Integers are the following numbers = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• The numbers that are less than zero are called negative.
• and the numbers that are more than zero are called positive.

The number line introduces negative numbers:

FACT: Each integer has its opposite. The opposite of 3 is -3, the opposite of -5 is 5, etc... and zero its its own opposite.

If I take the opposite of 3, it becomes -3. If I continue and take the opposite of -3, we get back to where we started, the original number. Therefore....

FACT: The opposite of the opposite is the original number!  - (-3) = 3

RULE:  - (-a) = a  for any integer a

### So, How Do You Add Integers?

FACT: With positive numbers, we already are used to counting up on the number line whenever we add.

Positive + Positive
•  7 + 2 = 9
• start at seven and "count up" 2

Negative + Positive
• -6 + 5 = -1
• start at -6 (on the number line) and "count up" 5

Positive + Negative

• 3 + (-8) = -5
• start on 3 and "count up" a  -8 . ("count up" -8 is the opposite of "count up" 8, meaning: "count down" 8)
• better stated: Start on 3 and "count down" 8

Negative + Negative
• -5 + -2 = -7
• start on -5 and "count up" a  -2 . ("count up" -2 is the opposite of "count up" 2, meaning: "count down" 2)
• better stated: Start on -5 and "count down" 2
• RULE:  a + (-b) = a - b  for any two integers a, b

### The +- Model is the Simplest Model for Addition of Integers

For this model, you need to know what a zero pair is. A zero pair is formed when a number and its opposite are added together. This always leads to an answer of zero. Three plus its opposite forms a zero pair.  3 + (-3) = 0

To make a zero pair, take the number and its opposite.  -5 + __ = 0   ->   -5 + 5 = 0

#### How the + - model works:

1. In the +- model, we use a "positive" sign to symbolize each +1 value. So, for the number 5 you would use  + + + + +
2. In the +- model, we use a "negative" sign to symbolize each -1 value. So, for the number -4 you would use  - - - -
3. Once both values are modeled with the appropriate sign, we start taking away all the zero pairs present (+ -) , as these cancel each other out (zero pairs = 0).
4. What is left is after you take away all zero pairs is the answer to the adding integers problem.

FACT: This is the simplest model to use as students can just use a pencil to model the + and the - . No manipulatives are needed.

### Modeling Adding Integer Problems with the +- Model

Remember that addition means combining two sets. When we combine two sets we sometimes find groups of opposite signs.
Find all zero pairs and get rid of them (as they cancel each other out).

+ 2 = 9
+ + + + + + +   + +

5 + (-4) = 1
+ + + + +
- - - -

-6 + 5 = -1
- - - - - -
+ + + + +

3 + (-8) = -5
+ + +
- - - - - - - -

-5 + (-2) = -7
- - - - -   - -