           ## Operations with Integers

Operations with integers follow the same rules as operations with whole numbers. The confusion comes in when you deal with the sign of the number. If you teach the rules for operating with signed numbers without helping students to understand what's going on, you may have quite a few confused students.

Materials: chart paper, a large classroom number line, individual number lines

Preparation: If you don't have a pre-made large number line visible from everywhere in the classroom, make one out of adding-machine tape. Include the integers 20 through +20 if possible.

Prerequisite Skills and Concepts: Students should be comfortable computing with whole numbers. The commutative properties of addition and multiplication are also important.

Subtracting, when some of the numbers involved are negative to start with, is sometimes confusing. Two ways to make this process easier are:

(1) Always remember that you can rewrite any subtraction problem as addition.
(2) If you're having trouble remembering the rules, use the Use Simpler Numbers problem-solving strategy. You may get confused with 762 + 1486, but if you can figure out 6 + 11, you'll be OK.
• Say: You all know that the number line includes numbers greater than zero and less than zero. Today, we'll talk about how to use the number line to think about addition and subtraction.
• Ask: Can someone show me, on the number line, how to add 6 and 5?
Students may think this is childish, but encourage them to play along with you. Work with the class to come up with a good description of the action required to show addition on the number line. Be sure the description includes both the addends and the action as well as the sum: Start at the first addend. Move to the right the number of units indicated by the second addend. You're now at the sum.
• Ask: Can someone show me, on the number line, how to subtract 5 from 6?
Students should be getting into the rhythm of the lesson by now. Their description of subtraction on the number line needs to include both the minuend and subtrahend (though you may not want to use those terms), the action, and the difference: Start at the minuend. Move to the left the number of units indicated by the subtrahend. You're now at the difference.
• Now lead a discussion of the similarities and differences between these two descriptions. The main difference appears to be in the direction of movement. When students compare the directions right for add and left for subtract, help them to see that the subtraction symbol tells them to go in the opposite direction from the addition symbol.
• Ask: Can someone show me, on the number line, how to subtract 5 from 6?
This will get tricky unless students have really grasped the concept that the subtraction symbol is telling you to do the opposite. Two subtraction symbols in a row, even if one is an operation symbol and one is the sign on a number, are a double negative. In English, a double negative results in a positive: I don't have no pencils means I do have some pencils. In the same way, a double negative in math results in a positive: 6 – 5 = 6 + 5.
• Now lead a discussion on how to rewrite any subtraction exercise as addition, using the concept of opposites. Here are some examples to try.
6 – 5 becomes 6 + 5
6 – 5 becomes 6 + 5 6 – 5 becomes -6 + 5 6 – -5 becomes -6 + 5
• Ask: Can someone show me, on the number line, each of the examples we've rewritten as addition examples?
Now students will need to learn to translate the messages they're getting from the symbols in these examples. The operation symbol, +, normally says go right. But, you can't do that in all of these examples because the sign on a number can counteract the operation symbol. If the number to be added is negative, the new direction is go left.
Terminating and Repeating
Decimals
Operations with Integers

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