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This page first made public: Feb 9, 2012

How Many Is A Million?

Roger Steinberg, Department of Natural Sciences, Del Mar College


To help students visualize the immensity of geologic time, or even the immensity of just one million years, I have created a very large sheet of paper that contains one million dots. The dots are regularly spaced, 100 per square inch. The dimensions of the sheet of paper are approximately 50 by 200 inches. I made it by cutting and taping together 200 pieces of 8 1/2 by 11 paper, each with 5000 dots. Because of the texture of the dots on the paper, you can kind of see or 'experience' all million of the dots while viewing from one end of the paper. Students both visually and viscerally 'get it'!

How this visualization is used in class:

I do this visual demonstration at the end of the semester, just before passing out the Final Exams, emphasizing that although the semester is over, learning never ceases. Plus, it helps relieve some pre-test tension. But you could present it earlier in the semester, if you wish.

I first hand out individual copies of the 8 1/2 by 11 pieces of paper containing 5000 dots to each student, and ask them the significance. Students usually look for a pattern in the dots (which I may actually encourage), until I inform them that it is the number of dots on the page that is significant. I ask them to guess the number of dots on the pages. Usually, someone will eventually correctly guess (or calculate) 5000. Then I select a student to help me unfurl the large sheet of paper, and ask the class to guess (or quickly calculate) the number of dots on it. Most commonly the responses are 4.6 billion (age of the Earth) or 13.7 billion (age of the Universe). Students are always shocked when I reveal that the number is only 1 million. (Yes, I actually cut and taped together 200 pieces of 8 1/2 by 11 paper, each with 5000 dots.)

Because I made the dots by hand on the original paper, using a grid that disappeared in the copying, my sheet of 1 million dots has some 'texture' that enables individual dots to be discerned from one end of the paper to the other. Students can see that a million is a very, very, large number, and a million years must be a very long time--yet only the blink of an eye compared to all of geologic time. (Computerized dots, even on a single piece of paper containing just 5000, are dizzying in a way that my hand-drawn paper is not—I know, I've tried it, and it doesn't work for that reason. I've included a pdf of my 5000 dots drawn by hand and 5000 dots drawn by computer with this visualization.)

As a final step, I ask the class to calculate the size of the sheet of paper, at the same scale, that would be required to contain 4.6 billion dots. I tell the students that I'm going to start working on creating that really, really large sheet of dot-covered paper after the semester ends, and ask for volunteers to help me with it. (It's not actually a project I intend to work on, of course.) To a first approximation, 4.6 billion is 5000 million. So imagine one of my 8 1/2 by 11 pieces of paper with 5000 dots. Now replace each dot with a sheet of paper as large as my 50 by 200 inch sheet containing 1 million dots. If you do the math, you will find that the new sheet of paper with 5 billion dots would be about 1.3 times as wide as a football field and 5.5 times as long! If only 4.6 billion dots, the paper would be the same width and still a bit longer than 5 football fields!

Advice on using this visualization in the classroom:

Good luck cutting and taping! Don't try to do it all at once, I spent several weeks on the effort, whenever I had spare time.

When using this visual demonstration for classes, make it fun as well as instructive. There are many other potential statements you could make or questions you could ask while presenting it. I sometimes make a point of emphasizing that there are 100 dots per square inch, and if a person is lucky (and pick their parents well), they may live to be the number of dots (years) contained in that one square inch of paper. This certainly puts a human life in perspective compared to all of geologic time, or even compared to just one million years.

As another example, you could ask students if they have ever seen a million of anything all at the same time and all in one place. Grains of sand on a beach may come to mind. But unlike my million dots, you really can't see the individual grains of sand on a beach unless you get quite close, so close that your field of view probably no longer encompasses one million grains. That's the beauty of my sheet of paper--you can both visually and viscerally understand that one million of anything is a lot because you can see the individual dots--and if a student had a million dollars, they probably wouldn't be sitting in my classroom!

Visualization and Supporting Materials

5000 Dots by Hand (Acrobat (PDF) 570kB Feb7 12)
5000 Dots by Computer (Acrobat (PDF) 411kB Feb7 12)


5000 Dots by Hand (Click image to enlarge and download.)
5000 Dots by Computer (Click image to enlarge and download.)

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