Download Adding and subtracting with negative numbers – Explanations, not

Adding and subtracting with negative numbers – Explanations, not rules
What does it mean to add or subtract in general? The number line is a good model for this.
Add two quantities of things and you get a larger quantity. This is shown on the number line by
moving to the right, in the more positive direction. When we introduce negative numbers, the
counting numbers (which represent actual quantities of things) are called positive numbers,
because when we add them, they make something more positive.
7+3
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
-5 + 3
-5 + 10
“Taking away” a quantity results in a smaller amount. This is represented on the number line by
moving to the left, in the more negative direction. When we try to take away an amount that is
larger than the starting value, we go “in the hole,” getting a negative number. This is familiar to
most students (in Michigan) from temperature and thermometers.
8–5
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
-10 -9 -8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13
8 – 10
-2 – 5
You can simplify the harder problems with a negative and a positive number. For 8 – 10,
recognize that you are subtracting 10 from 8. First subtract 8 from 8 (getting 0), then subtract 2
more (getting -2). For -5 + 10, first add 5 to -5 (getting zero), then add 5 more (getting +5).
What does it mean to add or subtract a negative?
Adding a negative amount means to make something more negative – like adding more grumpy
people, or taking on a debt, or adding more “cold cubes.” Making something more negative is
the same as subtracting – moving left on the number line. So adding a negative is the same as
subtracting.
7 + (-5) = 7 – 5
Subtracting a negative means to take away something that is negative, making the result more
positive. Examples include taking cold cubes out of a liquid, thus raising its temperature, or
removing grumpy people from a party, making the party more happy. The easiest example
involves starting with negative things and removing some of them:
-5 – (-2)
can be read as
5 negatives take away 2 negatives.
The answer is clearly 3 negatives, or -3.
This is the same result as starting at -5 degrees and getting warmer by 2 degrees: -5 + 2 = -3.
Subtracting a negative gives you the same result as adding a positive.
-5 – (-2) = -5 + 2
A problem such as -5 – (-8) is a little more difficult to visualize. It can be read as “5 negatives
take away 8 negatives.” However, you can’t take away 8 negatives from 5 negatives. The chip
model is helpful here (or hot and cold cubes). You can artificially introduce more negatives as
long as you balance any new negatives with positives. So add 3 negatives and 3 positives to the 5
negatives, resulting in 8 negatives and 3 positives; then take away the 8 negatives, leaving 3
positives.
Using the idea that subtracting a negative gives you the same result as adding a positive, then
-5 – (-8) = -5 + 8 = 3
You can also work this problem by adding 10 to the first and second numbers, then doing the
subtraction: [-5 + 10] – [-8 + 10] = 5 – 2 = +3.
A possible scenario for this involves debts. Say you have $20 in cash, but you also have $25 in
debts (you owe people $25). How much are you really worth? $-5. But then for some reason, $8
in debts are taken away from you: subtract (-8) from -5, written as -5 – (-8). You’re suddenly $8
richer, so your worth goes from $-5 to $+3. Subtracting $8 in debts is like becoming $8 richer –
i.e. subtracting a negative amount is like adding a positive amount.