Notation and Modeling Subtraction of Integers

Learning Outcomes

  • Model integer subtraction with color counters

You learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.

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 5−35−3 as five take away three. When we use counters, we can think of subtraction the same way.

We will model four subtraction scenarios using the numbers 55 and 33.

  • 5−35−3
  • −5−(−3)−5−(−3)
  • −5−3−5−3
  • 5−(−3)5−(−3)

example

Model: 5−35−3.

Solution:

Interpret the expression. 5−35−3 means 55 take away 33 .
Model the first number. Start with 55 positives. This figure has 5 blue circles with the number 5 underneath.
Take away the second number. So take away 33 positives. This figure has 5 blue circles. Under the last two circles, it say 2 positives.
Find the counters that are left. This figure has two blue circles.
5−3=25−3=2

The difference between 55 and 33 is 22 .

Now you can try a similar problem.

try it

Model the expression:

6−46−4

 

Model the expression:

7−47−4

In the previous example we subtracted 33 positives from positive 55. Now we will subtract 33 negatives from negative 55. Compare the results of this example to the previous example after you read through it.

example

Model: −5−(−3)−5−(−3)

You can try a similar problem.

try it

Model the expression:

−6−(−4)−6−(−4)

 

Model the expression:

−7−(−4)−7−(−4)

Notice that the previous two examples are very much alike.

  • First, we subtracted 33 positives from 55 positives to get 22 positives.
  • Then we subtracted 33 negatives from 55 negatives to get 22 negatives.

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

This figure has 5 blue circles and 5 red circles. Three circles from the blue group and red group are circled together. Above the blue circles it says 5 minus 3 equals 2. Above the red circles it says 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.

example

Model: −5−3−5−3

Solution:

Interpret the expression. −5−3−5−3 means −5−5 take away 33 .
Model the first number. Start with 55 negatives. This figure has 5 red circles labeled negative 5.
Take away the second number.

So we need to take away 33 positives.
But there are no positives to take away.

Add neutral pairs until you have 33 positives.

 This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.
Now take away 33 positives. This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.
Count the number of counters that are left. This figure has 8 red circles labeled 8 negatives.
−5−3=−8−5−3=−8

The difference of −5−5 and 33 is −8−8 .

 

Now you can try a similar problem.

try it

Model the expression:

−6−4−6−4

 

Model the expression:

−7−4−7−4

Now we will subtract a negative number from a positive number. Think of this as taking away the negative.

example

Model: 5−(−3)5−(−3)

Now you can try a similar problem.

try it

Model the expression:

6−(−4)6−(−4)

 

Model the expression:
7−(−4)7−(−4)

Now we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:

  • subtracting a positive number from a positive number
  • subtracting a positive number from a negative number
  • subtracting a negative number from a positive number
  • subtracting a negative number from a negative number

example

Model each subtraction.

  1. 8−28−2
  2. −5−4−5−4
  3. 6−(−6)6−(−6)
  4. −8−(−3)−8−(−3)

Now you can try a similar problem.

try it

Model each subtraction.

1. 7−(−8)7−(−8)
2. −7−(−2)−7−(−2)
3. 4−14−1
4. −6−8−6−8

 

Model each subtraction.

1. 4−(−6)4−(−6)
2. −8−(−1)−8−(−1)
3. 7−37−3
4. −4−2−4−2

Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression.  For example, in  −5−4−5−4, the result is −9−9, which is the same sign as −5−5. Now we will see subtraction where the sign of the result is different from the starting number.

example

Model each subtraction expression:

  1. 2−82−8
  2. −3−(−8)−3−(−8)

Now you can try a similar problem.

try it

Model each subtraction expression.

  1. 7−97−9
  2. −5−−9−5−−9

 

Model each subtraction expression.

  1. 5−85−8
  2. −7−(−10)−7−(−10)

When you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.

Watch the video below to see more examples of modeling integer subtraction with color counters.