Gnomons
See number sequences grow into geometric forms
Understand number sequences by seeing the pieces added at each growth step. Compare the geometric construction with its total and formula.
Key idea
A gnomon is an added shape that extends a figure while preserving a chosen form. Building successive squares reveals that consecutive odd numbers are the differences between consecutive square numbers.
Try this first
Choose square numbers and advance one growth step at a time. Compare the newly added pieces with the total already present. Then try triangular or cubic growth to see how a different geometric rule produces a different sequence.
How to get started
- Choose a family such as square, triangular, Fibonacci or a three-dimensional sequence.
- Add or remove steps, or play the growth animation. Compare the new layer with the total number of objects.
- Read the formula and sequence beside the figure. Switch families to compare how their growth rules differ.
Explore a little further
Try another experiment
Build the first few squares and count only the new pieces at each step. Compare those additions with the total number of pieces. Can you predict the next addition before advancing?
Good to know
The pieces illustrate a counting rule. Read the selected sequence and growth convention together with the formula rather than assuming every arrangement follows the same pattern.
Public Lite tools share 20 free minutes a day, without an account. New accounts receive a 10-day full-access trial without a card. Paid access continues under the selected plan.
Compare access options →