Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 28.1 When All the Chips are Down

At the end of Section 26.4, we talked about how certain subtraction problems do not work well with the basic idea of counting objects because you can be asked to subtract off a number that's larger than the number of objects that you have.

It turns out that we can still represent this idea, but we have to use a different type of manipulative. These are commonly called integer chips. These are plastic circles where the two sides are different colors (usually yellow and red). We are going to enhance these by also using a plus and minus sign in our diagrams. These objects allow for multiple alternative interpretations, such as electric charges or savings/debt. Here are some collections of chips and the numbers that they represent.

Notice that if we only focus on the positive chips, they behave exactly like we might expect.

If we were to add together negative chips, it turns out that the arithmetic would work out correctly as well.

But what happens when we have a mixture of the two types of chips? Since one step left followed by one step right would leave you right back where you started, we can see that a positive chip and a negative chip will cancel each other out. And by eliminating those pairs, we can see what the final answer would be.

Activity 28.1.1. Calculating Using Integer Chips.

Try it!

Calculate \(4 + (-5)\) using an integer chip diagram.

Solution.

As always, we want to avoid treating this as a system of "rules." We want to be able to understand these manipulatives as objects that represent familiar ideas. Specifically, we can relate these chips to movement on the number line. Thinking about the positive chips as movement to the right and negative chips as movement to the left, we can translate the calculation into a number line picture. This is a subtle shift from what we were looking at before. Initially, we were just starting from a position and making one movement. Now we're starting from zero and making two movements. But the primary concepts still remain the same.

Activity 28.1.2. Comparing Integer Chips and the Number Line.

Notice that this framework allows us to add any two numbers together, which is slightly more than what we did on the number line in the previous section. For that, addition always meant movement to the right. But this picture shows us that adding a negative number means moving to the left.

Try it!

Calculate \(3 + (-5)\) using an integer chip diagram. Then draw a number line picture to represent the calculation as movement on the number line.

Solution.

Now that we know how to represent addition of any two numbers with these chips, we will turn our attention to subtraction. Let's look at \(5 - 3\text{.}\) Instead of thinking of subtraction as "taking away" chips, we can get an equivalent form by flipping over the subtracted chips and using zero pairs.

It's easy for us to verify that this gives us the same final answer as any other method we use. But what is happening here? What we have done is replace the concept of "taking away" with a form of "undoing" the steps. How do we "undo" three steps to the right? By taking three steps to the left! The chips are merely a way of representing that concept.

Activity 28.1.3. Subtracting Negatives with Integer Chips.

There were calculations that we were unable to do with our previous concepts, but we can now do with the integer chips. Here is the calculation of \(-2 - (-3)\text{.}\)

Try it!

Calculate \(1 - (-3)\) using an integer chip diagram. Then draw a number line picture to represent the calculation as movement on the number line.

Solution.