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.

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

• Prepare a seating chart.
• Review PlayPosit and discussion questions and ask faculty, Gabriel, or Emlen any questions you have.

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.
• (20 min) Run through the gasoline spending activity.
• (45 min) Let the students work in small groups on the government spending activity.
• (5 min) Use any remaining time to work on the optional additional problems.
• (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?"
   1. Between zero and 100 trillion?
   2. Between 10 million and 10 trillion?
   3. Between 1 billion and 1 trillion?
   4. Between 100 billion and 1 trillion?
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 shout out 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

Additional Problems

(Optional given enough time) Have the students estimate in small groups any of the following quantities. Encourage them to give a lower and upper bound, as opposed to a single value.

1. Amount of food thrown in landfills in America every year.
2. The weight of all food an army battalion would have to bring on a 200-mile march across a rainforest.
3. The amount of water used to irrigate all the front lawns in Los Angeles in one year.
4. The total weight of tea (leaves) consumed by the British in one year.
5. The total amount of labour-hours needed to grade all the homework, projects, quizzes, and the final for L&S 22 in one semester. (How many 10 hrs/wk undergraduate readers do we have to hire to complete this task?)
6. As a laptop keyboard engineer, how many repeated key presses would you have to rate any individual key for, so that consumers do not typically encounter the malfunctioning of any key?

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].