Rectangle ratios inquiry 2

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; manipulation of algebraic expressions.

The prompt follows on from the rectangle ratios 1 inquiry. This time students might start the inquiry more independently, but will need support when it comes to the application of algebra (which includes the use of the quadratic formula). The prompt is made up of a square and rectangle, which together form a large rectangle. The ratio of the length to the width of the large rectangle equals the ratio of length to width of the smaller rectangle. The ratio in the prompt is satisfied by the proportions of the golden rectangle whose length and width are in the golden ratio (φ or phi).

φ = (1 + √5)/2 = 1.618 (accurate to 3 decimal places)

The inquiry starts again with a period of exploration. The prompt might seem easier than the last one because there are only two variables (a and b), although students still have to consider three lengths. Perhaps they start with the same suggestion as before, a = 8 and b = 4. However, that leads to a + b = 12 and gives (a + b):a = 12:8 = 3:2 and a:b = 8:4 = 4:2. We need to increase b to, say, 5. Now (a + b):a = 13:8 = 65:40 and a:b = 8:5 = 64:40. The next step is to substitute 5.1 or 4.9 for b. It is unlikely that the students will reach this position so quickly in the classroom, but the example illustrates how students move towards the value of phi as they adjust the values of a and b.

In the second phase, the inquiry could take on a more concrete approach (see below). Alternatively, the teacher might lead students in deriving phi algebraically, co-constructing as many of the steps as possible. Students can then apply the same steps to similar cases (see 'Extending the line of inquiry').

Extending the line of 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 = [(1 + √9)/2]:1 and a:b = [(1 + √13)/2]:1 respectively. Students should attempt to explain why, as they add rectangles to the design, the number in the square root increases by 4.

From abstract to concrete

Inquiry 1: Constructing a golden rectangle

Students could construct a golden rectangle by following the instructions below and then calculate the lengths of their construction lines.

1. Draw square ABCD with a side length of 1 unit.

2. Draw a line (EF) from the mid-point of AB to the mid-point of CD.

3. Draw the line EB.

4. Extend DC.

5. With EB as the radius of a circle, centre E, draw an arc BG where G is the point of intersection of the arc and the line extended from C.

6. Construct a perpendicular line GH from G.

7. Extend AB to the perpendicular line at H.

DE = EC = 1/2.

Using the right-angled triangle BCE, length EB is √(5)/2 (by Pythagoras' Theorem)

EG = EB (radii of the same circle)

DG = DE + EG = 1/2 + √(5)/2 = (1 + √5)/2

Inquiry 2: The golden rectangle in architecture and art

The Babylonians are said to have known about the golden rectangle in the ninth century BC, although it was not until 400 years later that Phidias, a Greek sculptor and mathematician, studied φ systematically. Ever since, there have been many examples of its use in architecture and art. Indeed, the dimensions of the golden rectangle are supposed to be pleasing to the eye. Two examples of its appearance in architecture and art are, firstly, in the dimensions of the plan, prayer space and minaret of the Great Mosque of Kairouan and, secondly, in the composition of Georges Seurat's 'Bathers at Asnières' (see pictures below). Students could research the use of the golden rectangle elsewhere in architecture and art and then extend their inquiry into the golden ratio's appearance in nature.