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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. Today’s Session. Welcome and introductions What students need to know well before operations with fractions

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multiplication of fractions thinking more deeply
Multiplication of Fractions: Thinking More Deeply

Steve Klass

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

today s session
Today’s 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
what students need to know well before operating with fractions
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 operations for whole numbers.
a context for fraction multiplication
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?

multiplication of fractions
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 Nadine’s brownie problem?
reasoning about multiplication
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
reasoning about multiplication7
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.
reasoning about multiplication8
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
models for reasoning about multiplication
Models for Reasoning About Multiplication
  • Area/measurement models

(e.g. fraction circles)

  • Linear/measurement (e.g paper tape)
materials for modeling multiplication of fractions
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
using an area model with fraction circles for fraction multiplication
Using an Area Model with Fraction Circles for Fraction Multiplication
  • How could you use these materials to model
materials for modeling multiplication of fractions13
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
using an area model with fraction circles for fraction multiplication15
Using an Area Model with Fraction Circles for Fraction Multiplication
  • How could you use these materials to model ?
mixed number multiplication
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?
mixed number multiplication17
Mixed Number Multiplication
  • Can you find out what each square is worth?
  • What about partial squares?
try it yourself
Try it Yourself
  • How can you use these materials to model ?
  • What contexts can you construct for these two problems?
other contexts for multiplication of fractions
Other Contexts for Multiplication of Fractions
  • Finding part of a part (a reason why multiplication doesn’t 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)
thinking more deeply about multiplication of fractions
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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