Multiplication and Division of Fractions: Helping Increase the Certainty of Understanding

1. Multiplication and Division of Fractions: Helping Increase the Certainty of Understanding Steve Klass and Nadine Bezuk NCTM Annual Conference, Salt Lake City Utah, April 2008

2. Today’s Session • Welcome and introductions • Meanings for division and multiplication • Models for division and multiplication of fractions • Contexts for division and multiplication of fractions • Discussion

3. What Students Need to Know Well Before Operating With Fractions • Meaning of the denominator (number of equal-sized pieces into which the whole has been cut); • Meaning of the numerator (how many pieces are being considered); • The more pieces a whole is divided into, the smaller the size of the pieces; • Fractions aren’t just between zero and one, they live between all the numbers on the number line; • A fraction can have many different names; • Understand the meanings for whole numberoperations

4. Solving a Division Problem With Fractions • How would you solve ? • How would you solve ? • How might a fifth or sixth grader solve these problems and what answers might you expect? • How can pictures or models be used to solve these problems?

5. What Does Elliot Know? • What does Elliot understand? • What concepts is he struggling with? • How could we help him understand how to model and reason about the problem?

6. What Do Children Need to Know in Order to Understand Division With Fractions?

7. What Does Elliot Know? • What does Elliot understand? • What concepts is he struggling with? • How could we help him understand how to model and reason about the problem?

8. Reasoning About Division • Whole number meanings for division 6 ÷ 2 = 3 • Sharing / partitive • What does the 2 mean? What does the 3 mean? • Repeated subtraction / measurement • Now what does the 2 mean and what does the 3 mean?

9. Now Consider 6 ÷ • What does this mean? • How can it be modeled? • What contexts make sense for • Sharing interpretation • Repeated subtraction interpretation

10. Reasoning About Division With Fractions

11. Reasoning About Division With Fractions • Sharing meaning for division: 1 • One shared by one-third of a group? • How many in the whole group? • How does this work?

12. Reasoning About DivisionWith Fractions • Repeated subtraction / measurement meaning 1 • How many times can one-third be subtracted from one? • How many one-thirds are contained in one? • How does this work? • How might you deal with anything that’s left?

13. Materials for Modeling Division of Fractions • How would you use these materials to model 1? • Paper strips • Fraction circles • You could also use: • Pattern blocks • Fraction Bars / Fraction Strips/ Paper tape

14. ? Using a Linear Model With a Measurement Interpretation 1 How many one-thirds are in one and one-half?

15. Using an Area Model With a Measurement Interpretation • Representation of with fraction circles.

16. How Many Thirds? ? ?

17. A Context For Division of Fractions • You have 1 cups of sugar. It takes cup to make 1 batch of cookies. How many batches of cookies can you make? • How many cups of sugar are left? • How many batches of cookies could be made with the sugar that’s left?

18. Reasoning About Multiplication With Fractions

19. Multiplication of Fractions Consider: • How do you think a child might solve each of these? • What kinds of reasoning and/or models might they use to make sense of each of these problems?

20. Reasoning About Multiplication • Whole number meanings - U.S. conventions • 4 x 2 = 8 • Set - Four groups of two • Area - Four units by two units

21. Reasoning About Multiplication • Whole number meanings - U.S. conventions • 2 x 4 = 8 • Set - Two groups of four • Area - Two units by four units • When multiplying, each factor refers to something different. One tells how many groups and the other, how many in each group. The representations are quite different.

22. Reasoning About Multiplication • Fraction meanings - U.S. conventions • Set - Two-thirds of one group of three-fourths • Area - Two-thirds of three-fourths of a unit • Set - Three-fourths of one group of two-thirds • Area - Three-fourths of two-thirds of a unit

23. Models for Reasoning About Multiplication • Area/measurement models (fraction circles) • Linear/measurement (e.g. paper strips)

24. Materials for Modeling Multiplication of Fractions • How would you use these materials to model ? • Paper strips • Fraction circles • You could also use: • Pattern blocks • Fraction Bars / Fraction Strips • Paper folding/ paper tape

25. How much is of ? Using a Linear Model With Multiplication © Professional Development Collaborative

26. Using an Area Model with Fraction Circles for Fraction Multiplication • How would you use these materials to model

27. How much is of ? Using a Linear Model With Multiplication © Professional Development Collaborative

28. Using an Area Model with Fraction Circles for Fraction Multiplication • How would you use these materials to model ?

29. Contexts for Multiplication • Finding part of a part (a reason why multiplication doesn’t always make things “bigger”) • Pizza (pepperoni on ) • Brownies ( is frosted, of the that part has pecans) • Lawn ( is mowed, of that is raked)

30. Thinking More Deeply About Multiplication and Division of Fractions • Estimating and judging the reasonableness of answers • Recognizing situations involving multiplication or division of fractions • Considering and creating other contexts where the multiplication or division of fractions occurs • Making thoughtful number choices when considering examples

31. Contact Ussklass@projects.sdsu.edunbezuk@mail.sdsu.eduhttp://pdc.sdsu.edu © 2007 Professional Development Collaborative