Fermi questions handouts_and_lesson_plan
- 1. Fermi Questions What would it take to fill this room with popcorn? How many people in the world are talking on their cell phones at this instant? If everyone in the country needed to be inoculated against a virulent strain of influenza, how quickly could this be done? Find the answers to these and other seemingly inaccessible questions by estimating and using knowledge gained from everyday experiences. Levels • Grades 3 through 12 • Adults will find this activity interesting • This activity can be extended to an undergraduate research project Topics • modeling • estimation • measurement • algebraic expressions • formulas • problem solving Goals • Identify and evaluate modeling strategies • Generate many potential solutions to a given problem • Apply principles and generalizations to new problems and situations • Analyze problems from different points of view • Make decisions as part of a group • Communicate mathematical ideas verbally and in writing • Work with a variety of measurement tools • Create and utilize formulas to tackle a problem • Use internet search tools to find information • Hone estimation skills and confidence Prerequisite Knowledge • familiarity with basic measurement skills • familiarity with common uses of addition, subtraction, multiplication, and division • familiarity with fractions, decimals, and percents Preparation Time 5 to 15 minutes Activity Time Usually 1 to 4 hours per Fermi question Materials & Preparation • journal or paper for recording findings • pencils • copies of handouts (optional) • appropriate measurement tools (optional) • appropriate equipment for gathering experimental evidence (optional) • access to the internet (optional) Authors Beth Marchant and Amanda Katharine Serenevy Riverbend Community Math Center This work is placed into the public domain by the Riverbend Community Math Center [email protected] http://riverbendmath.org (574) 339-9111
- 2. Credits Sample Fermi questions were gleaned from the following sources: • Teachers and students from the greater South Bend, Indiana area • Fermi Questions web page created by Louisiana Lessons in collaboration with the Math Forum; http://mathforum.org/workshops/sum96/interdisc/sheila1.html • Fermi Questions web page created for a Science Olympics competition by the Department of Physics at the University of Western Ontario; http://www.physics.uwo.ca/science olympics/events/puzzles/fermi questions.html • On Beyond a Million: An Amazing Math Journey by David M. Schwartz includes a broken popcorn machine in the story as a launch into large numbers. Elementary students might enjoy this book as a launch into popcorn Fermi questions. 2
- 3. Enrico Fermi Enrico Fermi (1901-1954) was an Italian physicist who made significant discoveries in nuclear physics and quantum mechanics. In 1938, he received the Nobel Prize in physics for his discovery of nuclear reactions caused by slow neutrons. This mechanism led directly to the development of atomic bombs and nuclear fission reactors. After receiving his Nobel Prize, he emigrated with his family to the United States to escape the fascist regime of Benito Mussolini, where he soon began contributing to the Manhattan Project. Fermi was famous for being able to make good estimates in situations where very little information was known. When the first nuclear bomb was tested, Fermi was nearby to observe. To get a preliminary estimate of the amount of energy released, he sprinkled small pieces of paper in the air and observed what happened when the shock wave reached them. (Being so close to the bomb on this and many other occasions exposed Fermi to dangerous radiation that led to his death by stomach cancer at the age of 53. Fermi was aware of the danger, but chose to work on this project anyway because he believed that the work was vital in the fight against Fascism.) Fermi often amused his friends and students by inventing and solving whimsical questions such as “How many piano tuners are there in Chicago?”. A “Fermi Question” asks for a quick estimate of a quantity that seems difficult or impossible to determine precisely. Fermi’s approach to such questions was to use common sense and rough estimates of quantities to piece together a ball-park value. For example, one way to estimate the number of piano tuners in Chicago is to break the process into steps: estimate the population; estimate the number of households in the population; estimate the fraction of households that have pianos; estimate how often each household has its piano tuned; estimate the time it takes to tune a piano; estimate how many hours a piano tuner would work each week. In this case, it is possible to check the estimate by looking in the phone book to see how many piano tuners are actually in Chicago. Fermi Questions in Everyday Life Here are a few examples of practical Fermi Questions. • In business: “How many teens live within a 30 mile radius of our proposed radio station?” • In environmental policy: “By how much would the amount of trash in landfills be reduced if it became illegal to throw away plastic grocery bags?” • In educational policy: “If the school district reduces the maximum class size to 20 students, how much would it cost to hire the extra teachers?” • In public health: “A virulent strain of influenza is spreading and everyone in our county needs to be vaccinated by a qualified health care professional. How quickly can this be done?” • Personal finance: “I am going to work in a fast food restaurant to cover my college tuition, books, and living expenses. Will I need to take out a loan? Will I have enough study time?” • Event planning: “Our city is organizing a parade with a mile-long route. About 150 organizations have expressed interest in being in the parade. For how much time will the streets need to be closed along the route?” It is empowering to cultivate your ability to think about these kinds of big picture questions. Thinking this way can enable you to dream big and accomplish your goals. Practicing this skill can equip you to identify opportunities and dangers that are not apparent to most people.
