Questions and answers

A website of interactive mathematics lessons. Each lesson is a short run of steps: you move the idea in a live lab (a figure to drag, a solid to turn, a number to scrub), predict what will happen, then check an answer. The aim is to understand why a result is true, not only how to use it.

20 courses, each one whole, in mathematics and physics:

Mathematics: Geometry, AP Calculus, Linear Algebra, Multivariable Calculus, Differential Equations, Probability and Statistics, Discrete Mathematics.

Further mathematics: Real Analysis, Complex Analysis, Abstract Algebra, Numerical Methods, PDEs and Fourier Analysis.

Physics: AP Physics 1, AP Physics 2, AP Physics C: Mechanics, AP Physics C: Electricity and Magnetism, Vibrations and Waves, Relativity, Quantum Physics, Statistical Physics.

Most have an extended track, such as Honors in Geometry or BC in Calculus, shown when the course map's switch is on. The courses page says what each covers.

Pick the course you're studying on the courses page; each says which courses are best taken before it. On its map, starting at 1.1 works, but you can also search for a topic (⌘K or Ctrl K), such as "similar triangles", and jump straight to it. Each topic lists the earlier topics it builds on, so you can see what to read first.

A laptop or desktop computer with a keyboard and a mouse or trackpad, and a current version of Chrome, Edge, Firefox or Safari. Lessons are laid out for a large screen, with the reading and the lab side by side.

The rail on the left lists the lesson's steps. Each step has something to try in the lab and one question: a prediction to commit to or an answer to type. When the last step is done, Finish lesson marks it finished and puts the topic into review.

What comes after a question stays hidden until the question is settled, so the answer is never one glance away. A question is settled when you get it right, make your prediction, use two tries, or open all three hints. Then the explanation appears and Continue (or ↵) opens the next step.

Type a number or an expression; a preview shows it typeset as you go. Any equal form counts: 3/4, 0.75 and 6/8 are all right. You can write

• fractions and powers: 3/4, x^2
• roots: sqrt(2), 2sqrt(3)
• π as pi or π: 4pi
• functions: sin(x), ln(2), e^(-t)
• angles in degrees where a question asks for them, with or without the sign: 72 or 72°
• complex numbers with i: 3 + 4i, -1/2 + sqrt(3)/2*i
• permutations in cycle notation: (1 3 2)(4 5)

A product can skip the *: 2x and 3(x + 1) work. A vector or a matrix gets a box for each entry, and a question with a unit shows the unit beside the box.

If it's a common mistake, the feedback is about that mistake, not just "wrong". You get two tries. Hints come in three steps, each giving away a little more, and you can open the next one at any time with H.

Yes. Tab to a point you can drag, then move it with the arrow keys. The measures in the lab update as it moves, exactly as when you drag it.

• 1–4 choose a prediction
• H the next hint
• ↵ continue to the next step
• S the scratch pad
• F focus mode, which hides the step rail
• ⌘K or Ctrl K search topics
• ? the list of shortcuts
• Esc close a panel

Three more problems on one topic, drawn from the lesson's templates with new numbers each time, so there is always another set and answers can't be memorised. Practice is for you only: it doesn't change when a topic comes back for review.

A finished topic comes back the day after, then further apart each time you get it right, and sooner when you don't. The timing follows FSRS, a spaced-repetition scheduler that estimates how likely you are to remember each topic.

A review session takes up to eight topics that are due, most overdue first, with two new problems each. A topic is graded Good if every problem was right at the first try, Hard if one came right after a miss, and Again if one never came right.

Each topic shows how well you know it: not started, in progress, learned (finished), strong (remembered over at least two reviews), fading (likely to slip soon) or shaky (many misses, or forgotten once and not yet back). An orange ring marks a topic that is due for review.

In this browser, on this computer. Another browser or computer doesn't see it. To move it, export it to a file on the progress page and import that file in the other browser.

Clearing the browser's site data clears it, and some browsers clear a site's storage after a while without a visit (Safari can after seven days). On the progress page, Keep my progress asks the browser not to, and an export keeps a copy you control.

Your progress never leaves your browser: it is stored there. Page views are counted with Cloudflare Web Analytics, which sets no cookies and doesn't follow you across sites. The privacy page has the details.

No. They follow the units of the AP course frameworks so you can study alongside a class, but every lesson, explanation and problem here is our own. AP and Advanced Placement are trademarks registered by the College Board, which is not affiliated with, and does not endorse, this site.

Write to support@pablin.me with the topic number (it's at the top of the lesson, like 5.1) and the step. Every formula and answer is checked by machine before it ships, but a person reading closely catches what a check can't.