Small Boards for Early Grades

                          

These boards have one important common feature. Each column forms a geometric progression with quotient one-half. The first historical record of such boards appears in John Napier's book titled Rabdology. Because all the numbers shown on the board are products and reciprocals of only three primes, 2, 3, and 5, regrouping rules for counters are simple and easy to learn.


On the last board each fraction with denominator 3, 5, 15, 30, or 60 can be represented as the sum of unit fractions.


Examples


Positive Whole Numbers


The board holds 6 + 1 = 7.



The board holds 10 + 5 + 4 + 4 = 23.



Negative Whole Numbers


The board holds - 30 - 5 - 12 - 12 - 3 - 4 = -66.



Fractions


The board holds 1/2 + 1/3 + 1/6 + 1/15 + 1/60 = 13/12.



The board holds 1/2 + 1/12 - 1/5 - 1/5 + 1/10 - 1/30 = 1/4.



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