In this activity, students are introduced to the type of thinking useful for solving Fermi problems. Students see different ways to break a Fermi problem down into smaller questions that can be measured, estimated, or calculated. Then, students work on answering those questions to solve the problem.

As students work, notice the range of their estimates and the sub-questions they formulate to help them solve the larger problem. Some examples of productive sub-questions might be:

• What information do we know?
• What information can be measured?
• What information cannot be measured but can be estimated?
• What information cannot be measured but can be calculated?
• What assumptions should we make?

Part of the appeal of Fermi problems is in making estimates for some things that in modern times we could easily look up. Challenge students to work without performing any internet searches.

If students are unsure how to break down the chosen Fermi problem into smaller sub-questions, consider demonstrating with a different example. Take this problem, for instance: “If you have a locker full of oranges, could you squeeze enough juice for everyone at your school?” Tell students that one thing we might want to know is the size of the locker: How big is it? Urge students to think of other questions that might be helpful to answer.

Here are some ideas:

• How many people are in the school?
• How many oranges are needed to get a cup of juice? 
• How large are the oranges?
• How much juice does each person get?
• Do adults and children get the same amount of juice?

Poll the class on their solution to the Fermi problem. Record and display the responses for all to see. It is highly unlikely that any two groups would make the exact same assumptions or use the exact same estimates, so different answers are to be expected. Discuss the variations in the solutions and likely reasons for them. If possible, display each group’s sub-questions and corresponding answers for all to see. 

A few key points to emphasize:

• There are multiple ways to go about solving a Fermi problem, all of which require making assumptions, estimating, measuring, calculating, relying on known facts, or a combination of these.
• The assumptions and estimates that we make about the situation in the problem affect how we solve the problem and the solution we get. 
• While the solutions to the same Fermi problem may vary a little or a lot depending on the solving process and the assumptions made, the range of solutions can help us see what answers might be realistic or reasonable.

This optional activity gives students an opportunity to apply the reasoning and tools developed in this unit to solve another Fermi problem. Students work with a partner to brainstorm a few new problems and select one to solve together.

To encourage ratio reasoning, look for problems that involve two or more quantities and that are related by several rates. The rates may be familiar (such as conversion across two units of measurement) or easy to find (such as the number of heartbeats in a unit of time). They may also be less straightforward and require estimation (such as the amount of time needed to paddle a unit of distance). 

When brainstorming, students may think of problems that involve a lot of steps but can be calculated with certainty. For instance, “How many minutes are in a decade with two leap years?” can be answered with multi-step multiplication. The question “How long would it take to show all the episodes of your favorite television series non-stop?” can be answered by adding up the durations of the episodes, information that is likely available online. Clarify that Fermi problems generally involve quantities that are unknown or impossible to measure directly, so it is necessary to make estimates and assumptions. This leads to a range of possible answers. Consider referring to a couple of Fermi problems in the Warm-up and discussing what needs to be estimated and assumed in each problem.

Ask some students (or all, if time permits) to present their problems and solutions to the class. 

If time allows, consider asking students to create a visual display of their work and then ask them to display them throughout the classroom. Invite students to quietly circulate and read at least 2 of the posters or visual presentations in the room. Ask students to consider what is the same and what is different about the sub-questions and solution methods for the different Fermi problems. 

Discuss any similarities and differences in the ways the Fermi problems were broken down and the smaller questions were answered. Highlight instances of ratio and rate reasoning, particularly productive use of double number lines or tables.