Cl ock Sums

The activity begins with an applet demo showing the hands of a clock moving in various positions while a number appears on the alarm bell at the top of the clock. The user is asked to determine what this number represents.

The next page allows the user to control the speed of the hands and to choose the base of the clock face. The following page allows the user to set the clock manually by clicking and dragging each hand. Whenever a time is set, a number appears on the alarm bell. Suggestions are given to help the viewer determine the relation between the time and the number on the alarm bell. Eventually the viewer discovers that the number on the alarm bell is the sum of the digits enclosed between the hands on the clock, starting from the hour hand and proceeding clockwise to the minute hand.

The next page allows the user to enter a number on the alarm bell (a candidate for the sum) and the computer automatically shows all positions of the hands corresponding to this number (if any exist), or announces that there are no solutions.

The next page presents a Clock Game. A number appears at random on the clock face, and the player's challenge is to place the hands in such a position that the sum of the digits enclosed between the hands is equal to the number on the clock face. The game can be played at different levels of difficulty. At level k the clock hands surround k clock digits. For example, at level 3 the number shown on a base 12 clock face could be 24, and there are two solutions: 7+8+9 and 11+12+1. The player will soon learn that the first solution can be obtained by averaging, because one-third of 24 is 8.

The ultimate goal of this activity is to have the user discover (with the help of tutorials) properties of sums of integers in arithmetic progression. Sometimes the numbers being added are consecutive, as in the foregoing example 7+8+9 = 24. The game becomes more interesting when the numbers being added overlap the clock base, as in the example 11+12+1 = 24.

Several variations of the Clock Game are also being considered. For example, the numbers on the clock face need not be consecutive but can be in any arithmetic progression.