### Why do this problem?

This problem provides an interesting context in which students can
apply algebraic techniques and ideas about modular
arithmetic. It also gives them a taste of an area of number
theory that they might study if they go beyond the school
curriculum.

### Possible approaches

"Mathematicians have been interested in which numbers can be
written as a sum of square numbers. Here are the numbers that
we can make as a sum of two squares."

"There don't seem to be any obvious patterns here. But the
numbers are only in ten columns because we're used to grids like
this. Perhaps we should try a different number of columns
instead, like nine."

"Can we see any patterns this time?"

[Students might notice the two empty vertical columns.
They might also notice a diagonal pattern (top right to bottom
left). Suggest that this diagonal pattern could become
vertical if we make each row one shorter.]

Hand out

this copy
of the grid and give students some time, working in pairs, to
look for patterns, make predictions, and explain those
predictions. You might want to encourage students to start by
looking at the three completely empty columns.

Possible prompts if students are having difficulties providing
convincing/rigorous explanations:

In which columns do the square numbers appear?

In which columns do the squares of even numbers appear?
Can you explain why?

And the squares of odd numbers? Can you explain
why?

How can we describe the numbers in a particular
column?

Bring the class together to pool ideas, and then offer

this grid with sums of
three squares for further investigation. Some students might
also like to consider what will happen when we add four
squares.

### Possible extension

Suggest that students look for patterns that they can explain
in the nine-column grid.

Students could also experiment with grids with different
numbers of columns.

### Possible support

Ensure that students have worked on

What Numbers Can We
Make?