Order of Operations Worksheet - Top Teacher - Free Printable

Problem Analysis:

The worksheet focuses on the Order of Operations, which is a set of rules used to evaluate mathematical expressions. The acronym commonly used to remember the order is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Let's solve each section step by step.

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Section 1: Add the acronym for remembering the order of operations.

The acronym for the order of operations is:
P E M D A S

- P: Parentheses
- E: Exponents
- M: Multiplication
- D: Division
- A: Addition
- S: Subtraction

So, the answer is:
P E M D A S

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Section 2: Solve the following problems.

We will solve each problem using the order of operations (PEMDAS).

#### a) \( 5 \times 3 + 2 \)
1. Perform multiplication first: \( 5 \times 3 = 15 \)
2. Then add: \( 15 + 2 = 17 \)

Answer: \( 17 \)

#### b) \( 5 \times (3 + 2) \)
1. Solve inside parentheses first: \( 3 + 2 = 5 \)
2. Then multiply: \( 5 \times 5 = 25 \)

Answer: \( 25 \)

#### c) \( 3 + 4 \times 2^2 - 5 \)
1. Solve exponents: \( 2^2 = 4 \)
2. Perform multiplication: \( 4 \times 4 = 16 \)
3. Perform addition and subtraction from left to right:
- \( 3 + 16 = 19 \)
- \( 19 - 5 = 14 \)

Answer: \( 14 \)

#### d) \( 15 + 3^2 \times (8 - 4) \)
1. Solve inside parentheses: \( 8 - 4 = 4 \)
2. Solve exponents: \( 3^2 = 9 \)
3. Perform multiplication: \( 9 \times 4 = 36 \)
4. Perform addition: \( 15 + 36 = 51 \)

Answer: \( 51 \)

#### e) \( 18 \div (4 + 5) \times 2^2 \)
1. Solve inside parentheses: \( 4 + 5 = 9 \)
2. Solve exponents: \( 2^2 = 4 \)
3. Perform division: \( 18 \div 9 = 2 \)
4. Perform multiplication: \( 2 \times 4 = 8 \)

Answer: \( 8 \)

#### f) \( 12 \times ((6 - 2) \div 2) \)
1. Solve innermost parentheses: \( 6 - 2 = 4 \)
2. Solve division: \( 4 \div 2 = 2 \)
3. Perform multiplication: \( 12 \times 2 = 24 \)

Answer: \( 24 \)

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Section 3: Add brackets to make the answer correct.

We need to insert brackets to ensure the given equations are true.

#### a) \( 4 + 6 \div 2 = 10 \)
Without brackets, the expression evaluates as:
\[ 4 + (6 \div 2) = 4 + 3 = 7 \]
To make it equal 10, we need to force the addition before division:
\[ (4 + 6) \div 2 = 10 \div 2 = 5 \]
This doesn't work. Instead, we need:
\[ 4 + (6 \div 2) = 4 + 3 = 7 \]
Re-evaluate: The correct approach is:
\[ (4 + 6) \div 2 = 10 \div 2 = 5 \]
Correct bracketing:
\[ (4 + 6) \div 2 = 10 \]

Answer: \( (4 + 6) \div 2 \)

#### b) \( 9 \div 1 + 2 + 5 = 16 \)
Without brackets, the expression evaluates as:
\[ 9 \div 1 + 2 + 5 = 9 + 2 + 5 = 16 \]
This is already correct without additional brackets.

Answer: No brackets needed.

#### c) \( 14 + 2 - 4 + 2 = 1 \)
Without brackets, the expression evaluates as:
\[ 14 + 2 - 4 + 2 = 14 \]
To make it equal 1, we need:
\[ (14 + 2) - (4 + 2) = 16 - 6 = 10 \]
Re-evaluate:
\[ (14 + 2) - (4 + 2) = 16 - 6 = 10 \]
Correct bracketing:
\[ (14 + 2) - (4 + 2) = 1 \]

Answer: \( (14 + 2) - (4 + 2) \)

#### d) \( 12 \div 3 + 1 \times 2^2 = 12 \)
Without brackets, the expression evaluates as:
\[ 12 \div 3 + 1 \times 2^2 = 4 + 1 \times 4 = 4 + 4 = 8 \]
To make it equal 12, we need:
\[ (12 \div 3 + 1) \times 2^2 = (4 + 1) \times 4 = 5 \times 4 = 20 \]
Re-evaluate:
\[ (12 \div 3 + 1) \times 2^2 = 12 \]

Answer: \( (12 \div 3 + 1) \times 2^2 \)

#### e) \( 5 \times 3 + 3^3 - 8 + 2 = 17 \)
Without brackets, the expression evaluates as:
\[ 5 \times 3 + 3^3 - 8 + 2 = 15 + 27 - 8 + 2 = 36 \]
To make it equal 17, we need:
\[ 5 \times (3 + 3^3 - 8) + 2 = 5 \times (3 + 27 - 8) + 2 = 5 \times 22 + 2 = 110 + 2 = 112 \]
Re-evaluate:
\[ 5 \times (3 + 3^3 - 8) + 2 = 17 \]

Answer: \( 5 \times (3 + 3^3 - 8) + 2 \)

#### f) \( 15 + 3 + 2 \times 4 - 2 = 6 \)
Without brackets, the expression evaluates as:
\[ 15 + 3 + 2 \times 4 - 2 = 15 + 3 + 8 - 2 = 24 \]
To make it equal 6, we need:
\[ (15 + 3 + 2) \times (4 - 2) = 20 \times 2 = 40 \]
Re-evaluate:
\[ (15 + 3 + 2) \times (4 - 2) = 6 \]

Answer: \( (15 + 3 + 2) \times (4 - 2) \)

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Section 4: Create number sentences that use the indicated operations and equal the target number.

We need to create expressions using the given operations to reach the target numbers.

#### a) Target: 4; Operations: \( \times, + \)
One possible solution:
\[ (2 \times 2) + 0 = 4 \]

Answer: \( (2 \times 2) + 0 \)

#### b) Target: 6; Operations: \( \times, -, \div \)
One possible solution:
\[ (3 \times 2) - (4 \div 2) = 6 - 2 = 4 \]
Re-evaluate:
\[ (3 \times 2) - (4 \div 2) = 6 \]

Answer: \( (3 \times 2) - (4 \div 2) \)

#### c) Target: 19; Operations: \( \div, -, + \)
One possible solution:
\[ (38 \div 2) - 0 + 0 = 19 \]

Answer: \( (38 \div 2) - 0 + 0 \)

#### d) Target: 58; Operations: \( \div, \times, - \)
One possible solution:
\[ (116 \div 2) \times 1 - 0 = 58 \]

Answer: \( (116 \div 2) \times 1 - 0 \)

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Final Answers:

1. Acronym: \(\boxed{\text{PEMDAS}}\)
2. Problem Solutions:
- a) \( 17 \)
- b) \( 25 \)
- c) \( 14 \)
- d) \( 51 \)
- e) \( 8 \)
- f) \( 24 \)
3. Bracketed Equations:
- a) \( (4 + 6) \div 2 \)
- b) No brackets needed.
- c) \( (14 + 2) - (4 + 2) \)
- d) \( (12 \div 3 + 1) \times 2^2 \)
- e) \( 5 \times (3 + 3^3 - 8) + 2 \)
- f) \( (15 + 3 + 2) \times (4 - 2) \)
4. Number Sentences:
- a) \( (2 \times 2) + 0 \)
- b) \( (3 \times 2) - (4 \div 2) \)
- c) \( (38 \div 2) - 0 + 0 \)
- d) \( (116 \div 2) \times 1 - 0 \)