Transcript 451 KB Microsoft PowerPoint

Division of Fractions:
Thinking More Deeply
Nadine Bezuk and Steve Klass
Session 502 CMC-N 2005
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
2
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.
3
Solving a Division Problem With
Fractions

How would you solve 1 31 ?

How would you solve 121  31 ?


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?
4
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?
5
What Do Children Need to Know in Order to
Understand Division With Fractions?
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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
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?
8
Now Consider 6 ÷
1
2

What does this mean?
 How can it be modeled?
 What contexts make sense for
– Sharing interpretation
– Repeated subtraction interpretation
9
Reasoning About Division
With Fractions

Sharing meaning for division:
1÷
1
3
• One shared by one-third of a group?
• How many in the whole group?
• How does this work?
10
Reasoning About Division
With Fractions

Repeated subtraction / measurement meaning
1÷
•
•
•
•
1
3
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?
11
Materials for Modeling
Division of Fractions

How would you use these materials to
model
1
1
?
12  3
• Paper tape
• Fraction circles

You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips
12
Using a Linear Model With a
Measurement Interpretation
1
1
2
1
3
How many one-thirds are in one and one-half?
1
0
1
1
3
1
3
1
3
1
3
1
2
?
13
Using an Area Model With a
Measurement Interpretation

Representation of 121  31 with fraction circles.
14
How Many Thirds?
?
0
1
3
1
1
3
1
3
1
3
1
3
?
15
A Context For Division of
Fractions

1
12
1
cup
3
You have cups of sugar. It takes
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?
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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 other contexts where
the division of fractions occurs

Using meaning as a springboard to understand
why “invert and multiply” works
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Questions/Discussion
18
Contact Us
[email protected]
[email protected]
http://pdc.sdsu.edu
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