The Keystone School · grades 6–12
High-School Math (Algebra → Calculus)
From the first unknown to the fundamental theorem — the long staircase, one honest step at a time.
Variables, equations, and lines — the language every later mathematics is written in.
The one high-school subject built entirely on reasons — where every claim must earn its because.
Syllabus · 4 units · ~30 hours
Unit I — Foundations & Proof
Points, lines, planes, and undefined terms · Definitions, postulates, and theorems · Writing a two-column proof · Angle relationships and the parallel postulate
Unit II — Triangles & Congruence
Triangle classification and angle sums · Congruence criteria: SSS, SAS, ASA, AAS · Isosceles and equilateral triangles · Triangle inequality and the hinge theorem
Unit III — Similarity & Right Triangles
Similar figures and scale factor · The Pythagorean theorem and its converse · Special right triangles · Introductory trigonometric ratios
Unit IV — Circles, Area & Solids
Chords, arcs, and inscribed angles · Area of polygons and circles · Surface area and volume of solids · Coordinate geometry and the distance formula
The function family tree — quadratics, exponentials, and logarithms, and the situations each one describes.
Syllabus · 4 units · ~34 hours
Unit I — Functions Reconsidered
Function notation and transformations · Domain, range, and end behavior · Composition and inverse functions · Piecewise-defined functions
Unit II — Quadratics & Complex Numbers
Vertex, standard, and factored forms · Completing the square · Complex numbers and the imaginary unit · Solving and graphing quadratic systems
Unit III — Polynomials & Rational Functions
Polynomial division and the remainder theorem · Finding roots and the factor theorem · Rational expressions and asymptotes · Radical equations
Unit IV — Exponential & Logarithmic Functions
Exponential growth and decay · The number e and continuous growth · Logarithms as inverses of exponents · Solving exponential and logarithmic equations
The mathematics of angle and rotation — from surveying a hillside to describing a wave.
Syllabus · 3 units · ~24 hours
Unit I — Right-Triangle Trigonometry
Sine, cosine, and tangent ratios · Solving right triangles · Angles of elevation and depression · The Pythagorean identity
Unit II — The Unit Circle
Radian measure · The unit circle and reference angles · Trigonometric functions of any angle · Periodic behavior
Unit III — Graphs & Identities
Graphing sine, cosine, and tangent · Amplitude, period, and phase shift · Fundamental identities · The laws of sines and cosines
The last mathematics before the calculus — where functions are stretched, combined, and pushed toward their limits.
Syllabus · 3 units · ~36 hours
Unit I — Advanced Functions
Polynomial and rational functions in depth · Exponential and logarithmic modeling · Transformations and symmetry · Systems and matrices
Unit II — Trigonometry Extended
Inverse trigonometric functions · Sum, difference, and double-angle identities · Polar coordinates · Vectors in the plane
Unit III — Sequences, Series & Limits
Arithmetic and geometric sequences · Summation notation and series · The idea of a limit · Continuity and behavior near a point
Two questions — how fast, and how much — answered by the derivative and the integral, and joined at last.
Syllabus · 4 units · ~44 hours
Unit I — Limits & Continuity
Limits, one-sided and infinite · Continuity and the intermediate value theorem · Limits at infinity and asymptotic behavior · The definition of the derivative as a limit
Unit II — The Derivative
Rules of differentiation · The chain rule and implicit differentiation · Derivatives of trigonometric, exponential, and log functions · Related rates
Unit III — Applications of the Derivative
Increasing, decreasing, and concavity · Optimization problems · The mean value theorem · Linear approximation
Unit IV — The Integral
Antiderivatives and indefinite integrals · Riemann sums and the definite integral · The fundamental theorem of calculus · Area between curves