Using Circle Geometry to Hunt Pokemon

If you know me in real life, it will come as no surprise to you that I have spent most of my summer holiday so far playing Pokemon Go. Those of you who don't know me in real life, if you see a pink-haired twenty-something with a phone precariously attached to a Pikachu lanyard wearing dungarees (optimal number of pockets) and Go Walk Sketchers (optimal walking shoe) walking at as close to 7km per hour as possible (optimal PoGo speed) then that's me.

After a few weeks of aimlessly walking around Coventry city centre or sitting on the steps of the bank where some kind stranger will always drop a lure, I decided that I had caught enough Drowzee and would like to make a bit of progress towards catching 'em all. My pokedex was hovering around the 80 mark for a long time, despite making trips to Leeds, Manchester and Birmingham in the hopes of finding exotic local pokemon and hatching eggs like Bernard Matthews. So I decided to give pokemon tracking a try.

In the bottom right-hand corner of the screen when you're in map view, there is a window you can expand that says "sightings" and lists some pokemon that have spawned nearby. Sometimes I would see a rare pokemon on this window, but learned helplessness has taught me that the rare ones never pop up when you want them to. By "pop up" I mean appear on your map as a tappable, catchable pokemon. So I decided that instead of wandering aimlessly and hoping for the rare ones to pop up, I would hunt them down strategically using geometry.

A pokemon will appear in "sightings" when you are within 200m of it. The pokemon will be catchable when you are within 70m of it. So when a rare pokemon, let's say a Charizard, appears on your sightings, you should be picturing the following diagram:

Of course, you could be anywhere in the purple circle (but not in the pink circle, or the Charizard would already be catchable). The fact that the pokemon just appeared in your sightings could either mean that you have just stepped inside the purple circle, or the pokemon has just spawned. Pokemon disappear after 15 minutes, so what you do next needs to be efficient and at a bit of a jog if possible. If you have a buddy with you, this is much easier, as you'll see in a minute.

Let's call the point you're at point A. What you need to do is identify a straight path that you can walk along that goes in both directions from point A. This can be very difficult, depending where you are. I have found that it is much easier in a park than in the city centre. Now, walk along that path, remembering where you started. Ideally, you would count your paces as you walk. Keep walking until the Charizard disappears from your sightings. The point where that happens we'll call point B. This will be a point on the circumference of the circle in your head. Of course, you could get lucky and walk right into the pink circle, in which case, get the razz berries and ultra balls ready! But let's assume the unluckiest situation.

Next, you need to turn around and walk in the exact opposite direction, back to point A and beyond it until the Charizard disappears from your sightings again. Call this point C. If you have a buddy with you, they can do this bit whilst you are doing step one, to save time. Again, it would be good if you could count your paces.

Now that you have identified two points on the circumference of the circle, and have walked a chord of the circle, this is where the geometry comes in. The perpendicular bisector of any chord of a circle always passes through the centre of the circle.

If you don't believe me, think about this: take a random chord of a circle and join up its end points to the centre as in this diagram:

You should be able to see that this makes an isosceles triangle, because two of the sides are radii.

This means that this triangle has a line of symmetry and if you cut it down this line, you get two right-angled triangles:

And clearly this line of symmetry goes through the centre of the circle.


So, back to our hunt. We're at point C, and we need to find the perpendicular bisector of the line segment BC which is the path we have just walked. The first thing we need to do is find the midpoint of B and C. In some places it is easy to do this by eye, if you have a good map or if you're in a very flat area. But if you have counted your paces, you will be able to find the midpoint much more accurately. I personally don't bother with counting. Because we're only trying to get inside the pink circle, not get to the exact centre, we don't have to be that accurate. So, walk to this midpoint, which we'll call D.

Now, turn ninety degrees and walk. But Emma! (I hear you cry) There's two ways of turning ninety degrees! Yes, you're right. At this point, you have not uniquely defined the purple circle. If you draw two random dots on a page, there are always exactly two different circles with a given radius that pass through those two points. You don't know which circle it is, so you have to guess. So turn ninety degrees in any direction and start walking (or running!) until one of two things happen: the Charizard pops up and you catch it, or the Charizard disappears from your sightings, in which case you do a 180 degree turn and run for it! If you have a buddy with you, you can take one direction each and invent some kind of signal for "I found it!" (smoke signal? A whistle? Make a sound like a dying giraffe?)

You have to do all this in the space of 15 minutes which can be tricky, and sometimes involves running and looking like a loon. But me and my husband went out to Coombe Abbey country park and Coventry's War Memorial Park last week and managed to use this method successfully several times. The handy thing is, pokemon can't spawn just anywhere, there are a set number of spawn points in a given area. So once we identified some of the spawn points, we didn't even have to use the method, we could run to the nearby spawn point we found earlier. There are still enough different spawn points around to keep the game challenging though.

In case you were wondering, my Pokedex is now up to 94. I promised my tutor group that I would have caught 'em all before term starts again in September, which is looking very unlikely. Then again, my tutor group promised me they wouldn't fail their AS levels, so I might have a bit of leverage there...

I hope this method helps you hunt down some rare pokemon and maybe understand the relevance of circle geometry a bit more. (Oh my gosh, did I just come up with a "real life" application of geometry??? Noooooo! Keep Maths pure, people!)

Emma x x x

The Mathematics of Voting

I'm going to do something completely out of character and write a TOPICAL post! Yes, I may live underneath a proverbial rock (what, there was an earthquake recently?) but even I am aware that there is a General Election today in the UK. In fact, I was the first person at my polling station to vote this morning! 

I've mentioned in a previous post that I love Lewis Carroll. He combines two of my favourite things: books and maths (Lewis Carroll is the pen name of mathematician Charles Dodgson. Please keep up!)

