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 \(c\)?
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 \(c\)?
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.