“Aragon, how many times a day do the small and large hands on the clock have a 45 Degree angle between them?”
I love these moments with children: a natural inquisitiveness about the world around them and the patterns that emerge. We spent the rest of breakfast guessing the answer and coming up with a plan to measure it, and simulate it!
Over Thanksgiving week, we’ll investigate this question. I encourage you to use this extra time with your children this coming week to explore our world with the kids in your family – encourage them to ask questions and investigate with you!
May try this project with your kids and send us your methods – I would be honored to place them up on our blog.
Stay tuned for our report in a week or so!
UPDATE: November 29th, 2016
We started by writing down the angle of the big and small hands as a funcation of minutes after 12 O’Clock:
Let’s define some variables first!
M = minutes after 12
BHA = Big Hand Angle
SHA = Small Hand Angle
M / 60 * 360 Degrees = BHA
BHA = 6 * M
(this simply divides minutes after 12 (M) by 60 (60 minutes in an hour) to get a fraction of the hour, then multiply by 360 to get an angle.
The Small Hand Angle is similar, except it measures in hours, not minutes.
SHA = (6 * M) / 12
SHA = M/6
Now, we go back to our original question, when do the two hands create an angle of 45 Degrees?
That happens when the difference betwwen the big and little hans is 54 Degrees:
BHA – LHA = 45
Lets substitute our formaulas above for BHA and LHA:
6M – M/6 = 45
Create a common denominator (that’s why I needed to learn how to add/subtract fractions!)
36M/6 – M/6 = 45
35M/6 = 45
M = 6*45/35
M = 7.71 minutes
M = 7 Minutes 42 Seconds
So, at 7 minutes 42 seconds after middady (or midnight), the angle between the big and little hands will be 45 Degrees.
Check your clocks this weekend and see if we got the right answer!
Bonus points: how many times does this happen every 12 hours? What are those remaining times? We tried it in Excel – send us your graphs and we’ll post it here!