## Formula Triangles

Formula Triangles, it would seem, are a much loved shortcut in the world of Mathematics teaching. You know the ones, it’s when you get a three term formula that is one thing = a ratio of two other things. They look like this:

I’ve mentioned my hatred for them in passing on the blog and on twitter a couple of times and come under fire for this, which has made me think about it them a bit deeper. I used the term “ban them” in a tweet, and this may have been the cause of the uproar – as with The Great Calculator Debate. The term is more important extreme than my actual viewpoint, so I figured I’d try to set my thoughts out here.

Formula Triangles were first shown to me by my GCSE IT/Electronics teacher Mr Walker. The formula he was teaching them for was V =IR (Voltage = Currently X Resistance if my memory serves me correctly). Mr Walker didn’t explain how they worked, or what was really going on. He said “I always have trouble getting them the right way round, so I use this triangle, and cover the one I need.” I’m fairly sure his algebra skills were a little lacking. I was good at algebra and quickly spotted why this worked. I had to explain these reasons to a number of classmates who weren’t happy with the “just do it like this” model and craved a deeper understanding.

I quite liked them as a short cut, and quickly realised they could be applied to any number of similar formula, including speed distance time formula and the trigonometric ratios for right angled triangles, or RAT Trig for short. I’m fairly sure I used these in my exam.

**So what’s the problem with them then?**

Well, since you asked…. It’s the way I’ve seen them taught. I’ve seen them taught in maths lessons the way Mr Walker taught them in IT. This misses the opportunity to cement the algebraic skills required to rearrange formulae, to see the links between different areas of maths and enables pupils with little to no algebraic knowledge to gain a good GCSE pass. This highlights the ineffective nature of the maths GCSE as a measure of mathematical ability, which surely it should be.

These formula triangles are taught as a replacement to algebra, the purpose of them is that you can cover the one you want and get the formula arranged the way you need it without having to rearrange. Becks (@becksta9) asked: “when finding an angle using the triangle you get sin x = o/h how do you make x the subject?” and this is a good question, unfortunately in my experience the use of formula triangles for Trig is normally coupled with the instruction “don’t forget that you press shift when finding an angle”, rather than “the opposite divided by the hypotenuse gives the sine ratio, so you need to use the inverse function to find the angle.”

**Is it ever ok to use them?**

I would say yes. I am fine with people who understand algebra using them as a shortcut to save time (although how long does it take to rearrange them properly? You must save milliseconds!) , I’m fine with teaching them to weaker students who have tried to learn algebra but are prone to mistakes after they’ve been shown how to rearrange them algebraically.

What I’m not fine with is the “do it like this and don’t worry about how it works” use of them. Especially when the learners in question want to go on to study Maths at A Level and beyond, it could damage their chances.

*Becks, Jo (@mathsjem), Martin (@letsgetmathing), Hannah (@missradders) and Colin (@icecolbeveridge) all came to the defence of formula triangles on twitter. There was some sense that I was personally attacking their methods, and that I was making generalisations about the use of formula triangles. Neither of these were my intention and I apologise if it seemed it was. I hope this post explains what I meant better that I could in 140 characters. I was surprised at the massive response the tweet got, and the massive, seemingly emotional, relationship some had to it. I would love to here how Formula Triangles are used to aid rearranging, instead of as a way to avoid it, as is the point. I’d love to hear more views on this in general, do you use Formula Triangles? If so how, and why?*