Foundations of Algebra: Variables and Expressions

TL;DR

Algebra uses letters, called variables, to represent unknown numbers. Expressions combine these variables and numbers with operation symbols like addition or multiplication. Understanding variables helps you translate everyday situations into mathematical statements, making problems easier to solve.

1. The Mental Model

Think of a variable as a placeholder, like an empty box, waiting for you to put a number inside it. An algebraic expression is simply a mathematical phrase built with these boxes, numbers, and operations.

2. The Core Material

When we talk about algebra, two of the very first things you'll encounter are variables and expressions. They're the building blocks!

What's a Variable?

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A variable is usually a letter (like x, y, a, or b) that stands for a number we don't know yet, or a number that can change. It's not a mystery letter; it's just a symbol representing a numerical value.

For example, if you say "I have some apples," the number of apples is unknown. In algebra, we could say "I have a apples," where a is our variable.

What's an Algebraic Expression?

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An algebraic expression is a combination of variables, numbers, and at least one operation (like addition, subtraction, multiplication, or division). It's like a mathematical phrase, but it doesn't have an equals sign.

Here are some examples of expressions:
* x + 5 (a variable x added to the number 5)
* 3y (the number 3 multiplied by the variable y - remember, 3y means 3 * y)
* a - b (the variable b subtracted from the variable a)
* z / 2 (the variable z divided by the number 2)
* 2x + 7 (the number 2 multiplied by x, then 7 is added)

Coefficients and Constants

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Inside an expression, you'll often see:

• Coefficient: The number multiplied by a variable. In 3y, 3 is the coefficient. In x + 5, the coefficient of x is 1 (because x is the same as 1x, we just don't usually write the 1).
• Constant: A number on its own in an expression. It doesn't change its value. In 2x + 7, 7 is the constant.

Here's how these pieces fit together:

graph TD
    A["Algebraic Expression"] --> B["Variables"]
    A --> C["Numbers"]
    A --> D["Operations (+, -, *, /)"]
    B --> E["Represent unknown values"]
    C --> F["Constants (fixed values)"]
    C --> G["Coefficients (multiplied by variables)"]
    D --> H["Combine variables and numbers"]

Translating Words to Expressions

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A big part of algebra is taking a word problem and turning it into an algebraic expression. This is where your understanding of variables and operations comes in handy.

Evaluating Expressions

When you know the value of the variable, you can evaluate an expression. This means you substitute the number in for the variable and then do the math.

Example: Evaluate x + 7 if x = 3.
1. Replace x with 3: 3 + 7
2. Do the math: 10

So, when x = 3, the expression x + 7 evaluates to 10.

3. Worked Example

Let's try translating a word problem into an expression and then evaluating it.

Problem: "Sarah has n stickers. Her friend gives her 8 more stickers. Then, Sarah gives half of her total stickers to her sister."

1. Start with the initial number: Sarah has n stickers.
2. Friend gives her 8 more: This means we add 8 to her current amount. So, she has n + 8 stickers.
3. She gives half of her total stickers to her sister: The word "total" here means we need to consider all the stickers she has after getting the 8 more. Then, she divides that total by 2.
   So, the expression is (n + 8) / 2. We use parentheses to show that n + 8 is a single quantity that gets divided.

Now, let's evaluate this expression if Sarah initially had 12 stickers (so, n = 12).

1. Substitute n = 12 into the expression: (12 + 8) / 2
2. First, do the operation inside the parentheses: 12 + 8 = 20
3. Now, divide: 20 / 2 = 10

So, if Sarah started with 12 stickers, she would give 10 stickers to her sister.

4. Key Takeaways

• A variable is a letter representing an unknown or changing number.
• An algebraic expression combines variables, numbers, and operations, but has no equals sign.
• Coefficients are numbers multiplied by variables (e.g., 3 in 3x).
• Constants are numbers alone in an expression (e.g., 5 in x + 5).
• Translating words into expressions involves identifying variables and the correct mathematical operations.
• Evaluating an expression means substituting known variable values and calculating the result.

Common Mistakes to Avoid:

• Confusing "less than" or "subtracted from": "5 less than x" is x - 5, not 5 - x.
• Forgetting multiplication: 2x means 2 * x, not "2 and x".
• Ignoring order of operations: When evaluating, always follow PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
• Mixing up expressions with equations: An expression (x + 5) doesn't have an equals sign; an equation (x + 5 = 10) does.

5. Now Try It

For 15 minutes, try the following:

1. Write two different algebraic expressions using at least one variable, one coefficient, and one constant in each.
   • Example: 3a - 7
2. Translate these word phrases into algebraic expressions:
   • "The product of a number and six, decreased by ten."
   • "Twelve more than a number, divided by three."
3. Evaluate both of the expressions you just created (from step 2) using n = 9.

What success looks like: You'll have two expressions you created, and for the two word problems, you'll have correct algebraic expressions and the correct numerical answers after evaluating them.