The Crusty Loaf of Bread: An Exploration of Area of a Surface of Revolution

To enable students to:

• 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

This activity for Calculus II students can be carried out in either a small class or a large lecture setting anytime after the application of integration known as area of a surface of revolution has been covered. The activity is comprised of five segments: 3-D graphing applet introduction, ConcepTest, Question of the Day, write-pair-share activity and 3-D graphing applet conclusion.

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).

Activity description:

• 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
  amath.colorado.edu/java/mvt3.0/
  Access path: Tools/Graphing Tools/Spherical Plotter


• MERLOT description of this resource
  https://www.merlot.org/merlot/viewMaterial.htm?id=89767&hitlist=keywords%3DMathematics%2520Visualization%2520Toolkit&fromUnified=true


• 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)

Students are often incredulous when their calculations reveal that all slices have the same surface area and they are inclined to think that they have made an inadvertent error. This is due to the facts that their visualization of the loaf naturally tends to give more importance to the slices at the center and at the ends and that they don't realize that the surface curvature adds a considerable amount of crust to slices of smaller radius. It may also be due to the fact that the slices at the ends of most normally-shaped loaves of bread appear to be much "crustier" than the other slices.

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.

I generally give the students a grade for participation in this activity and sometimes grade their final calculations concerning surface area.

MERLOT description of the "Mathematical Visualization Toolkit" that is used in this activity. Direct link to Mathematics Visualization Toolkit: amath.colorado.edu/java/mvt3.0/

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."