Objective To introduce division of fractions and relate the operation of division to multiplication. Assessment Management

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1 Fraction Division Objective To introduce division of fractions and relate the operation of division to multiplication. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find common denominators for pairs of fractions. [Number and Numeration Goal ] Use diagrams and visual models for division of fractions problems. [Operations and Computation Goal ] Solve number stories involving division of a fraction by a whole number, division of a whole number by a fraction, and division of a fraction by a fraction. [Operations and Computation Goal ] Write equations to model number stories. [Patterns, Functions, and Algebra Goal ] Key Activities Students use diagrams and visual models to divide fractions. They solve number stories involving division of a fraction by a whole number, division of a whole number by a fraction, and division of a fraction by a fraction. Students use visual fraction models and equations to represent the problem. Ongoing Assessment: Informing Instruction See page 68. Ongoing Assessment: Recognizing Student Achievement Use journal page 89. [Operations and Computation Goal ] Materials Math Journal, pp B transparency of Math Masters, p. 0B Student Reference Book, pp B Study Link 8 slate or half-sheets of paper Advance Preparation Math Boxes 8 Math Journal, p. 90 Students practice and maintain skills through Math Box problems. Study Link 8 Math Masters, p. 8 Students practice and maintain skills through Study Link activities. READINESS Playing Build-It Student Reference Book, p. 00 Math Masters, pp. 6 and 7 per partnership: six-sided die Students compare and order fractions and rename mixed numbers as fractions. ENRICHMENT Exploring the Meaning of the Reciprocal Math Masters, p. 9 calculator Students explore the meaning of the reciprocal. EXTRA PRACTICE Dividing with Unit Fractions Math Masters, p. B Students practice using visual models to divide fractions. Teacher s Reference Manual, Grades 6 pp Unit 8 Fractions and Ratios

2 Getting Started Mental Math and Reflexes Pose questions about unit fractions. Suggestions: How many _ s are in whole? How many _ s are in? 8 How many _ s are in _? How many _ s are in? 8 How many _ s are in _? 6 How many _ s are in _? How many _ s are in _? How many _ s are in _? How many _ s are in? Math Message Solve Problems and on journal page 88. Study Link 8 Follow-Up Have partners share answers and resolve differences. Ask volunteers to explain their solution strategies for Problem 9. Teaching the Lesson Math Message Follow-Up (Math Journal, p. 88: Math Masters, p. 8) WHOLE-CLASS DISCUSSION Discuss students solutions. Use a transparency of Math Masters, page 0B to illustrate Problems a c. Problem a 0 6 Problem b 0 6 Problem c Date 8 Math Message Fraction Division. Use the ruler to solve Problems a c. Student Page Time 0 6 Point out that each problem on the journal page asks: How many x s are in y? Ask students to translate each problem into a question of this form. Record the questions on the board.. a. How many s are in 6? b. How many _ s are in 6? c. How many _ s are in _ 8?. a. How many s are in 0? b. How many _ s are in 0? 0 6 segments a. How many -inch segments are in 6? segments b. How many _ -inch segments are in 6? 6 segments c. How many _ 8 -inch segments are in _ of an inch?. a. How many -pound boxes of nuts can be made from 0 pounds of nuts? Use the visual below to help you solve the problem. Sample answer: lb lb lb lb lb lb lb lb lb lb boxes b. How many _ - pound boxes can be made from 0 pounds of nuts? Draw a picture to support your answer. 0 boxes lb lb Sample answer: lb lb lb lb lb lb lb lb Math Journal, p. 88 Lesson 8 68

