Rectangle ratios inquiry 1

The prompt

Mathematical inquiry processes: Interpret; explore; test particular cases; reason; relate to context. Conceptual field of inquiry: Comparison and simplification of ratios; equivalent ratios; similarity; algebraic manipulation.

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).

Students can be convinced of the fact that 2:√2 = √2:1 by dividing both terms in the first ratio by √2.

Extending the inquiry

Once the class has gone through the first phase of the inquiry, students might be expected to work more independently to show that the ratios in the following two cases are a:b:c = 3:√3:1 and a:b:c = 4:√4:1 respectively.

The illustration (right) shows the approach of a year 9 student.

From abstract to concrete

The inquiry could become more concrete if the teacher decides to introduce A-sized paper. The ratio of the length and width of each size is the same as for the rectangles in the prompt.

Inquiry 1

Students might divide the length by the width to verify the ratio for all sizes from A0 to A8. As they do so, they could record the degree of accuracy (in decimal places) compared to √2, which is 1.4142 (rounded to four decimal places). For example, the calculation for A0 is 1189 ÷ 841 = 1.413793, which is accurate to 2 decimal places. This line of inquiry ends with a discussion about why the integer dimensions of A-sized paper can never give the exact value of √2.

Inquiry 2

Alternatively, students could work out the size of an A0 sheet given the dimensions of, for example, an A8 sheet. The A8 sheet is 74 mm by 52 mm, so the A7 sheet is (74 x 1.4142) mm by 74 mm. One question that arises in this pathway of the inquiry relates to the percentage error caused by rounding √2 in the calculations. How close are the dimensions of the students' A0 sheet to the ones given in the diagram below?