In response to participant interest, the final full day of the 1998 Math Forum Advanced Summer Institute was re-dedicated to project work and participant activities. After some initial time for project development in the Dupont lab, the group moved across the hall for two presentations:
Rob Rumppe posed this question:
"A student has to be able to weigh any object in integrals from 1 gram to 255. If each weight costs $1, what is the least he'll need to pay?"
While several people soon arrived at an answer of $8 by directly listing combinations and then extrapolating to find a pattern, Rob showed another solution and that the answer to some problems may not be as interesting as _why_ that solution answers the problem, stating the first few cases of the optimal solution and then motivating and arriving at a binary representation of integers, a method that appealed to many participants. A fuller exposition of Rob's activity is on the Web: see Weights and Binary Digits:
Marjorie Ader then distributed handouts she uses as an introductory activity in her Algebra III class, where she emphasizes groupwork and brainstorming multiple strategies to answer this problem:
"A camel is sitting by a stack of 3000 bananas at the edge of a 1000-mile-wide desert. He is going to travel across the desert, carrying as many bananas as he can to the other side. He can carry up to 1000 bananas at any given time, but he eats one banana every mile. What is the maximum number of bananas the camel can get across the desert?"
After some trial and error, groups independently began to refine strategies for solving this classic problem, also found in the Ask Dr. Math FAQ at
Marjorie gave some hints, suggesting an incremental approach and pointing out that 1) the camel does not have to go all the way across the desert in one trip, but can cache bananas along the route; 2) the answer contains a fraction; and 3) the camel need not be thought of as traveling one mile, then eating one banana, but rather as continuously eating bananas at the rate of one per mile. Soon participants became so involved in sensitivity analysis or considering extensions of the problem by varying the parameters that the activity ran well past the time alloted for it.
For more about this presentation, see Marjorie's Camel Crossing the Desert:
After lunch and some more work on participant projects, John Goebel showed how Mathcad can be used to do and share real math on the Internet. He sends his distance learning students Mathcad files, which they are able to modify and save, and they then do labs on Mathcad and e-mail John their papers. Current versions of Mathcad allow you to embed links to other files or URLs, making them very interactive.
John also shared some of the capabilities of Adobe Acrobat Writer for putting math on the Web, as live linkable documents and as high-quality printable documents. For more about his activity, see Mathcad: Doing and sharing real math on the Internet:
Finally, June Dershewitz, Chris Alexander, and Judy Ann Brown of the Forum staff held a break-out session for those interested in their Puzzler software, currently under development, to be used to administer the Problems of the Week (POWs) offered by the Math Forum: