A View from the Back of the Envelope top

Fermi Questions
On Being Approximate
Bounding uncertainty
Honesty in approximation
Deep understanding
Rough quantitative estimates

"... the estimation of rough but quantitative answers to unexpected questions about many aspects of the natural world. The method was the common and frequently amusing practice of Enrico Fermi, perhaps the most widely creative physicist of our times. Fermi delighted to think up and at once to discuss and to answer questions which drew upon deep understanding of the world, upon everyday experience, and upon the ability to make rough approximations, inspired guesses, and statistical estimates from very little data." [Philip Morrison [1]]

The idea is to use approximate calculation with landmarks and deep understanding, to explore the world and your understanding of it. Honesty is important.

Fermi questions are also called Fermi problems. I believe questions better captures the original usage and flavor, and my current intent.

Fermi problems are now being used from first grade (kindergarten?) through graduate school. And with families.

"What order of magnitude is ...?" game
A Pinocchio estimation game


Fermi Problems by Sheila Talamo
Back-of-the-Envelope Estimates from High School Mathematics at Work
Fermi Off-The-Wall Math League [grades 1-6]
"Plausible Estimation" Tasks (all in one page, .txt version)
Science Olympics Fermi Questions
Fermi Estimates II originally from Balanced Assessment in Mathematics. [more links below]
Fermi Problems from Fundamental Concepts of ?? Century Physics
Problem Solving in Engineering Technology: Creativity, estimation and critical thinking are essential skills [original .DOC]
Teaching Values Through A Problem-Solving Approach To Mathematics
How many ping-pong balls fit in a room? [PDF] from here.
Brain Teasers
`Back-of-the-Envelope' Calculations (The Seven Habits of Highly Effective Astronomers)
A Garden of Fermi Problems [with demo]
Week 3 Discussion Problems [with answers][link broken]
University of Maryland Fermi Problems Site
Old Dominion University Fermi Problems Site

Also:  ( stories: Colleagues, friends share memories (also here)
intros: and estimation | Fermi Problems | Back-of-the-Envelope Analysis [PDF] | M&M lesson | another[link broken] | another [at end] | The Back of the Envelope (Column 7 of Programming Pearls) | another | another
how to: plan | a short demo | another demo | solving | doing | writing | a few facts
questions: some 30 fermis | Fermi Problems | Physics 341 Homework Set #1 [PDF] | questions [PDF] from here | Estimation Problems | more[link broken] | more | more | Physics 309 homework | more | more
education: amusement park math/science day (rules[link broken]) | Mystery Halo Star etal in University Role-Playing Exercises for Teaching Astronomy and Physics, and a paper, from here | Something short but pictureful (a variant) | matthew's Physics 103 [some flaws, but innovative] | Metric Estimation Game | Fermi Estimates II's scoring, home, and other activities - search for Fermi [only student handouts are available without a password :(] [DOC] | possible issues | Order-of-Magnitude Astrophysics | Estimation Skills (with discussion) | FNAL's Enrico Fermi Problem-Solving Teacher Workshop (K-12), announcement , and family Fermi Problem-Solving Saturdays (families, grades 1-9) | Fermi Math - Grade 3 | pizza meeting
olympiad: Olympiad questions | different Olympic rules, and another
examples: Cooling a house with ice )
Feynman's Cargo Cult Science. On the need for scientific integrity, for bend-over-backwards honesty. Approximate reasoning can quickly become a farce without it.
An experiment with Mayonnaise, illustrating kitchen use of Fermi question-like reasoning.
How Big Are Things? [intended to help with "size" FermiQ's]
More generally: Quantitative Literacy/Reasoning | Math in the Real World


Some authentic Fermi questions (Fermi questions by Fermi):

[1] Philip Morrison, Letters to the Editor, Am. J. Phys., August 1963, v31n8 p626-627.

"[...] It is by no means possible to specify the training and readiness of a prospective graduate student by a mere list of topics. There is a kind of power over the theoretical and experimental studies in which he has engaged which is difficult to define, but whose presence is perhaps more important that much knowledge which is more formal and complete. There is one test for such power which is at the same time a remarkably apt method for its development. That is [... Fermi questions ...].

What is the photon flux at the eye from a faint visible star?
How far can a crow fly?
How many atoms could be reasonably claimed to belong to the jurisdiction of the United States?
What is the output power of a firefly, a French horn, an earthquake?

Such questions can of course be found for nearly any level of education. It should go without saying that no such question fulfills its purpose unless it is being heard for the first time. The accumulation of confidence and skill which such answers bring is a very good apprenticeship to research. Indeed, the conception of experiments and the formation of theoretical hypotheses are activities which are well simulated by asking and answering good Fermi questions."
[Morrison [1]]

A View from the Back of the Envelope
Comments encouraged. - Mitchell N Charity <mcharity@lcs.mit.edu>

  Re family Fermi Problem-Solving Saturdays,
    "They were fun and went over quite well."

  Introduce "FermiQ".
  Say numeracy.
  Link something about Problem Based Learning, Inquiry,
   using FermiQ's for Assessment, "number sense".
  www-ed.fnal.gov probably has something, but oy...
  Add http://www.simpson.edu/~gittinge/math_methods/Fermi_Problems.doc .
  soinc.org will "in the future" have a page on Olympiad trial/pilot events inc FermiQs.
  Pull links to related local pages into body of page, and gloss.
  "Fermi approximations" is used in [Programming Perls], but not elsewhere.

 2003-Feb-04  Flagged 4 newly broken links.
 2002-Oct-07  Added a link (actually 4).
 2002-Sep-14  Added 2+e links, note on breadth of current usage.
 2002-Sep-05  Fixed 5 links.  Removed 1.  Added many.  And reorganized.
              Removed ancient page clutter.  Fixed quote typo.
              All prompted by a reader's interest.
 2002-Jan-01  Updated a link.
 2001-Mar-13  Fixed a link.
 2000-Mar-14  Added `Order-of-Magnitude Astrophysics'.
 1999.Dec.19  Added phys-l link.
 1999.Nov.25  Added pfrancis link.
 1999.Nov.21  Added some links.  Fixed/removed others.
 1998.Aug.03  Added some links.
 1997.Aug.12  Added Mayonnaise link.
 1997.mid     Created.