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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Multiply the Addition Square

### Why do this
problem?

This problem is a good one for building on the learners'
ability to recognise number properties and reason about numbers. It
is easy for all at the start as it only requires simple
multiplication and subtraction and gives the satisfaction of
finding a pattern. If the results are generalised algebraically it
can prove a real challenge.

### Possible approach

### Key questions

### Possible extension

Learners could try to generalise, not only the results for $3
\times 3$ squares but also those for all squares on this grid.
Alternatively, they could try the same procedure with rectangles
rather than squares or use other grids such as multiplication
squares.

### Possible support

Suggest using a calculator for the multiplication and subtraction
or, alternatively, switch to
Diagonal Sums which is a similar, but easier, problem.

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

You could start by posing the problem to the whole group on a
computer or using a large-sized addition square.

After this learners could work in pairs on the problem so that
they are able to talk through their ideas with a partner. This
sheet gives two copies of the addition square which can be used for
rough work.

Many learners will tend to rush on and try $2 \times 2$, $4
\times 4$ and other squares and although an interesting pattern can
be found doing this, it may stop them from trying to generalise the
$3 \times 3$ example. Therefore, it may be wise to stop everyone at
an appropriate point and show all who need the help how the numbers
can always be named $n, n + 2$ and $n + 4$. (Alternatively, they
could be $n - 2$, $n$ and $n + 2$.)

At the end of the lesson learners could discuss the different
patterns they have discovered and the various generalisations
made.

How do the four numbers compare in size?

Try comparing the number in the top left corner of a square
with the other three numbers.

So what happens when these numbers are multiplied in the way
described?

Can you see a pattern there?

What sort of numbers are these?