Test on Order of Operations in Mathematics for Grade 3

Master order of operations: addition, subtraction, multiplication, division through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition and Subtraction
Exercise 1
Solve: 15 + 8 - 6 + 3
Definition:

Order of Operations: The sequence in which mathematical operations are performed

Addition/Subtraction Rule: When only addition and subtraction are present, perform operations from left to right

Order of Operations Method:
  1. Look at the operations in the expression
  2. Identify the order based on operation types
  3. Perform operations from left to right when same precedence
  4. Check your answer
Expression
15 + 8 - 6 + 3
Left to Right
23 - 6 + 3
Continue
17 + 3
Final Answer
20
Step 1: Perform first operation (leftmost)

15 + 8 = 23

New expression: 23 - 6 + 3

Step 2: Perform next operation (leftmost)

23 - 6 = 17

New expression: 17 + 3

Step 3: Perform final operation

17 + 3 = 20

15 + 8 - 6 + 3 = 20
Final answer:

The answer is 20

Applied rules:

• Left-to-Right Rule: Addition and subtraction have equal precedence

• Sequential Processing: Work from left to right when operators have same precedence

• Step-by-Step Calculation: Update expression after each operation

2 Multiplication and Division
Exercise 2
Solve: 24 ÷ 4 × 2 ÷ 3
Definition:

Multiplication/Division Rule: When only multiplication and division are present, perform operations from left to right

Higher Precedence: Multiplication and division have higher precedence than addition and subtraction

Expression
24 ÷ 4 × 2 ÷ 3
Left to Right
6 × 2 ÷ 3
Continue
12 ÷ 3
Final Answer
4
Step 1: Perform first operation (leftmost)

24 ÷ 4 = 6

New expression: 6 × 2 ÷ 3

Step 2: Perform next operation (leftmost)

6 × 2 = 12

New expression: 12 ÷ 3

Step 3: Perform final operation

12 ÷ 3 = 4

Step 4: Verify the calculation

Work backwards: 4 × 3 = 12, 12 ÷ 2 = 6, 6 × 4 = 24 ✓

24 ÷ 4 × 2 ÷ 3 = 4
Final answer:

The answer is 4

Applied rules:

• Left-to-Right Rule: Multiplication and division have equal precedence

• Higher Precedence: Multiplication/division before addition/subtraction

• Sequential Processing: Work from left to right when operators have same precedence

3 Mixed Operations
Exercise 3
Solve: 10 + 6 × 2 - 8 ÷ 4
Definition:

PEMDAS/BODMAS: Acronym for order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

Multiplication/Division Before Addition/Subtraction: Higher precedence operations are performed first

Expression
10 + 6 × 2 - 8 ÷ 4
First: Mult/Div
10 + 12 - 2
Then: Add/Sub
22 - 2
Final Answer
20
Step 1: Perform multiplication and division first (left to right)

6 × 2 = 12

8 ÷ 4 = 2

New expression: 10 + 12 - 2

Step 2: Perform addition and subtraction (left to right)

10 + 12 = 22

New expression: 22 - 2

Step 3: Perform final operation

22 - 2 = 20

Step 4: Verify the calculation

Following order: 10 + (6×2) - (8÷4) = 10 + 12 - 2 = 20 ✓

10 + 6 × 2 - 8 ÷ 4 = 20
Final answer:

The answer is 20

Applied rules:

• Precedence Order: Multiplication/Division before Addition/Subtraction

• Left-to-Right Within Same Precedence: Process operations of equal precedence left to right

• Sequential Simplification: Simplify expression step by step

Order of Operations Rules and Methods
PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Order of Operations
Multiplication/Division
Equal Precedence
Left to right
Addition/Subtraction
Equal Precedence
Left to right
Higher Precedence
Mult/Div vs Add/Sub
Mult/Div first
Key Definitions:

Order of Operations: The standardized sequence for performing mathematical operations

PEMDAS: Acronym for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Precedence: The priority level of operations (which to do first)

Left-to-Right Rule: When operations have equal precedence, work from left to right

Higher Precedence: Operations that must be done before others

Expression: A combination of numbers, variables, and operations

Order of Operations Methodology:
  1. Identify all operations: List all operations in the expression
  2. Determine precedence: Identify which operations have higher precedence
  3. Group same precedence: Identify operations that have equal precedence
  4. Process in order: Perform operations according to precedence rules
  5. Within same precedence: Work from left to right
  6. Check your work: Verify the final answer
Tip 1: Remember PEMDAS: Please Excuse My Dear Aunt Sally.
Tip 2: Multiplication and division have equal precedence.
Tip 3: Addition and subtraction have equal precedence.
Tip 4: Always work from left to right when operations have equal precedence.
Common Applications: Multi-step calculations, word problems, algebraic expressions, scientific calculations.
Key Concepts: Precedence, equality, sequential processing, verification.
Essential Rules:

• Precedence Hierarchy: Multiplication/Division > Addition/Subtraction

• Equal Precedence: Operations of same precedence processed left to right

• No Parentheses: Follow standard precedence order

• Verification: Always check your final answer

Solution: Exercises 4 to 5
4 Complex Mixed Operations
Exercise 4
Solve: 20 - 3 × 4 + 12 ÷ 6 × 2
Definition:

Complex Expression: An expression containing multiple operations of different precedence levels

Sequential Processing: Performing operations in the correct order step by step

Expression
20 - 3 × 4 + 12 ÷ 6 × 2
First: Mult/Div
20 - 12 + 2 × 2
Continue
20 - 12 + 4
Final Answer
12
Step 1: Perform multiplication and division first (left to right)

