Dan Meyer recently published a blog post on how to teach the idea that any number raised to the power of zero is simply one. This is an important result that crops up in many different areas and so it’s critical that students know it.
Meyer asked us to choose between two strategies. The first he described as ‘teaching tricks’ and I agree that it’s not very enlightening. The presenter of the video simply tells you that anything raised to the power of zero is one. And that’s it.
Meyer is keen on a different pattern-based activity that he describes as ‘sensemaking’. I think this approach is better than the first but it still doesn’t really explain anything. In my view, it’s more of a pattern-based trick. I’m not quite sure what ‘sensemaking’ means – which is ironic – but I thought this would be a good cue to explain the way that I would teach the concept.
The first important point is that I think this needs to be taught over and over again to students of different ages, whenever exponentials arise. So I’ve taught it in Years 8, 9, 10, 11 and 12. With older students studying Maths Methods – our abstract maths course – I would rely more on pronumerals in the explanation although I would still always illustrate with natural numbers.
Secondly, this would be a small part of my explanation of index laws rather than something that sits on its own. I just see the power of zero as an interesting result.
And I haven’t worked any of this out myself – it comes from an amalgam of textbook explanations, discussions with colleagues and so on. So this is not the patent Ashman method.
As ever with mathematics, there are some things that students need to understand in order to grasp the explanation. I won’t outline how I would teach these underpinning ideas because that might make the post rather long. Suffice it to say that there would be a process like the one I’m about to describe. It would be critical that they understood the following three points and so I would check this through questioning:
1. Fractions can be composed and decomposed in the following way (I might also discuss non-examples involving sums):
2. Indices are a shorthand for repeated multiplication of the same term:
3. Any number divided by itself is one – I usually remind students of this by saying something like, “How many fives [point to denominator] are there in five [point to numerator]”:
This enables us to work out some laws for indices. I would explicitly lead students through one of these and I would pepper the exposition with questions to the students as I went along. I don’t ask for them to raise their hands – as far as my students are concerned, I call on them randomly – and I tend to ask them to work out any of the component parts that relate to prior knowledge.
I would start by looking at something like this result:
With younger students, I would give a couple more examples with natural numbers and with older ones I might use a pronumeral such as ‘a’. To make it simple, I’d have more repeated terms in the numerator than denominator. The fact that we can then partner up each term on the bottom with one on the top means that, in order to get the final power, we can do a subtraction of the two original powers. This is similar to one of the ways that subtraction is explained to students in earlier number work.
As an aside, you might think that I should talk about ‘canceling’ here and I certainly introduce the shorthand of canceling at some point. However, I tend to persist with the kind of decomposition I’ve used above for a quite a while so that students can see where canceling comes from – even if I fade it out and only go through the whole process every third or fourth example. At this point, I might not rewrite as separate quotients but still talk of ‘three divided by three is one’ as I draw my cancelling lines through the terms.
I don’t think you can avoid the error of over-generalisation that many students eventually make that leads them to cancelling the two in an expression like this:
However, when they do this you can return them to decomposition as an explanation of why that doesn’t work.
Back to the original point, we have now derived a rule that looks something like this:
This is incredibly powerful and should also lead to a discussion and some examples and exercises that include negative powers, but that’s not the objective here.
Armed with this rule, we can reverse engineer a zero power. How can we construct one? Well, ‘m‘ and ‘n‘ would need to be equal:
If the students really need convincing then you can give a few more examples just to show that it always comes out as one.
I also use index laws to show the equivalence of fractional powers and roots.