- 4. Fermi Questions Lab Record your answers to each question on another page. Be sure to write your team name, list the members of your team, and write out the Fermi question you are investigating. 1. Question: State the question and discuss how you will interpret it. 2. Wild Guess: What is your answer without any calculating? 3. Educated Guess: List the pieces of information you will need to answer this Fermi question more precisely. Estimate the value of each quantity in your list. Based on your estimates, what is your solution to the Fermi question? Show all your steps and use words to explain them. 4. Variables and Formulas: Choose variable names for each quantity that you estimated. Write a series of formulas or a procedure that explains how you used the quantities to find the solution. Try to simplify the process into a single formula that answers the Fermi question if possible. 5. Gathering Data: Perform experiments, conduct surveys, make measurements, or search for informa- tion that would help you to obtain a more precise estimate. For each quantity, identify the smallest possible value, the largest possible value, and the most likely value (you will probably have to use your best judgement to estimate these values). Then use the formula you found in the previous step to find the smallest likely answer to the Fermi question, the largest likely answer to the Fermi question, and the most likely answer to the Fermi question. Show your work! 6. Conclusions: State your final answers to the question. Explain some possible sources of error in your procedure. List any interesting facts that you learned while seeking the answer to the Fermi question. Finally, describe a further direction that you could pursue if you wanted to extend your investigation into this topic.
- 5. Sample Fermi Questions 1. In Bendix Woods near the old test track, the word STUDEBAKER is spelled out in pine trees so that it is visible from the air (check out the satellite image online). How many pine trees were required? What would it take to spell your name or the name of your school in pine trees? 2. How many people in the world are talking on their cell phones at this instant? 3. If all the people of the world were crowded together, how much area would we cover? 4. If everyone in our city donated one day’s wages to a good cause, how much money could be raised? 5. How many dump trucks would it take to cart away Mount Everest? 6. How large a landfill would our county need to store 100 years of garbage? 7. How many square miles of paved surfaces are there in our city? 8. How much gasoline does a typical automobile use during its lifetime? 9. How many people are airborne over the US at any given moment? 10. How much money could the city of South Bend save this year by shortening the work day of all city employees by one hour? 11. How many port-a-potties should be planned for the next million-man-march? 12. How much carbon dioxide is converted into oxygen each day by the vegetation in a typical yard. 13. How big is the local market for home-made gourmet dog treats? 14. How many musical notes are played on your favorite radio station in a given year? 15. How many gallons of water move down the Mississippi River in one day? 16. How far does a bumblebee fly each day? 17. If there were no traffic, how quickly could a race car travel from Washington D.C. to Los Angeles? 18. How long would your hair be if it never broke or was cut from the time you were born until now? 19. If all the pizzas eaten by students in your school last year were laid out next to each other, what area would be covered? 20. What is the current population of mosquitos in our county? 21. How much milk is produced in the United States each year? 22. How many pencils would it take to draw a straight line along the entire Prime Meridian of the Earth (assuming that a suitable drawing surface could be placed along the entire route)? 23. If you took the thread from all the uniform shirts of the Notre Dame football team and laid them end-to-end, how many times could you wind it around the football stadium at Notre Dame? 24. How many people attend Art Beat in South Bend each year? 25. How many grains of sand are there on the beaches surrounding Lake Michigan? 26. If you posted an advertisement on a billboard on Lincolnway for one month, how many people would be likely to see it? 27. How many plastic flamingos still exist in the United States? 28. What portion of all students in the city/state/country/world attends this school? 29. If the top 10 Forbes businesses donated 10% of their annual proceeds to schools, how much money would each school receive? 30. What would it take to fill a swimming pool with Jell-O? 31. How much air would it take to fill all of the school’s basketballs, soccerballs, and volleyballs? 32. What is the total number of shots taken in one NBA season (including the tournament)? 33. How many square feet of toilet paper are in the school? 34. How much does it cost to leave a light on for an entire day/week/month/year? How much does it cost to power a refrigerator for a year? 35. How many hot dogs are bought at all the Major League Baseball games for one season? 36. How many water droplets make up fog? Stratus clouds? Cumulus clouds? Cumulonimbus clouds? 37. How many texts does the average student send per year? 38. How much food waste does the school have in a month? 39. What is the average lifetime of a pencil?
