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It is important to distinguish the difference between the astronomical movements of the earth, due to the influence of the moon and the sun, from the choices made by different cultures about the division of the day into 'hours' and the year into 'months' or 'seasons'. From the earliest times, people had difficulties in matching the motion of the moon (the lunar months) with the apparent duration of the year as measured by the equinoxes.
While we divide the year into 12 months, and use this for our international calendar, other cultures have had (and still have) different ways of organising the year. It would be interesting to explore the Chinese, Indian, Hebrew and Islamic calendars that are still in use today, and the Mayan calendar, where they calculated a slightly better value for the length of the year than the Gregorian Calendar For more information, see http://astro.nmsu.edu/~lhuber/leaphist.html and http://webexhibits.org/calendars/calendar-mayan.html
While we base much of our payment of bills and salaries on the calendar month, the state pension is paid into bank accounts every Lunar Month (so pensioners receive 13 instalments each year). What other transactions are managed in a similar way? Which cultures traditionally have different dates for commercial transactions?
What are the advantages and disadvantages of analogue and digital representations of time? Pupils often see sports events on TV with a timer on the screen. Which sports display times in minutes, in seconds, in tenths of a second, (or even smaller) and why is it important? Where do pupils regularly meet time represented as a 24 hour system? Many questions like this occur as soon as we start talking about the nature of time and our use of time measurement, rather than just the arithmetic of 'telling the time'.
Sun time and clock time are only the same at four dates during the year. The variation has two causes; the plane of the Earth's equator is inclined to the Earth's orbit around the Sun, and the orbit of the Earth around the sun is an ellipse and not a circle. The National Maritime Museum website shows two separate graphs for these causes, and a third graph where they are combined to give the full correction. If you have a sundial available nearby, check your clock time against the sundial time at different times of the year and plot the variation on a graph. For older pupils this is a good practical example of how two graphs can be 'added' together.