Fermi Problems

30 min

Teacher Prep
Setup
Students in groups of 1–4. Provide access to four-function calculators and, optionally, internet-enabled devices and tools for creating a visual display. Around 15 minutes of quiet work time followed by a discussion or gallery walk.
Required Preparation

Internet-enabled devices are necessary only if students will conduct research to find quantities that they need to know. As an alternative, you can supply the information when they ask for it.

Tools for creating a visual display are needed only if you would like students to present their work in an organized way and have the option of conducting a Gallery Walk.

Narrative

This activity is an opportunity for students to apply their understanding of rates to solve a Fermi problem estimating the time it would take an ant to run from Los Angeles to New York City. To do so, students need to consider various factors that could affect the time of travel, such as the distance between the two cities, the speed of travel, whether breaks are involved, and so on. As they consider relevant factors and make a plan to approach the problem, students make sense of problems and persevere in solving them (MP1). In making estimates and performing calculations involving large numbers, students practice attending to precision (MP6).

Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:

  • The distance students use as the basis for their calculations.
  • The assumptions students make about the ant's speed, how it travels, and other external factors.
  • The units of measurement students choose for speed and distance.
  • Whether or how measurements or estimates are rounded and converted in the process of problem solving.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Arrange students in groups of 2. Introduce the context of an ant running from Los Angeles to New York City. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.
  • Display only the statement “an ant is running from Los Angeles to New York City” without revealing the question. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, and then ask them to compare their questions with a partner.
  • Invite several partners to share one question with the class, and record the responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as finding reasonable estimates.
  • Reveal the question “How long will the journey take?” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
    • “Which of your questions is most similar to or different from the one given? Why?”

Tell students that they will now answer the question about how much time it would take an ant to travel between the two cities. Remind students that a problem like this is called a Fermi problem—a problem that cannot be solved by measuring directly but can be answered by estimating and reasoning.

Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need. Provide access to four-function calculators.

Vital information to have on hand includes:

  • The distance between Los Angeles and New York City is about 3,944 km.
  • An ant can run about 18 mm per second.

If conducting a Gallery Walk at the end, provide access to tools for making a visual display.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide a Fermi problem graphic organizer for students to use as a reference.
Supports accessibility for: Memory, Organization

Student Task

An ant is running from Los Angeles to New York City. How long will the journey take?

Sample Response

Sample response:

It will take the ant about 7 years to run from Los Angeles to New York City without stopping. The ant can run 18 mm per second. There are 60 seconds per minute and 60 minutes per hour, so this is 18⋅60⋅60=64,80018\boldcdot 60 \boldcdot 60= 64,800, or 64,800 mm per hour. 64,800 mm per hour is 64.8 meters per hour, or 0.0648 km per hour. A speed of 0.0648 km per hour translates to a pace of 1÷0.06481 \div 0.0648 hours per km. So to travel 3,944 km, it would take 3944÷0.06483944\div 0.0648 or approximately 60,864 hours. There are 24 hours per day and 365 days per year, so dividing by 24 and then by 365 tells us that it will take the ant about 7 years to make this trip without stopping.

Activity Synthesis (Teacher Notes)

Invite students or groups to share different solution approaches. Alternatively, consider asking students to create a visual display and conducting a Gallery Walk. Highlight explanations or visual displays that include keeping careful track of information such as:

  • How far the ant has to travel.
  • How fast the ant travels.
  • A step-by-step analysis changing mm per second to km per day and eventually km per year.

The goal of this discussion is to make sure students understand that many different estimates are expected and correct. Here are some questions for discussion:

  • “Why did groups end up with different answers? Is that expected?” (It is expected because any approach involves different assumptions. Even a small difference in ant speed, when applied over such a long distance, makes a large difference in the final answer.)
  • “Did you calculate only the time the ant is moving? Or did you include time for the ant to rest?” (Our calculations include only the time the ant is moving. For a real ant, it would take longer to allow things like eating and resting.)
  • “Did you round any numbers while working on the problem? Which quantities did you end up rounding and why?” (We rounded the distance between the cities, the number of hours, and the number of years. Rounding some numbers to powers of 10 or to larger units made calculation and estimation easier. Larger units could also be more informative. For example, expressing travel time in days, weeks, or months might be more helpful than in minutes or seconds.)

Also of interest is the fact that ants do not live long enough to complete this trip. Many ants live for only a couple months. If students realize this, ask them how many ant lifetimes it would take for an ant to make this journey.

Anticipated Misconceptions

If students have trouble starting with no given information, consider asking: 

  • “What are some things you need to know to solve this problem?”
  • “What subquestions do you need to answer to solve this problem?”
Standards
Addressing
  • 6.NS.B·Compute fluently with multi-digit numbers and find common factors and multiples.
  • 6.NS.B·Compute fluently with multi-digit numbers and find common factors and multiples.
  • 6.RP.A·Understand ratio concepts and use ratio reasoning to solve problems.
  • 6.RP.A·Understand ratio concepts and use ratio reasoning to solve problems.

20 min

20 min