These notes are intended to be read in conjunction with the resource files for the main task, Teddy bear, 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, Teddy bear, 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.

  • Video of students tackling the task and reflecting on how they solved the problem

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

Resource outline

This is a low-threshold, high-ceiling activity where learners are “simply” invited to match some circles on a graph with their equations. The circles are drawn to scale, but the axes are not labelled. The resource can be used to consolidate learning shortly after the Cartesian equations of circles are introduced, or as a revision task.

Student working on resource detail

Introducing the task and suggested ways of working

Students could work in pairs or small groups to encourage sharing ideas and justifying statements. Throughout the task encourage students to record their reasoning so that they can present or adapt their argument.

The circles diagram can be printed on A3. Students could be given 2 minutes to look at the diagram before they are given the problem sheet with the equations. Students should note what they see in the diagram (common centres, variation of radius or variation of the position of the centre) and share these ideas in pairs.

Part way through the task, invite all groups to share ideas. Students could suggest a circle that they have matched, make an observation, describe their strategy or ask a question.

Student working on resource detail
Student working on resource detail
  • Students categorising circles in some way.

  • Students building arguments, moving towards proofs.

  • “If this circle is … then this circle is…/could be….”. Awareness of how initial assumptions affect later deductions.

  • “If this is … then this is …, but then…”. Contradictions reveal false assumptions.

  • What strategies have you used?

  • Could you have solved the problem more efficiently?

  • What features of the circles were helpful for the matching? What did this tell you about circles?

  • What features of circles have you not used to solve this problem?

  • Could you use your strategy for a similar problem about other curves?

  • Could you make up a similar problem? How could you make it more difficult? How could you make it easier?

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


Cartesian equations of circles, completing the square

Skills involved in this task

Relating equations of a circle to the position of its centre, completing the square, sketching curves, estimation, reasoning and deduction