Course Curriculum
Math Research (MER)
Advanced Geometry (MG91), Combinatorics (ME91)
Textbooks
We will use parts of the following books at various points through the course, as well as many supplementary materials.
Geometry Revisited, Coxeter and Greitzer
A Vector Space Approach to Geometry, Hausner
Calculus, AP* Edition, Larson
A First Course in Probability, Ross
Elementary Number Theory, Dudley
Course Requirements
In addition to the standard array of homework, quizzes, and exams for the Analysis components of the course (Advanced Geometry, Combinatorics), the following work will be required for the Research component of the course:
- One major research paper (at least 20 pages) per term, that demonstrates a significant amount of original work
- One 10-minute presentation, plus Q&A, on your term paper topic
- One major group project/presentation requiring substantial technology use (Sketchpad, Geogebra, Excel, Maple, etc)
- In addition, several smaller writing assignments and projects shall be completed
Topics
Advanced Geometry
Review of Basic Geometry
Triangles; Quadrilaterals; Polygons; Circles; Coordinate Geometry; Elementary Solid Geometry
Methods of Proof
Synthetic Proof; Narrative Proof; Indirect Proof; Analytic Proof
Triangles
Ceva’s Theorem; Menelaus’ Theorem; Orthic Triangles; Excircles; Stewart’s Theorem;
Euler Line; Mass Point Geometry; Heron’s Formula
Quadrilaterals
Cyclic Quadrilaterals, Ptolemy’s Theorem; Brahmagupta’s Formula TheoremCircumscriptible Quadrilterals; Orthodiagonal Quadrilaterals
Trigonometry
Law of Sines; Law of Cosines; Cosine Rule; Circular Functions; Geometric Proofs of Trigonometric Identities
Using Trigonometric Substitutions to solve polynomial equations
Vector Geometry
Vector Proofs; 3-D Geometry with Vectors; Dot Product; Cross Product; Projections
Combinatorics
Counting
FCP; Permutations; Combinations; Permutations with Repetitions; Stars and Bars; Grid Walking;
Inclusion-Exclusion; Recursion; Generating Functions
Binomial Coefficients
Binomial Theorem; Proof by Induction; Binomial Identities; Mutlinomial Coefficients
Probability
Axiomatic Probability; Probability and Combinatorics; Condtional Probability; Bayes’ Theorem;
Experimental Probability
Elementary Statistics
Central Tendency; Measures of Dispersion; Projections; Representation of Data