Problem Teachers' Notes Hint Solution

This problem can be tackled at very different intellectual levels. It is an excellent context to practise adding and multiplying with decimals while trying to solve a problem.

Ask the students for various sets of numbers that add to 10 (exclude negatives!). Ask students to work out the products of some of these sets - students might demonstrate the methods on the board to help the group remember the algorithms.

Present the problem for the lesson - to find the biggest product, for numbers that add to $10$, to keep a sensible record of things that they have tried and what did and didn't work out.

After some time working, students could feed back on successful strategies they are using, and the class 'best' could be put up on the board for others to beat. Make sure that students know that there are ways of adapting the problem if they feel that they are making no further progress.

What about the sum being $0.5$ or $1.1$ or $-8$ etc., how would the strategy/solution change?

Students often work by reasoning from a few 'spot' values and showing that improvement occurs as they adjust towards a particular value, which they therefore declare to be the optimum. It is certainly good to draw attention to the logical possibility that an even better 'local maximum' might exist somewhere else. And also, if they are up to it, that adjustments are always incremental so how can we be sure that we haven't jumped right over an interval which contains something important.

Can we reason that this function is continuous ?

With students who aren't ready for working with these decimals, perhaps consider integer solutions only for various sums.