The Positives & Negatives of Working with Integers

Hello mathletes,

This week I had the task of presenting an activity dealing with integers and the pleasure to watch the presentations of 2 classmates on the same topic. During my preparation I learned about some common stumbling blocks experienced by students when they are first introduced to integers and also became more aware of integers in real life contexts that hadn't previously occurred to me. We all go through positive and negative experiences every day, and carry positive and negative perspectives about certain things so isn't life just a series of integers?


 Don't Worry Be Happy by Bobby McFerrin...Is this really a song about integers?


Real Life Examples

• Credits and debits to a bank account
• Negative acceleration experienced by a moving object that is slowing down
• Negative time when you count down to the new year or blast off
• Temperatures above and below zero
• Hockey plus/minus ratings
• Golf scores
• Heights above and below sea level
• Buildings with floors above and below ground level 

Big Ideas

Operation vs. Notation
Before being introduced to integers, students will be most familiar with the + and - symbols in an operational sense being used for the operations of addition and subtraction.  Attaching these symbols to the numbers themselves presents them in a notational sense, as a way to describe the number, which students are likely to be unfamiliar with and may cause some confusion, especially when performing the operations of addition and subtraction using negative numbers.

Value and Direction
When it comes to the value of negative and positive numbers, positive numbers are greater than negative numbers. It will be obvious to students that 5 > 3 BUT when those numbers become negative their values change and -3 > -5.  Integers are also associated with a direction on the number line.  Depending on what style of number line is used (vertical or horizontal), getting more negative will result in moving further left or down and getting more positive will result in moving further right or up. Think of the coordinates on a grid.

Opposites
Every positive number has its opposite in the form of a negative (1 and -1 ; 50 and -50, etc.).  Adding  opposites will always equal zero.  This is called the zero property of integers. In addition opposite operations can have the same effect.  I will try to explain this in detail below.

What I really wanted to focus on in my presentation was to try and conceptualize the idea of adding and subtracting negative numbers. While preparing, I read that a common point of confusion for students when adding and subtracting integers is that they are used to numbers getting bigger or greater in value when they add and smaller or lesser in value when they subtract.  For example, adding 5+5 always increases in value to 10 and subtracting 10-5 always decreases in value to 5. This is not the case when adding and subtracting negatives.  Adding -5 + -5 results in -10 which is lesser in value and subtracting (-5) - (-5) results in 0 which is an increase of value.

With this point of confusion in mind, I first found it easier to not even use the words 'plus' and 'take away'. It was easier for me to use the words 'join' and 'remove'. For example, -5 join -5 made it easier for me to understand that the result was a more negative number. For subtraction,  -5 remove -5 helped me understand that I was starting with -5 and by removing it, the resulting number was going to be more positive. This was a minor change but for myself and students like myself, will help them understand what's happening when we add and subtract negative numbers a little more clearly.

This was a big one for me:

• Adding something positive has the same effect as removing something negative, and
• Adding something negative has the same effect as removing something positive

Did I add milk or remove negative milk?
Still tastes good either way.
Shannon, Adam. (2016, October 28). Milk.

The example I used in class was to think of light and dark.  They are opposites the same way that every positive integer has its negative opposite.

Since darkness is just the absence of light, another way to say this is that darkness is 'negative light'. In class we had 6 rows of lights that were on.  I asked Courtney to turn off 2 and we were left with 4.  This can be represented as 6 - 2 = 4.

BUT, if darkness is just the absence of light, couldn't we say that instead of removing 2 rows of light we added 2 rows of negative light? We can represent this as 6 + (-2) = 4.

If we started with 6 rows of darkness aka negative light (-6) and added 2
rows of light (+2), it would look like (-6) + 2 = -4...we've added 2 rows of light and still have 4 rows of darkness aka negative light.

We could also say that we started with 6 rows of darkness aka negative light (-6) and removed 2 rows of negative light (-2) and that would look like (-6) - (-2) = -4.

Stay thirsty, mathletes.