MathNotations

[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.

Posted by Dave Marain at 3:48 PM 0 comments

Labels: clock problems, divisibility, investigations, middle school, modular arithmetic, periodic behavior

Have you submitted your vote yet in the MathNotations poll in the sidebar?

Target audience for this investigation: Our readers and algebra students (advanced prealgebra students can sometimes find a clever way to solve these).

Let's resurrect for the moment those ever popular rate/time/distance classics. Hang in there -- there's a more interesting purpose here!

We'll start by using fictitious presidential candidates running in a 'race.' Any resemblance to actual candidates is purely coincidental.

R and J are running on a huge circular track. J can run a lap in one month whereas it takes R twelve months to run the same lap. To be nice, J gives R a 3-month head start. After how many months will J 'catch up' to (overtake) R?

Are those of us who were trained to solve these feeling a bit nostalgic? Do you believe that our current generation of students has had the same exposure to these kinds of 'motion' problems or have most of these been relegated to the scrap heap of non-real world problems that serve no useful purpose. Well, they still appear on the SATs, a weak excuse for teaching them, perhaps, but I can certainly see other benefits from solving these. Can you?

Ok, there are many approaches to the problem above. Scroll down a ways to see a couple of methods (don't look at these yet if you want to try it on your own):







Method I: Standard Approach (using chart)

..............RATE ...x.......TIME .....=.....DISTANCE
............(laps/mo).....(months)............(laps)

R.........1/12....................t........................t/12

J................1....................t...........................t

Equation Model (verbal): At the instant when J 'catches up' to R:
Distance (laps) covered by J = Head Start + Distance covered by R

Equation: t = 1/4 + t/12 [Note: The 1/4 comes from the fact that R covers 1/4 of a lap in 3 months]

Solving: 12t = 3 +t --> 11t = 3 --> t = 3/11 months.

Check:
In 3/11 months, J covers 3/11 of a lap.
In the same time, R covers (1/12) (3/11) = 1/44 lap. Adding the extra 1/4 lap, we have 1/4 + 1/44 = 12/44 = 3/11. Check!

[Of course, we all know these fractions would present as much difficulty for students as the setup of the problem, but we won't go there, will we!]

Method II: Relativity Approach
Ever notice when you're zipping along at 65 mph and the car in the next lane is going the same speed, it appears from your vehicle that the other car is not moving, that is, its speed relative to yours is zero! However, if you're traveling at 65 mph and the vehicle in front is going 75 mph, the distance between the 2 cars is ever increasing. In fact, the speedier vehicle will gain 10 miles each hour! This 75-65 calculation is really a vector calculation of course, but, in relativity terms, one can think of it this way:
From the point of view of a passenger in the the slower vehicle, that person is not moving (speed is zero) and the faster vehicle is going 10 miles per hour. We can say the relative speeds are 0 and 10 mph.

Ok, let's apply that to the 'race':

If R's relative speed is regarded as zero, then J's relative speed will be 1 - 1/12 = 11/12 laps/month.
Since R is not 'moving', J only needs to cover the head-start distance to catch up:
(11/12)t = 1/4 --> t = (1/4)(12/11) = 3/11 months. Check!
[Note: Like any higher-order abstract approach, some students will latch on to this immediately and others will have that glazed look in their eyes. It may take some time for the ideas to set in. This method is just an option...]

There are other methods one could devise, particularly if we change the units (e.g., working in degrees rather than laps). Have you figured out how all of this will be related to those famous clock problems? Helping students make connections is not an easy task. One has to plan for this as opposed to hoping it will happen fortuitously.

Here is the analogous problem for clocks:

At exactly what time between 3:00 and 4:00, will the hour and minute hands of a clock be together?

Notes:
(1) I will not post an answer or solution at this time. I'm sure the correct answers and alternate methods will soon appear in the comments.
(2) A single problem like this does not an investigation make. How might one extend or generalize this question? Again, these are well-known problems and I'm sure many of you have seen numerous variations on clock problems. Share your favorites!
(3) Isn't it nice that analog watches have come back into fashion so we can recycle these wonderful word problems!
(4) For many problem-solvers, part of the difficulty with clock problems is deciding what units to use for distance (rotations, minute-spaces, some measure of arc length, degrees, etc.). This is a critical issue and some time is needed to explore different choices here.

Posted by Dave Marain at 6:20 AM 6 comments

Labels: algebra, clock problems, percent word problem, rate-time-distance problems

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