MW SysArcTools

Learn by calculating

Start with numbers. Build toward quantum mathematics.

Follow a practical path where every concept has a calculator, visible working, assumptions, meaning and limits. You do the mathematics instead of only memorising its language.

Start Level 1

The learning loop

Predict → calculate → explain → change

  1. Read the problem and predict what should happen.
  2. Enter values and calculate locally in your browser.
  3. Inspect the formula and every calculation step.
  4. Explain the answer's sign, size, units and assumptions.
  5. Change one input and predict the new result before recalculating.

Current curriculum

170 calculator lessons in dependency order

Level 1

Number foundations

Understand magnitude, factors, ratios, percentages, powers and roots.

Before starting: Comfort reading and entering whole numbers and decimals.
  1. Lesson 1 Absolute value Find a number's distance from zero and compare its original sign.
  2. Lesson 2 GCF and LCM Find the greatest common factor and least common multiple of two whole numbers.
  3. Lesson 3 Fraction simplifier Reduce a fraction to lowest terms using the greatest common divisor.
  4. Lesson 4 Ratio simplifier Simplify a two-number whole-number ratio and calculate its decimal relationship.
  5. Lesson 5 Percentage of a number Find a selected percentage of any number and see the decimal conversion.
  6. Lesson 6 Percentage change Calculate the percentage increase or decrease between an original and a new value.
  7. Lesson 7 Exponent Raise a base to a selected power and see the repeated-multiplication meaning.
  8. Lesson 8 Square root Find the principal square root and verify it by squaring the result.
  9. Lesson 9 Cube root Find the real cube root of a positive or negative number and verify it.
Level 2

Equations and algebra

Turn relationships into equations, isolate unknowns and understand functions.

Before starting: Complete Number foundations or understand arithmetic operations and powers.
  1. Lesson 1 Linear equation Solve a linear equation in the form ax + b = c and see the isolation steps.
  2. Lesson 2 Proportion solver Solve a proportional relationship a/b = c/x by cross multiplication.
  3. Lesson 3 Two-equation system Solve two linear equations in x and y using determinants.
  4. Lesson 4 Quadratic formula Solve ax² + bx + c = 0 and inspect the discriminant.
  5. Lesson 5 Logarithm Calculate a logarithm with any valid positive base.
Level 3

Coordinate and practical geometry

Connect equations and measurements to points, lengths, areas and volumes.

Before starting: Understand basic algebra and square roots.
  1. Lesson 1 Slope Calculate the slope of the line through two coordinate points.
  2. Lesson 2 Midpoint Find the point exactly halfway between two coordinates.
  3. Lesson 3 Distance between points Calculate straight-line distance between two coordinates.
  4. Lesson 4 Rectangle Calculate area, perimeter and diagonal from length and width.
  5. Lesson 5 Triangle area Calculate triangle area from a base and perpendicular height.
  6. Lesson 6 Circle Calculate a circle's diameter, circumference and area from its radius.
  7. Lesson 7 Trapezoid area Find trapezoid area from the two parallel base lengths and perpendicular height.
  8. Lesson 8 Ellipse Calculate ellipse area and an accurate perimeter approximation from its semi-axes.
  9. Lesson 9 Pythagorean theorem Find the hypotenuse of a right triangle from its two legs.
  10. Lesson 10 Cube dimensions Calculate a cube's volume, surface area and space diagonal from its side length.
  11. Lesson 11 Rectangular prism Calculate volume, surface area and space diagonal from length, width and height.
  12. Lesson 12 Cylinder Calculate cylinder volume and surface area from radius and height.
  13. Lesson 13 Cone volume Calculate the volume and base area of a circular cone from radius and height.
  14. Lesson 14 Sphere Calculate sphere volume and surface area from radius.
  15. Lesson 15 Square pyramid Calculate volume, slant height and total surface area from base side and vertical height.
Level 5

Statistics and probability

Describe datasets, standardise observations and reason about uncertain outcomes.

