Reasoning and Proof Geometry

Reasoning and Proof

Before your ever start a lesson on Reasoning and Proof you must understand 2 very important things. First, your students will not like proofs. For the most part even your absolute best students will strongly dislike proofs. Secondly, don't start with proving mathematical concepts!

Your students don't have to dislike proofs if you do a good job on the second must. Think about it. Your students love to think critically and they love to actually prove things to be right. Everyone does! Start with topics they are confident in. How about making them prove something they are sure of (which is not math by the way). Tell one of them to prove to everyone that the lights in the classroom are on (simply show the switch is on, then turn it to off as a way to show an "if then". If they are off then they are not on and vise versa), prove that chalk will write on the chalk board (write on the board), or prove that the football team won their last game (have the newspaper article with the score in it).

Then slowly start to lead into things that involve numbers. Not necessarily math proof, just things that use numbers. For example, prove that their are 20 people in the classroom (count them), prove that your desk is a rectangle (measure the dimensions and angles, this is also a great way to lead into proof by definition; in this case, of a rectangle), or prove that their are 52 cards in a deck. Nothing too crazy, ease them into it.

Finally, lead them through a review or lesson on the following concepts so they clearly understand what these things are before you ever go into a proof. We can't ask them to prove something using concepts for reasons that they don't understand. Once they have a grasp for what these things are and what they will be using them for you will lead them into the most important piece of the lesson. 

1. Given: This is generally either the problem (equation) we are trying to solve, or some piece of important information given in the problem.

2. Properties: These for the most part are the basic mathematical functions of adding, subtracting, multiplying, and dividing, such as the second reason in the example above (Property of Subtraction).

3. Definitions: Again, saying "Because it is" is not a reason. This sort of reasoning is not seen as often as other reasons. By using definitions, sometimes the answer or part of the working of a proof can be shortened. For example, by using the reason "definition of a bisector" (and being ALREADY able to prove through either given information or earlier parts of the proof), you can prove that that two adjoining angles are congruent without having to go through a more lengthy proof.

4. Postulates: Postulates hold the same value as theorems (explained next), except that they cannot be proven. However, these generalized rules have proven correct for a very long time and can be accepted with proof of their validity. An example is "Through any two points there is exactly one line". While it cannot be proven through a proof (although the authors dare anyone to disprove it), it is accepted as a reason. There are few of these, so as good as it may sound, if you make it up, someone will notice.

5. Theorems: Theorems are statements that have been proven true through proofs of their own. They are especially helpful shortcuts in their own right as by stating a theorem, a great many things are proven and you do not have to do all the work of re-proving the theorem. Theorems can be simple ("If two lines intersect, then they intersect in exactly one point.") or very complex ("The composite of two isometries is an isometry." [Don't panic if you don't understand; it will be explained later on]). Sometimes, you will be given the proofs for theorems; othertimes, as part of the exercises, you will be asked to prove it yourself.

6. Axioms: For most purposes, the same as Postulates. The difference is that Axioms are algebraic in nature, while Postulates come mainly from geometry.

7. Corollaries: These are statements that stem from what becomes proven in theorems and definitions and do not require (though usually have) separate proofs themselves.

You are ready to start proofs and this is the most important part. Do not start them into proofs by have blanks to fill in with whatever they want the reason to be! Give them multiple choice options for the reasons. Three options at most. You want them to master how a proof works and the connection between the two sides before they master being able to come up with the correct reasons on their own.

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