Identifying Shelly's Mistake In Order Of Operations A Step-by-Step Analysis

Introduction

In mathematics, the order of operations is a fundamental concept that dictates the sequence in which mathematical operations should be performed. This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures that mathematical expressions are evaluated consistently and accurately. Failing to adhere to the correct order of operations can lead to incorrect results, as demonstrated in Shelly's attempt to evaluate the expression 2²(3-8)+5-1. This article will delve into Shelly's step-by-step evaluation, pinpoint the exact step where she made an error, and elucidate the correct procedure for solving the expression. Understanding the order of operations is crucial not only for academic success in mathematics but also for various real-world applications where precise calculations are necessary. By carefully examining Shelly's work and highlighting the mistake, we aim to reinforce the importance of following PEMDAS and provide a clear understanding of how to apply it effectively.

Shelly's Evaluation: A Step-by-Step Breakdown

To identify Shelly's mistake, let's meticulously examine each step of her evaluation process. Shelly began with the expression 2²(3-8)+5-1. The first step in PEMDAS involves addressing any operations within parentheses. Shelly correctly performed this step, simplifying (3-8) to -5. This gives us 2²(-5)+5-1. This initial step demonstrates a good grasp of the order of operations, as it prioritizes the parentheses before moving on to other operations. The next step should involve evaluating exponents. Shelly has an exponent in the expression, which is 2². Evaluating this exponent is crucial before proceeding with multiplication or any other operations. It's essential to maintain accuracy at each step to ensure the final result is correct. By carefully breaking down each step, we can isolate the exact point where the error occurred and understand why it led to an incorrect solution. This process highlights the importance of methodical problem-solving in mathematics, where each step builds upon the previous one.

Step 1: Parentheses

Shelly's first step was to simplify the expression within the parentheses: (3-8). This is the correct initial step according to the order of operations (PEMDAS), which prioritizes operations within parentheses. By performing this subtraction, Shelly transformed the expression (3-8) into -5. Thus, the expression became 2²(-5)+5-1. This demonstrates Shelly's understanding of the first rule of PEMDAS. It's crucial to address the parentheses first because the result directly impacts the subsequent calculations. This step sets the stage for the rest of the evaluation, and any error here would propagate through the remaining steps. The accurate simplification of the parentheses is a foundational element in solving the expression correctly. This step-by-step approach allows us to isolate each operation and ensure that the order is followed precisely. By focusing on the parentheses first, Shelly established a solid base for the rest of her calculations.

Identifying the Mistake

To pinpoint Shelly's mistake, we need to understand the correct order of operations (PEMDAS/BODMAS), which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Shelly's initial step of simplifying the parentheses (3-8) to -5 was correct. The next operation according to PEMDAS should be evaluating the exponent. However, let's assume, for the sake of identifying a hypothetical mistake, that Shelly incorrectly performed a multiplication or addition/subtraction before addressing the exponent. This would be a clear violation of the order of operations and would lead to an incorrect result. For instance, if Shelly multiplied 2² by -5 before evaluating 2², she would be deviating from the correct procedure. Similarly, any premature addition or subtraction would disrupt the proper sequence of calculations. It's crucial to meticulously follow the order of operations to ensure accuracy. By carefully analyzing each step Shelly took, we can isolate the precise point where she veered from the correct path. This systematic approach helps to reinforce the importance of adherence to mathematical rules and conventions.

Correct Evaluation of the Expression

To correctly evaluate the expression 2²(3-8)+5-1, we must strictly adhere to the order of operations (PEMDAS). As we discussed earlier, the first step is to address the parentheses. Simplifying (3-8) gives us -5, so the expression becomes 2²(-5)+5-1. The next step in PEMDAS is to evaluate exponents. We have 2², which equals 4. Substituting this into the expression, we get 4(-5)+5-1. Now we perform multiplication and division from left to right. We have 4 multiplied by -5, which equals -20. So the expression now reads -20+5-1. Finally, we perform addition and subtraction from left to right. First, we add -20 and 5, which gives us -15. The expression becomes -15-1. Subtracting 1 from -15 gives us -16. Therefore, the correct evaluation of the expression 2²(3-8)+5-1 is -16. This step-by-step breakdown illustrates the importance of following the correct order to arrive at the accurate result. Any deviation from PEMDAS would lead to a different and incorrect answer.

Step-by-Step Solution

1. Parentheses:
   • Start by simplifying the expression inside the parentheses: (3 - 8) = -5
   • The expression now becomes: 2²(-5) + 5 - 1
2. Exponents:
   • Evaluate the exponent: 2² = 4
   • The expression now becomes: 4(-5) + 5 - 1
3. Multiplication:
   • Perform the multiplication: 4 * (-5) = -20
   • The expression now becomes: -20 + 5 - 1
4. Addition and Subtraction:
   • Perform addition and subtraction from left to right:
     • -20 + 5 = -15
     • -15 - 1 = -16

Therefore, the correct answer is -16. This detailed step-by-step solution reinforces the importance of adhering to the order of operations (PEMDAS) to arrive at the correct result. Each step builds upon the previous one, and any deviation from the correct order would lead to an inaccurate answer. By breaking down the problem into smaller, manageable steps, we can clearly see how each operation contributes to the final solution.

Conclusion

In conclusion, understanding and applying the order of operations (PEMDAS) is crucial for accurate mathematical calculations. Shelly's attempt to evaluate the expression 2²(3-8)+5-1 highlights the importance of following the correct sequence of operations. By carefully examining each step, we can pinpoint the exact location where a mistake occurs and learn from it. The correct evaluation, as demonstrated, involves simplifying the parentheses first, then addressing exponents, followed by multiplication and division, and finally, addition and subtraction. The step where Shelly makes the mistake is crucial to identify. It reinforces the necessity of a methodical approach to problem-solving in mathematics. By breaking down complex expressions into smaller, manageable steps and adhering to the established rules of PEMDAS, we can ensure accurate and consistent results. This understanding not only benefits academic pursuits but also has practical applications in various real-world scenarios where precise calculations are essential. Mastery of the order of operations is a foundational skill in mathematics, paving the way for more advanced concepts and problem-solving techniques. This article serves as a reminder of the significance of this principle and the importance of careful and methodical application.