Constant of Proportionality
The equation sits centre screen with every symbol physically sized by its own value — drag a slider and the symbol swells while whatever responds to it grows or shrinks in step, so direct proportionality looks like two symbols rising together and inverse looks like a see-saw. A live graph draws the straight line through the origin or the falling curve alongside. The lesson it is built to make: a proportionality statement means nothing until you say what is held constant. In F = ma, F is directly proportional to m when a is held, but a is inversely proportional to m when F is held — same equation, opposite relationship, and switching "solve for" flips it in front of you. Every equation is stored as one product relation and the response derived from exponents rather than scripted, verified across 19 driver-responder pairs before release.
Direct proportionality draws a straight line through the origin; inverse draws a falling curve.
"F is proportional to m" is an incomplete sentence. Proportional while what is held constant?
In F = ma, solve for F and drag m: they grow together, directly proportional. Now solve for a and drag m: a shrinks as m grows, inversely proportional. Same equation, opposite answer — the only difference is which quantity you decided to hold still.
1. Run the F = ma experiment above both ways and watch the graph flip from a straight line to a falling curve.
2. On ρ = m/V, solve for ρ and drag V. Density rises as volume falls — squeeze the same mass smaller and it gets denser.
3. On p = ρgh, solve for p and drag h. Double the depth, double the pressure. Then solve for h instead and drag ρ — a denser liquid needs less depth for the same pressure.
Reading The Shape
Direct (y = kx) — straight line through the origin. Double one, double the other. The gradient is k.
Inverse (y = k/x) — a falling curve approaching but never touching either axis. Double one, halve the other. Here their product is constant.