Lesson 1 Homework Practice Representing Relationships

Representing relationships in mathematical terms opens a gateway to understanding the world around us more profoundly. This fundamental skill, often introduced in early algebra, lays the groundwork for complex problem-solving and critical thinking. By learning how to translate real-world scenarios into mathematical equations, students can develop a more intuitive grasp of quantitative relationships.

Understanding the Basics

Before diving into the practice problems, it's crucial to solidify the basic concepts of representing relationships. This involves understanding variables, constants, and how to formulate equations that accurately reflect a given situation.

• Variables: Symbols, usually letters, that represent unknown or changing quantities. For example, 'x' might represent the number of apples in a basket, or 'y' could stand for a person's age.
• Constants: Fixed values that do not change within a given context. If a store sells apples for a fixed price of $2 each, then '2' is a constant in this scenario.
• Expressions: Combinations of variables, constants, and mathematical operations (addition, subtraction, multiplication, division). An example of an expression could be 3x + 5, where 'x' is a variable, '3' and '5' are constants, and the operations are multiplication and addition.
• Equations: Mathematical statements that show the equality between two expressions. Equations always contain an equals sign (=). For example, 2x + 3 = 7 is an equation, stating that the expression 2x + 3 is equal to 7.

Representing relationships often involves translating word problems into mathematical equations. This translation process requires careful reading and understanding of the context. Key phrases like "more than," "less than," "times," and "divided by" provide clues about the mathematical operations involved.

Key Strategies for Representing Relationships

Effectively representing relationships involves a systematic approach that helps to break down complex problems into manageable parts. Here are some key strategies:

1. Read Carefully: Thoroughly read and understand the problem statement. Identify the knowns, unknowns, and the relationship between them.
2. Define Variables: Assign variables to represent the unknown quantities. Clearly define what each variable stands for to avoid confusion.
3. Identify Key Phrases: Look for key phrases that indicate mathematical operations, such as "sum," "difference," "product," "quotient," "increased by," or "decreased by."
4. Translate into Equations: Use the identified information to formulate an equation that accurately represents the relationship described in the problem.
5. Check Your Work: After writing the equation, check it against the original problem statement to ensure it makes logical sense and accurately represents the given information.

Practice Problems: Representing Relationships

Let's work through some practice problems to apply these strategies and gain a deeper understanding of representing relationships.

Problem 1:

• Scenario: A taxi charges a flat fee of $3 plus $2 for every mile traveled. Represent the total cost (C) of a taxi ride in terms of the number of miles traveled (m).

• Solution:

  • Define Variables:

    • C = Total cost of the taxi ride
    • m = Number of miles traveled

  • Identify Key Phrases: "Flat fee of $3" (constant), "$2 for every mile" (multiplication).

  • Translate into Equations: The total cost is the sum of the flat fee and the cost per mile. Therefore, the equation is: C = 2m + 3

  • Check Your Work: If you travel 0 miles, the cost is $3 (the flat fee). If you travel 1 mile, the cost is $5 ($3 + $2). This aligns with the problem statement.

Problem 2:

• Scenario: Sarah has twice as many apples as John. If John has 'j' apples, write an expression for the number of apples Sarah has.

• Solution:

  • Define Variables:

    • j = Number of apples John has
    • s = Number of apples Sarah has

  • Identify Key Phrases: "Twice as many" (multiplication by 2).

  • Translate into Equations: Sarah has two times the number of apples John has. Therefore, the equation is: s = 2j

  • Check Your Work: If John has 3 apples, Sarah has 6 apples, which is twice as many. This makes sense.

Problem 3:

• Scenario: A rectangle's length is 5 units longer than its width. If the width of the rectangle is 'w', write an expression for the area of the rectangle.

• Solution:

  • Define Variables:

    • w = Width of the rectangle
    • l = Length of the rectangle
    • A = Area of the rectangle

  • Identify Key Phrases: "5 units longer than" (addition).

  • Translate into Equations:

    • The length is 5 units longer than the width: l = w + 5
    • The area of a rectangle is length times width: A = l * w
    • Substitute the expression for 'l' into the area equation: A = (w + 5) * w or A = w^2 + 5w

  • Check Your Work: If the width is 2, the length is 7, and the area is 14. Plugging w=2 into A = w^2 + 5w gives A = 2^2 + 5(2) = 4 + 10 = 14. This checks out.

Problem 4:

• Scenario: A store is having a 20% off sale on all items. Write an expression for the sale price (S) of an item with an original price of (P).

• Solution:

  • Define Variables:

    • P = Original price of the item
    • S = Sale price of the item

  • Identify Key Phrases: "20% off" (subtraction of 20% of the original price from the original price).

  • Translate into Equations:

    • The discount amount is 20% of the original price: 0.20 * P
    • The sale price is the original price minus the discount amount: S = P - 0.20P
    • Simplify the equation: S = 0.80P

  • Check Your Work: If the original price is $100, the discount is $20, and the sale price is $80. Plugging P=100 into S = 0.80P gives S = 0.80 * 100 = 80. This aligns with the problem statement.

Problem 5:

• Scenario: John is 5 years older than his sister, Mary. Write an equation representing John's age (J) in terms of Mary's age (M).

• Solution:

  • Define Variables:

    • J = John's age
    • M = Mary's age

  • Identify Key Phrases: "5 years older than" (addition).

  • Translate into Equations: John's age is Mary's age plus 5 years. Therefore, the equation is: J = M + 5

  • Check Your Work: If Mary is 10 years old, John is 15 years old, which is 5 years older. This makes sense.

Problem 6:

• Scenario: A train travels at a constant speed of 60 miles per hour. Write an equation for the distance (D) the train travels in terms of the time (T) in hours.