3 Refer students to the illustrations and questions for Problems and, and ask what division open number sentence fits the first question. 6 = s Continue for the other problems, writing the division open number sentence next to each question on the board. Ask students to refer to the visual models, if needed.. a. How many s are in 6? 6 = s b. How many _ s are in 6? 6 _ = s c. How many _ s are in _ 8? _ _ 8 = s. a. How many s are in 0? 0 = b b. How many _ s are in 0? 0 _ = b NOTE The division number models use b (for the number of boxes) and s (for the number of segments) to represent the unknowns. Students may prefer to use other letters or symbols. To avoid confusion in this introduction to division of fractions, the number models use rather than / to show division. Dividing with Unit Fractions (Math Journal, p. 89; Student Reference Book, pp. 79 and 80A) WHOLE-CLASS DISCUSSION Read and discuss the first example on page 79 of the Student Reference Book on dividing a whole number by a unit fraction. A unit fraction is a fraction with a numerator of. Briefly discuss the solution. Draw rectangles on the board, and ask students to copy the rectangles on a sheet of paper or slate. Date 8 _ = Time Dividing with Unit Fractions. Four pizzas will each be sliced into thirds. Use the circles to show how the pizzas will be cut. Find how many slices there will be in all. The drawings show that there will be slices in all.. Edith has of ribbon. She wants to cut the ribbon into _ -inch pieces. How many _ pieces can she cut? Edith can cut 8 pieces. So, _ = 8.. Two students equally share _ of a granola bar. Divide the rectangle at the right to show how much of the bar each will get. _ Each student will get 6 of a granola bar. _ = _ 6. Three students equally share _ of a granola bar. Divide the rectangle at the right to show how much of the bar each will get. _ Each student will get of a granola bar. _ = _ Student Page 0. When you divide a whole number by a unit fraction (less than ), is the quotient larger or smaller than the whole number? Explain. Sample answer: The quotient is larger than the whole number because you are finding how many small parts fit into something that is larger. Ask students to use the rectangles to illustrate the following problem: Jane has loaves of banana bread to share with her friends. If she cuts each loaf into _ s, how many quarter loaves will she have to share with her friends? Students should conclude that one way to illustrate the solution is to divide each of the rectangles into equal parts. Ask: _ = How many _ s are in? What number model represents this problem? _ = Ask students to write _ = below their rectangles. 6. When you divide a unit fraction (less than ) by a whole number, is the quotient larger or smaller than the fraction? Explain. Sample answer: The quotient is smaller than the fraction because you are dividing up the fraction into an equal number of smaller parts. Math Journal, p Unit 8 Fractions and Ratios

4 A unit fraction can also be divided by a whole number. Read and discuss the examples on page 80A of the Student Reference Book. Draw a rectangle on the board, and divide it into equal parts with _ shaded. Ask students to draw the same diagram on a piece of paper or slate. Tell students that you can represent a unit fraction (such as _ ) being divided by a whole number (such as ) by drawing a model for the fraction and then cutting it up into smaller equal parts. Pose the following problem: Three family members equally share _ of a loaf of corn bread. How much of the loaf of corn bread will each person get? Have students divide their rectangles to show how the corn bread can be divided to find the solution to the problem. Ongoing Assessment: Informing Instruction Watch for students who record the answer as _. Have students draw the line for thirds to extend across the rectangle in order to visualize the total number of parts out of. Have a volunteer come to the board to show the solution. The student should divide the shaded fifth into three equal parts using horizontal lines. If necessary, model the lines extended all the way across the larger rectangle, with one small part double shaded. Explain that because the family only has _ of a loaf to begin with, when it is divided into three equal parts, each part of the corn bread that is cut up is _ of the entire loaf. So each person will get _ of the loaf of corn bread. = Ask: What number model represents this problem? _ = _ Ask students to write _ = _ below their rectangles. Lesson 8 68

