Big Ideas for Growing Mathematicians

Exploring Elementary Math with 20 Ready-to-Go Activities

Ann Kajander

• Publisher: Zephyr Press
• Year: 2007
• Print ISBN:
  9781569762127
• eBook ISBN:
  9781613741337
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Introducing sophisticated mathematical ideas like fractals and infinity, these hands-on activity books present concepts to children using interactive and comprehensible methods. With intriguing projects that cover a wide range of math content and skills, these are ideal resources for elementary school mathematics enrichment programs, regular classroom instruction, and home-school programs. Reproducible activity sheets lead students through a process of engaged inquiry with plenty of helpful tips along the way. A list of useful terms specific to each activity encourages teachers and parents to introduce students to the vocabulary of math. This second Big Ideas book covers more advanced concepts, with projects including "One in a Million," where children use grains of rice to model the probability of astronomical odds; "Triangular Tessellations," in which students investigate the geometry and variations created by repeating patterns; and "Fractions of Salaries," where kids use a real-world scenario to multiply and divide fractions. 

1 One in a Million

The BIG Idea

Just how big is a million really?

Content Areas in This Activity

• Numeracy
• Estimation
• Basic probability

Process Skills Used in This Activity

• Creativity
• Reasoning

Prerequisite Knowledge and Skills

• Multiplication
• 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:

HELPFUL TERMS

Probability: a fraction between zero and one representing how likely an event is to occur. For example, the chances of getting a two facing up when rolling a six-sided die is 1/6.

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.

• a small bag of rice and a scale or measuring spoons may be useful to introduce the activity
• 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.

EXTENSION ACTIVITIES

Once students have finished their collections, they can experiment with trying to identify the winning “one” at random. For example, if you experiment and put your hand in a container of 1,000,000 objects and pulled out one object, how many times will it take before you pull out the winning object? Or will you give up before succeeding? Students can also compare this with their success with much smaller collections such as those of 100 or 1,000.

One in a Million Activity sheet

Have you ever seen a “one-in-a-million chance” advertised for winning a lottery or a contest and thought about entering? How does a million (1,000,000) compare with other large numbers, such as 1,000? In this activity you will investigate the number 1,000,000 to see for yourself just how big it really is!

Your task is to experiment with various materials to estimate and calculate the amount of each material you would need to make a collection of 1,000,000. For example, you could calculate the number of bricks on the wall of a building and then calculate the number of walls like that needed to use 1,000,000 bricks. You could paint one brick red to show the “one” in the million. But there is a catch: in this activity, your collection must be portable and affordable. You must find a creative solution to the problem that you can assemble yourself and actually see in front of you. Estimation is OK, but remember that it is 1,000,000 you are after, not a smaller number such as 100,000.

When you have finished your collection, identify one object as the “one” in your million—perhaps you might color it. Just how long do you think it would take to pick out the “one” at random? Try it!

Be sure you can explain how you knew there were 1,000,000 objects in your collection.

2 Math Magic

The BIG Idea

Lots of cool puzzles have math as their basis.

Content Areas in This Activity

• Numeracy

Process Skills Used in This Activity

• Problem solving
• Reasoning

Prerequisite Knowledge and Skills

• Place value for large numbers
• 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 pr...

Table of contents

• Cover
• Title
• Copyright
• Acknowledgments
• Contents
• Introduction
• 1. One in a Million
• 2. Math Magic
• 3. Cooperative Problems
• 4. Lots of Dots
• 5. Average Pentominoes
• 6. Distributing Tiles
• 7 Guess My Rule
• 8. The Rice Problem
• 9. Lines, Squares, Cubes, and Hypercubes
• 10. Fractional Salaries
• 11. A Temperature Experiment
• 12. Circular Reasoning
• 13. Coins for Compassion
• 14. Toothpick Rectangles
• 15. The Stairs Problem
• 16. Gumball Boxes
• 17. The Medicine Experiment
• 18. Fabulous Fractals
• 19. Triangular Tessellations
• 20. Kaleidocycles
• Glossary
• Index