← HomeIntroducing the Calculus AbacusA tool for building the idea of change by hand, before the formulas arrive. How the Abacus Works: Setup and MechanicsWhat every object and number on the board means, and why the machine was built this way. The Abacus Meets the TextbookFrom physical experience to the language of calculus: velocity at an instant, limits near a missing point, and the hunt for the base whose slope is exactly 1. Estimation Before Formalization: The Calculus Abacus and the Origins of CalculusA tool for seeing mathematical action: how the abacus's finite estimates connect with Wallis, Fermat, Leibniz, and Newton.
About the Calculus Abacus
Articles and exercises written for teachers.
Exercises
- The Shape of ChangeA visual, inquiry-based activity in which students build the change-curve of y = x² and discover that a bending curve can have a perfectly straight pattern of change.
- How Fast, Right Here?A visual, inquiry-based activity in which students narrow their attention to a single point on a curve and estimate how fast it is changing there.
- How Small Is Small Enough?A visual, inquiry-based activity in which students meet a smooth curve that impersonates a straight line at every ordinary scale, and discover that agreement among many readings is not the same as truth.
Exercises
- Be the AbacusA hands-on activity in which students reproduce the abacus's own arithmetic, pricing a size-stone, deriving every number in the left panel, and placing the stones themselves before letting the software check their work.
- Build the StripA construction activity for classrooms with a physical board and stones, in which students prepare a paper strip that anchors a physical board's meaning, then build both curves by hand.
- From Differences to RatesAn investigation in which students modify the value of the change-stones so that they represent the rate Δy/Δx instead of the difference Δy, and discover how the two differ.
Exercises
- The Ball and the TowerStudents estimate how fast a falling ball is moving at a particular moment by using average speeds over progressively shorter intervals, and discover that the estimates settle toward a value without ever using an interval of zero.
- The Window That Never ArrivesAn investigation of limits in which the Abacus shows what happens near a point even when the curve is undefined there.
- The Hunt for the Magic BaseA discovery hunt in which the class corners a famous number by measurement and bisection, without being told it exists.
Exercises
- From Columns to AreaStudents turn displayed heights into finite area contributions, compare two partitions on one board, and judge whether the estimates are stabilizing.
- From Secant to TangentStudents compare nearby values on both sides of x = 2, convert finite differences into slope estimates, and watch the estimates close around a stable value.
- Correct the GuessStudents use current error and a local slope estimate to construct tangent-style corrections toward √2, then judge whether the provisional roots are stabilizing.