Order Of Operations For 5th Graders

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to ensure a consistent and accurate result. For 5th graders, mastering this concept is crucial as it forms the bedrock of more complex mathematical problem-solving in the future. Understanding and applying the correct order of operations will help children avoid common errors and build confidence in their math skills.

What is the Order of Operations?

The order of operations is a convention used in mathematics to standardize how expressions are evaluated. Without this standard, different people could interpret the same mathematical expression in various ways, leading to multiple different answers.

The most common acronym used to remember the order of operations is PEMDAS, which stands for:

• Parentheses (and other grouping symbols)
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Alternatively, some use the acronym BODMAS, which is common in some countries and stands for:

• Brackets
• Orders (exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Both acronyms convey the same hierarchy of operations.

Why is the Order of Operations Important?

The order of operations ensures that everyone solves a mathematical problem in the same way, leading to the same correct answer. This is essential for:

• Consistency: In mathematics, consistent results are crucial. The order of operations provides a standardized method.
• Accuracy: Applying the correct order ensures that the answer obtained is mathematically accurate.
• Foundation for Higher Math: As students progress to more advanced topics like algebra, calculus, and beyond, understanding and applying the order of operations becomes increasingly important.
• Problem Solving: It helps in breaking down complex problems into manageable steps, making problem-solving easier and more systematic.

Breaking Down PEMDAS

Let’s explore each part of PEMDAS with examples suitable for 5th graders.

1. Parentheses (and Grouping Symbols)

Parentheses, brackets, and braces are used to group parts of an expression together. Operations inside these grouping symbols should be performed first. If there are multiple sets of parentheses, start with the innermost set and work outwards.

Example 1:

Solve: 2 × (3 + 4)

1. First, solve the operation inside the parentheses: 3 + 4 = 7
2. Then, multiply: 2 × 7 = 14

So, 2 × (3 + 4) = 14

Example 2:

Solve: 5 + (12 ÷ 3)

1. First, solve the operation inside the parentheses: 12 ÷ 3 = 4
2. Then, add: 5 + 4 = 9

So, 5 + (12 ÷ 3) = 9

2. Exponents

Exponents indicate how many times a number is multiplied by itself. For example, 2³ (2 cubed) means 2 × 2 × 2.

Example 1:

Solve: 3² + 5

1. First, solve the exponent: 3² = 3 × 3 = 9
2. Then, add: 9 + 5 = 14

So, 3² + 5 = 14

Example 2:

Solve: 2 × 4²

1. First, solve the exponent: 4² = 4 × 4 = 16
2. Then, multiply: 2 × 16 = 32

So, 2 × 4² = 32

3. Multiplication and Division

Multiplication and division are performed from left to right. It’s important to note that neither operation takes precedence over the other; the order is simply determined by which comes first as you read the expression from left to right.

Example 1:

Solve: 10 ÷ 2 × 3

1. First, divide: 10 ÷ 2 = 5
2. Then, multiply: 5 × 3 = 15

So, 10 ÷ 2 × 3 = 15

Example 2:

Solve: 4 × 6 ÷ 2

1. First, multiply: 4 × 6 = 24
2. Then, divide: 24 ÷ 2 = 12

So, 4 × 6 ÷ 2 = 12

4. Addition and Subtraction

Similar to multiplication and division, addition and subtraction are performed from left to right. Again, neither operation takes precedence; the order is determined by which comes first from left to right.

Example 1:

Solve: 8 + 5 - 2

1. First, add: 8 + 5 = 13
2. Then, subtract: 13 - 2 = 11

So, 8 + 5 - 2 = 11

Example 2:

Solve: 15 - 7 + 3

1. First, subtract: 15 - 7 = 8
2. Then, add: 8 + 3 = 11

So, 15 - 7 + 3 = 11

Examples Combining All Operations

Let's look at more complex examples that combine all the operations, applying PEMDAS step by step.

