Those BIG numbers fascinate, don’t they? I’ve watched 5^{th} and 6^{th} 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.

For your own classroom, I’ve looked for problems that may open discussion to large numbers. Please share your finds with your colleagues by posting comments.

**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.

**Making Your First Million**

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?

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