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The prompt
The prompt
Mathematical inquiry processes: Explore, verify, generalise and prove; extend to other curves. Conceptual field of inquiry: Polar coordinates and curves; integration; areas of enclosed regions.
Andrew Blair, a teacher of mathematics, designed the prompt for his year 13 A-level Further Maths class. The statement is false if the ambiguous phrase 'each time n increases' is defined more precisely as increasing by one each time.
However, there is a pattern if you consider two overlapping petals. For n = 1, 2, 3, and 4, the ratio of the areas of the regions contained within both petals is 12:6:4:3. Thus, in the first case the area halves but, in the next two cases, the areas are two-thirds and three-quarters of the preceding one.
The class had already studied integration to find the area under a curve on the Cartesian plane, coordinates on a polar grid, and sketching curves of polar equations. These are their questions in the question, notice, wonder phase of the inquiry:
How do you find the area of the rose petals?
For n = 1, the polar curves look like this:
If the region is contained within both curves, it must mean the overlap.
For n = 2 there are four petals, three petals when n = 3, eight when n = 4, 5 when n = 5 and then 12, 7, 16, 9 and so on. How can the area follow a regular pattern when the number of petals has an irregular pattern?
As the value on n increases, the width of the petals decreases so the area of the overlapping region decreases as well. As one curve can be transformed onto the other, there will never be an exact match of one petal over another.
How do you find the area of the overlapping region? Is it the same as for a Cartesian graph when you subtract one area from another?
Are we talking about the absolute area when r > 0 and r < 0? Is the 'negative' area included?
Does the overlap on normal (Cartesian) sine and cosine graphs halve as n increases?
If n tends to zero, does A tend to zero?
Preparing for inquiry
Before students start to explore, the class, under the teachers guidance, addressed two of the initial questions:
(1) Decide to which region or regions the prompt refers
In response to the question about the area following a regular pattern when the number of petals do not, the class agreed to restrict the focus of the inquiry to the region contained within the first petal (anticlockwise from the initial line) of both curves. As each curve is a transformation of the other, the region can be split into two congruent shapes with a line of reflection from the pole to the point of intersection of the two curves.
(2) Explain the integration formula
To answer the question about finding the area contained within polar curves the teacher should be prepared to explain the formula. In doing so, the teacher provides relevant knowledge when it is necessary, meaningful and connected - necessary to enter the field of inquiry, meaningful in the context of the inquiry, and connected to students' prior knowledge.
Acknowledgement
The inspiration for the prompt came from an investigation on Integral Maths.
January 2025
Lines of inquiry
Lines of inquiry
1. Exploration and proof
After the initial responses to the prompt, the students selected the regulatory cards Test different types of cases, Decide if the prompt is always true and Prove a generalisation. Their aim was to find the areas of the regions for n = 1, 2, 3, 4, ... , and then verify the prompt is true or make a different generalisation.
The class found that the areas in the first two cases are (π - 2)/8 and (π - 2)/16 (see below for illustrations of students' working on the exploration sheet). Even though the second area is half the first, the class was reluctant to accept the truth of the prompt without verifying other cases.
The third and fourth cases gave rise to a new generalisation: the area is in the form (π - 2)/8n. Members of the class went on to prove the generalisation. The picture shows a student's proof and the slides contain a more formal version.
2. Other curves
Another line of inquiry involves exploring curves that have equations in the form r = p + q cos(x) and r = p + q sin(x) where p = q, q < p < 2q, and p = 2q. Such equations form cardioids and limacons. See the area of the region when p = q = 1 in the student's working (pictured). A proof for the general case is given below. The inquiry can be extended further to consider lemniscates.
3. Different values of n
Using different values of n makes the inquiry more complex. As one curve is no longer a rotation of the other, the line from the pole to the point of intersection of the curves splits the region into two different shapes. It follows that there is no longer a line of reflection and the areas of both shapes have to be worked out separately (see the slides for an example).
Furthermore, as the regions contained within both curves are not all the same between 0 and 2π, the class might reconsider the decision to limit the inquiry to just the first region anti-clockwise from the initial line.
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