Multiplying a whole number by a fraction links directly to the division operation. But this fact will not be obvious to students when they first encounter it.

One suggestion for demonstrating to students that this is the case, and for helping them to construct this understanding for themselves, is to use a pictorial diagram in which a shape is divided into the relevant number of pieces, and a number is “shared” among the pieces. In the video, I show this method for the question “1/9 x 45″: a square with 9 parts can have a total of 45, evenly distributed, if there are exactly 5 in each piece.

The next step is to handle fractions in which the numerator is greater than 1. Following on from multiplying by a unit fraction, we can ask students to provide the answer based on the first step. In the video, I show that 6/9 of 45 must be “6 lots of 5″, or 30.

Note that we do not introduce an algorithm (such as “divide by the denominator and multiply by the numerator”) until much later, when the students have developed a good understanding of what it all means.