Build the Strip

Learning Objective

Students will prepare a complete paper strip, assign a value to (price) the size-stone, in the same way the machine does, and construct the size-curve and change-curve in stones, learning that a board without its strip is a stack of rocks and with it is a specific curve, one we associate with a particular equation.

Teacher Overview

This is the physical twin of the previous exercise, using the same configuration, and a mixed classroom can run the two side by side and compare boards at the end. The comparison itself teaches: same mathematics, two media. Budget one period, working in small teams around the board. A supply note before class: the full build uses 220 size stones across the eleven columns plus 55 change-size stones. If the class set is smaller, assign alternate columns to different teams, or build the left half of the window and reason out the right.

Student Activity

Setup

The same configuration as before: y = (x² + x)/2, midpoint 5, increment 1, 55 stones.

PREPARE THE STRIP: Compute the eleven x-values and label the strip's eleven columns with them. Add the equation, the value of one size-stone, the midpoint, and the increment.

Price the stone the way the machine does: the largest of the eleven y-values minus the smallest, divided by the stones available. Position the strip along the base of the board.

BUILD: Count out the size stones for each column, measuring above the floor (in this case, the floor is zero), and stack them. Step back after every few columns and look at the shape forming.

FORM THE CHANGE-CURVE: For each pair of neighboring columns, find the difference and set out change-size stones for it. Then arrange the change-size stones into their own curve above the size stones.

READ: Standing back, describe both curves in ordinary language. Then pick the middle column and estimate how fast the function is changing there, using the change-size stones, the stone's value, and the increment.

Reflection Questions

1. Remove the strip. What can someone still learn from the board, and what becomes impossible to know?
2. If a second team built y = x² on an identical board with its own strip, could the two boards look alike while meaning different things? What would settle it?
3. You placed the change-size stones by counting differences of size stones. Could you have found them without touching the size stone columns at all, straight from the y-values? Which way felt more trustworthy, and why?

Expected Student Discoveries

The strip is what turns stacks of stones into a curve associated with a particular equation; the labels, the stone's value, and the settings carry the meaning that the heights alone cannot. The triangular number pattern appears just as it does on the screen, which quietly confirms that the two media obey one rule. Estimating the middle column's rate by hand, students find the same 5-and-6 readings the virtual abacus reports, and averaging them lands on 5.5.

Common Misconceptions

Treating the strip as optional paperwork. It is the board's meaning; the reflection questions exist to make that felt rather than told. Assuming stone heights compare across boards. Two boards compare only after their strips are read.

Teacher Notes

The discipline built by constructing a strip here pays off later on the virtual abacus, where the same two anchors live in the top line of the left panel. If teams finish early, ask them to predict what the strip and stones would look like at midpoint 6 without rebuilding; the discussion covers most of the same ground as a second build at a fraction of the time.

Written for the Calculus Abacus Project by Shah Nawal, with AI assistance, 2026.