- 6. 40. How much popcorn is popped at the movie theater on an average Saturday? 41. If you played your favorite song continuously for a whole year, how many times would it play? 42. How many times does your heart beat per day? Per week? Per year? 43. How many hours of tv will you watch in your lifetime? 44. How many pennies in my jar? 45. How many times would my 22 inch rims go around if I could drive around the equator of the Earth? 46. How many leaves are on that tree? 47. How many calories does a student burn while switching classes? How does this compare with the number of calories in a school lunch? 48. How many steps would I need to climb to burn as many calories as there are in a bag of potato chips? 49. How many laps would I need to make around our classroom to go a mile? 50. How many laps around our school would burn enough calories to lose a pound? 51. How many pizza boxes would we need to cover the classroom floor? 52. How many sticky notes would it take to cover the chalkboard? 53. What would it take to make a paper chain long enough to go down the main hallway at school? 54. What is the weight of a building? 55. How many sheep would it take for every person in the world to have a wool sweater? 56. What is the average number of bricks used to build a building? 57. How many gum balls would it take to reach from the Earth to the Moon? 58. How many snowflakes would it take to completely cover a driveway? 59. How many trees would need to be planted to lower the average global temperature by one degree? 60. How many nerd ropes would it take to go around the Earth? 61. How many cells would fit in a gallon? 62. How many drops of lemon juice should be added to a liter of water to reach a pH of 5? 63. How many electrons would flow through an iPod in one day? 64. If the national debt were represented with a stack of $1 bills, how far would the stack reach? 65. What is the true cost of buying and owning a new car or truck for 5 years? 66. How much energy would be saved at our school each year if the lights were replaced by energy-efficient alternatives? 67. How many bees are needed to polinate an orchard? 68. What is the probability that you have a doppleg¨anger (someone who looks just like you, but who is not be closely related)?
- 7. Fermi Questions Lab For Pre-K through 2nd Grade Students Young students need concrete Fermi Questions. The students should actually accomplish the task implied by the question. This format works well for students who are 4 to 8 years-old, or who are in Pre-K through 2nd Grade. After writing (or pasting) the chosen Fermi Question and reading it, the group should have some discussion about what the question means and how they would like to interpret it. It is worth discussing whether the question is likely to have a definite answer or whether a range of answers might be acceptable. Students should then name some numbers that are probably too small to be the answer to the question, and other numbers that are probably too big to be the answer to the question. They should then make guesses that they think are close to the right number. Students should choose two pausing points when they would like to stop to revise their guesses. Usually, one of these should be soon after starting, and the other one should be about half way. After completing the worksheet (or doing the equivalent as a group), students should use pictures, numbers or equations, and words or sentences (as appropriate) to show what they did and what they learned. During the final discussion, which may be at a different time, students should share their posters with each other (through whole or small group discussion, or through a gallery walk). Students can discuss how accurate their various guesses were compared with the answer they found. They can also discuss whether the answer might be different if they repeated the experiment, and what factors or changes in interpretation might contribute to producing different answers. Students can conclude by discussing what they wonder now to generate more questions to investigate. Here are a few ideas for Fermi Questions for this age level. 1. How many blocks would we need to stack to reach your height? 2. How many crackers would cover this sheet of paper? 3. How many seconds can you balance on one foot? 4. How many stickers would cover this notebook? 5. How many post-it notes would it take to make a column reaching to the ceiling? 6. How many punctuation marks (or letter a’s, et cetera) are in this book? 7. How many tiles are on this floor? 8. How many apples (et cetera) do we eat in a week? 9. How many jumps would you need to travel across the carpet? 10. How many cotton balls would fill this cup? 11. How many stuffed animals would fit on this shelf? 12. How many seeds does a dandelion have? 13. How many snap cubes laid end to end would stretch across the room? 14. How many beads would it take to make a bracelet that fits perfectly on my wrist? 15. How many links do we need to make a paper chain that stretches across the bulletin board? 16. How many tablespoons (et cetera) of water would fill this container? 17. How many cars pass by here in a minute? 18. How many times do you blink in a minute? 19. How many sidewalk squares are in a block? 20. How many photos can fit in the photo album?
