# BIG Numbers

Those BIG numbers fascinate, don’t they? I’ve watched 5th and 6th graders gathered around the teacher just to hear more about the size of a million, or even a billion. Ths article, Thinking Involving Very Large and Very Small Quantities, shows how we, as adults, often fail to comprehend such quantities. The article begins: “Intuitively a million is a lot more like a billion than ten is like one hundred, because our intuition has some grasp of ten and one hundred, but we have little grasp of what millions and billions involve. Fortunately, there is often a way to make intelligent decisions involving big quantities. Use arithmetic!” Topics here, which can generally be dealt with through just multiplication and division, include national finances, terrorism, airplane crashes and lotteries among others.

How Much is a Million?
This lesson focuses students on the concept of 1,000,000. It allows them to see first hand the sheer size of 1 million while at the same time providing them with an introduction to sampling and its use in mathematics. Students will use grains of rice and a balance to figure out the approximate volume and weight of 1,000,000 grains of rice. The lesson, which involves solving an equation, can easily be adapted for pre-algebra middle school students.

Too Big or Too Small
This unit features three activities, but I’m recommending only the first of these. Here students explore whether one million dollars will fit into a standard suitcase. If so, how large would the suitcase need to be?  How much would it weigh? Figuring out real answers to these questions can promote number sense.

In this activity for grades 4-6, students attempt to identify the concept of a million by working with smaller numerical units, such as blocks of 10 or 100, and then expanding the idea by multiplication or repeated addition until a million is reached. Additionally, they use critical thinking to analyze situations and to identify mathematical patterns that will enable them to develop the concept of very large numbers.

The MegaPenny Project
This site illustrates the magnitude of large numbers by showing and describing arrangements of large quantities of U.S. pennies. It begins with 16 pennies that measure one inch when stacked and one foot when placed in a row. The next visual shows a thousand pennies, and in progressive steps the site builds to a quintillion pennies. All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area (if laid flat).

One Grain of Rice
Beginning with the famous story of the village girl trying to feed her people, the lesson involves students in the mathematics of exponential growth. Students work collaboratively to come up with a bargaining plan to trick a raja into feeding the village using algebra and estimation. The complete activity includes the development of an exponential equation, but just following the growth of the number of rice grains throughout the story gives a good introduction to exponential growth. Questions for students and ideas for assessment are provided.

Finally, from the Figure This! collection, developed especially for middle school students, come these short but interesting problems on working with large numbers. Each question contains a hint on how to get started and a complete mathematical set-up on how to solve it.

How Fast Does Your Heart Beat?
If you started counting your heartbeats at midnight on January 1, 2000, when would you count the thousandth beat? The billionth?

How Much is Your Time Worth?
Would you rather work seven days at \$20 per day, or be paid \$2 for the first day and have your salary double for every day of the week

How Much Water Do You Waste?
If the faucet leaks 2 drops every second for a week, how much water goes to waste—enough to fill a glass, a sink, or a bathtub?

# So You’re Teaching Algebra Next Year?

The prospect can sound daunting. You may even feel the need for a refresher in algebra content. If so, you may find these sites helpful.

Patterns, Functions, and Algebra
This college-level math course explores the “big ideas” in algebraic thinking. Created for elementary and middle school teachers, the online workshop consists of 10 two-and-a-half hour sessions. You begin with a session on algebraic thinking and go on to sessions on such topics as proportional reasoning, solving equations, and nonlinear functions. Each workshop meeting includes video of teachers working together on problems, interactive Web activities, homework exercises, and discussion questions. The final session explores ways to apply the algebraic concepts you’ve learned to your own K-8 classrooms. Graduate-level semester credits are available through registration at Colorado State University, or the sessions can be completed for free by any interested group of teachers.

Algebra in Simplest Terms
If you want a basic algebra review, take a look at this video series. Intended for high school classrooms and adult learners, the course offers 26 half-hour video programs and coordinated books—online and free. Offered by Annenberg/CPB: Teacher Professional Development.

Navigating through Algebra for Grades 6-8.
Written for the middle school teacher, this book outlines the main concepts to be covered in these critical years and presents full activities as teaching samples.

Finally, a general resource of ideas for the classroom is Algebraic Thinking: A Basic Skill.Topics range from algebraic expressions to solving equations to understanding graphs. Here you can find online activities at a click.

I hope you will find these sites helpful and enjoy next year’s class!