Dodgson became involved in college elections in the early 1870s at Oxford university where he was a professor. He became interested in the theory of voting, of the accuracy and fairness of different voting systems.

First Past The Post

Dodgson was not a fan of this voting system. He claimed "the extraordinary injustice of this Method may be very easily demonstrated". He then gives an example to show how stupid it is:

Suppose there are 11 electors and 4 candidates a, b, c and d. Each elector ranks the four candidates in order of preference. The 11 columns here show their choices:

a	a	a	b	b	b	b	c	c	c	d
c	c	c	a	a	a	a	a	a	a	a
d	d	d	c	c	c	c	d	d	d	c
b	b	b	d	d	d	d	b	b	b	b

It's easy to see that a is considered best by three of the electors and second best by the rest. But in actual fact, it is b who ends up winning, even though he/she was considered the worst by seven voters.

I don't think Dodgson looked at "Alternative Vote", although he did write about lots of other systems.

The Method of Elimination

In this method, each voter chooses their favourite, and then the one who gets the fewest votes is eliminated, and the process is repeated (a bit like Big Brother? The TV show, not the Orwellian thing). This method at first seems pretty flawless. However, consider the following situation:

b	b	b	c	c	c	d	d	d	a	a
a	a	a	a	a	a	a	a	a	b	c
d	c	d	b	b	b	c	c	b	d	d
c	d	c	d	d	d	b	b	c	c	b

Notice that a is everybody's first or second choice, and hence appears to be the best candidate. However, he/she will be eliminated first. c will be elected instead.

The Method of Marks

In this method, each voter is given a specified number of marks that they can divide between the candidates. Then the candidate who gets the most marks wins. Dodgson said that this method would be perfect as long as the voters divided their marks fairly: giving most to their favourite but some to the candidates that they wouldn't mind electing. But Dodgson commented that "since we are not sufficiently unselfish and would assign all our votes to our favourite candidate, the method is liable in practice to conicide with that of the simple majority [first past the post] which has already been shown to be unsound".

I hope you voted today! Let's get rid of the current education-ruining idiots!

 Emma x x x

All quotes are from Robin Wilson's "Lewis Carroll in Numberland", a book I highly recommend. 

The Chaos Game

Draw an equilateral triangle on a bit of paper. Draw it nice and big. Now pick a corner to start at. Next, get a die (a D-6) and assign two numbers to each corner. For example, the top corner can be 1 and 2, the left corner can be 3 and 4, and the right corner can be 5 and 6. Write these numbers near the corners on the outside of your triangle so you remember them.

Roll your die. The number you get tells you which corner you are heading towards. Find the midpoint of your current location and the corner you're heading towards. Use a ruler for this and try to be as accurate as you can. Mark this new point with a good-sized dot, that is your new location.

Repeat.

After about fifteen minutes your triangle should have lots of lovely dots, and at this stage you might even see a pattern emerging.

A pattern! (I hear you cry) How can there be a pattern, when I am moving randomly! Surely the dots will be scattered in a random manner, looking like the freckles on a pasty Irish face. But there is indeed a pattern. A very nice one in fact. A very familiar one, actually.

SPOILER ALERT

Please actually carry out this experiment before looking ahead.

I will now insert some line breaks to stop you seeing the pictures below. Don't scroll until you're ready.

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I didn't rub out my construction lines because I've been taught well :-)

Can you tell what it is yet?

Holy fractal, Batman! That there is Sierpinski's triangle! Or as my students would say: Illuminati confirmed.

OK, it doesn't look exactly like Sierpinski but it's definitely getting there. I need to do another thousand or so iterations.

I am a mathematician, which basically means I have turned being lazy into a career. So at this point I started thinking, why am I drawing this s*** when I could be running a simulation instead?

Here is the spreadsheet I used to simulate the Chaos game. It was kind of fun to set up, so I suggest you try it yourself before reading my formulae. You can probably find a much more elegant way to do it, but I'm a mathematician, dammit, not a programmer!

Teachers: this is a cool way to kill an hour with a group of students who know how to use a ruler and divide stuff by two. You could do it as I have suggested, with a triangle drawn on blank paper, but you could instead do it with a triangle drawn on a coordinate grid, with the corners having the coordinates I used in my spreadsheet. This way, students can practise finding the mid-points of two points (a skill that is needed for Higher tier GCSE and for AS Level). However, the numbers get nasty pretty quickly (as there is a root 3 involved). Even if you just use this as an exercise in measuring with a ruler, I think students can get a lot out of it. The pattern is so cool and unexpected, your students may even have that "wow" awe and wonder moment.

Have fun!

Emma x x x

Cheryl's Birthday

Photo credit: Kenneth Kong/Facebook. 

Imagine my delight when I found out last night that loads of people on the internet were talking about a maths puzzle! Some people are even saying #Cheryl'sBirthday was the new #TheDress. Not being a Twitter user (is it OK that I just don't get it?) I arrived a bit late to this party but I hope you don't mind.

So, this puzzle comes from a Singapore Maths competition which I think is probably comparable to the UKMT Intermediate Maths Challenge follow-up round (the Pink Kangaroo) which means it is supposed to be challenging. Can I just take this opportunity to brag and say I got the answer right in about three minutes? Please, if you haven't already, pause and work out the answer for yourself.

There are lots of explanations of the solution out there on the internet, but I'm a maths teacher and I can't help myself, I love explaining stuff!

Statement one:

Albert knows the month. This means he can't know when the birthday is, as there are no months with just one possible date.

However, Albert has deduced that Bernard cannot know either. The only way Bernard would be able to know is if the number he was told was 19th or 18th because they are the only dates with one possible month. So for Albert to know that Bernard does not know, these two options must not be possible. Albert only knows the month, so for him to know these are not possible, the months of these must not be correct. Therefore it is not May or June.

