Triangle Numbers: they aren't square!

I was sharing some web resources with some teachers recently and showed them the Who Am I puzzles from Solveme.edc.org . One such puzzle is shown below:

An honest question was asked: "I know what square numbers are but I don't know what triangle numbers are. Can you tell me what they are and why we need them?" As there were some blocks nearby, I decided to try a concrete-diagrammatic-symbolic approach with these adults.

Concrete
Triangle numbers are created in this way: start with a row of 1. This (1) is the first triangle number. (OK, some people say 0 is the first triangle number but I went for a natural number approach!). To get the second triangle number, add a row of 2 below the first (to get 3). Now to get the third triangle number, add a row of 3 (to get 6). To get the fourth, add a row of 4 (to get 10) and so on.

So the first few triangle numbers are 1, 3, 6, 10, 15, 21, 28, ...

So why do we need them? Well, they can pop up in certain problems, especially ones which involve finding the sum of the first n numbers. We know that the sum of the first four natural numbers, 1+2+3+4, is 10 which is also the 4th triangle number. I showed them this vine to help them visualise what is going on:

Diagrammatic
So how does this help us work out, say, the hundredth triangle number? Well, it's just the same as 1+2+3+...+100. This will take too long to build with blocks but a diagram will help (see right).
I asked them what size rectangle, these two 'triangles' would make and they quickly told me a 100 by 101 rectangle. This would have an area of 10100 units which means that one of the 'triangles' would be half of this. In other words:
1+2+3+...+100=(101×100)÷2=10100÷2=5050.
So far, so good, but can we now generalise this result?

Symbolic
I related one of my favourite math stories: of how a young Gauss was given a problem by a possibly hungover teacher in the hopes that it would keep Gauss and his classmates busy for the morning. The teacher had barely sat down when Gauss tossed his slate on his desk and in his peasant, Brunswickian dialect said, "Ligget se" or "'Tis there." I showed them his approach:

As I was doing so, the teacher who asked the initial question suddenly shouted out "Oh, shut up! He just added the two triangles like you showed earlier!"
Joyous math, for sure.
Now I could have explained triangle numbers by jumping straight to this formula, but I doubt very much if this would have led to the same sense of excitement and understanding that I saw. It also got me thinking that sometimes it doesn't matter whether or not triangle numbers have practical applications: triangle numbers even on their own are just incredibly cool.

The nrich site has a wealth of problems that involve triangle numbers. Triangle numbers (or near triangle numbers) can also be used when solving kakuros such as this one:

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.