Janelle's Errors In Expanding Algebraic Expressions A Detailed Analysis

Janelle attempted to expand the expression $-4(-3x + \frac{2}{7})$, but her solution, $-12x - 3\frac{5}{7}$, contains several errors. This article will delve into the mistakes Janelle made during the expansion process, providing a clear and detailed explanation of each error. By carefully examining her steps, we can identify the specific areas where she went wrong and understand the correct application of the distributive property and arithmetic operations. This analysis is crucial for anyone looking to improve their algebraic skills and avoid similar mistakes in the future.

Identifying the Errors in Janelle's Expansion

When expanding expressions, it’s vital to meticulously apply the distributive property and handle arithmetic operations accurately. Janelle’s attempt to expand the expression $-4(-3x + \frac{2}{7})$ resulted in $-12x - 3\frac{5}{7}$, which is incorrect. Let’s dissect her work to pinpoint the exact errors.

Error 1: The Sign of the First Term

One of the primary errors in Janelle's expansion lies in the sign of the first term. When distributing $-4$ across the expression $(-3x + \frac{2}{7})$, the first step involves multiplying $-4$ by $-3x$. According to the rules of multiplication, a negative number multiplied by another negative number results in a positive number. Therefore, $-4$ multiplied by $-3x$ should yield $+12x$, not $-12x$ as Janelle has it. This error indicates a misunderstanding or oversight of the basic rules of sign manipulation in algebra.

The sign error in Janelle's expansion significantly alters the correctness of the solution. In algebraic manipulations, maintaining the correct sign is paramount as it directly affects the value and direction of the terms. In this specific case, the incorrect negative sign on the first term leads to a fundamentally different expression. To accurately expand such expressions, it's essential to remember and apply the rule that the product of two negative numbers is positive. Neglecting this rule can lead to errors not only in this context but also in more complex algebraic problems.

The implications of this sign error extend beyond just this particular problem. In more advanced mathematical contexts, such as solving equations or graphing functions, incorrect signs can lead to drastically different outcomes. For instance, a positive term might indicate growth or movement in one direction, while a negative term might indicate decay or movement in the opposite direction. Therefore, a solid understanding of sign manipulation is crucial for building a strong foundation in algebra and mathematics in general. Students and practitioners alike should pay close attention to the signs of terms when performing algebraic operations to ensure the accuracy of their results. This focus on detail is what separates a correct solution from an incorrect one.

Error 2: The Sign and Value of the Second Term

The second significant error in Janelle's expansion is related to the second term, which involves multiplying $-4$ by $+\frac{2}{7}$. When we perform this multiplication, we are multiplying a negative number by a positive number, which should result in a negative number. Thus, the correct sign for the second term should be negative. However, the magnitude of the term is also incorrect. Multiplying $-4$ by $+\frac{2}{7}$ gives us $-\frac{8}{7}$, which can be expressed as $-1\frac{1}{7}$. Janelle's result of $-3\frac{5}{7}$ is significantly different, indicating an error in the multiplication and simplification process.

The error in the second term can be broken down into two components: the sign and the value. The incorrect sign, as discussed, stems from not properly applying the rules of multiplication with negative numbers. The incorrect value, on the other hand, suggests a flaw in either the multiplication of the fraction or its conversion into a mixed number. To prevent this type of error, it's helpful to review fraction multiplication and simplification. When multiplying a whole number by a fraction, the whole number can be treated as a fraction with a denominator of 1. In this case, $-4$ can be seen as $-\frac{4}{1}$, and multiplying it by $\frac{2}{7}$ involves multiplying the numerators and the denominators separately: $(-\frac{4}{1}) imes (\frac{2}{7}) = -\frac{4 imes 2}{1 imes 7} = -\frac{8}{7}$.

To convert the improper fraction $-\frac{8}{7}$ into a mixed number, we divide 8 by 7. The quotient is 1, and the remainder is 1. Therefore, $-\frac{8}{7}$ is equivalent to $-1\frac{1}{7}$. Janelle's result of $-3\frac{5}{7}$ is far from this value, suggesting a more fundamental misunderstanding of the arithmetic involved. This highlights the importance of not only understanding the distributive property but also having a solid grasp of basic arithmetic operations with fractions and negative numbers. Precision in these calculations is crucial for the correctness of algebraic manipulations. Students should practice these skills to ensure they can accurately perform these operations in more complex contexts.

Error 3: Incorrect Multiplication and Simplification

Janelle's third error is an umbrella error, encompassing the incorrect multiplication and simplification of the terms. As highlighted in the previous points, she made a mistake in both the sign and the numerical value of the terms. The correct expansion of $-4(-3x + \frac{2}{7})$ should start with multiplying $-4$ by $-3x$, resulting in $+12x$. Then, multiply $-4$ by $+\frac{2}{7}$, which equals $-\frac{8}{7}$ or $-1\frac{1}{7}$. Janelle's final expression, $-12x - 3\frac{5}{7}$, deviates significantly from this correct expansion, indicating a combination of errors in multiplication and simplification.

