# The prompt

Mathematical inquiry processes: Test particular cases, generalise, and prove. Conceptual field of inquiry: Gradient of the radius; equations of perpendicular lines; equation of a tangent to a circle; similar triangles.

Omar Khan, a teacher of mathematics in the west Midlands (UK), devised the prompt after one of his year 11 students, Taim, noticed that the solutions to an exercise on finding the tangents to circles had something in common.

Taim solved the questions below and realised that the y-intercept of the tangent is always the radius squared divided by the y-coordinate at the point where the tangent intersects with the circumference of the circle.

This not only fascinated Omar's year 11 students, but also the teachers in Omar's department. They devised proofs so that Omar could guide his students in proving the generalisation.

Orientation to the prompt

The prompt could be used at the start of a unit on the equation of a circle. Students add meaning to each element in the prompt up to the level of their existing knowledge. They would know about the form of the equation of a straight line, coordinates, and intercepts.

From the diagram they might be able to deduce that the other equation is that of a circle. If there was a question about the equation in the question, notice, and wonder phase of the inquiry, the teacher might deal with it by specifying that r2 = 16 and request possible values for x and y.

When required to select a regulatory card, the main choices are:

(1) Practise a procedure

Before addressing the contention in the prompt, students practise finding the equation of a tangent to a circle. The teacher is prepared to explain (or co-construct) the procedure and has a set of questions available. Only then would students check the y-intercept for each solution was in the form given in the prompt.

(2) Test different types of cases

Initially students use circles with a centre at the origin and find the tangents perpendicular to radii in all four quadrants. Afterwards, students explore circles with centres that are not at the origin.

(3) Decide if the prompt is always true or Prove a generalisation

Students who understand the prompt at the start of the inquiry look for ways to prove it is always true.

(See 'Lines of inquiry' below for details of what could follow after the selection of regulatory cards (2) and (3).)

Engagement and challenge

Omar reports that Taim's discovery led to a lot of interest and excitement among students and staff alike. In designing the generalisation as a prompt, he felt teachers could generate the same levels of engagement and challenge through inquiry.

August 2024

# Lines of inquiry

1. Proof

Omar's year 11 students used algebraic and geometric reasoning to prove the generalisation in the prompt is always true.

The algebraic approach follows the procedure for solving the questions in the exercise.

The geometric approach uses two right-angled triangles formed, in one case, by the y-axis, radius, and tangent and, in the other case, by the x-axis, radius, and the perpendicular line from the point on the circumference of the circle to the x-axis. The triangles can be shown to be similar, from which you can derive the ratio in the general form.

2. When the centre is not at the origin

What happens to the y-intercept when the centre is not at the origin?

The example (pictured below) shows that when the centre of the circle is translated from the origin to (2,3), the y-intercept is translated 5 units up.

Is it always the case that you sum the coordinates at the centre to find the vertical translation?

The general form for the y-intercept of a tangent to any circle is:

The y-intercept (c) takes the form it does in the prompt because the  coordinates at the centre of the circle (0,0) leave the radius squared over the y-coordinate of the point of intersection.