3.3 Subtract integers

Model subtraction of integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.

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

Perhaps when you were younger, you read $5-3$ as five take away three . When we use counters, we can think of subtraction the same way.

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

Model: $5-3.$

Solution

Interpret the expression.	$5-3$ means $5$ take away $3$ .
Model the first number. Start with 5 positives.
Take away the second number. So take away 3 positives.
Find the counters that are left.
	$5-3=2$ . The difference between $5$ and $3$ is $2$ .

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

Solution

Interpret the expression.	$\mathrm{-5}-\left(\mathrm{-3}\right)$ means $\mathrm{-5}$ take away $\mathrm{-3}$ .
Model the first number. Start with 5 negatives.
Take away the second number. So take away 3 negatives.
Find the number of counters that are left.
	$\mathrm{-5}-\left(\mathrm{-3}\right)=\mathrm{-2}$ . The difference between $\mathrm{-5}$ and $\mathrm{-3}$ is $\mathrm{-2}$ .

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Notice that [link] and [link] are very much alike.

• First, we subtracted $3$ positives from $5$ positives to get $2$ positives.
• Then we subtracted $3$ negatives from $5$ negatives to get $2$ negatives.

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

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

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

Solution

Interpret the expression.	$\mathrm{-5}-3$ means $\mathrm{-5}$ take away $3$ .
Model the first number. Start with 5 negatives.
Take away the second number. So we need to take away 3 positives.
But there are no positives to take away. Add neutral pairs until you have 3 positives.
Now take away 3 positives.
Count the number of counters that are left.
	$\mathrm{-5}-3=\mathrm{-8}$ . The difference of $\mathrm{-5}$ and $3$ is $\mathrm{-8}$ .

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