Bassima's Math Mistake Identifying The Error In Expression Evaluation
Hey guys! Let's dive into a mathematical puzzle where we'll dissect an expression evaluation performed by our friend Bassima. She tackled the expression:
and arrived at the answer 9. However, there's a little hiccup in her calculations. Our mission is to pinpoint where Bassima went wrong. We'll carefully examine each step, break down the order of operations, and uncover the mistake. So, let's put on our detective hats and get started!
Unraveling Bassima's Calculation Journey
To truly understand where Bassima's calculation went astray, we need to meticulously dissect each step of her mathematical journey. Think of it like tracing a detective's footsteps at a crime scene – every detail matters. Let's break down the expression and analyze her approach with a fine-tooth comb:
The Initial Expression: A Bird's-Eye View
At first glance, the expression might seem like a jumble of numbers and symbols. But fear not! We'll tackle it piece by piece. We have a combination of parentheses, exponents, multiplication, subtraction, and addition. The key to solving this puzzle lies in the order of operations, often remembered by the acronym PEMDAS (or BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Step 1: Taming the Parentheses (12 - 14)
Bassima correctly starts by simplifying the expression within the parentheses: (12 - 14). This gives us -2. So far, so good! Our expression now looks like this:
2(-2)^2-(-5)+12
This initial step showcases the importance of prioritizing parentheses, as they act as a container, demanding our attention first before we venture further into the mathematical landscape. Understanding the role of parentheses is crucial for maintaining the integrity of the expression and paving the way for accurate calculations.
Step 2: Exponentiation in Action (-2)^2
Next up, we encounter an exponent: (-2)^2. This means -2 multiplied by itself, which should result in a positive 4, not -4. This is where Bassima makes her crucial error! She incorrectly calculates (-2)^2 as -4. Remember, a negative number squared always results in a positive number. This is because multiplying two negative numbers yields a positive result.
Bassima's erroneous step transforms the expression into:
2(-4)-(-5)+12
Instead of the correct:
2(4)-(-5)+12
This seemingly small mistake has a cascading effect, throwing off the rest of her calculation. It's a stark reminder of how a single misstep in the order of operations can lead to a completely different outcome. The exponent is a powerful operator, and handling it with precision is paramount.
The Ripple Effect: How the Error Propagates
The consequences of Bassima's error become apparent in the subsequent steps. Let's trace the ripple effect:
- Incorrect Multiplication: Because she has 2(-4) instead of 2(4), she gets -8 instead of the correct 8.
- The Distorted Remainder: The rest of the calculation is performed correctly based on the incorrect value, but the final answer is inevitably wrong.
This highlights the importance of accuracy at each stage of the calculation. A single error early on can snowball, leading to a significantly different result. It's like a chain reaction – one faulty link weakens the entire chain.
Pinpointing Bassima's Mistake: The Exponent Snafu
After our in-depth analysis, it's clear that Bassima's error lies in the miscalculation of (-2)^2. She incorrectly evaluated it as -4 instead of the correct value of 4. This error stems from a misunderstanding of how negative numbers behave when raised to an even power.
The Correct Path: A Step-by-Step Recalculation
Let's walk through the correct solution to solidify our understanding:
- Parentheses: 2(12-14)^2-(-5)+12 becomes 2(-2)^2-(-5)+12
- Exponents: 2(-2)^2-(-5)+12 becomes 2(4)-(-5)+12
- Multiplication: 2(4)-(-5)+12 becomes 8-(-5)+12
- Subtraction (and handling the negative): 8-(-5)+12 becomes 8+5+12
- Addition: 8+5+12 becomes 25
Therefore, the correct answer is 25, not 9. By meticulously following the order of operations, we arrive at the accurate solution.
Why the Exponent Matters: A Deeper Dive
Understanding exponents is fundamental in mathematics. When a negative number is raised to an even power (like 2, 4, 6, etc.), the result is always positive. This is because the negative sign is essentially multiplied by itself an even number of times, canceling out the negative effect. On the other hand, when a negative number is raised to an odd power (like 1, 3, 5, etc.), the result remains negative.
This concept is not just a mathematical rule; it has real-world applications in various fields, from physics to finance. Grasping the behavior of exponents is crucial for building a strong foundation in mathematics.
Addressing the Answer Choices: Focusing on the Real Culprit
Now, let's take a look at the answer choices provided:
A. Bassima did not multiply 2 and 12 first. B. Bassima evaluated -(-5)
We can confidently rule out option A. Bassima correctly followed the order of operations by addressing the parentheses and exponent before multiplication. While it's true that option B is part of the process, it's not where Bassima made the actual error.
The real culprit is the exponent calculation. Bassima's mistake in evaluating (-2)^2 as -4 is the root cause of the incorrect answer. Therefore, neither of the provided answer choices directly addresses Bassima's error.
Crafting the Perfect Answer Choice
To accurately reflect Bassima's error, the correct answer choice should be something along the lines of:
- Bassima incorrectly evaluated (-2)^2 as -4 instead of 4.
This option directly pinpoints the specific mistake Bassima made, leaving no room for ambiguity. Clear and precise answer choices are essential for assessing understanding and avoiding confusion.
Key Takeaways: Lessons Learned from Bassima's Calculation
Bassima's mathematical journey, though slightly off-course, provides valuable learning opportunities for all of us. Here are the key takeaways:
- The Order of Operations is King: PEMDAS/BODMAS is not just a mnemonic; it's the golden rule for simplifying expressions. Deviating from it can lead to disastrous results.
- Exponents Demand Respect: Pay close attention to exponents, especially when dealing with negative numbers. Remember, a negative number raised to an even power becomes positive.
- One Error Can Cascade: A small mistake early on can have a ripple effect, impacting the entire calculation. Accuracy at each step is crucial.
- Dissect and Analyze: When troubleshooting mathematical errors, break down the problem into smaller steps and meticulously analyze each one.
By internalizing these lessons, we can avoid common pitfalls and become more confident and proficient problem solvers. Mathematics is a journey of learning and discovery, and even mistakes can serve as valuable stepping stones.
Conclusion: Mastering the Art of Expression Evaluation
In conclusion, Bassima's error in evaluating the expression highlights the critical importance of adhering to the order of operations and understanding the behavior of exponents. By carefully dissecting her steps, we identified the miscalculation of (-2)^2 as the root cause of the incorrect answer. This exercise serves as a reminder that accuracy and attention to detail are paramount in mathematics. So, let's keep practicing, keep learning, and continue to master the art of expression evaluation! Remember, every mistake is a chance to grow, and every solved problem is a victory!
