高中幾何：適合優秀學習者的直角三角形三角學

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 our prior knowledge of right triangles, starting with what we learned in grade 4 and expanding on concepts from grade 8 like the Pythagorean Theorem and slope. Now, we use that foundation to introduce trigonometric ratios in familiar contexts.

We begin by exploring special cases of similar right triangles, focusing on how right triangles with one congruent acute angle are similar. Though not required by the standards, these lessons offer valuable practice, especially for college exams. We then gather data on side length ratios, organizing it into a table to help us understand relationships before learning trigonometric terms like cosine, sine, and tangent.

After practicing with a calculator to find these ratios, we apply them to real-world problems, learning to check if our answers are reasonable and how precise they need to be. By the end of the unit, we also explore how mathematicians have historically refined the value of π using polygons.

Throughout the unit, we’ll notice patterns, such as the relationship between cosine and sine, before formally learning the concepts. Later, we use these patterns to prove that the ratios for complementary angles are equal. We’ll also estimate angles using our reference table before calculating them with inverse trigonometric functions like arcsine, arccosine, and arctangent.

As we progress, we’ll continue adding to our reference charts, which help us track key concepts and calculations