First multiplication: 3 × 4 = 12

New expression: 20 - 12 + 12 ÷ 6 × 2

Step 2: Continue with division

12 ÷ 6 = 2

New expression: 20 - 12 + 2 × 2

Step 3: Perform remaining multiplication

2 × 2 = 4

New expression: 20 - 12 + 4

Step 4: Perform addition and subtraction (left to right)

20 - 12 = 8

New expression: 8 + 4

Step 5: Perform final operation

8 + 4 = 12

20 - 3 × 4 + 12 ÷ 6 × 2 = 12
Final answer:

The answer is 12

Applied rules:

• Precedence Order: Multiplication/Division before Addition/Subtraction

• Left-to-Right Within Same Precedence: Process operations of equal precedence left to right

• Sequential Simplification: Simplify expression step by step

5 Verification Challenge
Exercise 5
Solve: 18 ÷ 3 + 4 × 5 - 10 ÷ 2
Definition:

Verification: Checking that the solution is correct by following the same order of operations

Multi-step Expression: An expression requiring multiple operations to solve

Expression
18 ÷ 3 + 4 × 5 - 10 ÷ 2
First: Mult/Div
6 + 20 - 5
Then: Add/Sub
26 - 5
Final Answer
21
Step 1: Perform all multiplication and division first (left to right)

18 ÷ 3 = 6

4 × 5 = 20

10 ÷ 2 = 5

New expression: 6 + 20 - 5

Step 2: Perform addition and subtraction (left to right)

6 + 20 = 26

New expression: 26 - 5

Step 3: Perform final operation

26 - 5 = 21

Step 4: Verify by rechecking each operation

18 ÷ 3 = 6 ✓, 4 × 5 = 20 ✓, 10 ÷ 2 = 5 ✓

6 + 20 = 26 ✓, 26 - 5 = 21 ✓

18 ÷ 3 + 4 × 5 - 10 ÷ 2 = 21
Final answer:

The answer is 21

Applied rules:

• Precedence Order: Multiplication/Division before Addition/Subtraction

• Left-to-Right Within Same Precedence: Process operations of equal precedence left to right

• Verification: Double-check each calculation

Comprehensive Guide: Order of Operations
PEMDAS: P-E-MD-AS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
Order of Operations Hierarchy
Key definitions:

Order of Operations: A standardized sequence for evaluating mathematical expressions

PEMDAS: Acronym for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Precedence: The priority level determining which operations to perform first

Left-to-Right Rule: When operations have equal precedence, evaluate from left to right

Higher Precedence: Operations that must be completed before lower precedence operations

Expression: A mathematical phrase containing numbers, variables, and operators

Evaluation: Finding the value of an expression by performing operations

Complete methodology:
  1. Scan the expression: Identify all operations present
  2. Apply precedence rules: Perform operations in order of precedence
  3. Handle equal precedence: Work from left to right for operations of same precedence
  4. Update expression: Replace completed operations with results
  5. Continue until complete: Repeat until a single value remains
  6. Verify your answer: Double-check each calculation
Tip 1: Always perform multiplication and division before addition and subtraction.
Tip 2: When multiplication and division appear together, work left to right.
Tip 3: When addition and subtraction appear together, work left to right.
Tip 4: Circle or highlight operations in the order you'll perform them.
Real-Life Applications: Calculating bills, following recipes, measuring materials, budgeting, time management.
Common Mistakes: Performing operations left to right regardless of precedence, forgetting that multiplication/division have equal precedence.
Essential formulas to remember:

• PEMDAS: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

• Multiplication and Division: Equal precedence, left to right

• Addition and Subtraction: Equal precedence, left to right

• Higher precedence operations performed first

• Verification: Check each step individually

Visualizing Order of Operations
Exercise 6: Operation Precedence Levels
Visual representation of operation precedence:
• Highest: Parentheses and Exponents
• Medium: Multiplication and Division
• Lowest: Addition and Subtraction

Analysis: The chart shows the hierarchy of operations in mathematical expressions.

  • Operations with higher precedence are performed first
  • Operations with equal precedence are performed left to right
  • This ensures consistent evaluation of expressions

Questions & Answers

Question: Why can't I just go from left to right for all operations?

Answer: Going left to right for all operations would give wrong answers! Let me show you:

Example: 3 + 4 × 2

• If you go left to right: 3 + 4 = 7, then 7 × 2 = 14

• But multiplication comes first: 4 × 2 = 8, then 3 + 8 = 11

So 3 + 4 × 2 = 11, not 14!

We need rules so everyone gets the same answer to the same problem.

Question: Is PEMDAS really necessary at this grade level?

Answer: Yes, establishing the foundation for order of operations is important in Grade 3 because:

• Students are working with mixed operations increasingly

• It prevents confusion and inconsistent results

• It builds critical thinking skills about mathematical relationships

• It prepares students for more complex math in higher grades

• It develops precision and logical reasoning abilities

Even simple problems like 5 + 3 × 2 require understanding order of operations.

Question: What does PEMDAS stand for? It's a funny word!

Answer: PEMDAS stands for:

• Parentheses: Do these first

• Exponents: Then these (we don't use these much yet)

• Multiplication: Then multiply

• Division: Then divide

• Addition: Then add

• Subtraction: Finally subtract

Some people remember it as "Please Excuse My Dear Aunt Sally"

Remember: Multiplication/Division are done together (left to right), and Addition/Subtraction are done together (left to right).