One in a Million

The BIG Idea

Just how big is a million really?

Content Areas in This Activity

[check] Numeracy

[check] Estimation

[check] Basic probability

Process Skills Used in This Activity

[check] Creativity

[check] Reasoning

Prerequisite Knowledge and Skills

[check] Multiplication

[check] Place value for large numbers

Age Appropriateness

This activity is appropriate for all ages.

The Mathematical Idea

While babies can distinguish small numbers — for example, they can tell one object from two — older children are able to perceive larger numbers, such as distinguishing three from five or ten from twenty. Very large numbers, however, are much harder. Just how far does our numerical judgment go?

Lotteries, population figures, and high finance are but a few of the applications of math that use dazzlingly large numbers. It can be difficult to develop any appreciation of these very large numbers. How big is a thousand? How big is a million? They both sound big, but a million is much larger than a thousand — one thousand times larger in fact! So to have the same chance of winning a lottery with one ticket in a million as you would with one ticket out of a thousand, you would need 1,000 tickets from the first lottery! You can verify this as follows: Assume your chances of winning are the number of tickets you own out of the number of tickets sold. So if you own 1,000 tickets in a 1,000,000-ticket lottery, your chances of winning are:

1,000/1,000,000 = 1/1,000 = 1 ticket owned/1,000 sold

In this activity students investigate concretely just how big one million is and explore the chances of winning a one-in-a-million draw.

Making It Work

Objectives

Students investigate the chances of picking one object out of a million by attempting to model one million concretely.

Materials

Materials will vary depending on students' choices of model.

[check] a small bag of rice and a scale or measuring spoons may be useful to introduce the activity

[check] calculators (optional but helpful)

Preparation

Counting out and measuring or weighing in advance 100 grains of rice may save time but is not necessary.

Procedure

1. Challenge students to think about the question of how big 1,000,000 really is and to come up with a tangible and concrete but affordable model of 1,000,000 objects.

2. Have students discuss their ideas in pairs or small groups.

3. Ask students to estimate the amount of various materials required to model their ideas.

For example, if they suggest grains of rice, have them do some quick counting and estimating that will show this idea to be unrealistic — depending on the size, it can take close to 30 bags of rice to make 1,000,000 grains!

Many ideas (such as rice) work well for 10,000 or 100,000 objects, but the number 1,000,000 makes things just that much more difficult.

4. Encourage students to make their models both affordable and transportable.

Suggestions

Since a main point of the activity is for students to devise and calculate the amount needed for various substances, suggestions of suitable materials should be withheld as much as possible. The investigation of unsuitable materials is fruitful in itself. When I first did the investigation, I was staggered at just how much rice 1,000,000 grains really was! However, here are a few suggestions to illustrate possibilities if really needed:

* Use a computer to print out a page of o's or any other character on the smallest possible printer font. Count the number of lines of print vertically, and the number of characters across, and use a calculator to multiply length times width of the characters to determine the number of characters on each page. This number can be divided into 1,000,000 to determine the number of such pages needed. Then, color in one o to show the "one" in one million. I keep a display like this in my classroom and vary the location of the "one" colored in; students love to search for the "one."

* Use Internet research to estimate the number of hairs the average person has on his or her head. (It's about 100,000.) Assemble 10 people — 10 times 100,000 is 1,000,000. Ask one person to volunteer one hair to serve as the "one."

Assessment

Students' models should be reasonably accurate but affordable and portable. I also like to see the entire 1,000,000 — as opposed to students saying, "Well, if you had 10 of these ..." — because it makes the activity more challenging. The "one" should be identifiable.

Math Magic

The BIG Idea

Lots of cool puzzles have math as their basis.

Content Areas in This Activity

[check] Numeracy

Process Skills Used in This Activity

[check] Problem solving

[check] Reasoning

Prerequisite Knowledge and Skills

[check] Place value for large numbers

[check] Basic algebra (to solve Puzzle 2)

Age Appropriateness

Children of all ages can enjoy doing the puzzles. Puzzle 1 is solvable with knowledge of multiplying by 1,000. Puzzle 2 may require algebraic equation solving to explain fully.

The Mathematical Idea

Puzzles 1 and 2 draw on the notion of place value. Puzzle 1 uses the fact that multiplying by 1,000 shifts the digits of a number left by three places. The numbers used in the "magic" of the puzzle have a product of 1,001 ... so multiplying by 7, then 11, and then 13 is essentially the same as multiplying by 1,001 (7 × 11 × 13 = 1,001). For example, 123 × 1,001 is the same as 1,000 × 123 plus 1 × 123, or 123, 123.

Puzzle 2 uses the fact that multiplying by 9 is the same as multiplying a number by 10, and then subtracting the number from the result: 9n = 10n — n.

Puzzle 3 uses the fact that binary numbers — numbers that are powers of 2, such as 1, 2, 4, 8, 16, etc. — can generate all other numbers by being added together. For example, 12 = 8 + 4 and 63 = 32 + 16 + 8 + 4 + 2 + 1.

One is a binary number because it is two to the power zero. (Recall that any number to the power zero is one.)

Details are found in the puzzle answers at the end of the Activity...