Lesson 5: Gradient Basics (Rise/Run)

Learning Objectives

By the end of this lesson, students will be able to:

  • Describe gradient as a rate of change using rise and run
  • Identify positive, negative, zero, and undefined gradients
  • Estimate gradient from a graph using grid squares

What Is Gradient?

Gradient describes how steep a line is. It compares the vertical change (rise) to the horizontal change (run).

\[\text{gradient} = \frac{\text{rise}}{\text{run}}\]

Interpreting Gradient

  • Positive gradient: Line slopes upward from left to right
  • Negative gradient: Line slopes downward from left to right
  • Zero gradient: Horizontal line
  • Undefined gradient: Vertical line

Example 1: Rise/Run on a Grid

A line goes up 3 squares and right 6 squares.

Solution: \[m = \frac{3}{6} = \frac{1}{2}\]

Example 2: Negative Gradient

A line goes down 4 squares and right 2 squares.

Solution: \[m = \frac{-4}{2} = -2\]

Example 3: Zero and Undefined Gradient

  • A horizontal line has \(m = 0\).
  • A vertical line has an undefined gradient.

Practice Exercises

  1. Use rise/run to find the gradient:
    • Up 2, right 5
    • Down 6, right 3
    • Up 0, right 4
  2. State whether the gradient is positive, negative, zero, or undefined:
    • A line sloping down left to right
    • A horizontal line
    • A vertical line
  3. A line rises 8 and runs 2. What is the gradient?

Real-World Application

Gradient represents a rate of change. For example: - In a distance-time graph, gradient represents speed. - In a cost-quantity graph, gradient represents unit price.

Homework

Complete the gradient practice set.

  1. \(\frac{2}{5}\), \(-2\), \(0\)
  2. Negative, zero, undefined
  3. 4

Previous Lesson: Lesson 4: The Midpoint Formula
Next Lesson: Lesson 6: Gradient Using the Formula