Spot The Math Mistake: A Student's Calculation

Hey math whizzes, let's dive into a common scenario we see in classrooms: a student's calculation. It's super important for us to be able to spot errors and understand why they happened. This helps us learn and grow, right? Today, we're going to dissect a specific problem to find the error. It’s a great way to sharpen our own math skills and become better problem-solvers. So, grab your thinking caps, and let's get started on this mathematical detective work! We'll be looking closely at the steps taken and figuring out where things might have gone a bit sideways. Understanding these nuances is key to mastering mathematical concepts and building a solid foundation for more complex problems down the line. This exercise isn't just about finding a single mistake; it's about understanding the process and reinforcing correct mathematical procedures.

The Problem at Hand: A Step-by-Step Breakdown

Alright guys, let's look at the calculation presented. We've got an expression that involves exponents, parentheses, multiplication, and addition. This is a typical problem designed to test order of operations (PEMDAS/BODMAS). The student's work is laid out step-by-step, which is awesome! Let's go through it together:

$$\begin{array}{c} 2^3(2-4)+5(3-8) \\ A. \\ 2^3(-2)+5(5) \\ B. \\ 8(-2)+25 \\ C. \\ -16+25 \\ D. \\ 9 \end{array}$$

So, the student started with the expression $2^3(2-4)+5(3-8)$ and ended up with $9$. Our mission, should we choose to accept it, is to figure out if this final answer is correct and, if not, where the mistake occurred. We need to be meticulous here, checking each step against the rules of arithmetic. This kind of problem is fantastic for reinforcing the importance of following a strict order when solving mathematical expressions. Making even a small slip-up early on can lead to a completely different and incorrect final answer. Let's analyze each line of the student's work critically, ensuring we're applying the correct mathematical principles at every stage. This detailed approach will help us pinpoint the exact moment the calculation deviated from the correct path.

Analyzing the Student's Steps: Where Did It Go Wrong?

Let's break down the student's work line by line, applying the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right).

Line 1: $2^3(2-4)+5(3-8)$ This is the original expression. Nothing to change here yet; it's just the starting point.

Line 2: $2^3(-2)+5(5)$

• Parentheses First: The student correctly simplified the expressions inside the parentheses. $(2-4) = -2$ and $(3-8) = -5$. Wait a minute! Look closely at the second parenthesis simplification. The student wrote $5(5)$ instead of $5(-5)$. This is a critical error right here. The difference between $3$ and $8$ is indeed $-5$, not $+5$. This single mistake in simplifying the second parenthesis will cascade through the rest of the calculation. It's a classic example of how a small oversight can lead to a major deviation from the correct answer. Guys, this is precisely the kind of detail we need to catch. It highlights the importance of absolute precision when dealing with negative numbers.

Line 3: $8(-2)+25$

• Exponents Next: The student correctly calculated the exponent: $2^3 = 2 imes 2 imes 2 = 8$.
• Multiplication: The student then performed the multiplication $8 imes -2 = -16$. This part is correct based on the previous line's calculation.
• The Addition: However, the student wrote $+25$. Based on their previous step where they incorrectly simplified $(3-8)$ to $5$, and then wrote $5(5)$, this implies they were adding $5 imes 5 = 25$. If we follow their incorrect path, this step seems consistent with their mistake in line 2. The crucial point here is that if line 2 had been correct ($5(-5)$), this line should have involved multiplying $5$ by $-5$, resulting in $-25$, not $+25$. So, while the calculation $8(-2) = -16$ is correct, the subsequent $+25$ is a direct consequence of the error in line 2. This is where the math starts to get really messy because of that early slip-up.

Line 4: $-16+25$

• Addition/Subtraction: The student performed the addition: $-16 + 25 = 9$. This calculation is correct based on the numbers in the previous line. If you have $-16$ and add $25$, you do indeed get $9$. But, as we've established, the numbers themselves are wrong due to the earlier error. This is a common pitfall: a student might do the arithmetic correctly on the wrong numbers and still arrive at an incorrect final answer, making it harder to spot the mistake if you're only looking at the final step.

Line 5: $9$ This is the student's final answer. It's the result of all the preceding steps, including the initial error.

Identifying the Core Error: A Deeper Dive

Now, let's pinpoint the exact error. We have two potential issues to consider based on the options provided:

A. The student should have simplified the exponent first. Let's check this. The order of operations (PEMDAS/BODMAS) dictates that we handle Parentheses/Brackets before Exponents/Orders. In line 1, the student correctly addressed the parentheses in line 2 ($2^3(-2)+5(5)$). The exponent $2^3$ was handled correctly in line 3 ($8(-2)+25$). So, the student did simplify the terms within the parentheses first, and then dealt with the exponent. Therefore, option A is incorrect. The student followed the correct order regarding exponents and parentheses in terms of when they should be simplified relative to each other.

B. The student did not subtract 3 and 8 correctly. Let's re-examine line 2: $2^3(-2)+5(5)$. The original expression had $5(3-8)$. The correct simplification of $(3-8)$ is $-5$. However, the student wrote $5(5)$. This means they incorrectly calculated $3-8$ as $5$, or they made a sign error when writing it down. This is a critical mistake. The calculation $3-8$ should result in a negative number, $-5$. The student produced a positive number, $5$. This error directly leads to the incorrect $+25$ in the subsequent step instead of the correct $-25$. Therefore, option B is correct. The error lies in the very first simplification within the parentheses, specifically the calculation involving $3-8$. It's a sign error that fundamentally altered the subsequent steps and the final answer.

The Correct Calculation: Let's See How It Should Be Done

To really drive home the point, let's perform the calculation correctly from start to finish:

Original Expression: $2^3(2-4)+5(3-8)$

1. Parentheses: Simplify inside the parentheses.

   • $(2-4) = -2$
   • $(3-8) = -5$ So the expression becomes: $2^3(-2)+5(-5)$

2. Exponents: Calculate the exponent.

   • $2^3 = 8$ So the expression becomes: $8(-2)+5(-5)$

3. Multiplication: Perform multiplication from left to right.

   • $8 imes (-2) = -16$
   • $5 imes (-5) = -25$ So the expression becomes: $-16 + (-25)$

4. Addition: Perform addition.

   • $-16 + (-25) = -16 - 25 = -41$

So, the correct answer should be -41, not $9$. This shows just how impactful that initial sign error was. It completely changed the outcome of the problem.

Conclusion: The Importance of Precision in Math

Guys, this exercise really underscores how vital it is to be super careful with every single step in a mathematical calculation. Even a tiny mistake, like a misplaced sign when subtracting, can lead to a wildly different and incorrect final answer. We saw that the student correctly handled the order of operations in terms of parentheses and exponents, but they made a critical arithmetic error when simplifying $(3-8)$, writing $5$ instead of $-5$. This single slip led to an incorrect final answer of $9$ when the correct answer is $-41$. Always double-check your work, especially with negative numbers and subtractions. Being meticulous pays off in mathematics, ensuring accuracy and a solid understanding of the concepts. Keep practicing, keep questioning, and keep that mathematical detective mind sharp!