Statement two:

Now that Albert has said that, Bernard has deduced that it is not May or June, just like we have. This information is enough for him to know the correct date. This means it can't be the 14th because there are two months with the 14th. So it must be 16th July, 15th August, or 17th August. .Bernard knows which one of these it is because he knows the number. We do not know.

Statement 3:

Albert has deduced the same as us, and narrowed it down to those three dates. But Albert knows the month, and by knowing this, he knows the answer. So it must be July, as if it was August he still wouldn't know.

Therefore the answer is 16th July.

What a wonderful question!

This question is similar to those questions where nobody knows anything but by saying "I don't know" enough times everyone suddenly knows everything. Do you know the kind of question I'm talking about? My favourite is probably the one with the island full of brown eyed and blue eyed people: The Blue Eyes Logic Puzzle. This is so difficult to wrap your head round it but once you do, you feel like you have understood the secrets of the universe and your brain suddenly enters this state of ultimate clarity. Unfortunately this only lasts for a few minutes and then you stop understanding it again. I have read about this puzzle so many times now I can hold onto this state for a whole evening. I always wake up ignorant again.

Maybe I will do a full post on the Island puzzle one day when I'm feeling brave and I have a large supply of stimulants to hand.

Let me leave you with a link to my favourite place to find logic puzzles. These are great for teaching the Logic chapter in D2. They are also nice and short so you can do one at the beginning of every maths faculty meeting just to get everyone's brains warmed up.

And also, a maths joke: Three logicians walk into a bar. The barman asks “does everyone want a drink?” The first logician says, “I don’t know”. The second logician says, “I don’t know”. The third logician says, “Yes”.

Ha ha ha ha ha ha!

Emma x x x

Why is 0! (zero factorial) equal to 1?

This post was originally written at the end of 2013.

Today I had a very typical Further Maths A Level lesson. Someone asked a very simple question, I started to answer it, and ten minutes later we were talking about how many imaginary sheep there were in the classroom.

Like I said, a typical lesson.

The question that I was asked was, "Why is zero factorial one?". This was asked by a female student I will refer to as H (to protect her identity- she probably doesn't want to be associated with this nerdy conversation). I'm not entirely sure why H asked me this, as she was supposed to be working on the Secant Method (otherwise known as the most painful mathematical process of all time).

But anyway, she asked me this, and my immediate answer was that very useful mathematical phrase: "by convention".

She responded with, "What do you mean?" to which another student, who I will call J, replied, "To make everyone happy", which pretty much sums it up. Zero factorial was defined as one to make everyone happy. What a lovely answer!

But I couldn't just leave it there, could I? Oh no, my geek sense was tingling. I tried to get my head back to where it should be (doing the register) but I just couldn't. Before I knew it a board pen had somehow leapt into my hand and I was on my feet.

Let's take a look at the factorial function.
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
1! = 1
0! =

Notice the deliberately blank space next to 0! =. Because that's what the answer is. A blank space.
"Three factorial is three times two times one".
"Two factorial is two times one".
"One factorial is one".
 "Zero factorial is ...[silence]".

So the question is, what number is "...[silence]"? By the way, when you say "...[silence]" you should accompany this with a hand movement kind of like "ta-da!" but less dramatic. I might post a video up here later so you can see what I mean.

Sorry, I was saying, what number is "...[silence]" *hand movement* ?

Well in my opinion, it's one. To me, it's obviously one. It's not zero. Zero has too much meaning. Zero is a very definite nothing. Zero is a dangerous number - it can ruin all kinds of calculations. I think the "blank" number is one.

Here's a reason why:

What's 3x - 2x?
Answer: x.
What's the coefficient of x?
Answer: 1.
But where's the 1?
Answer: you don't need it.

The blank space in front of the x means one.

Another example:

Say you had some algebraic fractions to simplify by cancelling common factors. Look at the first two examples. Using similar logic, surely the answer to c) is a blank space? But we know the answer is 1.

It is easy to see why 0! has to be 1 when we look at combinatorics. 5C0 ("5 choose 0") means how many ways are there of choosing zero items from a choice of 5. The answer to this is one. Why? Well if you have to pick zero items, how many ways are there to do this? Well the only way to do it is to not do it, which is one way, so the answer is one. The formula for the nCr (choose) function involves factorials, and the only way for nC0 to equal one is if 0! = 1. So 0! has to be 1, or the formula won't work.

In other words, it's 1 to keep everyone happy. I should have just listened to J.

And in case you're wondering, I never did get round to doing my register.


PS No I will not explain the imaginary sheep thing. You had to be there.

How Do You Round a Negative to the Nearest Whole Number?

How should you round -1.5 to the nearest whole number?

Almost anyone you ask this to will reply without thinking: -2, because 5 rounds up.

Spot the mistake!

Rounding -1.5 to -2 is not, in fact, rounding up, it is rounding down, because -2 < -1.5.

Of course, that doesn't mean rounding to -2 is necessarily wrong, but it does disobey the general rule that "5 rounds up". But this rule of thumb that we maths teachers use, have we actually thought it through?

For now, let's just consider positive numbers, and the reason we round things that have .5 up. Numbers whose decimal bit starts with .5 and then has loads of numbers after it, e.g. 3.532423765 would obviously round up, as they are more than half way between the two whole numbers. By deciding that 3.5 would also round up, it means if you are scanning a massive set of data, you only have to look at the first number after the decimal to know whether it will round up or down.

