Download Some notes about adding and subtracting integers

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.