Basic Math

Expressions | Sixth Grade

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

TermDefinitionExample
VariableA letter representing an unknown valuex, y, a, b
CoefficientThe number multiplied by a variableIn 5x, 5 is the coefficient
ConstantA number without a variableIn x + 7, 7 is the constant
TermParts of expression separated by + or −In 3x + 5, terms are 3x and 5
FactorNumbers/variables multiplied togetherIn 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

OperationKey WordsExample PhraseExpression
Addition (+)sum, plus, more than, increased by, total5 more than xx + 5
Subtraction (−)difference, minus, less than, decreased by, fewer7 less than yy - 7
Multiplication (×)product, times, multiplied by, of3 times a number n3n
Division (÷)quotient, divided by, ratiox divided by 4x/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

TermCoefficientFactors
5x55, x
-3ab-3-3, a, b
xy11, x, y

Quick Reference: Expression Rules

ConceptKey Points
ExpressionNo equal sign; contains variables, numbers, operations
VariableLetter representing unknown value (x, y, a, b, etc.)
CoefficientNumber multiplied by variable (5 in 5x)
TermParts separated by + or − signs
EvaluateSubstitute 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!

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