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'Largest Product' printed from http://nrich.maths.org/
Why do 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.
Part of the problem is to realise that there is, in fact, a
question to consider.
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.
How shall we start to make some progress with this challenge
Can you generalise your results somehow?
What is the same and what is different
about the solution if the $10$ is changed to another number?
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