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Squares in Rectangles

Age 11 to 14
Challenge Level

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.

Possible approach

This printable worksheet may be useful: Squares in Rectangles.

Start by displaying the $20$ and $50$ diagram and asking how many squares there are. It may be appropriate to give the answers and then ask pairs to go to the front to 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 without 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 what the results will be for other rectangles.

With a group who have not moved towards algebra, a final plenary could ask for observations about families of rectangles (e.g. with heights of 2, or heights of 3, or...), and discuss how the number of squares can be expressed algebraically for each family.
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.


Key questions

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

Consider rectangles with a height of $2$ units, and increase their width by $1$ unit at a time.
What effect does this have on the total number of squares?
Can a rectangle with a height of $2$ units contain exactly $100$ squares?

What about rectangles with a height of $3, 4, 5, \ldots$?

Possible support

Struggling students could shade squares on Worksheet 1 and Worksheet 2 with lots of small copies of 3 by 4, 3 by 5 and 4 by 6 rectangles. Encourage them to work systematically, in order to observe the structure, and then make conjectures about the number of the next size of square, or in the next rectangle.

Encourage students to start by considering rectangles with a height of $2$ units, increase their width by $1$ unit at a time, and in a table, record the number of squares in different rectangles with a height of $2$ units.
Do they notice anything special?

Encourage them to use their results to decide whether a rectangle with a height of $2$ units can contain exactly $100$ squares.

What about rectangles with a height of $3, 4, 5, \ldots$?

Possible extension

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.