Numerical Expressions

What is a Numerical Expression?

A numerical expression is a math problem with numbers and operations (+, −, ×, ÷) but NO equal sign.

Examples of numerical expressions:

• 5 + 3
• 12 × 4 - 7
• (8 + 2) ÷ 5
• 20 - 3 × 4 + 6

NOT expressions (these are equations):

• 5 + 3 = 8 (has equal sign)
• x + 7 (has a variable)

Key: Expressions show a calculation but don't have an answer written yet!

Parts of an Expression

Numbers: 3, 15, 100, etc. Operations: +, −, ×, ÷ Grouping symbols: ( ), [ ], { }

Example: (12 + 3) × 5 - 2

• Numbers: 12, 3, 5, 2
• Operations: +, ×, −
• Grouping: ( )

Evaluating Expressions

To evaluate means to find the value (solve it).

Example: Evaluate 7 + 3

7 + 3 = 10

The value of the expression is 10 ✓

Order of Operations (PEMDAS)

To evaluate expressions with multiple operations, use PEMDAS!

P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction

Example: Evaluate 6 + 3 × 2

Wrong way: 6 + 3 = 9, then 9 × 2 = 18 ✗

Right way: 3 × 2 = 6, then 6 + 6 = 12 ✓

Always multiply BEFORE adding!

Simple Expressions

One Operation

Example 1: 8 + 5 Value = 13 ✓

Example 2: 20 - 7 Value = 13 ✓

Example 3: 6 × 4 Value = 24 ✓

Example 4: 15 ÷ 3 Value = 5 ✓

Easy! Just do the one operation.

Two Operations (Same Level)

Example: 12 + 5 + 3

Work left to right: 12 + 5 = 17 17 + 3 = 20 ✓

Example: 20 − 8 − 4

Work left to right: 20 − 8 = 12 12 − 4 = 8 ✓

Expressions with Different Operations

Addition and Subtraction

Work left to right

Example: 15 + 7 − 3

15 + 7 = 22 22 − 3 = 19 ✓

Multiplication and Division

Work left to right

Example: 20 ÷ 4 × 2

20 ÷ 4 = 5 5 × 2 = 10 ✓

Mixed Operations (Use PEMDAS!)

Example: 5 + 3 × 4

Multiply first: 3 × 4 = 12 Then add: 5 + 12 = 17 ✓

Example: 18 ÷ 3 + 2

Divide first: 18 ÷ 3 = 6 Then add: 6 + 2 = 8 ✓

Using Parentheses

Parentheses tell you do this first!

Example: (5 + 3) × 4

Step 1: Parentheses first 5 + 3 = 8

Step 2: Multiply 8 × 4 = 32 ✓

Compare without parentheses: 5 + 3 × 4 3 × 4 = 12 5 + 12 = 17 ✓

Different answer! Parentheses matter!

Writing Expressions from Words

Key Words

Addition (+):

• sum
• total
• plus
• more than
• increased by

Subtraction (−):

• difference
• minus
• less than
• decreased by
• fewer than

Multiplication (×):

• product
• times
• multiplied by
• of

Division (÷):

• quotient
• divided by
• split into
• shared equally

Examples

Words: "The sum of 8 and 5" Expression: 8 + 5 ✓

Words: "The product of 6 and 4" Expression: 6 × 4 ✓

Words: "12 decreased by 7" Expression: 12 − 7 ✓

Words: "20 divided by 4" Expression: 20 ÷ 4 ✓

Words: "3 times the sum of 5 and 2" Expression: 3 × (5 + 2) ✓ (Parentheses show "sum of 5 and 2" happens first!)

