## An excellent puzzle – alternate methods

Yesterday I wrote this post looking at a nice puzzle I’d seen and how I solved it.

The puzzle again:

Lovely, isn’t It?

After I published my previous post I wondered if I may have been better using a vector approach or a coordinate geometry approach. So I gave them a try.

**Coordinate Geometry**

I started by sketching the figure against an axis.

I place the origin at the centre of the circle, worked out the equation if the circle and the right leg of the triangle and solved simultaneously for x. Giving x =1 and x=1/3. These x values correspond to half the base of each triangle, which shows the scale factor from the large triangle to the small one is 1/3. As the area of the large one is rt2 this gives the area of the small as rt2/9.

I like this method, probably a little better than the one prior to it.

**Vectors**

First I sketched it out and reasoned I could work it out easy enough with 4 vectors.

I saw that I could write AC as a sum of two others:

I knew that the length of AC was 1 so I used Pythagoras’s Theorem to calculate mu. It left me with the exact same quadratic to solve. This time mu was the fraction of DB needed so was automatically the length scale factor. The rest falls out as it did before.

As well as this, Colin Beveridge (@icecolbeveridge), maths god and general legend, tweeted a couple of 1 tweet solutions. First he used trig identities:

**Trig Identities **

I assumed this was right, but checked it through to ensure I knew why was going on:

We can see beta is 2 x alpha and as such the tan value is correct. The cos value (although it is missing a negative sign that I’m sure Colin missed to test me) follows from Pythagoras’s Theorem:

This is again the scale factor as it is half the base of the small triangle and the base if large triangle is 2.

**Complex Numbers**

Then Colin tweeted this:

At first I wasn’t totally sure I followed so I asked for further clarification:

And then saw where Colin was going. I tried to work it through, by way of explaining here in a better manner.

I sketched it out and reasoned the direction of lines:

Then I normalised that and equated imaginary parts to get the same scale factor:

I am happy that is is valid, and that it shows Colin is right, but I’m not entirely sure this as the exact method Colin was meaning. He has promised a blog on the subject so I will add a link when it comes.

I like all these methods. I dontvthink I would have though of Colin’s methods myself though. I’d love to hear another methods you see.

## 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?*

## The straight lines debate

For a long time I’ve been confused about straight lines. Not because they confuse me, or that I don’t understand them, but the prolific use of the formula *y-yo = m(x-xo). *I personally prefer the y = mx +c version, and I couldn’t see a benefit to y-yo at all. I certainly couldn’t fathom why at A Level is was so heavily favoured by teachers and markschemes alike. I had asked around a few times, and the only coherent point I had got that made any sense to me was that when the gradient is fractional students who weren’t any good with fractions could multiply up by the denominator and deal with integers more easily. But my experience showed me that students used y-yo in a mechanical manner, and weren’t building any deep understanding of the geometry of straight lines.

I then noticed, while reading his blog at flying colours maths, that Colin Beveridge (@icecolbeveridge) was very heavily on the side if y-yo, and called y=mx+c “the baby formula”! I thought, perhaps “Colin can explain why there’s such a preference.” I asked, and sparked another “blog off”.

Colin wrote this excellent piece on his blog setting the case for y-yo. He argued that y = mx + c is a special case of the other, also mentioned the aforementioned fractional gradient argument, and made a few other (occasionally compelling occasionally contradictory) points. I felt that it did give me a better understanding of why people preferred it, but still did nothing to persuade me that it was the better version to use. (If you haven’t read Colin’s post, you should!)

I then responded with a guest post on Colin’s blog setting out my counter arguments. Since the post was written I’ve discussed these formulae with teachers and students alike. I have noticed something about the students views, those who I taught in year 12 ALL prefer y=mx+c, but those my colleague taught in year 12 mostly prefer y-yo. So I wonder if I had somehow imparted my preference to them. A friend of mine who is also a maths teacher summed it up pretty well for me when he said:* *

*“Ok, well so full honesty then- y=mx + c is probably better, however, when working with weaker students I find it easier to use the mechanical form, rather than the one which provides the most information about the line itself!” *(Steve Atkinson 2014)

Which version do you prefer? Do you have any strong views? I’d love to hear them.

