35 Multiplying Fractions
Area Model
One of our models for multiplying whole numbers was an area model. For example, the product is the area (number of 1 × 1 squares) of a 23-by-37 rectangle:
So the product of two fractions, say, should also correspond to an area problem.
Example (4/7 × 2/3)
Let us start with a segment of some length that we call 1 unit:
Now, build a square that has one unit on each side:
The area of the square, of course, is square unit.
Now, let us divide the segment on top into three equal-sized pieces. (So each piece is .) And we will divide the segment on the side into seven equal-sized pieces. (So each piece is .)
We can use those marks to divide the whole square into small, equal-sized rectangles. (Each rectangle has one side that measures and another side that measures .)
We can now mark off four sevenths on one side and two thirds on the other side.
The result of the multiplication should be the area of the rectangle with on one side and on the other. What is that area?
Remember, the whole square was one unit. That one-unit square is divided into 21 equal-sized pieces, and our rectangle (the one with sides and ) contains eight of those rectangles. Since the shaded area is the answer to our multiplication problem we conclude that
Think / Pair / Share
- Use are model to compute each of the following products. Draw the picture to see the answer clearly.
- The area problem yielded a diagram with a total of 21 small rectangles. Explain why 21 appears as the total number of equal-sized rectangles.
- The area problem yielded a diagram with 8 small shaded rectangles. Explain why 8 appears as the number of shaded rectangles.
Problem 5
How can you extend the area model for fractions greater than 1? Try to draw a picture for each of these:
On Your Own
Work on the following exercises on your own or with a partner.
- Compute the following products, simplifying each of the answers as much as possible. You do not need to draw pictures, but you may certainly choose to do so if it helps!
- Compute the following products. (Do n0t work too hard!)
- Try this one. Can you make use of the fraction rule to help you calculate? How?
Think / Pair / Share
How are these two problems different? Draw a picture of each.
- Pam had of a cake in her refrigerator, and she ate of it. How much total cake did she eat?
- On Monday, Pam ate of a cake. On Tuesday, Pam ate of a cake. Both cakes were the same size. How much total cake did she eat?
When a problem includes a phrase like “ of …,” students are taught to treat “of” as multiplication, and to use that to solve the problem. As the above problems show, in some cases this makes sense, and in some cases it does not. It is important to read carefully and understand what a problem is asking, not memorize rules about “translating” word problems.
Explaining the Rule
You probably simplified your work in the exercises above by using a multiplication rule like the following.
Multiplying Fractions
Of course, you may