Concepts you understand by moving them.
Some theorems are learned by heart and stay strange for years. Almost always the formula is not to blame: you have simply never seen it move. Here every concept is a figure you can drag, step by step, with the explanation right beside it. The numbers are really calculated, not drawn by eye.
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Here the concepts move: try them!
First watch the story, then break it yourself.
1
Follow the steps
Each figure is built one piece at a time: assumptions, construction, conclusion. The caption always tells you what you are looking at.
2
Touch the elements
Hover (or tap) a curve, a point or a line: it tells you what it is and why it is there, without leaving the figure.
3
Try to make it fail
At the end you get to do the dragging. If the theorem is true you will not manage to disprove it, and understanding why you can't is the whole point.
The figures available
- Maths Year 5 (high school)
Rolle's theorem
Why a function that comes back to its starting height must have a horizontal tangent somewhere.
- Maths Year 5 (high school)
Derivative and tangent
Why the derivative is the slope of the tangent: a secant turning around P as Q gets closer, and a difference quotient that settles on one number.
- Physics Year 2 (high school)
Inclined plane
Why the weight splits into two components, where sine and cosine come from and what really changes when you raise the ramp.
- Maths Year 1 (high school)
Square of a binomial
Why (a+b)² is not a²+b²: a square cut into four regions that shows where the missing 2ab lives.
- Maths Year 1 (high school)
Difference of squares
a² − b² = (a+b)(a−b): a square with a corner removed, cut once and rearranged into a rectangle.
- Physics Year 4 (high school)
Independence of motions
A ball dropped and a ball thrown sideways from the same height hit the ground together, every time. Find out why.
- Maths Year 4 (high school)
Unit circle
Why sine and cosine are the coordinates of a turning point, where sin²θ + cos²θ = 1 comes from and how the signs change in the four quadrants.
- Maths Year 4 (high school)
Probability tree
Why you multiply along the branches and add across the paths, and why a filter that is right 90% of the time can be wrong about one flag in two.
Why an interactive figure and not a video
A video shows you one example: the one chosen by whoever recorded it. A figure you can move lets you choose your own, including the one you were about to use to prove the theorem false. That is where you really see what an assumption is for: when you try to remove it and see what happens.
If you want to practise the calculations after the figure, the practice tests give instant marking, and the blog explains the same topics at length.
The concept is clear but the exercise is not?
It happens a lot, and it is not a question of intelligence: a piece of method is missing. The first chat is free and is exactly for working out which one.