Friday, May 19, 2017

A Concrete-Diagrammatic-Symbolic Approach to Dividing Fractions

As quickly as you can, look at the question below and write down the answer:
This is a question I have shared with many adults (teachers and parents) and I often get the answer 1⅕. Likewise, if I ask students to do 6÷½, I often get the answer '3'. There is a major misconception here that needs to be addressed:
Dividing by a half ≠ Dividing in half
When I start students on the road to dividing fractions symbolically, I will avoid such sayings as:
'Ours is not to reason why, just invert and multiply.'
It can cause no end of misconceptions. Instead I want to ensure that they start by understanding that they know that dividing by a half is equivalent to doubling a number. This is something we can develop concretely. For example, for this question:
we can show using pattern blocks:
The trapezoid is one half of the hexagon, so how many of these are needed to make 5 hexagons?
By repeating this for similar questions and then recording our results, we are in a position to think about what dividing by a half is equivalent to:
By using the concrete, students are in a better position to see that dividing by a half is equivalent to multiplying by 2.

This can now be extended to dividing by a third:
Again, by using the concrete, students are in a better position to see that dividing by a third is equivalent to multiplying by 3. In fact, I often see students at this point not even 'complete' the division (as in the photo above) because they can see the solution. I can repeat this for dividing by a quarter, dividing by a fifth, dividing by a tenth etc. and record the results thus:
So what if we are dividing by fractions other than unit fractions? For example:
Here, I might use a diagrammatic approach with a number line:

The big idea I want to get at here, is not so much the answer (6) but the effect of dividing by two-thirds. Again, recording results might give something like:
When students notice that dividing by two-thirds is the same as multiplying by 1½, we can then show that this is equivalent to multiplying by three-halves. We can then extend this idea with other examples  and maybe get a set of results like this:
I find that with this progression, students are in a better position to see that dividing by a fraction is equivalent to multiplying by its reciprocal.
At this stage, I feel that students are ready to appreciate a symbolic approach to dividing by fractions. There are some prerequisites that students must be comfortable with though:
  • Dividing any number by 1 gives the same number.
  • Multiplying a fraction by its reciprocal gives 1.
  • Multiplying the numerator and denominator by the same number gives an equivalent fraction.

Now we can show:
I have yet to see a visual that shows that dividing by a generic fraction is the equivalent to multiplying by its reciprocal. If you know of one, please let me know!

I certainly want students to be able to divide fractions symbolically. However, I have found that I can't just jump to this stage. Students will greatly benefit from this concrete-diagrammatic-symbolic development.