High School Geometry 1: Similarity and Proofs for Exceptional Learners

This is the second course in a year-long sequence which covers standards and ideas in Geometry. These courses are taught in small-groups to provide individual instruction and social-learning opportunities aligned with a social constructionist or situated cognition view of learning. 

The curriculum is problem-based this means instead of lectures or videos students work together as a small group to solve problems to discover principles and strategies with teacher guidance, as necessary. Therefore, we will spend approximately 90% of each class period working on problems and discussing them as a group. The use of discussion and problem-solving leads to generalized mathematics or proofs and students will construct their own models or proofs of real-world scenarios in project based assignments. This process prepares students well who may seek advanced mathematics in high-school or beyond. 

In this unit, we build on students' previous experience with dilations and similarity from grade 8. Students have confirmed properties of dilations and informally justified similarity through sequences of rigid motions and dilations. They have practiced dilations on grids, which helped them understand how to determine or draw dilations precisely.

We start by using dilations and rigid transformations to justify triangle similarity. We prove that if triangles have three pairs of congruent angles and three pairs of sides in proportion, they are similar, leading to the Angle-Angle Triangle Similarity Theorem. We then apply these concepts to determine that in similar figures, corresponding angles are congruent, and sides are proportional.

The unit balances proofs with practical applications of similarity, including finding unknown side lengths and angles. Early lessons focus on proving theorems with transformations and dilations, while later lessons use shortcuts like the Angle-Angle Triangle Similarity Theorem for justification and calculation. Students may use calculators to ease computation and focus on reasoning or leave answers in non-computational forms.

The unit previews essential concepts for understanding trigonometry, including applications of similar right triangles and indirect measurement techniques that will be explored further in later units.