### Unit 1: Variables and Variable Expressions

When most people think of math, they think of numbers. However, with this math course, you will be working with letters. In algebra, letters are used to represent numbers. These letters are called variables, because the numbers they represent may vary. For example, if you are paid $10 per hour, your salary can be expressed as 10×*h*, where *h* is the number of hours you have worked. You can change the number that letter *h* stands for according to your particular case in order to calculate your salary.

In this unit, you will learn that the properties of numbers apply to letters as well, and you can use them to work with expressions containing variables. One of the main skills you have to master in this unit is recognizing like terms, because you can add and subtract them as if they were numbers.

**Completing this unit should take you approximately 8 hours.**

Upon successful completion of this unit, you will be able to:

- identify parts of algebraic expressions, such as terms, factors, and coefficients;
- evaluate algebraic expressions for the given values of the variables; and
- apply commutative, associative, and distributive properties of real numbers to simplify algebraic expressions.

### 1.1: Definition and Meaning of Variable Expressions

### 1.1.1: Variables and Variable Expressions

Watch the video and take notes. In this video, Sal Khan uses an everyday situation to explain how variables are used to represent numbers.

Read the article and complete practice problems 1-10. This article further explains the meaning and usage of variables. It also provides many examples of using variables to represent quantities that you often encounter in the real world as well as exercises to practice choosing appropriate variables and writing variable expressions to describe situations on your own.

### 1.1.2: Constant Terms and Coefficients

Read this page. This article points out various conventions used in algebra when writing variable expressions, such as always putting a numerical coefficient first and omitting 1 in front of a variable. Use the examples provided as a guide to practice identifying different parts of algebraic expressions. Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

Read the brief encyclopedia entry and complete the "Understanding Check" problems. You will receive immediate feedback on whether your response is correct or incorrect when you choose an answer choice. You will be using the terminology introduced here throughout the rest of this course.

### 1.2: Evaluating Variable Expressions

### 1.2.1: Replacing Variables with Their Values

Watch the video and take notes. This video gives a very detailed explanation of how to substitute the values of variables in the algebraic expressions. Note how the concept of order of operations is used in some examples. You should be familiar with this concept from your study of arithmetic.

Read this article, which explores various real-world situations where evaluating expressions with one or more variables is required. Then, complete practice problems 1-9 and 26-30. If you need help with problems 1-9, watch the first 2 minutes of the "Variable Expressions" video embedded in the article.

Complete this exercise set. It contains easy problems on evaluating simple algebraic expressions containing only one variable. You will encounter more complex problems later in this subunit. Substitute the given value of x into the given algebraic expression and perform the resultant set of numerical operations.

Complete this exercise set. The algebraic expressions here contain more than one variable and you might also see exponents. You should remember how to handle exponents from your study of arithmetic. Substitute the given values of the variables into the given algebraic expression and perform the resultant set of numerical operations.

### 1.2.2: Using Order of Operations Agreement to Simplify the Resulting Numerical Expression

### 1.2.2.1: Order of Operations Review

Read this page. Order of operations agreement is a convention used by mathematicians to ensure that expressions with many operations are always evaluated the same way, which is consistent with the properties of operations. Because we are used to reading from left to right, it is natural to add and multiply in the same direction, instead of thinking which operations should be performed first. With practice, you will get used to the correct order and will read the expressions accordingly.

Work through examples A and B and Guided Practice problem 1. Complete practice problems 1-7. Make sure to watch the "Order of Operations" video embedded in the text.

Complete this exercise set. It contains basic order of operation problems and will help you assess how well you remember this concept from your study of arithmetic.

### 1.2.2.2: Evaluating Expressions

Scroll down to Example C. In this activity, the examples and exercises require an extra step: substituting the values of the variables in the algebraic expressions. Then, they simply become numerical expressions, which can be evaluated by using order of operations agreement.

Work through Example C and Guided Practice problem 2, and then complete practice problems 8-10. Use the "Order of Operations Example" video embedded in the text for guidance. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to the Practice Problems section and complete practice problems 10-25. Watch the "Variable Expressions" video embedded in the text if you need help. Once you have completed the practice problems, check your answers against the answer key.

### 1.3: Simplifying Variable Expressions

### 1.3.1: Properties of Real Numbers

### 1.3.1.1: Commutative Property of Addition and Multiplication

Review the set of concepts outlined in the "Arithmetic Properties" series of videos, including the commutative and associative laws of addition and multiplication and the identity properties of addition and multiplication. You will study the algebraic applications of these properties in the next assignment.

Read this section through "Sample Set A," and work through the exercises in Practice Set A. The solutions to the practice problems are shown directly below each problem.

You know that changing the order in which two numbers are added does not change the result, and, likewise, changing the order in which two numbers are multiplied does not change the result. Because variables represent numbers, this is true for variables as well. With the help of this reading, you will learn to apply this property to any expression.

### 1.3.1.2: Associative Property of Addition and Multiplication

Scroll down to the section titled "The Associative Properties." Read this section through "Sample Set B," and work through the exercises in Practice Set B and Practice Set C. The solutions to the practice problems are shown directly below each problem.

The associative property states that several numbers can be added or multiplied in any order. Later in the course, you will be using this property often when you will have to switch terms around in order to simplify algebraic expressions.

### 1.3.1.3: Distributive Property of Multiplication over Addition/Subtraction

Scroll down to the section titled "The Distributive Properties." Read this section through "Sample Set D," and work through the exercises in Practice Set D. The solutions to the practice problems are shown directly below each problem.

The distributive property is another property that will be used extensively in simplifying algebraic expressions.

### 1.3.2: Combining Like Terms

### 1.3.2.1: Definition and Examples of Like Terms

Read this encyclopedia entry. Note that the like terms have the same variables in the same exponents but might have different numerical coefficients. You will need to recognize like terms in order to add and subtract them.

### 1.3.2.2: Simplifying Expressions by Combining Like Terms

Watch this video and take notes. In general algebraic expressions, you will need to open the parenthesis and then combine the like terms. Due to commutative and associative properties, you can move the like terms around in order to combine them.

Watch this video and take notes. Dr. Sousa explains a few slightly more complicated examples of simplifying algebraic expressions. Note that the last example in the video contains parenthesis within brackets. According to the order of operations, you need to first remove the inner grouping symbol of parenthesis and then simplify the expression within the brackets.

Attempt several exercises in this section until you feel comfortable with the material. Click on the "Show Solution" link next to each problem to see the correct solution.

Complete this exercise set. It will provide practice using the number properties, particularly commutative and distributive, to simplify simple algebraic expressions.