division of fractions: thinking more deeply nadine bezuk and steve klass session 502 cmc-n 2005
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Division of Fractions: Thinking More Deeply
Nadine Bezuk and Steve KlassSession 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 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?
11
2÷1
3
1÷1
3
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?
QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.
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
1
2
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?
1
3
11
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?
1
3
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
11
2÷1
3
13
Using a Linear Model With a Measurement Interpretation
11
2÷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?
14
Using an Area Model With a Measurement Interpretation
Representation of with fraction circles. 11
2÷1
3
15
How Many Thirds?
?
1
3
1
3
1
3
1
3
1 0
?
1
3
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?
1
2
1
3
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
