Proofs in the Coordinate Plane

My Geometry students are petrified of proofs; if I even mention the word they start to freak out.  First semester we worked with various formal, two-column proofs, which my students really struggled with.  They wanted to make huge leaps in logic and being required to state every piece of reasoning, along with an appropriate mathematical justification, really challenged them.  We have circled back to proofs again, but this time we are working in the coordinate plane so we are using distances, midpoints, and slopes to prove things about various figures.  In order to alleviate some of my students' stress over having to do "proofs" again I decided to give them some guidelines they could use when proving things in the coordinate plane.

Here is a list of the main things we will be proving in this unit: 

• Triangles: right, isosceles, equilateral, right isosceles
• Quadrilaterals: parallelograms, rectangles, rhombi, squares
• That a segment bisects a side of a figure
• That triangles are congruent according to the SSS Congruence Theorem or the HL Congruence Theorem

We will do a few other things, but these are the big ones that I wanted my kids comfortable with.  The added benefit of this is that in the process of doing these proofs we are reviewing a lot of material from first semester, which is great because the EOC will be here before we know it.  

I had each of my students get a piece of construction paper and two 1/4 pieces of graph paper.  We began by listing 4 major guidelines for our proofs.  

1) To prove segments congruent use the distance formula.

2) To prove sides/segments parallel show they have the same slope. 

3) To prove right angles exist prove the segments are perpendicular by showing they have slopes that are opposite reciprocals.

4) To prove segments have the same midpoint use the midpoint formula.

We then listed each of the figures we would be working with and some ways to prove that particular figure existed.  I did not give them every possible combination of ways to prove a figure is a parallelogram, but two or three options that they could work with.  The idea was to give my students who were intimidated by proofs a structure to follow, and still leave some things unsaid so that they were not overwhelmed by information and I have some options left to challenge my high achievers to figure out.  Finally, we used the two pieces of graph paper to do two 'sample proofs' as a class.  This gave them notes and examples on one piece of paper.  I gave my students a suggested format for their notes, but some deviated from my suggestions and came up with their own format.  In fact, I'm going to steal the format of one of my student's notes for next year.  Several of my students were gracious enough to allow me to take pictures of their notes.  Here are just a few of them.

Student #1 - This was my recommended setup.  Notes on the front side divided into 3 sections, then examples on the back.

Front - Notes

Examples

Student Sample #2  - This one is set up differently.  This student did her second example as a flip up section over the triangle strategies.

Notes with examples on top

Ex. 2 flips up to show triangle strategies

Student Sample #3 - This one is my favorite and the setup I will steal to use next year.  This student did all of his notes on the bottom, then taped his graph paper with examples over his notes to create flip-up sections.

Notes 

Examples that flip up to show strategies underneath

 After finishing these notes my students did practice on their own proving figures in the coordinate plane.  Based on an informal survey of my students, the notes served their purpose.  My students used them, and reported that they were extremely helpful.  This is definitely something I'll use again next year.