Clocks & Modular Arithmetic - A Middle School Investigation

[Did you think MathNotations was on hiatus? Actually, I've been working on a couple of investigations including an intro to the mathematics of circular billiard tables and the activity below -- hope you enjoy it...]

MathNotations has been invited to submit an article to Connect magazine. I'm considering something along the lines of the following investigation (the article would contain fuller explanations and additional teacher guidelines) and I would appreciate feedback particularly from middle school teachers. Feel free to suggest revisions, improvements, ...

If you have the time, as we approach the end of the school year, to implement some or all of the following, I would appreciate your observations. Also, what classroom organization (e.g., individual vs. small group) you used or what you would recommend. Thank you...

NOTE TO READERS OF MATH NOTATION: Your challenge is at the bottom!

CLOCK INVESTIGATION
Students are provided a handout with several clocks, numbered in the standard way from 1 through 12.

LEARNING OBJECTIVES/STANDARDS/TOPICS

• Divisibility concepts (remainders, lcm, factors)
• Repeating patterns (introduction to periodicity)
  NOTE: Later on, when students study the unit circle in trigonometry, they will encounter similar periodic behavior.
  
• Organizing data
• Developing effective communication - writing in mathematics
  

Part I
Place a marker at 3:00. This will be your START position. For the first part of this activity, you will be moving your marker FOUR hour-spaces in a clockwise direction from your starting point. So after your first move, you will be on 7:00. With your partner, record the results of each move up to 15 moves. You could of course mark it directly on the clock or you could make a table such as:

Start....3:00
Number of Move (N).................Position
1......................................................7:00
2......................................................11:00
...
15

Note: It's good experience for students to see that we often start indexing variables from zero, so instead of Start...3:00, one could start the table
0.....................................................3:00

Question 1: Try to answer the following without actually listing all the moves: What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves? Explain your reasoning or show your method.

Part II
Same starting point at 3:00, but this time you will move your marker FIVE spaces clockwise each time. Again, record the results of each move up to 15 moves.

Question 2: You should now have discovered that after 12 of these moves, you have returned to your starting point. Explain why at least 12 moves were needed (stating that you tried every move up to 12 isn't quite what we're looking for!).

Possible explanation (they may do better than this!): Starting position is repeated when the total number of hour-spaces moved is a multiple of 12. Since the the number of hour-spaces advanced after each move is also a multiple of 5, the position will repeat after 12 such moves. Note that 12⋅5 = 60 is the LCM of 12 and 5.

Question 3: Again, try to answer the following without actually listing all the moves:
What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves?
Explain your reasoning or show your method.

Question 4: In part I, you discovered that positions repeat after 3 moves. therefore, not all positions from 1 through 12 are reached. In Part II, you probably noticed that every location is reached. Explain both of these results in terms of divisibility.

Question 5: In both parts you started at 3:00. What results would be the same if you started from the 12:00 position? What results would be different?

Question 6: Devise at least one variation of your own for these clock problems. Extra points for most creative!
Sample: In addition to the obvious (changing starting position or number of spaces moved, you may want them to consider moving counterclockwise or changing the clock itself to 13 hours or some other variation).

Note: Students do not often consider generalizations (see challenge below) using variables to represent starting positions or the number of spaces moved each time. Middle schoolers may benefit from an introduction to such generalizations. I recommend only varying one of the parameters (either starting position or spaces). This would be appropriate for the prealgebra or more advanced student.

CHALLENGE TO READERS OF MATH NOTATION
Try to develop a general formula for the position of the marker after N moves given an initial position (S), number of hours on the clock (H) and the number of spaces moved (M). Also, an expression for the least number of such moves required to return to one's start position.