Math Placement Prep
Comprehensive training modules covering math, science, and engineering — from arithmetic through calculus, physics, and beyond.
120
Modules
556
Lessons
16255
Minutes
29
Categories
Start Learning
See how calculus, differential equations, and algebra propel rockets into orbit. From the Tsiolkovsky equation to orbital mechanics, every launch is powered by math.
Discover how differential equations and vector calculus govern the flow of water, air, and blood. Every wing that lifts, every pipe that flows, every weather pattern — it's all math.
Explore how math determines why steel bends, why glass shatters, and how we engineer materials from skyscrapers to smartphone screens.
Learn how CT scans, MRIs, and ultrasounds use linear algebra, Fourier transforms, and trigonometry to see inside the human body — saving millions of lives.
Explore how differential equations, statistics, and enormous numerical simulations predict weather and model Earth's changing climate.
Discover how linear algebra, trigonometry, and geometry power every video game, animated movie, and 3D visualization you've ever seen.
See how exponential functions, statistics, and calculus drive the global economy — from compound interest to options pricing to risk management.
Discover how stoichiometry, logarithms, and exponential decay are the heartbeat of every chemical reaction — from baking soda volcanoes to pharmaceutical design.
See how Newton's laws, kinematic equations, and energy conservation use algebra, trigonometry, and calculus to predict how every object in the universe moves.
Explore how engineers use algebra, geometry, and calculus to design bridges, calculate gear ratios, size electrical circuits, and build the modern world.
See how the laws of heat, energy, and entropy use algebra, calculus, and logarithms to govern everything from car engines to refrigerators to the fate of the universe.
Learn how every sensor in the modern world — from thermistors to GPS satellites — converts physical quantities into numbers using transfer functions, logarithms, and signal processing.
Discover how sine waves, differential equations, and resonance govern everything from musical instruments to earthquakes to radio signals.
Explore Einstein's Special Relativity — time dilation, length contraction, and E=mc². Watch space and time warp as you slide toward the speed of light.
Explore quantum mechanics through vivid interactive simulations — wave interference, wavefunctions, the Bloch sphere, and quantum computing circuits.
See how thermodynamics, algebra, and fluid mechanics power jet engines. From the Brayton cycle to thrust equations, every flight is a math problem solved at 35,000 feet.
Discover the math behind electricity generation — from steam-powered Rankine cycles to ultra-efficient combined-cycle gas plants. Every power switch you flip was enabled by thermodynamics.
Master the fundamentals of whole numbers — place value, rounding, and the four basic operations — which form the foundation for all higher mathematics.
Learn to work with fractions — simplifying, comparing, and performing all four operations — a critical skill for algebra readiness.
Understand decimal notation and perform arithmetic with decimals — essential for working with money, measurement, and scientific data.
Master percent concepts — conversions, percent equations, percent change, and real-world applications like tax, tip, discount, and interest.
Understand ratios, rates, and proportions — how to set them up, solve them, and apply them to real-world problems from unit pricing to scale drawings.
Learn to work with positive and negative numbers — essential for the transition from arithmetic to algebra.
Learn to solve one-variable linear equations and inequalities — the gateway skill for all of algebra.
Graph linear equations, understand slope and intercepts, and solve systems of two linear equations using substitution, elimination, and graphing.
Understand the concept of relations and functions — domain, range, function notation, and the vertical line test.
Master the laws of exponents, scientific notation, and polynomial factoring techniques — from GCF through grouping and special products.
Study quadratic equations, the quadratic formula, vertex form, and extend to higher-degree polynomials — end behavior, zeros, and graphing.
Master simplifying radicals, rationalizing denominators, operations with radical expressions, and the connection between radicals and rational exponents.
Understand exponential growth and decay, master logarithmic properties, and solve exponential and logarithmic equations — essential for science, finance, and advanced mathematics.
Master angle measurement, the six trigonometric functions, right-triangle and unit-circle trigonometry, identities, and solving trigonometric equations.
Review essential geometry concepts including angles, triangles, polygons, circles, perimeter, area, and volume — all commonly tested on placement exams.
Build a first-course foundation in limits, derivatives, integrals, and the core modeling ideas that power modern calculus.
Learn matrices, systems, vectors, determinants, and the language of linear structure used across mathematics, science, and computing.
