Angela's Multiplication: Correct Or Incorrect?

Hey math enthusiasts! Today, we're diving into a problem that's all about multiplication and simplifying expressions. Our friend Angela decided to use the distributive property to tackle the expression $\frac{1}{2}(6x + 2)$. Let's break down her steps and figure out if she aced it! We'll start by taking a look at her initial problem to see if we can find any clues.

Understanding the Problem: Angela's Expression

First things first, let's take a look at the expression Angela was working with: $\frac{1}{2}(6x + 2)$. This is where we need to apply our knowledge of the distributive property. Remember, this property tells us that when we have a number outside parentheses multiplied by an expression inside the parentheses, we need to multiply that number by each term inside. In Angela's case, $\frac{1}{2}$ needs to be multiplied by both $6x$ and $2$. This is a crucial step! Understanding the distributive property is the key to simplifying the expression correctly. Without it, we might end up with the wrong answer. So, think of it as a handy tool that allows us to break down complex problems into smaller, more manageable parts. By applying the distributive property, we can carefully solve this math problem. It’s like having a secret weapon that helps us make everything easier to understand.

Next, let's look at the correct steps needed to solve this problem. We're going to multiply $\frac{1}{2}$ by both $6x$ and $2$. Multiplying $\frac{1}{2}$ by $6x$ gives us $\frac{1}{2} * 6x = 3x$. Then, multiplying $\frac{1}{2}$ by $2$ gives us $\frac{1}{2} * 2 = 1$. When we combine these results, we get $3x + 1$. The distributive property is a fundamental concept in algebra. By correctly applying the distributive property, we simplify the expression and find the equivalent expression. This process is used across the world by mathematicians, students, and engineers.

Angela's Work: A Closer Look

Now, let's take a look at Angela's answer to see how she did. Her simplified expression is $3x + 2$. We need to carefully compare her answer with the correct answer we just calculated. Let's see how Angela reached her solution, by using the distributive property. We have $\frac{1}{2}(6x + 2)$. When we distribute $\frac{1}{2}$ we get $(\frac{1}{2} * 6x) + (\frac{1}{2} * 2)$. This simplifies to $3x + 1$. Looking at Angela's final answer of $3x + 2$, we can see that she didn't quite get the correct result. While she correctly multiplied $\frac{1}{2}$ by $6x$ to get $3x$, she made an error when multiplying $\frac{1}{2}$ by $2$. Instead of getting $1$, she somehow got $2$.

When we do math, we all make mistakes. Sometimes, these errors are due to a misunderstanding of a concept, or a simple miscalculation. It's not a big deal! The important part is that we learn from our mistakes and try again. It's the most important step in the process, and everyone makes mistakes at some point. It is critical to carefully double-check each step. This also allows us to become better problem-solvers. Remember that making mistakes is a natural part of the learning process! To help avoid errors, it's a good idea to write out each step. This way, we can see exactly where we might have gone wrong. Taking this extra moment to recheck each step can make all the difference.

Analyzing Angela's Solution

Here’s how we can break down Angela's work to identify where the mistake occurred. The initial expression is $\frac{1}{2}(6x + 2)$. The first step is to distribute the $\frac{1}{2}$ across the terms inside the parentheses. So, we multiply $\frac{1}{2}$ by $6x$ and then $\frac{1}{2}$ by $2$. Let's start with $\frac{1}{2} * 6x$. When we do this, we get $3x$, which is correct. The next step is to multiply $\frac{1}{2} * 2$. Ideally, this should result in $1$, but Angela’s final answer had a $2$ instead of a $1$.

So, the error happened in this second multiplication. It is the only area where she went wrong. The distributive property is quite important. Being able to use this property accurately can make a huge difference in your math skills. Also, it's not enough to simply know how to apply the distributive property; we need to do so correctly. It can seem straightforward, but it’s easy to make a small calculation error. That's why being careful and double-checking your work is so important. Make sure that you have an understanding of the concepts. Practice consistently so that you can avoid making errors, and improve your problem-solving skills.

Conclusion: Was Angela Correct?

Based on our analysis, Angela's work is not entirely correct. She made a minor error in the final step of her calculations. While she correctly applied the distributive property and handled the multiplication of $\frac{1}{2} * 6x$, she made a mistake in multiplying $\frac{1}{2} * 2$. Instead of getting $1$, her final answer was $2$, which led to an incorrect final expression of $3x + 2$. However, we can see that she understood the main concept of the distributive property. It's important to remember that math is all about practice and learning from mistakes. Angela was close, and with a little more attention to detail, she’ll be simplifying expressions like a pro in no time.

Remember, mastering math takes time and patience. It's important to stay positive, keep practicing, and ask for help when you need it. There are lots of resources available online and in your community to support you on your math journey. Don't be afraid to make mistakes; they are a valuable part of the learning process. The more you practice, the better you will become. Keep up the great work, math enthusiasts! With consistent effort and a positive attitude, you'll be well on your way to success.

For more information on the distributive property, check out this trusted resource: Khan Academy - The Distributive Property