Section 5: Explaining Multiplication Algorithms

Fraction Operations Unit Page 11

Goal

To help students be able to explain why the algorithm for multiplying fractions works using pictures and written explanations.

Big Ideas

The algorithm for multiplying fractions can be closely tied to reasoning with pictures of fraction multiplication. For example, suppose we wanted to show the following:

2 4 2! 4 ! = 3 5 3! 5

One way to show that this is true is by considering three different multiplication problems:

1 1 1 1. Show ! = . 3 5 3! 5

Explanation: To compute 1/3 ! 1/5 using pictures, we would first start with a picture of 1/5, and then divide the 1/5 into 3 equal parts, taking one of those parts to represent 1/3 of 1/5.

Then to find out what part of a whole the 1/3 of 1/5 is, we would divide the other fifths into thirds to create pieces of equal size throughout the entire whole.

Thus, in each fifth we get three pieces. This is the same as five groups of 3, or 5 ! 3 pieces. However, we could also group horizontally, which would give us three groups of 5, or 3 ! 5 pieces. Thus, each piece would be 1/(3 !5), because it takes 3 ! 5 of those pieces to make a whole.

1 4 4 1 1 1 2. Show ! = using the fact that ! = . 3 5 3! 5 3 5 3! 5

Explanation: We know that 1/3 of 1/5 is 1/(3 ! 5) from Step 1. Then what is 1/3 of 4/5? From the picture below, we can see that 1/3 of 4/5 is the same as four copies of 1/3 of 1/5.

1/3 of 1/5 1/3 of 4/5

Thus, 1/3 of 4/5 is four pieces of size 1/(3 ! 5), or 4/(3 ! 5).

2 4 2! 4 1 4 4 3. Show that ! = using the fact that ! = . 3 5 3! 5 3 5 3! 5

Explanation: 2/3 of 4/5 is the same as two 1/3s of 4/5:

1/3 of 4/5 2/3 of 4/5

Thus, since 1/3 of 4/5 is 4/(3 ! 5), then 2/3 of 4/5, or two 1/3s of 4/5, would be two groups of 4/(3 ! 5). So we have 2 ! 4 pieces of size 1/(3 ! 5), or (2 ! 4)/(3 ! 5).

In general, the multiplication algorithm can be explained by first looking at what size pieces result from multiplying unit fractions. Then we can use our understanding of what non-unit fractions mean, such as 4/5 means four 1/5s, to compute how many pieces of that particular size result from multiplying the two fractions.

Optional: Explaining Cross-Canceling

Students may want to know why cross-canceling works when multiplying fractions. The following is a brief explanation. You will have to supplement the ideas if you wish to discuss them with your students.

1. Canceling the denominator of the first fraction with the numerator of the second fraction.

5 3/ 5 1 Example: Show that ! = ! . 6 / 4 2 4

Explanation: When we can cross cancel between the denominator of the first fraction and the numerator of the second fraction, it means that there is an efficient way to partition the second factor so that we can make the requisite number of copies we need to make. For example, we can take 5/6 of 3/4 by taking each 1/4 in 3/4 and splitting it in half to make sixths in the 3/4, and then take 5 of those 1/6 of 3/4, as illustrated below:

This method requires much fewer partitions than a more standard approach to computing the product, such as the one illustrated below.

It is no accident that cross canceling with the denominator of the first fraction and the numerator of the second fraction corresponds to the existence of a more efficient way of partitioning our fractions than a standard method would suggests (try, for example, the problem 5/6 ! 8/9). Furthermore, it is the first picture, where we made as few partitions as possible, that helps us see that 5/6 ! 3/4 is equivalent to 5/2 ! 1/4. Notice that in this picture, the piece that is 1/2 of 1/4 is also 1/6 of 3/4. Thus, five of those pieces could be called either 5/2 of 1/4 or 5/6 of 3/4.

2. Canceling the numerator of the first fraction with the denominator of the second fraction.

3/ 5 1 5 Example: Show that ! = ! . 4 6/ 4 2

Explanation: Taking 3/4 of 5/6 is the same as taking 1/4 of 5/6 three times, because that's what 3/4 of 5/6 means — three 1/4s of 5/6. To see the connection between 3/4 ! 5/6 and 1/4 ! 5/2, it is helpful to take 1/4 of 5/6 from three copies of 5/6 rather than taking 1/4 three times from the same copy of 5/6. In the picture below, the

By rearranging the shaded portions, we get the following picture:

Notice that this is actually 1/4 of 5/2.

We can do this same process for any two fractions. However, we will not be able to perform the last step, where we simplified the resultant fraction, unless we can cross-cancel between the numerator of the first fraction and the denominator of the second fraction.