Mathematical Foundations (complete)

210 core topics · approximately 57 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers [E]	Multi-digit addition and subtraction.
Multiplication [E]	Multiplying whole numbers.
Division [E]	Dividing whole numbers.
Order of Operations [M]	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting [M]	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing [M]	Sign rules for products and quotients.
Exponents [M]	Repeated multiplication in shorthand.
Square Roots [M]	Undoing a square.

Fractions · 7 topics

Equivalent Fractions [M]	Different fractions can name the same amount.
Simplifying Fractions [M]	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators) [M]	Same-denominator addition.
Adding Fractions (Unlike Denominators) [M]	Rewrite over a common denominator first.
Multiplying Fractions [M]	Multiply straight across.
Dividing Fractions [M]	Multiply by the reciprocal.
Mixed Numbers & Improper Fractions [M]	Converting between forms.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction [M]	Line up the decimal points.
Fractions ↔ Decimals [M]	Converting between the two notations.
Percent of a Number [M]	Percent means per hundred.
Percent Increase & Decrease [M]	Applying a percent change to a quantity.
Ratios & Proportions [M]	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions [M]	Substituting a value for a variable.
Combining Like Terms [M]	Adding the coefficients of matching variable parts.
The Distributive Property [M]	Multiplying across a sum.
One-Step Equations [M]	Undoing a single operation.
Two-Step Equations [M]	Undo addition/subtraction first, then multiplication.
Multi-Step Equations [H]	Equations needing distribution or variables on both sides.
Linear Inequalities [M]	Solving with <, >, ≤, ≥.

Linear Functions · 10 topics

The Coordinate Plane [M]	Locating points with (x, y) pairs.
Slope of a Line [M]	Rise over run between two points.
Slope-Intercept Form [M]	y = mx + b describes a whole line.
Finding a Line from Points [H]	Reconstructing y = mx + b from data.
Systems of Equations (Substitution) [H]	Two equations, two unknowns.
Elimination: A First Look [H]	Add or subtract equations so one variable cancels.
Systems: Word Problems [H]	Translating two facts into two equations.
Absolute Value Equations [H]	Distance equations have two answers.
Arithmetic Sequences [H]	Add the same amount each step.
Geometric Sequences [H]	Multiply by the same ratio each step.

Quadratics & Polynomials · 13 topics

Adding & Subtracting Polynomials [E]	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL) [M]	Expanding products of binomials.
Factoring Out the GCF [M]	Undoing the distributive property.
Factoring Trinomials [M]	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns [M]	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring [M]	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k [M]	Taking square roots of both sides — remembering ±.
Completing the Square [M]	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula [H]	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
The Discriminant [M]	b² − 4ac tells you how many real solutions exist.
Vertex of a Parabola [M]	The turning point at x = −b/2a.
Graphs of Quadratics [M]	Intercepts and symmetry of a parabola.
Quadratic Models [H]	Projectile motion and other parabolic models.

Radicals & Exponentials · 10 topics

Product Rule for Exponents [E]	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules [E]	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents [M]	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Scientific Notation [M]	Writing very large or very small numbers as c × 10ⁿ.
Simplifying Radicals [M]	Pulling perfect-square factors out of a square root.
Operations with Radicals [M]	Adding like radicals and multiplying square roots.
Rational Exponents [M]	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Radical Equations [H]	Isolate the radical, then square both sides.
Exponential Growth & Decay [M]	Quantities that multiply by the same factor each time step: y = a·bᵗ.
Compound Interest [H]	Money growing exponentially: A = P(1 + r)ᵗ.

Geometry · 14 topics

Angle Relationships [E]	Vertical, complementary, and supplementary angle pairs.
Parallel Lines & Transversals [E]	Angle pairs formed when a transversal crosses parallel lines.
Triangle Angle Sum [E]	The three angles of a triangle always add to 180°.
The Pythagorean Theorem [M]	In a right triangle, a² + b² = c².
Distance & Midpoint [M]	Measuring segments in the coordinate plane.
Similar Triangles [M]	Same shape, different size: corresponding sides are proportional.
Perimeter & Area [E]	Measuring around and inside basic shapes.
Circles: Area & Circumference [M]	C = 2πr and A = πr².
Composite Areas [H]	Adding and subtracting simple shapes to measure a complicated one.
Volume: Prisms & Cylinders [M]	Volume = base area × height.
Volume: Cones, Pyramids & Spheres [M]	Pointed solids hold one third of the matching prism; spheres use 4/3 πr³.
Surface Area [M]	The total area of all the faces of a solid.
Special Right Triangles [H]	The 45-45-90 and 30-60-90 side ratios.
Arc Length & Sector Area [H]	A central angle takes the same fraction of the circumference and the area.

