Here are some prompts and suggestions for questions you could use to raise awareness of the overarching ideas, connections, common issues and misconceptions in this resource.
| Overarching ideas in this resource | Questions teacher could ask |
|---|---|
| Multiple representations | When do an image and equation represent the same circle? |
| Visualising | What does the equation tell you about what the circle looks like? Can you estimate the values of the irrational numbers? |
| Organising, categorising and ordering objects |
Do any circles or equations look particularly special? Can you break down the problem into some smaller problems? Can you work on small groups of circles/equations? What criteria could you use to group the circles? [radius; relative position of centres: left/right, up/down, quadrants] If you estimate the irrational numbers, what level of accuracy would be helpful? |
| Conjecturing, Logic, Proof |
What have you assumed? How can you decide between various possibilities? Which circle or equation could be a good one to start with? (e.g. 2, 7 and 9) Which are the easiest to match? Hardest? Why? |
| Talking about mathematics |
What are you sure about? What would you like to know? How did you approach the problem? Have you got the same matching as other students? Can you convince someone else that you’ve matched the circles correctly? |
| Making connections | Questions teacher could ask |
|---|---|
| Different forms of equation of a circle |
Would it be helpful to have all the equations in the same form? What is a useful form for this problem? Why? |
| Linking geometry and algebra |
How can we compare the size of the circles? How can we compare the positions of the centres of the circles? What aspects of this problem are specific to circles? Which relate more generally to equations and curves? |
| Common issues or misconceptions | How might these be revealed? | Teacher input |
|---|---|---|
| No idea where to start | Suggest making table of centre and radius for each equation. | |
| Thinking that \((x+10)^{2}+(y+15)^{2}=4\pi^{2}\) has centre at \((10, 15)\) | Protesting that there is not a circle centred there. | |
| Not square rooting to get radius | Talking about very large circles. | Where do the squares in the standard equation come from? |
| Estimation/lack of scale | Students get a ruler out | |
| Circles always have to be represented as \((x-a)^{2}+(y-b)^{2}=r^{2}\) | Not realising 6,11 are circles | Think about expanding this form. What terms will/won’t you get? |
| The constant in the expanded form of the equation is the square of the radius |
Not being able to find a circle of appropriate radius. Recognising there are too many circles that aren’t drawn. Commenting that rearranging the equation will give a negative \(r^{2}.\) |
What is the constant term if you expand equation 1? Or suggest another circle from the preliminary tasks. |
| Difficulty with irrational numbers in this context | Not recognizing that \(131\sqrt{5}\) is \(r^{2}.\) Discomfort with \(3\pi\) as a coordinate of the centre. |