Complex Expressions

Example 1: Evaluate (15 − 3) ÷ 4 + 5

Step 1: Parentheses 15 − 3 = 12

Step 2: Division 12 ÷ 4 = 3

Step 3: Addition 3 + 5 = 8 ✓

Example 2: Evaluate 20 − 4 × 3 + 6

Step 1: Multiply (M comes before A and S) 4 × 3 = 12

Step 2: Now we have: 20 − 12 + 6

Step 3: Work left to right 20 − 12 = 8 8 + 6 = 14 ✓

Multiple Grouping Symbols

Example: 5 × [(12 − 4) ÷ 2]

Step 1: Innermost parentheses 12 − 4 = 8

Step 2: Now: 5 × [8 ÷ 2]

Step 3: Division inside brackets 8 ÷ 2 = 4

Step 4: Multiply 5 × 4 = 20 ✓

Rule: Work from the inside out!

Equivalent Expressions

Different expressions can have the same value.

Example:

• 5 + 5 + 5 = 15
• 3 × 5 = 15
• 10 + 5 = 15
• 30 ÷ 2 = 15

All these expressions equal 15, so they're equivalent! ✓

Distributive Property

The distributive property lets you rewrite expressions.

Example: 3 × (4 + 2)

Method 1: Parentheses first 3 × (4 + 2) = 3 × 6 = 18 ✓

Method 2: Distribute (multiply each term inside) 3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18 ✓

Both give the same answer!

General form: a × (b + c) = (a × b) + (a × c)

Real-World Expressions

Shopping: You buy 3 shirts at $12 each and 2 hats at$8 each. Total cost? Expression: 3 × 12 + 2 × 8 Value: 36 + 16 = $52 ✓

Lunch: You have $20. You spend$7 on lunch and $5 on a snack. Money left? **Expression:** 20 − 7 − 5 **Value:** 13 − 5 =$8 ✓

Sharing: 24 cookies divided equally among 4 friends, then each friend gets 2 more. Expression: 24 ÷ 4 + 2 Value: 6 + 2 = 8 cookies each ✓

Common Mistakes

❌ Mistake 1: Ignoring order of operations Wrong: 5 + 3 × 2 = 8 × 2 = 16 ✗ Right: 5 + 3 × 2 = 5 + 6 = 11 ✓

❌ Mistake 2: Forgetting parentheses change everything (5 + 3) × 2 ≠ 5 + 3 × 2 16 ≠ 11

❌ Mistake 3: Working right to left instead of left to right For same-level operations, go LEFT to RIGHT!

Writing Your Own Expressions

Challenge: Write three different expressions that equal 24.

Possible answers:

1. 20 + 4
2. 6 × 4
3. 30 − 6
4. 48 ÷ 2
5. (5 + 1) × 4
6. 10 + 7 + 7

Many possibilities! ✓

Evaluating Strategy

Step-by-step process:

1. Look for parentheses → Do these first!
2. Check for exponents → Do these next
3. Find multiplication and division → Left to right
4. Complete addition and subtraction → Left to right
5. Write your final answer

Practice Problems

Problem 1: Evaluate 12 ÷ 4 + 5 12 ÷ 4 = 3 3 + 5 = 8 ✓

Problem 2: Evaluate 7 × (3 + 2) 3 + 2 = 5 7 × 5 = 35 ✓

Problem 3: Evaluate 18 − 6 ÷ 3 6 ÷ 3 = 2 18 − 2 = 16 ✓

Problem 4: Write an expression for "8 more than the product of 5 and 3" Expression: 5 × 3 + 8 Or: (5 × 3) + 8 ✓

Expressions vs. Equations

Expression: 5 + 3 × 2 (no equal sign)

• You can evaluate it (find its value)
• Value = 11

Equation: 5 + 3 × 2 = 11 (has equal sign)

• You can solve it (check if it's true)
• This equation is true! ✓

Remember: Expressions don't have = signs!

Summary

Numerical Expression:

• Math problem with numbers and operations
• NO equal sign
• NO variables (letters)

To Evaluate:

• Follow PEMDAS
• Do operations in the correct order
• Find the value

Key Points:

• Parentheses first
• Multiply/Divide before Add/Subtract
• Work left to right for same-level operations
• Different expressions can have the same value

Remember: Expressions are like recipes - they tell you what to do, step by step! ✓