The Crusty Loaf of Bread: An Exploration of Area of a Surface of Revolution
This write-pair-share activity for Calculus II students involves a hypothetical hemispherical loaf of bread with a 12-inch diameter that has been sliced into twelve one-inch-thick slices. The objective is to determine which slice contains the most upper crust (i.e., most area of its surface of revolution). Contrary to students' intuition and conjectures, the answer is neither the slices at each end of the loaf nor the two congruent slices at the center of the loaf.
- develop their understanding of the concept of area of a surface of revolution
- exercise their mathematical intuition and verify it via appropriate calculations
- bring to light and ultimately correct a common misconception concerning the surface area of a hemisphere
Context for Use
The time required for the entire activity is approximately 20-25 minutes but fewer segments can be offered as a shorter alternative (see activity description below for individual segment times).
Description and Teaching Materials
- The instructor displays a rotatable 3-D graph of a hemisphere by using the graphing applet listed below and the related Graphing Guide (Acrobat (PDF) 108kB Jul25 06). (~5 minutes)
- Mathematics Visualization Toolkit (MVT)-an excellent set of graphing tools
Access path: Tools/Graphing Tools/Spherical Plotter
- MERLOT description of this resource
- Afterwards, students are given a ConcepTest (Acrobat (PDF) 11kB Jul25 06) in the form of a straw poll (either show-of-hands or written) concerning a fanciful story about a crusty loaf of bread and which slices of bread contain the most upper crust. Each student is asked to make a conjecture and the instructor records the results for the class to see. (~2 minutes)
- After the straw poll, the question becomes the Question of the Day (Acrobat (PDF) 11kB Jul25 06) and students work in pairs to share and explain their reasoning in written form. (~5 minutes)
- Lastly, students are asked to carry out the necessary calculations to determine the surface area of the upper crust of each slice in a write-pair-share activity (Acrobat (PDF) 15kB Jul25 06) with the help of a graphing calculator to verify their conjectures. (~5 minutes)
- In conclusion, the instructor presents a summary of student experiences and points out the reasons behind the common misconception via use of the Mathematics Visualization Toolkit (MVT) and the related Graphing Guide (Acrobat (PDF) 108kB Jul25 06). (~3-8 minutes)
Teaching Notes and Tips
This is clearly a case where intuition does not match with reality and students are pleased to reconcile the two and arrive at an understanding of the truth of the matter. The 3-D graphing tool helps them to see the truth of the matter more clearly.
References and Resources
MERLOT description of the "Mathematical Visualization Toolkit" that is used in this activity. Direct link to Mathematics Visualization Toolkit.
MERLOT description of MIT OpenCourseWare -- Calculus by Gilbert Strang that is used in this activity. Direct link to MIT OpenCourseWare site. This online textbook provides an explanation and examples of how to find the area of a surface of revolution. Access path: Chapter 8, Sections 8.1-8.3, 8.3 Area of a Surface of Revolution, p.325-327.
MERLOT description of "Wolfram MathWorld: An Online Mathematics Encyclopedia" that is used in this activity. Alternatively link directly to "Surface of Revolution" resources by Eric W. Weisstein located within the "Wolfram MathWorld: An Online Mathematics Encyclopedia."