Lesson 25

This activity encourages students to apply ratio reasoning to solve a problem they might encounter naturally outside a mathematics classroom. The warm up invites open-ended thinking that is validated by mathematical reasoning, which is the type of complex thinking needed to solve Fermi problems in the following activities.

Arrange students in groups of 2. Display the image for all to see.

Optionally, instead of the abstract image, you could bring in a clear glass, milk, and cocoa powder. Pour 1 cup of milk into the glass, add 5 tablespoons of cocoa powder, and introduce the task that way.

Tell students to give a signal when they have an answer and a strategy. Give students 2 minutes of quiet think time.

Invite students to share their strategies with the class and record them for all to see. After each explanation, ask the class if they agree or disagree and how they know two hot cocoas will taste the same.

In this activity, students are introduced to the type of thinking useful for Fermi problems. The purpose of this activity is not to come up with an answer, but rather to see different ways to break a Fermi problem down into smaller questions that can be measured, estimated, or calculated.

Much of the appeal of Fermi problems is in making estimates for things that in modern times we could easily look up. To make this lesson more fun and interesting, challenge students to work without performing any internet searches.

As students work, notice the range of their estimates and the sub-questions they formulate to help them answer the large questions. 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 calculated?
• What assumptions should we make?

Open the activity with one or two questions that your students may find thought-provoking. Some ideas:

• “How many times does your heart beat in a year?”
• “How many hours of television do you watch in a year?”
• “Some research has shown that it takes 10,000 hours of practice for a person to achieve the highest level of performance in any field—sports, music, art, chess, programming, etc. If you aspire to be a top performer in a field you love—as Michael Jordan in basketball, Tiger Woods in golf, Maya Angelou in literature, etc., how many years would it take you to meet that 10,000-hour benchmark if you start now? How old would you be?”

Give students a moment to ponder a question and make a rough estimate. Then, share that the questions above are called “Fermi problems,” named after Enrico Fermi, an Italian physicist who loved to think up and discuss problems that are impossible to measure directly, but can be roughly estimated using known facts and calculations. Here are some other examples of Fermi problems:

• “How long would it take to paddle across the Pacific Ocean?”
• “How much would it cost to replace all the windows on all the buildings in the United States?”

Share the questions above or select a few other Fermi-type questions that are likely to intrigue your students. Have some resources on hand to support the investigation on your chosen questions (e.g., have a globe handy if the question about paddling across the Pacific is on your short list). As a class, decide on one question to pursue. For this activity, consider giving students the option to either work independently or in groups of two.

First, ask students to share their estimates. Note the lowest and highest estimates, and point out that it is perfectly acceptable for an estimate to be expressed as a range of values rather than a single value.

Ask students to share some of their smaller questions. Then, discuss how you might come up with answers to these smaller questions, which likely revolve around what information is known, can be measured, or can be computed. Also, discuss how our assumptions about the situation affect how we solve the problem.

This activity asks students to choose or pose a Fermi problem and solve it, with the aim of promoting the reasoning and tools developed in this unit. Students brainstorm potential problems, choose one, and—after your review—use a graphic organizer to help them formulate the sub-questions that could support their problem solving. They go on to solve their chosen Fermi problem.

To encourage ratio reasoning and the use of tools such as double number lines and tables, look for problems that involve two quantities. Questions that involve one quantity can be solved with multi-step multiplication and without ratio reasoning (e.g., “How many pens are there at the school?” involves only one quantity—the number of pens). But a problem such as “How much would it cost to replace all the windows on all buildings in the U.S.?” or “How long would it take to paddle across the Pacific Ocean?” involves accounting for two quantities at the same time (cost and number of windows, or time and distance across the Pacific) and is more likely to elicit ratio reasoning. Keep this in mind as you help students sift through their ideas.

Explain to students that they will now brainstorm some Fermi problems they are interested in answering and select one to solve. Consider sharing a few more examples of Fermi problems to jumpstart their thinking:

• How much would it cost to charge all the students’ cell phones in the school for a month?
• How much does it cost to operate a car for a year?
• How long would it take to make a sandwich for everyone living in our town?
• How long would it take to read the dictionary out loud?
• How long would it take to give every dog in America a bath?

Tell students that once they have a few good ideas, they should pause and get your attention so that you could help to decide on the one problem to pursue.

Arrange students in groups of 2, if desired. Provide tools for creating a visual display.

Students may think of problems that do not lend themselves to ratio reasoning because they only involve one quantity. If they have trouble coming up with any good options, offer them some examples. It may also be helpful to have a list of sample problems that students could refer to in creating their own problem.

Display students’ posters or visual presentations throughout the classroom. Consider asking some students (or all, if time permits) to present their problems and solutions to the class. Notice and highlight instances of ratio and rate reasoning, particularly productive use of double number lines or tables.