Study data summaries, probability, distributions, inference, and regression so you can read, model, and critique quantitative information.
Nassim Taleb's statistical worldview: Mediocristan vs Extremistan, power laws, black-swan risk, fragility/antifragility, and the barbell strategy for tail hedging.
Quantitative methods behind modern sports analytics: Pythagorean expectation, Elo ratings, regression to the mean, expected goals (xG), and advanced metrics like WAR and OPS+.
Master probability theory for random processes, Markov chains, Brownian motion, martingales, and Ito calculus.
Master Bayesian statistical inference, prior/posterior distributions, MCMC methods, and hierarchical modeling.
Study equations involving derivatives, learn standard first-order methods, and model growth, decay, and oscillation.
Coupled ODE/PDE systems where two or more physical domains interact: mechanical–mechanical, electromechanical, thermo-chemical, and reaction–diffusion.
Cover core discrete ideas including logic, sets, counting, recursion, modular arithmetic, and basic graph theory.
Explore counting techniques, probability rules, conditional probability, and expected value to analyze random events and combinatorial structures.
Study arithmetic and geometric sequences, summation notation, convergence tests, and power series to build a deep understanding of infinite processes.
Master the algebra and geometry of complex numbers including arithmetic, polar form, De Moivre's theorem, roots of unity, and poles and zeros of rational functions.
Study the four conic sections — circles, parabolas, ellipses, and hyperbolas — their standard equations, key features, and real-world applications.
Extend calculus to functions of several variables — partial derivatives, multiple integrals, gradient vectors, and an introduction to vector calculus.
Explore set operations, Venn diagrams, propositional logic, truth tables, and methods of mathematical proof.
Master the mathematics of money — simple and compound interest, annuities, loan amortization, and introductory investment analysis.
Approximate solutions when exact answers are impractical — root-finding, interpolation, numerical integration, and methods for ordinary differential equations.
Find the best solution — unconstrained and constrained optimization, linear programming, and Lagrange multipliers with real-world applications.
Explore Einstein's special relativity — time dilation, length contraction, relativistic momentum, and mass-energy equivalence. See how algebra and square roots reveal the surprising geometry of space and time at high speeds.
Mathematical navigation from coordinates and bearings to GPS trilateration and Kalman filtering for inertial/GPS fusion.
Rigorous foundations of calculus: the real number system, sequences, continuity, differentiation, Riemann integration, series of functions, and metric spaces.
Master the elegant theory of functions of a complex variable: analytic functions, Cauchy's theorem, residues, conformal mappings, and analytic continuation.
Master Fourier series, the Fourier transform, DFT/FFT, signal processing, distributions, and wavelet analysis — the mathematics of frequency and signal decomposition.
Master the algebraic structures underlying mathematics: groups, rings, fields, Galois theory, and representation theory — the language of symmetry and structure.
Master graph theory from connectivity and coloring to spectral methods and Ramsey theory — the mathematics of networks and combinatorial structure.
Explore the mathematics of shape, continuity, and global structure: topological spaces, homotopy, surfaces, homology, and De Rham cohomology.
Master strategic interaction through Nash equilibrium, mechanism design, evolutionary dynamics, and cooperative game theory — the mathematics of rational decision-making in multi-agent settings.
Master Shannon's mathematical theory of information: entropy, mutual information, channel capacity, source coding, rate-distortion theory, and the deep connections to statistics, physics, and computation.
Master the mathematics that drives chemistry — from balancing equations with algebra, to pH with logarithms, to reaction kinetics with exponentials and calculus.
PhD-level treatment of one-component and binary phase diagrams: Gibbs phase rule, chemical potential equality, Clausius-Clapeyron equation, Maxwell equal-area construction, van der Waals fluid, eutectic systems, lever rule, and three fully interactive explorers.
Mathematical modeling of biological systems: population dynamics, epidemiology, reaction-diffusion, evolutionary dynamics, and neuroscience.
Earth's interior structure, magnetic field, plate tectonics, heat flow, and climate cycles.
Explore how Newton's laws, kinematics, energy, and electromagnetism are built on algebra, trigonometry, and calculus.
Explore magnetic fields, forces on charges and currents, Faraday's law of induction, solenoids, and electromagnetic waves. Interactive tools let you visualize field lines, compute forces, and see how electricity and magnetism are two aspects of the same phenomenon.
