These alternative options for a preliminary task could be used in a lesson a few days before the main task, or set as 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.

Students could use mini-whiteboards in this task. The option to sketch is suggested to encourage all students to participate even if they struggle to suggest equations. Students can add equations to their sketches as ideas emerge through discussion.

Sketch or give equations of two quadratics that have \(x=2\) and \(x=-5\) as roots. What can you say about the coefficients in each case?

Sketch or give an equation of a quadratic with no real roots. What is the value of the discriminant? Can you change one coefficient to give an equation with real roots?

Give two quadratics where the coefficient of \(x\) is \(6\) but the equations have different numbers of solutions.

If \(a=2\) and \(c=5\), what can you say about \(b\) if \(ax^{2}+bx+c=0\) has two distinct real solutions? What if \(c=-5\) instead?

An interactive graph has been provided to support discussion of these ideas and connections between algebraic and graphical representations. As well as these questions, students could test their intuition by thinking about what happens to the roots if you vary \(a\), \(b\), or \(c\)?

Use Desmos or GeoGebra to investigate the effects on the graph and discriminant of a quadratic by changing coefficients using sliders. See the interactive in the resource.

Follow-up task

These could be used a few days after the main task to provide an opportunity to revisit key ideas from the main task.

Students add extra cards to Discriminating to try to challenge each other. They should include a written solution for each card they produce.

The resource Proving the quadratic formula asks students to order the steps in a derivation of the quadratic formula. To help build on ideas from the main task, students should focus on the following points

Why does the discriminant determine the nature of the roots?

Compare the steps in the proof with the statements on the discriminating cards. Does this help to explain the choice of MUST/MAY/CAN’T?

Prove that the line \(y=mx+c\) will touch the circle \(x^2+y^2=25\) if \(c^2=25(m^2+1)\). Hence or otherwise find equations of the two tangents to this circle from the point \((2,11).\)

We have suggested these preliminary and follow-up tasks as part of a sequence of teaching, but they are intended to be used flexibly. For example, you may prefer to use one of our suggested preliminary tasks as a follow-up task, or vice versa.