The compounding effect of these errors demonstrates how crucial each step is in algebraic manipulations. If the initial multiplication is incorrect, it sets off a chain reaction that leads to a completely wrong final answer. In this case, the incorrect sign in the first term and the incorrect value in the second term both contribute to a final expression that is far from the accurate result. This highlights the need for a systematic approach when expanding expressions, ensuring that each operation is performed correctly before moving on to the next. Students should be encouraged to double-check their work at each step, especially when dealing with negative numbers and fractions, as these are common sources of errors.

To avoid such mistakes, it's beneficial to break down complex expressions into smaller, manageable steps. First, distribute the number outside the parentheses to each term inside. Then, perform the multiplications carefully, paying close attention to signs. Finally, simplify the resulting expression by combining like terms or converting improper fractions to mixed numbers. This methodical approach minimizes the chances of making errors and makes it easier to identify mistakes if they do occur. Regular practice and a focus on accuracy in each step are the keys to mastering algebraic expansions.

Correct Expansion of the Expression

To ensure a clear understanding, let's correctly expand the expression $-4(-3x + \frac{2}{7})$.

1. Distribute -4 to both terms inside the parentheses:
   • $-4 \times -3x = 12x$
   • $-4 \times \frac{2}{7} = -\frac{8}{7}$
2. Combine the results:
   • $12x - \frac{8}{7}$
3. Convert the improper fraction to a mixed number:
   • $-\frac{8}{7} = -1\frac{1}{7}$
4. Final Expression:
   • $12x - 1\frac{1}{7}$

The correct expansion, $12x - 1\frac{1}{7}$, showcases the importance of accurately applying the distributive property and arithmetic rules. By following each step carefully and paying attention to detail, one can arrive at the correct solution. This correct expansion serves as a benchmark against which Janelle's errors can be clearly contrasted, emphasizing the specific areas where her approach diverged from the accurate method. The comparison between the incorrect and correct expansions is a valuable learning tool, highlighting the significance of precision in algebraic manipulations.

In summary, the accurate expansion of $-4(-3x + \frac{2}{7})$ involves distributing the $-4$ to both terms, resulting in $12x$ from $-4 imes -3x$ and $-\frac{8}{7}$ from $-4 imes \frac{2}{7}$. Simplifying $-\frac{8}{7}$ to $-1\frac{1}{7}$ gives the final correct expression of $12x - 1\frac{1}{7}$. This process not only corrects Janelle's errors but also reinforces the fundamental principles of algebraic expansion. Emphasizing this step-by-step approach will help students and practitioners avoid common mistakes and build confidence in their algebraic skills. The clarity and precision in this expansion serve as a model for how algebraic problems should be approached and solved.

Conclusion: Mastering Algebraic Expansions

In conclusion, Janelle's attempt to expand the expression $-4(-3x + \frac{2}{7})$ highlights several common errors that can occur during algebraic manipulations. These errors include incorrect application of sign rules, flawed multiplication of fractions, and improper simplification. By identifying and understanding these mistakes, we can reinforce the importance of careful attention to detail and a systematic approach to solving algebraic problems. The correct expansion, $12x - 1\frac{1}{7}$, serves as a clear illustration of the accurate application of the distributive property and arithmetic operations.

Mastering algebraic expansions is a critical skill in mathematics, forming the foundation for more advanced topics such as equation solving, calculus, and beyond. The errors Janelle made provide valuable learning opportunities, emphasizing the need for a solid understanding of basic arithmetic rules, especially those involving negative numbers and fractions. Students and practitioners should focus on developing a methodical approach, double-checking each step, and practicing regularly to build confidence and accuracy. The ability to correctly expand expressions not only improves problem-solving skills but also enhances mathematical fluency and critical thinking.

To further enhance understanding and prevent future errors, educators should emphasize the importance of breaking down complex problems into smaller, manageable steps. This approach makes the process less daunting and reduces the likelihood of mistakes. Additionally, providing ample opportunities for practice and feedback is essential. Students should be encouraged to work through a variety of examples, identify their own errors, and learn from them. This iterative process of learning and correction is key to developing a deep understanding of algebraic concepts. Ultimately, mastering algebraic expansions is not just about getting the right answer; it's about building a solid foundation in mathematical reasoning and problem-solving.

By addressing these errors and focusing on the correct methodologies, individuals can significantly improve their algebraic skills and achieve greater success in mathematics. Remember, the key to mastering algebra is practice, patience, and a meticulous approach to each problem.

Options that reflect the errors Janelle made:

• A. The first term should be positive.
• B. The second term should be positive.
• C. She did not multiply $-4$ by $\frac{2}{7}$ correctly.