5 Date 8 Time Relationship Between Multiplication and Division. Each division number sentence on the left can be solved by using a related multiplication sentence on the right. Draw a line to connect each number sentence on the left with its related number sentence on the right. Division Multiplication a. _ = n n _ = _ b. _ = n n = _ c. _ = n n _ = 9_ d. 0 _ = n n 9_ 0 = _ e. _ 9_ 0 = n n _ = Student Page Have students read through Problems on journal page 89. Ask them to describe how Problems and are different from Problems and. Sample answer: In Problems and, you are dividing a whole number by a unit fraction. In Problems and, you are dividing a unit fraction by a whole number. Have students solve Problems 6. Circulate and assist. Briefly discuss solutions. f. _ _ = n n _ = 9_ 0. Solve the following division number sentences (from above). Use the related multiplication sentences to help you find each quotient. a. _ = 0 b. _ = _ 0 c. _ = 0 d. _ _ =. How is dividing by _ different from dividing _ Sample answers: Dividing by? by _ results in a quotient that is greater than. Dividing _ by results in a quotient that is less than _. With _, you are finding how many _ s are in. With _, you are finding how many s are in _, which is a very tiny number.. Write a number story for _. Answers vary.. Write a number story for _. Answers vary. Ongoing Assessment: Recognizing Student Achievement Journal Page 89 Use journal page 89, Problems and to assess students ability to divide a whole number by a unit fraction using a visual model. Students are making adequate progress if they are able to solve Problems and. Some students may be able to solve Problems and, which involve dividing a unit fraction by a whole number. [Operations and Computation Goal ] Math Journal, p. 89A Relationship between Multiplication and Division (Math Journal, p. 89A) WHOLE-CLASS DISCUSSION Another way to solve a fraction division problem is to think about it as a related fraction multiplication problem. Remind students of the relationship between multiplication and division. For example, to solve 6 7, you can think: What number times 7 is 6? or n 7 = 6. 9 Write the following problems on the board to show how the relationship helps when dividing with fractions. To solve 0 0, think: What number times is _? Or n = _ 0 0. To solve 6 _, think: What number times _ To solve 6 _, think: What number times _ is 6? Or n _ = 6. 0 is 6? Or n _ = 6. 9 To solve _ 0 _, think: What number times _ is _ 0 0 0? Or n _ 0 = _ 0. _ Ask students to solve the problems on journal page 89A. Circulate and assist. Introducing Common Denominator Division (Math Journal, p. 89B) WHOLE-CLASS Draw four circles on the board, and ask students to copy these circles on a sheet of paper. Ask them to solve the problem _ and to illustrate their solution using the four circles. 68A Unit 8 Fractions and Ratios

6 After a few minutes, bring the class together to discuss their solutions. Use the students responses to emphasize that one method for obtaining the solution is to divide each of the circles into equal parts. As you illustrate this method on the board, point out that dividing each circle into equal parts is equivalent to renaming each whole as _. This shows that wholes is equivalent to _. _ _ = 6 Write _ _ = 6 under the circles on the board. Students will readily see that there are 6 groups of _. Emphasize that the answer is the result of dividing the numerators = 6. Guide the discussion toward the following algorithm for division with fractions: Step Rename the numbers using a common denominator. Step Divide the numerators, and divide the denominators. Discuss the examples at the top of journal page 89B. Point out that this method works for fractions divided by fractions or for mixed numbers or whole numbers divided by fractions. Use the following example to show that this method also works for fractions divided by whole numbers. _ 8 6 = _ 8 8_ 8 _ = = _ 8 = _ 8 Solve Problems on journal page 89B as a class. Ask students to come up to the board to record their steps. Problem Solution _ =? _ = 0 _ = 0 = _ 6 _ 8 =? _ 6 _ 8 = 8 _ 8 = = _ _ 6 =? _ _ 6 = 0 _ 6 = 0 6 _ 6 = 0 = Links to the Future The algorithm introduced in this lesson focuses students on the meaning of division with fractions. The standard algorithm that involves multiplying by the reciprocal will be introduced in Sixth Grade Everyday Mathematics. Date 8 Time One way to divide fractions is to use common denominators. This method can be used for whole or mixed numbers divided by fractions. Step Rename the fractions using a common denominator. Step Divide the numerators, and divide the denominators. Examples: Solve. Common Denominator Division _ = _ _ = _ = _, or Student Page _ _ 6 = _ 6 _ 6 = _ 6 6 = _, or _ _ = 8_ _ = _ 8. _ =. _ 6 _ 8 =. _ _ 6 =. _ _ 8 =. _ 6 = _ 6. 6 _ = 7. Chase is packing flour in _ -pound bags. He has 0 pounds of flour. How many bags can he pack? 0 bags 8. Regina is cutting string to make necklaces. She has feet of string and needs _ feet for each necklace. How many necklaces can she make? 0 necklaces 9. Eric is planning a pizza party. He has large pizzas. He wants to cut each pizza so that each serving is _ of a pizza. How many people can get a full serving of pizza? 8 people = 6_, or 6 0. A rectangle has an area of _ 0 m. Its width is _ 0 m. What is its length? meters Math Journal, p. 89B Lesson 8 68B

