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? |
| Communicating |
How could you convince someone that your function has these properties? |
| Making connections | Questions teacher could ask |
|---|---|
| Connecting a function and its derivatives |
What does the derivative tell you about the stationary points? |
| Connecting geometry and algebra, graph sketching and calculus |
What does the graph tell you about the equation? |
| Graph transformations |
How does transforming a graph affect its stationary points? |
| Factors of polynomials, completing the square |
What could the x-value of a stationary point be? |
| 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 |
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 |
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 |
‘Even’ as bottom row property |
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? |
