Time is divided into units of 60 because 60 was the basic unit of the Babylonian counting system. Why did they use 60? This motivates discussioninvestigation about factors and number bases. This can lead on to "clock" arithmetic, a very rich source of mathematics, which can provide useful mathematical activity from primary school to undergraduate level.
See www.shodor.orginteractivateactivitiesclock1NoNo and www-math.cudenver.eduwcherowiclockar.html.
Seven days in a week equals a quarter of a lunar month.
There is lots of mathematics in sundials, particularly geometry and measurement. See www.sundials.co.ukprojects.htmNoNoedial. A classroom project to make a sundial would be a worthwhile activity. See also Peter Ransom's piece on Sundials in Mathematics in School Vol. 27 No. 5.
The period of the pendulum is an excellent mathsscience investigation for secondary school. What effects the period? How long does a pendulum have to be to have a period of one second?
Navigation involves much geometry. Why is knowing the time accurately necessary to work out longitude? How do latitude and longitude define position on the surface of the earth? How is a sextant used to measure latitude and longitude? See www.mat.uc.ptheliosMestreNovemb00H61iflan.htm.
Given that the Julian calendar is out by 11 minutes a year, how many years before the error grows to one day?
Given that there are 24 time zones, why does each zone cover 15 degrees of longitude?