Before starting: Understand fractions, ratios, percentages and basic algebra.
  1. Lesson 1 Mean and median Calculate the arithmetic mean and median of six values.
  2. Lesson 2 Weighted mean Calculate the weighted mean of three values when observations have different importance or frequency.
  3. Lesson 3 Data range Find the minimum, maximum and range of six entered observations.
  4. Lesson 4 Standard deviation Calculate population and sample standard deviation for six values.
  5. Lesson 5 Z-score Standardise a value by measuring how many standard deviations it lies from the mean.
  6. Lesson 6 Probability Calculate event probability from favourable and total equally likely outcomes.
  7. Lesson 7 Odds to probability Convert odds in favour into probability, percentage probability and odds against.
  8. Lesson 8 Combinations and permutations Count selections when order either matters or does not matter.
Level 6

Precalculus and functions

Understand function families, sequences, series, exponential change and trigonometric values.

Before starting: Understand equations, powers, graphs, ratios and angle measurement.
  1. Lesson 1 Linear function Evaluate f(x) = mx + b and identify its slope and vertical intercept.
  2. Lesson 2 Quadratic function Evaluate ax² + bx + c and calculate the parabola's vertex coordinates.
  3. Lesson 3 Arithmetic sequence Find the nth term of a sequence with a constant difference.
  4. Lesson 4 Arithmetic series Calculate the sum of the first n terms of an arithmetic sequence.
  5. Lesson 5 Geometric sequence Find the nth term of a sequence with a constant multiplication ratio.
  6. Lesson 6 Geometric series Calculate the sum of the first n terms of a finite geometric sequence.
  7. Lesson 7 Exponential function Evaluate exponential growth or decay in the form f(x)=abˣ.
  8. Lesson 8 Sine, cosine and tangent Calculate sine, cosine and tangent for an angle entered in degrees.
Level 7

First calculus

Interpret instantaneous change and accumulated change using derivatives and integrals.

Before starting: Understand algebra, functions, powers, graphs and trigonometric notation.
  1. Lesson 1 Power-rule derivative Differentiate a monomial axⁿ and evaluate its derivative at a selected x-value.
  2. Lesson 2 Power-rule integral Integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
  3. Lesson 3 Definite power integral Evaluate the definite integral of axⁿ between lower and upper bounds.
Level 8

Linear algebra

Work with vectors and matrices, then understand determinants, inverses and eigenvalues.

Before starting: Understand algebra, coordinate geometry, trigonometry and functions.
  1. Lesson 1 Vector magnitude Calculate the length of a three-dimensional vector from its components.
  2. Lesson 2 Vector addition Add two three-dimensional vectors component by component.
  3. Lesson 3 Dot product Calculate the scalar dot product of two three-dimensional vectors.
  4. Lesson 4 Cross product Calculate the three-dimensional vector perpendicular to two input vectors.
  5. Lesson 5 Angle between vectors Find the angle between two three-dimensional vectors using their dot product.
  6. Lesson 6 2×2 determinant Calculate the determinant of a 2×2 matrix and identify whether it is invertible.
  7. Lesson 7 2×2 matrix inverse Invert a nonsingular 2×2 matrix and display all four inverse entries.
  8. Lesson 8 2×2 eigenvalues Calculate the real eigenvalues of a 2×2 matrix from its trace and determinant.
Level 9

Differential equations

Model growth, decay, equilibrium, numerical approximation and oscillating systems.

Before starting: Understand functions, exponentials, derivatives, integrals and basic linear algebra.
  1. Lesson 1 Exponential growth ODE Solve y′=ky from an initial value and evaluate the solution at time t.
  2. Lesson 2 Half-life decay Model exponential decay from an initial amount, half-life and elapsed time.
  3. Lesson 3 Logistic growth ODE Evaluate population growth with a carrying capacity using the logistic differential equation.
  4. Lesson 4 Newton cooling Estimate an object's temperature as it approaches a constant ambient temperature.
  5. Lesson 5 First-order linear ODE Solve y′+py=q with constant coefficients and an initial value.
  6. Lesson 6 Euler method step Approximate one step for y′=ay+b using the slope at the current point.
  7. Lesson 7 Harmonic oscillator Evaluate displacement and velocity for x″+ω²x=0 with amplitude and phase.
  8. Lesson 8 Damped oscillator Evaluate an underdamped oscillation x(t)=Ae⁻ᵝᵗcos(ωt+φ) and its decaying envelope.
Level 10

Complex and Fourier mathematics

Use the complex plane to represent rotation and waves, then measure frequency components with the DFT.

