The World of Extreme Numbers: Scientific Notation, Fermi Problems, and More

by Natalie Medolla

Natalie Medolla is an elementary Enrichment Program coordinator in New Jersey working with students in grades K-5. We began exchanging emails when she sent me a video link she thought I might like to add to POWERS of TEN and COSMIC EYE: Mind-Expanding for All Students. She mentioned that she and her students LOVE investigating extreme numbers and the challenge of Fermi problems, which are great for helping students deal with ambiguity. I wanted to learn more and the result is this post, in which Natalie generously shares the sequence she follows, videos she incorporates, and links to helpful resources!  Leave a comment below if you end up using some of Natalie’s ideas!

Why are we so fascinated by extremely large numbers? This attraction is even evident in ancient times when the Greek Philosopher Archimedes calculated how many grains of sand it would take to fill the entire universe, which came to 10⁶³.   

Maybe our fascination comes from the fact that our brains are not well-equipped to comprehend these large numbers, and our capacity to visualize these numbers is quite limited. This ambiguity seems to feed our student’s curiosity. Every year I witness my students passionately discussing, “What is a googol, a googolplex, Graham’s number, TREE(3), infinity…etc.”

For this reason, I love to begin our math enrichment class by quantifying REALLY BIG AND SMALL numbers. To get them excited, I introduce the topic by asking them, “What exists in really large and small numbers?”  

Their list becomes endless, often including world population, our national debt, the number of cells in our body, stars in the universe, etc.

Then I pose the question: “How can we quantify these extreme numbers? How can we visualize such extreme numbers?”

We start trying to put these large finite numbers in context. I share the videos below with them, and we use this resource, Measure of Things, which allows you to input a quantity. Then it creates lists of comparative and relative numbers measured in units you can conceptualize. Make sure to check it out. The students love it. 

We then discuss this question: How long will it take to count out loud to a million? (about 11 days)… A billion? (about 32 years) Students can strategize and use calculators. You may also share that it would take about 31,709 years to count out loud a trillion. This link is helpful, too, for Understanding Large Numbers. 

We usually watch a few videos on my YouTube playlist. (Some are worth checking out). My math kids love Beyond Infinity Number Comparison. 

Then we dedicate a few class periods to learning how to record big numbers using scientific notation.

We first talk about our base-ten number system relating all the way back to first grade when we counted the days by straws and the excitement of making a bundle of ten to then making our bundle of 100 on our 100th day of school.  This leads to the discussion about powers of ten and what happens when numbers get ten times bigger, ten times smaller, and how we can express that.  We talk about how scientists can use powers of ten to express a really large or really small number instead of writing all of those zeros.

I created this Google slide deck to help teach the concepts. I also love the free site Common Core Sheets for Practice.

Once the students are familiar with scientific notation, I show them the video below to introduce them to Fermi Problems:

I love Fermi problems, and my students do too!!! After the video, I make them put their iPads away and have them work in groups to figure out about how many steps a person takes in a lifetime. They usually begin discussing how many steps are in a mile. Usually, at least one student has a familiar reference relating to wearable technology that tracks steps. However, sometimes I may have to guide them in understanding a mile is about 2000 steps. Students then discuss how many average miles a day a person takes and multiply that by 365 days.

They also discuss the average life span, which they usually seem to guess to be anywhere from 75 to 80 years old. They use all these calculations to make an estimation and use scientific notation to record their answer. When time is up, each group records its final answer. Note that some groups may not finish, and this is a great chance to reflect on the challenges they faced. We google it at the end, and it is always surprising how relatively close some of their answers are. We then find ways to make that number relatable (like traveling 4 to 5 times around the equator).

We then extend to “Can any land mammal travel far enough to get to the moon, etc….” 

The students construct close-to-accurate measurements with limited information and learn the power of estimation. 

We do at least one Fermi problem a month. You can find a list of problems HERE. Some students also create their own.

If you have other ideas and resources to share, please add your comments below.

Additional Resources:

A Brief History of Numerical Systems – (YouTube video) Alessandra King 

From Nanoscopic to Astronomical ~ Introducing “Powers of Ten” Day (blog post)

Math for Smarty Pants – by Marilyn Burns / Chapter 7 is entitled “Thinking Big”

How Big is a Mole? (for older students)- TED Lesson