8.2 Fermi Problems

Learning Goals

[Link to PlayPosit]

[Link to instructional video]

More details: [link to topic on SSS website]

After this lesson, students should

1. Be confident in their ability to make a reasonable magnitude estimate of quantities for which one has no intuitive guess or direct knowledge.
2. Identify quantities that would and would not be appropriate to estimate with a Fermi calculation.
3. Provide rough estimates for real-world quantities using “back-of-the-envelope” (Fermi) approximations.
4. Evaluate the credibility of quantitative statements using “back-of-the-envelope” approximations.
5. Use Fermi estimates to identify first, second, third order causes for example problems, and estimate their effect sizes.

Definitions

• Fermi Estimates

  A systematic estimate of a quantity based on what you know. The typical goal is to get within an order of magnitude of the right answer. (This often proves possible even for topics about which you know very little.) The steps to do this are the following:

  • Decompose the problem into multiple components that you can estimate. (Break down unfamiliar components into familiar components).
  • Estimate components using approximations.
  • Combine estimated components to calculate Fermi estimate.
  • Compute upper and lower bounds (maximum and minimum quantities above/below between which you are fairly confident the correct estimate should be).

• Order of Magnitude

  A factor of ten: If A is approximately 10 times larger than B, the A is one order of magnitude higher than B.

Examples

• [item 1]

Common Misconceptions

• Mistaken quote

  Explanation.

Context

[Explanation of how the current topic into the larger context of the course by explaining how the previous topic(s) relate to the current topic and leaving cliffhangers for future topics where relevant throughout the lesson]

Before

X.X Lesson Name

Explanation.

• Itemized explanation.

After

X.X Lesson Name

Explanation.

• Itemized explanation.

Recommended Outline

Before Class

• [Any essential logistical things that need to be done for this class]
• Prepare a seating chart.
• Review PlayPosit and discussion questions and ask faculty, Gabriel, or Emlen any questions you have.
• (Optional) Prepare a presentation.

During Class

• (5 min) Come up with some fun way to assign the roles of spokesperson and notetaker (e.g. earliest birthday in the year, lives furthest from campus). Remind them of the responsibilities of these roles.
• ([#] min) [Description of some module from #Lesson Content]
• (5 min) Collect questions for plenary.

After Class

• [Any essential logistical things that need to be done as followup for this class]
• Collect answers from notetakers for the forum / plenary.

Lesson Content

Clicker Question

1. Question
   1. Option 1
   2. Options 2

American Spending on Gasoline

In this activity, you break down a complex Fermi problem into its constituent parts on the whiteboard so that the students get to see how these get broken down. See minute 22 of this video from a previous year to see a demonstration.

Instructions

1. (30 seconds) Ask the students to quickly guess answers to the question, "How much do Americans spend on gas each year?"
2. (10 min) Break down the actual problem as shown below. You may draw this tree diagram step by step on a whiteboard. As you go through the thought process, have the students share their estimates for each of the quantities listed.

Estimation Steps

You can break down the estimation in the following way.

Government Spending

[Brief description of and motivation for the activity]

Discussion Questions

1. Question 1
   1. Subquestion a

1. 1. Subquestion b

Food Thrown in Landfills

[Brief description of and motivation for the activity]

Discussion Questions

1. Question 1
   1. Subquestion a

1. 1. Subquestion b

Collect Questions for Plenary

(5 min) Collect remaining questions from the students for faculty in plenary (can be questions for clarification, extension, discussion, etc.), and add [ here].