Chapter 2 - The Clock Problem

You can ask students to find the number of degrees in angle A of triangle ABC if angle B equals 72 degrees and angle C is 34 degrees, and that is a reasonable and acceptable math problem.

But if you ask a student to find the number of degrees between the hands of a clock at 4:08, then this is more interesting math problem.

What if you pose the problem in this way: "Find the number of degrees between the hands of a clock at 4:08. Explain how to do this problem as if you were explaining it to someone who did not know how to do the problem, and came to you for help. Include diagrams and a complete explanation. Then find the number of degrees between the hands of a clock at 4:32, and explain how to do this also. Are there any differences in the method used for 4:08 and 4:32?"

Now you have a challenging question, an interesting answer, and "writing in mathematics"!

This was actually the first writing assignment I had every assigned, and I really didn't know what to expect, both in terms of the student's reaction to the idea of writing in a math class, and is terms of what they might actually write.

The students had done four simple "clock problems" from the text book, problems such as "Find the measure of the angle between the hands of a clock at 3 PM". If you consider that question, you will realize that the answer is simple, because the minute hand is "straight up; (vertical) and the hour hand is directly on the numeral 3, aso they hands simply form a right angle. What complicates the problem for times such as 4:08 and 4:32 is that the minute hand moves as the hour hand moves, and the angle between them changes every minute. A superficial glance at a clock might cause one to think that the calculations for other times might be also simple, but they are not, and it is a rather sophisticated math problem, one that the students found challenging. They not only rose to the challenge, but wrote excellent explanations, and presented their work in creative and imaginative ways! As a teacher, I believe that it is extremely important to challenge students in this way, and to give them opportunities to rise to the challenge!

They worked in groups of 3 or 4, and turned in either a "poster" or a "booklet" with illustrations and explanations. The work that my students did in answering this question was fascinating! They included wonderful diagrams, and interesting explanations. Some solved the problem in ways I had not anticipated, and all of them learned from the experience. Some of their work on this writing project is truly excellent,

In their explanation, one group wrote a very detailed explanation, complete with carefully drawn diagrams:

The Clock Problem

By Jenna, Reyn, and Elizabeth:

"The basic principle starts with the fact that there are 360 degrees of angular measurement in a circle. This is applied to the Clock Problem because a non-digital clock is a circle."

"Next we had to determine the measure in degrees between the increment of 1 hour. Since there are 12 hours on a standard clock face we divided the total 360 degree measurement by 12 to get 30. 360/12=30. This tells us that the space between any 2 hours is 30 degrees."

"A clock's full measure is 360 degrees. Therefore, between each pair of numbers (i.e. 10 and 11) it is 30 degrees. We figured this out because there are 12 pairs, and 360 divided by 12 = equals 30. At 4:08,the clock looks something like this:

The students went on to give a very detailed written explanation of the rest of the steps in solving this problem. I don't know whether they drew the diagrams themselves, or what computer software they used, but they certainly did an excellent job. Other groups also drew diagrams as part of the solution, but theirs were less "professional", but their written explanations were equally good.

Here is another example of student work on this interesting problem:

THE CLOCK PROBLEM

by Chris, Sam, Lance and Jared:

"1) A clock face is divided into 12 hours, and 60 minutes. There are 59 marks on the clock, with 60 spaces between them in total. They represent every minute in the hour. There are also 12 hour marks that show the hours, represented by numerals 1-12. The entire clock face is 360 degrees. 360 divided by 60 is 6, so you know that there are 6 degrees between one minute mark and the next. 2) At 4:08, the minute hand is 8 spaces between minute marks away from the twelve. 6 times 8 is 48, so the minute hand is making a 48 degree angle from the number twelve."

"3.) The numeral four on the clock is 20 minute mark spaces away from the 12 because there are (60/112) 5 minute mark spaces in every space between the hour marks. So it is making a (20x6) 120 degree angle from the twelve.

4.) The hour hand also moves between the hour marks as corresponding to the minute hand going around the clock. The minute hand is 40/360 of the way around the clock, or 2/15. Therefore, the hour hand is 2/15 of the way between the numerals 4 and 5, making a 30 degree angle (5 x 6 = 30). 2/15 times 30 degrees is 4 degrees. Add that to the angle the four is from twelve (120 degrees) and you get that the hour hand is 124 degrees away from the twelve.

5) To find the number of degrees between the two hands of the clock, just subtract their angles from the number 12. The hour hand has a 124 degree angle, and the minute hand has a 48 degree angle. 124 minus 48 is 76. The two hands make a 76 degree angle; they are 76 degrees apart.

To find the number of degrees between the hands a on a clock at 4:32, we can use steps similar to the previous steps above."

"1) There are 6 degrees in the space between every pair of minute marks, and the minute hand is 32 minute mark spaces away from the 12. 32 times 6 is 192, so the minute hand is making a 192 degree angle from the twelve.

2) The numeral makes a 120 degree angle from the 12, so we must also add in the movement between the hour marks. The minute hand is 192/360 of the way around the clock face, or 8/15. There are 30 degrees between the numerals four and five, and 8/15 times 30 degrees is 16 degrees. 16 degrees added to 120 degrees puts the hour hand at a 136 degree angle from the twelve.

3) To find the distance between the hands, we will subtract again. 192 minus 136 is 56. The hands make a 56 degree angle between them. (They are 56 degrees apart.)

There is no difference in the two methods, except that in the later example we subtracted the degrees of the hour hand from the degrees of the minute hand instead of the other way around. I you wanted both methods to be the same, you could just say that the distance between the two hands is the absolute value of the difference of the angles made by the two hands to the numeral 12 (going clockwise)."

I frequently asked the students to write reflections on the process of solving a problem, and have included some of their reflections below. "I liked this project because I gained a lot from doing it. I also learned I could explain things clearer when using diagrams. It made me feel real proud to see what we accomplished." Jon R

Working in groups was very helpful, in that they were not overwhelmed by the task when they could get help from one another, and the interaction was wonderful to listen to. It was great to hear them argue about the math concepts, come to an agreement, and learn from each other.

One student, Rachel, actually created a formula for solving this problem! Here is her explanation:

"let H= hour and M = minute; (for example, at 3:45, H = 3 and M = 45).

Then the measure of the angle between the hands of the clock at 3:45 =

30 (H + M/60) - 6M)

Shen added the comment "if you get an angle more than 180 degrees, subtract from 360. If you get a negative angle, don't worry about it, just ignore the negative sign."

In their reflections on this project, students made some very interesting comments:

"I think that the clock problem was important because it took teamwork. At first I didn't understand what my group was doing, but when I chipped in, I understood what the group and myself was thinking. I had my own way of doing it, but I found out that there is more than just one way of doing something." Shelton L.

In every project, I asked them to write what they thought about the project, which parts they found challenging, and what they learned in doing it. They never ceased to amaze me with their work, and their comments. Here is one student's response:

"This assignment represents the biggest AHA moment of the course for me. At first I thought I would never get it. Suddenly it hit me that the angle increased 5 1/2 degrees per minute, and I was on my way to answering the problem." David S.

In many of the projects and other assignments in this writing-intensive geometry class, I asked my students to write reflections on what they experienced, and what they learned. I found the results of this particular assignment so interesting that I wrote my own reflections on it. This was actually the first writing assignment, and I really didn't know what to expect, both in terms of their reaction to the idea of writing, and in terms of what they would actually write.

They had done four simple "clock problems", as a warm-up excercise - finding the angle between the hands of a clock at

"If we do not expect the unexpected, we will never find it."

Heraclitus