Mathematics courses often teach students how to solve problems, use algorithms, and number crunch. Mathematical proofs are often taught in Geometry, with a focus on form and exact detail over the elegance and excitement of deep understanding.
Our Special Topics and Mathematical Explorations courses teach students how to pose problems, develop algorithms, explore ideas, prove (both formally and informally) their methods and ideas work, and propose next steps. Students can use the skills learned in these classes to stretch their regular math curriculum, challenge their assumptions about mathematics, and truly think like a mathematician.
This course explores ideas in geometry, where they come from, how to come up with your own, how to apply them in interesting situations, and how to pose problems and think deeper. It is not based on your typical high school geometry course – the concepts we play with are usually taught prior to high school geometry, or are not part of the standard curriculum at all. The course is not intended to replace a typical curriculum, but rather to deepen and extend, to introduce new ideas, and to foster mathematical thinking.
We’ll look at the reasons behind the formulas and relationships you may already know, and largely derive them ourselves. We’ll think about shapes and figures deeply, considering how one would have to approach unknown shapes to determine their formulas, working on developing a spatial awareness and geometric reasoning rather than knowing and applying formulas. We will then explore making changes to the shapes (in two or three dimensions) and how we can change the formulas to deal with our new figures. We will design nets for the 3-dimensional figures, both to help improve our spatial awareness and to help us figure out the surface area. We’ll stretch everything as we go into non-euclidean spaces as well.
Students will do best in this class if they have done basic geometry formulas in the past – area, perimeter, surface area, and volume of the basic shapes. Variables will be used in this class, but not beyond the pre-Algebra level except for extra extensions and challenge work.
Syllabus is subject to change based on student interests and abilities.
One should not look at the syllabus and expect it to be too easy if you’ve encountered these topics before. The approach and the depth of the problem solving will engage even experienced geometer.
Prerequisites: None
All times are in Pacific Time. There will be one break week.
SYLLABUS: subject to change based on student interests and abilities
Week 1: Introduction – Euclidean and non-Euclidean geometry, the concepts of plane and space, what we’ll be doing, concept of axoims, postulates, etc. Is geometry an invention or a discovery?
Week 2: Area and perimeter formulas: deriving the why behind the formulas we have, do those still work in non-euclidean geometries, problem solving with our formulas/understanding.
Week 3: Triangle Inequality Theorem – deriving it, using it, applying it to polygons with more sides, considering if it works in non-Euclidean geometries.
Week 4: Pythagorean Theorem Proofs and problem solving – deriving it, exploring interesting ways of proving it, using it to help us approach the formulas we already know differently.
Week 5: Exploring shapes without simple formulas, including irregular shapes – pose problems about these.
Week 6: Similarity and congruence, and using it to solve problems, pose problems
Week 7: Angles, transversals, deep and extended problem posing and solving, non-Euclidean ideas
Week 8: Exploration of nets: Finding nets in pentominoes, drawing nets of rectangular prisms in many different ways.
Week 9 Prisms – right and oblique – nets, surface area, volume – how do we change a solid and how does that impact the formulas we know? Extended problem solving.
Week 10: Pyramids – right and oblique, deriving the formula for volume, pyramidal frustums. Introduction to using limits.
Week 11: Cylinders, cones, and spheres – deriving formulas for surface area and volume, exploration of nets. Conic frustums, spheric sections. Further use of limits.
Week 12: Platonic and Archimedean solids. Naming conventions, surface area and volume concepts.
Week 13: Surface area and volumes of unusual shapes
Week 14: Further explorations – extended problem solving, other ideas that have arisen during class
Week 15: Sharing of projects and wrap-up.