Overview

The instructor begins with a short lecture or summary of the relationship between the velocity, position, and area under a curve. Students are then assigned into groups of 3-4 and move to interactive whiteboards, at which they will open the notebook file included in the activity package. This file is used to help students in calculating the area under the curve, and includes a slider to highlight the desired area. The instructor should demonstrate the first few points of the first problem so that students understand how the notebook file works.

The students calculate the area under the curve at ...

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The instructor begins with a short lecture or summary of the relationship between the velocity, position, and area under a curve. Students are then assigned into groups of 3-4 and move to interactive whiteboards, at which they will open the notebook file included in the activity package. This file is used to help students in calculating the area under the curve, and includes a slider to highlight the desired area. The instructor should demonstrate the first few points of the first problem so that students understand how the notebook file works.

The students calculate the area under the curve at each interval, then use this information to fill out the table used to calculate the position as a function of time. Groups then plot the position as a function of time, and discuss the differences between the two plots and the physical implications. During this time the instructor should monitor discussion.

The instructor reviews the correct solution with the whole class, highlighting the main areas of difficulty identified from the groups' discussions and progress. The exercise is then repeated for more and more difficult problems.

After all problems have been completed, the instructor can review and discuss the material. This can lead into an introduction to the anti-derivative by exploring the fact that the intervals could be made smaller to increase accuracy.

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Objectives

Students will learn the relationship between position and the area under a velocity-time curve. Students gain a more intuitive grasp of this relationship, and build the groundwork for the later introduction of integration.

Context and requirements

Level Grade 12-U0
Discipline Physics
Course Mechanics
Activity Content Kinematics, Calculating the area under a curve
Technological Requirements Interactive whiteboards using notebook are strongly recommended. An adaptation could be made by printing the notebook files as handouts, however this loses the interactivity of the files.
Best Use Preparation

Author’s Notes

Benefits

This activity helps students identify the area under a curve and talk about what it means physically. Students often have difficulties visualizing what this means, and this activity allows them to visualize everything at once. Additionally, this activity creates an artefact around which they can construct knowledge. It lays the groundwork for the introduction of integration, and even provides an opportunity to introduce (at a very basic level) the idea of the anti-derivative.

Challenges

Without the instructor demonstrating at least the first few points, students will have a great deal of difficulty understanding the process. As long as you have clear instructions and model this activity, there aren’t many challenges. It is important, however, to consolidate the process at regular intervals, interrupting the class to give the correct answer and discuss difficulties.

Tips

It is important to stress the fact that this area does not have units of a physical area. This helps students understand that this is not a physical area, but is instead a distance.

You do not need to wait until all groups have finished each problem before reviewing it, nor do students have to wait before moving on to the next. It can be helpful to create a sense of urgency by reviewing them before all groups are done. You should nonetheless be sure to review each problem before the fastest group is halfway through the next.

At the end of the activity, it can be helpful to talk about reducing the intervals over which the area is calculated to gain more accuracy, and tie this into a very brief introduction to the anti-derivative.

Activity Pedagogical Components

Problem Solving

Working in GROUPS, students calculate the area under the curve in each interval, using the tools provided in the notebook file. This work is completed IN CLASS.

Analysis

With the collected data, students working in their GROUPS iterate through the table to fill it out, plotting this data and discussing the physical implications of the plot. Students attempt to describe the motion to each other. This is completed IN CLASS.

Iteration

In GROUPS, students repeat the exercise for examples of increasing difficulty. This is completed IN CLASS.

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