## 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?

1. October 12, 2014 at 10:17 am

This is an area where children often fail to connect subjects. I had some pupils last year who were not sure that they could use the distance speed triangle in Maths because they had been taught it in Science. AARRGGHH!

• October 12, 2014 at 10:21 am

Haha, that’s bizarre!

2. October 12, 2014 at 10:17 am

I used to use them in physics (admittedly alongside the mathematical version) until I saw a pupil try to generate a formula triangle to rearrange Efficiency = useful energy/energy input x 100. I haven’t used them since.
Damian

3. October 12, 2014 at 1:11 pm

Why not embrace them and get students to make as many formulae triangles as they can based on their maths knowledge? I had a student apply it to exterior angle = 360/number of sides in a regular polygon which I thought was neat. Fair enough, 360 isn’t a variable, but a constant of proportionality in this case, which took the lesson down another route.

• October 12, 2014 at 1:14 pm

Yes, so clearly the students in question were familiar with algebraic manipulation. This sounds like a cool idea and a nice lesson.

4. October 12, 2014 at 2:38 pm

“…enables pupils with little to know algebraic knowledge to gain a good GCSE pass…”

Is this a bad thing? Certainly I like to do things properly, whatever that is, but I’ll try anything to help a student get a C if that’s the best they can hope for, in the time available.

Nice typo by the way. 🙂

• October 12, 2014 at 2:45 pm

Haha, thanks for spotting that. And your question leads to the wider question: “are we teaching these kids maths? Or are we just getting them a piece of paper with a letter on it?” if it’s the latter, is there any real point? Also, I’ve known many pupils scrape bs using this and other methods then totally bomb a level as they have no grasp of algebra.

5. October 12, 2014 at 3:00 pm

“are we teaching these kids maths? Or are we just getting them a piece of paper with a letter on it?” if it’s the latter, is there any real point?

It depends on the student doesn’t it? Either way there’s certainly a point.

• October 12, 2014 at 3:29 pm

Ant that’s why we need a wider reform of education.

6. October 12, 2014 at 7:12 pm

Here, here! (Or hear, hear!)?

• October 12, 2014 at 7:13 pm

Hear here, I think. As in “I hear you here”.

7. November 28, 2014 at 11:12 pm

Damn formula triangles. To be fair, when I was taught, we only used them in Physics, which like the commenter said above, I didn’t initially link with algebra. I blame my autistic nature 🙂

It’s the same discussion every time though. Trying to teach people correctly versus enabling them to pass the exam. It’s probably the mark of what others would term ‘a good teacher’ that you know when and where to make the distinction but I can’t help but be snobbish that all attempts should be made to let the student discover these things themselves.

For my own sanity as well as theirs.

8. January 11, 2016 at 4:59 pm

I always prefer to omit this triangles. Overall because they don’t say anything. Perhaps with dot instead of vertical bar could improve reader experience. And with two bars (of fractions) instead of one bar (of what?).

With appropiate notation it could be a formula compressor!
@somenxavier

1. July 14, 2017 at 2:24 pm