7 Date 8 Math Boxes. Find the whole set. a. 0 is _ of the set. 0 b. is _ of the set. 6 c. 8 is _ 7 of the set. 8 d. is _ 8 of the set. e. 9 is _ of the set. Student Page Time. Complete the table. Fraction Decimal Percent _ % _ 0. 8.% _ 0 0. % _ _ % _ , % Practicing Common Denominator Division (Math Journal, p. 89B) PARTNER Assign Problems 0 on the journal page. Have students work with a partner. Circulate and assist. Briefly share solutions as needed Add. a. _ + _ = b. = _ 8 + _ 6 c. 6 _ + _ 9 _ = 6 7_ 8 d. = _ 8 + _ 8 0 _ e. = _ _. Write a fraction or a mixed number for each of the following: _ a. minutes = 60, or _ hour 0_ b. 0 minutes = 60, or _ hour _ c. minutes = 60, or _ hour _ d. minutes = 60, or _ hour _ e. minutes = 60, or _ hour Math Journal, p A worker can fill boxes of crackers in minutes. At that rate, how many can she fill in hour? 80 boxes 6. Measure the line segment below to the nearest _ inch. in Ongoing Learning & Practice Math Boxes 8 (Math Journal, p. 90) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-0. The skill in Problem 6 previews Unit 9 content. Study Link 8 (Math Masters, p. 8) INDEPENDENT Home Connection Students practice operations with fractions and mixed numbers. Differentiation Options Name Date Time STUDY LINK 8 Mixed-Number Review. a. Four pizzas will each be cut into eighths. Show how they can be cut to find how many slices there will be in all. b. The drawing shows that _ =, so there will be 8 slices in all.. a. Two families equally share _ of a garden. Show how they can divide their portion of the garden. b. The drawing shows that _ = _ gets _ 6 of the total garden. 6, so each family Common Denominator Division Step Rename the numbers using a common denominator. Step Divide the numerators, and divide the denominators. Solve. Show your work.. _ = _, or 7 _. _ 7 _ = 0_. _ 8 _ = _ Practice 7. _ = _ Study Link Master, or _ 6. 6 _ = 7, or 8 _ 7 8. _ = 7_ 9. _ + _ = _ 0. _ 8 - _ 8 = 6_ 8, or _. 7 _ 9-8_ 9 = _ 9. _ 7 + _ = 8_, or _ READINESS Playing Build-It (Student Reference Book, p. 00; Math Masters, pp. 6 and 7) PARTNER 0 Min To practice comparing and ordering fractions and renaming mixed numbers as fractions, have students play this variation of Build-It. If students did not keep their Fraction Cards, they will need to cut the cards from Math Masters, page 6. Students play the game as introduced in Lesson 8- except that at the end of each round, they toss a six-sided die to determine a whole-number part for each of their fractions. Students then rename the mixed numbers as fractions. For example, after tossing a, the fractions _, _, _, 7_, and _ would become _ 6, _, _, and _. Renamed as fractions, the list would be 6_, 7_, and 6. 6, _, 0_,. _ + _ 77_ =, or 8 _. - =. _ 99_ =, or _ 6. _ 6_ 8_ 7 = 7, or Math Masters, p Unit 8 Fractions and Ratios

