“Why do we need to learn this? What use is it?” This is pupils’ repeated refrain when it comes to maths and it’s not always easy to answer them. Here, however, is what you could say if they ask why they’re learning about negative numbers.

It may be worth mentioning temperature. Why do we have a temperature scale with temperatures below zero? Why is zero where it is? Is it a good idea to discuss the rather arbitrary labelling on the Celsius scale? It is helpful to label the freezing and boiling points of water as 0 and 100, respectively, and you can construct a scale from there. But perhaps it’s worth looking at other scales for temperature as well. Could we design a scale that doesn’t need negative numbers? Do we find negative numbers a useful idea for temperature?

Or you could look at money. For some children, the idea of borrowing money and paying it back might help them to ground negative numbers in something a little less abstract.

There may also be a useful link to geography: sea levels are a reason to use negative numbers, for example. Show some cross sections from maps to emphasise the reason for negative numbers and help pupils to visualise journeys that start below the sea and finish above it.

You could also come over all philosophical and ask your pupils: “What was the tallest mountain in the world before Mount Everest was discovered?” The answer, of course, is that Mount Everest has always been the tallest mountain in the world, which raises difficult questions for mathematicians. Should we think of certain aspects of mathematics as having been invented and not existing until someone created them? Or did they always exist but simply weren’t discovered yet? If it’s the latter, negative numbers - along with fractions and decimals and all the other numbers - have always existed and we just uncovered them. Does this inspire a little more awe and wonder than our usual functional and real-world approach?

Finally, there is a mathematical reason for negative numbers, to do with completeness. In the beginning, we want to count things, and we do lots of addition and multiplication, subtraction and division. Mathematically, we would like to have fractions so we can solve more multiplication problems, and negative numbers so we can solve more subtraction problems. If we’re tempted to say “You can’t take away 5 from 3”, negative numbers give us a great way to solve that - and to offer a whole new universe of mathematics. And then we never have to say “can’t” in a maths lesson again.

Peter Hall is an advanced skills teacher of maths at Beacon Community College, Crowborough, East Sussex

What else?

Lewi Hylton’s PowerPoint provides an introduction to negative numbers.

bit.lyNegNumbers

Help pupils to count their coins with the TES Money collection. bit.lyCountingCoins.