However, surely if we always round things ending in 5 up, we are creating an imbalance somewhere? This might seem minor, but if you consider all of the millions and billions of transactions that take place in, for example, bureaux de change, where currency is changed, this will add up to a lot of money that someone will be unfairly losing (or gaining).

I myself run into this problem every month when my husband and I sit down to pay off our shared credit card. Our Google Sheet halves the cost of all of our shared purchases and totals up how much we each have to pay. And when we share a purchase that is an odd number of pence, we run into a little problem. Google dutifully rounds our individual costs up, but then we would overpay our credit card by a penny for every such transaction. My husband, having the amazing qualities of both a mathematician and a computer scientist, fixed this so that one of the values rounds up and one rounds down. And my husband, also having the tight-fisted qualities of a Scotsman, fixed it so his costs always round down, and mine round up.

So always rounding .5 up (officially known as "Round Half Up") can be a bit of a problem. The whole ends up being less than the sum of the rounded parts. Maths teachers also know that this is incredibly annoying when it comes to pie charts and stratified sampling. You know exactly what I'm talking about.

There are some ways of fixing the unfairness of Round Half Up . A lot of these methods are actually used without you even being aware of them. I bet you didn't even know that the method you usually use has a name. I'd bet even more that you aren't aware their are eight types of commonly used rounding methods.

The Eight Main Rounding Methods

Round Half Up

When it's a 5 you round up. So 4.65 rounds to 4.7 and -2.5 would round to -2. This is known as "asymmetric rounding" because it is positively biased - that is, we round up slightly more often that we round down.

Round Half Down

When it's a 5 you round down. So 4.65 rounds to 4.6 and -2.5 rounds to -3. This is hardly ever used. This is also known (confusingly) as "asymmetric rounding".

Round Half Away From Zero

When it's a 5 you round away from zero. So 4.65 rounds to 4.7, and -2.5 rounds to -3. This is probably what most normal people probably assumes happens. This method is symmetric because half the time 5 rounds up and half the time 5 rounds down. However, this is only fair if positive and negative numbers are equally likely. There are some situations that deal only with positive numbers, and then the method would still be biased.

Round Half To Even

When it's a 5 you round towards an even number. So 3.5 rounds up to 4 but 6.5 rounds down to 6. -2.5 rounds to -2, -3.5 rounds to -4. This method of rounding should be unbiased because even and odd numbers are equally likely, right? But zero is even, so aren't there sort of more even numbers than odd? That's a debate for another post. This method of rounding is probably the most commonly used, as it is the default method used in IEEE 754 computing functions and operators.

Round Half To Odd

This should be obvious, having read the previous paragraph. This method, however, is almost never used.

Stochastic Rounding

When it's a 5, flip a coin, and use that to decide if it rounds up or down.This should be unbiased, as it really would be a 50/50 chance. However, if you let your students use this method in their maths homework, you would have thirty students with completely different sets of answers. Whilst the unbiasedness of this method appeals to me, the fact that you would get different answers every time would just be annoying. Some of my students (many of my year 11s) actually do apply this method of rounding, but without a coin. It's otherwise known as guessing. They have a 50% chance of being right, which is good enough for me.

Round Half Alternatingly

The first time you have a 5, you round up. The second time, round it down. So if you had this list of numbers: 3.5, 6.5, 2.5, -1.5, you would round these to: 4, 6, 3, -2. This will be free of bias as long as you have an even number of data that end in a 5.

So there you have it. Eight different rounding methods, six of which are commonly used (although some are more common than others). And many people (including many maths teachers) have absolutely no idea our money, our personal data, and the data we are presented with in newspapers, have been subjected to these methods. We could be missing out on half pennies all over the place!

Another method worth mentioning is Supermarket Rounding, which is where if something is half price, they always round the price up. So 99p becomes 50p when half price. Interestingly, when the same supermarket advertises 50% off, 99p still becomes 50p, even though the 50% that is taken off should be the bit that is rounded. Hey, these half pennies add up you know!


So back to my original question, how do you round -1.5 to the nearest whole number? The answer is either:
Round Half Up: -1
Round Half Down: -2
Round Half Away From Zero: -2
Round Half To Even: -2
Round Half To Odd: -1
Stochastic Rounding: *flips coin* -1
Round Half Alternatingly: -1
Supermarket Rounding: N/A

Simple.

Emma x x x

What Does the O Stand for in BODMAS?

I recently sat down to plan a lesson for year 7 students about the order of mathematical operations. Here in the UK, I believe this is most commonly known as"BODMAS". The B stands for brackets, D for division, M for multiplication, A for addition and S for subtraction.

But what does the O stand for?

I've heard several different answers to the above question, none of them satisfactory. One of my colleagues told me he taught it as "Orders". This random website I found agrees. But what the heck are orders? According to the aforementioned random website, they're "numbers involving powers or square roots". I have never heard this definition before, and after consulting the oracle (Wikipedia) I found no mention of indices or powers on the page for Order (mathematics). So why on earth would we teach students the word orders when we never call them that in lessons? We in the UK usually refer to these as "indices" although I believe the Americans prefer "exponents" (but I'll get to them later).

Another colleague told me he teaches that the O stands for "of" as in "powers of", and I'm ashamed to admit this was what I was taught in school. I think this one is faintly ridiculous. Firstly, O cannot stand for "Powers of" because "Powers of" clearly begins with a P not an O. Kids may be getting dumber every generation, but I have a feeling they will notice this. Also, why does the word "powers" even need an "of"? Can we not just call them powers? It reminds me a bit of learning French when we were always taught to write the following preposition after certain words like "decider de" or "je pense que" to help you form sentences. This was actually excellent advice for learning French, but this does not dilute my point.

