Basic Algebra Concepts
Master variables, expressions, and algebraic thinking for the ParaPro Assessment
Learning Objectives
Introduction to Variables
The foundation of algebra
What Are Variables?
Variables are symbols (usually letters) that represent unknown values or quantities that can change. They are the foundation of algebra and allow us to write general rules and solve problems.
Common Variables
- • x, y, z (most common)
- • a, b, c (often for constants)
- • n (often for counting numbers)
- • t (often for time)
Why Use Variables?
- • Represent unknown values
- • Create general formulas
- • Solve real-world problems
- • Show patterns and relationships
Variables in Action
Example 1: Age Problem
Sarah is 5 years older than her brother Tom. If Tom is x years old, how old is Sarah?
Tom's age = x
Sarah's age = x + 5
If Tom is 10, then Sarah is 10 + 5 = 15 years old
Example 2: Cost Problem
Pencils cost $0.50 each. How much do n pencils cost?
Cost = 0.50n dollars
If you buy 6 pencils: 0.50(6) = $3.00
Algebraic Expressions
Understanding the parts
Parts of an Expression
An algebraic expression is a mathematical phrase with numbers, variables, and operations.
3x + 2y - 5
Terms
3x, 2y, -5
Parts separated by + or -
Coefficients
3, 2
Numbers × variables
Constant
-5
Number without variable
Key Vocabulary
- Expression: Mathematical phrase without an equals sign
- Term: Part of an expression separated by + or -
- Coefficient: Number multiplied by a variable
- Constant: Number that doesn't change
Writing Expressions from Words
"Five more than a number"
x + 5
"Three times a number"
3x
"A number decreased by 7"
x - 7
"Half of a number"
x/2 or ½x
Evaluating Expressions
Substitution method
To evaluate an expression, substitute the given values for the variables and simplify.
Example 1: Single Variable
Evaluate 3x + 7 when x = 4
- Substitute 4 for x: 3(4) + 7
- Multiply: 12 + 7
- Add: 19
Example 2: Multiple Variables
Evaluate 2a - 3b + 5 when a = 6 and b = 2
- Substitute: 2(6) - 3(2) + 5
- Multiply: 12 - 6 + 5
- Simplify: 11
Simplifying Expressions
Combining like terms
Like Terms
Like terms have the same variable raised to the same power. We can add or subtract like terms.
Like Terms ✓
- 3x and 5x (both have x)
- 2y and -7y (both have y)
- 4 and 9 (both constants)
Not Like Terms ✗
- 3x and 3y (different variables)
- 2x and 2x² (different powers)
- 5 and 5x (one has variable)
The Distributive Property
a(b + c) = ab + ac
Multiply the outside term by each term inside the parentheses
Translation Keywords
Words to algebra
Addition (+)
- • sum, plus
- • more than
- • increased by
- • total
Subtraction (−)
- • difference
- • less than
- • decreased by
- • fewer
Multiplication (×)
- • product, times
- • of
- • twice, triple
- • double
Division (÷)
- • quotient
- • divided by
- • per, ratio
- • half, third
Common Mistakes to Avoid
Watch out for these
x + x = x² (Wrong!)
✓ x + x = 2x (addition)
✓ x × x = x² (multiplication)
"5 less than x" = 5 - x (Wrong!)
✓ "5 less than x" = x - 5
3(x + 4) = 3x + 4 (Wrong!)
✓ 3(x + 4) = 3x + 12 (distribute to all terms)
3x + 2y = 5xy (Wrong!)
✓ 3x + 2y stays as 3x + 2y (unlike terms)
Practice Problems
Test your skills
Writing Expressions
- Write: "Seven more than twice a number"
- Write: "The product of 5 and a number, decreased by 3"
- Write: "One-third of a number plus 8"
Evaluating Expressions
- Evaluate 4x - 3 when x = 5
- Evaluate 2a + 3b when a = 4 and b = -2
- Evaluate x² + 2x - 1 when x = 3
Simplifying Expressions
- Simplify: 5x + 3x - 2x
- Simplify: 2y + 4 + 3y - 1
- Simplify: 3(x + 2) + 2x
Show Answers
Writing Expressions:
1. 2x + 7 | 2. 5x - 3 | 3. x/3 + 8
Evaluating:
1. 17 | 2. 2 | 3. 14
Simplifying:
1. 6x | 2. 5y + 3 | 3. 5x + 6
Key Takeaways
Variables: Letters representing unknown values, making math flexible and general
Expressions: Mathematical phrases with numbers, variables, and operations
Like Terms: Terms with the same variable and power that can be combined
Distributive Property: a(b + c) = ab + ac - multiply all terms inside
Continue Learning
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Continue to solving equations or test your skills
