Regrouping


Regrouping is changing the pattern of counters on a board without changing the number represented on the board.


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In the left and right examples of regrouping, the number represented on the board remains the same despite different patterns of counters. Regrouping can be applied to boards with any number of columns, such as a board with 2 columns on the left or a board with 5 columns on the right.



The boards shown above are designed in such a way that one small set of rules is sufficient to do all the regrouping, not only on the simple boards shown above, but also on boards with any number of columns that may be used to carry out computations with any decimals.



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This set of images shows Rule 3 being performed in the rightmost column. All rules can be applied to any column.



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This set of images shows Rule 3 being performed in a different column. This time, the rule is done in the leftmost column.



Mastering the regrouping rules is a highly useful skill that is needed very often when one uses counting boards. Mastering this skill does not require any mathematical knowledge, so it can be compared to the typing skill needed to use a computer, or to the skill in writing that is needed to carry out paper and pencil computations.

Rules of Regrouping

stands for a counter.


Rule 1

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Rule 2

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Rule 3

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Rule 4

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Rule 5

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These five rules are valid for both red and white tokens, provided that all the tokens involved are of the same color. There is one more rule:

stands for a white counter.
stands for a red counter.


Rule 6

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Two tokens of different colors can be removed from any location or added to any location.

Explanations

1. These rules can be used on a board even when it is not labeled with numbers, similar to the rules of a simple game with the goal of transforming any configuration of counters into another one.



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The image shows Rule 2 being used on a board without labels. For reference, the orientation of the board remains the same throughout the example.



2. For children who are learning arithmatic, the rules should be justified by a variety of examples.



Rule 1 Example


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This set of images shows Rule 1 being used. In this case, two counters on the 10 location is the same as one counter on the 20 location.



Rule 2 Example


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This set of images shows Rule 2 being used. In this case, one counter on the 1 location and one counter on the 2 location is the same as one counter on the 3 location.



Rule 3 Example


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This set of images shows Rule 3 being used. In this case, one counter on the .01 location and one counter on the .03 location is the same as one counter on the .04 location.



Rule 4 Example


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This set of images shows Rule 4 being used. In this case, one counter on the .1 location and one counter on the .4 location is the same as one counter on the .5 location.



Rule 5 Example


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This set of images shows Rule 5 being used. In this case, one counter on the 100 location is the same as two counters on the 50 location.



Rule 6 Example


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This set of images shows Rule 6 being used. In this case, one white counter and one red counter on the 300 location is the same as no counters on the board.



3. The general theorem connecting these six rules with arithmetic states:

If, on a board of any size, as shown above, two configurations represent the same decimal number, then one can be obtained from the other by applying these six rules. The application of these rules never changes the number that is represented on a board.

The proof of this theorem requires college level mathematics, so it cannot be shown to children, but if such a board were used as a teaching aid in higher grades, students could be shown this theorem, and they could be told something about it.

Webpage Maintained by Owen Ramsey
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