Multiplication and Division of Fractions: Thinking More Deeply

1

• Multiplication and Division of Fractions
  Thinking More Deeply
• Nadine Bezuk and Steve Klass
• Session W106 GSDMC 2006

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Todays Session

• Welcome and introductions
• Meanings for division and multiplication
• How do we help children model and reason about
  the operations?
• Division and multiplication with whole numbers
• Division and multiplication with fractions
• Models for division of fractions
• Area, linear
• Models for multiplication of fractions
• Area, linear
• 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 arent 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.

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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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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?
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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?
• How can it be modeled?
• What contexts make sense for
• Sharing interpretation
• Repeated subtraction interpretation

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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 thats left?

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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?
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Using an Area Model With a Measurement
Interpretation

• Representation of with fraction
  circles.

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How Many Thirds?
?
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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 thats left?

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Multiplication of Fractions

• Consider
• How do you think a child might solve each of
  these?
• What kinds of reasoning and/or models might they
  use to make sense of each of these problems?

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Reasoning About Multiplication

• Whole number meanings - U.S. conventions
• 4 x 2 8
• Set - Four groups of two
• Array - Four rows of two columns
• Measurement - Four units by two units
• 2 x 4 8
• Set - Two groups of four
• Array - Two rows of four columns
• Measurement - Two units by four units

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Reasoning About Multiplication

• Fraction meanings - U.S. conventions
• Set - Two-thirds of one group of three-fourths
• Array - Two-thirds of a row of three-fourths of
  one column
• Measurement - Two-thirds of one unit by
  three-fourths of one unit
• Set - Three-fourths of one group of two-thirds
• Array - Three-fourths rows of two-thirds of one
  column
• Measurement - Three-fourths of one unit by
  two-thirds of one unit

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Models for Reasoning About Multiplication

• Area/measurement models (fraction circles)
• Linear/measurement (ribbon)
• Set models (eggs in cartons)

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Materials for Modeling Multiplication of Fractions

• How would you use these materials to model
  ?
• Paper tape
• Fraction circles
• You could also use
• Pattern blocks
• Fraction Bars / Fraction Strips
• Paper folding

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Using an Area Model with Fraction Circles for
Fraction Multiplication

• How would you use these materials to model
  ?

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Using an Area Model with Fraction Circles for
Fraction Multiplication

• How would you use these materials to model
  ?

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Contexts for Multiplication

• Finding part of a part (a reason why
  multiplication doesnt always make things
  bigger)
• Pizza (pepperoni on )
• Brownies ( is frosted, of the that part has
  pecans)
• Lawn ( is mowed, of that is raked)

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Thinking More Deeply About Multiplication and
Division of Fractions

• Estimating and judging the reasonableness of
  answers
• Recognizing situations involving multiplication
  or division of fractions
• Considering and creating other contexts where the
  multiplication or division of fractions occurs
• Using meaning as a springboard to understand why
  invert and multiply works

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Questions/Discussion
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Contact Usnbezuk_at_mail.sdsu.edusklass_at_projects.s
dsu.eduhttp//pdc.sdsu.edu