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OBJECTIVE - G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

INTERPRETATION OF OBJECTIVE - G.CO.C.10

There isn't much to interpret here - we are to prove the listed relationships. As I have looked at these relationships I would say that not all of them need to be done at this point of the year. There are a variety of ways to prove these relationships - informally through symmetry or formally through transformations, congruent triangles or coordinates. I believe there are a few other concepts that are not specifically stated in this list but still need to be discussed such as the exterior angle theorem or the triangle inequality theorem.

DIRECTLY IMPLIED SKILLS

(1) The student will be able to prove and apply that the sum of the interior angles of a triangle is 180°.

(2) The student will be able to prove and apply that the base angles of an isosceles triangle are congruent.

(3) The student will be able to prove and apply the midsegment (midline) of triangle theorem.

(4) The student will be able to prove that the medians of a triangle meet at a point, a point of concurrency.

INDIRECTLY IMPLIED SKILLS

(5) The student will be able to prove and apply that the exterior angle theorem.

(6) The student will be able to determine the conditions for forming a triangle, when given three lengths.

THE BIG IDEA

All polygons can be divided into triangles – thus the proof and use of properties and relationships of triangles is essential to geometric study.

The more we know about triangles -the more we know about all polygons. The listed relationships represent a few of the key theorems that get used later to build on other concepts.

TRAPS & PITFALLS

Proof.... is a trap!! Just kidding - that is what most students feel about them. I think a trap might be feeling that proofs must be presented in a two column organizational manner. I have been that teacher for 15+ years and now I find myself encouraging students to use more paragraph or flowchart proofs.

When our definition of congruent triangles is developed through transformations, then we need to develop proofs using that train of thought. The power of transformations is that they are visual and students connect with them easier than all of the technical jargon of the past. Expressing logical arguments in a written form, describing a process and a reason is a very powerful way to help students succeed with proof.

It takes time to teach students to language to use to match their visual understanding but the core unpinning logic is there because they see the transformations that would help move the proof along… this is a journey.. a new journey but I feel that more students are connected to proof through transformational point of view.

PAST CONNECTIONS

Students need to have a strong grasp on the definitions and properties of isometric transformations so they can establish relationships through transformations. They also need to know the newly established triangle congruence criteria so that they can be used in proof. They also need to know the relationships relating to parallel lines and a transversal. Notice that the early part of this unit has all been preparatory material so that students can use these concepts to prove new ideas and relationships.

FUTURE CONNECTIONS

These relationships that are specifically mentioned in this objective all become tools for later connections. The more we prove the more we can use to find other new relationships.

MY REFLECTIONS (over line l)

Proof has always been a difficult area to teach successfully. Before common core, textbooks always introduce conditional statements, properties of equality, algebraic proofs, etc.... all in preparation to prove triangles to be congruent. This was always a difficult journey for many students because of their limited exposure to logical reasoning.  Well the core curriculum changes that quite drastically. Relationships are not connected through a transformational approach which is visual and approachable.  The definitions have been altered to reflect congruence as a mapping between two shapes…. Yes I am still working out what do these things look like and how to write them correctly but I must admit I have way more students connected to the process and willing to try because they see it.

Again as I mentioned in a previous reflection, to aide in helping students create proofs I have developed a draft, revision and final approach.  I have them create proofs, have their peers read them and given them feedback and then attempt the writing again to clarify missed ideas or to fix the wording.  This has been very successful!!