Rates of Change

Most physical quantities change. The speed of a car changes with time; the size of a tumor changes with the quantity of radiation applied to it; and the price of oil changes as the supply of oil changes. Webster's New World Dictionary defines the word rate as the amount of anything in relation to something else.

In this first Pull-Out Lesson we will modify the dictionary definition to make it more precise mathematically: Rate is the change in the amount of anything in relation to the change in the amount of something else. We will look at rate as a measure of the change in distance relative to time. The student will be able to see how the familiar "rate= distance/time" is only one kind of rate of change of distance relative to time, the average rate; whereas, the speedometer reading on a car, plane, boat, etc. gives another form of rate of change of distance relative to time, the instantaneous rate. How does one determine the distance when only the ever-changing speedometer readings and time are known? An answer to this question is developed in detail.

This Pull-Out Lesson is designed to be used by teachers in the classroom. The material presented can be used at any level where the students have been exposed to some curve sketching, and reading material from a graph. With good students, even ninth graders should be able to handle the material here. 

Note: The information below was created with the assistance of AI.

Level of Mathematics
This lesson is suitable for secondary school students, specifically:

High school level (grades 9–12), depending on the student's prior knowledge.

Advanced middle school students (grade 8) may also access it with sufficient exposure to graphs and basic algebra.

The content does not assume knowledge of calculus, although it begins to introduce the concept of instantaneous rate of change, a foundational idea for calculus.

Application Areas
The lesson applies mathematics to several real-world domains:

Transportation: Speed and travel time scenarios using vehicles (cars, boats, airplanes).

Economics: Changes in per capita income, and company earnings over time (e.g., Volkswagen net earnings).

Physics and Engineering: Concepts related to motion, speed, and distance over time (inspired by Galileo's experiments).

Public Policy/Environment: Examination of U.S. gasoline production data over decades.

Prerequisites
Students should be familiar with:

Basic algebra: Particularly manipulation of formulas like 
rate = distance time rate = time distance.

Graph reading and interpretation: Understanding how to read both bar graphs and line graphs.

Concept of average vs. instantaneous rates: Even if intuitive rather than calculus-based.

Arithmetic operations and simple data analysis: Working with real data tables and extracting rates.

Subject Matter
Key mathematical ideas covered:

Rate of change: Defined rigorously as a change in one quantity relative to a change in another.

Average vs. instantaneous rate: Illustrated with speedometer readings and area under a curve approximations.

Graphical analysis: Visual interpretation of speed over time, use of bar and line graphs.

Units and dimensions: Emphasis on interpreting units in context (e.g., miles/hour, dollars/year).

Rectangular approximation method: Introduced informally as a way to estimate area under a curve.

Correlation to Mathematics Standards
While the document predates formal Common Core standards, it aligns well with several modern mathematical practices and content standards, particularly:

High School Algebra (Common Core)

HSA-CED.A.1: Create equations that describe numbers or relationships.

HSA-REI.D.11: Explain why the x-coordinates of the points where graphs of equations intersect are solutions of the equations.

HSA-APR.A.1: Interpret the structure of expressions.

Functions
HSF-IF.B.6: Calculate and interpret the average rate of change of a function over a specified interval.

HSF-LE.A.1b: Recognize that a quantity increasing linearly can be modeled with a linear function.

Mathematical Practices

MP2: Reason abstractly and quantitatively.

MP4: Model with mathematics.

MP5: Use appropriate tools strategically (e.g., graphs, tables).

MP6: Attend to precision, especially in interpreting units and context.