By Alessandra King, Posted January 5, 2015 – 

How many jelly beans does it take to fill a large jar? How many balloons would it take to fill the school gym? How many piano tuners are in your city? How many times does a human heart beat in a lifetime?

These are Fermi questions—estimation problems that foster quantitative understanding of the world around us and encourage creativity, communication, collaboration, and the ability to clearly identify initial assumptions. Discussion around this type of question is particularly effective when it highlights the weight that our assumptions have on the answer and the way modifying such assumptions affects our results. All too often our students are presented with word problems in which all the information is laid out for them; here, students have to not only look for the information they need but also find what information they need. In the process, they will be engaging in most of the CCSSM practices. There are plenty of examples of and resources for Fermi problems, including the NCTM Tips for Teachers (part 1), Math Forum, and the University of Maryland’s physics education site. 

Although the answers to some classic Fermi problems can be easily found online, any field that requires numerical estimates, such as biology, geography, environmental science, finance, economics, and even sports, can be a good source of problems. Fermi questions, therefore, can be used to encourage multidisciplinary, integrative problem solving and can help students discover connections among diverse content areas. In so doing, they will see mathematics and other disciplines “as permeating life and not just existing in isolation” (NCTM, Standards).

We usually have a short unit on simple Fermi questions when we study exponents and scientific notation. We then work on Fermi questions whenever the occasion arises during the school year because the students enjoy them. For example, when one student remarked that her Dad has told her something “a million times,” we jumped in to calculate how long that would in fact take.

And on a rainy day, we decided to estimate how many times the wipers had crossed the car windshield on the way to school.

Open-ended activities with multiple solutions teach students to work in teams by sharing their ideas, considering others’ perspectives, and holding discussions based on data and evidence. These activities also help students make sense of mathematics by exploring some important historical mathematical problems, such as Archimedes’s method for finding pi, the extension of the Pythagorean theorem (NCTM Problem SolvingStandards for Grade 6-8) (see fig. 1), the Lunes of Hippocrates (see fig. 2), and the Seven Bridges of Königsberg (see fig. 3). Other investigations that my students have enjoyed include a proportional representation of freshwater scarcity and a brief study of fractals. Finally, real-life applications, such as designing a vacation on a budget, scaling recipes for a class party (followed by a real class celebration, of course!), or budgeting for their first car (always a big hit!), nurture reasoning and communication of mathematical ideas.

It is hoped that these ideas will provide a Fermi basis for mathematical exploration in your classroom.

	Fig. 1 Extending the Pythagorean Theorem
	Fig. 2 Exploring the Lunes of Hippocrates
	Fig. 3 Fording the Königsberg Bridge problem

Alessandra King, [email protected], studies mathematics with her students at the Holton-Arms School in Bethesda, Maryland. She has taught mathematics and physics at the middle school and high school levels and is interested in creative problem solving, critical thinking, and quantitative reasoning.