Adding and Subtracting Integers - The Ice Cube Method

The first thing I want to say is that what I am about to explain is not scientifically valid (which I tell my students when I work with then) it is an excellent way to help students understand the concepts behind adding and subtracting integers.

So here is how it works...

in a simple addition or subtraction equation the first number will be the initial temperature of a container of water.  The symbol indicates whether something is to be added (+) or removed (-).  The second number indicates what is being added or removed.  

If the second number is positive then heat is either being added or removed, if it’s negative then ice is being added or removed.  When you add one scoop of warm water it increases the temperature by 1 degree and when you add an ice cube you decrease the temperature by one degree.  The opposite is true for subtracting.

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Adding Integers using the Ice Cube Method

Example 1: 3 + 4

The water starts at 3 degrees. 
4 scoops of warm water are added.  
The temperature increases by 4 degrees. 
The water finishes at 7 degrees. 

 I use this one just as an example of what is happening with the water.  


Example 2: (-6) + 11

The water starts at -6 degrees.  
11 scoops of warm water is added. 
The temperature increases 11 degrees. 
The water finishes at 5 degrees. 

Some students find it easier if they have a thermometer in front of them so they can count up.  (if you don’t have a thermometer you can use a printed version).  


Example 3: 4 + (-6)

The water starts at 4 degrees. 
Six ice cubes are added.  
The temperature drop 6 degrees
The water finishes at -2 degrees.  


Example 4: (-2) + (-4) 

The water starts at -2 degrees.  
Four ice cubes are added. 
The temperature go down 4 degrees
The water finishes at -6 degrees. 

This is by no means scientifically valid but the kids seem to understand the concept of adding negative temperature by adding ice cubes.  

Subtracting Integers using the Ice Cube Method 

I typically start subtraction by asking them what would happen to the temperature if I have a container of ice water and I start taking out ice cubes.  I have always gotten the answer that the temperature would rise.  So I explain that they are taking away negative temperature.  Here is what it looks like.  


Example 1: 4 - 6 

The water starts at 4 degrees. 
Remove 6 scoops of warm water. 
The temperature goes down 6 degrees. 
The water finishes at -2 degrees.

Some students may ask how you remove scoops of warm water from a container and the answer is you can’t, that we are doing it in theory not practice.


Example 2:  (-3) - 2 

The water starts at -3 degrees.  
Remove 2 scoops of warm water. 
The temperature of the water goes down 2 degrees.  
The water finishes at -5 degrees.  


Example 3: 5 - (-6) 

The water starts at 5 degrees.
Remove 6 six ice cubes. 
The temperature goes up 6 degrees. 
The water finishes at 11 degrees.  


Example 4: (-3) - (-1) 

The container starts at -3.  
Remove 1 ice cube. 
The temperature increase by 1 degree. 
The water finishes at -2 degrees.  



The ice cube method has worked wonders in my classroom with students who are struggling to grasp the concept of adding and subtracting integers.  With the visual aid of a thermometer, printed or physical, there are very few students who were unable to grasp the concepts.  


I have found that after students learn the laws for multiplying and dividing integers.  All to often they want to apply the rules to adding and subtracting integers which by no means works.  After students learn the easy rules for multiplication and division they want to make rules for adding and subtracting and are rarely successful.  I have not yet figured out a way to discourage this confusion.  If you have any ideas, I would love to hear them.