Slate · visual explanations you can check

Complex ideas. Crystal-clear understanding.

Explore concepts through beautiful animations, interactive experiments and evidence-based explanations. Don't just learn what happens. Discover why.

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12/12Speed

learning rate η0.4

start x₀-2.5

steps12

Why does a learning rate that is too large make training diverge?

Each step moves against the slope, by the learning rate times the slope. Near a minimum with curvature f'', the distance to the minimum is multiplied by 1 − η·f'' every step: inside (−1, 1) it shrinks, outside it grows.

Open the full lab

One idea, six ways in

Don't just explain something. Make people experience why it works.

Explore
Ask about anything. Slate picks the clearest way to show it.
Trace
Show your working. Find the exact step that breaks, with evidence.
Motion
Manim animations built from a checked scene plan, not a guess.
Lab
Move the variables and watch the idea respond, instantly.
Practice
New problems, checked. Progress only from work you did yourself.
Studio
Edit scenes, captions and timing. Share privately or publicly.

How it works

Understanding first. Decoration never.

01
Ask
A concept, a problem, an algorithm or your own notes. Slate reads the domain and the concept without a model call, and tells you what it read.
02
Slate picks the picture
An animation, an experiment you can change, a step-by-step derivation or a structured explanation: whichever shows the idea, not whichever looks busiest.
03
Checked before you see it
Formulas are tested, derivation steps verified, and every lab and template computes its numbers from the same function. What cannot be checked is labelled unverified.
04
Then you do it
Predict before the reveal, solve a new problem, and progress moves only when you get it right on your own.

Motion and Lab

The video and the experiment are the same computation.

This video was rendered by Manim from the gradient-descent lab's own settings, with no AI involved. Open the lab and the browser steps through exactly these iterates, live, as you move the learning rate.

Gradient descent with η = 0.4 on L(x) = ½(x − 1)² + ½. Rendered by the descent template (Manim Community 0.21), preview quality.

Scene plans, not random code

Every lesson is a typed, versioned storyboard. Scenes a trusted template covers compile with no model call; the rest are generated, sandboxed and checked before they render.

Preview first, final when you want it

Renders run in the background in an isolated worker with no network access. Identical scenes are cached by content, so a re-render redoes only what changed.

Interaction without re-rendering

Sliders never regenerate the video. The lab recomputes in your browser in milliseconds from the same parameter model the video was rendered from.

Open the gradient descent lab

Trace

Find the exact step where the reasoning breaks.

Edit the working below. Slate's own checker, not a language model, compares every step with the truth, and says whether its evidence is an exact proof or agreement at sampled points. Runs entirely in your browser.

LiveRuns in your browser · edit anything

The problem

The working (one step per line)

Try fixing the working: write x^2 + 4x + 4 and watch the verdict change. Or break it a new way.

1 step verified

\text{Expand} \; \left(x + 2\right)^{2}

Here's where it breaks: step 2

You squared each term on its own. Squaring a sum also creates the cross term $2 \cdot x \cdot 2 = 4 x$ .

Your working, step by step

1
$\left(x + 2\right)^{2}$
given
2
$x^{2} + 4$
breaks here
Not equal to the original expression: at $x = 1$ this step gives $5$ , the right value is $9$ .

You wrote

x^{2} + 4

It should be

x^{2} + 4 x + 4

Your step is exactly what this move produces: $x^{2} + 4$ .
Your $x$ term is $0$ ; it should be $4 x$ .

Explore it

$(x + 2)^2$ (whole square): 25
Your answer: 13
Gap: 12

Dashed tiles are missing from your answer: 2x + 2x = 12 at this x.

x

Drag $x$ : the whole square always has area $\left(x + 2\right)^{2}$ . Your answer covers only part of it; the uncovered part is exactly the cross term $4 x$ , whatever $x$ is.

Why

$\left(x + 2\right)^{2}$ means $(x + 2)$ multiplied by itself, so every term meets every other term.

Picture a square with side $x + 2$ . It splits into a $x$ -by- $x$ square, a $2$ -by- $2$ square, and two $x$ -by- $2$ rectangles. Those two rectangles are the cross term $4 x$ .

How this was checkedshow

step 2equals the original expressionfailed · counterexample foundpolynomial identity (term by term)

“Proved exactly” means the two sides were compared symbolically. “Agrees at sample points” is strong evidence, not a proof. A counterexample shows a step is wrong.

Now try one on your own

Expand it fully, without looking back.

\text{Expand} \; \left(2 x + 1\right)^{2}

Who it is for

For anyone who wants to see why, not just what.

Students
Paste your working and find the exact step that breaks; open a lab and move the variable that confuses you; practise until you can do it unaided.
Teachers and creators
Start from a lab or a sentence, edit the storyboard scene by scene, render a preview then the final video, and share a collection with your class.
Engineers and data people
See gradient descent diverge, watch sorting and graph search run step by step, and turn the same run into a video for a design review.
Researchers and the curious
Explain an idea at four depths, from intuition to expert, with every checkable claim checked and every unchecked one marked.

What works today

Every topic below has a working lab, template or checker.

Slate will try to explain anything, but it only claims what it has built and tested. Everything else is explained without live checking, and labelled that way.

Mathematics

algebra
derivatives
limits
integration (Riemann sums)
linear transformations
functions and graphs

Not yet: probability, statistics, differential equations, number theory.

Computer science

sorting algorithms
graph search (BFS, DFS)

Not yet: trees, dynamic programming, recursion, hashing.

AI and machine learning

gradient descent
learning rate
local minima

Not yet: neural networks, backpropagation, attention, embeddings.

Physics and engineering

projectile motion

Not yet: forces, waves, circuits, thermodynamics.

Privacy

Your learning belongs to you.

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Make the invisible understandable.

Watch it, change it, check it, then do it yourself.

Start exploring

Complex ideas. Crystal-clear understanding.

Don't just explain something. Make people experience why it works.

Explore

Trace

Motion

Lab

Practice

Studio