Sector of a circle inquiry
The prompt
Mathematical inquiry processes: Generate examples; verify; explore other cases; generalise and prove. Conceptual field of inquiry: Areas of a circle and a sector; radius; ratio.
Tariq Rasool, a teacher of mathematics at Drayton Manor High School (Ealing, UK) designed the prompt after one of his year 11 students attempted this question as part of an assessment.
The student mistakenly thought the radius of circle B was 7.5 cm and went on to show that the areas of shapes A and B are equal. Tariq was intrigued. Would the areas of all circles of radius x and quarter circles of radius 2x be equal? Tariq quickly established that they would.
He set about designing an inquiry prompt for his year 8 mixed attainment class who were learning about the area of a circle. The prompt would act as a bridge to thinking about the area of a sector.
Tariq considered different versions of the prompt. Firstly, he wanted to include the word 'quadrant' as new vocabulary to his year 8 class.
Secondly, Tariq decided to mark the right angle with a square even though the inclusion of 'quadrant' made it unnecessary. In doing so, Tariq aimed to focus attention on the angle in the sector as a precursor to changing the angle later in the inquiry.
Question, notice, and wonder
Tariq asked students to think, pair, share in the initial phase of the inquiry. Here we present the students' questions and observations:
What does the O mean? Is it the centre of the circle?
There is a right angle in the quadrant.
Is the quadrant a quarter of a circle?
The first x is the radius of the circle.
Is there a formula for the area of a quadrant like there is for a circle?
x could stand for any number.
To find the area of a quadrant, imagine it is a circle and divide by four.
If the quadrant is part of a circle, its radius is 2x.
The statement could be true because the overlap of the shapes is a quarter of the circle and it looks like a quarter of the quadrant.
In The teacher's response we outline how the teacher might respond to these different types of question and observation.
April 2024
Lines of inquiry
1. Generate examples to verify the prompt is true
The structured inquiry starts with students counting squares to estimate the areas of the shapes or using the formula to calculate the areas in terms of π.
They generate examples by using different values for x and, in this way, become convinced that the statement in the prompt is true.
2. Generalise and prove
However, the teacher reminds the class that generating more examples does not constitute a mathematical proof that the statement is always true. At best, the examples allow students to form a generalisation. The algebraic proof (illustrated) is accessible to most secondary school students.
3. Change the prompt
A new line of inquiry involves changing the angle in the sector to determine the new ratio between the lengths of the radii. By using factors of 360 for the angle students can follow the same approach as they did for the quadrant - i.e. divide by three for 120o.
When the angle is 180o, for example, the ratio of the radius of the circle to the radius of the semicircle is 1:√2. Students can use an inductive approach by exploring different lengths of radii or the deductive approach used to prove the case of the quadrant.
The ratios for a selection of cases are shown in the table. For proofs of the first two cases (120o and 180o) see the PowerPoint.
4. Derive the general form
If students have managed to prove particular cases, then they could attempt to derive the general form for any angle (right). This could be attempted individually or co-constructed under the teacher's guidance.
5. Extend into three dimensions
If the radius of the sphere is x and the volumes of the sphere and hemisphere are equal, what is the radius of the hemisphere? Is the pattern the same as for two-dimensional shapes?
