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.'*

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.

nice- love the diagrams and the logic/ patterns following on from the concrete.

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