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 |
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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 |
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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 |
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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? |