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 Lesson Plan: The Slope Formula

 

Lesson Summary

This lesson introduces students to the slope formula, an essential tool in algebra for determining the steepness and direction of a line between two points on a coordinate plane.

Building upon prior knowledge of slope concepts and types, students will:

  • Understand the Slope Formula: Learn that the slope (\( m \)) is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points, expressed algebraically as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  • Apply the Formula: Use the coordinates of two points to compute the slope, recognizing how changes in these coordinates affect the slope’s value and sign.
  • Analyze Different Slopes: Identify and interpret positive, negative, zero, and undefined slopes, understanding their graphical representations and real-world implications.

Lesson Objectives

  • Define the slope formula and how to use it
  • Develop an algebraic understanding of slope

Common Core Standards

  • CCSS.MATH.CONTENT.8.EE.B.6: Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
  • CCSS.MATH.CONTENT.8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.

Prerequisite Skills

  • Understanding the concept of slope
  • Identifying different types of slopes (positive, negative, zero, undefined)
  • Visualizing slope on a graph
  • Recognizing similar triangles and their properties

Key Vocabulary

  • Slope (\( m \)): A measure of a line’s steepness, representing the ratio of vertical change to horizontal change between two points.
  • Rise: The vertical change between two points on a line.
  • Run: The horizontal change between two points on a line.
  • Slope Formula: The equation \( m = \frac{y_2 - y_1}{x_2 - x_1} \) used to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
  • Positive Slope: A slope where \( m > 0 \), indicating the line rises from left to right.
  • Negative Slope: A slope where \( m < 0 \), indicating the line falls from left to right.
  • Zero Slope: A slope where \( m = 0 \), indicating a horizontal line.
  • Undefined Slope: A slope that occurs when the run (\( x_2 - x_1 \)) is zero, indicating a vertical line.
  • Coordinate Plane: A two-dimensional plane formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis).
  • Ordered Pair: A pair of numbers \( (x, y) \) that represent a point’s location on the coordinate plane.

 Multimedia Resources

 

 

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Visualizing Slopes

Review the previous lesson on Types of Slopes. Ask students to identify the slope of various line segments shown on the board or projector. Make the following observations:

  • Previous ways of calculating slope were more geometric and visual (coordinate grids, measuring the rise and run).
  • What if the only information you have about two points are their coordinates? How could you calculate the slope?

Activity 2: Review of Ratios

Since the slope formula is essentially a ratio, a brief review of ratios will help students better understand its structure. Start by asking students to define a ratio and provide examples from real life, such as:

  • The ratio of boys to girls in a classroom.
  • The ratio of red to blue marbles in a bag.
  • The ratio of wins to losses for a sports team.

Next, have students simplify given ratios and express them in fraction form. Examples:

  • Write the ratio 6:9 as a fraction and simplify it.
  • Express the ratio 12:16 as a fraction in simplest form.

Emphasize that a ratio compares two quantities and that the slope formula follows the same structure, comparing the vertical change (rise) to the horizontal change (run).

Activity 3: Using Desmos to Express Ratios as Fractions in Simplest Form

In this activity, students will use the Desmos Scientific Calculator to express given ratios as simplified fractions.

Steps:

  1. Have students open the Desmos Scientific Calculator.
  2. Provide a set of ratios and instruct students to enter them as fractions in Desmos. For example:
    • \( \frac{8}{12} \) → The calculator simplifies it to \( \frac{2}{3} \).
    • \( \frac{15}{25} \) → The calculator simplifies it to \( \frac{3}{5} \).
  3. Ask students to compare their manually simplified fractions with the Desmos output.
  4. Discuss how simplifying ratios connects to finding the slope between two points.

This warm-up will help students build fluency in simplifying ratios, reinforcing their understanding of slope as a ratio of changes in y and x.

 

Slope

 

 

Teach

Introduction

In this lesson, students will learn how to calculate the slope of a line using the slope formula. The concept of slope is fundamental in algebra and geometry, as it describes how a line changes as it moves along the coordinate plane. This lesson builds on students’ prior knowledge of ratios and coordinate points to help them understand how slope is determined and interpreted.

