Enrico Fermi is the father of “solving maths problems we will never know the exact answer to.” Such as how many leaves are on all the trees in Central Park. They are great for getting students to think mathematically and use problem-solving skills.

Fermi problems often require students to make reasonable assumptions and estimates about the situation to come up with an approximate answer.

Students should be reminded of the need to be able to explain and justify what they did when coming up with their solutions. Students’ answers may differ from each other, but if students have made sensible estimates and assumptions, then the different answers should be “close” to each other.

Take advantage of opportunities to discuss students’ different solutions and the effect of assumptions and estimates. You can also invent your own Fermi questions based on class experiences (e.g., after a trip to the zoo, you might ask students how many fish the seals consume in one year).

#### A COMPLETE FERMI TEACHING UNIT

Over 50 pages of engaging Fermi problems, teaching methods, tools and resources **NO PREP REQUIRED**. Your students will love the creativity, exploration and innovation of this unit.

*My students are super excited to start their first Fermi math question. I love this resource. Thank you.* **Tammy S**

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# Fermi Problems for Students

1) How many people could you fit into the classroom? How many soccer balls?

2) How old are you if you are a million seconds old? A million hours old? A million days old?

3) Could you fit $1,000,000 worth of $1 coins in your classroom? What about a billion dollars worth of $1 coins?

4) How much money is spent in the school canteen each day? In a week? Over the year?

5) If all the people in Australia joined hands and stretched themselves out in a straight line, how long would it reach?

6) How long would it take to count to a million?

7) If all the people in the world moved to Victoria, how crowded would it be?

8) How many cups of water are there in a bathtub? What about in an Olympic pool?

9) How many grains of rice are in a 10kg bag?

10) How many pages would be needed to show a million stars?

11) How many children are needed to have a mass the same as an elephant?

12) How many packets are needed to measure a single line of M&Ms to a distance of 100m?

13) How many jelly beans fill a bucket?

14) How long would it take to drive to the moon (if you could!)?

15) What is the total mass in kilograms of all the students in your school?

16) What is the weight of garbage thrown away by each family every year?

17) How many pizzas are eaten by our class in one year?

18) If you had a stack of $2 coins as tall as Mt Kosciusko, what would it be worth? Could you fit all the coins in your bedroom?

19) How far could you walk in one year?

20) How much water does your household use each week? Can you answer this without using a water bill?

21) How many blades of grass on a school oval?

22) Spend exactly $1,000,000 using things for sale in the newspaper

23) How much paper is used at our school each week?

24) Imagine the earth is at one end of the school oval, and the moon is at the other end. How far away is the sun?

25) How many beats will your heart make in a lifetime?

26) How many bricks are there in one wall of the classroom? The whole school?

27) How many books are read by children in our school/class in one year? About how many pages is that?

28) What distance will a ballpoint pen write?

29) How many times did the wheel of the bus turn on the class excursion?

30) How big a block of chocolate could you make using all the chocolate eaten by the class in a year?

31) How long would our class have to save to buy a car?

32) Get students to pose their own

questions …

Sharing and discussing strategies is paramount to this work.

**Some useful information:**

- The radius of the earth: about 6,400 km
- The distance of the earth from the sun: about 150 million km
- The distance of the moon from the earth: about 380,000 km
- The population of the world: about 6 billion
- The population of Australia: about 20 million
- The population of Melbourne: about 3.5 million
- Area of Tasmania: about 68000 square km
- Area of Victoria: about 228000 square km
- Area of Australia: about 7,700,000 sq. km
- Height of Mt Kosciusko: 2230m

## Fermi Question Examples with Answers

**How many piano tuners are there in New York City?****Answer:**Let’s estimate the population of New York City to be around 8 million people. Assuming a conservative estimate that 1 in 1,000 households owns a piano and that a piano needs tuning once a year, we can estimate there are approximately 8,000 piano owners. Given that a tuner can service about 100 pianos per year, we estimate there are about 80 piano tuners in New York City.

**How many basketballs can fit inside a school bus?****Answer:**A standard school bus is roughly 40 feet long, 8 feet wide, and 6 feet tall, giving it a volume of around 1920 cubic feet. A basketball has a volume of about 0.1 cubic feet. Estimating some empty space for stacking efficiency, we might fit around 15,000 basketballs inside the bus.

**How many liters of water does the Amazon River discharge into the ocean every second?****Answer:**The discharge rate of the Amazon River is estimated to be around 209,000 cubic meters per second. Converting this to litres gives us 209,000,000 litres per second.

**How many grains of sand are there on all the beaches in the world?****Answer:**Let’s estimate the total area of beaches worldwide and the average depth of sand. Assuming an average beach area of 100 meters by 100 meters and an average sand depth of 1 meter, we get 10,000 cubic meters of sand per beach. With approximately 372,000 miles of coastline worldwide, estimating an average beach density, we might arrive at a rough estimate of around 7.5 x 10^18 grains of sand.

**How many smartphones are currently in use worldwide?****Answer:**With an estimated global population of around 7.9 billion people and assuming smartphone penetration of around 60-70%, we can estimate the number of smartphones in use. Assuming 65% penetration, we’d have around 5.135 billion smartphones in use worldwide.

**How many words are spoken in a day worldwide?****Answer:**Let’s estimate the number of conversations people have daily and the average length of these conversations. Assuming each person has about 5 conversations per day, each lasting around 10 minutes, and estimating an average speaking rate of 150 words per minute, we’d have roughly 750 words spoken per conversation, totaling around 3,750 words per person per day. With a global population of approximately 7.9 billion people, we’d estimate around 29.6 trillion words spoken daily worldwide.

