You know more than you think.

Arithmetic & percentages

Predict the percentage for 8 shaded squares out of 20. Then change the count: how many squares would make 75%?

The denominator tells you the size of the whole. A half is 10/20 here, 5/10 in a smaller group, and 50/100 in a larger one. All are 50%. The absolute counts differ.

Do multiplication and division before addition and subtraction. For 2 + 3 × 4, first find 12, then add 2: 14. Parentheses change the grouping: (2 + 3) × 4 = 20. Negative numbers represent values below zero: 2 − 5 = −3.

Algebra is a rule with a placeholder

An equation is a rule with placeholders. Predict the output before moving the input, weight, or bias.

y = (3) × (2) + (2) = 8.

To solve y = 3x + 2 for an input when y = 14, undo the operations in reverse: subtract 2 to get 12, then divide by 3 to get x = 4. Substitution checks it: 3 × 4 + 2 = 14.

Graphs make a rule visible

Connect each row of the table to a point. Predict how the line changes when its weight becomes negative.

y = (3) × (2) + (2) = 8.

A steep-looking line can come from stretched axes. Read the numbered scales and calculate change in y ÷ change in x. A negative slope falls as x increases; zero slope is horizontal.

Averages and probability: combine values with care

The mean of 2, 4, and 6 is (2 + 4 + 6) ÷ 3 = 4. Each observation has one vote. Repeated values get more votes: below, each “2” counts once and each “8” counts once.

From observed counts to a probability model

Imagine a programmed spinner that awards one of two scores with the probabilities you set. Predict the long-run average before changing its probability. A 25% chance describes an uncertain single spin; it does not promise one success in every four spins.

A: 25% · B: 75%. The probabilities total 100%.

Expected score = 0.25 × (8) + 0.75 × (-2) = 0.50.

Expectation is a probability-weighted average. It may be a score that no single spin can produce. Repeated independent spins tend to have an average near it; small samples can differ. A measured success rate is evidence about a process, and treating it as a future probability requires assumptions about what stays the same.

Vectors and matrices: ordered numbers, repeated rules

A vector is an ordered list. The dot product multiplies corresponding coordinates and adds them. A matrix stacks several multiply-and-add rules; each row produces one output coordinate. Move the sliders and follow the arithmetic.

Functions and slopes: how fast does the output change?

A function turns an input into an output. Try f(x) = (x − 1)². Unlike a straight line, this curve has a different slope at each point. Predict the slope at the bottom, x = 1.

Shrink Δx and watch the segment slope approach the tangent slope. The derivative, written f′(x), is that local rate of change. Here f′(x) = 2(x − 1). A negative slope means increasing x lowers the output locally; a positive slope means it raises it. Zero slope means the first-order change is zero, not that the function is flat everywhere.

Gradient descent uses this local information to choose a direction. A large jump can cross the bottom and make the result worse; the derivative is not a guarantee for a distant point.

The chain rule: pass a change through the steps

One operation can feed another: x → h = wx + b → y = h². Predict what happens to the input sensitivity if w is zero. Then compare with the worked steps using your current settings.

For Δx = 0.01: derivative × Δx predicts a change of 0.1200. Recomputing y exactly gives 0.1204. These are close because the step is small, but need not be identical.

A gradient collects sensitivities to several adjustable numbers. Holding x fixed, dy/dw = 2hx = 6.00 and dy/db = 2h = 6.00. These are derivatives of this output; a learning model also passes sensitivity backward from its loss. Backpropagation calculates gradients, and an optimizer then changes parameters.

Information, surprise, and compression

Optional: this is not needed to start the language lessons. Imagine a source that independently emits A or B with a fixed probability. Your model predicts probabilities for the next symbol. Predict which outcome is more surprising when the model assigns A a probability of 75%.

Expected ideal code length using the model: 1.000 bits per symbol. Source entropy: 0.811 bits per symbol.

A bit is a binary choice. Halving a probability adds one bit of surprise: p = 1/2 gives 1 bit, p = 1/4 gives 2, and p = 1/8 gives 3. In this fixed independent source, matching the model probabilities to the source minimizes expected log loss and reaches its entropy.

These fractional lengths describe ideal averages under entropy-coding assumptions, not a literal fractional bit written for each symbol. A real compressor must encode blocks or a stream and include overhead. Zero-frequency source outcomes contribute zero to the expected value, even if their hypothetical surprise is infinite. This widget calculates probabilities and code-length estimates; it does not compress a file.

Cross-entropy uses the same negative-log idea. Machine-learning losses often use the natural logarithm instead, giving units of nats: divide nats by ln 2 to get bits. Low training loss alone does not show compression or prediction quality on new data.

Ready with just a browser.

The first lesson requires no account, downloads, or dedicated GPU. Progress is saved on this device when browser storage is available.

Optional: put the ideas into code

Learn variables, arrays, conditions, loops, and functions in six browser exercises, ending with a trainable neuron. No installation needed; coding is outside required completion.