Numerical Expressions | Fifth Grade

Complete Notes & Formulas

1. What is a Numerical Expression?

Definition: A numerical expression is a mathematical phrase that contains numbers and operation signs but does NOT have an equal sign.

🔑 Components of a Numerical Expression:

Numbers - whole numbers, decimals, or fractions
Operations - +, -, ×, ÷
Grouping symbols - ( ), [ ], { }

✏️ Examples:

✓ Numerical Expressions:

• 5 + 3

• 12 × (8 - 2)

• 45 ÷ 9 + 7

✗ NOT Numerical Expressions:

• 5 + 3 = 8 (has equal sign)

• x + 5 (has variable)

2. Write Numerical Expressions: One Operation

Definition: Write a mathematical expression from words using one operation.

📝 Key Words for Each Operation:

Operation	Key Words
Addition (+)	sum, plus, more than, increased by, total
Subtraction (-)	difference, minus, less than, decreased by, fewer
Multiplication (×)	product, times, multiply, of
Division (÷)	quotient, divided by, split equally, per

✏️ Examples:

Phrase: The sum of 12 and 5

Expression: 12 + 5

Phrase: The product of 8 and 6

Expression: 8 × 6

Phrase: 15 less than 40

Expression: 40 - 15

3. Write Numerical Expressions: Two Operations

Definition: Write expressions from words using two operations. Use parentheses to show which operation to do first.

🔑 When to Use Parentheses:

Use parentheses when you need to do one operation BEFORE another.

Operations inside ( ) are done FIRST

✏️ Examples:

Phrase: The sum of 9 and 5, multiplied by 3

Expression: (9 + 5) × 3

Parentheses show we add FIRST, then multiply

Phrase: 12 divided by the difference of 8 and 2

Expression: 12 ÷ (8 - 2)

Phrase: Subtract 4 from the product of 6 and 7

Expression: (6 × 7) - 4

4. Evaluate Numerical Expressions

Definition: To evaluate means to find the value of a numerical expression by following the order of operations (PEMDAS).

📐 Order of Operations (PEMDAS):

Parentheses - ( )
Exponents - ²
Multiplication and Division - left to right
Addition and Subtraction - left to right

✏️ Example: Evaluate 20 - 3 × 4

Step 1: Multiply first (3 × 4 = 12)

Step 2: Subtract (20 - 12 = 8)

Answer: 8

5. Evaluate Numerical Expressions with Parentheses

Definition: Always solve what's inside parentheses ( ) FIRST before doing other operations.

Parentheses ( ) = Do this FIRST!

✏️ Examples:

Example 1: (15 - 3) × 2 + 8

Step 1: Parentheses (15 - 3 = 12)

Step 2: Multiply (12 × 2 = 24)

Step 3: Add (24 + 8 = 32)

Answer: 32

Example 2: 40 ÷ (3 + 5)

Step 1: Parentheses (3 + 5 = 8)

Step 2: Divide (40 ÷ 8 = 5)

Answer: 5

6. Evaluate with Parentheses and Brackets

Definition: When expressions have both parentheses and brackets, work from the inside out.

📐 Order of Grouping Symbols:

Symbol	Name	Order
( )	Parentheses	Solve FIRST (innermost)
[ ]	Brackets	Solve SECOND
{ }	Braces	Solve LAST (outermost)

✏️ Example: 25 - [4 + (26 - 20)]

Step 1: Parentheses ( ) - innermost (26 - 20 = 6)

= 25 - [4 + 6]

Step 2: Brackets [ ] (4 + 6 = 10)

= 25 - 10

Step 3: Subtract (25 - 10 = 15)

Answer: 15

7. Identify Mistakes in Order of Operations

Definition: Look for common errors where operations were done in the wrong order.

❌ Common Mistakes:

Working left to right without following PEMDAS
Forgetting to solve parentheses first
Adding/subtracting before multiplying/dividing
Not working from innermost grouping symbols outward

✏️ Example: Find the Mistake

❌ Incorrect Solution:

8 + 2 × 5

= 10 × 5 (Added first - WRONG!)

= 50

✓ Correct Solution:

8 + 2 × 5

= 8 + 10 (Multiply first - CORRECT!)

= 18

8. Parentheses in Different Places

Definition: Moving parentheses changes which operation is done first, giving different answers.

✏️ Example: Same Numbers, Different Answers

Expression 1: (8 + 4) × 2

= 12 × 2 = 24

Expression 2: 8 + (4 × 2)

= 8 + 8 = 16

Expression 3: 8 + 4 × 2

= 8 + 8 = 16 (no parentheses, multiply first)

Parentheses placement matters!

9. Evaluate Expressions with Fractions

Definition: Follow the same order of operations (PEMDAS) when working with fractions.

✏️ Example: 1/2 + 1/4 × 2

Step 1: Multiply first (1/4 × 2 = 2/4 = 1/2)

Step 2: Add (1/2 + 1/2 = 2/2 = 1)

Answer: 1

10. Missing Operators

Definition: Find the missing operation symbols (+, -, ×, ÷) to make an expression true.

✏️ Examples:

Problem: 12 ___ 3 = 15

Solution: 12 + 3 = 15

Problem: 24 ___ 6 = 4

Solution: 24 ÷ 6 = 4

11. Comparison Statements with Numerical Expressions

Definition: Compare two expressions using <, >, or = symbols.

🔑 Comparison Symbols:

< Less than
> Greater than
= Equal to

✏️ Example:

Compare: 8 + 3 × 2 ___ (8 + 3) × 2

Left: 8 + 3 × 2 = 8 + 6 = 14

Right: (8 + 3) × 2 = 11 × 2 = 22

Answer: 14 < 22

12. Make the Largest Possible Quotient

Definition: Arrange given digits to create the largest quotient (answer to division).

🔑 Strategy:

To get the LARGEST quotient: Make dividend as LARGE as possible and divisor as SMALL as possible

✏️ Example:

Problem: Using digits 2, 5, 8, make the largest quotient

Strategy:

• Largest dividend: 85

• Smallest divisor: 2

• Division: 85 ÷ 2 = 42.5

Answer: 85 ÷ 2 = 42.5

Quick Reference Chart

Concept	Key Points
Numerical Expression	Numbers + operations (NO equal sign)
Order of Operations	PEMDAS: Parentheses, Exponents, MD, AS
Parentheses ( )	Solve FIRST
Brackets [ ]	Solve SECOND (after parentheses)
Braces { }	Solve LAST (after brackets)
Evaluate	Find the value using order of operations