Equation of a circle inquiry
Mathematical inquiry processes: Verify a particular case; test other cases; generalise and prove. Conceptual field of inquiry: Equation of a circle; equations of perpendicular lines; mid-points of straight lines; solving simultaneous equations.
Kwadwo Okyere, a Lead Practitioner in London (UK), devised the prompt for his year 12 A-level class. The students had previously learned about:
Circle theorems (including the properties of chords);
The gradients of perpendicular lines; and
The standard and general forms of the equation of a circle.
Kwadwo explained that he wanted to see how the students worked on a prompt that was just above their current level of understanding. They had all the mathematical tools required to decide if the prompt was true or not, but as Professor Schoenfeld says here, “It’s not only what you know, but how you use it (if at all) that matters.”
Kwadwo encouraged the class to start with any three points on the coordinate grid to see if it was possible to find the equation of a circle that passed through all of them. If students required more structure, he suggested A(2,1), B(6,5) and C(10,3). One pair of students joined up the points to form a triangle and realised that the triangle was inscribed inside the circle. At this point, Kwadwo told them that such a circle, if it existed, was called a 'circumcircle'.
Another pair, following the lead of the first, realised the sides of the triangle were chords of the circle. They checked the circle theorems to remind themselves that the perpendicular bisector of a chord passes through the centre of a circle. They reasoned that if they knew the coordinates of the centre (or 'circumcentre'), then they could find the equation of the circle.
Now the whole class was engaged in the discussion about points A, B and C. The suggestion of finding a perpendicular bisector by construction was rejected for a more precise approach. If the equation of the chord and its midpoint were known, then the students could deduce the equation of its perpendicular bisector. At this point, Kwadwo split the class into three groups, giving each one a chord to work on and present their findings (see 'Mathematical notes' below).
Each group reported back. Having deduced the equations, one student reminded the class that all of them would pass through the centre of the circle. The centre could be found at their point of intersection. This time it was another student who divided up the tasks. Each of the three groups took responsibility for finding the point of intersection of two lines.
With the centre located, one group found the length of the radius and presented the equation of the unique circle that passes through A, B and C to their peers. To verify the solution, Kwadwo demonstrated on the Desmos graphing calculator.
To end the collaborative inquiry, Kwadwo asked the students to reflect on the steps they had taken and wrote them on the board as a scaffold for individual inquiries. The students now chose their own three coordinates, which, Kwadwo stipulated, had to be in different quadrants.
The teacher of a year 12 class in an academy in the south-east of England sent three inquiries (below) to the Inquiry Maths website. He reports that the class had been studying the equation of a circle as part of their A-level course. Students discussed the prompt before deciding on the steps required to find the unique equation:
Find the midpoints of two chords.
Find the gradient of the chords.
Using the the negative reciprocal of the gradient, find the perpendicular bisector of one of the chords.
Repeat the process for the other chord.
Find the point of intersection of the two perpendicular bisectors (the centre of the circle).
Work out the length of the radius.
The teacher had prepared a list of three coordinates for each of the 24 students in the class. As students completed their working, a few individuals began to realise that pairs of students had been given the same coordinates.
They began to pair up to check their equations. Those pairs who had different equations compared each others' working, trying to identify inaccuracies. Those that had the same equations discussed the similarities and differences in the steps they had taken.
At the end of the inquiry, the teacher asked the pairs who could not resolve their differences to type the two equations into the Desmos graphing package, which was displayed on the board. They then checked whether the circles went through the coordinates they had been given.
Find the equation of the unique circle that passes through A(2,1), B(6,5) and C(10,3).