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Understand the idea, work through an example, then put it into practice.
61 lessons
Power Rule
Differentiate powers using the exponent and coefficient rule.
Open lessonPower Rule Integration
Integrate powers and evaluate definite integrals using antiderivatives.
Open lessonFormula Substitution and Variable Relationships
NJATC/ETA practice in substitution, signs, and how one variable changes when another increases.
Open lessonAlgebra Primer
A specially curated native introductory guide covering core algebraic concepts.
Open lessonCombinatorics I
An introduction to combinations, permutations, and discrete counting principles.
Open lessonCounting Specific Poker Hands
A focused lesson on the Rank-Then-Suit strategy for counting exact poker hand configurations using combinations.
Open lessonBurnside's Lemma
A step-by-step guide to Burnside's Lemma for counting distinct objects under rotational symmetry.
Open lessonProbability Fundamentals
A broad introduction to probability: sample spaces, events, axioms, and the classical formula. Covers complementary counting, uniform probability, and worked real-world examples.
Open lessonBinomial Distribution
The binomial model for repeated independent trials. Covers the binomial formula, mean, variance, and normal approximation with worked examples.
Open lessonConditional Probability & Bayes' Theorem
How new information updates probability estimates. Covers conditional probability, the multiplication rule, and Bayes' theorem with medical-test and dice examples.
Open lessonExpected Value & Linearity
The weighted average of a random variable's outcomes. Covers the definition, linearity of expectation, and applications to dice, games, and the coupon-collector problem.
Open lessonGeometric Distribution
Counts the number of trials until the first success. Covers the PMF, CDF, memoryless property, and expected value.
Open lessonPoisson Distribution
Models rare events over a fixed interval. Covers the Poisson PMF, relationship to the binomial, and the Poisson approximation theorem.
Open lessonNegative Binomial Distribution
Counts trials until the r-th success. Covers the PMF, relationship to the geometric distribution, and worked examples.
Open lessonHypergeometric Distribution
Sampling without replacement from a finite population. Covers the PMF, comparison with binomial, and quality-control applications.
Open lessonGeometric Probability
Probability defined by areas and lengths in continuous spaces. Covers uniform distributions on intervals and regions, and dart-board style examples.
Open lessonIndependent Events
When the occurrence of one event has no effect on another. Covers the product rule, mutual independence, and common misconceptions.
Open lessonThe Monty Hall Problem
A deep dive into the famous three-door paradox. Covers the conditional probability argument, simulation intuition, and generalizations.
Open lessonMartingales & the Gambler's Ruin
Fair-game stochastic processes and stopping times. Covers martingale definition, optional stopping theorem, and the Gambler's Ruin problem.
Open lessonPólya Urn & Exchangeability
A reinforcement model for probability. Covers the Pólya urn process, exchangeability, and the Beta-Binomial connection.
Open lessonProbability Generating Functions
A transform technique for discrete distributions. Covers the PGF definition, recovering probabilities and moments, and convolution of distributions.
Open lessonRandom Walks
Sequential coin-flip paths on the integer line. Covers simple random walks, return-to-origin probability, the reflection principle, and streak probabilities.
Open lessonThe Birthday Problem
How many people does it take before two share a birthday? Covers the complementary counting approach, the birthday paradox formula, and generalizations.
Open lessonAlgebra Essentials
Core algebraic techniques: solving quadratics, factoring, sequences, logarithms, and polynomial theorems. Includes the quadratic formula, Vieta's formulas, and the Factor Theorem.
Open lessonNumber Theory
Divisibility, primes, modular arithmetic, and number-theoretic functions. Covers GCD, LCM, Euler's totient, Wilson's theorem, and the Chinese Remainder Theorem.
Open lessonDerivatives of Trigonometric Functions
An introduction to the derivatives of sine, cosine, and tangent. Covers the geometric definition of trig derivatives, key formulas, and applications.
Open lessonThe Unit Circle
Exact values of sine, cosine, and tangent at the fundamental angles using the unit circle. Covers the construction of the unit circle and all quadrant signs.
