Western Illinois University
Rules for Adding and Subtracting Integers with Manipulatives[1]
Definitions:
➢ Add means join. This means put the two addends on the work mat and push them together.
➢ Subtract means take-away. For a – b, you put a on the work mat and take-away b. Take-away means remove from the work mat. Note that you might take-away a positive or you might take-away a negative[2].
Two rules are important:
• You may remove a neutral pair at any time. That is, if you have a –1 and +1, these can be removed. This is important when you want to simplify to get the answer.
• You may introduce neutral pairs as often as you wish. That is, if you may put –1 and +1 on the work mat anytime. This is important when you want to take-away something you do not yet have on the work mat.
The other way to handle subtraction is to “add the opposite” (the “change the sign” technique) which follows the rule [pic]. Under this scheme you do not take away. You add the opposite and then remove neutral pairs as needed. While the [pic] rule is important and should be learned eventually, by itself it is procedural and rule-based (and is therefore, not conceptual and may not have much meaning for students). The take away method (above) should be used first and then the [pic] rule can follow logically.
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[1] Manipulatives: Two options for modeling positive and negative numbers are:
• Colored Counters
o Use two different colors - One color (for example, black) stands for +1 and the other color (for example, red) stands for –1.
o Two-color counters – Each chip has a different color on each side. One color stands for +1 and the other color (side) stands for –1.
• Positive and negative regions of a work mat. Use a work mat divided into two regions. Objects in one region stand for positive quantities and objects in the other region stand for negative quantities.
[2] Using the take-away definition, we do not use the “change the sign” technique.
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