#MTBoSBLaugust - Thinking About Magnitude and Number Relationships

A few days ago I found myself down an internet rabbit hole and came across this reply to a tweet about random facts:

Of course, this tweet is referring to the geologic time scale, a perfect vehicle for consider large numbers.

This got me thinking about magnitude and wondering how we can help students move beyond basic place value knowledge to a true understanding of large numbers. As we work to provide opportunities for students to build number sense, developing an understanding of magnitude and how large quantities relate to one another can be a challenge for some students. The fact that teachers and textbooks often emphasize digits (place value names) to the exclusion of understanding relative size makes ideas of magnitude challenging for students. 

For many years I have asked preservice teachers to place 1 million on a number line with 0 and 1 billion marked. The results are always eye-opening. A few years ago, Mark Chubb wrote a post entitled How Big is Big? in which he described asking teachers to show where 1 billion would go on a number line with 0 and 1 trillion marked. Go read it. He describes all the issues that come to mind when grappling with these ideas. 

It's not just school age students who struggle with issues of magnitude. In the 1982 article entitled On Number Numbness, Douglas Hofstadter wrote:

It is fashionable for people to write of the appalling illiteracy of this generation, particularly its supposed inability to write grammatical English. But what of the appalling "innumeracy" of most people, old and young, when it comes to making sense of the numbers that run their lives? 

. . . . .  

This kind of thing worries me. In a society where big numbers are commonplace, we cannot afford to have such appalling number ignorance as we do. Or do we actually suffer from number numbness? Are we growing ever number to ever-growing numbers?. . . . .  

. . . . .  

Combatting number numbness is basically not so hard. It simply involves getting used to a second set of meanings for small numbers-namely, the meanings of numbers between say, five and twenty, when used as exponents. It would seem revolutionary for newspapers to adopt the convention of expressing large numbers as powers of ten, yet to know that a number has twelve zeros is more concrete than to know that it is called a "trillion". 

. . . . . 

The world is a big place, no doubt about it. There are a lot of people, a lot of needs, and it all adds up to a certain degree of incomprehensibility. That, however, is no excuse for not being able to understand, or even relate to, numbers whose purpose is to summarize in a few symbols some salient aspects of those huge realities. . . . . For people whose minds go blank when they hear something ending in "illion" all big numbers are the same, so that exponential explosions make no difference. Such an inability to relate to large numbers is clearly bad for society. It leads people to ignore big issues on the grounds that they are incomprehensible.  

I'm seeing a lot of this "incomprehensibility" in articles and data visualizations related to COVID-19, though I worry using these examples may be too traumatic for some students. I want to use authentic problems and real-world contexts in examining large quantities. There is a nice overlap here with science, as considering the solar system leads to examinations of size and distance. Here are a few resources I like to use to consider large quantities and measures.

Powers of 10 Video (1977) - It's old, but oh so powerful.

The Powers of Ten video takes viewers on an adventure in magnitudes. Starting at a picnic by the lakeside in Chicago, viewers are transported to the outer edges of the universe. Every ten seconds the starting point is viewed from ten times farther out until the Milky Way galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, the view moves inward- into the hand of the sleeping picnicker- with ten times more magnification every ten seconds. The journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell. 

Pair this video with the book Looking Down by Steve Jenkins.

Powers of 10 - Interactive Java Tutorial
Soar through space starting at 10 million light years away from the Milky Way down through to a single proton in Florida in decreasing powers of ten (orders of magnitude). Explore the use of exponential notation to understand and compare the size of things in our world and the universe.

How Big Are Things?
This site tries to make size something you can remember, use, and optionally calculate with.

Is That a Big Number?
A web site designed to "restore some number sensitivity," you can enter any number and receive a bunch of relevant comparisons to put the number in context. 

I few years ago I heard a story on NPR. It began with the question "How many trees are there on the planet?" I started to think about how one would make such an estimation. My guess was 100 billion. Was I close? No. The actual answer is closer to 3 trillion. That's TRILLION, or 3 x 1,000,000,000,000. You can hear the story at NPR in the post entitled Tree Counter is Astonished By How Many Trees There Are.

And while this sounds like a huge amount (no, we don't have enough), the researchers found that the Earth has lost nearly half its trees since the start of human civilization. We also know that we are losing 10 Billion trees every year. All of this is pretty disturbing. 

These numbers teach us a lot about habitat loss, how much carbon dioxide is being absorbed from the atmosphere, how water is recycled in an ecosystem, and how we can preserve and replenish our forests. Take a minute to learn more in this video.

You can read more about the study that produced these data at Live Science in the article Earth Lost Half Its Trees to Humans.

There is so much math here to consider!

That's it for day 12 of #MTBoSBlaugust. I hope you'll come back tomorrow to see what else I have to share.