Division of Fractions: Thinking More Deeply Division of Fractions: Thinking More Deeply Steve Klass National Council of Teachers of Mathematics Kansas City Regional Conference, October 25, 2007 National Council of Teachers of Mathematics Kansas City Regional Conference, October 25, 2007

2 Southern California Fires

3 Today’s Session Welcome and introductions What students should know before operating with fractions Watching a student use division with fractions Reasoning about division Models for division of fractions Contexts for division of fractions Questions

4 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 number operations

5 Solving a Division Problem With Fractions 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?

6 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?

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

8 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?

9 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?

10 Now Consider 6 ÷ What does this mean? What does the answer mean? How could the problem be modeled? What contexts make sense for –Sharing interpretation –Repeated subtraction interpretation

11 Reasoning About Division With Fractions

12 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?

13 Reasoning About Division With 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?

14 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

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

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

17 How Many Thirds? ? ?

18 Contexts for Division With Fractions

19 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?

20 Another Context For Division of Fractions You have 1 yards of licorice rope. It takes yard to make one package of licorice.  How many packages can be made?  How much of a yard of licorice is left?  How much of the original amount of licorice is left?

21 Model Using Your Materials Use your materials to model

22 Thinking More Deeply About Contexts for Division of Fractions Which contexts work for division of fractions? What aspects allow some contexts to work better than others? Develop your own new context for the problem we just modeled,.

23 Thinking More Deeply About Division of Fractions Estimating and judging the reasonableness of answers Recognizing situations involving division of fractions Considering and creating contexts where the division of fractions occurs Using a reasoning approach to consider why “invert and multiply” works

24 Questions/Discussion

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