Fun Mathematical Modeling Problems for Halloween
Are you looking for something fun to do with your students on Halloween that still hits the math standards? Why not give my Halloween Fermi Problems a try?
What a Strange Name for a Problem? What is a Fermi Problem Anyways?
A Fermi Problem is named after an Italian physicist who won a Nobel Prize for physics. It is a problem that requires students to estimate and make assumptions for a nearly impossible problem—one that you can never know the exact answer to. Or perhaps a problem where the exact answer is difficult to practically find. For example, how many school busses are in the United States? Or how many leaves are on a tree?
Note: This does not necessarily mean that students have to do mental math. Although they could if you require it; it depends on your math goals. If students are dealing with big numbers, you could consider letting them use calculators as long as it aligns with your math goal for the day.
Why Fermi Problems?
Fermi problems are interesting and engaging. They require students to make reasonable mathematical assumptions and accurate estimations. They promote problem solving, critical thinking, decision making, numeracy, proper use of units, and of course estimation. Students may not get an exact answer, but they should get a reasonable answer based on their assumptions. Additionally, they are easy ways to incorporate mathematical modeling in the classroom.
What is a Mathematical Assumption?
A mathematical assumption is something you assume to be true. It’s something that you need to do in order to solve complex real-world problems. It makes the problem simpler and solvable. The problem solver or mathematical modeler decides what to keep and what to ignore about a problem. For example, you might make the assumption that all pumpkin patches are rectangular in shape? Or one group might assume that all public-school kids go trick or treating but to ignore kids in different schooling situations (parochial, homeschool, online etc.) Another assumption could be that each house gives out the same number of pieces of candy per kid or even that each house gives out two pieces of candy per kid. Of course, the sophistication of the assumption should vary by grade level. Whereas in the early grades, the teacher may help or define the assumptions, in later grades the students may be required to make and justify their own assumptions.
What Grade-Level standards to Fermi Problems Align to?
Math Practice Standards
Naturally Fermi problems align to the math practice standards:
- SMP.1: Make sense of problems and persevere in problem solving. Students need to step back and understand the problem. They need to break the problem down into a simpler problem(s) and find entry points.
- SMP.2: Reason abstractly and quantitatively. Reasoning quantitatively involves making sense of units, and in Fermi problems units are essential.
- SMP.3: Construct viable arguments and critique the reasoning of others. Since it’s impossible to find a truly “correct” answer to a Fermi problem, students need to be assessed based on their reasoning skills. Does their answer make sense? Can they convince others? What assumptions did they make? Note: Although, there is no one correct answers, there are definitely incorrect answers.
- SMP.4 Model with Mathematics. Fermi problems are a great introduction to mathematical modeling. They introduce several steps of the modeling process such as making assumptions, but they are short and manageable. They can be used for a class period or part of a class period unlike more complex modeling problems which may take days or weeks to complete.
Subscribe to gain access to a student-facing problem solving rubric for grades 3-8 based on SMP.1. By subscribing you will also gain access to student-facing perseverance rubric for grades K-8.
Math Content Standards
The content standards that you use will depend on the grade level that you are teaching and what each problem lends itself. The level of precision required from students will also vary based on grade level. However, Fermi problems align nicely with many OA, MD, EE, Geometry standards and N.Q.1 in the high school standards.
What about Calculators?
Whether you give students a calculator or not depends on your educational goals for the lesson. For example, if one of your goals is to have students become fluent in multiplication of whole numbers, you may not want to give students a calculator. However, if your goal is N.Q.1, it would be perfectly find for students to use a calculator as long as they were using units correctly.
Halloween Fermi Problems
Here are some ideas I had on Fermi Problems that you can use around Halloween or in the Fall. Feel free to add to the list or suggest some new ones in the comments. You may also modify the ideas I have it works better for your classroom.
K-2
- How many Jack O’Lanterns are on your street?
- How much candy do you eat on Halloween (assuming your parents don’t give you a limit)?
- If you lined up all your Halloween candy, how long would it be?
Grades 3+
- How much candy was given out in your neighborhood trick-or-treating?
- How many kids went trick-or-treating in your city?
- How many steps do you take trick-or-treating?
- How many seeds does a pumpkin have? (You may need to do a little data collection: Bring in several pumpkins of various weights and count the seeds.)
- How many pumpkins are sold in your city in October?
- What is the average number of pumpkins in a pumpkin patch?
- How many leaves are the leaf pile you raked up?
- What is the typical amount of leaves that are raked up in your neighborhood?
Process
If your students are unfamiliar with Fermi Problems, you may need to model solving one of these types of problems. Here is one way to approach Fermi Problems with your students. I gave a sample process below, but this can be shortened quite a bit depending on the classroom time that you want to spend on it. Check out the resources below for even more (and perhaps better 😊) ideas about integrating Fermi problems into your classroom. This is an especially great resource for implementing Fermi Problems for Grades K-2: fermi_questions_handouts_and_lesson_plan.pdf
Question of Interest:
How many kids went trick-or-treating in your city?
Step 1: Individually have students write a wild guess on a piece of paper and collect it. Also, have them guess a number they know for sure is too high and a number they know for sure is too low!
Step 2: Just give them the problem for 5 minutes and 5 minutes only and see what they come up with.
Step 3: Show students a picture of Fermi and explain what a Fermi problem is.
Step 4: Ask them to pause and not do any calculations but spend 5 minutes writing down all the questions they would need to explore in order to solve the problems.
Step 5: Write down each group’s questions on the board. Tally those that appear more than once.
Step 6: Discuss what a mathematical assumption is and give an example that pertains to the context of the problem. For example, you are going to assume that kids in your school whose neighborhoods don’t have trick-or-treating (for example, because they live in apartment) travel to a nearby neighborhood to trick-or treat.
Step 7: Discuss the importance of units.
Step 8: Discuss reasonableness. Use the guesses from Step 1 help establish some type of reasonableness. You may even want to write it as an inequality depending on the grade level.
Step 9: Give each group sticky chart paper and have them solve the problem. Emphasize that they need to write down their assumptions and label their units.
Step 10: Once all the groups are finished, have them adhere their chart paper to the wall. Do a gallery walk where students can go around and look at other group’s solutions.
Step 11: Discuss the reasonableness of people’s solutions. Have students critique their classmate’s conclusions. Make sure to model how to politely critique another group. Discuss as a group whether different assumptions led to different answers. Don’t forget to emphasize the importance of units.
Assessment
Rubrics are one of the best ways to assess modeling problems. Assessment in modeling should focus on the process, not the product. The GAIMME report has a few examples that you could use or you can pull from some of the resources below. You could also choose one or two mathematical practices to assess. I have developed a Student-Facing Problem Solving (SMP.1) Rubric for Grades 3-8 that is shown above. It is available free to subscribers. For high schoolers, the Ohio Department of Education and Workforce has rubrics for all the math practices posted on their website.
Other Resources
Here are some resources on Fermi Problems:
- An excellent collection of Fermi problems for your class – Innovative Teaching Ideas
- Fermi Problems: Estimation
- Fermi Problem: Fermi Problems: Breaking Down Complex Questions – FasterCapital
- fermi_questions_handouts_and_lesson_plan.pdf (Great resource that breaks Fermi problems down based on grade level)
- fermi problems oapt 2018.pptx is a PowerPoint that you could use in class or to coach teachers.
- GAIMME report
Subscribe
Subscribe to get more great math ideas sent directly to your inbox. Share this post with your math teacher friends or other homeschooling parents!
Comment
Do you have your own idea for a Halloween Fermi Problem? Please share it in the comments!
