We ended the previous post with a bit of a cliffhanger, with two possible diagrams to represent
The first of these diagrams is more familiar to students because it reflects their past work, but the second is more productive for understanding “dividing by a unit fraction is the same as multiplying by its reciprocal.”
Why is the first one more familiar? In grades 3 and 4, students study both the “how many in each (or one) group?” and “how many groups?” interpretations for division with whole numbers (see our last blog post for examples). In grade 5, they study dividing whole numbers by unit fractions and unit fractions by whole numbers. But, as we mentioned in that post, in grade 5 the “how many groups?” interpretation is easier when dividing whole numbers by unit fractions because students do not have to worry about fractions of a group. Going from
The main intellectual work here is seeing that
So the “how many groups” interpretation is useful for understanding important aspects of fraction division and has an important role in students’ learning trajectory. It enables students to see that dividing by
The “how much in each group” interpretation shows why. Here are diagrams using that interpretation showing In fact, the structure of this context is so powerful, we can see why dividing any number by
This is true for dividing by any unit fraction, for example In the diagram above, we can see that
With a little more work to make sense of it, we can use this interpretation to see why we multiply by the reciprocal when we divide by any fraction, for example In the diagram above, we can see that
Now, just as before, to find the full container, we multiply by 5:
This shows that dividing by
There is nothing special about these numbers, and a similar argument can be made for dividing any number by any fraction. Now students, instead of saying “ours is not to reason why, just invert and multiply,” can say “now I know the reason why, I’ll just invert and multiply.”
Next time: Beyond diagrams.
More amazing to me is that fraction division works without inverting and multiplying. 9/20 divided by 3/5 results in 3/4. Just divide the top lines and the bottom lines. Exploring this further will also explain why we invert and multiply.
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