Iterative Processes
Iterative Processes is repeating the same steps over and over to see how a result grows or changes each time.
Each time you add a spoonful of sugar to a bowl of water, the water line rises a little more. You repeat the same step — add, stir, check — and the level climbs higher each time. After ten spoons, the bowl is nearly full. Each pass through the same steps builds on the last one.
Explaining iterative processes by grade level
Think about turning a Rubik's Cube the same way again and again. Each turn mixes the colors more. But after enough turns, the colors come back to where they started. Repeating one move shows you a pattern that only shows up over time.
Projects that explore iterative processes
When you feed a result back into the same equation over and over, small differences can compound. In the logistic equation x(n+1) = rx(n)(1-x(n)), some constants produce sequences that settle on one number, while others create chaos with no pattern at all. Even a tiny change in precision can shift the outcome from a steady pattern to total unpredictability.
Repeating the same move sequence on a Rubik's Cube does not steadily increase disorder — it eventually brings the cube back to its solved state. The number of repetitions needed to return to the start is called the order of the sequence. A computer program written in QBASIC tracks the cube's average variegation (degree of disorder) after each repetition. When you graph the results, the data points fit a 4th-degree polynomial equation, suggesting that larger orders produce higher-degree polynomials. What looks like random scrambling at each step turns out to follow a predictable mathematical pattern — one that further testing could confirm holds across all sequences.