fraction model

7/29/2019 Fraction Model

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Session 9, Part A:

Models for the Multiplication and Division of Fractions (45 minutes)

In the past, you may have learned particular algorithms for the multiplication and division of fractions.We are now going to use some of the visual models we've employed earlier in this course to betterunderstand what is actually happening when we perform these operations.Note 2

First we'll use an area model -- one that superimposes squares that are partitioned into the appropriatenumber of regions, and shaded as needed -- to clarify what happens when you multiply fractions. Forexample, here's how we would use the area model to demonstrate the problem 3/8 2/3:

Shade one square,partitioned vertically, torepresent 3/8 (shown belowin pink):

Shade another square,partitioned horizontally, torepresent 2/3 (shown belowin blue):

Superimpose the twosquares. The product is thearea that is double-shaded(shown below in purple):

What is the value of this purple area? There are 3 2, or 6, purple parts out of 8 3, or 24, parts in all,so the value of the purple area is 6/24.

This model visually demonstrates the familiar algorithm: To multiply two fractions, multiply thenumerators and then multiply the denominators. This algorithm "counts" both the purple parts (theproduct of the two numerators) and the total number of parts (the product of the two denominators).

We can also use this model to "reduce" the fraction. First we swap the positions of some of the purpleparts. Two of the purple parts can be moved to the top, and thus, two of the eighths are now shaded.These two eighths are the same area as one quarter:
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Multiply and Divide Fractions

Area Model of Multiplication

Multiplication of two fractions can be modeled using area. Remind students that one way to understand

the product of 3 and 4 is as the area of a rectangle that is 3 units by 4 units.

Similarly, equals the area of a rectangle that is units by units. Draw this rectangle inside a unit

square. Since the area of the whole square is 1 and it is made up of 3 4 = 12 equal rectangles, the area

of each little rectangle is . There are 2 3 shaded rectangles, so the area of the shaded region is 2

4 = = .

In general, a rectangle that is by can be separated into a cnon-overlapping rectangles, each with

area d. Thus, the area of the rectangle is

a

d.

Multiplication of Fractions

The multiplication rule for fractions is quite easy to remember: Multiply numerators and then multiply the

denominators.

where b 0, d 0

The development of the rule for multiplying and actually requires two steps: multiplying by a and

dividing by b. Multiplication by a counting number involves repeated addition.

Example:

The general rule is where d 0.

Division of a fraction by a counting number takes more thought. What is 3? Think of as two unit

fractions of . Three does not divide into 2 evenly. However, we can write as 2 3 = . This is 6 unit

fractions of , and 3 divided by 3. So the quotient equals 2 unit fractions of , or .


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The general rule is b = dwhere b 0, d 0.

Notice that when d= 1, this shows that c b = . Using what has been shown above, = a dcan

be derived. First divide the fraction by b and then multiply by a.

b = dand a d= a d.

Division of Fractions

The division rule is easy to statemultiply by the reciprocalbut it requires a careful explanation.

where b 0, c 0, d 0

The reciprocal of the fraction is the fraction . A number times its reciprocal equals 1.

Multiplication and division are inverse operations, so 6 2 = 3 because 3 2 = 6. Similarly, divided by

can be thought of as the number of fractions in (the solution to = m ). By multiplying both sides of

this equation by the reciprocal of , the solution is m = . So = .

Fractions can also be divided by using a common denominator. How many 6-inch ribbons can be cut from

a ribbon that is 15 inches long? The answer can be found by dividing 15 by 6: 15 6 = = = 2 ribbons.

This same idea can be used to show the rule for division of fractions. Recall that the numerator of a

fraction tells how many parts there are and the denominator tells the size of each part. How many s are

there in 1 ? Find 1 . The problem 1 is the same as . Since the denominators are the same,

divide the numerators. The answer is 7 2 = = 3 . The quotient of fractions with the same denominator

is the quotient of the numerators.

At some point, students will need to begin the transition from a hands-on approach

to a more symbolic approach to fractions. Before completely abandoning the use of

concrete manipulatives in favor of pure symbolic notation, a transitional step of

using drawings of fractions is recommended.

In this transitional period, students are required to master three models for drawing

fracitons: area, line, and set.

Examples:


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[microfiche] / John Arul Phillips

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Peningkatan daya pemikiran kanak-kanak
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