Before starting: Understand trigonometry, vectors, exponentials, oscillation and radians.
  1. Lesson 1 Complex addition Add two complex numbers by combining their real and imaginary components.
  2. Lesson 2 Complex multiplication Multiply two complex numbers and display the rectangular-form result.
  3. Lesson 3 Complex division Divide one complex number by another using the denominator's conjugate.
  4. Lesson 4 Complex polar form Convert z=a+bi into magnitude and phase angle.
  5. Lesson 5 Euler formula Evaluate eⁱᶿ as cosine and sine components on the unit circle.
  6. Lesson 6 Principal complex root Calculate the principal nth root of a complex number in polar form.
  7. Lesson 7 Phasor addition Add two magnitude-phase phasors and return the combined magnitude and phase.
  8. Lesson 8 Four-sample DFT Calculate one discrete Fourier transform bin from four real-valued samples.
Level 11

Mathematical physics

Apply calculus, vectors, fields and waves to motion, energy and fundamental forces.

Before starting: Understand algebra, trigonometry, vectors, derivatives, differential equations and oscillation.
  1. Lesson 1 Constant acceleration Calculate final velocity and displacement under constant acceleration.
  2. Lesson 2 Projectile motion Calculate ideal launch components, flight time, maximum height and horizontal range.
  3. Lesson 3 Kinetic energy and momentum Calculate classical linear momentum and kinetic energy from mass and velocity.
  4. Lesson 4 Work and force angle Calculate mechanical work done by a constant force over a displacement.
  5. Lesson 5 Newton's second law Calculate net force and weight from mass, acceleration and gravitational field strength.
  6. Lesson 6 Gravitational force Calculate Newtonian gravitational attraction between two point masses.
  7. Lesson 7 Coulomb force Calculate the signed electrostatic force magnitude between two point charges.
  8. Lesson 8 Wave speed and wavelength Calculate wave speed and period from frequency and wavelength.
Level 12

Quantum mathematics

Connect complex amplitudes and probability to matter waves, uncertainty, qubits and quantized energy.

Before starting: Understand probability, complex numbers, vectors, waves, energy and introductory mathematical physics.
  1. Lesson 1 Photon energy Convert a photon wavelength into frequency, energy in joules and energy in electronvolts.
  2. Lesson 2 De Broglie wavelength Calculate the matter wavelength associated with a non-relativistic particle's momentum.
  3. Lesson 3 Uncertainty principle Calculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
  4. Lesson 4 Wavefunction probability Convert a complex wavefunction amplitude into magnitude, phase and probability density.
  5. Lesson 5 Two-state normalization Normalize two real state amplitudes and calculate their measurement probabilities.
  6. Lesson 6 Bloch sphere state Convert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
  7. Lesson 7 Infinite well energy Calculate a particle's quantized energy level in a one-dimensional infinite square well.
  8. Lesson 8 Hydrogen transition Calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.
Level 13

Number patterns and practical algebra

Use discrete sequences, remainders, roots, logarithms, rates and weighted allocation confidently.

Before starting: Understand arithmetic, powers, roots, ratios and elementary equations.
  1. Lesson 1 Factorial Calculate n factorial and see its multiplicative expansion.
  2. Lesson 2 Fibonacci number Calculate the nth Fibonacci number by recurrence.
  3. Lesson 3 Triangular number Calculate the number of objects in a triangular arrangement.
  4. Lesson 4 Sum 1 to n Add every whole number from one through n.
  5. Lesson 5 Modulo Find quotient and Euclidean remainder after integer division.
  6. Lesson 6 Percentage difference Compare two values symmetrically using their average magnitude.
  7. Lesson 7 Nth root Calculate the real nth root of a number.
  8. Lesson 8 Change-of-base logarithm Evaluate a logarithm using natural logarithms.
  9. Lesson 9 Distance, rate and time Calculate distance from constant speed and elapsed time.
  10. Lesson 10 Ratio allocation Split a total according to three ratio weights.
Level 14

Extended geometry

Calculate circular regions, arbitrary triangles, polygons, three-dimensional distances and torus geometry.

