When they have completed the questions, display and direct students' attention to Questions 17–20 from the Divide Fractions pages in the Student Guide.
- What do you notice about the problems in Questions 17–20? (Possible response: There's a pattern. Each question uses the same numbers. It's a whole divided by a fraction and then it's a fraction divided by a whole number.)
- The numbers in the problems are the same, but do they mean the same thing? What does 3 ÷ 1/2 mean in Question 17? Attach a story to it. (Possible responses: The number sentences do not mean the same thing. Three divided by 1/2 means how many halves are in 3. For example, how many 1/2 - sandwich servings are in 3 whole sandwiches.)
- What does 1/2 ÷ 3 mean? What is being divided? (Possible responses: One-half divided into 3 equal shares. 12 divided into 3 parts. One-half is being divided.)
- In which problem will the quotient be larger? 3 ÷ 1/2 or 1/2 ÷ 3? Why? (Possible response: The quotient for 3 ÷ 1/2 is bigger because you are finding how many halves are in 3 wholes. In 1/2 ÷ 3, you are starting with only 1/2 and then you are dividing that up into 3 smaller parts.)
- Does division with fractions always lead to a quotient that is smaller than the numbers you started with? Why? Can you give an example? (No. Possible response: For example, 3 ÷ 1/2 = 6. The quotient 6 is bigger than 1/2 or 3. You are starting with 3 and cutting it into smaller parts so you will have more than 1/2 or 3.)
- Can you give an example when division of fractions leads to a quotient smaller than the numbers you started with? ( 1/2 ÷ 3 = 1/6 )
Direct students' attention to Question 23.
- What division number sentence did you use to describe one-third of a sheet of paper divided into 4 parts (Question 23A)? ( 1/3 ÷ 4 = 1/12 )
- Is the quotient larger or smaller than the numbers in the sentence? (smaller)
- Does this make sense? How do you know? (Possible response: Yes, it makes sense because you are starting with only 1/3 a sheet of paper and then are dividing that into 4 parts. Each of those parts is going to be smaller than 1/3 or 4.)
- What is 1/3 × 1/4 ? ( 1/12 )
- What do you notice? (I get the exact same answer.)
- Does it mean the same thing? Does 1/3 ÷ 4 = 1/3 × 1/4 ? (yes)
Direct students' attention to the number sentences in Question 24:
1/3 ÷ 4 = 
× 4 = 1/3
Students can use the relationship between multiplication and division to explain that 1/3 ÷ 4 = 1/12 because 1/12 × 4 = 1/3 . Remind them that division is the inverse or opposite of multiplication. Give them an example involving whole numbers: 21 ÷ 7 = 3 because 3 × 7 = 21. Guide students to use this relationship when describing their answers for Questions 24–28. Ask them to make connections between the symbols in the number sentences and the drawings on the page for each problem.
- How is the multiplication sentence related to the division sentence? (See Figure 3 for an example. Possible response: The division sentence is 1/2 divided into 5 equal parts is 1/10 . The multiplication sentence is opposite of that. 5 times 1/10 is 1/2 .)
- Use the drawing next to the problem to help you explain what is happening in the number sentences. (Possible response: For 1/2 ÷ 5, the half is cut into 5 parts. Each little part is 1/10 , so 1/2 ÷ 5 = 1/10 . For 1/10 × 5, each little part is 1/10 and if you have 5 of them, they cover 1/2 of the whole.)
Finally, ask students to discuss the patterns they see in the problems in Questions 29–30. By writing the corresponding division and multiplication sentences, students will start to see the important connection between the two operations. In Question 29, students will notice pairs of numbers such as 3 × 1/3 = 1 or 1/3 × 3 = 1 that when multiplied, result in a product of 1. In Question 30, students solve problems like 1 ÷ 3. There is not a complete group of 3 to measure out of 1. Instead there is exactly one-third of a group of three, 1 ÷ 3 = 1/3 . Students should notice a pattern. Since 1 ÷ 3 = 1/3 , 1/3 × 3 = 1. Help students apply this pattern to solve the following problem:
2 ÷ 1/3 =
- How many groups of 1/3 are in 1? (3)
- How many groups of 1/3 are in 2? How do you know? (6; Since there are 3 groups of 1/3 in 1, there must be twice as many groups of 1/3 in 2.)
Display the following problem:
4 ÷ 1/3 =
- How many groups of 1/3 are in 4? How do you know? (12; Since there are 3 groups of 1/3 in 1, there must be four times as many groups of 1/3 in 4.)
Practice with problems such as these will help students begin to see the connection between multiplication and division and make better sense of the invert-and-multiply algorithm for division of fractions they will learn in the future.
Assign the Multiply and Divide Fractions Quiz Assessment Masters for students to complete individually to check progress with multiplying and dividing fractions.
Use the Multiply and Divide Fractions Quiz Assessment Masters with the Feedback Box to assess students' abilities to represent and identify the simplest form of a fraction [E2]; represent multiplication and division of fractions [E3]; multiply and divide fractions using area models, drawings, and number lines [E4]; solve word problems involving multiplication and division of fractions [E5]; explain the effects of factors less than and greater than 1 on the product or quotient of fractions [E6]; choose appropriately from among estimation and computation strategies [E7]; and find a strategy [MPE2].
Use the Workshop in Lesson 11 as targeted practice.
