These notes are intended to be read in conjunction with the resource files for the main task, Two-way functions, including the teacher notes and solution.

These notes have been produced as part of a research project in collaboration with colleagues at the University of Cambridge, Faculty of Education. We are researching how teacher notes and video clips can support teachers to use Underground Mathematics resources.

In these notes you will find

  • Suggestions of how the main task, Two-way functions, could be used, indications of mathematical behaviour to look out for and reflection questions to draw together ideas.

  • Details of opportunities for learning in the main task, including identifying big ideas, connections, common issues and misconceptions. We also suggest questions and prompts that you could use to raise awareness of these while students work on the main task.

  • Suggestions of preliminary and follow-up tasks for the main task.

  • Sample student work

  • Video of students tackling the task

For more information on this project please e-mail us.

Resource outline

Two-way functions gives students an opportunity to explore their understanding of properties of functions, such as asymptotes and roots. Students identify properties that certain functions have in common and also seek functions that have particular properties. In the process, students may sketch lots of functions, and develop their sense of how the equations relate to the sketches. Some of the cells of the table may be quite hard to fill in with the set of functions that students may be particularly familiar with, so students may like to use the list of functions in the suggestion, which includes enough functions to complete the table plus some extras. This could help to extend the set of example functions that students are familiar with.

Students working on resource

Introducing the task and suggested ways of working

When introducing the task, emphasise that there is not one ‘right’ answer. Allow individual thinking time at the start of the activity, then students could work in pairs or small groups to encourage sharing, challenging and justifying ideas. Include an opportunity for groups to share ideas: students could suggest a function or property or ask a question. Alternatively, groups could swap tables to see how other groups have completed cells.

  • Print the table onto A3 paper to give space for more than one answer or sketches.

  • Students should try to think of example functions themselves before using the suggested functions. They can then use the suggested functions if necessary (available in a list or as cards to cut out).

  • Encourage students to sketch graphs to develop their ability to visualise. After students have tried to sketch graphs, software such as Desmos may help them to construct examples.


  • Complete the table using a different function in every cell.

    This may encourage students to move beyond the examples they typically use. Some students may also start to modify their column and row headings.

  • How few different functions can you use in the table?

    This may encourage reflection on multiple properties of the functions already chosen. Again, some students may modify column and row headings.

Student work with sketch graphs
Students working on resource
  • Asking questions in both directions, e.g. “Does this function have this property?” and “Which functions do I know that have this property?”

  • Connecting information about roots or asymptotes with the equation and graph.

  • Constructing functions that combine properties by modifying familiar functions.

  • Constructing arguments, suggesting counter-examples to eliminate possible properties or functions, e.g. “If this is … then this is …, but then…”.

  • Awareness that there are many ways to complete the table and decisions affect their options later.

  • Which cells were easy to fill? Which were more difficult?

  • Could you generalise any of the functions in the cells?

  • Did you have any choice about the column and row headings?

  • What types of function have you used? Are there other families of functions that you could have used?

  • What can you do well in this topic? What do you need to think about?


Some familiarity with the language of graphs of functions, asymptotes, roots or zeros of functions; some knowledge of quadratic, rational and modulus functions

Skills involved in this task

Thinking about properties of functions and their graphs, sketching curves, reasoning and deduction, counter-examples, modifying a function to give new functions