We have suggested alternative options for a preliminary task that could be used in a lesson a few days before the main task, or set as a homework. The purpose of the preliminary task is to

remind students of certain ideas or skills related to the main task so that they do not become an artificial barrier in the main task, and

help to inform the way you use the main task by assessing students’ familiarity or confidence with these ideas or skills.

These questions could be used in a mini-whiteboard session.

Sketch or give an equation of…

… a circle that does not cross the axes

… a circle whose centre lies on an axis

… a circle that lies completely inside the previous circle you drew

… a circle with the same centre as \(x^{2}+y^{2}-6x+8y-75=0\)

… a circle with the same radius as \(x^{2}+y^{2}-6x+8y-75=0\)

… a circle which has the \(x\)-axis as a tangent

The option to sketch is suggested so that all students can start to give examples of circles satisfying the suggested conditions. Students may add equations to their previous sketches as ideas emerge through discussion.

Show that the circles having equations \(x^{2}+y^{2}=25\) and \(x^{2}+y^{2}-24x-18y+125=0\) touch each other. Calculate the coordinates of the point at which they touch.

Follow-up task

We have also suggested options for a follow-up task, which could be used a few days after the main task. This provides an opportunity to revisit key ideas from the main task.

Students add circles and equations to the original set in the Teddy bear resource

Add a circle or equation of a circle which

is tangent to an axis

is an enlargement of a circle already given

is contained within a circle already given

is a reflection of a given circle in the y-axis

is a reflection of a given circle in the line y=x

passes through the centre of a circle already given

has the same constant term as a circle already given

The distance between two non-intersecting circles: Prove that the points whose coordinates satisfy the equation \(x^{2}+y^{2}+2gx+2fy+c=0\) lie on a circle. State the coordinates of the centre of the circle and the length of its radius. Prove that the circles \(x^{2}+y^{2}-20x-16y+128=0\) and \(4x^{2}+4y^{2}+16x-24y-29=0\) lie entirely outside each other, and find the length of the shortest distance from a point on one circle to a point on the other.

Pairs of circles is another problem that will reinforce links between the equation of a circle and its graphical representation. This problem presents a simple image that is surprisingly rich when considered in detail. Students can take an algebraic approach but they will find that a geometrical approach can make the problem much simpler.

We suggest these tasks as part of a sequence of teaching, but they can be used flexibly. For example, a preliminary task could be used as a follow-up, or vice versa.