## Unit 3 – Angles

20 Oct

We learned our lesson last year when we tried to cram Angles & Triangles together in a unit that we needed to take a step back & slow down…at least until we have students who have actually experienced the geometry standards in the lower grades as prescribed by CCSS-M. Since this is the first real year of 6-8 implementation by our teachers, we still have a few years before we might be able to make more adjustments. For now, this one unit covers what has traditionally been covered by 3 chapters of our geometry textbook. 3 chapters have been reduced to one CCSS-M standard. The 3 chapters? Points, Lines & Planes; Deductive Reasoning & Logic; Parallel Lines.

Level 1: I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

Level 2: I can use and prove theorems about special pairs of angles. G-CO 9

Level 3: I can solve problems using parallel lines. G-CO 9

Level 4: I can prove theorems about parallel lines. G-CO 9

Congruence G-CO

G-CO 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.

I know that I have read statements about de-emphasizing the structure and building of a deductive system in CCSS-M. I wish I could remember in which document I read that. I’ve only been emphasizing the structure and building of a deductive system for about 8 years now – ever since I took a graduate course on geometry and realized its importance. So last year, we tried to jump right in to proving theorems without thinking about logic, conditionals, converses, inverses, etc. It was a disaster. Even though conditional statements aren’t explicitly listed in CCSS-M and hence will not be assessed, I believe they are important. I think it is a big deal that we accept the following as a postulate in our system: If two parallel lines are cut by a transversal, then corresponding angles are congruent. That’s our equivalent to Euclid’s 5th Postulate, and our geometry would look differently if we didn’t accept it without proof. The Triangle Sum Theorem relies on this postulate, as do other important results, and I want my students to know that it is our choice of accepting this postulate that brings about results that wouldn’t be true without it.

So we started with a lesson on logic. I asked students to create a word morph – go from WARM to COLD, changing one letter at a time, creating a real word each time. This short exercise emphasizes that we all start with WARM and we all end with COLD, but we don’t all get there the same way.

I read someone’s tweet about having students create a conditional statement for which the statement, converse, inverse, and contrapositive were always true, which we did. I wish I could remember whose it was. Thanks for the idea.

We also gave student groups cards with a conditional statement, its converse, inverse, and contrapositive. The students were to choose a card as the conditional and then decide which of the other cards was the converse, inverse, and contrapositive. Once they have those statements straight, then I choose one of their other cards (like their inverse) and tell them it is now their conditional statement. If it is now the conditional, then which of the other cards are now the converse, inverse, and contrapositive? This short exercise emphasizes that it doesn’t matter how the conditional is written, from that conditional, you can still create the converse, inverse, and contrapositive.

Then we thought about the game of basketball being a deductive system. What would be the undefined terms? Defined terms? Postulates? Theorems? Monopoly and soccer also work for this.

Then we played The Letter Game as an introduction to proofs. I wish I knew the source of The Letter Game. A quick search on the internet did not produce any results (except my Canvas notes from class), so I’ll include it below.

The “Letter Game”

Undefined terms: Letters M, I, and U

Definition: x means any string of I’s and U’s

Postulates:             1. If a string of letters ends in I, you may add U at the end.

2. If you have Mx, then you may add x to get Mxx.

3. If three I’s occur, that is III, then you may substitute U in their place.

4. If UU occurs, you drop it.

Example. Given: MI

Prove: MIIU

1. Given: MIII

Prove: M

2. Given: MIIIUUIIIII

Prove: MIIU

3. Given: MI

Prove: MUI

4. Given: MI

Prove: MIUIU

5. Given: MIIIUII

Prove: MIIUIIU

I like The Letter Game because it introduces students to the idea that we start with some given information, we have a goal of what to prove, and we use the rules (postulates, theorems, and definitions) in our deductive system to get there. I do not care that students produce a formal two-column proof. I care that students can create a valid, logical argument. I also care that they notice that their valid, logical arguments don’t have to look exactly the same as everyone else’s in the class. There is almost always more than one way to get from the given information to what we are trying to prove. And more than a few students admit they like figuring out a Letter Game proof.

Then we do a few jumbled proofs about segments and angles before we prove our first theorem: Vertical angles are congruent.

And so, right or wrong, the journey continues…