A third colleague (it is amazing how many of them are willing to contribute to my inane Monday-morning conversations) said that he teaches that the O stands for "Other" as in, any other operations not mentioned. This is quite nice actually, because it includes not just powers and roots but also sines, logs, factorials, etc. Very handy.

I then went into my year 7 lesson and asked them what they thought the O stood for. Interestingly, the most common response was one I had not heard yet: "operations". This is perhaps the one that annoys me the most. BODMAS is the tool we use to remember in which order we should do operations. If O stands for "operations", then we are basically saying, do the bit in the brackets first, then do the operations. Oh wait, what order do I do the operations in? Use BODMAS. So I do the brackets and then the operations. But what order do I do those operations in? etc etc. Thank you Primary school teachers. Thanks a bunch. You have just created an infinite loop in my head. You have given my eleven year-old students an acronym to learn that is actually a recursive formula. After infinite iterations they will still not have found the value of 3 + 2 x 5.

So I bet you're dying to know what I taught them in the end, right? Well I told them about the conversations I'd had in the maths office. I also told them about the American version: PEMDAS. Seriously. That's what they call it. The MDAS is obvious enough. The P is for "parentheses" which my students had never heard of but which is quite useful to know I suppose, and the E is for "exponents" as I mentioned above. My year 7s were not happy that our friends across the Atlantic do their multiplication before their Division though. "Surely they'll get different answers from us and then spaceships won't work!!" they cried. (I must have told them about the metric/imperial satellite mix up in a previous lesson). This led to a nice discussion about how those two operations are interchangeable and you would still get the same answer (or would you? I have just thought of a topic for a future post).

Anyway, in the end, I taught them the O stands for Indices. That's right, I'm on Team BIDMAS. All you BIDMAS haters out there can hate hate hate but if we refer to powers as "indices" the rest of the time why not in this? And if you have a problem with me not including trig functions or logarithms or whatever in my acronym well you shouldn't because by the time you're learning that sort of stuff you shouldn't need a mnemonic to help you remember which order to do stuff in anyway!

Over to you: what did you learn at school, and, if you're a teacher, what do you teach now?

Emma x x x

Confession: I Hate Drawing Graphs

Shocker! Emma, the girl who loves maths, hates drawing graphs. This includes: cumulative frequency curves, histograms, graphs of parametric equations, roots of unity, and those log ones from C2 chapter 10.

Students are always surprised to find out there's an aspect of maths I find boring. They exclaim, "but you're a maths teacher!" Well, the nicer ones do. The not-so-nice ones exclaim, "but you're a massive nerd!"

I have a huge problem with this. Think of your music teacher from school. They loved music, didn't they? Do you think they loved all types of music? Classical, rap, indie, ska, dubstep, One Direction? Probably not. You wouldn't be surprised if you found out there was a genre of music they hated.

Think about an art teacher. They love art. But in their house, would they hang impressionists, surrealists, er... etc... on their walls? Would they love every medium, every era, every subject? Of course not. If they did, you would probably think less of their love of art. "Proper" art lovers usually know what they like, and are dismissive of, or even angered by, artwork they don't like.

So why shouldn't mathematicians be the same? And more importantly, why shouldn't our students see this? I let my students know when we're studying a bit of maths I'm not interested in. Maybe this puts them off it. But who cares? Of course, when my favourite bits come up, I make that known too. I also encourage students to have favourites in maths. I like my students to feel they have a specialism. It makes them feel good when that topic comes up. I want them to see mathematics as being similar to music and art: subjective.

Do you admit to students that there are things in maths you don't enjoy?

Emma x x x

When Will I Use Algebra in Real Life?

There is one question my students ask me that I hate far more than all others. Far more than "Why are you just a teacher?" and "Do you have a boyfriend? What's his name? What's his job..." etc.

That question is: "When will I use simultaneous equations/ the laws of indices/ completing the square in real life?"

A good maths teacher, of course, would have a bank of answers to such questions. "Satellite dishes are in the shape of a parabola!" etc. Urgh.

Even though I do happen to know a few applications of algebraic principles to "real life" (picture me making air quotes with my fingers), I never tell these to my students. I refuse. 

First of all - real life? I'm sorry, are my maths lessons not real? When you enter room 204, are you entering some kind of alternate universe? Is a  maths lesson merely a state of mind? Some kind of lucid dream that looks real and feels real, but can't possibly be real because instead of English the teacher is speaking in an alpha-numeric jumble?

Secondly, where did students (and, for that matter, teachers) get the absurd idea that everything one learns has to have some kind of practical "use"? Can't we simply enjoy learning maths for its own sake? Does everything we do have to have a useful purpose? What kind of depressing life would that be?

Maths is beautiful. It is deep and interesting. It is a language. It is an art. It is a self-contained world with its own rules, patterns, and mysteries. So don't try to spoil my beautiful maths with your ugly "applications". 

I will leave you with some profound quotes from mathematician G H Hardy: 

"Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics". 

“Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.”

"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

Of course G H Hardy was wrong with that last one - his work was actually widely applied to genetics and thermodynamics. But the point is, that wasn't why he did it. And he didn't need these applications as motivation for producing this work.

Do you agree with me or are you someone who sees maths as a "tool" with which to get things done?

Emma x x x

What is x?

How do you introduce "x" to your students?

I'm guessing you start teaching informal algebra in this kind of way:

10 + ? = 20

Where the question mark obviously represents the mysterious-sounding "unknown". Except it's not really unknown, because it's obviously 10.

Eventually you replace the question marks (or empty boxes) with letters. Not just the letter x, obviously, but you have to admit x is a popular one.

In this number sentence, what is x? I don't mean what is it's value, I mean, what is it?

2x + 4 = 12

It's an unknown. It is a number that definitely exists and has one particular value which at this very moment is unknown to us but in a matter of seconds will be completely known. Two quick steps and we and x will be on first-name terms.

