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OBJECTIVE - G.CO.C.9

Prove theorems about lines and angles.  Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

INTERPRETATION OF OBJECTIVE - G.CO.C.9

Ah ha.. we are here finally - Proofs. These proofs are about lines and angles and for the most part emphasize the relationships found between parallel lines and a transversal. In the past, we were given as a postulate that corresponding angles are congruent and then ask to begin work from there. This is not the case now. We can establish corresponding angles to be congruent through a simple transformation, the translation. All of the listed proofs in this objective can be easily done through transformations. This is why we have spent the time building expertise in transformations.

DIRECTLY IMPLIED SKILLS

(1) The student will be able to prove and apply that vertical angles are congruent.

(2) The student will be able to prove and apply the angle relationships formed when two parallel lines are cut by a transversal.

(3) Students will be able to prove that all points on a perpendicular bisector of a segment are equidistant from the segment endpoints.

INDIRECTLY IMPLIED SKILLS

(4) Students will be able to know all of the relationships between pairs of angles.

THE BIG IDEA

Proof of many of the geometric relationships can be done from a symmetry/transformational approach. For many years we have approached proof through a very algebraic methodology and now that we have built the definitions of transformations and determined their properties and symmetries we can use these facts to establish many new truths. It's new and different and that doesn't make it wrong or even better… it is just an alternate way to approach these concepts. It will just take some time to know what relationships to develop stronger so that they can be used in proofs later.

TRAPS & PITFALLS

Students struggle to remember all of the names of the relationships dealing with parallel lines and a transversal. They know the relationships but often struggle with remember the name associated to that relationship so spend a bit more time with vocabulary.

This is the beginning of proof and the buildup to this hasn't come with a usual chapter of material all talking about logic, conditional statements, algebraic proof, properties of equality and all the other usual things, so you need to allow for a transition from weaker proofs with limited reasoning to stronger proofs with clear reasoning. The key to this process is to highlight and discuss proofs of students that have well done, that have a logical structure to their writing, that have clear reasons for each of their actions and that those reasons use geometric vocabulary. This takes time. I will say the more time you spend on G.CO.A.2 - G.CO.A.4 where you define the symmetry, isometric transformations, and transformations definitions and properties the easier this will go.

PAST CONNECTIONS

Students will need to bring with them a good understanding of the symmetry, transformation definitions and transformation properties. We will use these two unlock many of these proofs.

FUTURE CONNECTIONS

Proof gives us new relationships, and new relationships lead to more new relationships. With each thing we prove the more tools we have to prove other new relationships. The relationships found in parallel lines and a transversal are relationships used ALL YEAR LONG!!

MY REFLECTIONS (over line l)

This was really the first time I started to clearly see the logical structure created by the earlier objectives to build enough content to be able to prove these angle and line theorems. I find the more I research these objectives the more I find the wisdom of those that put these objectives together. In this objective we did the proofs from a different standpoint that I had ever done before (because they came from a transformational approach) but the solving and application work was very similar to what I had always done. Once the theorem is proved, then the use of it differs none.

For example, for the last 1000 years we have proved vertical angles to be equal using the linear pair relationships, a substitution or two, and then the addition/subtraction property of equality.... but now we can speak about how they are congruent because a rotation of 180 degrees maps one vertical angle onto another.

I discuss both the transformational and algebraic approaches to these proofs … it can only make my students stronger to have options and to see how things can be approached from multiple areas.  Also one of the things I did for the first time was with the transformational proofs where they were written in a paragraph/explanatory style I provided a DRAFT PROOF and then gave feedback through peers and myself so that they could go home and create their FINAL PROOF.  I modelled it after what English teachers do with writing essays.  This was very effective.