# Integers

## Add Integers


### Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.
Most students are comfortable with the addition and subtraction facts for positive numbers.  But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.
The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of  Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.
Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.
We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.
If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in .
We will model four addition facts using the numbers 
 and  are very similar. The first example adds  positives and  positives—both positives. The second example adds  negatives and  negatives—both negatives. In each case, we got a result of  positives or  negatives. When the signs are the same, the counters are all the same color.
Now let’s see what happens when the signs are different.

### Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers. 
For example, if you want to add  you don’t have to count out  blue counters and  red counters. 
Picture  blue counters with  red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because  there are  more negative counters.
Let’s try another one. We’ll add  Imagine  red counters and  more red counters, so we have  red counters all together. This means the sum is 
Look again at the results of  - .
The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

### Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.
Next we'll evaluate an expression with two variables.

### Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

### Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of  could be represented as  Let’s practice translating and solving a few applications.
Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

### Key Concepts

1. Addition of Positive and Negative Integers


### Practice Makes Perfect

Model Addition of Integers 
In the following exercises, model the expression to simplify.
Simplify Expressions with Integers 
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers 
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions 
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Add Integers in Applications 
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
