This assignment is based on the work that we have done in class from Section 2.1. However, the assignment is not simply to give answers to the exercises. Note that you can discuss this assignment with other people in the class, but you should write up your own answers to turn in. The assignment is as follows:

Part A)  In class, we came up with an answer for Exercise 4 that had to do with first "reducing" the board size and then checking whether the width and height of the reduced board size are even or odd. State the procedure carefully: Starting with two numbers h and w that give the height and width of the board, describe exactly what you do with the numbers to determine which corner the ball ends up in. Apply your procedure to a 660-by-550 board. (Don't draw the board, just figure out where the ball ends up!) Also turn in the drawings of boards that you made for Exercise 2; for each board, write the reduced board size next to the board, and circle the corner where the ball ends up.

Part B)  Exercise 6 asked you to figure out a rule for the number of times the ball touches the sides of the board. The rule should take the height and width of the board, and say exactly what to do with those numbers to answer the question. Write a short essay that describes the rule that you came up with and also the process that you went through when looking for the rule. Apply your rule to a board of size 660-by-550.

Part C)  Exercises 9, 10, 11, and 12 ask you to come up with a rule for boards that have fractional widths and heights. The rule can do this by first transforming the board size into another board size that has the same answer and that you already know how to deal with. Find the rule, and show how you use the rule to answer exercises 9, 10, and 11. Also, explain why the rule is valid.

Part D)  We looked in class at "unfolded trajectories" and used them to explain the "odd/even" rule from Part A, above. Do the same for the "number of hits" rule from Part B. That is, write an essay discussing why the rule that you came up with for Part B is valid, based on the unfolded trajectory. Hint: how many times does the unfolded trajectory hit a fold, and why? Illustrate your answer with one or two pictures of unfolded trajectories. (I'll hand out graph paper for this purpose.)