Illuminations: How Much Is a Million?

How Much Is a Million?

This lesson focuses students on the concept of 1,000,000. It allows students 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.

Learning Objectives

By the end of this lesson, students will:

Find the weight of rice needed to make up 1,000,000 grains of rice
Learn how to extrapolate from a smaller group of data to a larger conclusion
Solve proportions

Materials

Small cups (1 per group of 2 or 3 students)
1 larger cup or glass jar
Rice
Poster Paper (optional)
Balance – either a beam balance or a inexpensive postal scale
Calculator (optional)

Instructional Plan

Setting Up the Lesson

Start by asking students how much a million grains of rice really is and what they think it would look like. Inform students that they are going to see for themselves, by visualizing whaht a million grains of rice looks like through an investigation. For younger students, reading How Much Is a Million? by David M. Schwartz is a great introduction. Ask students how long it would take to count to a million themselves. It would take a very long time to count to a million if each number takes 1 second. Ask the students how long they think it would take. [A million seconds is a little over 11 days!] As a class, they will figure out how much rice constitutes 1,000,000 grains, and determine how much a million grains of rice would weigh.

Counting Rice

Ask the class for ideas about techniques for counting a million grains of rice. Hold up a package or cup of rice. Ask students, Do you think that the package contains a million grains? Ask how knowing the number of grains in the package or the cup would help a person know how much a million is.

For the class to count a significant sample:

Put students into groups of 2 or 3.
Give each group a cup that contains some rice (100–200 grains).
Have each group count the grains of rice in their cup. Ask students to write the number of grains on the cup. For the first few groups that finish, you may want them to count a second cup. Ask if any groups are finding efficient ways to count the numbe of grains of rice in their cups.
After each group has counted theier grains, have students record their data in some way for the class to see (e.g., draw a chart on the board or poster paper, or enter the data into a spreadsheet).
Review the results as a class. Encourage students to estimate the total number of grains of rice that have been counted so far. Add the estimates to the data. Ask students how they obtained their estimates.
Add up the total number of grains and compare the actual number to the estimates. Which estimates are close?

Here is an example:

Ask students to discuss and answer the following questions in their groups:

How many more cups about this size do we need to count in order to reach 1 million grains?
[Answers will vary.]
If each student in our school had a cup of rice about the size of ours to count, would the total reach a million?
[Answers will vary.]
What does a million grains of rice look like? Where could it be stored?
[Students may not yet have an idea of the volume of rice, so answers will vary greatly]
Hold up a 907‑gram (2‑;pound) package of rice. Ask students to estimate how many grains of rice are in the bag.

Show the class that a package of rice is measured by weight, and in order to determine the weight of a million grains of rice, we'll have to weigh some rice and do some calculations.

Weight of a Million Grains of Rice
Use a balance or a scale to determine the weight of the total number of counted grains of rice using a balance. Be sure to exclude the weight of the cups that you are using to hold the rice. You can weigh each cup seperately, or pour the rice into a larger cup or jar and weigh the whole thing at once.

Once students have the weight of the rice they counted, they can calculate the weight of 1 million grains. This problem can be solved by using a proportion, since the relationship between the number of grains and their weight is a direct proportion. Be aware that some rate problems cannot be solved using a proportion because the relationship is not directly proportional.

To find the weight of a million grains of rice, use the known ratio of 517 grains of rice to 14.5 grams. Set this ratio equal to 1,000,000 grains of rice to the unknown weight.

The number of grains (517 in this case) is obtained by students counting the rice in their cups. The weight of the grains that you counted is found using a balance. An electronic postal scale is another inexpensive way of finding the weight. To solve for the weight of 1,000,000 grains of rice, first cross multiply:

to obtain the equation:

And now, divide both sides by 517:

This means that 1,000,000 grains of rice weighs about 28,046 grams.

Your calculations will be different, but the process is the same to obtain an estimate in the neighborhood of 28,000 gm for a million grains of rice.

Use the Weight to Determine What a Million Grains Looks Like

Conclude the class by discussing how much space students think would be needed to store a million grains of rice. The 2 lb bag is about 907 grams, so 28,046/907 ≈ 31.

This means that 31 packages of rice would make a million grains. A discussion of what that looks like can follow.

Questions for Students

About how many pounds (kilograms) of rice does it take to make 1 million grains of rice? Explain how you got your answer?
How exact is your answer? Explain your reasoning.
Explain and show what you could do to make your answer more precise.
Explain how you could use a sampling technique to estimate the number of people who are watching a football game.

Assessment Options

What is the weight of a million M&M’s? Suppose every person in the United States (approximately 300,000,000) receives 1 M&M candy. What is the approximate weight of 300,000,000 M&M's?
What is the weight of a million pennies? Explain how you would move 1,000,000 pennies from a store to the bank.

Extensions

Environmental Science: Parts per million (ppm) is a common unit of measurement for pollutants. The concept of ppm can be difficult for students to visualize. The Environmental Protection Agency (http://www.epa.gov) lists acceptable levels of different pollutants. One such pollutant is mercury. The EPA states that 1 ppm of mercury is an acceptable level. To demonstrate this to students, take 1 grain of rice from a pile of 1 million grains. Use a marker and color this grain black and then put the grain back into the bucket. This is truly 1 ppm. Students can take turns looking to see if they can find the 1 dark colored grain of rice. Then place the bucket of rice on a table and allow students to search during off times during the day.
Social Science: How much does something cost? How much is our school district's annual budget? If each grain of rice was a dollar, how many pounds of rice would it take to equal that amount? How many pounds of rice will be “spent” in a year?
Consider an making an example out of the national debt. How much is the debt? How fast is it growing?

Teacher Reflection

In what ways did this lesson allow students to make connections to their world?
At what level did students interact and discuss their solutions and strategies in this lesson? What di you observe regarding student interaction?
In what ways did this lesson allow students who struggle with math access the mathematics involved with this lesson?
How effectively did this lesson address the needs of my students who require a tactile approach to learning?

NCTM Standards and Expectations

Number & Operations 6-8

Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.
Understand and use ratios and proportions to represent quantitative relationships.
Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.