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

Introduction to Mechanics: Lesson Plan and a Speed Riddle

First, the riddle:

You are driving along a 2-mile long bridge. You drive the first mile at an average speed of 30mph. You want your average speed for the whole bridge to be 60mph. What speed do you need to drive at for the second mile?

This is how I began my mechanics taster session with next year's AS students. Have you worked it out yet?

The first  answer I got from the class was 90mph. Is that what you think the answer is? If you do, you're wrong. Sorry.

(30+90)/2 = 60, but this is not the average. You would spend longer driving at 30mph than at 90mph, and this average does not take that into account.

Hint Number One:

What is the definition of "average speed"? Total distance covered divided by total time taken.

At this point the calculators started to come out. Pens and paper had yet to make an appearance. I then started to get some bizarre answers like 2mph. This is an example of why a calculator without pen and paper is a dangerous thing.

Hint Number Two:

To average 60mph, how long should the entire journey take?

Speed = distance/time. So 60 = 2/t. So t = 2/60 hours. This would be 2 minutes. Have you worked out the answer yet?

Hint Number Three:

How long have they been travelling so far?

Speed = distance/time. So 30 = 1/t. So t = 1/30 hours which is... 2 minutes.

So it is impossible.

Surprised? I was. It still don't really see how it can be impossible. Surely if you go fast enough you can catch up? It seems wrong somehow.

Now, onto the rest of the lesson.

I borrowed from the science prep room a mechanical weighing scale. The kind you have in your bathroom. Side note: what is it about science prep room technicians that makes them so formidable? I have never returned a borrowed item so promptly!

Anyway, I put the scales on the floor and stood on them. I asked the class to look at the number displayed. I then pointed out I was wearing heavy clothes and block heels and a big watch and I'd just eaten lunch. Then I asked the class what would happen to the number if I put my hands on the back of the chair in front whilst standing on them. They correctly told me the scales would say I've lost weight. Then I asked if there was a way for me to make the scales think I've gained weight. That's a more interesting question. I'll leave you to think about that.

Then I asked them to consider a person weighing themself whilst in a lift. What would happen to your weight when the lift is going up? The class was split almost exactly in half on this. Some thought you would gain weight because the lift is pushing the scales into your feet. Some said you would lose weight because the lift is pushing your feet off the scales. They were also divided over what happens when the lift is going down.

Mechanics is a very sciency bit of maths, and when there's a debate in science there's only one way to settle it: an experiment! So I simply declared, "To the lift!" The students were surprised and I think a little bit excited. We all went to the lift and took it in turns to go in groups of four up to the third floor then down to the ground floor, then back.

We returned to the classroom and discussed our findings and tried to explain them.I won't tell you the result of our experiment, but if you ever get the opportunity to try it out, please do!

Next I held up a tennis ball and a basket ball (borrowed from the PE department, who are a lot less scary than the lab techs) and told them we were going to do another experiment. I asked them which ball they thought would hit the floor first if we dropped them at the same time from the same height.

Again, the class was divided. We discussed mass, surface area, rigidness, air resistance... It was a good discussion. And then we gleefully left the classroom to do our experiment. Half of us went to the top floor (the third floor, or the fourth floor if you're American), and the rest went down to the bottom. Our school is kind of open so that from the top floor you can see all the way to the bottom if you lean over the balcony on the inside. This made it ideal. (The building won an award recently for its awesome architecture).

There were a few students in the corridor working on the computers or printing stuff so we drew a bit of an audience. And when the two balls hit the ground (at the same time? Well, that would be telling...) we definitely drew some attention to ourselves! It was loud.

So we went back to the classroom and discussed our findings. We talked about how the experiment was kind of rubbish because there were too many variables. So I told them we were going to watch a better experiment where these variables were controlled. That's when I showed them this clip from Brainiac. Sorry about the Arabic subtitles.

That concluded the lesson. I think it was a great way to introduce the mechanics module and give them a bit of a taster. Hopefully it has made them excited to start their A level maths in September!

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 New Way to Teach Dividing Fractions

How do you divide a fraction by another fraction?

For example, how would you do something like this:

My guess is you would flip the second fraction upside down and then multiply like so:

If you are a maths teacher, is this how you teach students?

Do you think your students understand why this method works? And, be honest, do you understand why it works?

Well today in the maths office at my academy one of my colleagues showed us a new method he'd thought of.

It works like this:

I think this is a little bit more intuitive.

My colleague got the idea from one of his year sevens who had answered this question without showing any working out:

The answer is quite obviously three. How many quarters are there in three quarters? Three, duh. But I am quite certain most of my A* students would perform the technique of flipping and timesing without even thinking.  My colleague was impressed that this student had used some common sense. He wondered whether the same idea could be applied to fractions with different denominators. It is a little bit less obvious that 21/28 divided by 20/28 is 21/20, but it's not entirely unbelievable. Whereas the "trick" of flipping and timesing can look a little bit like magic to some students.

I haven't tried teaching this method so I can't comment on its effectiveness yet. But as a mathematician it appeals to me. It's quite neat. And in case you were wondering, yes this works with algebraic fractions too.


If you're going to be teaching fractions soon, why not try this out? If you do, please let me know how it goes.

Do you think this is a good method?

Emma x x x