Subtracting integers using models

Subtracting Integers

[[Integer graph1-blue-yellow.png]]


  1. Understand how to use models to subtract integers
  2. Evaluate integer subtraction problems using mental math
  3. Develop methods for using integer tiles
  4. Identify and apply zero pairs


  • Writing activity: Explain to a friend how to add: -8 + -7
  • What is the definition or another way to say subtraction?
  • What is the definition of an expression?

Subtracting integers using models

How can we use models when we subtract integers?

Well, the answer to this question is slightly different from when we added integers.

In the warm up, I asked for the definition or another way to say “subtraction.” The definition is: the process or skill of taking one number or amount away from another. Another way to say subtraction: Take away

Statement: If you start with eight apples and you take away four, how many are left? We could write an expression in order to solve this.

Expression: 8 - 4 To solve we simply do the subtraction.

Using Models: To use integer tiles with this solution, we would start with 8 white tiles. Since the expression tells us to subtract 4 (positive) white tiles, we simply take these four away and the remaining tiles tells us our answer.

8 White tiles: 8 White Tiles.jpg

Take away 4: 8 white tiles take away 4.jpg

Here is the challenge... How do we subtract integers when the second number is larger than the first?

For example: Evaluate: 5 - 8

Using our tile method, we start with five white tiles.

But we need to take away 8 positive tiles. Since we don’t have 8 to take away, we need to somehow add in three more positive tiles. But we can’t simply add white tiles. The equation needs to be balanced. In other words, we can no change the value of the equation.


How might we do this?

Hint: How could we use zero pairs?

Remember our goal, to be able to remove 8 white tiles. We already have 5 white tiles, so we need another 3 while tiles. So, we want to add three zero pairs into our problem. Remember a zero pair adds up to zero and contains one positive and one negative tile. Now our tiles should look like this:

5-8 zero pairs.jpg

Now we have enough positive white tiles to take away. Go ahead and subtract or “take away” eight tiles. Our answer is the remaining tiles.

5-8 zero pairs -8.jpg

Answer; 5 - 8 = -3

Now, let’s try another example using our tiles.

Evaluate: - 4 - 3

Set up your initial tiles for -4.

-4 tiles.jpg

Again, we need to take away three positive white tiles. So we must add in some zero pairs because we don’t not have any white tiles yet.

- 4 - 3.jpg

Now, we can take away our three positive white tile.

What is the final answer? 7 red tiles or -7.


Let’s think about why this makes sense. If you owe you dad 4 dollars, then you are in debt, or you have $ -4. Then you realize you also owe your sister another three dollars.Now you are even further in debt! So our answer does make sense!

Therefore, -4 - 3 = -7

One final example using our integer tiles.

Evaluate: 6 - (-4)

Set up your tiles 6 positive tiles.

6 white tiles.jpg

Add in 4 zero pairs.

6 - -4 tiles.jpg


Why do we have to add in 4 zero pairs? Answer to adding in 4 zero pairs

Now we can take away our negative four tiles. We are left with 10 white tiles. Therefore, our answer is positive 10!

6 - (-4) = 10

There is a rule that we can follow for subtracting integers. Whenever we see a subtraction problem, we can change it to an addition problem by “adding the opposite.”


Add the opposite.jpg


Try these on your own! You may use your integer tiles to help you!

  1. -10 - (-8)
  2. 9 - (-9)
  3. 15 - (- 5)
  4. 5 - 9
  5. -7 - 7

Answers for # 1-5

  • Try to use mental math for the following problem...
  1. -4 - (-4)
  2. -9 - 10
  3. 8 - 10

Answers to mental math subtraction

For even more practice, try these websites:

  • [1]
  • [2] <-- this websites uses models to help you subtract!