Variable Expressions - Fifth Grade
Complete Notes & Formulas
Understanding Variables
What is a Variable?
A variable is a letter or symbol that represents an unknown number or a number that can change.
| Term | Definition | Example |
|---|---|---|
| Variable | A letter representing a number | x, y, n, a, b |
| Constant | A fixed number | 5, 12, 100 |
| Expression | Numbers, variables, and operations | x + 7, 3n − 5 |
| Equation | Expression with equal sign | x + 7 = 15 |
| Coefficient | Number multiplying a variable | In 5x, the coefficient is 5 |
Common Variables: x, y, n, a, b, c (but you can use any letter!)
1. Write Variable Expressions
What is a Variable Expression?
A variable expression (or algebraic expression) is a mathematical phrase that contains numbers, variables, and operation symbols (+, −, ×, ÷).
Key Phrases and Their Meanings
| Operation | Key Phrases | Expression |
|---|---|---|
| Addition | plus, sum, more than, increased by, total, added to | x + 7 |
| Subtraction | minus, difference, less than, decreased by, subtract from | n − 5 |
| Multiplication | times, product, multiply, of, twice (×2) | 4n or 4 × n |
| Division | divided by, quotient, split, per | x ÷ 3 or x/3 |
How to Write Variable Expressions
Step 1: Identify the unknown number → choose a variable
Step 2: Look for key words (sum, product, difference, etc.)
Step 3: Translate the phrase into math symbols
Step 4: Write the expression
Examples
| Phrase | Expression |
|---|---|
| A number plus 8 | n + 8 |
| 15 less than a number | x − 15 |
| 5 times a number | 5n |
| A number divided by 4 | n ÷ 4 or n/4 |
| Twice a number plus 3 | 2n + 3 |
Remember: In multiplication, write the number before the variable (5x, not x5)
2. Write Variable Expressions: Word Problems
Real-World Applications
Variable expressions can represent real-life situations like money, age, measurements, and more.
Steps to Solve Word Problems
Step 1: Read the problem carefully
Step 2: Identify what the variable represents
Step 3: Find key words for operations
Step 4: Write the expression
Step 5: Check if it makes sense
Examples
Problem 1: "Sarah is 5 years older than her brother Tom. If Tom is t years old, write an expression for Sarah's age."
Variable: t = Tom's age
Key phrase: "5 years older" → add 5
Expression: t + 5
Problem 2: "A pizza is cut into n slices. If 4 slices are eaten, write an expression for the slices remaining."
Variable: n = total slices
Key word: "remaining" → subtract
Expression: n − 4
Problem 3: "Each book costs $12. Write an expression for the total cost of b books."
Variable: b = number of books
Key idea: Total = price × quantity
Expression: 12b or 12 × b
3. Evaluate Variable Expressions
What Does "Evaluate" Mean?
To evaluate an expression means to find its value by substituting a given number for the variable and then calculating the result.
Steps to Evaluate
Step 1: Write the expression
Step 2: Substitute (replace) the variable with the given number
Step 3: Follow the order of operations (PEMDAS)
Step 4: Calculate to find the answer
Order of Operations (PEMDAS)
P - Parentheses
E - Exponents
M/D - Multiply and Divide (left to right)
A/S - Add and Subtract (left to right)
Examples
Example 1: Evaluate x + 7 when x = 5
Expression: x + 7
Substitute: 5 + 7
Calculate: 12
Answer: 12
Example 2: Evaluate 3n − 8 when n = 10
Expression: 3n − 8
Substitute: 3(10) − 8
Multiply first: 30 − 8
Subtract: 22
Answer: 22
Example 3: Evaluate 2x + 5 when x = 7
2(7) + 5
14 + 5
Answer: 19
Tip: Always use parentheses when substituting numbers to avoid mistakes!
4. Write Variable Equations: Word Problems
Expression vs Equation
Expression: A mathematical phrase (no equal sign)
Example: 2x + 5
Equation: Two expressions equal to each other (has = sign)
Example: 2x + 5 = 13
Key Words for Equations
• "is" or "equals" → use = sign
• "the same as" → use = sign
• "the total is" → use = sign
Examples
Problem 1: "Five more than a number is 12. Write an equation."
Unknown: Let n = the number
"Five more than n": n + 5
"is 12": = 12
Equation: n + 5 = 12
Problem 2: "Twice a number minus 3 equals 15. Write an equation."
Let x = the number
"Twice x": 2x
"minus 3": 2x − 3
"equals 15": = 15
Equation: 2x − 3 = 15
Problem 3: "The sum of a number and 8 is 20."
Equation: n + 8 = 20
5. Find a Value Using Two-Variable Equations
What is a Two-Variable Equation?
A two-variable equation has two different variables (like x and y) that depend on each other.
