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Multiplication and Division of Fractions: Thinking More Deeply

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How do we help children model and reason about the operations? ... Questions/Discussion. 27. Contact Us. nbezuk_at_mail.sdsu.edu. sklass_at_projects.sdsu.edu ... – PowerPoint PPT presentation

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Title: 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

2
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

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
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?

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?
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

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?

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 thats left?

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

13
Using a Linear Model With a Measurement
Interpretation
How many one-thirds are in one and one-half?
14
Using an Area Model With a Measurement
Interpretation
  • Representation of with fraction
    circles.

15
How Many Thirds?
?
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 thats left?

17
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?

18
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

19
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

20
Models for Reasoning About Multiplication
  • Area/measurement models (fraction circles)
  • Linear/measurement (ribbon)
  • Set models (eggs in cartons)

21
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

22
Using an Area Model with Fraction Circles for
Fraction Multiplication
  • How would you use these materials to model
    ?

23
Using an Area Model with Fraction Circles for
Fraction Multiplication
  • How would you use these materials to model
    ?

24
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)

25
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

26
Questions/Discussion
27
Contact Usnbezuk_at_mail.sdsu.edusklass_at_projects.s
dsu.eduhttp//pdc.sdsu.edu
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