Planning tool

Students:

• recognise and prove that two triangles are congruent
• know the conditions for two triangles to be congruent
• know the conditions for two triangles to be similar
• use knowledge of angles and plane shapes to prove two shapes are congruent or similar
• solve problems using knowledge of congruence and similarity
• construct a logical geometric proof
• begin to use deductive reasoning to proofs involving plane shapes
• use geometric theorems to solve spatial problems.

There are many misconceptions and misunderstandings about mathematical proof. It is a difficult concept for students. AAMT Top Drawer offers a number of articles and activities that address these.

It is difficult for students to grasp logical arguments and make connections with a series of logical progressions of thought. Students may confuse the difference between deductive and inductive reasoning. Explain that inductive reasoning begins with a particular scenario and makes generalisations about a conclusion; that is, you make inducements to arrive at a conclusion. The problem with this is the conclusion may be incorrect.

Deductive reasoning requires more evidence presented in a logical progression that naturally leads to a conclusion, so that the conclusion is true without the possibility of being incorrect. Asking students to use reasoning is difficult to apply, so begin with easy problems with known geometric knowledge and gradually increase the difficulty of the problems. 

Students also make errors when they are constructing proofs. For example, students:

• assume the conclusion in order to prove the conclusion
• are unsure on how to start the proof
• provide unnecessary examples to supplement their valid proofs
• misunderstand definitions
• do not use enough explanations in their steps of reasoning
• use incorrect steps.

Explain that every line of argument needs to be true and not just a claim. Explain that the layout of the proof needs to be able to be followed in a neat and logical manner. Encourage students to use reasoning phrases and vocabulary, for example, 'Statement B follows Statement A because …'.

Give feedback and encourage fellow students to give feedback on clarity of statements.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

• We are learning how to write a formal proof of a geometric property using the correct mathematic language and notation.
• We are learning how to apply geometrical properties to solve geometry problems.

Why are we learning about this?

• There are many areas where knowledge of shape and angle are useful.
• It is important that you can communicate your ideas using the correct mathematical terminology and notation.
• If you are going on to study mathematics at a higher level, a knowledge of formal proof is important.

What to do

Proving congruence with support

1. Open this webpage, Proving triangles congruent
2. Work through the three sections:

• SSS: Dynamic Proof!
• SAS: Dynamic Proof!
• ASA Theorem?

In each section you can view videos and do short exercises to build and test your understanding.

Success criteria

• I can use knowledge of angles and plane shapes to prove two shapes are congruent or similar.
• construct a logical geometric proof.