In the prompt we are told that the ratio of the length to the width of the large rectangle equals the ratio of length to width of the smaller rectangles. Moreover, the length of the bigger rectangle is twice the width of a smaller rectangle (a = 2c) and the width of the bigger rectangle equals the length of a smaller one. These constraints are satisfied by the square root of two - that is, both ratios equal √2:1 in their simplest forms.

Once students have understood the meaning of the prompt during the question and observation phase of the inquiry, they might explore by using a kind of iterative process. To start the process, the teacher might encourage the class to speculate about the lengths and widths.

For example, if a = 8 and b = 4, then a:b = 8:4 = 2:1 and b:c = 4:2. As a = 8 and c = 2, this does not satisfy the constraint that a = 2c. Let's try a = 8, b = 6 and c = 4, which gives a:b = 8:6 = 4:3 = 1:0.75 and b:c = 6:4 = 3:2 = 1:0.67 (accurate to 2 decimal places).

Alternatively, we could could compare the ratios using a multiplier and common part, which gives a:b = 16:12 and b:c = 18:12. This is closer, but we need to reduce b in the first ratio, say to b = 5.5. Now a:b = 8:5.5 and b:c = 5.5:4, which simplify to a:b = 32:22 and b:c = 30.25:22. Let's change b to 5.4 or 5.6 and so on.

Ultimately, the teacher uses algebraic reasoning, co-constructing as many of the steps as possible, to show that the solution involves an irrational number. In so doing, students encounter important algebraic manipulations. In the first two steps of the approach that follows, a ratio is converted into a fraction and then the fractions are 'cross multiplied'. Both manipulations will need an explanation before students develop the confidence to reproduce the procedure in extending this line of inquiry (see below).