Expressions - Sixth Grade
Complete Notes & Formulas
1. What is an Algebraic Expression?
Definition
An expression is a mathematical phrase
containing numbers, variables, and operations
NO equal sign (=)
Examples
• 3x + 5 (expression)
• 2y - 7 (expression)
• 4a + 3b (expression)
• x² + 2x + 1 (expression)
Remember: Expression = No equal sign | Equation = Has equal sign (=)
2. Parts of an Expression
Key Vocabulary
Term | Definition | Example |
---|---|---|
Variable | A letter representing an unknown value | x, y, a, b |
Coefficient | The number multiplied by a variable | In 5x, 5 is the coefficient |
Constant | A number without a variable | In x + 7, 7 is the constant |
Term | Parts of expression separated by + or − | In 3x + 5, terms are 3x and 5 |
Factor | Numbers/variables multiplied together | In 4xy, factors are 4, x, y |
Example: Identify parts of 5x + 3y - 7
Terms: 5x, 3y, and -7
Variables: x and y
Coefficients: 5 (for x), 3 (for y)
Constant: -7
3. Writing Expressions with One Operation
Key Words and Operations
Operation | Key Words | Example Phrase | Expression |
---|---|---|---|
Addition (+) | sum, plus, more than, increased by, total | 5 more than x | x + 5 |
Subtraction (−) | difference, minus, less than, decreased by, fewer | 7 less than y | y - 7 |
Multiplication (×) | product, times, multiplied by, of | 3 times a number n | 3n |
Division (÷) | quotient, divided by, ratio | x divided by 4 | x/4 or x ÷ 4 |
Examples
• "8 more than a number x" → x + 8
• "10 less than y" → y - 10
• "6 times n" → 6n
• "a divided by 5" → a/5
4. Writing Expressions with Two Operations
Order Matters!
Read the phrase carefully to determine the order of operations
Use parentheses when needed to group operations
Examples
Example 1: "5 more than twice a number x"
• "Twice a number x" = 2x
• "5 more than" means add 5
Expression: 2x + 5
Example 2: "The sum of x and 7, multiplied by 3"
• "Sum of x and 7" = (x + 7)
• "Multiplied by 3" means multiply the sum
Expression: 3(x + 7)
Example 3: "10 less than the product of 4 and y"
• "Product of 4 and y" = 4y
• "10 less than" means subtract 10
Expression: 4y - 10
5. Evaluating Expressions
What Does "Evaluate" Mean?
To evaluate means to SUBSTITUTE
Replace the variable with a given value
Then CALCULATE the result
Steps to Evaluate
Step 1: Write the expression
Step 2: Substitute (replace) the variable with its value
Step 3: Follow order of operations (PEMDAS)
Step 4: Calculate the answer
Example 1: Evaluate 3x + 7 when x = 4
Step 1: Expression: 3x + 7
Step 2: Substitute x = 4: 3(4) + 7
Step 3: Multiply first: 12 + 7
Step 4: Add: 19
Answer: 19
Example 2: Evaluate 2y² - 5 when y = 3
Step 1: Expression: 2y² - 5
Step 2: Substitute y = 3: 2(3)² - 5
Step 3: Exponent first: 2(9) - 5
Step 4: Multiply: 18 - 5
Step 5: Subtract: 13
Answer: 13
6. Evaluating Multi-Variable Expressions
When There Are Multiple Variables
Substitute EACH variable with its given value
Use parentheses to keep values organized
Follow order of operations carefully
Example: Evaluate 2a + 3b when a = 5 and b = 4
Expression: 2a + 3b
Given: a = 5, b = 4
Substitute: 2(5) + 3(4)
Multiply: 10 + 12
Add: 22
Answer: 22
7. Evaluating with Decimals, Fractions, and Mixed Numbers
Same Process, Different Numbers!
Follow the same steps as with whole numbers
Be careful with fraction and decimal operations
Use parentheses when substituting fractions
Example with Decimals: Evaluate 5x - 2.5 when x = 3.2
5x - 2.5
5(3.2) - 2.5
16.0 - 2.5
= 13.5
Answer: 13.5
Example with Fractions: Evaluate 3n + 1/2 when n = 1/3
3n + 1/2
3(1/3) + 1/2
1 + 1/2
= 1½ or 3/2
Answer: 1½
8. Expression Word Problems
Steps for Word Problems
Step 1: Read the problem carefully
Step 2: Identify the variable (what's unknown?)
Step 3: Look for key words
Step 4: Write the expression
Step 5: Check if it makes sense
Example 1: Writing from Word Problem
Problem: Sarah has 5 more marbles than John. If John has n marbles, how many does Sarah have?
• John has: n marbles
• Sarah has: 5 more than John
• Key word: "more than" = addition
Expression: n + 5
Example 2: Evaluating from Word Problem
Problem: A taxi charges $3 plus $2 per mile. What is the cost for a 7-mile trip?
Step 1: Write expression: 3 + 2m (m = miles)
Step 2: Substitute m = 7: 3 + 2(7)
Step 3: Calculate: 3 + 14 = 17
Answer: $17
9. Identifying Terms and Coefficients
Special Cases
• If no number is shown, the coefficient is 1
• Example: x = 1x (coefficient is 1)
• Example: -y = -1y (coefficient is -1)
• A constant has NO coefficient (it's just a number)
Practice Example: 7x - 2y + 9 - x
Terms: 7x, -2y, 9, -x (4 terms)
Coefficients:
• 7 (coefficient of x in 7x)
• -2 (coefficient of y)
• -1 (coefficient of x in -x)
Constant: 9
10. Factors in Variable Expressions
What are Factors?
Factors are numbers or variables
MULTIPLIED together in a term
Example: Find factors of 6xy
6xy means 6 × x × y
Factors: 6, x, and y
We can also say: 6xy = (2)(3)(x)(y)
All factors: 1, 2, 3, 6, x, y, and combinations
Example: Factors vs Coefficients
Term | Coefficient | Factors |
---|---|---|
5x | 5 | 5, x |
-3ab | -3 | -3, a, b |
xy | 1 | 1, x, y |
Quick Reference: Expression Rules
Concept | Key Points |
---|---|
Expression | No equal sign; contains variables, numbers, operations |
Variable | Letter representing unknown value (x, y, a, b, etc.) |
Coefficient | Number multiplied by variable (5 in 5x) |
Term | Parts separated by + or − signs |
Evaluate | Substitute value for variable and calculate |
💡 Important Tips to Remember
✓ Expression ≠ Equation (no equal sign!)
✓ Coefficient of x is 1 if no number shown
✓ "More than" = Addition
✓ "Less than" = Subtraction (reverse order!)
✓ "Times" or "Product" = Multiplication
✓ "Quotient" = Division
✓ Always use parentheses when substituting
✓ Follow PEMDAS when evaluating
✓ Terms are separated by + or −
✓ Factors are multiplied together
🧠 Memory Tricks & Strategies
PEMDAS for Evaluating:
"Please Excuse My Dear Aunt Sally"
Parentheses, Exponents, Multiply/Divide, Add/Subtract
Writing Expressions:
"Key words are the key to the code - they show what operation mode!"
Terms:
"Plus and minus split them apart - those are the signs where terms start!"
Coefficient:
"The number in front is what you see - that's the coefficient's identity!"
Substitution:
"Replace the letter, calculate the rest - that's how you ace the test!"
Master Algebraic Expressions! 🔢 ✏️ 🎯
Remember: Expressions tell a math story without an ending!