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

next © Math As A Second Language All Rights Reserved 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. next © Math As A Second Language All Rights Reserved It is quite possible that they have been attracted to sales that say, 1 / 2 off our regular low prices! next 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.

© Math As A Second Language All Rights Reserved next 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.

© Math As A Second Language All Rights Reserved 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.

© Math As A Second Language All Rights Reserved next 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.

Hence, no matter what the item cost originally, the new price would be 1 / 4 of the original price. © Math As A Second Language All Rights Reserved next 1 / 2 of 1 / 2 of $100 = 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 next

To help students see this more concretely, have them imagine that they separately purchased 5 of those devices. © Math As A Second Language All Rights Reserved 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. next

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. © Math As A Second Language All Rights Reserved Mathematically… next 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 ( 1 / 2 of 1 / 2 ) of $500 = next

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. © Math As A Second Language All Rights Reserved 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… next © Math As A Second Language All Rights Reserved 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$200$ / 200 = 1 / 4 $50$300$ / 300 = 1 / 4 $75$400$ / 400 = 1 / 4 $100$500$ / 500 = 1 / 4 $125 next

The table suggests some observations that students might want to explore… © Math As A Second Language All Rights Reserved 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 $200$ / 200 = 1 / 4 $50 $300$ / 300 = 1 / 4 $75 $400$ / 400 = 1 / 4 $100 $500$ / 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 next © Math As A Second Language All Rights Reserved note 1 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 in which the entries are multiples of 4. next © Math As A Second Language All Rights Reserved Using the Corn Bread Corn Bread The entire corn bread represents the regular price of the electronic device. $100 next

In the diagram below… © Math As A Second Language All Rights Reserved 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 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). next

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… © Math As A Second Language All Rights Reserved Notes next $40 and if the regular price had been $360, it becomes… $90

The corn bread model is also helpful as a segue to algebra. For example, suppose we were told that after the 2 nd 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… © Math As A Second Language All Rights Reserved next And since all 4 pieces have the same size, the regular price had to be 4 × $60 or $240. $60

Our previous models interpreted the times sign as meaning “of”. The area model shows us how to visualize why we really are multiplying. next © Math As A Second Language All Rights Reserved 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). next

Suppose we now subdivide the unit square into 4 smaller rectangles by drawing the horizontal and the vertical line as shown below. © Math As A Second Language All Rights Reserved next

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. © Math As A Second Language All Rights Reserved next 1/41/4 1/21/2 1/21/2 ×

If we had wanted to, we could have used this diagram as our corn bread model. © Math As A Second Language All Rights Reserved next 1/41/4 1/41/4 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. © Math As A Second Language All Rights Reserved 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? next

As a starting point, students might first compute 3 / 4 of $120 to determine that your friend paid $90 for the camera. © Math As A Second Language All Rights Reserved next 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.

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. © Math As A Second Language All Rights Reserved next 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. Corn Bread = $

And 3 / 5 of these 15 pieces (9 pieces) represents the price (9 × 6 = $54) you paid. next © Math As A Second Language All Rights Reserved next 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 $ note 2 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 next

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… © Math As A Second Language All Rights Reserved next 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…

From the diagram, we see that the regular price of the camera was 20 × $20 or $400. next © Math As A Second Language All Rights Reserved next 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

next In our next presentation, we will discuss division of fractions. © Math As A Second Language All Rights Reserved ×