Key Concepts

  • Definition of Slope: Slope (\( m \)) measures the steepness and direction of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points.
  • Slope Formula: The formula for finding slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • Interpreting Slope:
    • Positive Slope: A line that rises from left to right (\( m > 0 \)).
    • Negative Slope: A line that falls from left to right (\( m < 0 \)).
    • Zero Slope: A horizontal line (\( m = 0 \)).
    • Undefined Slope: A vertical line where \( x_2 - x_1 = 0 \), making division by zero undefined.
  • Finding Slope from a Graph: Given a graph with two points, students can apply the formula by counting the rise and run visually before substituting values into the equation.
  • Real-World Applications: Understanding slope is essential in various fields, such as physics (motion and velocity), engineering (ramp inclines), and economics (rate of change in cost or revenue).

By the end of this lesson, students will be able to accurately compute the slope between two points and understand its significance in different contexts. This foundational knowledge will prepare them for more advanced topics such as linear equations and graphing lines in slope-intercept form.

Show students the definition of the slope formula: https://www.media4math.com/library/22185/asset-preview

Elaborate on the following points:

  • The slope formula provides an algebraic way of calculating slope.
  • The algebraic symbol for slope is m.
  • The slope formula uses the coordinates of the two points to calculate the slope.
  • The slope formula is the ratio of the change in the y-coordinates (the rise) to the change in the x-coordinates (the run)

Relate this to the concept of slope as the rise over the run (https://www.media4math.com/LessonPlans/WhatIsSlope).

Engage

Explain why someone would use the slope formula:

  • To precisely calculate the slope between any two points on a line
  • To determine if two lines are parallel (same slope) or perpendicular (negative reciprocal slopes)
  • To write the equation of a line when given two points it passes through

Explore

Show students the following math examples using the slope formula:

Additional examples can be found here: https://www.media4math.com/MathExamplesCollection--SlopeFormula to guide students through solving examples using the slope formula. Project or distribute the step-by-step examples, and have students follow along as you solve them together. 

You can also show the following video tutorials using the slope formula, which go through a step-by-step process for using the slope formula:

 Explain

Break down the slope formula and discuss how to plug in coordinates to find the slope. Emphasize the importance of the order of the points and how it affects the sign of the slope.

Example 1: Positive Slope

Find the slope of the line passing through the points \( (2,3) \) and \( (5,7) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (2,3) \)
    • \( (x_2, y_2) = (5,7) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{7 - 3}{5 - 2} = \frac{4}{3} \]
  4. Since \( m \) is positive, the line rises from left to right.

Example 2: Negative Slope

Find the slope of the line passing through the points \( (4,8) \) and \( (-2,12) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (4,8) \)
    • \( (x_2, y_2) = (-2,12) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{12 - 8}{-2 - 4} = \frac{4}{-6} \]
  4. Simplify the fraction: \[ m = -\frac{2}{3} \]
  5. Since the slope is negative, the line falls from left to right.

Example 3: Zero Slope

Find the slope of the line passing through the points \( (-3,5) \) and \( (4,5) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (-3,5) \)
    • \( (x_2, y_2) = (4,5) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{5 - 5}{4 - (-3)} = \frac{0}{7} = 0 \]
  4. Since the slope is zero, the line is horizontal.

Example 4: Undefined Slope

Find the slope of the line passing through the points \( (6,2) \) and \( (6,9) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (6,2) \)
    • \( (x_2, y_2) = (6,9) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{9 - 2}{6 - 6} = \frac{7}{0} \]
  4. Since division by zero is undefined, the slope is undefined, meaning the line is vertical.

Example 5: Real-World Application

Using a grid, a wheelchair ramp starts at point \( (0,0) \) and reaches a height of 3 feet at a horizontal distance of 12 feet. What is the slope of the ramp?

  1. Identify the coordinates:
    • \( (x_1, y_1) = (0,0) \) (starting point)
    • \( (x_2, y_2) = (12,3) \) (end point of the ramp)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{3 - 0}{12 - 0} = \frac{3}{12} \]
  4. Simplify the fraction: \[ m = \frac{1}{4} \]
  5. The ramp has a slope of \( \frac{1}{4} \), meaning it rises 1 foot for every 4 feet of horizontal distance.

