Solved Exercises on Using Order of Operations in Grade 3

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

Solution: Exercises 1 to 3
1 Addition and Subtraction
Exercise 1
Solve: \(12 + 5 - 3\)
Definition:

Addition and subtraction: Operations performed from left to right when they appear together

Method for addition and subtraction:
  1. Perform operations from left to right
  2. Work sequentially through the expression
  3. Complete each operation as you encounter it
Expression
\(12 + 5 - 3\)
First Operation
\(12 + 5 = 17\)
Second Operation
\(17 - 3 = 14\)
Result
\(14\)
Step Operation Result
1 12 + 5 17
2 17 - 3 14
Step 1: Identify the operations

We have addition (+) and subtraction (-)

Both operations have the same precedence level

Step 2: Apply the left-to-right rule

Perform operations from left to right when they have the same precedence

Step 3: Calculate the first operation

12 + 5 = 17

Step 4: Calculate the second operation

17 - 3 = 14

\(12 + 5 - 3 = 14\)
Final answer:

\(12 + 5 - 3 = 14\)

Applied rules:

• Left-to-right rule: When operations have the same precedence, perform them from left to right

• Same precedence: Addition and subtraction have the same precedence level

• Sequential evaluation: Work through the expression in order

Tip 1: Always work from left to right when operations have the same precedence.
Tip 2: Addition and subtraction are performed at the same level.
2 Multiplication and Division
Exercise 2
Solve: \(24 ÷ 4 × 2\)
Definition:

Multiplication and division: Operations performed from left to right when they appear together

Expression
\(24 ÷ 4 × 2\)
First Operation
\(24 ÷ 4 = 6\)
Second Operation
\(6 × 2 = 12\)
Result
\(12\)
Step Operation Result
1 24 ÷ 4 6
2 6 × 2 12
Step 1: Identify the operations

We have division (÷) and multiplication (×)

Both operations have the same precedence level

Step 2: Apply the left-to-right rule

Perform operations from left to right when they have the same precedence

Step 3: Calculate the first operation

24 ÷ 4 = 6

Step 4: Calculate the second operation

6 × 2 = 12

\(24 ÷ 4 × 2 = 12\)
Final answer:

\(24 ÷ 4 × 2 = 12\)

Applied rules:

• Left-to-right rule: When operations have the same precedence, perform them from left to right

• Same precedence: Multiplication and division have the same precedence level

• Higher precedence: Multiplication and division come before addition and subtraction

Tip 1: Multiplication and division have higher precedence than addition and subtraction.
Tip 2: Always work from left to right when operations have the same precedence.
3 Mixed Operations (Multiplication First)
Exercise 3
Solve: \(5 + 3 × 4\)
Definition:

Mixed operations: Expressions containing operations of different precedence levels

Expression
\(5 + 3 × 4\)
First Operation
\(3 × 4 = 12\)
Second Operation
\(5 + 12 = 17\)
Result
\(17\)
Operation Type Precedence Level Order of Execution
Multiplication (×) High First
Addition (+) Low Second
Step 1: Identify the operations and their precedence

5 + 3 × 4

Multiplication (×) has higher precedence than addition (+)

Step 2: Perform the highest precedence operation first

Calculate 3 × 4 = 12

Step 3: Substitute and continue with lower precedence operations

Replace 3 × 4 with 12: 5 + 12

Step 4: Calculate the remaining operation

5 + 12 = 17

\(5 + 3 × 4 = 17\)
Final answer:

\(5 + 3 × 4 = 17\)

Applied rules:

• Precedence order: Multiplication and division before addition and subtraction

• Higher precedence first: Perform operations in order of their precedence level

• Left-to-right for same level: When operations have the same precedence, go left to right

Tip 1: Multiplication and division always come before addition and subtraction.
Tip 2: Do multiplication first, then addition in expressions like this.
Rules and methods, laws,...
\(\text{Multiplication/Division} > \text{Addition/Subtraction}\)
Precedence Order
\(\text{Left to Right for Same Precedence}\)
Evaluation Direction
Level 1
\(\text{Multiplication/Division}\)
Higher precedence
Level 2
\(\text{Addition/Subtraction}\)
Lower precedence
Rule
\(\text{Left to Right}\)
Same level
Key definitions:

Order of operations: The sequence in which mathematical operations should be performed

Precedence: The priority level of operations (which ones are performed first)

Multiplication and division: Operations with higher precedence than addition and subtraction

Addition and subtraction: Operations with lower precedence than multiplication and division

Left-to-right rule: When operations have the same precedence, perform them from left to right

PEMDAS: A memory aid for order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

Sequential evaluation: Performing operations in the correct order to get the right answer

