division of fractions: thinking more deeply division of fractions: thinking more deeply steve klass...
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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
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
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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?
11
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?
÷13
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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.
11
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 [email protected]
http://pdc.sdsu.edu
Contact [email protected]
http://pdc.sdsu.edu
© 2007 Professional Development Collaborative