How Much Work is Required: Intuition vs. Mathematical Calculation
This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with a simple question concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations.
Students are given opportunities to address this question by means of a ConcepTest, a Question of the Day and a write-pair-share activity. The results are quite revealing and show that while students may have learned how to perform the necessary calculations, their conceptual understanding concerning work may remain faulty.
- develop their understanding of the concept of work as a product of force and distance
- exercise their mathematical intuition and verify it via appropriate calculations
- recognize and correct a common misconception concerning work
- recognize and correct a common calculation error involving work
Context for Use
The activity is comprised of five segments: Flash tutorials, ConcepTest, Question of the Day, write-pair-share activity and conclusion. The time required for the entire activity is approximately 45-50 minutes but fewer segments can be offered as a shorter alternative (see activity description below for individual segment times).
Description and Teaching Materials
- Instructor presents Flash tutorials on work and pumping liquids. (~10-15 minutes) Note: This segment may be assigned for homework rather than presented in class.
- Applications of Integration-Work http://archives.math.utk.edu/visual.calculus/5/work.1/index.html
- MERLOT description of this resource http://www.merlot.org/artifact/ArtifactDetail.po?oid=1010000000000322872
- Afterwards, a ConcepTest (Acrobat (PDF) 13kB Jul25 06) in the form of a straw poll (either show-of-hands or written) is presented concerning the amount of work required to pump the liquid out of each of two right circular cones of the same shape and size that are filled with water. One of the cones has its narrow end pointed downwards while the other has its narrow end pointed upwards. Each student is asked to make a conjecture and the instructor records the results for the class to see. (~5 minutes)
- After the straw poll, the question becomes the Question of the Day (Acrobat (PDF) 16kB Jul25 06) and students work in pairs to share and explain their reasoning in written form. (~10 minutes)
- Lastly, students are asked to carry out the necessary calculations in a write-pair-share activity (Acrobat (PDF) 19kB Jul25 06) with the help of a graphing calculator to verify their conjectures. (~10 minutes)
- In conclusion, the instructor presents a summary of student experiences and points out the reasons behind the common misconceptions and calculation errors. (~10 minutes)
Teaching Notes and Tips
Another common error comes to light in the write-pair-share activity when the students perform the calculations necessary to find the amounts of work required. Some who had the correct intuitive understanding may be baffled by the seeming fact that their calculations don't agree with their intuition. What they don't realize is that they have made a very common error in assuming that the ratios involving the radius and height in each cone are the same when in actuality they are different.
This is a good opportunity to allow students to wrestle with the contradiction between their intuition and the results of their calculations and to enter into a discussion about this issue in more general terms and about how to resolve such contradictions. Bringing these misconceptions to light and correcting them are both eye-opening and rewarding for students.
References and Resources
MERLOT description of the "Visual Calculus: Applications of Integration-Work" resource that is used in this activity.
Direct link to Visual Calculus which includes tutorials and animations (Flash and Java) that are helpful for students who are visual learners.