# Back-of-the-Envelope Calculations

Barbara Tewksbury, Hamilton College

Many numbers that we use in the geosciences are outside a student's frame of reference and personal experience. Using back-of-the envelope calculations that help put very large or very small numbers into perspective by analogy with something a student can visualize can accomplish a number of things:

• deepens a student's level of understanding (you know you've succeeded when a student sits back at the end of a calculation and says, "Wow..." in an overwhelmed voice!).
• gives students practice in estimating or seeking out reasonable starting numbers.
• gives students practice in simple quantitative tasks.
• encourages students to do this on their own in the future because it is simple and easy.

### Tips for using back-of-the-envelope calculations effectively:

• Choose calculations that allow students to gain a perspective on something (e.g., size, distance, rate) that is outside their normal frame of reference.
• Pick problems that are relevant to the course topic. Having students estimate the number of snowflakes that have ever fallen on Earth or the number of ping pong balls to fill a squash court might give them practice in estimation, but they're not problems that have much relevance.
• Ask students to predict an answer before calculating. This gives the instructor a sense of students' preconceptions.
• Choose calculations that involve at least some estimation. Otherwise, it's just an ordinary quantitative problem.
• Be sure that students have reasonable values for items that can't be estimated successfully. For example, if students are asked to estimate how high the Himalayas would be on an Earth scaled down to the size of a basketball or a person's head, students can successfully estimate the size of a basketball or a head. They need, however, reasonable numbers for the radius of the Earth and the height of the Himalayas. You can either provide the values or give students practice in looking them up.
• After the calculations are done, take the opportunity to have students explore how much difference it makes in the results to have estimated values differently.

### References and Resources

A View From the Back of the Envelope : "Fermi Questions" is another term for what we call Back of the Envelope Calculations. This site has more information on approximate calculations as well as many links to further resources.