# 3.3 Subtract integers  (Page 2/6)

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Model the expression:

$-6-4$

−10

Model the expression:

$-7-4$

−11

Model: $5-\left(-3\right).$

## Solution

 Interpret the expression. $5-\left(-3\right)$ means $5$ take away $-3$ . Model the first number. Start with 5 positives. Take away the second number, so take away 3 negatives. But there are no negatives to take away. Add neutral pairs until you have 3 negatives. Then take away 3 negatives. Count the number of counters that are left. The difference of $5$ and $-3$ is $8$ . $5-\left(-3\right)=8$

Model the expression:

$6-\left(-4\right)$

10

Model the expression:

$7-\left(-4\right)$

11

Model each subtraction.

1. 8 − 2
2. −5 − 4
3. 6 − (−6)
4. −8 − (−3)

 ⓐ $8-2$ This means $8$ take away $2$ . Start with 8 positives. Take away 2 positives. How many are left? $6$ $8-2=6$
 ⓑ $-5-4$ This means $-5$ take away $4$ . Start with 5 negatives. You need to take away 4 positives. Add 4 neutral pairs to get 4 positives. Take away 4 positives. How many are left? $-9$ $-5-4=-9$
 ⓒ $6-\left(-6\right)$ This means $6$ take away $-6$ . Start with 6 positives. Add 6 neutrals to get 6 negatives to take away. Remove 6 negatives. How many are left? $12$ $6-\left(-6\right)=12$
 ⓓ $-8-\left(-3\right)$ This means $-8$ take away $-3$ . Start with 8 negatives. Take away 3 negatives. How many are left? $-5$ $-8-\left(-3\right)=-5$

Model each subtraction.

1. 7 - (-8)
2. -2 - (-2)
3. 4 - 1
4. -6 - 8

Model each subtraction.

1. 4 - (-6)
2. -8 - (-1)
3. 7 - 3
4. -4 - 2

Model each subtraction expression:

1. $\phantom{\rule{0.2em}{0ex}}2-8$
2. $-3-\left(-8\right)$

## Solution

 ⓐ We start with 2 positives. We need to take away 8 positives, but we have only 2. Add neutral pairs until there are 8 positives to take away. Then take away eight positives. Find the number of counters that are left. There are 6 negatives. $2-8=-6$
 ⓑ We start with 3 negatives. We need to take away 8 negatives, but we have only 3. Add neutral pairs until there are 8 negatives to take away. Then take away the 8 negatives. Find the number of counters that are left. There are 5 positives. $-3-\left(-8\right)=5$

Model each subtraction expression.

1. $\phantom{\rule{0.2em}{0ex}}7-9$
2. $\phantom{\rule{0.2em}{0ex}}-5--9\right)$

4

Model each subtraction expression.

1. $\phantom{\rule{0.2em}{0ex}}4-7$
2. $\phantom{\rule{0.2em}{0ex}}-7-\left(-10\right)$

3

## Simplify expressions with integers

Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.

• Subtract $-23-7.$
Think: We start with $23$ negative counters.
We have to subtract $7$ positives, but there are no positives to take away.
So we add $7$ neutral pairs to get the $7$ positives. Now we take away the $7$ positives.
So what’s left? We have the original $23$ negatives plus $7$ more negatives from the neutral pair. The result is $30$ negatives.
$-23-7=-30$

Notice, that to subtract $\text{7,}$ we added $7$ negatives.
• Subtract $30-\left(-12\right).$
Think: We start with $30$ positives.
We have to subtract $12$ negatives, but there are no negatives to take away.
So we add $12$ neutral pairs to the $30$ positives. Now we take away the $12$ negatives.
What’s left? We have the original $30$ positives plus $12$ more positives from the neutral pairs. The result is $42$ positives.
$30-\left(-12\right)=42$

Notice that to subtract $-12,$ we added $12.$

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

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