Using Order of Operations Answer Key for Grade 3

Complete answer key: order of operations rules, step-by-step solutions, and detailed explanations.

Answer Key: Exercises 1 to 3
1 Basic Addition and Subtraction
Exercise 1
Solve: 12 + 5 - 3 + 6
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

Using Order of Operations Method:
  1. Identify all operations in the expression
  2. Check if there are different types of operations
  3. Since only addition and subtraction are present, work from left to right
  4. Perform each operation sequentially
Expression
12 + 5 - 3 + 6
First Operation
17 - 3 + 6
Second Operation
14 + 6
Final Answer
20
Step 1: Perform first operation (leftmost)

12 + 5 = 17

New expression: 17 - 3 + 6

Step 2: Perform next operation (leftmost)

17 - 3 = 14

New expression: 14 + 6

Step 3: Perform final operation

14 + 6 = 20

12 + 5 - 3 + 6 = 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 Basic Multiplication and Division
Exercise 2
Solve: 20 ÷ 4 × 3 ÷ 5
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
20 ÷ 4 × 3 ÷ 5
First Operation
5 × 3 ÷ 5
Second Operation
15 ÷ 5
Final Answer
3
Step 1: Perform first operation (leftmost)

20 ÷ 4 = 5

New expression: 5 × 3 ÷ 5

Step 2: Perform next operation (leftmost)

5 × 3 = 15

New expression: 15 ÷ 5

Step 3: Perform final operation

15 ÷ 5 = 3

Step 4: Verify the calculation

Work backwards: 3 × 5 = 15, 15 ÷ 3 = 5, 5 × 4 = 20 ✓

20 ÷ 4 × 3 ÷ 5 = 3
Final answer:

The answer is 3

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 with Multiplication First
Exercise 3
Solve: 8 + 4 × 3 - 6
Definition:

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

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

Expression
8 + 4 × 3 - 6
First: Mult/Div
8 + 12 - 6
Then: Add/Sub
20 - 6
Final Answer
14
Step 1: Perform multiplication first (higher precedence)

4 × 3 = 12

New expression: 8 + 12 - 6

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

8 + 12 = 20

New expression: 20 - 6

Step 3: Perform final operation

20 - 6 = 14

Step 4: Verify the calculation

Following order: 8 + (4×3) - 6 = 8 + 12 - 6 = 14 ✓

8 + 4 × 3 - 6 = 14
Final answer:

The answer is 14

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

Using 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

Answer Key: Exercises 4 to 5
4 Complex Mixed Operations
Exercise 4
Solve: 15 - 2 × 4 + 10 ÷ 5 × 3
Definition:

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

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

Expression
15 - 2 × 4 + 10 ÷ 5 × 3
First: Mult/Div
15 - 8 + 2 × 3
Continue
15 - 8 + 6
Final Answer
13
Step 1: Perform multiplication and division first (left to right)

First multiplication: 2 × 4 = 8

New expression: 15 - 8 + 10 ÷ 5 × 3

Step 2: Continue with division

10 ÷ 5 = 2

New expression: 15 - 8 + 2 × 3

Step 3: Perform remaining multiplication

2 × 3 = 6

New expression: 15 - 8 + 6

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

15 - 8 = 7

New expression: 7 + 6

Step 5: Perform final operation

7 + 6 = 13

15 - 2 × 4 + 10 ÷ 5 × 3 = 13
Final answer:

The answer is 13

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: 24 ÷ 4 + 3 × 6 - 12 ÷ 3
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
24 ÷ 4 + 3 × 6 - 12 ÷ 3
First: Mult/Div
6 + 18 - 4
Then: Add/Sub
24 - 4
Final Answer
20
Step 1: Perform all multiplication and division first (left to right)

24 ÷ 4 = 6

3 × 6 = 18

12 ÷ 3 = 4

New expression: 6 + 18 - 4

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

6 + 18 = 24

New expression: 24 - 4

Step 3: Perform final operation

24 - 4 = 20

Step 4: Verify by rechecking each operation

24 ÷ 4 = 6 ✓, 3 × 6 = 18 ✓, 12 ÷ 3 = 4 ✓

6 + 18 = 24 ✓, 24 - 4 = 20 ✓

24 ÷ 4 + 3 × 6 - 12 ÷ 3 = 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

• Verification: Double-check each calculation

Complete Guide: Using 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 do I have to follow the order of operations? Can't I just go left to right?

Answer: Great question! Following the order of operations ensures everyone gets the same answer to the same problem. 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!

Without these rules, different people would get different answers to the same problem.

Question: How can I help my child remember the order of operations?

Answer: Here are some helpful strategies:

• Use the acronym PEMDAS: "Please Excuse My Dear Aunt Sally"

• Create visual aids like posters with the order

• Practice with simple examples before moving to complex ones

• Use games and activities to make it engaging

• Remind them that multiplication and division are equals (work left to right)

• Remind them that addition and subtraction are equals (work left to right)

Practice regularly with fun problems to build confidence.

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

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

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).