Comparing Numbers

In this unit, children learn to tell how to use a counting board to determine which of two whole numbers in the range of 1 to 99 is bigger, and to determine what the difference is between the two numbers.

This picture shows the board used in the lesson.


Children work in groups of two, three, or four. The teacher gives to each child (verbally or in writing) a number in the range from 1 to 99. Each child represents her/his number on the board using white counters. Then, a pair of children in the same group compares their numbers and finds their difference, using the method described below.

This task should be repeated many times until children can do it with ease and without errors. This task should not be taught as "discovery learning". The teacher should show how to do it and encourage children to learn from and help each other.

An Explanation of the Method Using an Example

1. Assume that children need to compare the two numbers 81 and 57, which they have already represented on their boards with white counters. They should use as few counters as possible to represent each number. They may represent the same number in different ways. For example, the number 7 can be represented on the board as 5+2 or 4+3.

First Child's Board Second Child's Board

The pictures above show the boards of two children comparing the numbers 81 and 57. The first child represents the number 81 with counters on the 50, 30, and 1 squares. The second child represents the number 57 with counters on the 50, 4, and 3 squares.

2. One student changes the color of his/her counters to red. Then, each student copies the configuration from the other board onto his/her own. White and red counters "cancel" each other. So now the counters on each board represent the difference of the original numbers.

First Child's Board Second Child's Board

The pictures above show the children's boards after copying the counters. After this step, both children's boards should show the same configuration.

3. After removing red and white counters from the same locations, each student regroups the remaining counters until all counters on the board are the same color.

Example of Regrouping

  30 = 20+5+4+1   4+-4 = 0   5+-3+1+1 = 4  

The pictures above show the previous configuration being regrouped to result in one color of counter. The regrouping rules used are also shown in between each regrouping step.

Conclusion: 81 is bigger than 57 and their difference is 24.

Information for Teachers

Regrouping is a skill that is acquired by practice. So the teacher has to achieve enough practical skill to provide help in every situation that students may encounter.

The mathematical justification for the rules is simple: each instance of a rule is an arithmetic equality. Each general rule is an algebraic equality involving only two operations: addition, x+y, and opposite, -x.

For example the rule:

corresponds to the algebraic equality x+x = (x+1) + (x-1), and it is valid in every column of a counting board used in these units. Note that the rule also corresponds to the algebraic equality x+x = (x+y) + (x-y), and is not exclusive to the first expression where y=1.

This explanation does not make sense for children in early grades, so the teacher has to make her own decision about how to teach regrouping, and whether to provide any explanation for the rules involved.

Webpage Maintained by Owen Ramsey
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