Functions & Algebra II · 24 topics

Function Notation & Evaluation [E]	Reading f(x) notation and plugging in inputs.
Domain & Range [E]	Which inputs a function accepts, and which outputs it can produce.
Function Composition [M]	Feeding one function's output into another: f(g(x)).
Inverse Functions [M]	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Transformations of Functions [M]	How f(x − h) + k slides a graph around the plane.
Piecewise Functions [M]	Functions defined by different rules on different intervals.
Absolute Value Equations [M]	|x − a| = b splits into two linear equations.
Systems by Elimination [H]	Adding or subtracting equations to cancel a variable.
Nonlinear Systems [H]	Where a line meets a parabola: set the two formulas equal.
Complex Numbers [M]	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers [M]	Multiplying complex numbers with FOIL and i² = −1.
Quadratics with Complex Roots [H]	When the discriminant is negative, the roots come in a conjugate pair a ± bi.
Polynomial Division [H]	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems [M]	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials [H]	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials [E]	Far from the origin, only the leading term matters.
Simplifying Rational Expressions [M]	Factor top and bottom, then cancel the common factor.
Operations on Rational Expressions [H]	Multiplying and dividing algebraic fractions.
Rational Equations [H]	Clearing denominators to solve equations with x below the line.
Logarithms [M]	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms [M]	Logs turn products into sums, quotients into differences, powers into multiples.
Exponential & Log Equations [M]	Matching bases and rewriting between exponential and log form.
Arithmetic Sequences [M]	Sequences that grow by a constant difference each step.
Geometric Sequences [M]	Sequences that grow by a constant ratio each step.

Trigonometry · 15 topics

Right-Triangle Trigonometry [M]	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Solving for Sides with Trig [M]	Using a known angle and one side to find another side.
Degrees & Radians [E]	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle [M]	Exact sine, cosine, and tangent values at the special angles.
Trig of Any Angle [H]	Reference angles plus quadrant signs extend trig beyond 90°.
Graphs of Sine & Cosine [M]	Reading amplitude, period, and midline from y = a sin(bx) + c.
Phase Shifts & Other Trig Graphs [M]	Horizontal (phase) shifts of trig graphs, and the period of tangent.
The Pythagorean Identity [M]	sin²θ + cos²θ = 1 links sine and cosine of the same angle.
Basic Trig Identities [M]	Quotient, reciprocal, and even-odd identities.
Sum & Difference Formulas [H]	Expanding sin(A ± B) and cos(A ± B) to reach non-special angles.
Double-Angle Formulas [H]	sin 2x = 2 sin x cos x and cos 2x = 1 − 2 sin²x.
Trig Equations [H]	Isolating a trig function and reading solutions off the unit circle.
Law of Sines [H]	In any triangle, each side over the sine of its opposite angle is constant.
Law of Cosines [H]	c² = a² + b² − 2ab cos C generalizes the Pythagorean theorem.
Inverse Trig Functions [M]	arcsin, arccos, and arctan undo the trig functions on restricted ranges.

Precalculus · 12 topics

Asymptotes of Rational Functions [M]	Where rational functions blow up and where they level off.
Graphs of Rational Functions [M]	Holes, asymptotes, and intercepts tell the whole story of the graph.
Polynomial Inequalities [M]	Sign charts: zeros split the number line into test intervals.
Vectors: Components & Magnitude [M]	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations [M]	Scaling, adding, and dotting vectors — all component by component.
Parametric Equations [M]	Describing a moving point by giving x and y as functions of time.
Polar Coordinates [H]	Locating points by distance from the origin and angle from the x-axis.
Polar Graphs [M]	Recognizing circles, lines, and rose curves from polar equations.
Circles & Ellipses [M]	Reading centers, radii, and intercepts from conic equations.
The Binomial Theorem [H]	Expanding (x + a)ⁿ without multiplying it out term by term.
Sigma Notation & Series [M]	Σ compresses a sum: read the limits, add up the terms.
Average Rate of Change [M]	The slope of the secant line: (f(b) − f(a)) / (b − a).

