottle divided into 6 equal parts. Each part is of a liter. Five shaded parts show how much water is in the bottle. 6 Alberto drank parts of the water in the bottle, so take away shaded parts of the bottle. There is part of the bottle left with water in it. L6: Add and Subtract Fractions sixths sixths sixth Model It You can use a number line to help understand the problem. The following number line is divided into sixths, with a point at 6. 0 6 6 6 6 6 Start at and count sixths to the left to subtract. 6 6 0 6 6 6 6 6 Mathematical Discourse What is the difference between adding fractions and subtracting fractions on a number line? Responses may indicate direction, moving to the right to add and moving to the left to subtract. What is another way to solve this problem? Students may mention using fraction strips or writing an equation. L6: Add and Subtract Fractions 6

Part : Guided Instruction Students revisit the problem on page 6 to learn how to subtract fractions using equations. Then, students solve subtraction word problems. Read page 7 as a class. Be sure to point out that Connect It refers to the problem on page 6. SMP Tip: Discuss with students how important it is to communicate clearly and precisely by reviewing the meanings of numerator (the number of equal parts you re talking about) and denominator (the total number of equal parts in the whole). Ask, If Alberto s water bottle was divided into equal parts, what fraction would represent of those parts? (SMP 6) Remind students that subtracting fractions is like subtracting whole numbers. Say, When finding the number of parts of the water bottle that still have water, you subtract the numerators of the fractions and write the difference over the denominator. Hands-On Activity Use paper plates to subtract fractions. Materials: paper plates, markers, rulers, scissors Distribute paper plates, markers, and scissors to each student. Model how to use the ruler to divide the plate into 8 equal sections. Students should draw lines. Direct students to color of the plate and then 8 cut out that fraction of the plate. Ask students to name the fraction of the plate they have. 8 Tell students to subtract more eighths. Guide students to cut more sections from the color portion of the plate they are holding. Ask students to name the fraction of the plate they are left with. 8 Write 8 8 on the board. 8 If time allows, repeat for other subtraction problems. Part : Guided Instruction Connect It L6: Add and Subtract Fractions Lesson 6 TRY IT SOLUTIONS Solution: ; Students may show on a number line divided into fourths and count marks to the left. They also may write the equation. ERROR ALERT: Students who wrote ( or ) subtracted from a full carton of eggs ( ) rather than the of a carton that Mrs. Kirk had. Lesson 6 Now you will solve the problem from the previous page using equations. 9 How do you know that each part is of a liter? 6 Possible answer: The denominator tells the number of equal parts the bottle is divided into. The numerator tells the number of parts you are talking about. 0 What do the numerators, and, tell you? Possible answer: tells the number of parts that have water. tells the number of parts that Alberto drank. How many parts of water are left in the bottle after Alberto drank parts? Write equations to show what fraction of the bottle has water left in it. Use words: sixths sixths sixth Use fractions: 6 6 6 Explain how you subtract fractions with the same denominator. Possible answer: Subtract the numerators and leave the denominator as is. Try It Use what you just learned to solve these problems. Show your work on a separate sheet of paper. Mrs. Kirk had of a carton of eggs. She used of the carton to make breakfast. What fraction of the carton of eggs does Mrs. Kirk have left? Carmen had 8 of the yard left to mow. She mowed of the yard. What fraction of the yard is left to mow? Solution: ; Students may show 8 on a number line divided into tenths and count marks to the left. They also may write the equation 8. 7 6 L6: Add and Subtract Fractions

Part : Guided Practice Lesson 6 Part : Guided Practice Lesson 6 Part : Guided Practice Lesson 6 The student used labels and jump arrows to show each part of the hike on a number line. It is just like adding whole numbers! Study the model below. Then solve problems 6 8. Student Model Jessica hiked mile on a trail before she stopped to get a drink of water. After her drink, Jessica hiked another mile. How far did Jessica hike in all? Look at how you could show your work using a number line. before drink after drink 7 Mr. Chang has a bunch of balloons. of the balloons are red. of the balloons are blue. What fraction of the balloons are neither red nor blue? Show your work. Possible student work using a model: r r r b b I think that there are at least two different steps to solve this problem. Pair/Share How else could you solve this problem? Solution: mile 0 red Solution: blue neither red nor blue Pair/Share How is this problem different from the others you ve seen in this lesson? What fraction represents the whole fruit smoothie? 6 Ruth made a fruit smoothie. She drank of it. What fraction of the fruit smoothie is left? Show your work. Possible student work using an equation: 8 Emily ate of a bag of carrots. Nick ate of the bag of carrots. 6 6 What fraction of the bag of carrots did Emily and Nick eat altogether? Circle the letter of the correct answer. A 6 B C 6 D To find the fraction of the bag Emily and Nick ate altogether, should you add or subtract? Pair/Share How did you and your partner decide what fraction to start with? of a smoothie Solution: Rob chose D as the correct answer. How did he get that answer? Rob added both the numerators and the denominators. Pair/Share Does Rob s answer make sense? 8 L6: Add and Subtract Fractions L6: Add and Subtract Fractions 9 Students use models, number lines, or equations to solve word problems involving addition and subtraction of fractions. Ask students to solve the problems individually and label fractions in their drawings. When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. SOLUTIONS Ex A number line is shown as one way to solve the problem. Students could also solve the problem by drawing a model that is divided into fifths and shading sections ( sections out of plus sections out of ). 6 Solution: of a smoothie; Students could solve the problem by using the equation. (DOK ) 7 Solution: ; Students could solve the problem by drawing a picture of 0 balloons and labeling as red and as blue. (DOK ) 8 Solution: C; Rob added the numerators correctly, but he mistakenly added the denominators together, too. Explain to students why the other two answer choices are not correct: A is not correct because you are not subtracting from ; this is an addition problem. 6 6 B is not correct because is not equivalent to. 6 (DOK ) L6: Add and Subtract Fractions 6