Example 1:

Solve: 2 × (5 + 3) - 4 ÷ 2

1. First, solve the operation inside the parentheses: 5 + 3 = 8
2. Then, multiply: 2 × 8 = 16
3. Next, divide: 4 ÷ 2 = 2
4. Finally, subtract: 16 - 2 = 14

So, 2 × (5 + 3) - 4 ÷ 2 = 14

Example 2:

Solve: 15 ÷ (7 - 2) + 3² - 1

1. First, solve the operation inside the parentheses: 7 - 2 = 5
2. Then, solve the exponent: 3² = 3 × 3 = 9
3. Next, divide: 15 ÷ 5 = 3
4. Then, add: 3 + 9 = 12
5. Finally, subtract: 12 - 1 = 11

So, 15 ÷ (7 - 2) + 3² - 1 = 11

Example 3:

Solve: 4 × (2³ + 1) ÷ 5 - 2

1. First, solve the exponent inside the parentheses: 2³ = 2 × 2 × 2 = 8
2. Then, solve the addition inside the parentheses: 8 + 1 = 9
3. Next, multiply: 4 × 9 = 36
4. Then, divide: 36 ÷ 5 = 7.2
5. Finally, subtract: 7.2 - 2 = 5.2

So, 4 × (2³ + 1) ÷ 5 - 2 = 5.2

Common Mistakes and How to Avoid Them

Understanding common mistakes can help students avoid them and reinforce the correct application of the order of operations.

1. Forgetting PEMDAS:

   • Mistake: Performing operations from left to right without considering the correct order.
   • Solution: Always write out PEMDAS (or BODMAS) at the top of the page as a reminder.

2. Incorrect Order of Multiplication and Division:

   • Mistake: Always doing multiplication before division, regardless of their order from left to right.
   • Solution: Remember to perform multiplication and division from left to right.

3. Incorrect Order of Addition and Subtraction:

   • Mistake: Always doing addition before subtraction, regardless of their order from left to right.
   • Solution: Remember to perform addition and subtraction from left to right.

4. Misunderstanding Parentheses:

   • Mistake: Ignoring parentheses or solving them out of order (e.g., solving outer parentheses before inner ones).
   • Solution: Always start with the innermost set of parentheses and work outwards.

5. Skipping Steps:

   • Mistake: Trying to do multiple steps at once, leading to errors.
   • Solution: Break the problem down into smaller, manageable steps and write out each step clearly.

Tips for Teaching Order of Operations to 5th Graders

Teaching the order of operations can be made more effective with the right strategies and techniques. Here are some tips:

1. Start with Simple Expressions:

   • Begin with expressions that only involve a few operations. For example, start with addition and subtraction only, then introduce multiplication and division, and finally include parentheses and exponents.

2. Use Visual Aids:

   • Create posters or charts that display PEMDAS or BODMAS. These visual reminders can help students remember the correct order.

3. Work Through Examples Step-by-Step:

   • Solve problems on the board, explaining each step clearly and emphasizing the order of operations.

4. Hands-On Activities:

   • Use manipulatives or interactive games to make learning more engaging. For example, create cards with different operations and have students arrange them in the correct order.

5. Real-World Examples:

   • Relate the order of operations to real-world scenarios. For example, calculating the total cost of items with discounts and taxes involves multiple operations that need to be performed in the correct order.

6. Practice Worksheets:

   • Provide plenty of practice worksheets with varying levels of difficulty. This will help students reinforce their understanding and build confidence.

7. Group Work:

   • Encourage students to work in pairs or small groups to solve problems together. This allows them to discuss their thinking and learn from each other.

8. Regular Review:

   • Regularly review the order of operations to reinforce the concept. This can be done through quick quizzes, warm-up exercises, or review games.

Fun Activities and Games to Reinforce Learning

Engaging students with fun activities and games can make learning the order of operations more enjoyable and effective.

1. PEMDAS Bingo:

   • Create bingo cards with mathematical expressions. Call out the answers, and students mark off the corresponding expressions on their cards.