8 ENRICHMENT PARTNER Exploring the Meaning of the Reciprocal (Math Masters, p. 9) 0 Min To explore the relationship between a number and its reciprocal, have students use what they know about fractions, fraction multiplication, and their calculators to find the reciprocals of numbers. When students have finished the Math Masters page, ask them to describe the pattern for finding the reciprocal of a number. Guide students to see that the reciprocal of a fraction is the fraction with the numerator and denominator interchanged, or inverted. For example, the reciprocal of _ is 9_, and _ 9 9 9_ = 6_ =. The 6 reciprocal of a whole number is a unit fraction that has the whole number as its denominator. For example, the reciprocal of 8 is _ 8, so 8 _ 8 = 8_ 8 =. Name Date Time 8 Exploring the Meaning of the Reciprocal Lamont and Maribel have to divide fractions. Lamont doesn t want to use common denominators. He thinks using the reciprocal is faster, but he s not sure what a reciprocal is. Maribel looks it up on the Internet and finds this: One number is the reciprocal of another number if their product is. Example : Example : º? = _ º? = º _ = _ = _ º = _ = _ is the reciprocal of is the reciprocal of _ is the reciprocal of is the reciprocal of. Find the reciprocals. _ Teaching Master a. 6 6 b. _ 7 c. 0 0 d. _ 9. What do you think would be the reciprocal of _ 6? 7 Reciprocals on a Calculator On all scientific calculators, you can find a reciprocal of a number by raising the number to the - power.. Write each number in standard notation as a decimal and a fraction. _ a , 8 b , c. 0. -, 8. Write the key sequence you could use to find the reciprocal of 6. 6 ( ) F D, or 6. Write the key sequence you could use to find the reciprocal of _ 7. n 7 d ( ) F D, or 7 6. What pattern do you see for the reciprocal of a fraction? Once the original number is written as a fraction, the reciprocal is the original fraction written with the numerator as the denominator and the denominator as the numerator. Math Masters, p. 9 _ 6_ 9 EXTRA PRACTICE PARTNER Dividing with Unit Fractions (Math Masters, p. B) Students practice using visual models to divide fractions. Min Name Date Time 8 Number Stories: Division with Fractions. Five pies will each be sliced into fourths. Ira would like to find out how many slices there will be in all. a. Show how the pies will be cut. b. The drawings show that _ = 0, so there will be 0 slices in all.. Jake has a -inch strip of metal. He would like to find out how many _ -inch strips he can cut. 0 Jake can cut 6 strips. So, _ = 6.. Two students equally share _ of a granola bar. They would like to know how much of the bar each will get. a. Show how the piece of granola bar will be cut. 8, so each _ student will get 8 of a granola bar.. a. Drawing A can be used to find _. Drawing B can be used to find _ of _, or _ _. Use the drawings to show that _ = _ _. A B b. The drawing shows that _ = _ b. Complete. _ = = = _ Teaching Master = _ Math Masters, p. B Lesson 8 68

9 Name Date Time STUDY LINK 8 Mixed-Number Review. a. Four pizzas will each be cut into eighths. Show how they can be cut to find how many slices there will be in all. b. The drawing shows that _ 8 slices in all. =, so there will be. a. Two families equally share _ of a garden. Show how they can divide their portion of the garden. b. The drawing shows that _ =, so each family gets of the total garden. Common Denominator Division Step Rename the numbers using a common denominator. Step Divide the numerators, and divide the denominators. Solve. Show your work.. _ =. _ 7 _ =. _ 8 _ = 6. 6 _ 7 _ 9 = Practice 7. _ = _ 8 8. _ = 7 _ 9. _ + _ = 0. _ 8 - _ 8 =. 7 _ 9-8 _ 9 =. _ 7 + _ =. _ + _ =. - _ =. _ = 6. _ 6 _ 7 = Copyright Wright Group/McGraw-Hill

10 Name Date Time 8 Solving Mixed-Number Addition Problems Add. Write each sum as a mixed number in simplest form. Show your work.. _ + _ =. _ + _ 0 =. _ + _ =. _ + _ =. Josiah was painting his garage. Before lunch, he painted _ walls. After lunch, he painted another _ walls. How many walls did he paint during the day? 6. Julie s mom made muffins for Julie and her friends to share. Julie ate _ muffins. Her friends ate _ muffins. How many muffins did Julie and her friends eat altogether? Without adding the mixed numbers, insert <, >, or =. Explain how you got your answer. 7. _ _ 8. _ + 7 _ 8 8 Copyright Wright Group/McGraw-Hill A

11 Name Date Time 8 Number Stories: Division with Fractions. Five pies will each be sliced into fourths. Ira would like to find out how many slices there will be in all. a. Show how the pies will be cut. b. The drawings show that _ =, so there will be slices in all.. Jake has a -inch strip of metal. He would like to find out how many _ -inch strips he can cut. 0 Jake can cut strips. So, _ =.. Two students equally share _ of a granola bar. They would like to know how much of the bar each will get. a. Show how the piece of granola bar will be cut. Copyright Wright Group/McGraw-Hill b. The drawing shows that _ =, so each student will get of a granola bar.. a. Drawing A can be used to find _. Drawing B can be used to find _ of _, or _ _. Use the drawings to show that _ = _ _. A B b. Complete. _ _ = _ = _ = _ = B

12 Name Date Time Rulers Copyright Wright Group/McGraw-Hill 0B