Discriminating - teacher support

Excerpt: Detailed notes for Discriminating, produced as part of an ongoing research project in collaboration with teachers.

Outline

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.

About these notes

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 …”

Mathematical behaviour to look out for

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?

Reflection questions

Prerequisites

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.

Opportunites for learning

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
Generalising [and specialising]	Can you try some values or find particular examples? (If students sketch graphs.) Can you think of specific examples with similar sketch graphs?
Conjecturing, logic, proof	What have you assumed? How can you decide between MUST/MAY/CAN’T? How can you demonstrate truth/otherwise in general? When is it enough to give an example?
Talking about mathematics	Have you convinced someone else? Have you asked someone to convince you of their ideas?
Visualising	What does the graph look like? How are the discriminant and the roots connected? What happens to the graph if you change the sign of $a$ ? What is the effect of increasing or decreasing $c$ ?

Overarching ideas

Making connections	Questions teacher could ask
Quadratics and the discriminant	How can we tell if a quadratic has a solution?
Algebraic and graphical representations	How could we show this on a graph? What is another quadratic that has a similar graph to this?
Links between $a$ , $b$ , $c$ and the graphs	What are the key features of the graph of a quadratic? What do the coefficients tell us about the curve?
Quadratic inequalities	How could we show this on a graph? What happens if you complete the square?
Link to complex numbers	What happens if the discriminant is negative?
Graph transformations	Card 3: Why does the new equation have two roots? Why does $c$ move the graph up/down?
Symmetry	What if you replace $b$ by $–b$ in the equation or discriminant? What happens if you swap $a$ and $c$ ?

Opportunities for making connections

Common issues or misconceptions	How might these be revealed?	Teacher input
No idea where to start.	Not being able to process the statements on the cards. Asking “What are $a$ , $b$ , and $c$ ?”	Ask students to work with specific examples which satisfy the first part of a statement. Can you tell how many roots a quadratic has without solving it?
What is a proof? How do you prove something?	Examples given as proofs.	Can you generalise your example? What would happen if…? Can you vary the example to make it fail?
What $<$ and $>$ mean and how to solve inequalities.		Try positive and negative values for $a$ , $b$ , and $c.$
What roots are and the link with $x$ -intercepts.	“What is a root?”	Use of graphing software. “What do you notice about this graph?”
Thinking a change in $b$ only translates horizontally.		Compare graphs, e.g. $y=x^2$ and $y=x^2+2x$
$(-b)^{2}=b^{2}$ Assuming $–b$ is always negative.	Card 8: $ax^{2}-bx+c=0$ Unable to work with $–b$	Look at the graphs of $x^2-6x=0$ or $x^2+4x-6=0$ . What do you notice if you change the sign of $b$ ?
Thinking $x^{2}-9=0$ only has one root.	Card 5	What is $(-3)^{2}?$
Square rooting negative values, calculator mistakes.	Getting error message	Talk through the steps. Look at the graph.
Thinking $x^{2}+4=0$ has real solutions.		What happens if you substitute your solutions in?
Limited examples used.	Examples all U-shaped/have $a>0$ or all crossing $x$ -axis	What does $y=x^2$ look like? What about $y=-x^2$ or $y=x^2+3$ ? Try to sketch graphs of your algebraic examples.

Common issues and misconceptions

Preliminary and Follow-up tasks

Preliminary task

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 working at boards

student work

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 $c$ ?

Option 1 - Give me an example of…

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.

Option 2 - Investigation: roots and coefficients

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.

Option 1 - Extending Discriminating

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?

Option 2 - Proving the quadratic formula

Can we find the tangents to a circle from a point?

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 $(2,11).$

Option 3 - Review question

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.

Sample work

This sample work is provided to help prompt discussion and encourage students to reflect on alternative approaches.

Please note that to give you more flexibility in how you use the sample work, there is an extra printable copy without questions.

Card 1

Two hand-drawn graphs showing a parabola crossing the x-axis twice and another that does not cross the x-axis

What do these graphs tell you about the statement on card 1?
What do these examples tell you about the statement on card 3?

Questions

Card 3

Two hand drawn parabolas where the first does not cross the x-axis and has a y-intercept of +c and the second crosses the x-axis twice and has a y-intercept of -c

Are there any other cases to be considered?
Can you use the discriminant to explain what you see in these graphs?

Questions

Card 4

Two hand-drawn graphs showing a parabola that does not cross the x-axis and another that crosses the x-axis twice

Based on these examples, would you choose MUST, MAY or CAN’T for this statement?
Can you relate this to the discriminant?
Why has this student chosen two examples?

Questions

Card 5

Three hand-drawn parabolas where the first does not cross the x-axis and has a y-intercept of +c, the second crosses the x-axis at one point and c=0 and the third is a negative parabola which does not cross the x-axis and has a y-intercept of -c

What do these graphs tell you about the statement on card 5?

Questions

Card 7

graphs with c=0

Try to explain the connection between graph and the written argument. How could you suggest the student makes this clearer?
What are the solutions of $ax^2+bx+c=0$ if $c=0$ ?

Questions

Card 8

reflections of parabolas in y axis

Has this student considered a general example or a special case?
How could you use this idea to give a convincing argument for this card?

Questions

Card 9

algebraic work

What other inequalities can you obtain from $b^2-4ac>0$ ?

Questions

Card 11

Two hand-drawn sets of axis. The first has 2 parabolas drawn where they both cross the x-axis at two points but the seocnd had a higher y-intercept and a steeper gradient. The second axis has a parabola and a straight line where the parabola crosses the x-axis twice and has a y-intercept of c=0 and the straight line has a positive gradient with a +c y-intercept

What do these examples show?
How could you modify the statement on card 11 so the two equations must have the same number of solutions?

Questions

Card 12

“Because they’ve all been turned to negatives, the $b^2$ will still be a positive, but you times the $-a$ and $-c$ together to make a positive and times that by $-4$ and then it’s a positive, minus negative times negative which makes negative.” Explanation by Student 1

Work by Student 2

Work by Student 2

What do you think student 1 and student 2 are trying to communicate?
What advice would you give them to help improve their solutions?

Questions