Complete methodology for order of operations:
  1. Identify operations: Look at all the operations in the expression
  2. Determine precedence: Group operations by their precedence level
  3. Perform highest precedence first: Do multiplication and division before addition and subtraction
  4. Work left to right: For operations at the same precedence level
  5. Continue sequentially: Move to the next precedence level
  6. Check answer: Verify the result makes sense
Tip 1: Remember: Multiplication and division come before addition and subtraction.
Tip 2: When operations have the same precedence, work from left to right.
Tip 3: Always perform multiplication/division before addition/subtraction.
Tip 4: Write out each step to avoid mistakes.
Common errors: Performing operations in the order they appear instead of following precedence, doing addition before multiplication, forgetting the left-to-right rule for same-level operations.
Memory aid: "Multiply/Divide first, then Add/Subtract" for precedence levels.
Essential rules:

• Precedence order: Multiplication/Division before Addition/Subtraction

• Left-to-right rule: For operations at the same precedence level

• Sequential evaluation: Work through operations in order of precedence

• Higher precedence first: Perform operations in order of their importance

Solution: Exercises 4 to 5
4 Mixed Operations (Division First)
Exercise 4
Solve: \(15 - 12 ÷ 3 + 4\)
Definition:

Complex mixed operations: Expressions containing multiple operations of different precedence levels

Expression
\(15 - 12 ÷ 3 + 4\)
First Operation
\(12 ÷ 3 = 4\)
Substitute
\(15 - 4 + 4\)
Left to Right
\(15 - 4 = 11\), then \(11 + 4 = 15\)
Step Operation Precedence Result
1 12 ÷ 3 Highest 4
2 15 - 4 Same level 11
3 11 + 4 Same level 15
Step 1: Identify operations and their precedence

15 - 12 ÷ 3 + 4

Division (÷) has higher precedence than subtraction (-) and addition (+)

Step 2: Perform the highest precedence operation first

Calculate 12 ÷ 3 = 4

Step 3: Substitute and continue

Replace 12 ÷ 3 with 4: 15 - 4 + 4

Step 4: Handle remaining operations (same precedence)

For subtraction and addition (same level), work from left to right

First: 15 - 4 = 11

Then: 11 + 4 = 15

Step 5: Final answer

15 - 12 ÷ 3 + 4 = 15

\(15 - 12 ÷ 3 + 4 = 15\)
Final answer:

\(15 - 12 ÷ 3 + 4 = 15\)

Applied rules:

• Precedence hierarchy: Perform higher precedence operations first

• Left-to-right for same level: When operations have the same precedence, go left to right

• Sequential substitution: Replace completed operations with their results

Tip 1: Always do division/multiplication before addition/subtraction.
Tip 2: After completing higher precedence operations, work left to right with remaining operations.
5 Multi-step Expression
Exercise 5
Solve: \(8 × 2 + 6 ÷ 3 - 5\)
Definition:

Multi-step expression: A complex expression requiring multiple applications of order of operations rules

Expression
\(8 × 2 + 6 ÷ 3 - 5\)
Multiplication
\(8 × 2 = 16\)
Division
\(6 ÷ 3 = 2\)
Substitute
\(16 + 2 - 5\)
Left to Right
\(16 + 2 = 18\), then \(18 - 5 = 13\)
Operation Expression Part Result Order
Multiplication 8 × 2 16 1st
Division 6 ÷ 3 2 2nd
Addition 16 + 2 18 3rd
Subtraction 18 - 5 13 4th
Step 1: Identify all operations and their precedence

8 × 2 + 6 ÷ 3 - 5

Multiplication (×) and Division (÷) have higher precedence than Addition (+) and Subtraction (-)

Step 2: Perform all multiplication and division (highest precedence)

Calculate 8 × 2 = 16

Calculate 6 ÷ 3 = 2

Step 3: Substitute results back into expression

Replace 8 × 2 with 16 and 6 ÷ 3 with 2: 16 + 2 - 5

Step 4: Handle remaining operations (same precedence)

For addition and subtraction (same level), work from left to right

First: 16 + 2 = 18

Then: 18 - 5 = 13

Step 5: Final answer

8 × 2 + 6 ÷ 3 - 5 = 13

\(8 × 2 + 6 ÷ 3 - 5 = 13\)
Final answer:

\(8 × 2 + 6 ÷ 3 - 5 = 13\)

Applied rules:

• Precedence hierarchy: Perform higher precedence operations first

• All same level together: Complete all multiplication and division before moving to addition and subtraction

• Left-to-right for same level: When operations have the same precedence, go left to right

Tip 1: Complete ALL multiplication and division before starting addition and subtraction.
Tip 2: Work through the expression systematically, following precedence rules.
Key Concepts, Laws, Methods, and Definitions for Order of Operations
\(\text{Multiplication/Division} > \text{Addition/Subtraction}\)
Precedence Order
\(\text{Left to Right for Same Precedence}\)
Evaluation Direction
Key definitions:

Order of operations: The sequence in which mathematical operations should be performed

Precedence: The priority level of operations (which ones are performed first)

Multiplication and division: Operations with higher precedence than addition and subtraction

Addition and subtraction: Operations with lower precedence than multiplication and division

Left-to-right rule: When operations have the same precedence, perform them from left to right

PEMDAS: A memory aid for order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

Sequential evaluation: Performing operations in the correct order to get the right answer