In this number sentence, what is x?

2x + 4 = y

Suddenly, x is no longer an "unknown". It is a "variable". Meaning, its identity is still a secret, but it's not one specific number, it could be any (any any?) number in the world.

2x + 4 = y and y = 5.

Now, suddenly, although x is a variable, it has been forced to stop varying, and simply be unknown.

Can you see how the dual nature of x (or any letter really) could be very confusing for students? If students think of a letter as representing one particular number (even if they realise that number can change on a daily basis), this might hinder them when it comes to studying linear graphs, or functions.

Maybe we should try to introduce x as a variable instead of an unknown. Think about how you could do this, perhaps with your brand new, untainted year sevens with their clean-blanket-of-snow brains.

Let me know how you get on,

Emma x x x

What is a Regular Quadrilateral?

Have you missed me? It's been over a whole glorious month since I last posted. I've been lying on my sofa reading nineties teen romance novels and leaving the house only to visit Ikea for free tea and to pretend I live there. 

But anyway, I'm back and have I got a mathematical ponderance for you!

First, grab a piece of paper and a pen (or open a notepad file, for the more evolved among you) and write down (type) the definition of regular (as in, a regular polygon). 

Done? OK. Hands up who wrote this:

"All the sides are the same length".

And hands up who wrote this:

"All the sides are the same length and all the angles are equal".

Did any of you just write this:

"All the angles are equal".?

A more important question, perhaps, for teachers, is this: what do you tell your students?

Now another little exercise for you. Is this statement always, sometimes, or never true:

"If a polygon has all equal sides then all of the angles must be the same size".

Let's think for a moment. We know it is true for triangles, although you might not be able to prove it, or even justify it, beyond the fact that your Year Five teacher told you it was true during a particularly soul-crushing numeracy hour. We have also always assumed it was true for "big polygons" like octagons, decagons, etc. But is it true?

There is one type of polygon for which the statement is definitely not true. Quadrilaterals. Sorry if the title spoiled this major reveal for you. A square is equilateral and equiangular. However, a rhombus is equilateral but the angles are not all the same size. 

OK, here's another one for you:

"If a polygon has all equal angles, then all of its sides must be the same length". Always, sometimes or never true?

Again, think about triangles first, and then think about hexagons etc. Can you prove your conjectures?

Again, the statement is clearly false for quadrilaterals. A rectangle is equiangular but not equilateral. 

So what is a regular quadrilateral? Is it a square, a rhombus, or a rectangle?

My initial thought (because I admit, I didn't actually know the correct answer), was that a rectangle is regular. I thought this because geometry mostly comes from Greece, and in Greece, they're mostly bothered about angles. Hence the word polygon: "poly" meaning many and "agon" meaning angles. So a hexagon is literally a shape with six angles. In the UK, we're more likely to say a hexagon is a shape with six sides. So I thought a regular polygon would mean a shape with regular angles. 

However a quick tussle with my favourite search engine revealed that a regular polygon must be both equilateral and equiangular. 

I'm ashamed to admit I think I might have taught students that regular just means equilateral. Or, even worse, I think I might have even implied that equilateral shapes were always equiangular! It's funny how such a simple little definition can be messed up because you think you understand it perfectly (after all, I was taught it in year five). Maths teachers like me need to make sure we are completely clear about these things. Maths doesn't leave much room for error, and these definitions need to be water-tight. 

On that note, have a good first day at school!

Emma x x x 

PS I am of course talking about convex regular polygons. Non-convex (star) polygons are of course a whole other kettle of fish! 

Is Zero a Factor of Zero?

Inspiration for this post came from an unlikely source: a bottom set year seven class and a non-specialist maths teacher. He was teaching them factors, and was asked by a student, "is zero a factor of zero?"

This question would have just about made me explode with excitement. I think my colleague's reaction was a bit different. He is a humanities teacher by trade, so naturally he was very good at fobbing them off with an answer that sounded impressive without actually answering the question.

This colleague has obviously been working with me for too long now, because after this lesson he came and asked me the same question, because he was actually interested in the answer. I even caught him reading my post Is Zero a Square Number?  at lunchtime. He's one of us now.

So, is zero a factor of zero? Well, as is often the case in maths, the answer is: it is if you want it to be.

It all depends on how you define a factor. Here are some possible ways.
For n and m natural numbers, n is a factor of m if:
1) n divides m with no remainder.
2) n x p = m, where p is a natural number.

[Side note: obviously natural numbers have negative factors too, and factors can be defined on integers rather than just the natural numbers, but negative factors aren't interesting, they're just the same as the positive ones but with a minus sign.]

Let's look at the two definitions:

1) From this definition, for zero to be a factor of zero, zero would have to divide zero with no remainder. What is zero divided by zero? That's my all-time favourite maths debate (and you all know I love to maths debate). Here are three possible answers:

-Anything divided by zero is infinity, therefore the answer is infinity.
-Zero divided by anything is zero, therefore the answer is zero.
-Anything divided by itself is one, therefore the answer is one.

In fact the answer could be anything you want it to be:
0 x pi = 0
Therefore 0/0 = pi.

So we say the question (and therefore the answer) is undefined. Or "MA ERROR" on your old Casio.

So by this definition, zero is not a factor of zero, in fact it can't be a factor of anything. However, every other number must be a factor of zero. Zero divided by anything other than zero is zero, which is a whole number with no remainder.

2) I think we'd all agree there exists a p such that 0 x p = 0. There are infinitely many such p! So by this definition, zero is a factor of zero.

So what is the answer? Well I'm going to solve this mathematical mystery the way mathematicians solve most of the really puzzling mathematical mysteries. I'm going to use the magic words: "by convention".