Common Form
y = (rule involving x)
Examples: y = x + 3, y = 2x, y = 4x − 5
How to Find a Value
Step 1: Look at the equation
Step 2: Identify which variable you're given
Step 3: Substitute that value into the equation
Step 4: Calculate to find the other variable
Examples
Example 1: If y = x + 7 and x = 5, find y
Equation: y = x + 7
Substitute x = 5: y = 5 + 7
Answer: y = 12
Example 2: If y = 3x − 4 and x = 6, find y
y = 3(6) − 4
y = 18 − 4
Answer: y = 14
Example 3: If y = 2x and y = 10, find x
10 = 2x
x = 10 ÷ 2
Answer: x = 5
6. Complete a Table for a Two-Variable Relationship
What is a Function Table?
A function table (or input-output table) shows the relationship between x and y values based on a rule.
Steps to Complete a Table
Step 1: Identify the rule (equation)
Step 2: For each x value, substitute into the rule
Step 3: Calculate the y value
Step 4: Write the result in the table
Example
Rule: y = 2x + 1
Complete the table:
| x | y = 2x + 1 | y |
|---|---|---|
| 0 | 2(0) + 1 | 1 |
| 1 | 2(1) + 1 | 3 |
| 2 | 2(2) + 1 | 5 |
| 3 | 2(3) + 1 | 7 |
7. Complete a Table from a Graph
Reading Points from a Graph
Each point on a graph represents an ordered pair (x, y) that can be written in a table.
Steps to Complete Table from Graph
Step 1: Look at each point on the graph
Step 2: Find the x-coordinate (horizontal position)
Step 3: Find the y-coordinate (vertical position)
Step 4: Write (x, y) in the table
Example
If a graph shows these points: (1, 3), (2, 6), (3, 9), (4, 12)
| x | y |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |
| 4 | 12 |
Pattern: y = 3x (each y is 3 times the x)
8. Graph Patterns Using Rules
From Rule to Graph
You can graph a pattern by using a rule to create ordered pairs and then plotting them on a coordinate plane.
Steps to Graph a Pattern
Step 1: Use the rule to create a table of values
Step 2: Write ordered pairs (x, y)
Step 3: Plot each point on a coordinate plane
Step 4: Connect the points (if they form a pattern)
Example
Rule: y = x + 2
Step 1: Create table
| x: 0, 1, 2, 3, 4 |
| y: 2, 3, 4, 5, 6 |
Step 2: Ordered pairs
(0,2), (1,3), (2,4), (3,5), (4,6)
Step 3: Plot points on graph
The points will form a straight line going up!
9. Graph a Two-Variable Relationship
Visual Representation
Graphing a two-variable relationship shows the visual pattern of how x and y are connected.
Types of Relationships
Linear Relationship: Points form a straight line
Example: y = 2x + 3
Proportional Relationship: Line passes through origin (0,0)
Example: y = 4x
Non-Proportional: Line doesn't pass through origin
Example: y = x + 5
Real-World Example
"Each pencil costs $2. Graph the relationship between number of pencils (x) and total cost (y)."
Rule: y = 2x
Points: (0,0), (1,2), (2,4), (3,6), (4,8)
This creates a straight line through the origin
10. Write a Two-Variable Equation
From Pattern to Equation
When you see a pattern in a table or graph, you can write an equation to describe the relationship.
How to Find the Equation
Step 1: Look at the pattern in the table
Step 2: Ask: What operation connects x and y?
Step 3: Test your rule with multiple points
Step 4: Write the equation: y = ___
Common Patterns
| Pattern | Equation |
|---|---|
| y is always 3 more than x | y = x + 3 |
| y is always double x | y = 2x |
| y is 5 less than x | y = x − 5 |
| y is triple x plus 1 | y = 3x + 1 |
Example
Look at this table and write an equation:
| x | y |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | 15 |
| 4 | 20 |
Analysis:
1 × 5 = 5 ✓
2 × 5 = 10 ✓
3 × 5 = 15 ✓
4 × 5 = 20 ✓
Equation: y = 5x
Quick Reference: Variable Expressions
Key Formulas
Expression: Variable + numbers + operations (no = sign)
Equation: Expression = Expression
Evaluate: Substitute value and calculate
Two-Variable: y = (rule with x)
Operation Keywords
+ : sum, plus, more, increased, total
− : difference, minus, less, decreased, fewer
× : product, times, of, twice
÷ : quotient, divided, per, split
💡 Important Tips to Remember
✓ Variables are letters that represent unknown numbers
✓ Expressions don't have equal signs; equations do
✓ To evaluate, substitute the number and calculate
✓ Write coefficients before variables (3x, not x3)
✓ Always follow PEMDAS when evaluating
✓ In y = rule, x is input, y is output
✓ Tables show the relationship between x and y
✓ Graphs give a visual picture of patterns
✓ Test your equation with all values in the table
✓ Check your work by substituting back!
🧠 Memory Tricks
Variable Memory:
"A variable varies - it can change!"
Expression vs Equation:
Expression = No equal sign (like a phrase)
Equation = Has equal sign (like a complete sentence)
PEMDAS for Order:
Please Excuse My Dear Aunt Sally
Coefficient Before Variable:
"The number is the boss - it goes first!"
Two-Variable Relationship:
"X is the input (what you put IN)"
"Y is the output (what comes OUT)"
Master Variable Expressions! ✏️📊
Variables are the building blocks of algebra - practice every day!