Question

Dear Mitch,

As I'm sure you know, Friday, February 22^{nd}, is the anniversary of George Washington's birthday.

Would you happen to have any ideas for some math that would tie in to some aspect of our first president?

Sincerely,

Judy Elfenbach

Resource Room Instructor for Gifted Children with learning differences that make an "ordinary" classroom a difficult place for them to learn.

Baltimore, MD

Answer

Dear Judy,

Yes, of course!

One of the most important mathematical skills to develop for intelligent functioning as students move on in life is **estimation**. As I have said before (perhaps even somewhere on this website), it is usually far more important to be able to *estimate* how much gas you need in your car's tank before embarking on a journey than to be able to calculate the exact rate at which your vehicle burns that fuel.

So, for George Washington's Birthday, try this:

Break the class into groups, and provide each group with a one-dollar bill. If possible, it is a great help to provide each group with a magnifying glass and flashlight as well.

NEXT: Point out the obvious: The centered portrait.

(If you or another teacher in your school has an overhead projector transparency of the U.S. dollar bill, that would eliminate the awkward set-up of providing dollar bills to students for even a brief class period, and it would eliminate the need to form separate groups, but, of course, having such an item is not necessary.)

Educational companies make artificial dollar bills for classroom use, but for this particular activity the currency has to be realistic enough to be printed with certain details that such products often lack.

Of the many options, my personal favorite is to have super-sized copies of the dollar bill, but the problem is that a lot of commercial copiers now have a piece of built-in software that stops all operation of the machine and including the typical "restart" switches, and refuse all restart efforts until the 'matter' of someone attempting to make a copy of any form of money has been investigated. Why? Because the photocopying of legal tender is not a joke, and has never been taken lightly by any thinking person I know. In fact, in recent years, as the quality of color copiers has risen, the duplication of money has become a real concern, for even the best intentioned teacher could, with pure educational intent, end up after a long day mixing up real bills with the counterfeit. In fact, most educational companies have gradually moved away from authentic-looking products and have redesigned their artificial money to be less realistic than previous versions. Though understandable, it is unfortunate because part of its educational success requires that it be close enough to make the learning easy to transfer from the model used in school to real bills students will be using in the real world. And if the copier you have is the type to even temporarily self-destruct when sensing any form of counterfeiting, one must be very careful. One could have the best motive for the activity, but counterfeit is counterfeit; and confusing one of yours with one of the nation's can happen in a single distracted moment -- thereby adulterating the integrity of the currency in circulation. When I directed a large mathematics department, my compromise was this: I played it safe by either making the bills ENORMOUS or printing them on fluorescent green paper and labeling each and every bill "NOT REAL MONEY/ FOR EDUCATIONAL USE ONLY."

So, now that we've included a disclaimer, here's the George Washington Day Math of Estimation:

Direct the class' attention to the side showing a pyramid and an eagle. Give them five seconds to study the eagle, then have them TURN THE BILL OVER so they can no longer see that image.

Have them raise hands to volunteer estimates for the number of feathers depicted on the eagle's tail. Without drawing attention to HOW you are categorizing their estimates, write them on the board in two columns or clusters. Make one group of the estimates that are odd numbers, and make the other group from estimates that are even.

Watch to make sure that no one is turning their bill back to 'cheat'.

When all estimates are in, announce that you are not sure how many feathers there are, but you strongly suspect that if it is up on the board, it is in the cluster of numbers that are odd. Ask: WHY?

Have them turn the bill over again so they can take a second look (or if it is on the wall from an overhead, simply turn it on and off for each activity).

Socratically help the class develop and understand the concept that the tail appears symmetric, with one feather in the center. This would necessitate their number to be odd. Discuss.

Provide about ten seconds for the class to observe and then estimate the number of feathers comprising both wings.

Bills are then turned over again to conceal the eagle.

Cluster estimates into two groups: one odd, the other even.

Develop the idea that since the two wings appear to be mirror images of each other, their combined feather-number is likely to be even. Ask whether it would matter if each wing is comprised of an odd or even number of feathers. Discuss pattern:

Odd number plus odd number equals even number,

Even number plus even number equals even number, and

Odd number plus even number equals odd number.

Have students take the time necessary to count feathers and assess their previous estimates.

NEXT: Allow ten seconds for the observation of the shield in front of the eagle.

Take estimates for the number of vertical stripes that form it.

Odd or even?

Odd.

Why?

Same reasoning as was applied to the tail...

Have students take the time necessary to count the stripes and assess their estimates.

To be continued...

Hope this helps!

Mitch