## Circle Theorems: Should we bother?

On the most recent edition of “Wrong, but useful” cohost Colin Beveridge (@icecolbeveridge) had a bit of a rant about circle theorems. He feels they are pretty pointless, and he is in a fairly good position to discuss this, as he spent a decade researching the topology of the sun, basically circular in nature, and never used any of them. He says that he has only ever used one once, and that was to find the centre of a circle (this use is the most practical use if a circle theorem I can think of).

The discussion came about because the other cohost, Dave Gale (@reflectivemaths) was talking about when he trained and the things he hadn’t encountered before. His experience reminded me very much of my own. While I was training to teach I was also working to ensure my subject knowledge was entirely up to scratch, and that I was familiar with the syllabus. There I discovered Circle Theorems, and they were pretty new to me. I don’t know if I’d ever been taught them, I did know that diameters make right angles at the circumference, and that chords make the same angle in the same segment, so I suppose I may have learned then forgotten them. The one in particular that I was certain I’d never met was “Alternate Segment Theorem”, infact it was something that at first confused me and I spent a long time investigating it during my PGCE year before I was completely satisfied that I understood it fully and could teach it.

These Circle Theorems seem to stand alone in the syllabus, they seemingly have no direct link to any other area of the maths GCSE, and they certainly seem to have no real practical use at all and if there is an answer (other then “never”) to the question, “when will we ever use Alternative Segment Theorem in ‘real life’?” I’d love to here it!

**So, should we be bothering with them?**

The usual pro-circle theorem argument goes along these lines: “It’s a nice introduction to mathematical proof which gives students a good grounding for future proofs.” This doesn’t really wash with me. The questions we ask are not very stringent as far as proving goes and more often than not the teaching is focused around “this is how you get the marks on this question,” than any actual proof.

However, I do feel that they have a place on the GCSE syllabus. I used to sit in the anti-circle theorem camp, but my views on this have changed. The more I get into circle theorems the more I love them. And the fact that they don’t have a point just adds to it. The circle theorems are beautiful. They show geometry at its finest, and they have been derived purely because someone wondered about the properties of a circle, not because there was a problem that needed fixing.

Students should be exposed to this kind of maths. They should be allowed to investigate these theorems, and allowed to conjecture about them, before trying to prove them. I think they very definitely should stay in our syllabus, but that we need to address the way they are taught, and assessed. Take the mechanical nature away from the topic and allow the beauty of the maths to prevail.

## The great calculator debate

Back in December I posted this blogpost about calculators. It caused quite a stir and prompted many responses. Dave Gale, aka reflective maths, tweeted back with this video, Colin Beveridge, of flying colours maths, responded with this blogpost and there was a much wider debate on twitter with tons of people getting involved on each side. It was brilliant to see. I thought though, that I needed to write a further post to clarify and review what had been said.

In the first instance, I selected a sensationalist title which was intended to catch the eye. I do think, though, that the title may have led people to think my stance was a little more hardline than it actually was. And having seen the views set forward by the alternate position, I think my view has softened further still.

When I wrote the original post I was certainly advocating the banning of calculators in primary classrooms, and I would stick by this now. The opposing case to this was that an inability to subtract two and three digit numbers from 360 was causing a barrier to teaching angles. I would counter this with the statement that subtracting two and three digit numbers from a three digit number is such a basic skill that it needs to be mastered either before moving on to angles, or with angles providing a great opportunity to hone this skill. The other argument was that the government were banning calculators from KS2 tests, but using the same test. On the face of it, this is silly, but I don’t think it is a valid argument for keeping calculators. Rather it is valid argument for altering the tests.

Colin wrote in his post that calculators are not the enemy, but rather it is their misuse. I can see his point here, but I wasn’t advocating we destroy them all, I was advocating that we eliminate their use in primaries and cut it down radically in secondaries. He questioned the necessity of adding 4 or 5 numbers with 5 or more digits together, and this is a point I will concede. My hard line of only using it for trigonometry was perhaps too hard. But I still feel a vast reduction in their use would produce better mathematicians in the long run.

