Multiplication of Common Fractions © Math As A Second Language All Rights Reserved next #6 Taking...

Multiplication of

Common Fractions© Math As A Second Language All Rights Reserved

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#6Taking the Fear

out of Math

1 3×1 3

Applying

In order to attract student interest, we have to find ways of making the topic relevant to them, and that is not always a simple task. In that context, knowing how to multiply fractions is certainly important.

However, knowing when we have to use multiplication of fractions in the real-world is equally as important.

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© Math As A Second Language All Rights Reserved

Real World Applications

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The problem is that when it comes to the multiplication

of fractions there aren’t many applications that are

relevant to students in elementary school.

One way that we feel is effective for attracting the students’ attention is to

show them something that makes them wonder.

However, even at their age, they are often buying electronic devices of one kind

or another and therefore they might feel comfortable discussing sale prices.

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It is quite possible that they have been attracted to sales that say, 1/2 off our

regular low prices!

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For example, a store reduces the price of an item by 1/2. Later when the item is still

not selling, the store reduces the sale price by another 1/2 off.

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A store reduces the price of an item by 1/2 , and later when the item is still not

selling, the store reduces the sale price by 1/2. Therefore, since 1/2 – 1/2 = 0, the item

should now be free.

Problems of this type may lead to the flawed argument that is shown below.

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next Since students prefer to think concretely rather than abstractly, to

analyze this flaw, it is probably a good idea to start with a specific number, one that is

relatively easy to work with.

Thus, we might ask them to assume that the original price of the item was $100. After it is reduced by 1/2 the sale price is $50. And when this price is reduced by 1/2 again, the new price is $25. Hence, the item, rather

than being free, now costs $25.

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At this point, it is not unusual for a student to ask, “But what if the item costs a

different amount?” In fact, if students do not raise this question perhaps you should.

The point is that answering this question leads to explaining what was actually

demonstrated is that your savings are $75 per each $100 the item originally cost.

In the language of fractions, since $25 is 1/4 of $100, the final price is 1/4 of the

regular price.

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Hence, no matter what the item cost originally, the new price would be

1/4 of the original price.


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1/2 of 1/2 of $100 =

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Thus…

This example illustrates taking a fractional part of a fractional part, in particular that

1/2 of 1/2 = 1/4.

1/2 of (1/2 of $100) =1/2 of $50

1/2 of 1/2 of $100 =

(1/2 of 1/2) of $100 =1/4 of $100 = $25 = $25

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To help students see this more concretely, have them imagine that theyseparately purchased 5 of those devices.

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Then all in all, they would have paid $500 for the 5 items, if there had been no sale.

However, each time they purchased one of the items, they paid only $25.

Hence, they would have paid $25 five times, or a total of $125.

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Then as a check, have them see that if the original price of the item was $500, then

at 1/2 off, the sale price would have been $250, and after this price was reduced by 1/2, the new price would be $125, which agrees

with the previous result.

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Mathematically…

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1/2 of 1/2 of $500 =1/2 of (1/2 of $500) =

1/2 of $250

1/2 of 1/2 of $500 =

1/4 of $250= $125 = $125(1/2 of 1/2) of $500 =

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It might seem “babyish” but something as simple as making a table often helps students internalize concepts that otherwise seem too

abstract for them to handle. So as a teacher you might want to have the students make a table

showing in a systematic way what happens for various prices of the item.

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Have them make a table in which the price of the item is a multiple of 4 (this avoids the students having to deal with fractions but

otherwise there is no reason to do so).

Using a Table

The table might look something like the one below…

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Regular Price Price after first 1/2 off reduction

Fractional PriceYou Paid

Price after second 1/2 off reduction

$100 $50 25/100 = 1/4$25

$200 $100 50/200 = 1/4$50

$300 $150 75/300 = 1/4$75

$400 $200 100/400 = 1/4$100

$500 $250 125/500 = 1/4$125

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The table suggests some observations that students might want to explore…

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For example, every time the regular price increases by $100 the cost of the item when

it is on sale increases by $25.

Regular Price Price after first 1/2 off reduction

Fractional Price You Paid

Price after second 1/2 off reduction

$100 $50 25/100 = 1/4$25

$200 $100 50/200 = 1/4$50

$300 $150 75/300 = 1/4$75

$400 $200 100/400 = 1/4$100

$500 $250 125/500 = 1/4$125

The table only shows us the results for prices that are multiples of $100. However, what we do know is that the final sale price will always be 1/4 of the original price. So if the original price was $280, the final sale

price would be 1/4 of $280 or $280 ÷ 4 or $70.1

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note1 As an estimate, since $280 is between $200 and $300 but closer to $300, the

table shows us that the sale price is more than $50 but a “little less” than $75.