Before starting: Understand algebra, angles, circles, area, volume and the Pythagorean theorem.
  1. Lesson 1 Annulus area Calculate the area between two concentric circles.
  2. Lesson 2 Sector area Calculate the area enclosed by a central angle.
  3. Lesson 3 Arc length Calculate circle arc length from radius and central angle.
  4. Lesson 4 Heron's triangle area Calculate triangle area from all three side lengths.
  5. Lesson 5 Parallelogram area Calculate parallelogram area and perimeter.
  6. Lesson 6 Rhombus area Calculate rhombus area from its diagonals.
  7. Lesson 7 Regular polygon Calculate regular polygon area and perimeter from side count and side length.
  8. Lesson 8 Polygon interior angles Calculate the interior-angle sum and regular interior angle.
  9. Lesson 9 3D distance Calculate Euclidean distance between two points in three dimensions.
  10. Lesson 10 Torus geometry Calculate the volume and surface area of a circular torus.
Level 15

Extended trigonometry

Solve non-right triangles, oscillations, rotations, coordinate conversions and bearings.

Before starting: Understand sine, cosine, radians, vectors and coordinate geometry.
  1. Lesson 1 Cosine law side Find a triangle side from two sides and their included angle.
  2. Lesson 2 Cosine law angle Find an angle from three triangle sides.
  3. Lesson 3 Sine law side Find a corresponding triangle side from one side and two angles.
  4. Lesson 4 Sine wave Evaluate a sinusoid from amplitude, frequency, time and phase.
  5. Lesson 5 Angular to linear speed Convert angular speed at a radius into tangential speed.
  6. Lesson 6 Polar to Cartesian Convert a radius and angle into x and y coordinates.
  7. Lesson 7 Cartesian to polar Convert x and y coordinates into radius and direction.
  8. Lesson 8 Bearing components Resolve a distance along a compass bearing into north and east components.
Level 16

Extended statistics

Measure distribution shape, relative spread and linear relationships between paired variables.

Before starting: Understand means, variance, standard deviation, coordinates and basic probability.
  1. Lesson 1 Population variance Calculate population variance for six observations.
  2. Lesson 2 Quartiles Find lower quartile, median and upper quartile for six values.
  3. Lesson 3 Interquartile range Calculate the middle-50-percent spread for six values.
  4. Lesson 4 Root mean square Calculate the quadratic mean of six values.
  5. Lesson 5 Geometric mean Calculate the geometric mean of six positive values.
  6. Lesson 6 Harmonic mean Calculate the harmonic mean of six positive rates.
  7. Lesson 7 Coefficient of variation Compare population standard deviation with the mean for six values.
  8. Lesson 8 Covariance Calculate population covariance for three paired observations.
  9. Lesson 9 Pearson correlation Calculate Pearson's correlation coefficient for three pairs.
  10. Lesson 10 Linear regression Fit a least-squares line to three paired observations.
Level 17

Extended probability

Combine events, condition evidence and work with expected, binomial and Poisson models.

Before starting: Understand fractions, percentages, combinations, averages and independent events.
  1. Lesson 1 Probability complement Calculate the probability that an event does not occur.
  2. Lesson 2 Independent event union Calculate the chance that at least one of two independent events occurs.
  3. Lesson 3 Conditional probability Calculate P(A given B) from the intersection and conditioning event.
  4. Lesson 4 Expected value Calculate the weighted mean of three possible outcomes.
  5. Lesson 5 Binomial exact probability Calculate exactly k successes in n independent equal-probability trials.
  6. Lesson 6 Binomial mean and variance Calculate expected successes, variance and standard deviation.
  7. Lesson 7 Poisson probability Calculate the probability of exactly k events at an average rate λ.
  8. Lesson 8 Bayes theorem Update a prior probability after a positive observation.
Level 18

Applied derivatives and integrals

Use derivatives for rates and tangent models, and integrals for exact and approximate accumulation.

Before starting: Understand functions, quadratics, exponentials, logarithms and first calculus.
  1. Lesson 1 Quadratic derivative Differentiate ax²+bx+c and evaluate its slope.
  2. Lesson 2 Exponential derivative Differentiate Ae^(kx) at a selected x.
  3. Lesson 3 Logarithm derivative Evaluate the derivative of A ln(x) at positive x.
  4. Lesson 4 Quadratic tangent line Find slope and intercept of a tangent to ax²+bx+c.
  5. Lesson 5 Quadratic average rate Calculate secant slope of a quadratic over an interval.
  6. Lesson 6 Definite linear integral Integrate mx+b over a finite interval.
  7. Lesson 7 Definite quadratic integral Integrate ax²+bx+c over a finite interval.
  8. Lesson 8 Trapezoidal rule Approximate an integral from two endpoint function values.
Level 19

Growth, decay and compounding

Compare linear, periodic and continuous growth and solve inverse timing and rate problems.

