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Multiplication of Fractions: Thinking More Deeply

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Nadine is baking brownies. ... Which one best represents Nadine's brownie problem? 6. Reasoning About Multiplication ... Brownies ( is frosted, of the that part ... – PowerPoint PPT presentation

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Title: Multiplication of Fractions: Thinking More Deeply


1
  • Multiplication of Fractions Thinking More Deeply

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

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 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.

4
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?

5
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?

6
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

7
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.

8
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

9
Models for Reasoning About Multiplication
  • Area/measurement models
  • (e.g. fraction circles)
  • Linear/measurement (e.g paper tape)

10
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

11
Using a Linear Model With Multiplication
12
Using an Area Model with Fraction Circles for
Fraction Multiplication
  • How could you use these materials to model

13
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

14
Using a Linear Model With Multiplication
15
Using an Area Model with Fraction Circles for
Fraction Multiplication
  • How could you use these materials to model
    ?

16
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?

17
Mixed Number Multiplication
  • Can you find out what each square is worth?
  • What about partial squares?

18
Making Connections
19
Try it Yourself
  • How can you use these materials to model
    ?
  • What contexts can you construct for these two
    problems?

20
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)

21
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

22
Questions/Discussion
23
Contact Ussklass_at_projects.sdsu.eduhttp//pdc.s
dsu.edu
24
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