Fermi Problems in Mathematics
Using the 5E Model
Named for Enrico Fermi, a physicist, who was able to make educated guesses at problems with little to no information given him. His most famous question asked was, "How many piano tuners are in Chicago?" A series of assumptions and estimations are needed to arrive at some sort of guess. Today, Fermi problems are a great way for students to practice estimating and reasonableness in math classes.
Mathforum.org states that Fermi Problems:
Mathforum.org states that Fermi Problems:
- are posed with limited information given.
- How many water balloons would it take to fill the school gymnasium?
- How many piano tuners are there in New York City?
- What is the mass in kilograms of the student body in your school?
- require that students ask many more questions.
- How big is a water balloon?
- What are the approximate dimensitons dimensions of the gym?
- What measurment must be estimated using the dimensions of the gym?
- ... and the list goes on.
- demand communication.
- utilize estimation.
- emphasize process rather than "the" answer.
Standards and Objectives
Texas Essential Knowledge & Skills and Common Core State Standards
3rd Grade 4th Grade 5th Grade 6th Grade All
- Apply mathematics to problems arising in everyday life, society, and the workplace (TEKS: 3.1A, 4.1A, 5.1A, 6.1A).
- Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution (TEKS: 3.1B, 4.1B, 5.1B, 6.1B).
- Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems (TEKS: 3.1C, 4.1C, 5.1C, 5.1D).
- Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate (TEKS: 3.1D, 4.1D, 5.1D, 6.1D).
- Create and use representations to organize, record, and communicate mathematical ideas (TEKS: 3.1E, 4.1E, 5.1E, 6.1E).
- Analyze mathematical relationships to connect and communicate mathematical ideas (TEKS: 3.1F, 4.1F, 5.1F, 6.1F).
- Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication (TEKS: 3.1G, 4.1G, 5.1G, 6.1G).
- Round to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems (TEKS 3.4B - SUPPORTING).
- 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 (this standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order) (CCSS.MATH.CONTENT.3.OA.D.8).
- Round whole numbers to a given place value through the hundred thousands place (TEKS 4.2D - SUPPORTING).
- Use place value understanding to round multi-digit whole numbers to any place (CCSS.MATH.CONTENT.4.NBT.A.3).
- Estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division (TEKS 5.3A - SUPPORTING).
- Apply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving rates and ratios (TEKS 6.4B - READINESS).
- Represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting (TEKS 3.4E - SUPPORTING).
- Use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit numbers. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties (TEKS 3.4G - SUPPORTING).
- Represent one-step and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations (TEKS 3.5A - READINESS).
- Represent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations (TEKS 3.5B - READINESS).
- 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 (CCSS.MATH.CONTENT.3.OA.A.3).
- Add and subtract whole numbers and decimals to the hundredths place using the standard algorithm (TEKS 4.4A - READINESS).
- Solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders (TEKS 4.4H - READINESS).
- Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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 (CCSS.MATH.CONTENT.4.OA.A3).
- Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit umbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models (CCSS.MATH.CONTENT.4.NBT.B.5).
- Multiply with fluency a three-digit number by a two-digit number using the standard algorithm (TEKS 5.3B - SUPPORTING).
- Fluently multiply multi digit whole numbers using the standard algorithm (CCSS.MATH.CONTENT.5.NBT.B.5).
- Add, subtract, multiply, and divide integers fluently (TEKS 6.3D - READINESS).
- Multiply and divide positive rational numbers fluently (TEKS 6.3E - READINESS).
- Determine the area of rectangles with whole numbers side lengths in problems using multiplication related to the numbers of rows times the number of unit squares in each row (TEKS 3.6C - READINESS).
- Determine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problem (TEKS 3.7B - READINESS).
- Multiply side lengths to find areas of rectangles with whole-numbers side lengths in the context of solving real world and mathematical problems, and represent whole-numbers products as rectangular areas in mathematical reasoning (CCSS.MATH.CONTENT.3.MD.C.7.B).
- Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems (CCSS.MATH.CONTENT.3.MD.C.7.D).
- Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters (CCSS.MATH.CONTENT.3.MD.D.8).
- Solve problems related to perimeter and area of rectangles where dimensions are whole numbers (TEKS 4.5D - READINESS).
- Solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate (TEKS 4.8C - READINESS).
- Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor (CCSS.MATH.CONTENT.4.MD.A.3).
- Represent and solve problems related to perimeter and/or area and related to volume (TEKS 5.4H - READINESS).
- Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems (CCSS.MATH.CONTENT.5.MD.C.5.C).
- Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles (TEKS 6.8D - READINESS).
- Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers (TEKS 6.8C - SUPPORTING).
This is a great tool to use to collaboratively solve problems. It is one of the best ways to get kids to think about steps, procedures, reasoning,
Example Using Solvr as a Problem Solving Tool
Questions to Explore
- What shortcuts have you discovered?
- How many yards long is an American football field (trick question for students - it is NOT 100 yards)?
- What assumptions are you making? How are you justifying them?
- What happens if you lost your "measurement tool?" Would work have to stop? (have they thought of converting to standard units?)
- How long do you anticipate it taking before you have an answer?
- The biggest question to explore is "why" with your students. Answering a Fermi question is all about explaining the process.
Explain, Elaborate, Evaluate
- Critical writing - Expository or procedural
- Mind-mapping (written or virtual tools like Popplet)
- VIP (Visual Instructional Poster - words and pictures)
- Virtual Posters (Edu.Glogster)
- Video [Jellycam (stop-motion), Animoto (music video)]
- Cartoons (Bitstrips)
- Slideshows (PowerPoint, Vcasmo, Prezi)
- Screencasts (Screencast-o-matic)
- Movie (iMovie, Moviemaker)
- Figure the volume if a 1-ft. wall was built around the perimeter of the field. How many "measurement tools" would it take to fill the entire field, up to the top of the wall?