- 8. Fermi Question: Numbers That Are Probably Too Small: Numbers That Are Probably Too Large: Numbers That Are Probably Close: 1st Part-Way Answer: Revised Guess: 2nd Part-Way Answer: Revised Guess: Final Answer:
- 9. Fermi Questions Lab For 3rd through 5th Grade Students Students in this age range usually need Fermi Questions that allow them to touch the objects under discussion. The students should physically begin the task implied by the question, though it may not be practical to complete it. This format works well for students who are 9 to 11 years old, or in 3rd through 5th grade. It is usualy best to break up the steps over several sessions. However, it is possible to do a single Fermi Question as a group within an hour-long session, and sometimes it helps to do this the first time. After writing the chosen Fermi Question and reading it, the students should discuss what the question means and how they would like to interpret it. It is worth asking whether the question is likely to have a definite answer or whether a range of answers might be acceptable. Students should then name some numbers that are probably too small to be the answer to the question, and other numbers that are probably too big to be the answer to the question. They should then make guesses that they think are close to the right number. Next students should list some questions they would need to answer (or measurements they would need to make) to figure out the question. They then make their best guess about the answer to each question. Next, they use the guessed quantities to answer the Fermi Question, and they should explain their thinking. The next step is to make a plan to gather more precise information about the problem. Students then carry out their plans, record their data, and then use that information to answer the Fermi Question, again explaining their thinking. During the final discussion, which may be at a different time, students should share their posters with each other (through whole group discussion, or small group jigsaws, or through a gallery walk). Students can discuss how accurate their various guesses were compared with the answers they found and the data they gathered. They can also discuss whether the answer might be different if they repeated the experiment, and what factors or changes in interpretation might contribute to producing different answers. Students can conclude by discussing what they wonder now to generate more questions to investigate. Here are a few ideas for Fermi Questions for this age level. 1. How many pennies are needed to equal your height, the height of the school, the tallest building in the world, Mount Everest, outer space? 2. How many times would this hula hoop roll to travel down the hall? 3. How many sheets of paper could be stacked from the floor to the ceiling? 4. What would it take to fill this room with popcorn? 5. How many people would be needed to surround this city if they held hands? 6. If you prepared a tank with all of the air you need to breathe in one day, how large would it be? (Hint: you could use a balloon to start thinking about this question.) 7. How many blades of grass are in a typical lawn in your neighborhood? 8. How many minutes does the average student play video games per day, week, year? 9. What would it take to make a paper chain that runs the length of our school hallway? (How much time, paper, tape). 10. How much water is wasted by a leaky faucet in one day?
- 10. Fermi Questions Lab 1. Write the Fermi Question. Explain how you will interpret it. 2. Guess the answer. a. Guess some numbers that are probably less than the real answer. b. Guess some numbers that are probably greater than the real answer. c. Guess some numbers that are probably close to the real answer. 3. List some questions you would need to answer to figure out the Fermi Question. 4. Write your best guess for each question on your list. 5. Based on your guesses, what do you think is the answer to the Fermi Question? Explain your thinking using pictures, equations, and sentences. 6. Make a plan for gathering data that will allow you to answer the questions on your list more precisely. You might perform experiments, conduct surveys, make measurements, or search for information. Describe your plan to find the answer to each item on your list. 7. Carry out your plan. Record the data you obtain. 8. Use the data to answer the Fermi Question. Explain your thinking.
