division of fractions: thinking more deeply division of fractions: thinking more deeply steve klass...

Division of Fractions: Thinking More DeeplyDivision of Fractions: Thinking More Deeply

Steve KlassNational Council of Teachers of Mathematics

Kansas City Regional Conference, October 25, 2007National Council of Teachers of Mathematics

Kansas City Regional Conference, October 25, 2007

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Southern California Fires

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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

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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

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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?

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2÷1

3

1÷1

3

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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?

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

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

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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?

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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?

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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

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Reasoning About Division With Fractions

Reasoning About Division With Fractions

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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?

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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

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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

12÷1

3

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Using a Linear Model With a Measurement Interpretation

12÷1

3

?

1

3

1

3

1

3

1

3

1

1

2 1 0

How many one-thirds are in one and one-half?

1

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Using an Area Model With a Measurement Interpretation

Representation of with fraction circles.

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2÷1

3

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How Many Thirds?

?

1

3

1

3

1

3

1

3

?

1

3

1 0

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Contexts for Division With Fractions

Contexts for Division With Fractions

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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?

1

2

1

3

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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?

1

2

2

3

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Model Using Your Materials

Use your materials to model

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2÷2

3

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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, . 11

2÷2

3

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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

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Questions/Discussion

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Contact Ussklass@projects.sdsu.edu

http://pdc.sdsu.edu

Contact Ussklass@projects.sdsu.edu

http://pdc.sdsu.edu

© 2007 Professional Development Collaborative