Common Core Sheets: Order of Operations for Grade 3

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

Common Core Solution: 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

Common Core 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

Common Core Order of Operations Standards
PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Common Core 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

Common Core 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:

• Common Difference: d = Second Term - First Term

• Common Ratio: r = Second Term ÷ First Term

• nth Term: an = a1 + (n-1) × d

• Verification: Check that all consecutive differences are equal

Common Core Solution: 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 the calculation

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

Comprehensive Guide: Order of Operations and Common Core Standards
an = a1 + (n-1) × d
nth Term Formula
Key definitions:

Arithmetic Sequence: A sequence where each term differs from the next by a constant amount

Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant amount

Common Difference (d): The constant amount added in an arithmetic sequence

Common Ratio (r): The constant amount multiplied in a geometric sequence

First Term (a₁): The starting number of the sequence

Position (n): The place of a term in the sequence (1st, 2nd, 3rd, etc.)

Term (aₙ): The value of the term at position n

Complete methodology:
  1. Identify the sequence type: Determine if it's arithmetic or geometric
  2. Find the first term (a₁): Identify the starting number
  3. Calculate the common difference/ratio: Subtract for arithmetic, divide for geometric
  4. Determine the position (n): Know which term you're looking for
  5. Apply the formula: Use the appropriate formula for the sequence type
  6. Verify your answer: Check that the pattern continues correctly
Tip 1: Always write down the first few terms to see the pattern clearly.
Tip 2: For arithmetic sequences, subtract consecutive terms to find the common difference.
Tip 3: For geometric sequences, divide consecutive terms to find the common ratio.
Tip 4: Always double-check that your pattern works for multiple consecutive terms.
Real-Life Applications: Saving money, plant growth, counting days, age progression, building floors, stair steps.
Common Mistakes: Confusing arithmetic and geometric sequences, calculation errors, not identifying the correct pattern.
Essential formulas to remember:

• Arithmetic Sequence: an = a1 + (n-1) × d

• Geometric Sequence: an = a1 × r^(n-1)

• Common Difference: d = a₂ - a₁

• Common Ratio: r = a₂ ÷ a₁

• Position Counting: Count terms carefully (1st, 2nd, 3rd, etc.)

Visualizing Arithmetic vs Geometric Sequences
Exercise 6: Comparing Growth Patterns
Compare arithmetic and geometric sequences:
• Arithmetic: 2, 4, 6, 8, 10 (d = 2)
• Geometric: 2, 4, 8, 16, 32 (r = 2)
Notice how geometric grows much faster!

Analysis: The chart shows how geometric sequences grow exponentially while arithmetic sequences grow linearly.

  • Arithmetic sequences grow at a constant rate
  • Geometric sequences grow at an increasing rate
  • Geometric growth becomes much faster than arithmetic growth over time

Questions & Answers

Question: What is the difference between arithmetic and geometric sequences?

Answer: The main difference is how the numbers change!

Arithmetic sequences: You ADD the same number each time

Example: 2, 4, 6, 8, 10... (add 2 each time)

Geometric sequences: You MULTIPLY by the same number each time

Example: 2, 4, 8, 16, 32... (multiply by 2 each time)

Think of arithmetic like climbing stairs with equal steps, and geometric like a ball bouncing where each bounce is half the height of the previous bounce.

Question: Are these concepts appropriate for 3rd grade? They seem quite advanced.

Answer: Yes, these foundational concepts can be introduced at 3rd grade level through:

• Concrete examples using familiar objects like money, plants, and counting

• Visual representations and hands-on activities

• Simple integer relationships and counting patterns

• Focus on recognizing patterns rather than complex calculations

The goal is to build number sense and pattern recognition skills that will support later formal learning of sequences and series.

Question: Why do geometric sequences grow so much faster than arithmetic ones?

Answer: It's because of how they grow!

In arithmetic sequences, you add the same amount each time:

Start with 2, add 2: 2 → 4 → 6 → 8 → 10...

Each step adds the same amount (2).

In geometric sequences, you multiply by the same amount each time:

Start with 2, multiply by 2: 2 → 4 → 8 → 16 → 32...

Each step multiplies by the same number (2), so the amounts being added get bigger and bigger!

It's like the difference between getting $10 every day versus getting twice as much as yesterday!