Subtracting integers

Using a model: balloons and sandbags

We will be modeling subtracting with positive and negative numbers by talking about balloons and sand bags that are tied to a basket.

Imagine a basket tied to several little balloons floating in the clouds. How many balloons does it require to get a basket to float up into the sky? We don't know, but surely the number is rather large.

Also imagine, that the person in the basket doesn't want to float too high, since at very high altitudes there is no air to breathe. So, surely the basket must also have several sand bags tied to it, to prevent the basket from going too high.



Example 1

(+3) – (+8) means three balloons are tied to the basket which raises the basket three feet above the clouds. Then eight balloons are removed from the basket, which lowers the basket eight feet. In all, the basket is now five feet below the clouds. Therefore, the answer to (+3) – (+8) is -5.

Example 2

What does the subtraction problem (-5) – (+7) mean?

Example 3

(-4) – (-9) =

Example 4

(+5) – (-4) =

Finding the rules of subtraction

Remember that we can use balloons and sand bags to help explain subtraction. Previously, we learned that when "subtracting +3" we imagine removing three balloons, which would lower the basket. We also learned that when "subtracting –5" we imagine removing five sand bags, which would raise the basket.

For each of the following examples we will draw the changes on number lines. Beneath the problem we will also record what changes actually occurred to the basket.

Example 1
(-7) – (+3) =
Example 2
(-7) – (-3) =
The problem (-7)–(+3)= -10 The problem (-7)–(-3)= -4
What actually happened (-7)+(-3)= -10 What actually happened (-7)+(+3)= -4
Example 3
(+7) – (-3) =
Example 4
(+7) – (+3) =
The problem (+7)–(-3)= +10 The problem (+7)–(+3)= +4
What actually happened (+7)+(+3)= +10 What actually happened (+7)+(-3)= +4

 By looking at the four examples above, we can see a pattern that occurs in each of the examples:

  1. The first integer stays the same;
  2. The subtraction sign turns into an addition sign;
  3. The second integer turns into its opposite.


In fact, the rule for subtracting positives and negatives is:

To subtract two integers: The first integers stays the same and add the opposite of the second integers.



  1. Use the blue and purple sliders to create a problem.
  2. Notice how the arrows change as you move the sliders.
  3. Use the "hint" slider to reveal various types of hints.


Duane Habecker, Created with GeoGebra




Question 1




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Question 2




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Question 3




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Question 4




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