Part : MAFS Practice Lesson 6 Part : MAFS Practice Lesson 6 Part : MAFS Practice Lesson 6 Solve the problems. Liang bought some cloth. He used } 8 of a yard for a school project. He has } of a yard 8 left. How much cloth did Liang buy? A B C D } of a yard 8 7 }} of a yard 6 7 } of a yard 8 8 } of a yard 8 Carmela cut a cake into equal-sized pieces. She ate }} of the cake, and her brother ate }} of the cake. What fraction of the cake is left? A }} of the cake B }} of the cake C 7 }} of the cake D }} of the cake Lee s muffin mix calls for } cup of milk, } cup of oil, and } cup of sugar. How much more milk than oil does she need for the muffin mix? cup Lucy and Margot are mowing the lawn. They divided the lawn into 8 equal sections. Lucy mowed sections and Margot mowed sections. Which model can be used to find the total fraction of the lawn they mowed? Circle the letter of all that apply. A B C D 6 7 0 8 8 8 8 8 8 8 6 7 0 8 8 8 8 8 8 8 In all, Cole and Max picked }} 9 0 of a bucket of blueberries. Cole picked }} of the 0 bucket of blueberries. What fraction of the bucket of blueberries did Max pick? Possible student work using a number line: Show your work. Answer Max picked of the bucket of blueberries. 6 A pizza is cut into 8 equal slices. Together, Regan and Juanita will eat } of the pizza. 8 What is one way the girls could eat the pizza? Show your work. Possible student work using a model: Answer Regan could eat Juanita could eat 6 7 8 9 0 0 0 0 0 0 0 0 0 0 6 8 8 J R R J J of the pizza, and of the pizza. Self Check Go back and see what you can check off on the Self Check on page 9. 0 L6: Add and Subtract Fractions L6: Add and Subtract Fractions Students add and subtract fractions to solve word problems that might appear on a mathematics test. SOLUTIONS Solution: C; Possible student work using an equation: 8 8 7 (DOK ) 8 Solution: C; Possible student work using equations: ; 7 (DOK ) Solution: cup; Possible student work using an equation: (DOK ) Solution: A; The model shows shaded in light gray 8 for Lucy s sections and shaded in dark gray for 8 Margot s sections. The total shaded sections represent the total fraction of the lawn they mowed. D; The number line starts at Margot s fraction ( ) 8 and adds for Lucy s fraction, for a total of. 8 6 8 (DOK ) Solution: 6 ; Possible student work using an equation: 9 6 (DOK ) 6 Solution: Possible student work using equations: 0 8 8 8, 8 8 8, 8 8 8, 8 8 8, 8 8 8, 8 0 8 (DOK ) 8 66 L6: Add and Subtract Fractions

Differentiated Instruction Lesson 6 Assessment and Remediation Ask students to find. 6 or For students who are still struggling, use the chart below to guide remediation. After providing remediation, check students understanding. Ask students to explain their thinking while finding. or If a student is still having difficulty, use Ready Instruction, Level, Lesson. If the error is... Students may... To remediate... 6 0 have added both the numerators and the denominators. have added numerators, added denominators, and then simplified. have subtracted the fractions. have subtracted the fractions and simplified. Remind students that the denominator tells the kind of parts you are adding. Explain that just as apples apples 6 apples, tenths tenths 6 tenths. Remind students that the denominator tells the kind of parts you are adding. Explain that just as apples + apples = 6 apples, tenths + tenths = 6 tenths. Remind students to read the problem carefully to be sure they re using the correct operation. Remind students to read the problem carefully to be sure they re using the correct operation. Hands-On Activity Use fraction strips to add fractions. Challenge Activity Write a problem for a given sum. Materials: strips of paper, markers Distribute paper and markers to each student. Direct students to fold a strip of paper in half, and then in half again. Tell them to unfold the strips and use the marker to show the equal sections. Tell students to color of the strip. Then have them color another of the strip. Write on the board. Challenge them to use their fraction strips to show that the sum is or. If time allows, repeat for other denominators by folding another strip of paper three or four times. Tell students that the sum of two fractions is. However, the original fractions did not have denominators of. Challenge students to write a fraction addition problem that has a sum of. Possible answer: L6: Add and Subtract Fractions 67