1 Activities for this section:

Where have the integers been?

2 Integers

Up to this point, we have been working with only positive numbers. Negative numbers don’t show up in Ohio’s standards until Grade 6, meaning that younger children work only with positive numbers. We then extend the number system to include both positive and negative numbers. Here are a few definitions that might help you organize how you talk about numbers.

It might seem silly to have a different name for the whole numbers than the counting numbers, but zero can be a difficult concept for children to grasp. Why do we use something to represent nothing? If you are interested in the history of zero, here is a video that goes through the history of zero in mathematics.

But if zero is a difficult concept for children, negative numbers can be even more challenging. While some children can easily conceptualize negative numbers as “owing” someone something, other children can be confused as to why we need to introduce more numbers for this concept. In fact, negative numbers were given their full status as “numbers” later than many other kinds of numbers in Western European mathematics. We can find evidence of whole numbers, fractions, and decimals starting as early as $3000$ BCE. But Western European mathematicians didn’t start really using negatives in their calculations until the $17$th century AD, and didn’t allow them as answers to problems or give them equal status amongst the positive numbers, decimals, and fractions until the $19$th century! (It’s worth noting that negative numbers appear earlier in some other cultures, like China and India, so the ideas were out there even if the Western European mathematicians didn’t embrace them.) If the history of numbers is interesting to you, you can read more in a textbook like A Brief History of Numbers by Leo Corry.

Much of mathematicians’ hesitation to treat negative numbers as actual numbers came from the idea that mathematics was a subject that should make sense in the real world and model real-world phenomena. People didn’t have a very good representation for negative numbers that made sense all the time – and children can often feel the same way!

As we work with modeling negative numbers, pay close attention to times when the models don’t really make sense. There will be times when our models feel like they don’t make sense, or they feel a bit contrived. Remember that mathematicians throughout history have struggled with these ideas, so try to give yourself (and your future students) some grace! We will work with several models to give you several choices so that hopefully something connects for you and for the kids you’ll teach.

But why work with negative numbers at all? If we work only with whole numbers, we can run into two major problems.

• We can add any two whole numbers and get another whole number, but the same isn’t true for subtraction.
• We can multiply any two whole numbers and get another whole number, but the same isn’t true for division.

We fix the second issue by introducing the set of fractions – we can now divide any whole number by any other whole number which is not zero and get a meaningful answer. In fact, the same is true for the set of integers and fractions together. To fix the first issue, we will need to do something similar: expand the set of numbers we’re working with to include negative numbers.

In situations with an artificial zero, or a zero arbitrarily set, it makes sense to be talking about amounts that are less than zero. Here are some examples.

• Temperature. In the Fahrenheit and Celsius scales, zero is set at a chosen level, but it is possible for temperatures to be colder than that set zero. Notice that the Kelvin scale is different: zero is set at the point there is completely no heat, so a temperature less than zero Kelvin is not possible.
• Finances. In finances, a person is said to have zero financial worth when they don’t have any money. Also, it makes sense for a person to have a negative financial worth in the sense of owing someone money. So, having a worth of $-5$ dollars means you need to somehow gain $5 before you can say you are “even”, with no debt and no profit. Notice that this can get complicated! If oranges are the currency, one could say one has $-5$ oranges instead of $-5$ dollars even though it is impossible to physically have $-5$ oranges or $-5$ dollar bills. The idea of owing someone oranges or dollars makes this idea work.
• Sports. In football, we can take zero to be the line of scrimmage (or starting line for the play), or where the play begins, and we report the result of the play based on this zero. For instance, if a player who is running gets tackled behind the starting line, we say that the play has resulted in negative yardage.
• More sports. In golf, “par” is defined as the number of shots experts think a golfer should need in order to be able to get the golf ball into the hole. We can take par as our zero in this situation. However, if the golfer takes fewer shots to get into the hole, we say the golfer is so many shots “under par”, and if the golfer takes more shots to get into the hole, we say the golfer is so many shots “over par”.

Keep in mind, however, that there are physical situations in which it doesn’t make sense to extend to negative numbers, just as we had situations where it didn’t make sense to extend to fractions. For example, it doesn’t make sense to say “I’m taking $-3$ classes this semester” or “I have $-4$ pictures on my wall”. The models below are designed so that we always have a way of modeling positive and negative whole numbers as well as addition and subtraction. Let’s get started!

3 Red and black chips

Our first model is using red and black chips. We use one black chip to represent one positive unit and one red chip to represent one negative unit. If I have one black chip together with one red chip, I have the same amount in value as if I have no chips at all. We might say that one red chip cancels one black chip.

One of the most important things to keep in mind while working with chips is that we can easily change the representation of our number without changing the value of our number. Let’s investigate this phenomenon.

4 Checks and Bills

Our next model for integers is a particular story problem scenario called checks and bills.

If we are using chips to represent the checks and bills in our stories, a black chip will mean $$1$, and a red chip will mean $$-1$.

5 Number Lines

Previously, we marked zero and one on our number lines, and used the spacing between zero and one to mark all of the other positive numbers to the right of zero. With integers, we extend the number line to the left of zero as well, using the same spacing between zero and one.

When we model our operations using number lines, we will use the following conventions.

• An addition sign means to walk or move forward. (Addition is an action I take.)
• A subtraction sign means to walk or move backward. (Subtraction is an action I take.)
• A positive number means to face right, or towards the positive numbers.
• A negative number means to face left, or towards the negative numbers.

The result that you see after following these procedures should answer the question, “Where on the number line are we now?” We can also often use our story situation, whether it is checks and bills or something else, to understand how to move on the number line. Some people think of this as asking the question, “Did we get good news or bad news?”

In our course, we will primarily use chips, checks and bills, and number lines to represent operations with negative numbers. While each model has its difficulties, we hope that they give you a chance to explain why the rules for operating with negative numbers aren’t random: they make sense! While you can also try writing other story problems with negative numbers, we caution you to be very careful to always make sure that you are modeling positive and negative numbers as well as addition and subtraction. Many models you find online don’t make a distinction between negatives and subtraction, but we want to emphasize that numbers and operations are different things!