1. Some notes about adding and subtracting integers
Math 123

2. Terminology
Number like 6 and -6 are called additive inverses. We can call -a the additive inverse of a, or we can call it the opposite of a, but we should not call it “negative a,” since –a could be positive.

3. Ordering of integers Which number is bigger: -5 or -6?
Whichever number is farther to the right is bigger (as with positive numbers), which makes ____ bigger.
This is confusing for both kids and adults.

4. Absolute value
Distance from zero needs special consideration. The distance of a number a from 0 is called its absolute value. Absolute value is a useful concept when giving rules for adding and subtracting integers.

5. The rules for adding integers
If a and b are positive, the sum is a+b.
If a and b are negative, the sum is –(|a|+|b|).
If one number is positive, and one negative, then if |a|>|b|, a+b=|a|-|b|, and if |b|>|a|, a+b = - (|b|-|a|). We can also say this in words: the sum of a positive and a negative number is found in the following way: first we subtract the smaller absolute value from the bigger absolute value, and then put in front the sign of the number with a bigger absolute value.

6. The number line model
Reminder: Adding means walking forward, and subtracting means walking backward. A positive number means facing forward, and a negative number means facing backward.
Justify the rule for adding integers using the number line.

7. The pattern model
Justify the rule for adding integers using patterns. For example, consider: 3+3=6 3+2=5 3+1=4 3+0=3 3+(-1)=? 3+(-2)=?

8. 3+(-1)=2 2+(-1)=1 1+(-1)=0 0+(-1)=-1 -1+(-1)=? -2+(-1)=?

9. The chip model
Use the chip model to justify the rules for adding integers.

10. The rules for subtracting integers
Positive – positive: Note that a-b = a +(-b) so the same rule applies as for adding a positive and a negative. Or, we can say, as before, subtract the smaller number from the bigger number and, if a>b, then the answer is positive, and if a<b, then the answer is negative.
Negative – positive: Note again that a-b = a+(- b), so the same rule applies as for adding two negatives, so –(|a|+|b|).

11. Any number – negative. As before a-b = a+(-b)
Any number – negative. As before a-b = a+(-b). Since b is negative, -b will be positive, so we say that subtracting a negative is like adding a positive.

12. The number line
Use the number line model to justify the rules for subtracting integers. You can’t just flip a chip over. That operation does not exist.

13. The pattern model
Use patterns to justify the rules for subtracting integers. For example: 5-3=2 5-4=1 5-5=1 5-6=? 5-7=?

14. 5-7=-2 4-7=-3 3-7=-4 2-7=-5 1-7=-6 0-7=-7 -1-7=? -2-7=?

15. 3-2=1 3-1=2 3-0=3 3-(-1)=? 3-(-2)=?

16. The chip model
Use the chip model to justify the rules for subtracting integers.