- 11. Fermi Questions Lesson Plan Introducing Fermi Questions 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. Fermi Question Lab The lab outlined here asks students to complete six steps for each Fermi Question: 1. Question: State the question and clarify the interpretation. 2. Wild Guess: Make a wild guess involving no calculations. 3. Educated Guess: Make an educated guess involving a chain of reasoning and calculations based on everyday experiences and estimates. 4. Variables and Formulas: Define variables and create a formula to solve the Fermi question. 5. Gathering Information: Perform experiments, conduct surveys, make measurements, and search for information to improve estimates and to find a smallest reasonable value, a largest, reasonable value, and a most likely value for the answer to the Fermi Question. 6. Conclusions: Summarize the overall conclusions, possible sources of error, interesting facts learned, possible directions for future investigation. Providing guidance 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. Presentations A project of this type is a great opportunity to have students practice their written and verbal communica- tion 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. 3
- 12. Examples Illustrating How To Make Educated Guesses 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. How many bricks are in the exterior of our school building? 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. What is the volume of air that I breathe in one day? 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. How many kernels of popcorn would it take to fill this classroom? 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. How many people in the world are talking on their cell phones in any given minute? 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. If I divide the 3 billion people with phones by 24, I should obtain a rough count of the number of people using a phone in any given minute. This means that about 125,000,000 people world wide are using a phone during any given minute. 4
- 13. How many pennies would need to be stacked to reach your height, the height of the school, the tallest building in the world, Mount Everest, outer space? • I think that it takes about 4 penny rolls to equal one foot. I am about 5 feet tall, so that is about 5 × 4 = 20 penny rolls. Each roll has 50 pennies. So that is about 50 × 20 = 1000 pennies to equal my own height. (About $10.00 in pennies.) • I think that the school is two stories tall and that each story is about 15 feet tall. That means that the school is 6 times as tall as I am and so it would take about 6000 pennies to be the size of the school (about $60.00 in pennies). • One hundred stories is more than most tall buildings have, so 3000 × 100 = 300, 000 (about $3,000 in pennies) is probably enough pennies to surpass the tallest building. • Denver is the mile high city, and I know that the Rockies go up at least another mile. I don’t know how the Rockies compare with the Himalayas where Mount Everest is, but let’s say that they are twice the elevation. That would be four miles high. There are more than 5,280 feet in a mile, so I would need more than 21,000 feet of pennies to reach four miles. That means about 84,000 penny rolls or about 4,200,000 pennies ($42,000 in pennies). • How far up is outer space? I am not really sure, but I think 100,000 feet (roughly 19 miles) is about right. So that would be about 20,000,000 pennies to reach to outer space (a mere $200,000 in pennies). 5
- 14. Example of a Full Fermi Lab Solution 1. Question: 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. 2. Wild Guess: 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. 3. Educated Guess: Assumptions: I can make a more educated guess by making some assumptions. • Today, I had 4 mugs of coffee (about one and one half pints) two glasses of orange juice (half a pint), a can of soda (about half a pint), some milk on my cereal (about a third of a pint). I must have missed something . . . so I shall write down this assumption: On a typical day I drink about 3 pints of liquid. • Now I also need to know about bathtubs. I am 6 feet tall, and when I soak in the bathtub, I can reach the taps with my toes while keeping my head above water. So the bath must be about 5 feet long. The tub is about 2 feet 6 inches wide on the inside, and about 1 foot deep. Using the formula for the volume of a rectangular solid, I can make my second assumption: A full bath holds about V = L × W × H = 5 × 2.5 × 1 = 12.5 cubic feet. • One last assumption: I will live about 75 years. Calculations: 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: • I drink about 3 × 29 ≈ 90 cubic inches per day. (Notice that I rounded my answer because I am approximating anyway.) • My bath holds 12.5 × 1728 ≈ 22, 000 cubic inches. • In 75 years that is 90 × 365 × 75 ≈ 2, 500, 000 cubic inches. So that means I will drink about 2, 500, 000 ÷ 22, 000 = 113 bath fulls of liquid. 6
- 15. Answer: 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 11 2 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. 4. Variables and Formulas: Here are the variables I used while estimating the answer. • Let CP be the average number of pints I consume each day. • Let C be the average number of cubic inches I consume each day. • Let T be the total number of cubic inches I will consume over my lifetime. • Let L be the length of the inside of the bathtub in inches (since I ended up converting). • Let W be the width of the inside of the bathtub in inches. • Let H be the height of the inside of the bathtub in inches. • Let V be the volume of the bathtub in cubic inches. • Let Y be the number of years that I will live. • Let D be the number of days that I will live. • Let B be the number of bathtubs of liquid I will consume over my lifetime. Now I can use these variables to write the formulas used to calculate the answer. • C = 29CP (This formula converts the number of pints I consume each day to the number of cubic inches I consume each day.) • V = LWH (This formula finds the volume of the bathtub in cubic inches.) • D = 365Y (This formula gives the total number of days that I will live.) • T = CD (This formula gives the total number of cubic inches I will consume over my lifetime.) • B = T V (This formula divides the total amount of liquid I will consume over my lifetime by the amount of liquid held by one bathtub to get the number of bathtubs of liquid I will consume over my lifetime.) Notice that I could combine all these small formulas together into a single formula, B = (29CP)(365Y ) LWH which simplifies to B = 10585CPY LWH . 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. 7
