## Thursday, 28 October 2010

### Change for £5 solution

Why is the question I asked in the previous post a good example of a mathematical problem for primary school children? Here are some reasons.

a) It is not answered simply by applying a routine process or algorithm, so it is not just an exercise or a practice question – for the solver there is a gap between the givens and the goal, and it is not immediately obvious how to fill that gap, how to move from the given information to the goal of having enough change to pay the Dartford toll.
b) It calls upon a range of mathematical knowledge, concepts and skills – such as mental multiplication and subtraction of money, knowledge of coinage and the calculation of the various combinations of coins to make up a given amount of change.
c) All this mathematics is accessible to, say, a Year 5 or 6 child – the challenge is to determine what you have to do.
d) The solution is multi-step, requiring the solver to determine what steps to take and in what order.
e) As is often the case with real-world problems, there is more than one solution! Let's not always ask questions with only one solution in mathematics.
f) The initial problem gives rise to other similar problems. For example, what if I had only a £10 note? Would the solutions to the £5 question still work?
g) It is a genuine, real-life problem!

The solution, incidentally, is that I could buy either 3 stamps (which is what I did) or 8 stamps with my £5 note and guarantee getting £1.50 in the change.

I calculated mentally that buying 3 stamps at 41p each would give me £3.77 in change. I then have to consider all the different ways this £3.77 could be made up of various coins and check that in each combination I would have coins that make up £1.50! I'll leave the reader to confirm that this is the case.

The same is true if I buy 8 stamps, for which I would get £1.72 in change.

For any other number of stamps you could get change from which you cannot make up exactly £1.50 in coins. For example, if you buy 6 stamps the change could be a £2 coin, a 50p and two 2p coins.

However, to be honest, the post office could undermine my cunning plan by giving me all the change in 1p and 2p coins – which are not acceptable at the Dartford crossing toll booth. And, in the latest Spending Review, the government has announced plans to increase the toll in 2011 and 2012, changing the givens of the problem significantly.