Before starting: Understand percentages, exponentials, logarithms and sequences.
  1. Lesson 1 Simple interest Calculate linear interest growth without compounding.
  2. Lesson 2 Compound interest Calculate periodic compound growth.
  3. Lesson 3 Continuous compounding Calculate exponential growth under continuous compounding.
  4. Lesson 4 Doubling time Calculate time required to double under continuous exponential growth.
  5. Lesson 5 Exponential half-life Calculate half-life from a positive continuous decay-rate magnitude.
  6. Lesson 6 Required growth rate Find the periodic compound rate needed to move from a start to a target value.
Level 20

Computational number theory

Investigate primes, divisors, coprimality and modular arithmetic with efficient finite algorithms.

Before starting: Understand factors, powers, remainders, whole numbers and finite iteration.
  1. Lesson 1 Prime checker Determine whether a positive whole number is prime.
  2. Lesson 2 Divisor count Count the positive divisors of a whole number.
  3. Lesson 3 Sum of divisors Add every positive divisor of a whole number.
  4. Lesson 4 Euler totient Count positive integers up to n that are coprime with n.
  5. Lesson 5 Modular exponentiation Calculate base^exponent modulo m without constructing the full power.
Level 21

Analytic algebra and geometry

Read quadratics structurally, invert linear maps and model elliptical and spherical geometry.

Before starting: Understand functions, quadratics, coordinates, trigonometry and radians.
  1. Lesson 1 Quadratic vertex Find the vertex and axis of symmetry of ax²+bx+c.
  2. Lesson 2 Quadratic discriminant Calculate a quadratic's discriminant and classify its real roots.
  3. Lesson 3 Inverse linear function Invert f(x)=mx+b and evaluate the inverse at y.
  4. Lesson 4 Ellipse eccentricity Calculate focal distance and eccentricity from semiaxes.
  5. Lesson 5 Great-circle distance Calculate spherical surface distance between two latitude-longitude points.
Level 22

Matrix computation

Add, scale, multiply and apply 2×2 matrices while connecting trace and determinant to transformations.

Before starting: Understand vectors, systems of equations and the first linear-algebra level.
  1. Lesson 1 2×2 matrix addition Add corresponding entries of two 2×2 matrices.
  2. Lesson 2 2×2 matrix multiplication Multiply two 2×2 matrices in order A times B.
  3. Lesson 3 Matrix–vector product Apply a 2×2 linear transformation to a two-component vector.
  4. Lesson 4 Matrix trace Calculate trace, determinant and eigenvalue sum for a 2×2 matrix.
  5. Lesson 5 Matrix scalar multiplication Multiply every entry of a 2×2 matrix by one scalar.
Level 23

Statistical inference and numerical calculus

Estimate normal probabilities and confidence intervals, then approximate roots and integrals numerically.

Before starting: Understand normal distributions, standard error, derivatives, integrals and approximation error.
  1. Lesson 1 Normal CDF Estimate the probability below a value in a normal distribution.
  2. Lesson 2 Normal interval probability Estimate probability between two bounds in a normal distribution.
  3. Lesson 3 Mean confidence interval Calculate a z-based confidence interval for a mean with known standard deviation.
  4. Lesson 4 Newton square-root step Perform one Newton iteration toward the square root of a positive number.
  5. Lesson 5 Simpson's rule Approximate an integral from endpoint and midpoint function values.

The road ahead

From calculus to quantum mathematics

New calculators will be added where this curriculum has a missing prerequisite. Advanced tools will not skip the mathematical staircase beneath them.

Clear answers

Learning mathematics with calculators

Where should I start?

Start at Level 1 unless every concept there already feels familiar. Each level states its prerequisite and learning outcome.

How do I learn with a calculator?

Predict first, calculate second, inspect the working, then change one input and explain why the result changed.

Can this pathway lead toward quantum mechanics?

Yes, by building the required mathematics in order: algebra, trigonometry, probability, calculus, linear algebra, differential equations, complex numbers and Fourier methods before quantum models.