By convention, zero is not a factor of itself.

Done.

Emma x x x

The Terrible Twos

This week marks my two year anniversary as a qualified teacher, and also the two year anniversary of NQTpi. Woo!

As a third-year teacher, I look forward to:
-A slight pay rise (the last automatic one I'll have. Cheers for that, government).
-The authority that comes with the phrase "I used to teach your brother" (of course this is far less impressive than "I used to teach your father", but I've got a good few years until that one I hope).
-Possibly having a TLR (teaching and learning responsibility). I am interviewing for this next week.

Now that the "terrible twos" are behind me and I enter my third year, I thought I'd talk about the terrible twos that appear in mathematics. That is, the things in maths that are always taught together, but perhaps shouldn't be.

Word association test (please join in at home):

Area and ...

HCF and ...

Differentiation and ...

Volume and....

Here's what I think you said: perimeter, LCM, integration, and surface area. Am I right? If you didn't, then I'm guessing you're not a maths teacher.

These things are always taught in pairs. And these are all things that get confused.

My year 9 class are not completely stupid. But every single time they are asked to find the area of a shape, most of them give me the perimeter instead. Why?! I think it is fairly obvious that the word "area" means the amount of space inside the shape. I don't see how this can be confused with the length of the border. But students always get these confused.

Area and perimeter are always taught at the same time. I have heard many maths teachers say that they shouldn't be. They are two entirely different concepts, after all. If we taught them separately, would this confusion be avoided?

Similarly with HCF and LCM. My top-set students always get these confused. I think it's because they think that the HCF must be higher than the LCM, because of the name.

For me, the really interesting one is integration and differentiation. Obviously these are opposites. They're inverse operations, according to the Fundamental Theorem of Calculus. But when you think about what they actually do, they don't seem to be that linked at all. Finding the gradient and finding the area don't seem that similar. I think what many maths teachers do is teach differenriation, then teach un-differentiation, and announce that this is called integration, and then teach the application of integration to finding areas. I believe it should be the other way round: teach integration in its own right, and then discover that, holy sh*t, it's the opposite of differentiating! By the way, if your students swear in maths lessons it's a sign that you're doing something right.

Where do you stand on the area/perimeter: together or apart debate?

And congrats to all PGCE/GTP/PGDE teachers that have just qualified! Enjoy your NQT year!
And also congrats to all NQTs who have just passed their probation year! Enjoy your terrible twos!

Emma x x x

Happy Palindrome Day to Me

Today I am 8888 days old.

Enough said.




Emma x x x x x x x x

A Mathematician's Lament

Sorry it has been so long since I last wrote. Exam season is a very busy and stressful time for teachers as well as students! (Perhaps even more so). Now that the wonder of "release time" is upon us, I should be able to write more.

I stumbled upon a piece of writing called A Mathematician's Lament. I think it is one of the most beautiful pieces of writing I have ever read. As I was reading it, I found myself screaming (inside my head, I don't want to make a scene - I'm British remember) "that's exactly what I think!". It's like Paul Lockhart looked inside my head, stole my thoughts, and wrote them down but in a nicer order and with better punctuation.

I urge you to read it if you can spare the time. At least read page one. For those of you who really can't be bothered, here is a TL;DR summary for you:

*Mathematics is an art.

*Mathematics should be taught the same way that music and art are taught.

*Maths is not about acquiring skills that you might apply "one day".

*Much like music lovers have certain types of music they like and certain types they dislike, mathematicians can dislike some types of maths. It is a matter of taste.

*Maths is about imagination, you do not deal with real things because real things are inaccurate and messy; maths is simple and beautiful.

My favourite quote:

"there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics."

I have shared this with my department and I am considering sharing it with my students as well. I think it is especially important to share with year 11s who are looking to take A Level maths. Those students who have thought they were good at maths all these years because they got all the answers right might not realise that rote learning won't get them very far in A level.


Read the whole thing and then tell me what you think in the comments below.

Emma x x x

Factorising: a Divisive Topic

See what I did there?

We had a faculty meeting last week, which can only mean one thing: a raging argument about the best way to teach something. We're an opinionated bunch.

The topic in question was quadratic equations. More specifically, factorising, although we touched on solving them in general.

What method would you use to factorise this?
6x² + 26x + 20 = 0


The mathematician answer, of course, is I wouldn't, I would just use the quadratic formula.
The more observant amongst you will have divided by two first and then found the answer quite easily.
For the purpose of this exercise, dividing by two is not allowed, and neither is taking two out as a factor in the beginning.

This is the method that was put up on the board for us to discuss:

6x² + 26x + 20 = 0
20 * 6 = 120
Two factors of 120 which add to make 26 are 20 and 6.
6x² + 20x + 6x + 20 = 0
2x (3x + 10) + 2(3x + 10) = 0
(2x + 2)(3x + 10) = 0

I would estimate that about a third of our faculty looked at this and immediately said, yes, that's how I teach it. Another third said, I know that method but I don't really teach it, and the last third said, I've never understood that method, I think it's stupid.

OK, that last group was probably smaller than the other two, but it felt bigger because I was in it.

I hate this method! It took me ages to get my head round it. I finally got round to proving it to myself so I feel happier, but there is no way I would teach this to a class, because first I'd have to prove it to them to show them where it comes from. I don't think they'd be able to use this method without understanding how it works.

But apparently, I am wrong. There are a lot of students who are successful using this method, and many teachers swear by it. The alternative, trial and error kind of method that I use is obviously quite annoying (hence why in those cases where the coefficient of the x squared term is not 1 or a prime, I would never bother factorising) and it probably puts some students off.

So here's the big question: should we teach students a method they do not understand if it makes getting the answer faster?

