These notes are intended to be read in conjunction with the resource files for the main task, Discriminating, including the teacher notes and solution.

These notes have been produced as part of a research project in collaboration with colleagues at the University of Cambridge, Faculty of Education. We are researching how teacher notes and video clips can support teachers to use Underground Mathematics resources.

In these notes you will find

  • Suggestions of how the main task, Discriminating, could be used, indications of mathematical behaviour to look out for and reflection questions to draw together ideas.

  • Details of opportunities for learning in the main task, including identifying big ideas, connections, common issues and misconceptions. We also suggest questions and prompts that you could use to raise awareness of these while students work on the main task.

  • Suggestions of preliminary and follow-up tasks for the main task.

  • Sample student work

For more information on this project please e-mail us.

Resource outline

Discriminating consists of several statements about the number of solutions of a quadratic equation. Students are asked to decide whether MUST, MAY or CAN’T is the correct choice of word within the statement. The statements are available as cards to be cut up and sorted according to which word should be used. Some cards may be simpler when tackled with an algebraic approach, whereas others may be more suited to a graphical approach.

Discriminating can be used to consolidate understanding of the discriminant and make connections between graphs and algebra. Students are required to justify their arguments by giving proofs or examples as appropriate. There’s the potential for some quite subtle thinking here.

Introducing the task and suggested ways of working

If possible, provide mini-whiteboards to help encourage graph sketching.

students working

Allow individual thinking time at the start of the activity for students to think about one or two cards. To help process the statements on the cards, students could think of examples that satisfy the first part of the statement (not cards 6, 8, or 11) and then think about which is the correct word for the special case of their examples. Note that students will need to recognize the distinction between these examples and the general results.

In pairs, students should sort the cards into ‘MUST’, ‘MAY’, ‘CAN’T’ and ‘Unsure’ piles, and record their reasoning. Pairs could then compare and contrast solutions and approaches.

After several cards have been sorted, ask students to write down their argument for at least one card in each of the MUST, MAY and CAN’T piles, or present their arguments to each other. How can they use sketches to support their arguments? They should think about what makes a convincing argument – a general proof or specific example. The interactive graph in the resource could be used to support discussion.

As an alternative the statements could be used as a sequence of problems over a series of lessons.

  • Using the discriminant to justify the choice of MUST, MAY or CAN’T.

  • Distinguishing between a proof, an example and a counter-example.

  • Generalising from sketches and concrete examples to general cases, e.g. “This would be also true for any positive \(a\).”

  • Exploring through variation and finding limiting cases, e.g. “This MUST be true as \(c\) gets bigger, but not if \(c\) goes below …”

  • What strategies have you used?

  • If you used algebra, could you have used a graph? If you used a graph, could you have used algebra?

  • Were any cards surprising?

  • What features of quadratics have you not used to solve this problem?

  • Could you make up a further card for each of ‘MUST’, ‘MAY’ and ‘CAN’T’?

  • What can you do well in this topic? What do you need to think about?


The discriminant, manipulating inequalities, sketching quadratics.

Skills involved in this task

Calculating the discriminant, relating roots of a quadratic to the coefficients in the equation, completing the square, manipulating inequalities, sketching curves, graph transformations, reasoning and deduction, proving results, generating examples.