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Some notes about adding and subtracting integers Math 123 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. 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. 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. 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. 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. 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)=? 3+(-1)=2 2+(-1)=1 1+(-1)=0 0+(-1)=-1 -1+(-1)=? -2+(-1)=? The chip model • Use the chip model to justify the rules for adding integers. 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|). • 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. 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. 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=? 5-7=-2 4-7=-3 3-7=-4 2-7=-5 1-7=-6 0-7=-7 -1-7=? -2-7=? 3-2=1 3-1=2 3-0=3 3-(-1)=? 3-(-2)=? The chip model • Use the chip model to justify the rules for subtracting integers.