Multiplying & Dividing Fractions – The Challenge of Computation vs. Conceptualization

1. Multiplying & Dividing Fractions – The Challenge of Computation vs. Conceptualization Math Alliance July 27, 2010 DeAnn Huinker, Melissa Hedges, Chris Guthrie, & Beth Schefelker

3. Learning Intentions & Success Criteria • Learning Intention – We are learning to: • Deepen our conceptual understanding of division and multiplication with fractions. • Success Criteria – You will be able to: • Use real-life problem-solving situations to surface the meaning of division and multiplication of fractions.

4. Popcorn PartyServing Size: 2 cups How many servings from: • 8 cups of popcorn • 5 cups of popcorn

5. Summarize Popcorn Party: Serving Size 2 cupsHow many servings from 8 cups; 5 cups? • Is this a measurement or partitive context? • Write the equation for each situation (8 cups; 5 cups). • Identify the meaning of each number in the equations. • Draw a picture, make a diagram, or use paper strips to show how you got your answer.

6. SummarizeHow many 2–cup servings? 8 ÷ 2 = 4 cups cup servings serving 12 5 ÷ 2 = 2 cups cup serving servings

7. 12 What does the represent? • One-half of a cup of popcorn? • One-half of a serving? • Are you sure?

8. 12 12 Popcorn PartyServing Size: cup How many servings from: 1 cup of popcorn 4 cups 2 cups

9. 12 12 SummarizeHow many 2–cup servings? 1 ÷ = 2 cup servings 4 ÷ = 8 cups cup serving servings cup serving

10. TaskPopcorn Parties #1 & #2 • Facilitator • Paper Strip Demonstrator

11. Algorithm

12. TaskPopcorn Parties #1 & #2 • Facilitator poses one problem at a time. • Each individual silently solves it. • On facilitator’s cue: State answer. • Take turns as the demonstrator who models with paper strips. • Take turns to justify your reasoning.

13. 1 4 Popcorn Party #1 • Serving Size: cup of popcorn • How many servings can be made from: • 1 cup of popcorn • 2 cups of popcorn • 3 1/2 cups of popcorn

14. Popcorn Party #2 3 4 • Serving Size: cup of popcorn • How many servings can be made from: • 3/4 cup of popcorn • 6 cups of popcorn • 2 1/4 cups of popcorn • 4 1/2 cups of popcorn

15. 34 Popcorn PartyServing Size: cup How many servings from 4 cups? Using your paper strips, work individually to solve the problem. THEN compare results.

16. 34 4 ÷ = cups cup serving Just as with whole numbers, it is important to understand the meaning of the answer and how to interpret and relate ways of reporting “remainders.” • Think about a “leftover or extra” amount • Think about the “number of groups” • Think about the “size of a group”

17. Interpreting Remainders For each problem, • Place it in a context • Solve it with paper strips • Interpret the solutionwith both a “leftover” and as it relates to the “groups.” ÷ = ? 3 ÷ = ? 78 38 58 Think about • “leftover or extra” amount • the “number of groups” • the “size of a group”

18. Discussion In reviewing these division situations and your solutions, what are you noticing?

20. Consider ⅔ × ¾ Describe a real-world situation that can be modeled by this equation.  What could the ⅔ represent?  What could the ¾ represent? Turn and Share As you crafted a real-world situation, what struggles emerged?

21. Back to the basics… Consider for a moment 2×3.  What could the 2 represent?  What could the 3 represent? • What insights surfaced? • In what ways are 2×3 and ⅔ × ¾ similar? • In what ways are they different? • Does the meaning of multiplication change when moving from whole numbers to fractions?

22. What do you understand about 2/3 x 3/4? “Algorithms for multiplication of common fractions are easy for teachers to teach and students to use, but their meanings are elusive.” -- Zhijun Wu

23. Julie is making the family dinner. She buys 4 packages of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy? Turn and talk –  What does the 4 represent?  What does the 5/8 represent? • Individually – • Work through this problem. • Record your thinking on note cards. • Represent your thinking using numbers, pictures, & words. • Place face down in the middle of your table when done.

