EDSS 444 — Marks Fall 2007

PROBLEM OF THE WEEK (POW)

(adapted from Fendel et al., Interactive Mathematics Program, Year 3, Key Curriculum, 1999, pp. 100-101)

Problem

The king is planning a new rectangular courtyard in the palace, which will be laid out using square tiles. He has chosen some very pretty, but very inexpensive, tiles. However, his advisor has informed him that some of the tiles will have to be a more expensive type.

You see, visiting dignitaries will always walk along a certain diagonal, from one corner of the courtyard to the opposite corner, so the tiles along the diagonal will get lots of wear and tear. Therefore, those tiles must be more durable (and hence more expensive). That is, every tile that contains a segment of the diagonal must be an extra-strength, expensive tile. (If a tile touches the diagonal only at a corner, it can be a regular tile.)

For example, the diagram above shows what the situation would look like if the king made a 4-by-6 courtyard. In this case, eight tiles (shaded darker than the rest) include some portion of the diagonal, and so these would need to be the more expensive kind.

1. The king wants to know how many of these special tiles he needs to order. At the moment, he is planning to have a 63-by-90 courtyard. That is, he will have 63 rows of tiles with 90 tiles in each row. If he stays with this plan, how many special tiles will he need?

2. The king keeps changing his mind about the dimensions of the courtyard, so it would be expecially helpful if you could find a general formula for him. Suppose the courtyard has r rows with c tiles in each row. How many special tiles will the king need?

Writeup

Your writeup must include:

* The problem statement, rewritten in your own words.

* The process you went through to solve the problem; include false starts and dead ends as well as successful ideas.

* A clear statement of your solutions to parts 1 and 2.

* An argument explaining why you think each solution is correct.

* A brief reflection on how you felt about this problem.

The writeup must be neat, clear, and well written.

Assessment

Your writeup will be evaluated using the accompanying 6-point rubric.

LEVEL	STANDARD TO BE ACHIEVED
6	Fully achieves the purposes of the task, and in addition provides an interesting insight or extension. Specifically: * Presents a correct solution of both the general and specific cases, with clear and complete justification. * Explains how the solutions were obtained. * Communicates very clearly, using text, graphics, and symbols effectively. * Appears well-organized and neat. * Extends the problem in an interesting way or provides unusual depth of analysis.
5	Fully achieves the purposes of the task. Specifically: * Presents a correct solution of both the general and specific cases, with solid justification. * Explains how the solutions were obtained. * Communicates effectively, using some combination of text, graphics, and symbols. * Appears well-organized and neat.
4	Substantially completes the purposes of the task. Specifically: * Presents a correct solution of the specific case, with reasonable justification. * Solution of the general case may be not quite correct or complete, or may contain minor gaps in justification. * Explains how the solutions were obtained. * Communicates successfully. * Appears organized and neat.
3	Partially achieves the purposes of the task. Specifically: * Presents a reasonable solution of the specific case, with perhaps slight errors in the solution or justification. * Solution of the general case may contain significant errors but still shows reasonable understanding of the problem and progress toward solution. * Justifications may be logically weak or have missing steps. * Explains how the solutions were obtained. * Communication and organization are reasonably effective.
2	Important purposes of the task are not achieved. Specifically: * Solution of the specific case contains significant errors and indicates weak understanding of the problem or limited progress toward its solution. * Solution of the general case may be very limited, misdirected, or missing. * Justifications are incomplete or unclear or contain significant errors. * Shows some attempt at communication and organization.
1	Purposes of the task are not achieved. Specifically: * Shows little evidence of understanding of the problem or progress toward a solution. * Does not communicate relevant ideas successully.
0	No work submitted.