Divide Fractions

Est. Class Sessions: 2–3

Developing the Lesson

Introduce and Discuss Division of Fractions. Division of fractions can be difficult to conceptualize and understand. In this lesson, students will use real-world contexts, concrete models, drawings, and invented strategies to help them learn when and how to divide fractions. They will apply their prior understanding of division to divide unit fractions by whole numbers (e.g., 1/3 ÷ 4) and whole numbers by unit fractions (e.g., 4 ÷ 1/5 ). This will eventually lead to the construction of a meaningful procedure for dividing fractions where students will be able to make sense of the symbols and steps in an algorithm.

The lesson begins with division word problems that become increasingly more complex. The numbers in the word problems vary. They include a whole number divided by a whole number (24 ÷ 3), a whole number divided by a whole number that results in a fraction (2 ÷ 5), a whole number divided by a fraction (3 ÷ 1/2 ), and a fraction divided by a whole number ( 1/2 ÷ 4).

Direct students to the Divide Fractions pages in the Student Guide and ask them to read the problem in Question 1. Students should be able to connect this measurement problem to the question of “How many groups of 3 cups are in 24 cups?” and to the division sentence 24 ÷ 3 = 8. This will enable students to use their understanding of whole number concepts to build understanding of division with fractions. In Question 2, 2 cups of frosting are divided among 5 cupcakes, 2 ÷ 5. The question is, “How much of a group of 5 is there in 2?” Since there is not a complete group of 5 to measure out of 2, each cupcake will get exactly 2/5 of a cup of frosting. In Question 3, guide students to thinking “How many halves in 3?” and connect that to the division sentence 3 ÷ 1/2 . Question 4 presents a fraction divided by a whole number, 12 of a cake divided by 4 people, 1/2 ÷ 4. The question is, “How much of a group of 4 is in a half?” or “How can I divide 1/2 into 4 equal parts?” Remind students to answer the question, “How much of the whole cake will each person get?”

For each problem in Questions 1–4, student pairs will read the question carefully, determine what they need to find out, decide how they will label their answer, and draw pictures as a part of their invented solution strategies. Assign and discuss one question at a time, using the Student Guide questions to guide your discussion. When students have solved a problem, ask several to display their pictures and solutions. Have students give their answers verbally first before recording their steps with symbols. Be sure to discuss the reasonableness of each answer. Based on their experiences with whole number division, students might think that division with fractions will always result in answers smaller than the original numbers. That is true in the problem in Question 4, 1/2 ÷ 4, but in Question 3, the quotient is greater than the dividend or the divisor: 3 ÷ 1/2 = 6. Discuss solution strategies and help students make connections between the drawings, the words in the problem, and the following number sentences after they find a solution:

Question 1: 24 ÷ 3 = 8

Question 2: 2 ÷ 5 = 2/5

Question 3: 3 ÷ 1/2 = 6

Question 4: 1/2 ÷ 4 = 1/8

See Figure 1 for possible solution strategies to the problems in Questions 1–4.

Invert-and-Multiply Algorithm. “To invert the divisor (find the reciprocal) and multiply is probably one of the most mysterious rules in elementary mathematics. We want to avoid this mystery at all costs. It makes sense to examine division with fractions from a more familiar perspective. As with the other operations, go back to the meaning of division with whole numbers. ” (Van de Walle, 2014)

In this lesson, students should focus on using different models of division (partitive and measurement) to model division problems that involve unit fractions. Students do not need to develop or discover an algorithm but might notice the patterns across problems.

How many 2s are in 10? 10 ÷ 2 = 5

How many 5 are in 10? 10 ÷ 5 = 2

How many 1/3 are in 4? 4 ÷ 1/3 = 12

How many 1/2 are 8? 8 ÷ 1/2 = 16

There are 2 halves in each 8. 2 × 8 = 16.

Use Invented Strategies to Model and Solve Problems. After discussing a variety of solution strategies, assign Questions 5–16 on the Divide Fractions pages to student pairs. Before students begin to work, display and direct their attention to the Math Practices page in the Student Guide reference section. Focus their attention on MPE1, 2, 5, and 6.

