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LearnMathora

No grades. No wrong moves.

The playground

Nineteen stations from across the curriculum — calculus, linear algebra, statistics, optimization, signals — each one idea made touchable. Drag things. Break nothing. Intuition built here makes every lesson land harder.

Station 01

Ride the wave

Slope is a living thing: positive on climbs, zero at crests, negative on descents.

InteractiveTangent explorer
0
At x = 0, the curve is changing at a rate of 1 — the slope of the tangent line.

Station 02

Shrink the gap

The derivative being born: averages over a closing gap settling on the instant.

InteractiveFrom average to instant
1.6
Average rate between the two points: 3.6. As the gap shrinks toward 0, it closes in on 2 — the instant rate at x = 1.

Station 03

Slice and stack

Riemann sums in your hands — rectangles melting into exact area.

InteractiveArea accumulator
4
8
Distance covered from t = 0 to 4: 9.06 — the area under the speed curve. More slices hug the curve more closely.

Station 04

Function vs. derivative

Two graphs, one truth: the bottom one's height is the top one's slope.

InteractiveA function and its derivative

The function

Its derivative — the slope, graphed

0.6
Slope of the top curve at x = 0.6 is 1.2 — and that is exactly the height of the bottom curve there.

Station 05

The moving cart

Position, velocity, acceleration — one slider, three linked stories.

InteractivePosition, velocity, acceleration

Position s(t) — where it is

Velocity v(t) = s′(t) — how fast

Acceleration a(t) = v′(t) — how the speed changes

0.6
At t = 0.6 the cart is moving forward — velocity 0.96, acceleration -2.8. Each graph is the slope of the one above it.

Station 06

Find the peak

Optimization by feel: walk uphill until the slope reads zero.

InteractiveOptimization: the best box
max 16 at x = 1
0.35
Cut squares of x = 0.35 from a 6×6 sheet → volume 9.8. Slope: 20.7. Uphill — a bigger cut still helps.

Station 07

The +C family

Infinitely many originals, one shared rate — why antiderivatives come with a constant.

InteractiveWhy +C exists
1
1
Every curve x² + C has slope 2 at x = 1 — the derivative can't tell them apart, so the antiderivative must carry a +C.

Station 08

The grand connection

Watch accumulation grow at exactly the rate of the curve above — the fundamental theorem, live.

InteractiveThe fundamental theorem, watched live

A rate f(t) — area shading in

Total so far A(x) — the area, graphed

3
Accumulated area so far: 6.49. The bottom curve grows at rate 1.64 — the exact height of the top curve. Accumulating undoes rate-taking.

Station 09

The circle that makes waves

Walk the unit circle and watch your height trace a sine wave — trigonometry's one essential picture.

InteractiveThe circle that makes waves
0.9
At angle 0.9 rad: height sin θ = 0.78, horizontal cos θ = 0.62. Walk the full circle and the height traces one complete sine wave — triangles and waves are the same picture.

Station 10

Move all of space

Rotate, stretch, squash, shear: a matrix is a verb, and the grid is its sentence.

InteractiveA matrix is a verb
Rotate 45°: The whole plane turns together. Every game camera move is a cousin of this matrix. Matrix: [0.71, -0.71; 0.71, 0.71]

Station 11

The bell curve

Slide the mean, stretch the spread, and watch the 68–95–99.7 rule count its share.

InteractiveThe bell curve
0
1
1
Mean 0, spread σ = 1. Within ±1σ of the mean lives 68.3% of everything. (±1σ ≈ 68%, ±2σ ≈ 95% — the most useful rule of thumb in statistics.)

Station 12

Euler's bridges

The 1736 puzzle that founded network theory — toggle bridges and let degree-counting decide the walk.

InteractiveThe seven bridges of Königsberg
3353
Toggle bridges
4 regions have an odd number of bridges. No such walk exists — Euler proved it needs 0 or 2 odd regions. The original seven bridges? Four odd regions. Impossible — and graph theory was born from the proof.

Station 13

Square waves from circles

Fourier's scandalous claim, performed: stack smooth harmonics until corners appear.

InteractiveBuilding a square from circles
2
2 harmonics: corners are forming. Each added harmonic is smaller and faster, sharpening the shoulders.

Station 14

Catch the impostor

Sample a wave too slowly and a phantom frequency appears — the aliasing that haunts all digital audio.

InteractiveThe sampling theorem, felt
3 Hz
8 Hz
Sampling at 8 Hz — more than twice the signal's 3 Hz — the dots pin down exactly one sine. The signal survives digitization.

Station 15

Compress with SVD

Rebuild a data matrix from its strongest rank-1 layers — the low-rank approximation behind PCA, compression, and recommendations.

InteractiveSVD low-rank approximation

Rank-2 approximation

Full data (rank 8)

singular values σ₁ … σ₈ — kept layers are solid

2
Keeping the top 2 of 8 layers captures 83.4% of the data's energy while storing 98 numbers instead of 576 (83% smaller). The biggest singular values already carry the broad structure.

Station 16

Roll downhill

Tune the learning rate and watch gradient descent converge, crawl, or diverge — exactly how models learn.

InteractiveGradient descent on a loss surface
0.3
-3.2
8
After 8 steps at learning rate 0.3: loss 0.52, gradient -0.18. Still descending — add steps or raise the rate to reach the minimum.

Station 17

Count without listing

The fundamental counting principle as a branching tree, and permutations vs combinations as filling slots.

InteractiveThe counting lab
9 leaves = 3^2
2
3
2 stages, each with 3 choices → 3^2 = 9 outcomes. The fundamental counting principle: independent choices multiply.

Station 18

The bell from chaos

Average samples from any lopsided population and watch the sample mean go normal — the central limit theorem, live.

InteractiveCentral limit theorem

distribution of the sample mean

1
Population shape: Skewed. Averaging n = 1 draws at a time, the sample means have spread σ/√n = 0.273. At n = 1 you're just seeing the raw population — often nothing like a bell.

Station 19

3D Loss Landscapes & Gradients

Rotate surfaces in 3D and watch the gradient vector ∇f point along the path of steepest climb or descent.

Interactive3D Loss Landscapes & Gradient Vectors
∇f = [1.60, 1.20]
Point: (0.80, 0.60, z=1.00)Gradient Magnitude |∇f| = 2.00