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