**How many breaths does a person take in a lifetime?****Answer:**Assuming an average lifespan of around 80 years and an average breathing rate of 12-20 breaths per minute, we can estimate the total number of breaths. Let’s use 15 breaths per minute for simplicity. With 60 minutes in an hour, 24 hours in a day, and 365 days in a year, we can calculate approximately 31,536,000 breaths per year. Over 80 years, this would amount to around 2.5 billion breaths in a lifetime.

## What is the Fermi Paradox?

The Fermi Paradox refers to the tension between estimates suggesting our galaxy is likely to harbor many interstellar civilizations and the fact that we have observed no such civilizations.

The solution to this paradox has implications for how we perceive the likelihood of us surviving as a civilization. It is far deeper than a simple math challenge.

The expected number of interstellar civilizations observable from Earth is the product of the number of planets in the galaxy and the following three probabilities:

- The probability that any given planet produces intelligent life
- The probability that any given intelligent species develops an interstellar civilization
- The probability that any given interstellar civilization would be detectable from Earth at this time

Since the number of planets in the galaxy is in the hundreds of billions, even very small values of the three probabilities would imply that we should see many interstellar civilizations. However, we observe none. This means that at least one of the probabilities must be *extremely* low.

You can learn more about the Fermi Paradox here

### 5 Ways Fermi Problems Help Students Become Problem Solvers

**The Power of Fermi Problems in Student Development**

In the realm of education, there are few tools as potent or transformative as the **Fermi problem**. Named after the prodigious Italian physicist Enrico Fermi, these seemingly improbable questions—like “How many piano tuners are there in Chicago?”—can transform students into a new breed of problem solvers and critical thinkers.

#### The Magic of Fermi Problems

Fermi problems are unique—they blur the line between science, math, and reality in an obscure yet fascinating way. It’s here, in this nebulous juncture, that students shine their analytical prowess, squirrelling out data, making educated guesses, and calculating odds—all skills necessary for effective problem-solving.

#### Encouraging Estimation and Logical Reasoning

We’ve all heard the age-old adage—”Estimate, don’t calculate.” Well, Fermi problems breathe life into this wisdom. Students are compelled to rely on intuition and logic rather than precise calculations. They learn to make educated guesses—a skill that goes a long way in lateral thinking.

#### Fostering Mental Flexibility

With Fermi problems, there isn’t necessarily a ‘right’ answer but rather a range of possible answers. It’s a game of ballpark figures and probabilities. This openness encourages mental flexibility, allowing students to view problems from multiple perspectives.

#### Unleashing Creativity With Fermi Problems

The beauty of Fermi problems lies not in finding the correct answer, but in the journey of discovery. This path nudges students towards creative thinking. It encourages them to devise their own methods and routes to reach plausible solutions—an essential trait in innovative problem-solving.

#### Encouraging Interdisciplinary Connections

While seemingly rooted in mathematics, Fermi problems defy categorization. They weave a web across various disciplines, touching upon geography, population statistics, and physical principles. Students are encouraged to draw from their existing knowledge across different subjects, fostering a sense of interconnected learning.

#### Shaping Independent Thinkers

By introducing Fermi problems in classrooms, we’re not just honing problem-solving skills; we’re molding independent thinkers. While working on these problems, students become adept at dissecting complex questions, identifying crucial elements, and conducting their research. These transferable skills will help them tackle real-world issues with confidence.

#### Fermi Problems as A Stepping Stone

It’s clear that Fermi problems bring an abundance of benefits for students. By embracing these challenges, they develop vital skills—logical reasoning, mental flexibility, creativity, and independence—all cornerstones of an effective problem solver. It might seem like we’re merely estimating the number of piano tuners in Chicago, but in reality, we’re paving the way for better problem solvers, critical thinkers, and bright minds.

## Characteristics of Fermi Problems

**Rough Estimation**

Fermi problems involve making rough estimates rather than precise calculations. The goal is to obtain an order-of-magnitude answer rather than an exact value.

**Limited Information**

**Limited Information**

Fermi problems typically provide limited information, requiring problem solvers to make reasonable assumptions and fill in the gaps using their knowledge and intuition.

**Logical Assumptions**

**Logical Assumptions**

Solvers are encouraged to make logical and justifiable assumptions to simplify the problem. These assumptions should be based on common sense and prior knowledge.

**Quick Thinking**

**Quick Thinking**

Fermi problems require a rapid thought process. Solvers should be able to make decisions quickly, employing their intuition and knowledge to arrive at reasonable estimates without lengthy calculations.

**Real World-Applicability**

**Real World-Applicability**

Fermi problems model real-world situations where complete information is often unavailable. The skills developed through Fermi problem-solving are applicable in a wide range of fields and practical scenarios.

## Q/A Section

**Q: What is a Fermi problem?**

A: Named after physicist Enrico Fermi, a Fermi problem is a question that requires estimation skills and logical reasoning to solve. The solution isn’t exact but lies within a range of probable answers.

**Q: How does solving Fermi problems foster mental flexibility?**

A: Fermi problems don’t have one ‘right’ answer, pushing the students to consider multiple scenarios and possibilities. This encourages them to view problems from various perspectives, fostering mental flexibility.

**Q: Why are Fermi problems considered interdisciplinary?**

A: Although they are often tied to math or physics, Fermi problems require knowledge from various disciplines like geography, population statistics, or general world facts. Hence, they promote an interdisciplinary approach.