A View from the Back of the Envelope top

For Children
If you can't explain it to a nine year old...

[Another page in need of overhaul. The basic idea is order of magnitude calculation and perspective seems potentially accessible quite young.]

A hypothetical presentation sequence

  1. The concept of orders of magnitude
    Counting by tens
    "What order of magnitude is...?" game
  2. Simple approximate reasoning
    If a box is this wide, then its area and volume are... (m, m2, m3)
    Well, we know it has to be atleast this, and less than that. Now lets see if we can tighten these a bit more...
  3. Foundations and examples
    Scaling the universe to your desktop

Counting by tens

"One, ten, a hundred,
a thousand, ten thousand, a hundred thousand,
a million, ten million, a hundred million,
a billion, ten billion, a hundred billion."

Counting by tens
... on your fingers.

A feel for big numbers

Getting a feel for big numbers using grains of salt.

David M. Schwartz and Steven Kellog's children's picture book "How Much is a Million?" tries this...
"If one million kids climbed onto one another's shoulders, they would be...
taller than the tallest buildings,
higher than the highest mountains,
and further up than airplanes can fly.
If you wanted to count from one to one million...
it would take you about 23 days.
If a goldfish bowl were big enough for a million goldfish...
it would be large enough to hold a whale."
(Though I wonder whether one can really count on, and build up from, mental images of airplane hight, stadium and harbor volume, etc...)

Order of magnitude math (just + and -)

Can be done with just one or two digit addition and subtraction.

Exponential notation math

A place to practice single/few digit arithmatic.


"What order of magnitude is...?"


I once read a story which went vaguely like this (I would appreciate any pointers towards the original):

"A child asks an adult. "What is that?" The adult says, "I don't know, but it is good you asked. Asking questions is very important. It is how you learn things.". Later the child asks "What does that do?". "I don't know." says the adult. "How does that work?" "I have no idea, but it is good you are asking questions." "How big is that?" "I don't know." the response. How long was it before the child stopped asking questions?"

One advantage of the back of the envelope is that it makes many things accessible. One can often estimate things to an order of magnitude just by thinking about them. Things where obtaining the "real" values might take days or months of research.

How young?

It seems much of this could be accessible in 3rd grade (8 years old), and some of it earlier.
(Just from looking at the skill sets in a mathematics curriculum.[link broken])

The Fermi Off-The-Wall Math League includes grades 1-6.


Education and Outreach by ATLAS Institutions, Rutherfoord of Arizona


David M. Schwartz and Steven Kellog's children's picture book "How Much is a Million?".
(see Resources page)

Comparisons of distance, size, area, volume, mass, weight, density, energy, temperature, time, speed
Diagram Group.
(see Resources page)

Beyond Facts & Flashcards: Exploring math with your kids
Mokros, Jan; and TERC
"... offers parents a rich collection of games and activities that help children become successful math learners. The activities are engaging, easy to understand, and designed so that parents can incorporate them into the busy schedules of their daily lives."[back cover]
[One learns learn reading by "being read to every day and from having grown-ups listen to [your] reading." And one learns math the same way...]
(Not on the Resources page)

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

  2003-Feb-03   Added link to Fermi League.
  2003-Feb-03   Repaired links - 1 fixed, 1 flagged.
  1998.Jun.01   Added link to Rutherfoord's Atlas project.
  1997.Aug.27   Added `A presentation sequence'.
  1997.May/Jun  Previous significant revision.