Open lessonTrigonometric Ratios
SOH-CAH-TOA and the definitions of the six trig functions via right-triangle ratios. Covers the 30-60-90 and 45-45-90 special triangles.
Open lessonACT Trigonometry
A focused ACT Math lesson covering right-triangle ratios, unit-circle signs, identities, interval equations, the Laws of Sines and Cosines, arc length, and sinusoidal models.
Open lessonQuadrant-Constrained Double-Angle Identities
An exact-value trigonometry subtopic where quadrant information determines signs before a double-angle identity is applied.
Open lessonReciprocal Trig Functions
Definitions and values of cosecant, secant, and cotangent. Covers their relationship to sin, cos, tan, and exact values at standard angles.
Open lessonPythagorean Identities
The three fundamental Pythagorean identities and how to derive and apply them. Covers proofs from the unit circle and uses in simplifying expressions.
Open lessonEvaluating Trigonometric Expressions
A step-by-step guide to evaluating expressions involving sec, sin, and tan at special angles. Covers reciprocal trig functions, Pythagorean identities, and the 30-60-90 triangle.
Open lessonIdentity-First Exact Trig Expression Evaluation
A focused lesson for expressions that pair secant-squared minus tangent-squared with squared cosecant at a special angle, using identity recognition before exact-value evaluation.
Open lessonEvaluating Trig Expressions (Practice 1)
A detailed lesson on evaluating sine, cosine, and tangent expressions using the sine-cosine Pythagorean identity.
Open lessonEvaluating Trig Expressions (Practice 2)
A detailed lesson on evaluating sine, cosine, and tangent expressions using the fundamental Pythagorean identity and special right triangles.
Open lessonEvaluating Trig Expressions (Practice 3)
A detailed lesson on evaluating cosecant and cotangent expressions using the cosecant-cotangent Pythagorean identity.
Open lessonComplementary Sine and Cosine Values
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonTangent and Cotangent Reciprocal Sum
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonDouble-Angle Sine at 45 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonDouble-Angle Cosine at 30 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonSecant–Tangent Pythagorean Identity
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonCosecant–Cotangent Pythagorean Identity
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonSine–Cosine Pythagorean Identity
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonTangent Squared at 60 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonCosecant Minus Cotangent at 30 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonSine Double Angle at 30 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonCosine of a Sum of Angles
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonSine with a 150-Degree Reference Angle
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonCosine in Quadrant II at 135 Degrees
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonTangent as a Sine–Cosine Ratio
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonSecant as the Reciprocal of Cosine
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Open lessonMatrix Addition & Scalar Multiplication
Element-wise matrix operations. Covers matrix addition, subtraction, scalar multiplication, and the zero matrix.
Open lessonThe Matrix Trace
The sum of the diagonal entries of a square matrix. Covers the definition, linearity, cyclic property, and relationship to eigenvalues.
Open lessonThe Determinant (2×2)
Computing and interpreting the determinant of a 2×2 matrix. Covers the ad-bc formula, geometric meaning as area scaling, and conditions for invertibility.
Open lessonDeterminants (General)
Determinants of n×n matrices via cofactor expansion. Covers properties, the identity matrix determinant, and the effect of row operations.
Open lessonMatrix Properties & Powers
Key algebraic properties of matrices including the determinant of a matrix power, transpose rules, and invertibility conditions.
Open lessonThe Dot Product
The inner product of two vectors. Covers the algebraic and geometric definitions, projection, angle between vectors, and the Cauchy-Schwarz inequality.
Open lessonOrthogonal Vectors
Vectors that meet at a right angle. Covers the dot-product test for orthogonality, orthogonal complements, and the Gram-Schmidt process.
Open lessonSingular Matrices & Linear Dependence
When a matrix has no inverse. Covers linear dependence of columns, zero determinant, and the connection to null spaces and inconsistent systems.
Open lessonVisual Regression Tests
Quality assurance test book for verifying layout, styles, and KaTeX math equation rendering.
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