Distribute the Fermi Questions handouts to the students. Briefly introduce Enrico Fermi and Fermi Questions by reading and discussing the introductory page together. Explain that they will relate seemingly complicated questions to their everyday experiences. They will estimate by making a series of simple assumptions to arrive at a reasonable solution.
You can opt to choose an activity ahead of time, allow students to choose a topic as a whole group, allow small groups to create their own questions, or give students a limited set of options to choose from. These choices have different implications for the amount of time the activity will take and what materials might be needed.
The lab outlined here asks students to complete six steps for each Fermi Question:
Walk around and listen to students as they discuss and work through the problems, providing guidance as necessary. If students need more support, stop them after each step and have them share their work so far. Depending on the level of the students, it may be helpful to have each group turn in their work following each step so that you can verify that they are on the right track. This can also break up the process into smaller chunks of time.
A project of this type is a great opportunity to have students practice their written and verbal communication skills. Students often enjoy making a poster showing their findings, making a power point presentation, or creating a group report using a blog or a collaborative editor.
Here are sample reasoning processes for several Fermi Questions. Note that some of the estimates may not be accurate. The people making these estimates will need to gather additional information.
I think that each brick is about 6 inches long and about 3 inches high. I think that the school is about the length of a football field on each side. A football field is 100 yards or 300 feet. It would take 600 bricks to equal this length on each of the four sides of the building. I think the school is about 30 feet tall. It would take 4 bricks for each foot, so that means the school is about 120 bricks high. So each of the four sides of the school needs about 600 × 120 = 72,000 This means there are about 288,000 bricks.
It takes about 10 breaths to blow up a balloon the size of a two-liter bottle. So, that means I breathe about one liter of air for every five breaths.
I breathe about 10 times every minute, so I breathe about two liters of air every minute. This means that in an hour, I breathe about 120 liters of air. So each day, I breathe about 2,880 liters of air.
I think that a puffed kernel of popcorn occupies less than a cube which is a half inch on each side. This means that 8 pieces of popcorn should occupy each cubic inch. There are 12×12×12 cubic inches in a cubic foot. I will approximate that as 10×10×10 cubic inches since I am just estimating anyway. That means that there are about 1000 cubic inches in a cubic foot and about 8 pieces of popcorn in each cubic inch, so I have about 8,000 pieces of popcorn in each cubic foot.
The square ceiling tiles in our classroom seem to measure about 2 feet on each side. The room is 25 tiles long and 25 tiles wide, so the length and width of the classroom is approximately 50 feet by 50 feet for 2500 square feet. I think the classroom is probably about 2 of me tall, so the ceiling might be about 10 feet high. This gives a volume of about 25,000 cubic feet.
So about 25,000 × 8,000 = 200,000,000 kernels of popcorn would be required to fill the room.
I think that about half the people in the world have cell phones and my guess is that there are about 6 billion people in the world now. That means that 3 billion people have cell phones.
I use my cell phone for a total of about one hour each day. Some people use their phones more than I do and some people use their phones less than I do. That means that I use my phone about 1/24 of the minutes in a day, which I will round to 1/25 of the minutes in a day. If I divide the 3 billion people with phones by 25, I should obtain a rough count of the number of people using a phone in any given minute. This means that about 120,000,000 people world wide are using a phone during any given minute.
The question I am choosing is "If I combine all of the liquid I will drink over my lifetime, how many baths would it fill?" I am interpreting this to mean liquid that I drink from a cup of some kind. I am not including liquid in soup, fruit, or other foods. I am assuming that the bath tub is filled completely.
If I just make a wild guess, I think that the answer might be about 5,000 bath tubs of liquid. Making the wild guess is not very satisfying because I have no idea whether it is reasonable or not.
I can make a more educated guess by making some assumptions.
