Lesson 6

Fermi Problems

Lesson Narrative

This lesson is optional. The activities in this lesson plan are sometimes called “Fermi problems” after the famous physicist Enrico Fermi. A Fermi problem requires students to make a rough estimate for quantities that are difficult or impossible to measure directly. Often, they use rates and require several calculations with fractions and decimals, making them well-aligned to grade 7 work. Fermi problems are examples of mathematical modeling (MP4), because one must make simplifying assumptions, estimates, research, and decisions about which quantities are important and what mathematics to use. They also encourage students to attend to precision (MP6), because one must think carefully about how to appropriately report estimates and choose words carefully to describe the quantities.

In determining your exact age, the level of accuracy depends on how exact you know your moment of birth: to the day? the minute? the second? In determining the number of heartbeats in your lifetime, it is impossible to know the exact answer because we do not have access to all of the necessary information (and even if we did, the numbers involved are too large to count). In determining the number of hairs on your head, there is no method available other than counting, yet there are no tools to do this accurately. These two scenarios are far more difficult to estimate with the same degree of accuracy as your age. You can reliably determine your age within a day: it would be very difficult to estimate heartbeats or hairs with this level of accuracy.

Any of these tasks can stand on its own. Choose those that you have time for. It is likely that it would take more than a single day to do all of the tasks in this lesson. Make sure to leave plenty of time for discussion. Important topics of discussion should include why the quantities in question are difficult to measure and the level of precision we should use to record our estimates.

As with all lessons in this unit, all related standards have been addressed in prior units. This lesson provides an optional opportunity to go deeper and make connections between domains.

Learning Goals

Teacher Facing

  • Calculate a rough estimate for quantities that are difficult or impossible to measure directly and explain (orally) the reasoning.
  • Choose an appropriate level of accuracy when reporting estimates of quantities.
  • Make simplifying assumptions to solve problems about estimating quantities.

Student Facing

Let’s estimate some quantities.

Required Preparation

During the A Heart Stoppingly Large Number activity, students have the option of measuring one another’s pulse rate. If measuring pulse rate, students will need access to stopwatches.

During the All Hairs on Your Head activity, students have the option of measuring their head. If measuring heads, students will need access to string and measuring tape marked in centimeters. 

CCSS Standards

Building On


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