- 16. 5. Gathering More Information There are several measurements and pieces of information that I could gather to improve the estimate. • I could measure the length, width, and height (in inches) of my bathtub. • I could look up the average lifespan (in years) of people in the United States. • I could keep track of how much liquid I consume every day for a week and take an average. The formula I found in the previous step would then make it easy to obtain a revised answer. Data The actual length in inches of my bathtub is L = 52. The actual width in inches of my bathtub is W = 21. The actual height in inches of my bathtub is H = 13. The average lifespan (in years) of people in the US is Y = 78.11. (Source: CIA World Factbook) 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. Best Computed Answer 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: B = 10585CPY LWH . So my best estimate for the number of bathtubs of liquid I will consume during my lifetime is: B = 10585 · 3.7 · 78.11 52 · 21 · 13 ≈ 215.5 Largest and Smallest Possible Values 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. 8
- 17. 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 B = 10585CPY LWH = 10585 · 2 · 35 72 · 72 · 36 = 740950 186624 ≈ 4 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 B = 10585CPY LWH = 10585 · 16 · 110 30 · 30 · 12 = 18629600 10800 ≈ 1725 So the largest possible answer should be 1725 (very small) bathtubs of water consumed. 6. Conclusions 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: 0.5 ounces × Body Weight in Pounds = Daily Fluid Requirement in ounces 0.034 ounces × Daily Caloric Intake = Daily Fluid Requirement in ounces. Another direction that I could take this investigation is to consider how much fluid I receive from foods. 9
- 18. Common Core State Standards This lesson incorporates all eight of the standards for mathematical practice described in the Common Core State Standards. In addition, the following content standards may be covered (depending on the specific Fermi Questions chosen and the level of the students). 3.Focus.1 Develop an understanding of the meanings of multiplication and division. 3.Focus.3 Recognize area as an attribute of two-dimensional regions. 3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56÷8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. 3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 3.MD.7 Relate area to the operations of multiplication and addition. 4.Focus.1 Develop understanding and fluency with multi-digit multiplication and division. 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, capacity, masses, and money, including problems involving simple fractions or decimals and problems that require use of simple unit conversions. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. 5.Focus.2 Fluently add, subtract, multiply, and divide multi-digit numbers and make sense of standard algorithms with decimals using models for fractions and decimals. 5.Focus.3 Develop an understanding of volume. 5.OA.1 Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols. 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expres- sions without evaluating them. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.MD.1 Convert among measurement units within a given measurement system. 10
- 19. 5.MD.3 Recognize volume as an attribute of solid figures and understand the meaning of volume in terms of unit cubes. 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world problems involving volumes of rectangular prisms. 6.Focus.1 Solve ratio and rate problems and understand related concepts. 6.Focus.2 Exhibit conceptual understanding and algorithmic fluency with rational numbers and the four operations. 6.Focus.3 Write, interpret, and use expressions and equations. 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems (make tables, solve unit rate problems, work with percents, and convert measurement units). 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithms. 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.4 Identify when two expressions are equivalent and understand what equivalence of expressions implies. 6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problems. 7.Focus.1 Develop understanding of and apply proportional relationships. 7.Focus.3 Solve problems involving scale drawings and informal geometric constructions, and solve prob- lems involving surface area and volume. 7.RP.2 Recognize and represent proportional relationships between quantities. 7.RP.3 Use proportional relationships to solve multi-step ratio and percent problems. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems. 7.SP.2 Use data from a random sample to draw inferences about a population with an unknown charac- teristic of interest. 8.Focus.1 Formulate and reason about expressions and equations. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. HS.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. HS.Modeling Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. 11
- 20. HS.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. 12