3.3 Subtract integers  (Page 2/6)

Model: $5-(\mathrm{-3}).$

Solution

Interpret the expression.	$5-\left(\mathrm{-3}\right)$ means $5$ take away $\mathrm{-3}$ .
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 $\mathrm{-3}$ is $8$ .	$5-\left(\mathrm{-3}\right)=8$

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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?	$6$
	$8-2=6$

ⓑ
	$\mathrm{-5}-4$ This means $\mathrm{-5}$ 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?
	$\mathrm{-9}$
	$\mathrm{-5}-4=\mathrm{-9}$

ⓒ
	$6-\left(\mathrm{-6}\right)$ This means $6$ take away $\mathrm{-6}$ .
Start with 6 positives.
Add 6 neutrals to get 6 negatives to take away.
Remove 6 negatives.
How many are left?
	$12$
	$6-\left(\mathrm{-6}\right)=12$

ⓓ
	$\mathrm{-8}-\left(\mathrm{-3}\right)$ This means $\mathrm{-8}$ take away $\mathrm{-3}$ .
Start with 8 negatives.
Take away 3 negatives.
How many are left?
	$\mathrm{-5}$
	$\mathrm{-8}-\left(\mathrm{-3}\right)=\mathrm{-5}$

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Model each subtraction expression:

1. ⓐ $2-8$
2. ⓑ $\mathrm{-3}-(\mathrm{-8})$

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.
	$2-8=\mathrm{-6}$

ⓑ 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.
	$\mathrm{-3}-\left(\mathrm{-8}\right)=5$

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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 $\mathrm{-23}-7.$
  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.
  
  $\mathrm{-23}-7=\mathrm{-30}$
  
  Notice, that to subtract $\text{7,}$ we added $7$ negatives.
• Subtract $30-(\mathrm{-12}).$
  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.
  
  $30-(\mathrm{-12})=42$
  
  Notice that to subtract $\mathrm{-12},$ 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.