ESCOT Summer Workshop 2001Yoni Kahn's Thoughts on Extending Problems
Hispanola: Allow teachers to customize the sizes of the containers. To keep things interesting, don't allow a container size of 1. Also, add a third container that water can be dumped into, and make it display the amount of water it contains. However, do not allow water to be taken from this container. The third container allows students to get larger amounts of water, even with smaller cups. These modifications allow students to investigate the questions, "As long as one container is even and one is odd, can you get all amounts regardless of the actual size of the container?" and, "Which container sizes make getting certain amounts the hardest?"

Galactic Exchange: Allowing the customization of the coin values is possible, but since there are already so many different values on the practice applet, it doesn't really seem necessary.

Scale 'n Pop: Keep a version of the original problem, but add a more advanced problem on a separate page. In the more advanced version, the balloon expands at a definite rate as it moves (in units of: "new size = x initial diameters/foot moved * distance traveled") The balloon can be launched from two stations, one a given distance back from the other. The object is for students to figure out what initial diameter allows the balloon to be popped, and from that, to figure out the distance to the nails and the distance between the nails. All distances, rates, etc. should be customizable. This makes the problem much more appropriate for upper middle school/ high school students, as it involves solving simultaneous equations, etc, while still retaining the original objective of allowing students to experiment with an applet.

Marathon Graphing: Allow teachers to change the data to be graphed. Suggest using functions such as exponential functions and periodic functions to illustrate how, with certain sets of data, a best-fit line might not be very helpful. In addition, use data points that are relatively linear, and allow teachers to choose and "turn on" different objects on the graph based on different ways of measuring closeness with statistics (line of least squares, etc.)

Rumors: Allow teachers to increase the number of students in the school, to observe the long-term behavior of the graph. Make a graph and a simulation that models the situation in the bonus (i.e. Shanika tells 3 people, and each of the people she tells, tells 3 people, etc.), allow teachers to change how many people that each person tells, and add a Functions component to allow customizable graphs to be shown alongside the simulation graph. This allows an investigation of how the simulation resembles the behavior of exponential functions. In addition, students can discover just how fast rumors spread if each person tells a certain number of other people. Another suggestion: Allow teachers to program the simulation so that it meets certain parameters about how the rumor is spread. Students can suggest ways to program the simulation so that the graph of the simulation looks as similar as possible to a target graph. (This seems a bit complicated, but it's worth a try)

Search and Rescue 1: Make the map at the bottom an interactive applet, and allow the base and camps to be moved to anywhere on the map. This prompts an investigation into where the best base location is for minimum flying distance, given a certain configuration of the camps. Another suggestion: Bring the simulation closer to home by allowing teachers to make the background map a map of a familiar city, state, country, or whatever. Students can then investigate the headings and distances needed to get from one place to another in the real world. For more advanced math classes, make the simulation 3-dimensional, and all the locations customizable. Try to answer the question, "What are some good ways to specify locations in 3-D space?"

Search and Rescue 2: Allow more camps to be placed on the map in various locations. Have students experiment geometrically with finidng the best base location for minimum response time to each camp, based on the configuration of the camps. If similar configuations to those in Search 1 are used, students can compare the difference when base locations are used to minimize time, instead of minimizing total flying distance.

Fish 1: Along the lines of Jody's suggestion, have different "levels." Add two more levels from the ones suggested in her e-mail: one of them will have more complicated ratios such as 3 males: 5 females, 7 males: 2 females, etc. The last one will ask students to predict a solutoin to a fairly complicated situation algebraically, then test it using the applet. All equations will be customizable.

Fish 2: Make the number of fish in the lake customizable. Make it possible to add different types of fish (instead of male/female, the fish in the pond could be salmon, carp, koi, minnows, etc.). One possible activity could involve giving the students the ratios of fish in the lake (suggest making it fairly simple) and having students investigate different methods of getting their data samples to match the actual ratio.

Fractris: Make the size of the falling blocks customizable. Then blocks that don't fit with the given group of fractions, like the example of 1/5 in question 2, can be sent down, and students can investigate how close they can get to 1 with those fractions.

Mosaic: Allow the width of the mosaic blocks, as well as the length, to be set by the teacher or student. This encourages experimentation regarding how much better the estimate is when the width of the blocks is changed. Also, allow different images to be placed as the background image, to be filled up by Mosaic blocks, or allow students to actually draw their own images as a background.

Polyrhythms: Allow teachers to add as many rhythm lines as they want, and allow them to choose the available options for rhythms (1:2, 1:3, etc.). Allow teachers to create their own mystery rhythm for students to experiment with. Along those lines, students could also make up their own rhythm and trade it with other students in the class, who will try to discover what it is.

Pythagoras: Allow teachers to change the square and side length inputs to a circle and radius inputs, allowing students to determine the value of pi.