Preliminary and follow-up tasks

Preliminary task

This could be used in a lesson a few days before the main task, or set as homework. The purpose of the task is to

  • remind students of certain ideas or skills related to the main task so that they do not become an artificial barrier in the main task, and

  • help to inform the way you use the main task by assessing students’ familiarity or confidence with these ideas or skills.

The options proposed below involve the suggested function cards from Two-way functions. (You could choose to use a combination of the three options.)

In pairs, students sort the functions on the cards into those they think they can sketch and those they’re not sure about. Then discuss those they’re not sure about in fours.

  • Why are these more difficult to sketch?

  • What features of these functions could help with sketching them?
Students working on preliminary task

This uses the cards from Two-way functions and the tables from the main task of Function squares. Ask students to place the functions in appropriate squares, according to whether or not they have certain properties. Does this help students to sketch some of the less familiar functions?

Students sort the cards into families in some way and think about

  • How can they choose to define families?

  • Are there some functions that belong to more than one family?

  • Once the functions are in families, can we identify what criteria were used?

Encourage students to think about the graphs of the functions as well as the algebra.

Sorting cards in preliminary task

Follow-up task

One of these tasks could be used a few days after the main task to provide an opportunity to revisit key ideas.

Can we sketch this function with its asymptotes?

Write down the equations of the two asymptotes of the curve \(y=x+1-\dfrac{4}{(x-2)}\) and find the coordinates of the points where the curve meets the axes. Sketch the curve.

How many real roots does \((x-1)(x+2)(x+3)=3x\) have?

Express \(\dfrac{3x}{(x-1)(x+2)}\) in partial fractions.

Show that \(\dfrac{dy}{dx}\) is negative at all points on the graph of \(y=\dfrac{3x}{(x-1)(x+2)}.\)

Sketch this graph, showing the two asymptotes parallel to the \(y\)-axis and the asymptote perpendicular to the \(y\)-axis.

By sketching on the same diagram a second graph (the equation of which should be stated), or otherwise, find the number of real roots of the equation \((x-1)(x+2)(x+3)=3x.\)

Can you find … asymptote edition presents a sketch graph and students are asked to find a function which could have this graph.

We have suggested these preliminary and follow-up tasks as part of a sequence of teaching, but they are intended to be used flexibly. For example, you may prefer to use one of our suggested preliminary tasks as a follow-up task, or vice versa.