Why use this resource?

Counter example is an important tool in the proof armoury. Many students, when asked, are able to explain what a tangent is, in this resource are some common suggestions. Students are asked to identify examples where these “definitions” fail and try to come up with a better definition: the solution gives a detailed explanation of what a tangent really is.


  • Printouts of, or facilities for displaying the solutions could be useful.
  • Statements printed out or displayed on the board.

Possible approach

  • Each student decides which they think is the correct “definition.”
  • Students then pair up and decide, discussing why if their original choices didn’t match and choosing one between them.
  • Pairs join up and repeat the process of coming to a consensus opinion. Then, as a whole group, feed back which have been chosen.
  • Challenge students to think of an example where their chosen definition would fail.
  • Present refutation to the first choice. (Or a popular choice in the class.)

Key questions

  • Can you sketch a curve that would make your choice fail?
  • Can you improve on the definition?

A version of this resource has been featured on the NRICH website. You might like to look at some students’ solutions that have been submitted there.