Rate of Change or Slope

This page introduces the concept of slope as a rate of change in mathematics. It explains various ways to express and calculate slope, focusing on the "rise over run" method.

Definition: Slope is the ratio of a line's vertical change compared to its horizontal change.

The page presents multiple representations of slope: • As a ratio of rise to run • As "change in y over change in x" • Using the mathematical formula m = Δy / Δx

Vocabulary: Rise refers to the vertical change, while run refers to the horizontal change.

The document then provides a step-by-step guide for finding slope using two points:

1. Label the first ordered pair as (x₁, y₁)
2. Label the second ordered pair as (x₂, y₂)
3. Substitute values into the slope formula and solve

Example: For points (4,4) and (0,1), the slope is calculated as m = (1-4) / (0-4) = -3/4

The page concludes with two more examples, demonstrating how to calculate slope for different sets of points, including cases resulting in negative slopes.

Highlight: Understanding how to calculate slope is crucial for analyzing linear relationships and graphing lines.

Practice Problems and Notes

This page offers a series of practice problems to reinforce the concepts of slope calculation and point-slope form equations.

The problems cover various scenarios:

1. Given a point and slope, write the equation in point-slope form and convert to slope-intercept form
2. Given two points, find the slope and write the equation in both point-slope and slope-intercept forms
3. Apply the concepts to a real-world situation involving temperature change over time

Example: For the point (0, -9) and slope 4, the solution progresses from y + 9 = 4(x - 0) to y = 4x + 9

The page provides step-by-step solutions for each problem, demonstrating the process of: • Identifying given information • Calculating slope when necessary • Writing equations in point-slope form • Converting equations to slope-intercept form

Highlight: Practice problems help solidify understanding of how to find slope using two points and how to apply this knowledge to create and manipulate linear equations.

The final problem presents a real-world application, asking students to model the temperature change in a pond over time using a linear equation. This demonstrates the practical utility of understanding slope and linear equations in scientific contexts.