Extending the Columns of a Counting Board

If the numbers in each column of a counting board form a geometric progression, the column can be extended both up and down. Regrouping rules for the board created in this way are the same as for the original board. So using the board doesn't require learning any new skills.

Even small extensions can significantly enlarge the range of application of the expanded board. We show this using an example of an extended fraction board, where three out of four columns of the original board have been extended up.



On the extended fraction board 11/12 can be represented by 2/3 + 1/4. And 14/15 can be represented by 2/3 + 4/15 + 1/15.

2/3 + 1/4 = 11/12

2/3 + 4/15 + 1/15 = 14/15

Whole Class Task

Represent each of the 59 proper fractions, 1/60, ..., 59/60, with the smallest possible numbers of white counters. Does any fraction require two counters in any location? Do you get simpler representations when you are also allowed to use red counters (representing negative numbers)? Make a chart showing what you found out.

Optional Task
(only for students interested in this topic)

Would adding one more location to the last column simplify some representations of the fractions, 1/60, ..., 59/60? If so, which ones?

The new boards:


Using the New Boards

8/15 can be represented using 2 tokens on the original board (left), and 1 token on the new board (right).

7/15 can be represented using 3 tokens on the original board (left), and 2 tokens on the new board (right).

Webpage Maintained by Owen Ramsey
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