Here is a picture that illustrates the rule for multiplying fractions

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

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.

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.

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.

and

so

Since

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.

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

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

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

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.

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.

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

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

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

This will work; because if we check we see

We conclude

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.

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.

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

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

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

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.

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.