Multiplying and Dividing with Unit Fractions

Dividing and Multiplying with Unit Fractions

In this video, we will explore how models can help us divide and multiply unit fractions.

\(\mathbf{ \frac{1}{3} \div 3 }\)

Our first step is to model our dividend. We make a whole, cut it into thirds, and shade one of the thirds. Now we are going to divide the \(\mathsf{ \frac{1}{3} }\) into 3 even groups. To do this, we need to divide each of the \(\mathsf{ \frac{1}{3} }\)s into three groups.

If we look at the shaded pieces, these are each \(\mathsf{ \frac{1}{3} }\) divided into 3 groups. The quotient is the size of each of these pieces. There are now 9 pieces in this whole, so one of these individual pieces is \(\mathsf{ \frac{1}{9} }\). \(\mathsf{ \frac{1}{3} \div 3 = \frac{1}{9}}\).

\(\mathbf{ \frac{1}{3} \times \frac{1}{3} }\)

Fraction multiplication translates to “of,” so we can think of this multiplication expression as \(\mathsf{ \frac{1}{3} }\) of \(\mathsf{ \frac{1}{3} }\). To find \(\mathsf{ \frac{1}{3} }\) of \(\mathsf{ \frac{1}{3} }\), we model \(\mathsf{ \frac{1}{3} }\). This shaded piece is \(\mathsf{ \frac{1}{3} }\). To find \(\mathsf{ \frac{1}{3} }\) of this shaded \(\mathsf{ \frac{1}{3} }\), we will cut the \(\mathsf{ \frac{1}{3} }\) into three equal-sized pieces. Now we will shade one of them; I'm shading in blue. The double-shaded piece is our quotient. To determine the size of this double-shaded piece, we need to create equal-sized pieces throughout the whole. So, we cut each of these \(\mathsf{ \frac{1}{3} }\) into 3 pieces. Now we have a total of 9 pieces in our whole, and 1 of them is double-shaded. Our product is \(\mathsf{ \frac{1}{9} }\).