 

 

Review

Summary of the Lesson

In this lesson, students learned how to determine the slope of a line using the slope formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The slope measures how steep a line is and whether it increases or decreases from left to right. The four types of slope are:

  • Positive slope: The line rises from left to right (\( m > 0 \)).
  • Negative slope: The line falls from left to right (\( m < 0 \)).
  • Zero slope: The line is horizontal (\( m = 0 \)).
  • Undefined slope: The line is vertical (division by zero, slope is undefined).

Students practiced finding slope from given coordinate pairs, interpreting slopes from graphs, and applying the concept to real-world problems.

Key Vocabulary

  • Slope (\( m \)): A measure of the steepness of a line, found using the ratio of vertical change (rise) to horizontal change (run).
  • Rise: The vertical change between two points on a line.
  • Run: The horizontal change between two points on a line.
  • Slope Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), used to calculate slope from two points.
  • Positive Slope: A line that rises from left to right.
  • Negative Slope: A line that falls from left to right.
  • Zero Slope: A line that is completely horizontal.
  • Undefined Slope: A line that is completely vertical, resulting in division by zero.
  • Coordinate Plane: A two-dimensional grid defined by an x-axis and a y-axis.
  • Ordered Pair: A set of values \( (x, y) \) that represent a point’s location on the coordinate plane.

Additional Examples

Example 1: Finding a Positive Slope

Find the slope of the line passing through the points \( (1,2) \) and \( (5,8) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (1,2) \)
    • \( (x_2, y_2) = (5,8) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{8 - 2}{5 - 1} = \frac{6}{4} \]
  4. Simplify the fraction: \[ m = \frac{3}{2} \]
  5. Since the slope is positive, the line rises from left to right.

Example 2: Finding a Negative Slope

Find the slope of the line passing through the points \( (3,5) \) and \( (-1,9) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (3,5) \)
    • \( (x_2, y_2) = (-1,9) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{9 - 5}{-1 - 3} = \frac{4}{-4} \]
  4. Simplify the fraction: \[ m = -1 \]
  5. Since the slope is negative, the line falls from left to right.

Example 3: Finding an Undefined Slope

Find the slope of the line passing through the points \( (6,3) \) and \( (6,-2) \).

  1. Identify the coordinates:
    • \( (x_1, y_1) = (6,3) \)
    • \( (x_2, y_2) = (6,-2) \)
  2. Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  3. Substitute the values: \[ m = \frac{-2 - 3}{6 - 6} = \frac{-5}{0} \]
  4. Since division by zero is undefined, the slope is undefined, meaning the line is vertical.

Multimedia Resources

Review slope and the slope formula with this presentation: https://www.media4math.com/library/21537/asset-preview.

 

 

Quiz

Answer the following questions.

  1. Find the slope of the line passing through the points (2, 3) and (5, 7).
  2. Calculate the slope of the line passing through the points (-4, 2) and (3, -5).
  3. Determine the slope of the line passing through the points (0, 0) and (6, 0).
  4. Find the slope of the line passing through the points (1, 4) and (1, -2).
  5. Calculate the slope of the line passing through the points (-2, 5) and (4, 5).
  6. Determine the slope of the line passing through the points (3, -1) and (3, 4).
  7. Find the slope of the line passing through the points (0, 0) and (0, 0).
  8. Calculate the slope of the line passing through the points (2, -3) and (-1, 4).
  9. Determine the slope of the line passing through the points (5, 2) and (-3, -4).
  10. Find the slope of the line passing through the points (4, 0) and (0, 3).

Answer Key

  1. Question 1: \( m = \frac{2}{5} \)
  2. Question 2: \( m = -3 \)
  3. Question 3: \( m = 0 \) (Zero Slope)
  4. Question 4: Undefined Slope
  5. Question 5: \( m = \frac{4}{7} \)
  6. Question 6: \( m = -\frac{5}{2} \)
  7. Question 7: \( m = 1 \)
  8. Question 8: \( m = \frac{3}{4} \)
  9. Question 9: Undefined Slope
  10. Question 10: \( m = -\frac{7}{3} \)