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
Multiple representations Can you sketch the graph of a function with this property?
What is \(f(0)\) if the graph of \(f(x)\) passes through the origin?
Symmetry What makes some functions symmetrical and others not?
What do we mean by symmetry on a graph? Can you express it in algebra?
Visualising Can you describe visual features of these functions? What are the ‘interesting’ features of their graphs?
Could you rewrite the algebra to help you?
What would happen to the graph if you changed the function in this way? [e.g. if you change from \(x\) to \(x-1\) in the denominator, or change the type of quadratic denominator]
Categorising objects What is the same about these functions? What is different?
Talking about mathematics, conjecturing, justifying Which examples/properties are you sure about?
How did you approach the problem?
Which cells did you try to fill in first?
Have you got the same properties or examples as other students?
How can you convince someone else that you have a correct solution?
Making connections Questions teacher could ask
Graphical and algebraic representation of functions What does this property ‘look’ like on a graph?
What does it ‘look’ like in the formula?
How can we construct examples of functions with these properties?
Domain and range of functions What are the possible \(x\)-values or inputs? What are the possible \(y\)-values or outputs?
Connecting task with graph transformations Can you suggest other functions that are like this one?
How could you generalise this example?
Common issues or misconceptions How might these be revealed? Teacher input
A function can’t be one-many Sketches of curves which don’t represent functions Can you describe what a function is?
Not connecting \(y\)-intercept with \(f(0)\) or substituting \(x=0\) Don’t know how to find the \(y\)-intercept. Asking how other students found the \(y\)-intercept. Thinking \(y=1+\tfrac{1}{x}\) has \(y\)-intercept \(1.\) Can you give me the coordinates of any point on the \(y\)-axis? If you have a graph \(y=mx+c\), how can you find \(c\)?
Not connecting roots with \(x\)-intercepts What is a root?
Not understanding what it means for the denominator to approach \(0.\) Trying to substitute an \(x\)-value, for which the function isn’t defined e.g. if the function isn’t defined when \(x=2\), what happens if you work out \(y\) when \(x\) is \(2.1,\) or \(2.01,\) or \(2.001\) on a calculator?
Not having a concrete example of a function with certain properties Gaps in table Can you sketch a graph of a function that would do? Can you suggest a function that looks a bit like this?
The graph of a function can’t cross an asymptote Struggling to complete the column for “Passes through the origin” Saying there can’t be roots if the \(x\)-axis is an asymptote Refer to examples from suggestion. (The function \(f(x)=\tfrac{x}{(x-1)^{2}}\) has \(x\)-axis as an asymptote but passes through the origin.)
The difference between horizontal and vertical asymptotes.
Asymptotes limited to axes
Sketching only examples such as \(y=\tfrac{1}{x}\) and \(y=\tfrac{1}{x^{2}}.\) What does ‘horizontal asymptote’ mean?
What does ‘vertical asymptote’ mean?
When can they occur?
Thinking that the \(y\)-axis is an asymptote for \(y=\sqrt{x}\) Sketch graph or function written down The \(y\)-axis is a vertical tangent to the graph \(y=\sqrt{x}.\)