Master the mathematics of orbital motion — from Kepler's laws and circular orbits to Hohmann transfers, escape velocity, and gravitational potential energy. Learn how Newton's law of gravitation, conic sections, and energy conservation govern the trajectories of satellites, planets, and spacecraft.
A PhD-level exploration of quantum mechanics and quantum computing — from Hilbert spaces and the Schrödinger equation through entanglement, quantum algorithms, and error correction. Develop deep intuition through interactive visualizations of wavefunctions, the Bloch sphere, and quantum circuits.
Master the elegant reformulations of Newtonian mechanics through Lagrangian and Hamiltonian approaches, variational principles, phase space analysis, and canonical transformations.
Maxwell's equations, electromagnetic waves, radiation, and modern electrodynamics.
From microstates to macroscopic thermodynamics via probability, partition functions, and quantum statistics.
Physical optics from first principles: wave optics, diffraction, interference, coherence, laser physics, nonlinear optics, and fiber optics.
Einstein's geometric theory of gravitation: spacetime curvature, the Einstein field equations, black holes, gravitational waves, and cosmology.
The Standard Model, quantum field theory, and the fundamental structure of matter and forces.
Learn the algebra, geometry, and trig that engineers use to design circuits, size beams, calculate gear ratios, and analyze control systems.
Apply math to structural analysis, beam loading, soil mechanics, fluid flow in civil systems, and surveying calculations.
Explore circuit analysis, AC phasors, signals and systems, transfer functions, and control theory including poles and zeros.
Apply math to orbital mechanics, aerodynamic lift and drag, rocket propulsion, and flight dynamics.
Apply math to digital logic, binary arithmetic, Boolean algebra, algorithm complexity, and data structures.
Apply math to operations research, quality control, scheduling, inventory management, and process optimization.
Apply math to system modeling, reliability engineering, feedback systems, requirements analysis, and decision theory.
Explore the mathematics of nuclear engineering — radioactive decay, chain reactions, reactor criticality, neutron diffusion, radiation shielding, and nuclear energy. From exponential decay laws to the four-factor formula, learn how algebra, calculus, and differential equations govern nuclear power, medical isotopes, and radiation safety.
Feedback control foundations: transfer functions, step response, stability, PID tuning, and frequency-domain design (Bode, margins, root locus).
Master the math of heat, energy conversion, and entropy — from the ideal gas law to Carnot efficiency to calorimetry.
Master steady-state heat conduction through single and composite walls, cylindrical systems, and the thermal resistance network method.
Learn forced and natural convection, Newton’s law of cooling, and the key dimensionless numbers (Re, Nu, Pr, Gr, Ra) used in convection correlations.
Analyze and size heat exchangers using the Log Mean Temperature Difference (LMTD) method for parallel-flow, counter-flow, shell-and-tube, and cross-flow configurations.
Analyze heat exchangers using the effectiveness-NTU method, which avoids iteration when outlet temperatures are unknown. Covers all major configurations with worked problems.
Analyze extended surfaces (fins) used to enhance heat transfer. Covers rectangular, pin, and annular fins with efficiency, effectiveness, and fin array calculations.
A rigorous graduate-level treatment of the Carnot cycle: exact state-point thermodynamics, entropy generation, availability/exergy, endoreversible engine theory, and the foundations of the second law.
Explore the math behind modern sensors and measurement systems — transfer functions, signal processing, error analysis, and the decibel scale.
Platinum RTDs, thermistors, strain gauges, potentiometers and photoresistors — the classical resistive transducers, treated in the analytical style of Webster and Pallàs-Areny.
Capacitive and inductive transducers — parallel-plate sensors, LVDTs, and eddy-current probes — in the signal-conditioning tradition of Pallàs-Areny and Webster.
Thermocouples, piezoelectric elements, and photodiodes — passive transducers that produce their own signal without external excitation.
Wheatstone bridges, instrumentation amplifiers, anti-alias filters, and the ADC — the complete analog front-end for modern sensors.
Learn how sine waves, differential equations, and Fourier analysis govern vibrations in structures, musical instruments, and modern electronics.
Turn numbers into insight — choose the right chart, design effective graphics, and master the scientific computing tools (MATLAB, R, Scilab, LabVIEW, Python) used to visualize and analyze data.
