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: