Transformation Sequences

Pre-Algebra · Axiom Academy

Slide, flip, and turn a shape one move at a time — and discover why the order of those moves can change where you land. A single transformation moves a shape once. But the real power comes from chaining them: do one move, then apply the next move to the result , then the next. Each image becomes the input for the move that follows — and a few simple moves in a row can carry a shape anywhere on the grid. Watch one triangle run a three-step sequence: first slide it, then flip the slid copy across the x-axis, then turn that copy a quarter-turn. The faded ghosts mark where it has been; the coordinates at the corner update at every step. A sequence is just "apply one move, then the next" — the output of each step is the input of the next. Compose your own two-step move Point P starts at . Step 1 is fixed: slide it right by 2. You choose Step 2 . Then walk the chain forward and watch each step take the previous result as its input — the running coordinate is shown at every stage. Every choice of Step 2 lands somewhere different — but the recipe is always the same: feed each result into the next move. Take the same two moves — rotate 90° and slide right by 3 — and run them on point (2,1) in each order. Flip the order and compare the two endpoints. Do you land in the same place? Same two moves, two orders, two different endings — order is part of the recipe.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.