Ask:

What does it mean to “know the problem” [MPE1]? (Possible response: It means you have read the problem carefully so you know what you need to figure out and what questions to answer. You need to know what information in the problem is important and what kind of operation to perform to get the answer.)
What kinds of strategies can you use to solve fraction division problems [MPE2]? (Possible responses: I can use invented strategies, draw pictures, use area models like rectangles or fraction circle pieces, use repeated subtraction, and use number lines.)
How can you clearly show your work [MPE5]? (Possible response: I can draw a picture and clearly show the steps I took to solve the problem.)
How do you know which labels to use [MPE6]? (Possible response: You have to read the question carefully to see what it is asking you to find out and then label the quotient appropriately.)

Remind students to read each question carefully, determine what they need to find out, decide how they will label their answer, and draw pictures as a part of their invented solution strategies. Again, the division word problems become increasingly more complex, beginning with a whole number divided by a whole number in Question 5. Questions 6 and 13 involve a whole number divided by a whole number that results in a fraction (e.g., 4 ÷ 6). Questions 7, 9–12, and 15 involve whole numbers divided by fractions (e.g., 8 ÷ 1/3 ). Questions 8, 14, and 16 involve fractions divided by whole numbers (e.g., 1/2 ÷ 3). Circulate as students work and observe the strategies they use to divide fractions. Select a few students to explain their solution strategies to the class upon completion of the problems in Questions 7–16. Try to select a variety of problem types and solution strategies.

Discuss Solution Strategies. Upon completion, ask the selected students to discuss their solutions to the problems in specific questions. Have students give their answers verbally first (e.g., 8 feet of wood cut into 1/3- foot pieces will give Romesh 24 pieces of wood) before helping them to record their steps with symbols (e.g., 8 ÷ 1/3 = 24).

As students share, ask questions such as the following:

What is Question [15] asking you to find out? (How many 1/3 - foot pieces of wood will Romesh have?)
Show us how you solved the problem in Question [15]. (Possible response: First, to understand the problem, I sketched a long piece of wood and divided it into 8 sections. Then I divided each of those sections into thirds. To solve the problem, I drew a number line to represent the 8 feet of wood. I started at 8 and took 1/3 - long hops backwards until I got to 0. It was 24 hops.) [See Figure 2.]
Why did you choose this strategy? (Possible response: The long line of wood made me think of a number line. I knew I could take away 1/3 - foot sections at a time using backwards hops of 1/3 on the number line.)
Did your picture help you solve the problem? How? (Possible response: When I drew the length of the wood, it helped me picture a number line.)
How does your picture match with your solution? (Possible response: 3 pieces in each of the 8 sections is 24 pieces. Starting at 8 on the number line, I repeatedly subtracted 1/3 and I did that 24 times.)
How did you know how to label your answer? (Possible response: The question was “How many pieces of wood?” so I knew the answer was going to be some number of pieces.)
How do you know that your answer makes sense? (Possible response: 8 feet is long, and 1/3 of a foot isn't that long, so I knew it could be cut into a lot of pieces. If there are 3 one-third pieces to every foot, it would be 24.)

Use Check-In: Questions 15–16 on the Divide Fractions pages in the Student Guide and the corresponding Feedback Box in the Teacher Guide to assess students' abilities to represent division of fractions with drawings [E3]; solve word problems involving division of fractions [E5]; divide fractions using area models, drawings, and number lines [E4]; know the problem [MPE1]; find a strategy [MPE2]; show work [MPE5]; and use labels [MPE6].

Use the Workshop in Lesson 11 to provide targeted practice.

Assign Questions 17–30. Students will solve division number sentences and look for patterns in the problems. They will use the relationship between the multiplication and division to make connections between dividing fractions and multiplying fractions. This begins to lay a foundation for students to develop an understanding of the invert-and-multiply algorithm.