Division of Fractions: Thinking More Deeply Nadine Bezuk and Steve Klass Session 502 CMC-N 2005

2 Today’s Session Welcome and introductions Meanings for division How do we help children model and reason about division? Division with whole numbers Division with fractions Models for division of fractions Area, Linear Contexts for division of fractions Questions

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 operations for whole numbers.

4 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?

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 Sharing meaning for division: 1 ÷ One shared by one-third of a group? How many in the whole group? How does this work?

11 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?

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

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

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

15 How Many Thirds? ? ?

16 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?

17 Thinking More Deeply About Division of Fractions Estimating and judging the reasonableness of answers Recognizing situations involving division of fractions Considering and creating other contexts where the division of fractions occurs Using meaning as a springboard to understand why “invert and multiply” works

18 Questions/Discussion

19 Contact Us