Cut Once, Counted Twice

Where it usually goes wrong

a(x + c) = ax + ac is practiced as a move — "multiply the thing outside by both things inside" — with no picture of what is actually being claimed: that a rectangle's total area does not care where you decide to stop counting one piece and start counting the next. The rule survives being applied correctly for years without that ever being obvious.

Try this

• Drag c out to 5 or 6 without touching a. The counted grid on the right grows, the strip on the left does not, and both labels — a·x and the number — update on their own.
• Drag a taller. BOTH pieces grow by the same number of new rows at once, because the cut runs the full height — a still multiplies everything on either side of it, together, not one piece first.
• Press "2(x + 1)" and then "5(x + 4)". Read the total two ways each time: as one rectangle, a rows of (x + c), and as two added pieces, a·x and a·c. The cut changes nothing about which one is true.

Where this goes next

Combining like terms is this same cut read backwards: 3x + 5x is two separate strips of width x sitting side by side, which is one rectangle of width x and height 8 the moment the cut between them is erased — addition of the areas, not a rule about the letters.