Jalen's Slope Calculation Error Analysis And Correction

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Calculating the slope of a line given two points is a fundamental concept in algebra. However, it's easy to make mistakes if the formula isn't applied carefully. In this article, we will analyze a step-by-step calculation of the slope, identify the error made, and understand the correct approach. This detailed explanation aims to provide clarity and ensure you can confidently calculate slopes in the future.

Problem Analysis

Jalen attempted to calculate the slope of a line passing through the points (−4,0)(-4, 0) and (2,3)(2, 3). Let's break down Jalen's steps:

Step 1: Points are identified as (−4,0)(-4, 0) and (2,3)(2, 3).

Step 2: −4−20−3\frac{-4 - 2}{0 - 3}

Step 3: −6−3\frac{-6}{-3}

Step 4: The slope is 2.

Our goal is to pinpoint the error in Jalen's calculation. The possible error provided are:

A. Jalen computed the ratio of change in xx to change in yy.

B. Jalen used the ordered pairs in the wrong order when calculating the differences.

Understanding the Slope Formula

The slope of a line, often denoted by m, measures its steepness and direction. It represents the change in the vertical coordinate (y-coordinate) for every unit change in the horizontal coordinate (x-coordinate). The formula for calculating the slope (m) given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2−y1x2−x1\qquad m = \frac{y_2 - y_1}{x_2 - x_1}

This formula represents the change in y (the rise) divided by the change in x (the run). It's crucial to maintain the order of subtraction consistently in both the numerator and the denominator.

Key Concepts in Slope Calculation

Before diving deeper into Jalen's mistake, let's reinforce the key concepts involved in calculating the slope:

  • Ordered Pairs: Points on a coordinate plane are represented as ordered pairs (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
  • Change in y (Rise): The difference between the y-coordinates of the two points, i.e., y2−y1y_2 - y_1.
  • Change in x (Run): The difference between the x-coordinates of the two points, i.e., x2−x1x_2 - x_1.
  • Consistency in Subtraction Order: The order of subtraction must be consistent. If you subtract y1y_1 from y2y_2 in the numerator, you must subtract x1x_1 from x2x_2 in the denominator.
  • Slope Interpretation: A positive slope indicates that the line is increasing (going upwards) from left to right. A negative slope indicates the line is decreasing (going downwards) from left to right. A slope of zero represents a horizontal line, while an undefined slope represents a vertical line.

Understanding these concepts is essential for accurate slope calculations and for interpreting the meaning of the slope in various contexts.

Identifying Jalen's Error

Let's revisit Jalen's steps and apply the slope formula correctly to identify the mistake.

Given points: (−4,0)(-4, 0) and (2,3)(2, 3).

Let's assign the coordinates:

  • (x1,y1)=(−4,0)(x_1, y_1) = (-4, 0)
  • (x2,y2)=(2,3)(x_2, y_2) = (2, 3)

Now, we apply the slope formula:

m=y2−y1x2−x1\qquad m = \frac{y_2 - y_1}{x_2 - x_1}

Substitute the values:

m=3−02−(−4)\qquad m = \frac{3 - 0}{2 - (-4)}

Simplify:

m=32+4\qquad m = \frac{3}{2 + 4}

m=36\qquad m = \frac{3}{6}

m=12\qquad m = \frac{1}{2}

The correct slope is 12\frac{1}{2}.

Now, let's examine Jalen's calculation again:

Step 2: −4−20−3\frac{-4 - 2}{0 - 3}

Here, Jalen calculated the difference in x-coordinates in the numerator and the difference in y-coordinates in the denominator. This is the reverse of the correct slope formula, which should have the difference in y-coordinates in the numerator and the difference in x-coordinates in the denominator.

Jalen computed the ratio of change in x to change in y, instead of change in y to change in x. This is the fundamental error in Jalen's calculation. The subsequent steps are arithmetically correct based on this initial error, but the result is incorrect because the formula was misapplied.

Detailed Analysis of Jalen's Steps

To further clarify Jalen's error, let's break down each step and compare it to the correct approach:

  1. Identifying Points: Jalen correctly identified the points as (-4, 0) and (2, 3). This is the first and crucial step in calculating the slope.
  2. Applying the Slope Formula (Incorrectly):
    • Jalen's Approach: −4−20−3\frac{-4 - 2}{0 - 3}. This step is where the primary error occurs. Jalen placed the difference in x-coordinates (-4 - 2) in the numerator and the difference in y-coordinates (0 - 3) in the denominator. This is the inverse of the slope formula.
    • Correct Approach: 3−02−(−4)\frac{3 - 0}{2 - (-4)}. The difference in y-coordinates (3 - 0) should be in the numerator, and the difference in x-coordinates (2 - (-4)) should be in the denominator.
  3. Simplifying the Expression:
    • Jalen's Approach: −6−3\frac{-6}{-3}. Based on the incorrect setup in Step 2, Jalen correctly simplified the expression. -4 - 2 = -6 and 0 - 3 = -3. The arithmetic here is correct, but it's based on the wrong initial setup.
    • Correct Approach: 32−(−4)=32+4=36\frac{3}{2 - (-4)} = \frac{3}{2 + 4} = \frac{3}{6}. The correct simplification follows from the correct application of the slope formula.
  4. Calculating the Slope:
    • Jalen's Approach: The slope is 2. Jalen correctly divided -6 by -3 to get 2. However, this result is incorrect because the initial formula application was wrong.
    • Correct Approach: The slope is 12\frac{1}{2}. Simplifying 36\frac{3}{6} gives the correct slope of 12\frac{1}{2}.

By carefully examining each step, it's clear that Jalen's error lies in inverting the slope formula. This detailed breakdown emphasizes the importance of understanding and correctly applying the slope formula to avoid similar mistakes.

Correcting Jalen's Error

To correct Jalen's error, we must use the correct slope formula:

m=y2−y1x2−x1\qquad m = \frac{y_2 - y_1}{x_2 - x_1}

Using the points (−4,0)(-4, 0) and (2,3)(2, 3), we have:

  • x1=−4x_1 = -4
  • y1=0y_1 = 0
  • x2=2x_2 = 2
  • y2=3y_2 = 3

Substitute these values into the slope formula:

m=3−02−(−4)\qquad m = \frac{3 - 0}{2 - (-4)}

Simplify the expression:

m=32+4\qquad m = \frac{3}{2 + 4}

m=36\qquad m = \frac{3}{6}

m=12\qquad m = \frac{1}{2}

The correct slope is 12\frac{1}{2}. This positive slope indicates that the line is increasing from left to right. For every 2 units we move to the right along the x-axis, we move 1 unit up along the y-axis.

Emphasizing the Importance of the Correct Formula

The mistake Jalen made highlights the critical importance of using the correct formula and understanding what each part of the formula represents. The slope formula is a ratio that compares the vertical change (rise) to the horizontal change (run). Inverting this ratio leads to an incorrect calculation of the slope. By consistently applying the formula and double-checking the setup, errors like this can be avoided.

Conclusion

Jalen's error was in computing the ratio of change in x to change in y instead of the correct ratio of change in y to change in x. Understanding and correctly applying the slope formula is crucial for accurate calculations. By remembering that the slope is the rise over the run (y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}), we can avoid this common mistake. The correct slope for the line passing through points (−4,0)(-4, 0) and (2,3)(2, 3) is 12\frac{1}{2}. This comprehensive analysis not only identifies the error but also reinforces the correct method for calculating slope, ensuring a solid understanding of this fundamental concept in mathematics.