2. Order of Operations Maze:

   • Design a maze where students must solve mathematical expressions to find the correct path from start to finish.

3. Math Card Game:

   • Create a deck of cards with numbers and operations. Students draw cards and create expressions that follow the order of operations, aiming to reach a target number.

4. Online Games:

   • Utilize online platforms that offer interactive games and quizzes on the order of operations. Many websites and apps provide engaging and educational content.

5. Worksheet Competitions:

   • Organize worksheet competitions where students solve problems individually or in teams, with prizes for the fastest and most accurate solutions.

6. Create-a-Problem Activity:

   • Have students create their own mathematical expressions and challenge their classmates to solve them using the order of operations.

Adapting Instruction for Different Learning Styles

Students have different learning styles, and adapting instruction to accommodate these differences can enhance their understanding of the order of operations.

1. Visual Learners:

   • Use diagrams, charts, and color-coded notes to illustrate the order of operations. Visual aids can help visual learners grasp the concept more easily.

2. Auditory Learners:

   • Provide verbal explanations and encourage students to talk through the problem-solving process. Auditory learners benefit from hearing and discussing the concepts.

3. Kinesthetic Learners:

   • Use hands-on activities and manipulatives to engage kinesthetic learners. For example, use blocks or counters to represent numbers and operations.

4. Read/Write Learners:

   • Provide written instructions and encourage students to take notes and write out the steps of each problem. Read/write learners benefit from reading and writing the concepts.

Real-World Applications of the Order of Operations

Connecting the order of operations to real-world applications can help students understand its relevance and importance.

1. Cooking:

   • Explain how recipes require following specific steps in a certain order to achieve the desired result. For example, mixing dry ingredients before adding wet ingredients.

2. Shopping:

   • Calculate the total cost of items with discounts and taxes. This involves multiple operations that need to be performed in the correct order.

3. Budgeting:

   • Create a budget that includes income, expenses, and savings. This requires performing addition and subtraction in the correct order to determine the balance.

4. Construction:

   • Measure materials and calculate quantities for building projects. This involves performing multiplication, division, and addition in the correct order.

5. Travel:

   • Calculate travel time, distance, and cost. This involves performing multiple operations, such as multiplying speed by time to find distance.

Assessment and Evaluation

Regular assessment and evaluation are essential to monitor students' understanding of the order of operations and identify areas where they may need additional support.

1. Formative Assessments:

   • Use quick quizzes, exit tickets, and class discussions to assess students' understanding of the concepts during the learning process.

2. Summative Assessments:

   • Administer tests and exams to evaluate students' overall understanding of the order of operations at the end of a unit or lesson.

3. Diagnostic Assessments:

   • Use diagnostic assessments to identify specific areas where students are struggling and tailor instruction accordingly.

4. Performance-Based Assessments:

   • Assign projects or tasks that require students to apply the order of operations to solve real-world problems.

5. Self-Assessments:

   • Encourage students to reflect on their own learning and identify areas where they feel confident or need further practice.

Resources for Teaching Order of Operations

There are numerous resources available to support teaching the order of operations to 5th graders.

1. Textbooks:

   • Refer to math textbooks for explanations, examples, and practice problems.

2. Worksheets:

   • Use online resources to find printable worksheets with varying levels of difficulty.

3. Online Tutorials:

   • Utilize online tutorials and video lessons to provide additional explanations and demonstrations.

4. Educational Websites:

   • Explore educational websites that offer interactive games, quizzes, and activities on the order of operations.

5. Teacher Resource Books:

   • Consult teacher resource books for lesson plans, activities, and assessment ideas.

Conclusion

Mastering the order of operations is a fundamental skill for 5th graders, laying the groundwork for success in higher-level mathematics. By understanding the importance of PEMDAS (or BODMAS) and practicing regularly, students can develop confidence in their ability to solve complex mathematical expressions accurately. Through engaging activities, real-world applications, and tailored instruction, educators can make learning the order of operations an enjoyable and rewarding experience for all students.