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Why do this problem?

This problem is a context for systematic number work, geometrical thinking and problem solving.

It is an excellent example of a situation where the thinking involved in analysing one rectangle can be applied directly to other rectangles. These transferable insights about the structure of the problem can then be expressed as algebraic statements about all rectangles.

As students enter, display the $20$ and $50$ diagram, asking how many squares there are. It may be appropriate to give the answers then ask pairs to come and explain - one to talk, and one to write/draw/record on the board. Well laid out number work will help with the algebra later, so the students' boardwork should prompt more suggestions about how to record working for this problem.

Present the problem, give students an opportunity to share first ideas. Several approaches (working backwards, trial and error, building up from smaller ones, systematic searching) might be suggested, and advantages/disadvantages discussed.

Encourage students to compare results with peers, and to resolve discrepancies by mathematical argument, rather than relying on the teacher's spreadsheet (see below). It might be useful to gather the results of the students as they work, to help them to see patterns and encourage them to conjecture the results for other rectangles.

With a group who have not moved towards algebra, a final plenary could ask for observations about the rectangles, and discuss how each can be expressed algebraically.

Is there an obvious rectangle which contains $100$ squares?

How might you organise a search for rectangles with exactly $100$ squares?

Is what you're describing specific to this rectangle? how does it generalise?

Prove that you have all the rectangles.

Can you find an algebraic rule for the number of squares contained in an '$m \times m$' square? an '$m \times n$' rectangle?

For a given area, which rectangle gives the largest total number of squares? Can you show this in general?

If the original question didn't say $100$, what other numbers (under $100$) would give a problem with non-trivial solutions? Is there a pattern to these?

Set up a spreadsheet to calculate numerical solutions to these problems.

Struggling students could shade squares on worksheets (2nd sheet ) with lots of small copies of the rectangles. Encourage them to work systematically, in order to observe the structure, and then make conjectures about the numbers of the next size of square, or in the next rectangle.

Once they, either individually or as a group, have worked out the full counting they could be asked to do this activity: In a pair, each guess a rectangle which might have $40$ squares. Swap squares and each work out the total. Whose guess was closest to $40$? Between you, can you come up with a better guess?

This spreadsheet contains a calculator to work out the totals and also lists all of the possibilities; you might find this helpful for checking students' work quickly.