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Fermi Problems - From Pianos to m&m'sBy Barbara Romack, Kaneland Elementary School, Elburn, IL
Fermi problems are questions which help students learn to think for themselves. They do not contain all the information you need to solve the questions exactly, but you are still expected to arrive at a reasonable answer. Fermi problems are named for Enrico Fermi, Nobel Prize winning physicist, who had the reputation for asking his students unusual and seemingly impossible questions. One classic Fermi problem is: How many piano tuners are there in Chicago?
Fermi's advice to his students was (1) break the problem into smaller, more manageable questions, and (2) have the courage to make some estimates and assumptions. Thus, in solving a Fermi problem, a number of possible strategies can lead to one reasonable answer. This is different from an open-ended problem which has a number of reasonable solutions.
Cooperative learning, problem-solving skills, estimation skills and number sense are important components of all Fermi problems. These activities would be appropriate for grades K-12. Using classroom activities built around Fermi problems will assist students in meeting NCTM Problem Solving Standards:
- Develop and apply strategies to solve a wide variety of problems
- Verify and interpret with respect to the original problem
- Formulate problems for everyday and mathematical situations
- Acquire confidence in using mathematics meaningfully
Objectives and Materials:
- Develop and apply strategies to solve a given problem; a 1-liter bottle, 3-4 large packages of m&ms
- Improve estimation skills; Paper & pencil
- Make decisions as part of a group; Ruler (may be needed)
- Communicate with group members and other groups; Scissors (may be needed) Calculator (may be needed)
Before class teacher fills the 1-liter bottle with m&ms. Save the remaining m&ms for use in finding possible solutions. Show bottle to class and ask:
How many m&ms fill a one-liter bottle?
Students look at bottle and write their estimates (guesses) on paper. The teacher asks students to give reason for the number they wrote down. The teacher and students would then count to see how reasonable their estimates were. (K-1 may wish to use a smaller bottle.)
- Use other candies or similar objects.
- Try other Fermi problems* - How many golf balls will fit into a suitcase?
- Have students create their own Fermi problems; then try to solve them.
* See resources and benchmarks.
- Students look at the bottle and work in groups of three or four to develop a strategy for solving one possible solution:
- Have students construct or visualize a paper cube that measures 1 cubic centimeter.
- How many m&ms will fit in the cube?
- Remind students 1 cc = 1 ml, and 1000 ml = 1 liter.
- How many m&ms in the 1-liter bottle?
____ cc = ____ ml = ____ m&msMiddle/High School Students:
In addition to previous activity, encourage students to use formulas and equations to arrive at estimates. Break down the big question into a series of smaller questions.
- What is the approximate size of an m&m?
- Do m&ms completely fill the liter bottle?
- The number of m&ms is the occupied volume of the jar divided by the volume of a single m&m.
- The volume of one m&m is approximated by the volume of a (shape)__cm long and ____ cm in diameter.
- Thus the approximate number of m&ms in the bottle is:
(this series of questions based upon the solution of "How many jelly beans fill a 1-liter bottle?" at http://forum.swarthmore.edu/workshops/sum96/interdisc/classicfermi.html)
MEANINGFUL m&m STATISTICS
"Mars, the maker of M& Ms, says that it produces the colored candies in the following proportions: 30 percent brown, 20 percent red, 20 percent yellow, 10 percent green, 10 percent orange, and 10 percent blue. The different colors are then all mixed together before packaging. In a perfect bag of 50, you'd have 15 brown, 10 red, 10 yellow, 5 green, 5 orange, and 5 blue."
Peterson, Ivars, "Food Counts," Muse, volume 3, no. 7, September 1999, p. 34.
Burns, Marilyn, The Book of Think, Little, Brown and Company, Boston, 1976. (ISBN 0-316-11742-0)
Burns, Marilyn, The I Hate Mathematics! Book, Little, Brown and Company, Boston, 1975. (ISBN 0-316-11740-4)
Burns, Marilyn, Math for Smarty Pants, Little, Brown and Company, Boston, 1982. (ISBN 0-316-11738-2)
Gittinger, Jack, Dept. of Education at Graceland College, "Methods of Teaching Elementary Math Education," @ http://www.graceland.edu/~jackg/math/links.html.*
Gleeson, Austin, Dept. of Physics, University of Texas at Austin, "Fermi Problems," Dimensional Analysis: The Beginning of Physics," @ http://www.ph.utexas.edu/~gleeson/httb/section1_3_3_.html.
Schwartz, David M., How Much Is a Million?, Lothrop, Lee & Shepard Books, New York, 1985. (ISBN 0-688-04049-7)