## Dr. Wilson

Here is a picture that illustrates the rule for multiplying fractions

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2/3 x 3/4 = 6/12

The result can be expressed simply by saying that we multiply the tops, and we multiply the bottoms. Of course the answer can be reduced

6/12 = 1/2

We could tell that 3 would go into the top and bottom because we can see 3 going into the top and going into the bottom in the original problem. We could multiply first and then cancel or we could cancel first and then multiply.

Such a picture can also be used to illustrate multiplication of mixed numbers.

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There are two ways of seeing this. One is to notice that there is a 1 x 2 = 2 in the upper left corner, there is a 1 x 1/2 = 1/2 in the upper right, there is a 2 x 3/4 = 6/4 = 11/2 in the lower left and a 1/2 x 3/4 = 3/8 in the lower right.

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This illustrates that to correctly multiply mixed numbers you need to use FOIL. It also illustrates that it is probably simpler to express the numbers using improper fractions.

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and

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so

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Since

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we see that one gets the same results.

If you look at the picture, there are 8 little rectangles in each square unit, so each rectangle is 1/8 of a square unit. There are 5 rectangles going horizontally in the picture and 7 rectangles going vertically, so there are 5x7 = 35/8 in the picture.

Therse types of pictures are not easily adapted for use in illustrating division of fractions. There is a type of picture which can be used to illustrate both multiplication and division of fractions.

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We have 2 and 1/2 1 and 3/4s laid end to end coming out to 4 and 3/8. It is also a picture of

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since 4 and 3/8 is divided up into 2 and 1/2 equal pieces. On the other hand, since we see that 1 and 3/4 goes into 4 and 3/8 2 and 1/2 times it is also a picture of

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As a general rule, one of these types of pictures are good for all three exercises in a family. A real advantage of using this type of picture to illustrate this type of division problem is with something like

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Since 3/4 is smaller than 1, it takes more 3/4s to get out to 2 and 1/2 than it does units. This illustrates how, when the divisor is less than 1, the answer to the division problem is larger than the dividend.

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We could also use the idea of missing addends to explain this. The unknown factor approach is quite popular. Let us consider the following problem.

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Let us pretend that we do not know what the answer is.

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The answer to a division problem is what one needs to multiply the divisor by to obtain the dividend.

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It is possible to produce a ? which will work, viz.

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This will work; because if we check we see

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We conclude

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There is a short cut for getting from the left side of the above equation to the right side: invert the divisor and multiply.

Once the students are familiar with the invert and multiply technique, pictures can be used to illustrate why it works. Here is a picture of the following exercise.

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We could ask, how many times does 3/4 go into 21/2?

In the picture we see that 3 2and 1/2s would come out to 2 and 1/4, leaving 1/4 more which is 1/3 of a 3/4 to get to 2 and 1/2.

The invert and multiply technique is actually illustrated here. Notice that we have 2 and 1/2 groups of 4 to determine how many quarters there are in 2 and 1/2, so we are multiplying 2 and 1/2 by 4, and that these quarters are divided up into groups of 3 when we see how many 3/4s would need to be laid out end to end to get to 2 and 1/2.

In this approach the 3/4 is the rate, and the answer, the 3 and 1/3 is the base. It is possible to draw a picture of the problem where the 3/4 is the base and the 31/3 is the rate.

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If 3/4 is the base, the number of pieces then the answer is the rate, the amount in one piece. The answer is then found over the 1 on the second number line. In this picture we can see that we are dividing by 3 and multiplying by 4. This picture illustrates another explanation. Change from mixed numbers to improper fractions and express the division problem as a fraction

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Next multiply top and bottom of the fraction by 4/3.

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The bottom cancels out and we are left with an answer of

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In the picture we can see that the 2 and 1/2 and the 3/4 are being multiplied by 4/3. It looks like we are first dividing by 3 and then multiplying by 4. While students may find it more natural to multiply by 4 first and then divide by 3, the picture is a little more ungainly.

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If you multiply both the 2 and 1/2 and the 3/4 by 4, 2 and 1/2 x 4 = 10, and (3/4) x 4 = 3. At this point we see the 10 lined up above the 3. Now to find out how much is in 1 piece, we would need to divide by 3.