Modeling Phrases With Expressions A Guide To Translating Word Problems
In the realm of mathematics, translating verbal phrases into algebraic expressions is a fundamental skill. This ability forms the bedrock for solving a myriad of real-world problems. Today, we will dissect a seemingly simple phrase, "Wally mowed two more lawns than he mowed last week," and determine the algebraic expression that accurately represents it. This exercise will not only reinforce your understanding of algebraic translation but also highlight the importance of careful reading and interpretation. We will explore each option, A. n + 2, B. n - 2, C. 2 - n, and D. 2n, to methodically arrive at the correct answer.
Breaking Down the Phrase
To begin, let's meticulously dissect the given phrase: "Wally mowed two more lawns than he mowed last week." The core of this phrase revolves around comparing the number of lawns Wally mowed this week to the number he mowed last week. The key words here are "two more than." This indicates an addition operation. We are adding two to the number of lawns Wally mowed last week to find out how many he mowed this week.
Identifying the Variable
In algebraic expressions, we often use variables to represent unknown quantities. In this scenario, the number of lawns Wally mowed last week is the unknown quantity. Let's represent this unknown quantity with the variable n. Therefore, n symbolizes the number of lawns Wally mowed last week. This crucial step of assigning a variable is fundamental in translating verbal phrases into mathematical expressions. Without this variable, we cannot proceed with constructing the algebraic representation.
Constructing the Expression
Now that we have identified the variable n as the number of lawns mowed last week, we can proceed with building the expression. The phrase states that Wally mowed "two more" lawns this week than he did last week. This directly translates to adding 2 to the number of lawns mowed last week, which is n. Thus, the expression becomes n + 2. This expression accurately captures the relationship described in the phrase, where the number of lawns mowed this week is two greater than the number mowed last week.
Evaluating the Options
Now, let's evaluate the given options to confirm our understanding and eliminate any potential confusion.
Option A: n + 2
As we deduced earlier, n + 2 perfectly encapsulates the phrase "Wally mowed two more lawns than he mowed last week." Here, n represents the number of lawns mowed last week, and adding 2 to it gives us the number of lawns mowed this week. This option aligns perfectly with our analysis and the key phrase "two more than."
Option B: n - 2
The expression n - 2 suggests that Wally mowed two fewer lawns this week than he did last week. This is because we are subtracting 2 from the number of lawns mowed last week (n). However, the original phrase explicitly states "two more lawns," indicating addition, not subtraction. Therefore, option B is incorrect.
Option C: 2 - n
The expression 2 - n implies that we are subtracting the number of lawns mowed last week (n) from 2. This interpretation does not align with the given phrase. It suggests that the number of lawns mowed this week is the difference between 2 and the number mowed last week, which is not what the original statement conveys. Thus, option C is also incorrect.
Option D: 2n
The expression 2n represents the product of 2 and the number of lawns mowed last week (n). This means Wally mowed twice as many lawns this week as he did last week. While this is a valid mathematical expression, it does not accurately represent the phrase "Wally mowed two more lawns than he mowed last week." The key difference is that 2n signifies multiplication, whereas the phrase implies addition. Therefore, option D is incorrect.
The Correct Expression
After a thorough examination of all options, it is evident that option A, n + 2, is the correct expression. It accurately represents the phrase "Wally mowed two more lawns than he mowed last week." The variable n symbolizes the number of lawns mowed last week, and adding 2 to it reflects the "two more" aspect of the phrase. This methodical approach to dissecting the phrase and evaluating each option solidifies our understanding of translating verbal expressions into algebraic ones.
In conclusion, the algebraic expression that models the phrase "Wally mowed two more lawns than he mowed last week" is n + 2. This exercise underscores the significance of careful interpretation and the precise application of mathematical operations. Translating verbal phrases into algebraic expressions is a cornerstone of mathematical problem-solving, enabling us to represent real-world scenarios in a concise and mathematical form. The ability to correctly identify variables, understand key phrases like "more than" or "less than," and apply the appropriate operations is essential for success in algebra and beyond. This skill allows us to move from words to mathematical symbols, which is crucial for solving complex problems and making informed decisions based on quantitative data. The process of breaking down the phrase, identifying the unknown, and selecting the right operation helps in translating any verbal problem into its mathematical equivalent, opening doors to a deeper understanding and application of algebraic concepts.
