Foundations of Algebra and Expressions

TL;DR

Algebra is like a language for solving puzzles, using symbols to represent unknown numbers. We'll learn to translate word problems into mathematical expressions and simplify them. Understanding these basics is key to tackling more complex math problems later on.

1. The Mental Model

Think of algebra as detective work: you're given clues (known numbers and relationships), and you need to find the missing piece (the unknown number). We use letters as placeholders for these unknowns, and then we use mathematical rules to figure out their values.

2. The Core Material

What's an Expression?

Photo by Alexander Krivitskiy on Pexels

An expression in algebra is a combination of numbers, variables (letters that represent unknown values), and operation symbols (like +, -, ×, ÷). It's a mathematical phrase, but it doesn't have an equals sign.

Examples:
* 5x + 3
* y - 7
* 2(a + b)

Variables, Constants, and Coefficients

Photo by Markus Spiske on Pexels

• A variable is a letter that stands for a number. Its value can change. Common variables are x, y, a, b.
• A constant is a number whose value never changes. In 5x + 3, 3 is a constant.
• A coefficient is the number multiplied by a variable. In 5x + 3, 5 is the coefficient of x.

Operations and Order of Operations (PEMDAS/BODMAS)

Photo by EqualStock IN on Pexels

We perform operations in a specific order to get the correct answer. Remember PEMDAS:
1. Parentheses (or Brackets)
2. Exponents (or Orders)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Translating Words to Expressions

Photo by Pixabay on Pexels

One of the first big steps in algebra is taking a word problem and turning it into a mathematical expression. This helps us set up problems to solve them.

graph TD
    Start["Word Problem"] --> IdentifyUnknown["Identify Unknowns (use variables)"]
    IdentifyUnknown --> IdentifyOperations["Identify Operations (keywords like 'sum', 'difference', 'product', 'quotient')"]
    IdentifyOperations --> WriteExpression["Write the Algebraic Expression"]
    WriteExpression --> End["Check for Clarity"]

Keywords to look for:
* Addition: sum, total, plus, increased by, more than
* Subtraction: difference, minus, decreased by, less than, subtracted from
* Multiplication: product, times, multiplied by, of, twice, triple
* Division: quotient, divided by, per, ratio of

Example translations:
* "5 more than a number" $\rightarrow$ x + 5
* "The product of 3 and a number" $\rightarrow$ 3x (or 3 * x)
* "A number decreased by 7" $\rightarrow$ n - 7
* "The quotient of a number and 2" $\rightarrow$ y / 2 (or y ÷ 2)

Evaluating Expressions

To evaluate an expression, you substitute a given numerical value for each variable and then perform the operations using PEMDAS.

Example: Evaluate 2x + 5 when x = 3.
1. Substitute 3 for x: 2(3) + 5
2. Multiply: 6 + 5
3. Add: 11

3. Worked Example

Let's evaluate the expression (y + 4) * (x - 2) when x = 5 and y = 1.

1. Substitute the values: Replace y with 1 and x with 5.
   (1 + 4) * (5 - 2)

2. Perform operations inside parentheses first (PEMDAS):

   • For the first parenthesis: 1 + 4 = 5
   • For the second parenthesis: 5 - 2 = 3
     Now the expression looks like: 5 * 3

3. Perform the multiplication:
   5 * 3 = 15

So, the evaluated expression is 15.

4. Key Takeaways

• Algebra uses variables (letters) to represent unknown numbers in mathematical expressions.
• An algebraic expression combines numbers, variables, and operation symbols without an equals sign.
• Always follow the order of operations (PEMDAS/BODMAS) when evaluating expressions to get the correct result.
• Translating word problems into expressions requires identifying keywords for mathematical operations.
• To evaluate an expression, substitute given numbers for variables and then simplify using PEMDAS.

Common Mistakes to Avoid:
* Not following the order of operations, especially multiplication/division before addition/subtraction.
* Confusing "less than" with "minus"; "5 less than x" is x - 5, not 5 - x.
* Forgetting that a number next to a variable (like 3x) means multiplication.
* Mixing up variables and constants; variables change, constants stay the same.

5. Now Try It

You're a chef, and you're trying to figure out how much you spend on ingredients. Apples cost $2 per pound, and bananas cost $1.50 per pound.

Your Task:
1. Write an algebraic expression for the total cost if you buy A pounds of apples and B pounds of bananas.
2. Evaluate your expression if you buy 3 pounds of apples and 4 pounds of bananas.

What success looks like: You'll have an expression like (something * A) + (something_else * B) and a single numerical answer for the total cost after substitution.