Exploring Grade 8 Mathematics Module 2

The Grade 8 Mathematics morning session focused on the lessons that are in Module 2: The Concept of Congruence. In the first lesson, students explore why and how we move things around in a plane and what a transformation is. They describe their motions intuitively using language such as “slide,” “flip,” and “turn.”  They see that when an object is moved, it doesn’t change; it remains rigid.

Lesson 2 provides a bridge to high school geometry.  The bridge needs to be built well here, and participants in the session agreed that definitions and the use of language are very important.  Students formalize their definitions and properties of translations, reflections, and rotations.  Students are introduced to the concept of congruence through the sequencing of these motions in Lessons 8 and 9.  If students move something, how can they move it back? Students also explore inverse transformations and the concept of mapping a figure onto itself.

The culmination of all the previous lessons takes place in Lesson 10 where the precision of language is crucial.  Students will need to define and describe the sequence of motions needed to map a figure onto its given image.  This formalizing of language once again is helping to build the bridge to high school geometry.  Students need to translate their thinking process into precise mathematical language.  They get to compare and critique the reasoning of others and they see that more than one sequence can produce a successful result.  They might discover that getting the figures to touch if they are not touching eases the composition.

In Lesson 11, students are given as little information as possible, forcing them to provide what they need to have first in order to correctly identify the transformations needed.  This is a step up from Lesson 10.  In Lesson 11, we end with the official definition of congruence.

In Lesson 12, students will utilize rigid motions to understand the relationships of angles formed when two parallel lines are cut by a transversal.  They first explore and conjecture using a protractor, but then confirm “why” this happens based on what they have learned about rigid motions. The rest of the module involves introducing students to the concept of proof with proving the sum of the three angles of a triangle is 180 and the exterior angle of a triangle is equal to the sum of its two remote interior angles.

Lesson 15 is optional and deals with the proof of the Pythagorean Theorem.  It fits in nicely here because of the connection to the major themes of the unit: rigid motions, congruent triangles, and triangle sum.  If teachers choose not to teach the lesson now, they should go back to it before teaching Module 7.

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