Understanding Large Numbers

By Bob Peterson

SOURCE: War resisters league (www.warresisters.org) based on data from the center for defense information

It’s a zillion!” calls out one student in response to my question of how to read the number 1,000,000,000 I’ve written on the overhead. “No, it’s not, it’s a million,” argues another.

Despite the fact that my fifth graders have been taught place value throughout their elementary years, there is something about big numbers that lends itself to guessing. Perhaps it’s the omnipresent state lottery advertisements that tend to blur big numbers together. Or more likely it’s the fact that big numbers are just difficult to read, much less understand. Imagining a billion boggles my mind, whether I’m trying to fathom that number of galaxies swirling around the universe or the number of H 2 O molecules in a drop of water.

Kids are fascinated with big numbers, especially if they connect with the real world. Thanks to the U.S. government’s addiction to military spending, students have an endless stream of large numbers to study. And those numbers just get bigger and bigger! The growth in the military budget comes as schools face massive budget cuts. Teaching about these matters provides students an opportunity to improve their understanding of large numbers, and even more importantly, understand the power of math in debates about the future of our communities and world.

Before I delve into budget issues, I do a couple of activities to help children put meaning behind place value. This year, the night before I was to start my mini-unit on big numbers and budgets, the students’ homework was to ask family members what they thought one million and one billion meant. The next day students shared responses. They ranged from the precise “one million is one thousand thousands” to the comical “it’s what you get when you win the lottery,” to the practical, “it’s enough money to buy everything we ever need and still have some left over.”

I asked the students, “How many days equals a million seconds? After some initial guesses, the students worked in groups with calculators using different strategies to solve the problem. Eventually they came up with about 11.8 days.

I then asked how long it would take for us to count to a million. Some students suggested we just do it and time ourselves. Others were more skeptical. After some practice with six-digit numbers we estimated that it would take, on the average, about two seconds a number. Some more calcuation and the class realized it would take a little over 23 days. “I’m not going to be wasting my time doing that,” one student proclaimed.

To visualize a million I asked the students to look closely at a strand of their hair and then I told them that if one piled a million of those on top of each other it would reach up to a seven story building. I also showed the students the book How Much Is a Million? by David Schwartz (Scholastic, 1985). Some of the pages are filled with tiny stars — 14,364 per page. The book encourages students to guess how many pages of stars it would take to reach a million, and they are surprised to find it would take 70 pages.

I then repeated some of these activities with a billion. The students soon discovered that their calculators did not go that high and so we did some whole-class work. We calculated that one billion seconds equals about 32 years. After timing the counting of a series of very large numbers, we estimated it would take about three seconds a number if we were to count to a billion — leading us to conclude it would take almost 96 years to count to a billion. We examined the star-filled pages of How Much Is a Million? and calculated that it would take 70,000 such pages!

Next I wrote $80,000,000,000 on the chalk board and had a student read it. I then wrote the number $10,000,000 and did the same. I had students guess the numbers’ significance. I then explained to them that the former is the estimated cost of war and occupation of Iraq for a year, and the latter is approximately the amount of money that is going to be spent renovating and adding to our 100-year old building. I asked, “How many schools like ours could get a major renovation for the cost of just one year of the Iraqi war?” After a wide range of guesses, I asked how we could know for sure.

The class decided that we’d go around the room counting by 10 million and that Markese would keep track of the number of school construction projects that could be bought. As each student added another 10 million, Markese made a mark on a piece of paper. After we went around the class two times, we had only reached 460 million and it was clear our effort to count to 80 billion was going to take a while. When we finally reached 1 billion, Markese announced we could rebuild 100 schools. I stopped the counting and suggested the students use mental math to figure out the final solution. “8,000!” one student called out. “That’s a lot of schools that could be rebuilt!” added another. “That’s more than in all of Milwaukee!” added a third. In fact, a quick check on the Internet showed that there are just over 2,000 public schools in Wisconsin.

Later, as part of this mini-unit, the students discussed graphs from United for a Fair Economy (UFE) that contrast the U.S. military budget with federal social spending and also with military budgets of other countries around the world. Using UFE’s data, students figured out that one Stealth Bomber, at the cost of $2.1 billion, could have paid for the annual salary/benefits of 38,000 teachers. This was of extra significance to my students because our school principal had just informed their parents we’d be losing two teachers (gym and music), half our librarian time, and two paraprofessionals. “Just for a little part of one of those bombers, we could have all our teachers back,” one student said.

I did this unit at the very end of the school year in the midst of news of mounting budget cuts and continuing conflict in Iraq. I wanted to give my departing fifth-graders a different perspective on the cuts. So often the talk about budgets is filled with hopelessness and inevitability. I knew that in just a few days we’d only touch the surface of a complicated issue. Next school year, I will cover this earlier so that we can spend more time looking at how policymakers make budget decisions, the relative merits of various types of spending, and what social action groups are doing on these issues.

Holding students’ attention during the waning days of the school year is always a challenge. This lesson held their attention, and it was a fitting way to end the school year as the students prepared to go to middle schools that also faced drastic cuts.

“This really isn’t fair,” one student wrote when I asked the class to reflect on the matter in their journals. “So much money is being spent when our schools need so little.” Actually our school — like most schools — needs a lot. But it’s just a little compared to the more than $400 billion this nation spends each year on the military budget.

Hmmm, I wonder how many schools that could rebuild . . . and how many jobs that would create?

Bob Peterson ( repmilw@aol.com ) is an editor of Rethinking Schools and teaches fifth grade at La Escuela Fratney in Milwaukee.