Multiplication of Fractions: Thinking More Deeply

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• Multiplication of Fractions Thinking More Deeply

Steve Klass
48th Annual Fall Conference of the California
Mathematics Council - South Palm Springs, CA,
Nov. 2, 2007
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Todays Session

• Welcome and introductions
• What students need to know well before operations
  with fractions
• Contexts for multiplication of fractions
• Meanings for multiplication
• Models for multiplication of fractions
• Discussion

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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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A Context for Fraction Multiplication

• Nadine is baking brownies. In her family, some
  people like their brownies frosted without
  walnuts, others like them frosted with walnuts,
  and some just like them plain.
• So Nadine frosts 3/4 of her batch of brownies
  and puts walnuts on 2/3 of the frosted part.
• How much of her batch of brownies has both
  frosting and walnuts?

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Multiplication of Fractions

• Consider
• and
• How do you think a child might solve each of
  these?
• Do both representations mean exactly the same
  thing to children?
• What kinds of reasoning and/or models might they
  use to make sense of each of these problems?
• Which one best represents Nadines brownie
  problem?

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Reasoning About Multiplication

• Whole number meanings - 2 U.S. textbook
  conventions
• 4 x 2 8
• Set - Four groups of two
• Area - Four rows of two columns

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Reasoning About Multiplication

• 2 x 4 8
• Set - Two groups of four
• Area - Two rows of four columns
• When multiplying, each factor refers to something
  different. One factor can tell how many groups
  there are and the other, how many in each group.
  The end result is the same product, but the
  representations are quite different.

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Reasoning About Multiplication

• Fraction meanings - U.S. conventions
• Set - Two-thirds of a group of three-fourths of
  one whole
• Area - Two-thirds of a row of three-fourths of
  one column
• Set - Three-fourths of a group of two-thirds of
  one whole
• Area - Three-fourths rows of two-thirds of one
  column

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Models for Reasoning About Multiplication

• Area/measurement models
• (e.g. fraction circles)
• Linear/measurement (e.g paper tape)

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Materials for Modeling Multiplication of Fractions

• How could 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 a Linear Model With Multiplication
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Using an Area Model with Fraction Circles for
Fraction Multiplication

• How could you use these materials to model
  

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Materials for Modeling Multiplication of Fractions

• How could 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 a Linear Model With Multiplication
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Using an Area Model with Fraction Circles for
Fraction Multiplication

• How could you use these materials to model
  ?

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Mixed Number Multiplication

• Using a ruler and card, draw a rectangle that is
  by
• inches, and find the total number of
  square inches. Find your answer first by
  counting, then by multiplying.
• Compare your answers, are they the same?

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Mixed Number Multiplication

• Can you find out what each square is worth?
• What about partial squares?

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Making Connections
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Try it Yourself

• How can you use these materials to model
  ?
• What contexts can you construct for these two
  problems?

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Other Contexts for Multiplication of Fractions

• 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)
• Ribbon (you have yd , of the ribbon is used
  to make a bow)

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Thinking More Deeply About Multiplication of
Fractions

• Estimating and judging the reasonableness of
  answers
• Recognizing situations involving multiplication
  of fractions
• Considering, creating and representing contexts
  where the multiplication of fractions occurs
• Making careful number choices

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Questions/Discussion
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Contact Ussklass_at_projects.sdsu.eduhttp//pdc.s
dsu.edu
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