Writing Proofs - need help?One of the most difficult aspects of a course like geometry is the writing of proofs. Why? Because to write a proof successfully requires you to think! Most of your math classes in the past teach you skills and techniques to solve problems. Master the techniques and you can solve any problem. Hardly any thinking needed! Unfortunately, herein lies the problems with proofs - no techniques needed but plenty of thinking required.
Take a deep breath. Before you think writing proofs is hopeless, quit and decide that English class is more appealing than geometry, I do have some strategies that can be used to help you develop your "proof-writing abilities". My strategies involve four main ideas: understanding, planning, writing and checking. Use these strategies often and they will quickly become part of your thinking process.1) Understand the Proof (study the parts)
- Do you understand what the given statement is telling you? What does it mean?
- Do you understand the diagram? Should you redraw it by removing some of segments, etc.? (See "more on diagrams" below for more ideas for working with the diagrams.)
- Do you understand what the prove statement is telling you to prove? What does it mean?
2) Plan the Proof (take notes on the parts)
- What information can we get from the given statement?
- What information can we get from the diagram?
- What information do I need to get in order to complete the proof (the prove statement)?
3) Write the Proof (translate the notes into statements/reasons)
- Which notes are useful in writing the proof?
- Which notes are not useful?
- What statements do I still need to write - ones that may not come from the notes?
4) Check the Proof (make sure that it makes sense)
More on diagrams (from Mrs. MacMinn): Draw the diagram twice.
- Did I miss any statements that are needed to write a complete proof?
- Did I include any unnecessary statements?
One last thing to consider: While practicing the writing of proofs can help you become an expert, an often overlooked strategy is the reading/studying of proofs that have already been written. (These proofs can be found in your notes, old homework and the textbook.) By studying how these proofs were written - structure, ordering of statements, reasons - you can develop specific thinking strategies that can help you in writing proofs.Common Mistakes when Writing ProofsUsing the Transitive vs. Substitution PropertyUsing the Definition vs. the Postulate/Theorem
- On the first diagram: mark it up while brainstorming different ideas.
- On the second diagram: draw on this diagram AS you write the proof (with congruent marks, tick marks, etc). This will help you to keep track of the brainstorming ideas that you have already written in your proof as well as to see which brainstorming ideas you still need to formally record as a proof statements.