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Writing a proof

Be aware that some students take much longer than others to appreciate the demands of a proof. They may need to practise with one or two step proofs, rehearsing the reasons verbally.

Once students are ready, constructing a proof is much like writing an essay.

  • Plan the sequence of ideas
    Allow students time to explore the diagrams and experiment without interruption. Following unsuccessful pathways often teaches more about deductive thinking than achieving instant success.
  • Provide time for discussion of ideas
    Students can evaluate a variety of strategies and select the most efficient or elegant sequence.
  • Model the formal writing stage
    Demonstrate how to write the reasons either by providing the outline of a proof or the lines of proof which need to be put in order.
    Highlight instances when the order of the ideas is important. For example, in a similarity proof, the order of the angles does not matter. However, in the SAS test for congruence, the angle argument should be stated between the side statements to show that the angle is included.

Providing alternative proofs builds student confidence and develops their capacity to construct a proof without assistance.

Proving Pythagoras' theorem

Pythagoras' theorem will be familiar to students. There are many different ways to verify the theorem visually, and to prove it.