A Penny for Your Thoughts
By Michael Naylor

Penny flash cards feature pennies grouped in ways that help develop addition skills.
Artwork: Laura Bethel-Sehn

Beans or washers may be cheaper, but there's nothing like a bowl of shiny pennies to grab kids' attention. Ask your bank for rolls of new, shiny pennies, add some interesting number and spatial reasoning puzzles and your small investment in learning will pay off big!

Penny flash cards (Grades K-2)
Glue arrangements of three to 10 pennies on cards. Show one card to your students for one or two seconds, then hide the card. Ask for guesses as to how many pennies are on the card. Show the same card again and ask for more guesses. Have students explain how they knew the number of pennies. You'll find that your students have many different ways to see each arrangement. Discussing different students' methods leads to developing number sense and addition facts.

Penny jar (Grades K-8)
Keep a jar for pennies in your classroom and have the kids bring in any pennies they find. Every week or two, have students estimate the number of pennies in the jar. Small groups can take turns counting the pennies each week. At the end of the year, use the money to buy something for the entire class.

Penny nim (Grades 3-8)
First demonstrate the ancient game of "nim" on the overhead projector. Fifteen pennies are spread on the table between two players. Players take turns taking one or two pennies at a time. The player who takes the last penny wins. Have your students try to find a winning strategy. They'll find that it's good to leave your opponent with three pennies, because no matter if the opponent takes one penny or two, you'll be able to take the remaining pennies and win. This strategy can be extended: if you can leave six pennies for your opponent, then you can guarantee you'll be able to leave them three pennies on your next turn, which will enable you to win the game. (Now, how can you guarantee you'll leave your opponent six pennies?)

When students have discussed strategies and understand how to win, challenge them with one of these variations:

1. What if the person who takes the last penny loses?
2. What if a player may take one, two or three pennies on a turn?

Penny puzzles (Grades 4-8)
Why not add penny puzzles to your math activity center? Here are some of my favorites:

Rows of three. Nine pennies may be arranged as shown below to make eight rows of three. How can your students arrange nine pennies to create 10 rows of three?

Here's the solution.

U-turn. Arrange 10 pennies in an upward-pointing triangle, as shown at right. How can students move three pennies to make the triangle point in the opposite direction?

Here's the solution:

Celestial challenge. How can students arrange 10 pennies to make five rows of four? This puzzle is very challenging, with a pleasing solution:

Penny payout (Grades 5-8)
Pose the following math riddle and give your students some time to figure out the answer. Some students may discover an algebraic way to solve this riddle:

A famous story tells of a king who was so pleased about having learned how to play chess that he offered his chess tutor any reward the man desired.

The tutor replied, "I ask only for one penny on the first square of the chessboard, two on the next, four on the next, then eight, and so on, doubling the number of pennies each time, until every square is covered."

"Done!" the king agreed, thinking the tutor had made a very modest request indeed. How much money does the king owe his tutor?

Solution: The nth square will contain 2^n-1 pennies. The 27th square will contain 2²⁶ pennies, which is over one million dollars. The total amount of money on the chessboard will be about 184 quadrillion dollars – more than any king could ever have.

I hope you and your students enjoy "making cents" of these activities. I'll be back in the autumn, with more math ideas for your classroom. Have a great summer!

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

May, 2004, Vol.34, No.8