Higher precedence: Operations that must be performed before others

Complete methodology for order of operations:
  1. Step 1: Identify operations - Look at all the operations in the expression
  2. Step 2: Determine precedence - Group operations by their precedence level
  3. Step 3: Perform highest precedence first - Do multiplication and division before addition and subtraction
  4. Step 4: Work left to right - For operations at the same precedence level
  5. Step 5: Continue sequentially - Move to the next precedence level
  6. Step 6: Substitute results - Replace completed operations with their results
  7. Step 7: Check answer - Verify the result makes sense
Tip 1: Remember: Multiplication and division come before addition and subtraction.
Tip 2: When operations have the same precedence, work from left to right.
Tip 3: Always perform multiplication/division before addition/subtraction.
Tip 4: Write out each step to avoid mistakes.
Tip 5: Complete ALL multiplication and division before starting addition and subtraction.
Common errors to avoid: Performing operations in the order they appear instead of following precedence, doing addition before multiplication, forgetting the left-to-right rule for same-level operations, not completing all higher-precedence operations before moving to lower levels.
Exam preparation: Practice identifying precedence levels, work on multi-step problems, memorize the order of operations, and always check your work.
Essential formulas and rules:

• Precedence order: Multiplication/Division before Addition/Subtraction

• Left-to-right rule: For operations at the same precedence level

• Sequential evaluation: Work through operations in order of precedence

• Higher precedence first: Perform operations in order of their importance

• Complete same level: Finish all operations at one precedence level before moving to the next

Detailed Summary: Using Order of Operations

Definitions of Key Concepts:
Order of operations: The sequence in which mathematical operations should be performed (e.g., in 5 + 3 × 4, we multiply first then add).
Precedence: The priority level of operations (which ones are performed first).
Multiplication and division: Operations with higher precedence than addition and subtraction.
Addition and subtraction: Operations with lower precedence than multiplication and division.
Left-to-right rule: When operations have the same precedence, perform them from left to right.
PEMDAS: A memory aid for order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
Sequential evaluation: Performing operations in the correct order to get the right answer.
Higher precedence: Operations that must be performed before others.
Core Rules and Principles:
Precedence Hierarchy: Multiplication and division come before addition and subtraction (e.g., 5 + 3 × 4 = 5 + 12 = 17, not 8 × 4 = 32).
Left-to-Right Rule: When operations have the same precedence level, work from left to right (e.g., 12 ÷ 4 × 2 = 3 × 2 = 6, not 12 ÷ 8 = 1.5).
Complete Same Level: Finish all operations at one precedence level before moving to the next.
Sequential Evaluation: Work through operations in order of precedence.
Substitution Principle: Replace completed operations with their results.
Step-by-Step Methods and Procedures:
Order of Operations Method: 1) Identify all operations, 2) Group by precedence level, 3) Perform highest precedence first, 4) Work left to right for same-level operations, 5) Substitute results, 6) Continue with next precedence level.
Simple Expression Method: 1) Look for multiplication or division, 2) Perform them left to right, 3) Then perform addition and subtraction left to right.
Complex Expression Method: 1) Complete ALL multiplication and division, 2) Then complete ALL addition and subtraction.
Verification Method: 1) Check each step, 2) Verify that precedence rules were followed, 3) Confirm the final answer makes sense.
Concrete Examples:
Simple Example: 12 + 5 - 3 = 17 - 3 = 14 (left to right for same precedence).
Intermediate Example: 5 + 3 × 4 = 5 + 12 = 17 (multiplication before addition).
Advanced Example: 8 × 2 + 6 ÷ 3 - 5 = 16 + 2 - 5 = 18 - 5 = 13 (complete all high precedence first).
Tips, Tricks, and Common Pitfalls:
Tip 1: Always do multiplication and division before addition and subtraction.
Tip 2: When operations have the same precedence, work from left to right.
Tip 3: Write out each step to avoid mistakes.
Tip 4: Complete ALL multiplication and division before starting addition and subtraction.
Pitfall 1: Performing operations in the order they appear instead of following precedence.
Pitfall 2: Doing addition before multiplication.
Pitfall 3: Forgetting the left-to-right rule for same-level operations.
Key Notes for Memorization:
Mnemonic: "Multiply/Divide first, then Add/Subtract" for precedence levels.
Mnemonic: "Please Excuse My Dear Aunt Sally" for PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
Key Point: Multiplication and division have equal precedence and are performed left to right.
Key Point: Addition and subtraction have equal precedence and are performed left to right.
Explanation for Students:
Order of operations is like a rulebook for math! Just like in sports where there are rules about what you can and can't do, in math there are rules about which operations to do first. The most important rule is that multiplication and division come before addition and subtraction. Think of it this way: multiplication and division are like VIPs at a party - they get to go first! Only after they're done can addition and subtraction join the fun. And if you have multiple VIPs (multiple multiplication or division operations), they take turns from left to right. The same goes for addition and subtraction - they wait for their turn and then go left to right too!