Maths Teaching Idea #7
Working with my Y10s on straight lines and coordinate geometry we have been looking at equations of circles not centred at the origin and tangents to them. Lots of rich interconnected ideas appear and lots of opportunities for students to investigate.
We started in our first lesson after half term reminding ourselves of how to write an equation of a circle and how that related to the distance formula for two points. I asked students to write down the equation for three circles with given centres and radii. I could have varied these questions more, giving the equations themselves and asking for the radius and centre, giving diameters, giving areas etc. I could also have asked students for their centres and radii when setting the questions but part of me wanted circles that had one ordinate negative and both negative (structured variation).
We went through the answers to these and then I asked them to demonstrate whether (1,1) was inside, outside or on the circumference of the three circles. I asked them how they demonstrated this. We talked about how substitution into the circle formula and using the distance formula were functionally the same thing.
At this point I asked for them to be more creative: ‘Can you find a nice coordinate on each of the circles that is neither horizontally nor vertically aligned with the centre of the circle?’ I purposefully left nice undefined and let students attempt to find some points on the circle. There were a number of different methods for this with talk of Pythagorean Triples and vectors — we got some answers and discussed why one of the circles was not producing any nice results (it’s radius squared was not the sum of two squares).
After three students had shared a point for each of the circles I asked them to calculated the equation of the tangent at these points. I then asked them whether there were any other tangents parallel to the first. Revision of circle theorems and perpendicular gradients came about here. Finally the class had to find the area of the triangle formed by the tangent, the extended radius the x-axis. This was mainly to encourage them to sketch their diagrams.
The lesson allowed my students to revise and consolidate a series of interconnected ideas but more than this allowed them to take ownership and direct the lesson in some ways. Between the points on the circumference and the tangents I’m aware that I could have challenged students to verify that the points were indeed on the line rather than doing it myself — this would have given them practice in demonstrating this property (as (1,1) was not on any of the earlier circles).
In general, I was aware of how starting from an idea and an end point allowed the lesson to respond directly to learner response. In a sense each section of the lesson were ending or beginning with Hinge Questions that informed the next steps. I will look to set the students a quiz soon to see how confident they are with the content.
PS a note on the image at the top: this was an extension that came from a discussion with a teacher in my department. We considered what we could work out if we knew two lines were both tangential to a circle and whether that uniquely defined a circle. We realised that we needed to add something else (e.g. the circle area) to limit the number of solutions. This can also be increased to three lines being tangential, do we need any more info to uniquely define a circle here?
