Introducing algebraic proofs

Today I introduced algebraic proofs as a lead in to geometric proofs.

I started by having the expression, "Oh yeah, prove it!" on the board. I then asked students what it means to prove something. The discussion brought out using evidence to demonstrate a theory or statement was correct.

I wanted to have students consider rules and properties they work with when calculating or working with expressions. The idea was to pull out some of the properties that we would use as building blocks for algebraic proofs. This turned out to be a bit tougher than I expected.

I did provide an example:

For any two real numbers a and b, if a equals b then b = a. The meaning is shown in writing by "If a = b, then  b = a." (This is the symmetric property of equality.)

It may work better to start with something even more simple such as the reflexive property, a = a.

Students slowly started putting things on the board, such as the additive equality, a x 0 = 0, PEMDAS (an acronym for order of operations), and a couple more. I then pointed out how these are accepted properties and rules. I used order of operations as a way to explain that the accepted order guarantees that any two people making a calculation from an expression will reach the same result.

I next put up the following nine properties:
   1) addition property of equality
   2) subtraction property of equality
   3) multiplication property of equality
   4) division property of equality
   5) distributive property
   6) substitution property
   7) reflexive property
   8) symmetric property
   9) transitive property

I provided an example for the first property of writing it symbolically: if a = b then ac = bc for any other value c.

I asked students to write expressions for the other eight properties. I told them they could use their cell phone to search for assistance.

The students completed most of the other properties.  These properties are the building blocks that we use for algebraic proofs.

The first piece I worked with was from an algebraic proof worksheet that a colleague found online. I talked through the first proof, asking the class at each step what allowed us to write the statement. I then wrote in the appropriate property. I told the class to use Q.E.D. to show that their proof was concluded.

Students were then turned loose on the next two proofs. I checked with students to verify they completed the second proof correctly. The remaining proofs were assigned as homework, although some students were able to finish these before leaving class.