Primary schools have moved away from teaching un-understandable methods like column subtraction, bus-stop division, long multiplication in columns, etc towards using methods that students can understand how they work, like chunking, partitioning, open numberlines, the grid method, etc. Surely we should be doing the same higher up the key stages?

I would love to know your opinions on this. If you're a maths teacher, which method do you teach? If you're not, what method were you taught when you were in school?

Emma x x x

Appendix: Explanation of above method

ax² + bx + c
= (kx + l)(mx + n)
=kmx² + knx + lmx + ln
=kmx² + (kn + lm)x + ln

So we are looking for k, l, m and n such that:
km = a
kn + lm = b
ln = c

Aim to split b up into kn + lm
ac = kmln = kn * lm
So we can factorise ac such that the two factors sum to make b.
Working forwards:
ax² + bx + c
= kmx² + knx + lmx + ln
=kx(mx + n) + l(mx + n)
= (kx + l)(mx + n)

Minus versus Negative: Some Mathematical Grammar

Ooh, today you're getting a discussion of maths and grammar, aren't you lucky?

We had a "moderation day" last week. It's an INSET day where teachers moderate their coursework. As you can probably imagine, the maths department was incredibly swamped that day. NOT!

Maths doesn't have coursework, so theoretically, we didn't have anything to do. In practice, however, we had absolutely loads to do, because we are still teachers, and a teacher's work is never done. That sentence has far too many commas. Should I really be writing a post about grammar? You can always put bad grammar down to style, can't you? My style is to use too many commas. Like, this.

Anyway, the maths department decided to take a long lunch on this moderation day, and went to a popular pizza restaurant armed with multiple two-for-one codes. The joke "how many maths teachers does it take to split a restaurant bill?" comes to mind.

We spent most of the meal maths debating. This often happens to us. Luckily the restaurant was almost empty, or it could have been quite embarrassing. We were scrawling equations on napkins using board markers (the only pen we ever have on us) by the end.

The subject of the debate? Should the word "minus" only ever be used as a verb?

Read the following sentence out loud: The weather today is -2 degrees. How did you say it? Did you say "minus 2 degrees"? Or did you say "negative 2"? I would guess that if you are from the UK you probably said minus. I know that's what I say. It's definitely what the weather people say on TV.

Now read this out loud: x - 5 = 7. Did you say "minus" again? You might have said "take-away", possibly "subtract", or even "less".  I would say minus, probably because this is what all of my maths teachers used to say to me.

Third test: what rule were you told about why -5 x -2 = 10 not -10? Say this rule out loud. Did you just say something like "a minus times a minus makes a plus" or "two minuses makes a plus"?

Can you see a slight issue? We're using "minus" as an adjective, meaning negative, and we're also using it as a verb* meaning subtract. And thirdly, we're using it as a noun when we say "a minus" meaning a number less than zero.

We're all quite comfortable with this word having several meanings. But what about students? When they first learn about "directed numbers" (as they're known), does this odd quirk of English confuse them?

I can see why some might think this. I understand that the language of mathematics should be used very carefully. I've always been very interested in grammar, which was why I did A level French (which, incidentally, everyone thought was weird: most of my teachers assumed I would be studying maths and the three sciences). At uni I took some modules that were about logic, which is pretty much just another word for language. I've taught enough EAL students to know that you need to choose your words carefully. But to be honest... I'm not completely convinced.

In a "number sentence" (a wonderful expression, thank you primary school teachers), I will always pronounce a dash (or hyphen, or em dash, or en dash) as "minus". Let me tell you why: that little symbol represents two things at once; it's the operation of taking away, and it's also to indicate something that is being negated (notice that both of these things are actions: I'm not saying it represents a negative number, I'm saying it represents something that is being negated). You absolutely need this symbol to represent both at once, because you want to be able to swap between the two meanings depending on how you feel.

Take this example:

3 - 4 (x - 7) = 10

If you wanted to solve the above equation, there are a few ways you could do it. Before reading ahead, please solve it.

How did you treat the minus before the 4? Did you see it as indicating something you're taking away from 3? Or did you see it as attached to the 4, making it a negative 4?

Did you do this:
3 - [ 4x - 28] = 10 (expanding the bracket with 4 as the multiplier)
31 - 4x = 10
etc

or this:
3 [- 4x -- 28] = 10 (expanding the bracket with -4 as the multiplier)
3 - 4 x + 28 = 10
etc

Or something different?

Can you see that if I, as a teacher, had indicated in some way that the minus before the 4 was a negative symbol, the first method wouldn't really make sense? And if I had indicated it was a take away, the second method doesn't really make sense, because students aren't taught that subtraction follows the distributive law.

The duplicity of the minus is one of those mathematical things that makes sense when you are mathematically fluent. Just like in English, how we have words that look the same and sound the same but mean two different things. As a fluent speaker of English, I don't even notice these. Look, I just used one! Notice! I didn't have to think: wait, is this the verb to notice, or a kind of sign stuck on a wall? I just used the word. And guess what, when I learnt French, my professeurs didn't just remove all homophones from the syllabus so that, as a learner, I wouldn't get confused, they left them in, so that I could aim to become fluent. Why should maths teachers do this? Don't we want our students to become fluent in maths?

Yes, we should be careful with our language in maths lessons. We should make sure when we say "line" we don't mean "line segment". But we cannot protect our students from the difficult to understand bits. We need to expose them to these things.

What do you think?

Emma x x x

*Technically it is a preposition rather than a verb. But these days we use it as a verb, saying things like "minusing" and "you minus the five from both sides". I know technically these uses are wrong, but it's what we say. Just like how we say "timesing" and "timesed" because we use the word "times" as a synonym for multiply now.