The video Dave sent was of Conrad Wolfram talking about why the future of maths should basically be entirely computational. Conrad feels that we need to stop teaching hand calculation and start teaching only computational mathematics. I feel that this would be an entirely wrong move. Computers can only do as they are told. If we are looking to prove a theorem generally, then we need to be able to hand calculate. Computers can check case after case, but this is not enough for a “proof”, as it is impossible to check an infinite amount of cases. A computer would not have been able to come up with mathematical induction or infinite decent.

A number of people responded along the lines of “What’s the point in learning how to do this when you can use a calculator?” This seems to me to be a ridiculous argument, like saying “Why learn to write when you can use a word processor?” or “Why learn to walk when you can use a mobility scooter?” If we head down that path is won’t be long before we are like the fat oafs in “Wall.E” (see this video) or even completely plugged in, a la “The Matrix”.

When teaching my further maths class numerical methods, I often have to field questions as to why we are doing this when if it were needed in the research world a computer would just do it. My answer is always simple, and always the same. “If no-one learns the theory, it will be forgotten, and no-one will be able to programme the computers to do it.”

No, calculators are not the enemy. But if the world becomes too reliant on them then we lose the skills we have built up over the centuries, we lose the ability to construct proofs for general cases, and we lose the beauty and the satisfaction one can get from solving a problem with nothing more than a pencil and paper.

Since the original post, I have realised that this is a wider issue than just calculators. Discussions with colleagues have highlighted that this problem occurs in other subjects when scaffolds are used. Thesauruses can lead to nonsensical sentences in English, for example. Scaffolds can also just mask a problem, pupils can get round something they cant do in lesson (ie subtract 197 from 360), but if it comes up on a non-calculator exam then they will not be able to obtain the correct answer.

Further reading:

From Mark Miller: Removing the cues

and Revision before redrafting (which includes the “greatest” sentence known to man: “a quantity of the most evil inscription is fashioned subsequently to a lexicon”.)

## Why calculators should be banned.

“*Sir, there’s a fraction in it. Have I got it wrong?”*

*“Sir, what do I do with this fraction one?”*

*“Sir, I get confused when there’s a fraction.”*

All these phrases are far too common in my year twelve class at the moment. We’ve just finished c1 and are doing some past papers, and I’m fairly worried by the way some of them baulk at fractions. This isn’t a problem that is solely theirs though. Some of my year 13s sometimes have trouble with fractions too. It’s not an isolated problem either. I think it’s symptomatic of the “calculator culture” which we live in.

Students of all ages have become far too reliant on the infernal contraptions! My year 13s think I’m obsessed with triangles (so do my year 10s, 11s and 12s. Perhaps I am?! They are amazing shapes with endless possibilities though.). The reason for my year 13s is that I try to encourage them to calculate trig functions using triangles, rather than using calculators.

*“Why? When you’re allowed a calculator in an exam?”*

Because it’s quicker, because there’s less chance of error, and because it will ultimately make you a better mathematician.

This isn’t a problem that is limited to the sixth form either. I was observing a year 9 lesson yesterday on pie charts. The class are quite bright, and the tasks involved dividing 360 (ugh, degrees) by some nice numbers like 90, 60 and 12. When one of the girls near me reached for her calculator to divide 360 by 60 I took it off her. She looked at me in shock and I simply asked “what’s 360 divided by 60?” she said “6” without even thinking. I then asked her why she had reached for the calculator and she said “because it was there.” I then circulated the room and all the pupils were at it.

Recently I wrote a post on multiplication methods which was inspired by a twitter chat on the subject and itself inspired a further chat. During one of them the someone inevitably suggested “just use a calculator”.

I don’t agree. I think calculators are responsible for a major decline in basic maths skills. I think they are responsible for creating lazy A-level mathematicians. And I’m sure they will have cost many gcse students many marks in exams.

A while ago the government announced a ban on the use of calculators in primary maths tests. Perhaps I should have written this then. I thought about. I’m in complete agreement on this one. I’d go further and at least encourage against them for most things across all key stages. I don’t allow my pupils to use them unless it’s necessary. I want them to be fluent in the maths, not good at following instructions to type stuff into a calculator.