Planning a Proof Geometry

Planning a Proof

Planning a Proof in Geometry is one of the most nerve racking subjects in the course. For the average student who will never be a math major or professor, being able to write a mathematical proof from scratch is not all that important. All of the standardized test only require that the students be able to fill in missing pieces to proofs that are already assembled for them. However, the most important thing these non- math major kids will learn in your geometry course is how to do a proof. Allow me to explain...
 
We have the responsibility of not only teaching our students our content area but also how to grow as an individual mentally, socially, and so on. Planning a proof is more about learning to reason and logically think through a process, to draw a conclusion, to create a plan than it is about the algebra or geometry behind the proof. 
 
Now, let's discuss why proofs are so difficult for students. First, the way we are taught to teach proofs is to cram a bunch of theorems and postulates into a short time frame in which they are supposed to master, some of which were even based off of content they learned in the previous chapter on angles. So the students are already lost and uncomfortable with the material they are supposed to be using as facts and reasons for the proofs, which they do not know how to construct yet either! 
 
What we have to understand as their teacher is that it is not the actual process of planning a proof that is the issue. It is there prior knowledge of the topics they are trying to construct the proofs on. Try this, take a day and work only on one topic. For example, vertical angles. Teach them and lead a discussion on vertical angles. Do enough examples so that they are all comfortable with the topic. Then at the end of class lead them through planning a  proof (a two column proof) that shows that vertical angles are congruent. When you do this don't tell them they are planning a proof, let them walk you through it by you asking leading questions. Then once the proof is over let them know they as a class just created their first proof.
 
They need to start with that confidence. Instead of teaching a chapter on proofs mix them in with the topics as you teach them. Meaning when you teach triangles go through the proofs with each triangle, when you teach complementary and supplementary go through each of the proofs with them. The problem is when we teach just a unit on proofs we are also asking them to recall things they learned long ago. Not that this is a bad thing but it does destroy the confidence of the students who naturally struggle with math. 
 
Rework your lessons so that they see proofs throughout their entire geometry class. If you don't have the time to rework your lesson plans you can download ours by CLICKING HERE. The stress, time, and frustration this will save you is well worth the effort. 
 
Below is an excerpt from Proofs for Dummies. This is a great tool / reference to go over with your students as well.
 
  • Make a game plan. Try to figure out how to get from the givens to the prove conclusion with a plain English, commonsense argument before you worry about how to write the formal, two-column proof.

  • Make up numbers for segments and angles. During the game plan stage, it's sometimes helpful to make up arbitrary lengths for segments or measures for angles. Doing the math with those numbers (addition, subtraction, multiplication, or division) can help you understand how the proof works.

  • Look for congruent triangles (and keep CPCTC in mind). In diagrams, try to find all pairs of congruent triangles. Proving one or more of these pairs of triangles congruent (with SSS, SAS, ASA, AAS, or HLR) will likely be an important part of the proof. Then you'll almost certainly use CPCTC on the line right after you prove triangles congruent.

  • Try to find isosceles triangles. Glance at the proof diagram and look for all isosceles triangles. If you find any, you'll very likely use the if-sides-then-angles or the if-angles-then-sides theorem somewhere in the proof.

  • Look for parallel lines. Look for parallel lines in the proof's diagram or in the givens. If you find any, you'll probably use one or more of the parallel-line theorems.

  • Look for radii and draw more radii. Notice each and every radius of a circle and mark all radii congruent. Draw new radii to important points on the circle, but don't draw a radius that goes to a point on the circle where nothing else is happening.

  • Use all the givens. Geometry book authors don't put irrelevant givens in proofs, so ask yourself why the author provided each given. Try putting each given down in the statement column and writing another statement that follows from that given, even if you don't know how it'll help you.

  • Check your if-then logic.

    For each reason, check that

    • All the ideas in the if clause appear in the statement column somewhere above the line you're checking.

    • The single idea in the then clause also appears in the statement column on the same line.

    You can also use this strategy to figure out what reason to use in the first place.

  • Work backward. If you get stuck, jump to the end of the proof and work back toward the beginning. After looking at the prove conclusion, make a guess about the reason for that conclusion. Then use your if-then logic to figure out the second-to-last statement (and so on).

  • Think like a computer. In a two-column proof, every single step in the chain of logic must be expressed, even if it's the most obvious thing in the world. Doing a proof is like communicating with a computer: The computer won't understand you unless every little thing is precisely spelled out.

  • Do something. Before you give up on a proof, put whatever you understand down on paper. It's quite remarkable how often putting something on paper triggers another idea, then another, and then another. Before you know it, you've finished the proof.

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