Modeling Subtraction of Integers

Remember the story 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 \(3.\)

\(5-3\phantom{\rule{1em}{0ex}}-5-\left(-3\right)\phantom{\rule{1em}{0ex}}-5-3\phantom{\rule{1em}{0ex}}5-\left(-3\right)\)

Example

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 \(2\).

Example

Model: \(-5-\left(-3\right)\text{.}\)

Solution

Interpret the expression.	\(-5-\left(-3\right)\) means \(-5\) take away \(-3\).
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.
	\(-5-\left(-3\right)=-2\). The difference between \(-5\) and \(-3\) is \(-2\).

Notice that the examples above 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.

This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.

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.

Optional Video: Subtracting Integers with Color Counters (Extra Zeros Needed)

Example

Model: \(-5-3.\)

Solution

Interpret the expression.	\(-5-3\) means \(-5\) 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.
	\(-5-3=-8\). The difference of \(-5\) and \(3\) is \(-8\).

Example

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 \(8\). \(5-\left(-3\right)=8\)

Example

Model each subtraction.

8 − 2
−5 − 4
6 − (−6)
−8 − (−3)

Solution


	\(8-2\) 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?
	\(-5\)
	\(-8-\left(-3\right)=-5\)

Example

Model each subtraction expression:

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

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.
	\(-3-\left(-8\right)=5\)

Modeling Subtraction of Integers