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.

Monday, March 27, 2017

A Concrete-Diagrammatic-Symbolic Approach to Multiplying Fractions

Look at this fraction and ask yourself, how do you say it:
When I give this question to colleagues, I get a range of answers: One-quarter, one-fourth, one divided by four. And often, 'one over four'. Admittedly, this is something I have said over the years. At a talk given by Cathy Bruce last year, I realised that this might be problematic: for the number 25 we don't say 'two beside 5' so we shouldn't use positional language ('one over four') to describe a fraction. A simple switch in the language we use can help our students from developing misconceptions.

I've been thinking a lot about how to use the Concrete-Diagrammatic-Symbolic approach to multiplying fractions to get help students learn how to multiply fractions. My concern is that if I jump straight to a symbolic explanation (multiply numerators, multiply denominators) students are at risk of developing more misconceptions. 
Starting with multiplying a whole number by a fraction, I'll use pattern blocks to model things like 5 x half:
or 7 x third:

8 x two-thirds:

Diagrammatically, I could use a number line to show these, or an interactive fraction tool:

Now I can develop this to symbolic notation:
At this point my aim is to get students to articulate what happens symbolically when we multiply a whole number by a fraction. I'm finding that this shift from Diagrammatic to Symbolic is smoother if I develop the Symbolic alongside the Diagrammatic and not treat them as separate domains. All the time I am doing this I must remember to say the fractions using the correct terminology (e.g. eight x two-thirds as opposed to eight times two over three). Also, I need to make sure that students transfer their knowledge of commutativity so that they understand that 5 x one-half is the same as one-half of 5.

When looking at multiplying a fraction by a fraction, I will introduce this concretely by folding Post-it notes. For example to do 1/3 of 3/4 , fold a post-it note into thirds and shade one-third blue. Now fold into quarters and shade three-quarters yellow. Notice how the post-it note is now divided into 12 parts (why?) and since 3 of those parts are shaded blue AND yellow, then1/3 x 3/4 equals three-twelfths or, more simply, one-quarter. (How might a student use this post-it note to prove that three-twelfths is equivalent to one-quarter?)
This method is not so easy for 3/5 x 5/6 though, so a diagram will be more useful now:
Notice that the overlap here has 15 units shaded (why?) and that the whole comprises of 30 units (again, why?). Hence three-fifths times by five-sixths is fifteen-thirtieths which simplifies to one-half.
As students do more of these diagrammatically, they will be in a better position to generalise by using this diagram:

This then gives reason to our symbolic rule.

Interestingly, having shared these strategies with colleagues and parents, a surprising number of people comment how they now understand when fractions are multiplied. If they understand it better, they will have less misconceptions, and, as a result, they are less likely to make mistakes.