Problem 81 - College Algebra

Question

A. Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned} -3(x-6) &=2-x \\ -3 x-18 &=2-x \\ -2 x-18 &=2 \\ -2 x &=-16 \\ x &=8 \end{aligned}$$ You should note that 8 checks, and the solution set is \(\\{8\\} .\) The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.

Step-by-Step Solution

Answer

The student made a crucial mistake in the very first step by incorrectly distributing the -3 within the parenthesis. Therefore, they wouldn't receive full credit. If each step worth 2 points, the student might receive 2 out of 10 points for correctly isolating x at the end after considering the magnitude of error.

Step 1: Check The First Step

The first line of the equation is given as -3(x - 6) = 2 - x. When you distribute -3 inside the parentheses you get -3x + 18 not -3x - 18 as the student has written.

Step 2: Check The Second Step

Since there is an error in step 1, further steps derived from this error are incorrect. The correct continuation should have been: -3x + 18 = 2 - x. If we add x to both sides we get -2x + 18 = 2.

Step 3: Check The Third Step

Now solving -2x + 18 = 2 we subtract 18 from both sides yielding -2x = -16.

Step 4: Check The Fourth Step

Finally solving for x, we divide by -2 on both sides, getting x = 8.

Step 5: Points Handling

Applications of the penalty should depend on the allocated points for each step. However, since the first error carried on to succeeding steps, the student should not get full credit. Also, as the error was due to a misunderstanding of algebra, it's quite significant.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements where two expressions are set equal to each other by an equals symbol. They play a central role in algebra because they allow us to solve for unknown variables. In the exercise, the equation begins as -3(x - 6) = 2 - x. To solve algebraic equations like this:

• First, simplify each side of the equation if necessary.
• Use inverse operations to isolate the variable on one side of the equation.
• Ensure that any operations applied to one side of the equation are applied to the other side as well.

The student correctly identified the goal to solve for the variable, but made an error during simplification. Understanding the rules governing algebraic equations helps prevent and identify such mistakes.

Distribution Property

The distribution property is a useful tool in algebra that enables us to simplify expressions and solve equations. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. This property is crucial in the exercise when determining how to manage the expression -3(x - 6). Instead of: -3(x - 6) = -3x - 18 The correct application is: -3(x - 6) = -3x + 18. To properly distribute a multiplication:

• Multiply the term outside the parenthesis by each term inside the parenthesis separately.
• Ensure that both terms inside are influenced by the outside term's sign (positive or negative).

Familiarity with the distribution property ensures accuracy when simplifying and equating mathematical expressions.

Point Deduction Rubric

When evaluating algebraic solutions, teachers often use a point deduction rubric to ensure fairness and consistency. The rubric outlines how many points are awarded or deducted based on specific criteria or steps in problem-solving. In this exercise:

• The error propagated from the incorrect distribution affects multiple steps, indicating a more pronounced misunderstanding.
• In grading, it's important to assess not just the final answer but also the logical process leading to that answer.
• Given the solution's progression, a significant penalty, such as deducting a substantial portion from the full 10 points, might be appropriate.

Using a deduction rubric helps clearly communicate errors to students and offers guidance on how they can improve their understanding in future problems.

Solution Validation

Solution validation involves confirming that the solving process is correct and the final answer is accurate. While the student eventually arrives at the right answer, it’s crucial to verify every step for validity. Here’s how solution validation can be effectively conducted:

• Review all arithmetic operations and steps for any errors or omissions.
• Check each transformation of the equation to ensure they adhere to algebraic rules.
• Recalculate any derived equations if steps are interdependent to catch trickle-down errors.

Despite reaching the correct answer, validation would uncover that the path to the solution contained errors. It's this thorough review process that emphasizes accurate methods, not just right answers.