Opportunities 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 What other function would fit?
Is there a general form of this function?
What is the simplest function that would fit in each cell?
Conjecturing, logic and proof What can/can’t be filled in straight away?
Are any of these properties incompatible?
Can you predict what types of stationary points the function has?
Justify your answers (graphically or algebraically)
Visualising, multiple representations

Can you sketch a function that would satisfy these criteria?
Can you sketch the function and the gradient function?


How could you convince someone that your function has these properties?
Have other groups used the same properties and functions?

Making connections Questions teacher could ask
Connecting a function and its derivatives

What does the derivative tell you about the stationary points?
What do the stationary points tell you about the equation of a function?

Connecting geometry and algebra, graph sketching and calculus

What does the graph tell you about the equation?
What does the equation tell you about the shape of its graph?

Graph transformations

How does transforming a graph affect its stationary points?
What could a function with one stationary point look like?
How could you transform the function to move the stationary point to e.g. \((1,1)\)?

Factors of polynomials, completing the square

What could the x-value of a stationary point be?
What if you write the derivative in factorised form?
[Start with the factorised form of derivative, e.g. \((x-1)^2 (x+a)\) and work backwards to find a suitable value of \(a.\)]

Common issues or misconceptions How might these be revealed? Teacher input
Not knowing where to start No progress

What should I do?
What can you say about the row/column headings?
What types of stationary points do these printed functions have?
Misunderstanding of definitions

Limited progress
Misuse of terms in discussion

Can you draw sketches to show what the row/column headings mean?
Can you give an example?
Misunderstanding conditions in headings What does “increasing for \(x>1\)” mean?
Saying that a function can’t be “increasing for \(x>1\)” if it is decreasing somewhere else

Is \(x^2\) increasing for \(x>1\)?

Form of the given functions is too difficult

Errors in differentiation

Struggling to sketch the graphs or comment on common features
Avoidance of certain cells
Printed functions don’t ‘fit’ their cells
What are the most important features of this function?
How do you expect the gradient function to behave?
Points of inflection
Thinking all stationary points are turning points
Not knowing how to deal with \(\tfrac{d^2 y}{dx^2}=0\) Look at the gradient either side of the stationary point

Assuming a degree \(n\) polynomial has \(n-1\) stationary points
How do we count stationary points?
Confusing even/odd functions, even/odd degree and even/odd number of stationary point

‘Even’ as bottom row property
Confusion over whether repeated root of \(\tfrac{dy}{dx}=0\) gives more than one stationary point

How many points are there where the gradient is \(0\)?
How many stationary points does \(y=x^3\) have?
How many distinct roots does the gradient function have?
Are all quadratics even functions?
Difficulty finding an equation to match given criteria Not using factorised form of derivative Could you transform a function you know about?
How can you rewrite the derivative to use what you know about its roots?