The mathematical theory of algorithms: asymptotic analysis, sorting, dynamic programming, NP-completeness, randomized algorithms, and network flow.
The rigorous mathematical foundations of machine learning: optimization, statistical learning theory, kernel methods, neural networks, and PAC learning.
Number theory, public-key cryptography, symmetric ciphers, hash functions, and post-quantum security.
A mathematically rigorous tour of economics — from supply and demand to game theory, behavioral biases, growth models, monetary policy, and international trade. Every topic is grounded in equations and quantitative reasoning.
Explore David Ricardo's theory of differential rent — how land-quality differences create economic surpluses, how cultivation expands at the extensive and intensive margins, and how the concept of economic rent extends to modern resources, urban land, and talent markets.
PhD-level treatment of Programmable Logic Controller internals: CPU organisation, memory maps, I/O subsystems, the deterministic scan-cycle model, real-time OS scheduling, and the zero-order-hold (ZOH) sampling framework used to design stable discrete-time control loops on PLCs.
PhD-level treatment of IEC 61131-3 programming languages for PLCs: Ladder Diagram (LD), Boolean algebra reduction via Karnaugh maps, Structured Text (ST), timer/counter function blocks, and formal verification of safety-critical logic using Boolean satisfiability and temporal logic.
PhD-level treatment of industrial robot mechanics: Denavit–Hartenberg parameterisation, forward and inverse kinematics for serial manipulators, the velocity Jacobian, singularity analysis, workspace computation, and trapezoidal/S-curve trajectory generation for smooth, jerk-limited motion.
PhD-level operational amplifier analysis: ideal and non-ideal op-amp models, negative feedback stability (gain-bandwidth product, phase margin), input-referred noise analysis, and systematic active filter design using the Butterworth and Chebyshev approximations in Sallen–Key and multiple-feedback topologies.
PhD-level analysis of bipolar junction transistor (BJT) and MOSFET amplifiers: DC bias analysis using the Thevenin voltage-divider equivalent, small-signal hybrid-π and T models, mid-band gain derivation, and frequency response including Miller capacitance and the short-circuit current-gain bandwidth product f_T.
PhD-level digital design: from Boolean algebra and Karnaugh map minimisation through combinational hazards, CMOS gate implementation, D/JK flip-flops, synchronous and asynchronous finite state machines, metastability and clock-domain crossing, with formal verification concepts using temporal logic.
PhD-level combustion science: stoichiometric and non-stoichiometric fuel-air ratios, Orsat and NDIR flue-gas analysis, first-law adiabatic flame temperature calculation using sensible enthalpy integration, chemical equilibrium at high temperature, and extended Zeldovich NOx kinetics with implications for burner design and emissions compliance.
PhD-level solar thermal engineering: the Hottel–Whillier–Bliss (HWB) equation for flat-plate and evacuated-tube collector performance, stagnation temperature and instantaneous efficiency, solar geometry (declination, hour angle, altitude, azimuth), irradiation models (clear-sky beam + diffuse + reflected), and thermal storage sizing using the f-chart method.
PhD-level geothermal engineering and power cycle analysis: geothermal resource classification, single-flash and binary ORC cycle thermodynamics using second-law (exergy) analysis, working fluid selection criteria (critical temperature, ODP, GWP, flammability), turbine design constraints, and brine reinjection for sustainability of hydrothermal systems.
PhD-level pharmacokinetics: one- and two-compartment models for intravenous bolus, infusion, and oral dosing; derived PK parameters (volume of distribution, clearance, bioavailability, half-life, AUC); multiple-dose accumulation and steady-state concentration; population PK with nonlinear mixed-effects models and the implications for individualised dosing.
PhD-level pharmacodynamics: receptor occupancy theory, Hill–Emax dose-response model and its derivation from the law of mass action, Schild analysis for competitive antagonists, time-delayed PK/PD models with effect-compartment linking, tolerance and sensitisation, and applications to therapeutic window design and clinical trial endpoint selection.
PhD-level analytical chemistry: potentiometric acid-base titration curves with buffer region analysis (Henderson-Hasselbalch, polyprotic systems), Beer-Lambert law and UV-Vis spectrophotometry (calibration, limit of detection, matrix effects), high-performance liquid chromatography (HPLC) resolution equation, the van Deemter equation for plate-height optimisation, and method validation (linearity, precision, accuracy, specificity).