For the more visual learners, it might be helpful to use a corn bread model. In this

case, our chart would be replaced by a corn bread that is sliced into 4 pieces of equal

size. This takes the place of a chart inwhich the entries are multiples of 4.

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Using the Corn Bread

Corn Bread

The entire corn bread represents the regular price of the electronic device.

$100

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In the diagram below…nextnext


The 2 pieces that are shaded in blue represent 1/2 of the corn bread

(that is, 1/2 of the regular price of the item).

$25 $25 $25 $25

The 1 piece in yellow represents 1/2 of the remaining part of the corn bread (that is,

1/2 of the sale price).The 1 piece in red represents the final cost

of the item ($25).

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The previous diagram applies, no matter what the regular price of the item is. For example, if the regular price of the time is $160, the diagram becomes…


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$40 $40 $40 $40

and if the regular price had been $360, it becomes…

$90 $90 $90 $90

The corn bread model is also helpful as a segue to algebra. For example, suppose

we were told that after the 2nd price reduction a person bought the item for $60, and we wanted to determine from this what

the regular price of the item was. In that case, we could draw the corn bread as

shown below…


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And since all 4 pieces have the same size, the regular price had to be 4 × $60 or $240.

$60$60 $60 $60 $60

Our previous models interpreted the times sign as meaning “of”. The area model

shows us how to visualize why we really are multiplying.

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The Area Model

For example, suppose we have a unit square (that is, no matter what unit of measurement we choose to use the

square is 1 unit by 1 unit, and hence, its area is 1 square unit).

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Suppose we now subdivide the unit square into 4 smaller rectangles by

drawing the horizontal and the vertical line as shown below.


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The shaded region is a rectangle whose dimensions are 1/2 of a unit by 1/2 of a unit. Hence, its area is 1/2 × 1/2 square units; and since it also 1 of the 4 smaller rectangles, we see that the area is also 1 fourth of the

square unit.


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1/41/2

1/2×

If we had wanted to, we could have used this diagram as our corn bread model.


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1/4

1/4

Both diagrams are equivalent ways to view the corn bread.

Let’s end this presentation by going through a similar example that might

make it easier to internalize the different ways to approach solving applications of

when we multiply two fractions.

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A camera, when not on sale, costs $120. Your friend buys it when it is being sold at

3/4 of its regular price. Later your friend sells it to you for 3/5 of the price he paid for

it. What fractional part of the original price did you pay for the camera?

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As a starting point, students might first compute 3/4 of $120 to determine that your

friend paid $90 for the camera.

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Next they might compute 3/5 of $90 to determine that you paid $54 for the camera.

Hence, the fractional part of the price you paid was $54/$90 or 9/20 of the regular price of

the camera.

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Using the corn bread model we can assume that the corn bread is presliced into 20 pieces (that is, a multiple of 4 and 5) and that the whole corn bread represents $120.

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Each of the 20 pieces represents $6 and therefore, 3/4 of 20 pieces is 15 pieces, each of which represents $6. These 15 pieces, shaded in red above, represent

the price your friend paid for the camera.

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Corn Bread = $1206 6 6 6 6 6 6 6 6 6 6 6 6 6 6

And 3/5 of these 15 pieces (9 pieces) represents the price (9 × 6 = $54) you paid.

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We may then make sure that the students see that no matter what the regular price of the camera was, the same diagram would apply, except that it would no longer be

true that each pieces represented $6.

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6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6

note2 3/5 and 3/4 are modifying different amounts. Specifically, notice

that 3/5 is modifying $90 but 3/4 is modifying $120.

This is illustrated in blue below.2

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For example, if the regular price had been $180, we would have divided 180 by 20 to determine that each of the 20 pieces represented $9 and the diagram would have become…

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If we had been told that you paid $180 for the camera, it would mean that the 9 blue pieces

represented $180 and therefore each blue piece represented $180 ÷ 9 or $20.

Hence, the diagram would become…

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 96 6 6 6 6 6 6 6 6 9 9 9 9 9 99 9 9 9 9 9 9 9 9

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 2020 20 20206 6 6 6 6 6 6 6 6 2020 20 2020 202020 20 2020 20 2020 20

From the diagram, we see that the regular price of the camera was 20 × $20 or $400.

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Keep in mind that the above methods are segues for helping students internalize what multiplication of fractions really

means. Eventually, we want students to understand that no matter what the regular price of the camera was, you paid 3/5 of 3/4

or 9/20 of the regular price.

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 20 20 20 20 206 6 6 6 6 6 6 6 6 2020 20 2020 2020 20 2020 20 2020 20 20

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In our next presentation, we will discuss division

of fractions.


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