Limits & Continuity · 10 topics

Limits: Graphical & Numerical [E]	What value a function approaches — which need not be the value it takes.
Evaluating Limits Algebraically [M]	Direct substitution — and the factor-and-cancel fix for 0/0.
One-Sided Limits [M]	Approaching from the left or right — and when the two disagree.
Limits by Rationalization [H]	Clearing a 0/0 form by multiplying by the conjugate.
Infinite Limits & Vertical Asymptotes [M]	Where a function blows up: reading the sign of an infinite limit.
Limits at Infinity [M]	End behavior of rational functions: compare the degrees.
Trig Limits [M]	The two special limits sin(x)/x → 1 and (1 − cos x)/x → 0.
The Squeeze Theorem [M]	Trapping a wild function between two tame ones with the same limit.
Continuity [H]	No jumps, holes, or blow-ups: the limit equals the value.
Intermediate Value Theorem [M]	A continuous function can't skip values: sign changes force roots.

Differentiation · 16 topics

The Limit Definition of the Derivative [M]	The derivative is the limit of average rates of change.
Derivatives Graphically [M]	Reading slopes off a graph.
The Power Rule [M]	d/dx xⁿ = n·xⁿ⁻¹ for any real n.
Sum & Constant-Multiple Rules [M]	Differentiate term by term.
The Product Rule [M]	(fg)′ = f′g + fg′.
The Quotient Rule [M]	(f/g)′ = (f′g − fg′)/g².
Derivatives of Trig Functions [M]	d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x.
Derivatives of Exponentials & Logs [M]	eˣ is its own derivative; (ln x)′ = 1/x.
The Chain Rule [H]	d/dx f(g(x)) = f′(g(x)) · g′(x).
Combining Differentiation Rules [H]	Product, quotient and chain rules together.
Implicit Differentiation [H]	Differentiating equations that mix x and y.
Derivatives of Inverse Trig [M]	(arcsin x)′ = 1/√(1−x²), (arctan x)′ = 1/(1+x²).
Derivatives of Inverse Functions [M]	(f⁻¹)′(b) = 1 / f′(f⁻¹(b)).
Higher-Order Derivatives [M]	Differentiating again: f″, f‴, …
Differentiability & Continuity [M]	Differentiable ⇒ continuous, but not conversely.
Tangent Lines & Linear Approximation [M]	The tangent line is the best local linear stand-in for f.

Applications of Differentiation · 10 topics

Motion: Position, Velocity, Acceleration [M]	v = s′, a = v′; at rest when v = 0.
Related Rates [H]	Differentiating a geometric relationship with respect to time.
Critical Points & Extrema [M]	Where f′ = 0 or is undefined — the candidates for extrema.
The Mean Value Theorem [M]	Somewhere, instantaneous rate equals average rate.
Increasing & Decreasing Intervals [M]	Sign of f′ decides the direction of f.
Concavity & Inflection Points [M]	f″ > 0 bends up, f″ < 0 bends down.
Curve Sketching & f, f', f'' [M]	Reading the shape of f from its derivatives.
Optimization [H]	Maximizing or minimizing with calculus.
L'Hôpital's Rule [M]	For 0/0 or ∞/∞, differentiate top and bottom.
Indeterminate Forms [M]	0·∞ and repeated applications.

