Objective: The goal of this demo is to provide students with an opportunity to use information on average rates of change to create a story about the workout of a jogger from a time vs. distance graph of the jogger's run. Level: Precalculus and calculus courses in high school or college. Prerequisites: Familiarity with the concept of slope of a line, computing the slope of a line, and average rate of change. Platform: Included is an Excel worksheet that will work on both a PC and a MAC, together with an animation in both Flash and QuickTime. Instructor's Notes: Understanding and working with average rates of change are stepping stones to the basic calculus concepts of instantaneous rates of change and the fundamental concept of a limit. Thus it is important to provide a variety of learning experiences about average rates of change so that students understand this fundamental notion of change. This demo provides visual experiences which connect time vs. distance graphs for a jogger's run to the creation of short story that describes the jogger's workout. Average rate of change is often introduced by saying it is the change in distance over the change in time:
Let s denote distance and t denote time, then we use the symbols for change in distance and for change in time. Thus we have
In this demo we use such a computation step to compute the average velocity along segments of a jogger's path. Imagine that a jogger is going for a run along a path or around a track. Different joggers will vary the distance covered per unit of time and thus will cover portions of the route at different speeds. To keep things simple we will generate a time vs. distance graph which reflects different constant speeds over intervals of time. Thus the jogger's velocity over these sections is computed as an average rate of change. As the graph is generated we want to determine the average velocity for the various segments of the run. From this data students are asked to create a short story that describes the jogger’s workout using the changes in velocity. For the story they can use various situations such as jogging around a track, following a trail through a wooded area, or just running through a neighborhood. Example 1. In Figure 1 we show a typical time vs. distance graph from which the average velocity of the jogger over the various straight segments can be computed . With this information a story about the jogger's workout can be constructed.
■ Example 2. Here we provide an animation involving a time vs. distance graph for a jogger. We compute the average velocity along straight segments, and provide a possible story based on the information derived from the graph. This animation has both audio and captions. It is available in Flash and QuickTime formats. Figure 2 shows a typical portion of the graph with information about the average velocity of a segment.
Click one of the following to view the animation which contains audio. (To get the Flash player click here; to get the QuickTime player click here. ) ■ We have developed an Excel file that contains 6 time vs. distance graphs for joggers. In order to select a path enter a number from 1 to 6 in the box as shown in Figure 3.
Click here to execute or download the Excel file. Auxiliary Resources: The demo Average Rates of Change (click to go to this resource) provides students with a concrete understanding of the average rate of change for physical situations and for functions described in tabular or graphic form. It includes animations and examples of a falling ball and a moving car, as well as five interactive Excel programs for classroom or student use. Credits: This demo, the Excel files, and the animation were developed by David
R. Hill
and is included in Demos with Positive Impact with his permission. 

1/11/2007 Last updated 1/13/2007 DRH