# What are Fermi problems and how can they be brought into the mathematics classroom?

My biggest frustration is when presenting a problem to students and then watching them give up when they can’t see their way to the end of it. Encouraging students to take some initial steps in solving a problem is half the battle towards enabling them to solve it entirely.

I’ve recently been working with students on this aspect of mathematics and have enjoyed much success using Fermi problems.

The solution to a Fermi problem could be termed a ‘back of envelope calculation’: more formally, a problem that requires estimates to be made to arrive at an answer which is difficult or impossible to measure exactly. An example of a Fermi problem is:

“How many trucks would you need to move Mount Fuji?”

An essential feature of a Fermi problem is that to make progress towards solving the problem, it has to be split into parts and relevant areas of mathematics selected. To arrive at a final answer, all of the parts are then combined. These skills are essential to building the resilience required for success in the problem-solving element of the GCSE mathematics course and for Core Maths.

In the case of the Mount Fuji problem, I started by asking students to think about how they could model the shape of a mountain. There was a range of ideas but in the end, we settled on using a pyramid.

Other topics we used to arrive at a solution included calculating a volume, using compound measures, converting between units and standard form. At each stage, students were required to make estimates of quantities.

This led to a wide range of answers, but that’s part of the beauty of Fermi problems, there isn’t a single correct answer. It was interesting to see though that in most cases the final answers were of the same order of magnitude: around 1010 trucks.

### Bringing Fermi problems into the classroom

You can find a collection of Fermi problems, provided by the University of Bristol, on our website. Having seen the positive benefits of using Fermi problems I would thoroughly recommend using them to help develop resilience.

To experience ideas for teaching through understanding, join us on the following bursary-supported CPD activities.

###### Mastering mathematics at key stage 3

It is important that mathematics at key stage 3 builds upon the mathematical experiences students experience at primary school. Explore what is meant by mastery, consider the transition between primary and secondary school and the mapping of progression through key stage 3.

###### Teaching mathematics GCSE content with understanding

​​Explore the content of the mathematics GCSE and gain an understanding of the importance of mathematical reasoning and problem-solving. Develop problem-solving skills and resilience in the context of hard to teach GCSE topics. Topics covered include trigonometry, linear graphs, proportional reasoning, standard form and powers, frequency trees, Venn diagrams, equations of circles, turning points, iterative processes, quadratic sequences, vectors and proof.

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