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Current Unit - Shadows

What is the length of a shadow? We begin this unit with Euclid's 5 postulates as a foundation for all Geometry on a plane. The concept of similarity is the central theme of this unit. Through this concept, students explore the following important ideas from geometry and algebra.

Similarity and Congruence

- Developing intuitive ideas about the meaning of “same shape” and learning the formal definitions of similar and congruent
- Understanding the role of similarity in defining the trigonometric functions of sine, cosine and tangent Proportional Reasoning and the Algebra of Proportions
- Understanding the meaning of proportionality in connection with similarity
- Developing equations of proportionality from situations involving similar figures
- Understanding the role of proportionality in nongeometric situations
- Discovering the triangle inequality and investigating its extension to polygons

- Working with the concept of counterexample in understanding the criteria for similarity
- Understanding the role of the parallel postulate in proofs

- Learning standard terminology for triangles, including hypotenuse, leg, opposite side, and adjacent side
- Learning the right triangle definitions of sine, cosine, and tangent
- Using sine, cosine, and tangent to solve
real-world problems

Unit 3 - Pit and Pendulum

This unit opens with an excerpt from The Pit and the Pendulum, by Edgar Allan Poe. In the story, a
prisoner is tied down while a pendulum with a sharp blade slowly descends. If the prisoner does not
act, he will be killed by the pendulum. When the pendulum has about 12 swings left, the prisoner
creates a plan for escape and executes it. Students are presented with the problem of whether the
prisoner would have enough time to escape. To resolve this question, students construct pendulums
and conduct experiments to find out what variables determine the period of a pendulum and what the
relationship is between the period and these variables. In the process, students are introduced to the
normal distribution and the standard deviation as tools for determining whether a change in one
variable really does affect another. They make and refine conjectures, analyze data collected from
experiments, and learn about quadratic equations and explore curve fitting. Finally, after deriving a
theoretical answer to the problem, students actually build a 30-foot pendulum to test their theory.