24. Debriefing Strategies • Each person picks a card to study. • After 30 seconds pass the card to the right. • Study the strategy on each card. • After all cards have been passed, share comments, questions or ah-ha’s with the table group. Table Discussion How do these strategies demonstrate an understanding of multiplication?

25. Looking at student work... Julie is making the family dinner. She buys 4 packages of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy? In what ways do the students show understanding of the situation as multiplication? How do students show that they understand the “number of groups” and the “size of groups” in their representations?

26. “These types of situations can be modeled by the repeated-addition interpretation. The link between multiplication and addition is clearly seen here. The repeated-addition model offers a satisfying interpretation in this case.” --Zhijun Wu How does your thinking change when you consider this next problem . . .

27. Taking a run... I wanted to run 4 miles. I ran 5/8 of the distance before I stopped for water. How many miles did I run before I had to stop for water? Put your pencils down. Turn and talk –  What does the 4 represent?  What does the 5/8 represent? • Individually – • Work through this problem. • Record your thinking on note cards. • Represent your thinking using numbers, pictures, & words. • Place face down in the middle of your table when done.

28. Taking a run... Strategy Debrief 1. Each person picks a card to study. 2. After 30 seconds pass the card to the right. 3. Study the strategy on each card. 4. After all cards have been passed, share comments, questions or ah-ha’s with the table group. Table Discussion How do these strategies demonstrate an understanding of multiplication?

29. Thinking about parts and wholes Combining parts: Problem 1 4 = packages of meat 5/8 = weight per package (quantity)‏ “4 parts of 5/8 lbs each” Finding part of a group: Problem 2 4 = miles OR the whole run 5/8 = part of the 4 miles (5/8 is now the operator – we do not need a complete whole but we need a part of that whole.)‏ “5/8 parts of 4 miles”

30. Taking a Run – Looking at Student Work In what ways do the students show understanding of the situation as multiplication?

31. What does multiplication of fractions encompass? • Multiplication of fractions involves: •  Combining equal parts •  Finding a part of a whole or part of a group • Doing both – combining equal parts and finding part of a whole. • Problem Sort • Read through each problem. • As a table, decide which problem type is represented in each context. • Solve using pictures, diagrams, and numbers.

32. From Whole Numbers to Fractions What experiences do students need to extend the meaning of multiplication from whole numbers to multiplication of fractions? • Experience with real-life problem solving situations. • Use of concrete and pictorial representations to support students as they reason. • Opportunities to explore the meaning of multiplication through a variety of problem formats involving fractions.

33. Homework To help you prepare for the exam next week it is recommended that you complete the following: Division of Fractions • Read Beckmann pp. 326-329 • Do Practice Problems for Section 7.4 pp. 335-339 #1, #3, #8a, b, c Multiplication of Fractions • Read Beckmann pp. 263-268 • Do Practice Problems for Section 6.1 pp.268-269

34. Problems for Card Sort Mrs. Smith has 120 books in her fourth grade classroom. 4/5 of the books are fiction. How many books are fiction? All notebooks at the local store are discounted by ¼ A notebook originally cost $0.96. How much do you save on one notebook if you buy it today? Julie bought 4/5 of a yard of fabric for her class project. Later she found that she needed only ¾ of the material. How much material did Julie use for her project? At the supermarket potatoes are bagged in ¾ pound bags. Mom bought 3 bags of potatoes. How many pounds of potatoes did mom buy? Red cabbage cost $0.39 a pound. Julie bought 3 1/3 pounds of red cabbage to prepare her dish. How much did she pay for the red cabbage? I put a container holding a half gallon of ice cream into the freezer. Two days later the ice cream container is 2/3 full. How much ice cream is in the container? Melissa is planning on making several batches of cookies. She needs 2/3 cup of sugar for every batch she makes. She plans on making 2 ½ batches. How much sugar will she need to make these batches.