Unraveling Calvin's Math Mistakes: A Step-by-Step Analysis

Let's dive into Calvin's mathematical journey and pinpoint where things went a bit sideways. We'll meticulously examine each step, highlighting the correct approach and the errors Calvin made. Understanding these errors is crucial for mastering mathematical concepts and avoiding similar pitfalls. We'll use a clear, step-by-step approach, making sure everything is easy to follow. Calvin's problem involves a sequence of calculations, and our goal is to identify exactly where the mistakes occurred and why.

Decoding Calvin's Mathematical Steps

Before we pinpoint the errors, let's lay out the steps Calvin presented. Each step represents a specific operation, and it's essential to understand the intended calculation to identify any deviations. We'll start with the initial expression and then analyze each subsequent step. The original problem is designed to test understanding of the order of operations and the correct application of arithmetic rules. Calvin's work provides a learning opportunity to clarify any misunderstandings. This exercise isn't just about finding mistakes; it's about solidifying our understanding of the fundamental principles of mathematics. By carefully reviewing each step, we can identify specific areas where Calvin faltered and understand the underlying reasons.

Step 1: -rac{2}{3} imesig(5+rac{1}{6}ig)

Step 2: ig(-rac{2}{3}+5ig) imes ig(-rac{2}{3}+rac{1}{6}ig)

Step 3: 4 rac{1}{3} imes-rac{1}{2}

Step 4: -2 rac{1}{6}

Step-by-Step Analysis: Uncovering Calvin's Errors

Now, let's dissect each step to reveal where Calvin's calculations went astray. This meticulous examination is key to understanding the nature of the errors and preventing them in the future. We'll compare Calvin's steps with the correct mathematical procedures to pinpoint the exact locations of the mistakes. This process is not just about finding errors; it's about reinforcing our knowledge of mathematical principles. By carefully examining each step, we gain valuable insights into common misconceptions and how to avoid them. Let's start with Step 1 and gradually move through the subsequent steps, revealing the critical errors Calvin made. It's a great opportunity to sharpen your understanding and improve your mathematical skills.

Step 1: Evaluating the Initial Expression

In Step 1, Calvin presented the initial expression: -rac{2}{3} imesig(5+rac{1}{6}ig). The correct approach here involves following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). First, you must resolve what's inside the parentheses. So, let's focus on the expression ig(5+rac{1}{6}ig). To add these, you need a common denominator, which in this case is 6. So, we rewrite 5 as rac{30}{6}. Now, we add the fractions: rac{30}{6} + rac{1}{6} = rac{31}{6}. Then, we multiply this result by -rac{2}{3}. Therefore, the correct calculation should be -rac{2}{3} imes rac{31}{6}. The next step involves multiplying the numerators together and multiplying the denominators together. This gives us -rac{62}{18}. Finally, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. The simplified form becomes -rac{31}{9}, or as a mixed number, -3rac{4}{9}. This is the correct result of Step 1.

Step 2: The Incorrect Application of Operations

Step 2 showcases a significant deviation from the correct mathematical procedure. Calvin presented the expression ig(-rac{2}{3}+5ig) imes ig(-rac{2}{3}+rac{1}{6}ig). This step incorrectly applies the distributive property or misinterprets the order of operations. The original expression in Step 1 does not involve distributing the multiplication over the addition inside the parentheses. Instead, it involves resolving the parentheses first and then multiplying. In Step 2, Calvin seems to have attempted to distribute the multiplication, but this is not the correct approach. The correct way is to first simplify the expression inside each set of parentheses separately. For the first part, -rac{2}{3} + 5, this becomes rac{-2+15}{3}, which simplifies to rac{13}{3}. For the second part, -rac{2}{3} + rac{1}{6}, we need a common denominator of 6. Rewriting, we get rac{-4+1}{6}, which simplifies to -rac{3}{6}, or -rac{1}{2}. The next step involves multiplying these two simplified fractions together: rac{13}{3} imes -rac{1}{2}. This yields -rac{13}{6}, which equals -2rac{1}{6}. However, the original step was incorrect in its initial approach.

Step 3: Calculation of Intermediate Results

In Step 3, Calvin provided 4rac{1}{3} imes -rac{1}{2}. This step appears to be related to the expression in Step 2. However, it seems to have taken a new direction. Before we assess this step, let's clarify how to approach the correct calculation. Convert the mixed number 4rac{1}{3} into an improper fraction. 4rac{1}{3} is equivalent to rac{13}{3}. Now, multiply this fraction by -rac{1}{2}: rac{13}{3} imes -rac{1}{2} = -rac{13}{6}, which is also -2rac{1}{6}. The result in Step 3 is correct. However, this step is derived from the incorrect steps, so the approach Calvin used, although leading to the correct answer, is not the correct approach to solve the original problem.

Step 4: Final Answer and Conclusion

Step 4: -2rac{1}{6}. As mentioned earlier, this result is correct. Calvin arrived at the correct final value. However, the path taken to reach this answer was not entirely correct. While the final answer matches what would be obtained from Step 2, the methodology used was flawed, specifically in how the initial expression was treated. This final step is based on an incorrect application of mathematical principles. It underscores the importance of not only reaching the right answer but also using the correct methods. It's crucial to understand why specific approaches are taken to solidify mathematical understanding and avoid future errors.

Identifying Calvin's Error

To identify Calvin's error, we must examine the transformations between each step and determine where the logic faltered. Analyzing the differences allows us to pinpoint the source of the mistakes and understand how to avoid similar errors. By carefully comparing each step with the correct methodology, we can definitively determine the nature of the error. We have to understand that, in mathematics, the process is as important as the answer.

Calvin's primary error lies in Step 2, where he incorrectly applied operations. Specifically, he appears to have misapplied the order of operations or perhaps attempted to distribute multiplication in a way that is not mathematically valid. While the numerical value in Step 4 is correct, the steps leading up to it are flawed, particularly in the transformation from Step 1 to Step 2. The mistake involves a misunderstanding of how to correctly manipulate the original expression. Calvin did not correctly follow the rules of PEMDAS and didn't correctly resolve the mathematical expression.

Final Assessment

Calvin's mistake is not just a calculation error. It demonstrates a misunderstanding of fundamental mathematical principles. The correct answer is that Calvin's error is in Step 2, where he incorrectly transformed the initial expression. It highlights the importance of mastering the order of operations and understanding how to correctly apply different mathematical operations. By thoroughly understanding where the mistakes happened, students can build a solid foundation in mathematics. This understanding allows the student to correctly solve mathematical problems and promotes logical reasoning skills. The ability to identify and correct errors is an essential skill in mathematics and in various other fields.

For further understanding of mathematical concepts, you can check out these trusted resources:

• Khan Academy: https://www.khanacademy.org/