# A Profound Respect for 1 Million

By Chris Robinson (@isomorphic2crob), The Blake School

How long will it take for your heart to beat ONE MILLION times?

Before you read on, imagine that the ball just dropped, January 1st, 2015, when will your one millionth heart beat occur? Will it be later that day? Some point in January? In July? In 2016? 2017? How long does it really take to reach ONE MILLION heartbeats? Before doing any figuring, take a guess, then after you have committed to that conjecture, go ahead and calculate it.

This question was posed by Marjory Baruch (Syracuse University) to a group of high school math teachers attending PROMYS for Teachers at Boston University this summer. The initial conjectures for this group of math teachers ranged from a couple months to a couple years. All of us were astounded when we calculated the result to be slightly less than two week’s time.

The questions stuck with me as I began to prepare for my year at Blake, and so I decided to ask my class of Algebra 1B ninth graders, my Honors Geometry ninth graders, and my Honors Algebra II tenth graders. Their answers were a much wider range, including a student or two in each class that guessed less than a month. But not too many predicted it to take more than a couple of years. There was very little difference in the guessing from one class to the next leading me to believe from a very small sample size that students ability to estimate the size of large numbers was not necessarily connected with their current success in mathematics. There was however a difference between the teachers and the students.

Why do math teachers have such a profound respect for the size of 1 million? Maybe its related to our salaries…. But I digress.

Here’s Hank Green discussing a very similar idea (How long was it a million seconds ago?) on his YouTube channel.

I also had my students place the number 1 million on a number line from 0 to 1 billion, like the one shown below.

Students struggled to see how much larger 1 billion was compared to 1 million, with one student commenting that they always felt like millionaires and billionaires were the essentially the same.

OK, so what’s the big deal (get it? Big.)? I am not sure I have a punch line here, so I will leave you with some observations, and the hope that you find this intriguing as well.

• For most students, and perhaps most teachers, our initial impression of numbers beyond 10,000 take on the same amount of “bigness” until they are compared to other large numbers.
• Asking students to reflect on why they guessed what they did is a useful exercise, and having them write about it for a few minutes can help them develop habits of explaining their mathematical reasoning.