1.3 Add and subtract integers  (Page 5/10)

The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!

Subtract integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read $\text{“}5-3\text{”}$ as $\text{“}5$ take away $3.\text{”}$ When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers $5$ and $3.$

To subtract $5-3,$ we restate the problem as $\text{“}5$ take away $3.\text{”}$

We start with 5 positives.
We ‘take away’ 3 positives.
We have 2 positives left.
The difference of 5 and 3 is 2.	2

Now we will subtract $\mathrm{-5}-\left(\mathrm{-3}\right).$ Watch for similarities to the last example $5-3=2.$

To subtract $\mathrm{-5}-\left(\mathrm{-3}\right),$ we restate this as $\text{“}\mathrm{–5}$ take away $\mathrm{–3}\text{”}$

We start with 5 negatives.
We ‘take away’ 3 negatives.
We have 2 negatives left.
The difference of −5 and −3 is −2.	−2

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both white and red counters as well as some neutral pairs . Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

• To subtract $\mathrm{-5}-3,$ we restate it as $\mathrm{-5}$ take away 3.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

	−5 − 3 means −5 take away 3.
We start with 5 negatives.
We now add the neutrals needed to get 3 positives.
We remove the 3 positives.
We are left with 8 negatives.
The difference of −5 and 3 is −8.	−5 − 3 = −8

And now, the fourth case, $5-\left(\mathrm{-3}\right).$ We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.

	5 − (−3) means 5 take away −3.
We start with 5 positives.
We now add the needed neutrals pairs.
We remove the 3 negatives.
We are left with 8 positives.
The difference of 5 and −3 is 8.	5 − (−3) = 8