Integration · 12 topics

Antiderivatives [M]	Reversing differentiation: the power rule backwards.
Antiderivatives: Trig & Exponential [M]	∫cos = sin, ∫sin = −cos, ∫eˣ = eˣ, ∫1/x = ln|x|.
Riemann Sums [M]	Approximating area with rectangles.
The Trapezoidal Rule [M]	Averaging left and right sums.
Properties of Definite Integrals [M]	Linearity, additivity, and orientation.
The Fundamental Theorem: Evaluating Integrals [M]	∫ₐᵇ f = F(b) − F(a).
Accumulation Functions & FTC Part 1 [H]	d/dx ∫ₐˣ f(t) dt = f(x).
u-Substitution [H]	The chain rule in reverse.
Integration by Parts (BC) [H]	∫u dv = uv − ∫v du.
Partial Fractions (BC) [H]	Splitting rational functions to integrate them.
Improper Integrals (BC) [H]	Integrals to infinity, defined by limits.
Integrals Yielding Inverse Trig [H]	1/(1+x²) → arctan, 1/√(1−x²) → arcsin.

Applications of Integration · 8 topics

Average Value of a Function [M]	f_avg = (1/(b−a)) ∫ₐᵇ f.
Motion: Displacement & Distance [H]	Displacement is ∫v; distance is ∫|v|.
Accumulation & Net Change [H]	Final amount = initial + ∫(rate).
Analyzing Accumulation Functions [H]	Reading g(x) = ∫f from the graph of f.
Area Between Curves [H]	∫(top − bottom) between the intersections.
Volumes: Disc & Washer [H]	V = π∫R² dx (discs), π∫(R² − r²) dx (washers).
Volumes by Cross-Section [H]	V = ∫A(x) dx for known cross-sectional areas.
Arc Length (BC) [H]	L = ∫√(1 + (y′)²) dx.

Differential Equations · 6 topics

Differential Equations: Verifying Solutions [M]	A solution is a function that satisfies the equation.
Slope Fields [M]	A picture of dy/dx at every point.
Euler's Method (BC) [H]	Stepping along tangent lines.
Separation of Variables [H]	Move all y's left, all x's right, integrate both sides.
Exponential Growth & Decay Models [M]	dy/dt = ky means y = y₀e^{kt}.
Logistic Growth (BC) [H]	Growth limited by a carrying capacity L.

Parametric, Polar & Vector Calculus · 8 topics

Parametric Derivatives (BC) [H]	dy/dx = (dy/dt)/(dx/dt).
Parametric Second Derivatives (BC) [H]	Differentiate dy/dx with respect to t, divide by dx/dt again.
Parametric Arc Length (BC) [H]	L = ∫√((dx/dt)² + (dy/dt)²) dt.
Vector-Valued Functions (BC) [H]	Differentiate component by component.
Motion in the Plane (BC) [H]	Speed is the magnitude of velocity.
Slopes of Polar Curves (BC) [H]	Convert to parametric: x = r cos θ, y = r sin θ.
Area in Polar Coordinates (BC) [H]	A = ½∫r² dθ.
Area Between Polar Curves (BC) [H]	Subtract the inner sweep from the outer sweep.

Infinite Series · 15 topics

Convergence of Sequences (BC) [M]	A sequence converges if aₙ approaches a limit.
Geometric Series (BC) [M]	Σarⁿ = a/(1−r) when |r| < 1.
The nth-Term Test (BC) [M]	If terms don't go to 0, the series diverges — but 0 proves nothing.
Integral Test & p-Series (BC) [M]	Σ1/nᵖ converges iff p > 1.
Comparison Tests (BC) [M]	Compare with a series you already understand.
Alternating Series (BC) [M]	Alternating + decreasing to 0 ⇒ converges.
Alternating Series Error Bound (BC) [H]	|error| ≤ first omitted term.
The Ratio Test (BC) [H]	L = lim|aₙ₊₁/aₙ|: L<1 converges, L>1 diverges, L=1 says nothing.
Absolute vs Conditional Convergence (BC) [M]	Does it still converge with all terms made positive?
Radius of Convergence (BC) [H]	The ratio test gives |x − c| < R.
Interval of Convergence (BC) [H]	Check both endpoints separately.
Taylor Polynomials (BC) [H]	Matching derivatives at a point: Pₙ(x) = Σ f⁽ᵏ⁾(a)(x−a)ᵏ/k!.
Taylor & Maclaurin Series (BC) [H]	The big four: eˣ, sin x, cos x, 1/(1−x).
Manipulating Known Series (BC) [H]	Substitute, multiply, differentiate, integrate known series.
Lagrange Error Bound (BC) [H]	|Rₙ| ≤ M|x−a|ⁿ⁺¹/(n+1)!.

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