Understand why it works, not only how to use it.
Mathematics, physics and biology explained by moving the idea yourself. Predict, see, check, and review just before you would forget.
On f(x) = x², with h = , the secant from x = 1 to x = 1 + h has slope 2.800.
Drag the orange point toward (1, 1), or the orange number toward 0, and watch the slope.
Every lesson moves the same way
Short steps, each with one thing to try and one question. The next step opens once you've answered, so nothing is skimmed.
Each idea starts as something to drag: a point, a number, a solid to turn. Formulas come after you have seen it happen.
Commit to a guess before the answer appears. The measures stay hidden until you have.
Type an answer in any equal form. A wrong one gets feedback on that exact mistake, and hints come in three steps.
Topics come back as they start to fade, with new numbers each time, further apart as you get them right.
Start from what you are studying
Every course is a map of topics and what each builds on. Search for "similar triangles", find it on the map, and see which earlier topics it leans on. The map fills in as you learn, and fading topics glow until you review them.
Open the Geometry map →22 courses, from geometry to quantum physics and biology
1055 lessons across mathematics, physics and biology, each course a map of topics, with its first unit free.
Start free. Open everything when you're ready.
The first unit of every course is free, with its practice and review. A subscription opens every lesson in every course, from $10 a month or $70 a year. Cancel any time.
What is ConvergerLab?
A website of interactive lessons in mathematics, physics and biology. 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.
What can I learn right now?
22 courses, each one whole, in mathematics, physics and biology:
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.
Further physics: Classical Mechanics.
Biology: Biology.
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.
What do I need?
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.
How do I type an answer, and which forms count?
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
piorπ:4pi - functions:
sin(x),ln(2),e^(-t) - angles in degrees where a question asks for them, with or without the sign:
72or72° - 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.
How does review work?
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.
Try one lesson and see how it feels.
Angle pairs, in a few short steps: turn a line and watch which angles stay equal. It's in the free first unit.