In the morning session for Grade 8 Mathematics, participants picked up with Lesson 9 in Module 3: Similarity. As a group, participants worked on the exploratory challenge problem that focused on the transitivity of similarity. It was a great exercise that once again demonstrated how the problems in the modules force students to pull from their learning experiences in previous lessons and apply their knowledge to new situations. The embedded ideas here were:

- being able to define what makes triangles similar;
- how to find a scale factor;
- ordering the composition of transformations;
- using the multiplication effect on coordinates; and
- precision of language, i.e., identifying the center of rotations clearly.

In Lesson 10, participants discussed what is meant by an informal argument. In eighth grade, are numeric calculations enough, or do they need to be justified with mathematical statements and evidence? Participants talked about the importance of building that bridge to high school gand all agreed that students need to be justifying numeric answers with the mathematical facts behind them (i.e., triangle sum).

Another strength of the modules is that most all of the problems force students to use the language of the discipline; if they talk the math, they know the math. In Lessons 11 and 12, students needed to first determine if there was enough information provided to solve the problem at hand, once again providing an opportunity to critique the reasoning of others, which is one of the mathematical practices.

The modeling problem in Lesson 12 generated a lot of discussion, in particular with interpreting the distance from eye-level “straight down” to the ground. Pythagorean relationships and where the longest side of a right triangle is found was a helpful guide for the labeling of the “straight down” distance. The vague description in the problem created great classroom dialogue.

Participants ventured on to prove the Pythagorean Theorem using similar triangles, something traditionally seen in high school geometry. Proof by contradiction is a challenging task and is used to prove the converse. Teachers might need to walk through some simple “non”-mathematical examples of proof by contradiction first.

Participants spent the remainder of the time looking at the new Module 4 which created a lot of excitement. Connections to similar triangles pop up in Lesson 17 to show y=mx+b. Great ending to a great morning!

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