These notes are intended to be read in conjunction with the resource files for the main task, Two-way calculus, 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 the classroom.

In these notes you will find

  • Suggestions of how the main task, Two-way calculus, 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

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

Resource outline

Two-way calculus provides an opportunity to explore the use of calculus to determine stationary points, increasing and decreasing behaviour of functions, and relate this to the graph of the function. The resource can provide a framework for discussion of these ideas. Alternatively, the resource could be used to introduce ideas of turning points or integration.

Introducing the task and suggested ways of working

students working together

It might be helpful to have the table printed on A3 to allow room for sketching graphs. Providing mini-whiteboards could help to encourage sketching and adapting functions.

When introducing the task, emphasise that there are many different ways to complete the table. Give students some individual time for reading the table and thinking about any questions they may have. Then invite them to share these questions in pairs or small groups to encourage discussion and further progress with the task.
Some students may not understand the structure of a two-way table. Using the top left empty cell as an example could help to emphasise that the function in the cell has the properties given by its row and column heading.

students sharing ideas

Part way through the task, invite all groups to share some thoughts or exchange their work with other groups to see if they agree with each other. This may encourage students to revisit some of their suggested functions and headings to see if they can find more specific headings, or more general or surprising functions. It also offers an opportunity to check that students have understood what the properties in the row and column headings mean as they may have interpreted them differently.

Throughout the task, encourage students to think about how the curves should look as well as using algebra.


  • 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.

  • Complete the table with a smaller number of functions.

    This may encourage students to reflect on multiple properties of the example functions they have chosen, rather than on simply completing the table. Again, some students may modify column and row headings.

  • Anticipation: doing enough working to check what’s required, but not more.

  • Making conjectures and testing them out. Building an argument to support ideas and finding counter-examples to eliminate cases.

  • Using sketches to clarify terms such as “increasing”, “decreasing” and “stationary point”.

  • Using multiple approaches to test ideas. Moving fluently between graphs and algebra or between functions and derivatives. Describing properties of gradient functions.

  • What strategies have you used?

  • Did you have any choice over the row or column headings?

  • What types of functions did you use to complete the table? Could you have used different types of functions?

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


Differentiation of polynomials, nature of stationary points, some familiarity with the language of stationary points and increasing or decreasing functions

Skills involved in this task

Differentiation (and integration) of polynomials, relating roots of polynomials to stationary points, classifying stationary points, sketching curves, reasoning and deduction