This problem solving activity has a geometry focus.
Hinea glances at her watch. At first she sees only one hand!
Then she realises that one is on top of the other.
She wonders how many times a day the hour hand and the minute hand on her watch are in the same position.
How many do you think?
This problem involves taking a problem about time, looking at it geometrically, and solving it algebraically. The problem is about when hands on a watch coincide. This is easily seen as a geometry problem as a diagram can be drawn to show the situation. Finding the algebraic approach may be more challenging. This requires knowing about the relative speed of the hands of a watch and the significance of angle.
Hinea’s Other Watch, Geometry, Level 6 is a similar problem.
Hinea glances at her watch. At first she sees only one hand! Then she realises that one is on top of the other. She wonders how many times a day the hour hand and the minute hand on her watch are in the same position.
How many do you think?
One way to do this is to turn the hands of a real clock around and count the number of times the hands overlap. Then ask how you can be sure that you have the right answer.
The minute hand goes through 360° in 1 hour. The hour hand goes through 360° in 12 hours, or 30° an hour. So the minute hand moves 12 times as quickly as her hour hand. While the hour hand is moving through an angle α, the minute hand is moving through an angle of 12α.
On the other hand, because they are on top of each other, 12α - 360° = α. So 11α = 360° or α = 360/11 = 32.72. This represents (32.72 / 360) x 60 minutes. This is approximately 5 minutes and 27.3 seconds.
The hands are therefore on top of each other at 0:00:00; 1:05:27; 2:10:55; 3:16:22; 4:21:49; 5:27:16; 6:32:44; 7:38:11; 8:43:38; 9:49:05; 10:54:33; and every 12 hours afterwards.
This therefore happens 22 times in a day.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/hinea-s-watch-s-hands at 9:00pm on the 26th February 2024