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.

Thursday, April 6, 2017

Joyous Maths: Dan's Favourite Pattern

The wonderful James Tanton speaks of the importance of getting our students (indeed everyone) to experience 'joyous maths'. One of the most satisfying sounds for any teacher is the squeal of delight and the "Oh, that is so neat!" when a problem has been solved. My colleague Dan Allen (past-president of OMCA) recently confessed that this is his favourite visual pattern.

A typical question to accompany a question like this might be 'How many squares are in the hundredth term?' or even 'what is the nth term?' In the past, I have dutifully created a table of values and used all sorts of algebraic techniques to come up with such an algebraic rule. This is fine, to an extent, but if this is all we do (or get our students to do) then we are missing out on a grand opportunity to do some beautiful maths.
So why does Dan get so excited about this pattern? Think about how many different ways it can be seen:
Perhaps you see it as an inner rectangle and two squares. Generalising this suggests an nth term of (n-1)(n+1)+2 

Maybe you see an inner square and two identical rectangles. Generalising, this suggests an nth term of (n-1)2+2n
Maybe you see an opportunity to complete an 'outer' square with two identical rectangles. Generalising, this suggests an nth term of (n+1)2– 2n
Or maybe you transformed the pattern by taking the top layer of each term, rotating it 90˚ to create a large square and a small square. 

Generalising, this suggests an nth term of n2+1.

In fact, much fun can be had showing that all these general terms do simplify to n2+1.

Visual patterns like these are so important as they allow us to be able to decompose shapes in different ways. This in turn helps us generalise by moving along the Next, Near, Far, Any continuum. Indeed, I am noticing how effective it is to get students (and adults) to generalise by asking them to 'draw' the hundredth term. When I did so recently in  Grade 9 class, the students moved on from drawing individual tiles to thinking about the dimensions of the inner rectangle and thence how to write this algebraically:

So I set myself a challenge: create a new visual pattern that will become Dan's new favourite puzzle. This in itself involved much problem solving and eventually I came up with this:

Using decomposing strategies like those shown above, how can we describe the hundredth term? the nth term? How many ways can we see this?

I showed it to Dan. He liked it (how much I'm not sure) and funnily enough, he saw it in a different way than I. He then set about creating a 'decomposable' visual pattern of his own:

So, be honest, which of these is the most joyous?

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These diagrams were created using the Colour Tiles from the site.