The units I have chosen are incompatible. I've got pints and cubic feet. This is where I need a reference book. It says that 1 US pint = 29 cubic inches I know that 1 cubic foot = 12 × 12 × 12 = 1728 cubic inches. (12 inches are in a foot.) So, lets change all the units to cubic inches:
In a lifetime I will drink a little over 100 bath fulls. That answer strikes me as surprisingly low because that means that I only drink about 1½ bath fulls of liquid each year. Perhaps some of my estimates were off a bit or perhaps my sense of how many bathtubs of liquid I drink is not accurate. On the other hand, I see now that my wild guess of 5,000 bathtubs of liquid is too high, since that would mean that I drink 5000 ÷75 ≈ 67 bathtubs of liquid each year or one bathtub full every 5 or 6 days.
Here are the variables I used while estimating the answer.
Now I can use these variables to write the formulas used to calculate the answer.
Notice that I could combine all these small formulas together into a single formula,
This formula requires that I estimate or measure how many pints we drink each day on average, how many years I will live, and the three dimensions of a bathtub to find the total number of bathtubs of liquid I will consume over my lifetime.
There are several measurements and pieces of information that I could gather to improve the estimate.
The formula I found in the previous step would then make it easy to obtain a revised answer.
I tracked my liquid consumption for a week and obtained the following data.
Sunday: 10 cups of liquid = 5 pints
Monday: 6 cups of liquid = 3 pints
Tuesday: 7 cups of liquid = 3.5 pints
Wednesday: 12 cups of liquid = 6 pints
Thursday: 4 cups of liquid = 2 pints
Friday: 8 cups of liquid = 4 pints
Saturday: 5 cups of liquid = 2.5 pints
Based on this sample, I drink an average of about 3.7 pints each day, so Cp = 3.7.
The formula that I found earlier tells me that the number of bathtubs of liquid consumed in my lifetime can be computed using the following formula:
So my best estimate for the number of bathtubs of liquid I will consume during my lifetime is:
|B =||10585 · 3.7 · 78.11||≈ 215.5|
|52 · 21 · 13|
Now I will investigate what the largest and smallest values
Bathtubs come in different sizes, but I could decide what the smallest and largest dimensions would be. It should at least be possible to sit down in a bath tub, so the smallest dimensions might be 30 inches by 30 inches by 12 inches deep. Large bathtubs can be pretty big, but let's just say for the sake of argument that the bathtub is at most 6 foot by 6 foot by 3 feet, or 72 inches by 72 inches by 36 inches.
People live different numbers of years, but I know that I have already lived 35 years and I know that I am unlikely to live longer than 110 years.
The largest amount of liquid I can imagine drinking in one day is 2 gallons (or 16 pints). I think that I would need to drink at least 2 pints a day on average.
To find the smallest possible answer, I should use the smallest possible numbers for the number of pints consumed and the number of years lived, and the largest possible numbers for the length, width, and height of the bath tub. If I do this, I find that
|=||10585 · 2 · 35|
|72 · 72 · 36|
So the smallest reasonable estimate is 4 (very large) bathtubs of water consumed over the course of my lifetime.
To find the largest possible answer, I should use the largest possible numbers for the number of pints consumed and the number of years lived, and the smallest possible numbers for the length, width, and height of the bath tub. This gives
|=||10585 · 16 · 110|
|30 · 30 · 12|
So the largest possible answer should be 1725 (very small) bathtubs of water consumed.
Based on this analysis, I conclude that I will drink between 4 and 1,725 bathtubs of liquid over the course of my lifetime. The most likely estimate for the answer to this question is 215.5 bathtubs of liquid.
One possible source of error in my computations is that my bath tub is not a perfect rectangular solid.
One interesting fact that I learned during this investigation is that (according to MyFoodDiary.com) most doctors recommend drinking 8 to 12 glasses of water per day.
Two other formulas for calculating the amount of liquid (according to the web page) are:
Another direction that I could take this investigation is to consider how much fluid I receive from foods.