• Solution:

  • Define Variables:

    • D = Distance traveled by the train
    • T = Time in hours

  • Identify Key Phrases: "Constant speed of 60 miles per hour" (multiplication).

  • Translate into Equations: Distance is equal to speed multiplied by time. Therefore, the equation is: D = 60T

  • Check Your Work: If the train travels for 2 hours, it covers a distance of 120 miles (60 * 2). This aligns with the problem statement.

Problem 7:

• Scenario: The sum of two consecutive integers is 25. Represent this relationship using an equation, where the first integer is 'n'.

• Solution:

  • Define Variables:

    • n = The first integer
    • n + 1 = The next consecutive integer

  • Identify Key Phrases: "Sum of two consecutive integers" (addition).

  • Translate into Equations: The sum of the first integer and the next consecutive integer is 25. Therefore, the equation is: n + (n + 1) = 25

  • Check Your Work: Simplifying the equation gives 2n + 1 = 25, then 2n = 24, and finally n = 12. The two integers are 12 and 13, and their sum is indeed 25.

Problem 8:

• Scenario: A phone company charges $0.10 per minute for calls plus a monthly service fee of $15. Write an equation for the total monthly cost (C) in terms of the number of minutes (m) used.

• Solution:

  • Define Variables:

    • C = Total monthly cost
    • m = Number of minutes used

  • Identify Key Phrases: "$0.10 per minute" (multiplication), "monthly service fee of $15" (constant).

  • Translate into Equations: The total cost is the sum of the service fee and the cost per minute. Therefore, the equation is: C = 0.10m + 15

  • Check Your Work: If you use 0 minutes, the cost is $15 (the service fee). If you use 100 minutes, the cost is $25 ($15 + $10). This aligns with the problem statement.

Problem 9:

• Scenario: The perimeter of a square is four times the length of one side. If the side length is 's', write an equation for the perimeter (P) of the square.

• Solution:

  • Define Variables:

    • P = Perimeter of the square
    • s = Length of one side of the square

  • Identify Key Phrases: "Four times the length of one side" (multiplication by 4).

  • Translate into Equations: The perimeter is four times the side length. Therefore, the equation is: P = 4s

  • Check Your Work: If the side length is 5, the perimeter is 20 (4 * 5). This makes sense.

Problem 10:

• Scenario: Lisa earns $12 per hour at her job. She also receives a bonus of $50 each week. Write an equation to represent her total weekly earnings (E) in terms of the number of hours she works (h).

• Solution:

  • Define Variables:

    • E = Total weekly earnings
    • h = Number of hours worked

  • Identify Key Phrases: "$12 per hour" (multiplication), "bonus of $50 each week" (constant).

  • Translate into Equations: Her total earnings are the sum of her hourly earnings and her bonus. Therefore, the equation is: E = 12h + 50

  • Check Your Work: If she works 0 hours, she earns $50 (the bonus). If she works 10 hours, she earns $170 ($120 + $50). This aligns with the problem statement.

Common Mistakes and How to Avoid Them

Representing relationships can be challenging, and several common mistakes can hinder accurate equation formulation. Being aware of these pitfalls and knowing how to avoid them is crucial for success.

1. Misinterpreting Key Phrases: Incorrectly interpreting phrases like "more than," "less than," "times," or "divided by" can lead to incorrect equations. Always carefully analyze the context and ensure the mathematical operation accurately reflects the described relationship. Example: Confusing "5 more than x" (x + 5) with "5 times x" (5x).
2. Incorrectly Defining Variables: Vague or unclear variable definitions can cause confusion and errors in the equation. Always clearly define what each variable represents. Example: Using 'a' to represent both the number of apples and the total cost, leading to a mixed-up equation.
3. Forgetting Constants: Neglecting to include constants in the equation when they are part of the relationship. Example: In a scenario with a flat fee and a per-unit cost, forgetting to add the flat fee to the equation.
4. Reversing the Relationship: Incorrectly placing variables on the wrong side of the equation or reversing the operation. Example: If John is twice as old as Mary, writing J = M/2 instead of J = 2M.
5. Not Checking the Equation: Failing to check the formulated equation against the original problem statement can result in undetected errors. Always substitute values and ensure the equation logically reflects the described scenario.

Advanced Applications of Representing Relationships

Mastering the skill of representing relationships opens doors to more advanced mathematical concepts and real-world applications. Here are a few examples:

• Linear Equations and Graphing: Representing linear relationships as equations allows for graphing them on a coordinate plane, providing a visual representation of the relationship between two variables. This is crucial for understanding slope, intercepts, and the behavior of linear functions.
• Systems of Equations: Many real-world problems involve multiple relationships between variables. Representing these relationships as a system of equations allows for solving for multiple unknowns simultaneously.
• Modeling Real-World Phenomena: From physics to economics, mathematical models are used to represent and predict real-world phenomena. Representing relationships is the foundation for building these models. Example: Modeling population growth, predicting the trajectory of a projectile, or analyzing economic trends.
• Computer Programming: In programming, representing relationships is fundamental for creating algorithms and solving problems. Variables, expressions, and equations are used extensively to manipulate data and control program flow.

Conclusion

Representing relationships is a fundamental skill that forms the bedrock of mathematical understanding and problem-solving. By mastering this skill, students gain the ability to translate real-world scenarios into mathematical language, enabling them to analyze, interpret, and solve complex problems. Through careful reading, clear variable definitions, and a systematic approach, anyone can develop proficiency in representing relationships and unlock the power of mathematical thinking. The practice problems presented here serve as a starting point for honing this skill, and continued practice will lead to greater confidence and expertise. Remember to always check your work and be mindful of common mistakes. With dedication and perseverance, you can master the art of representing relationships and unlock a deeper understanding of the world around you.