Sometimes the most interesting math projects are staring us right in the face. For me, it was a series of realizations about the calendar.
- The days-per-month are a mess. As a grown-up, I still cannot remember which months have 30 vs 31 days.
- Weeks don’t fit into months evenly.
- Weeks don’t fit into the year evenly (there are not an even 52 weeks per year!)
- The calendar is human-created… so why doesn’t any of this work nicely?
- Months like September, October, November, and December have roots that don’t line up with their number. Sept means 7, Oct means 8, Non means 9, and Dec means 10. They’re all off by two! 🤔
These questions led to more questions:
- Why do some months have a number root (Sept, Oct, etc) and some don’t?
- Where the heck did the names of the weekdays come from? I get Sunday and Mo(o)nday, but what’s a Wednesday?
- 365-days-per-year comes from the earth’s movement. But why are there seven days in a week?
Eventually, all of this turned into a messy math project about factoring with loads of cross-disciplinary options.
Any Calendar’s Problem
The unchangeable, core problem we have is that, because of the earth’s movement, we’re stuck with 365 days in a year. For some reason (another great investigation opportunity) we have ended up with 12 months. I’d point out to students that this is a factoring problem.
Because 12 is not a factor of 365 (ask students to calculate 365 ÷ 12 to see the problem), we end up with 30-day months and 31-day months. Plus weird February (another great investigation opportunity).
So my challenge to students is to create a better system than 12 months of 30/31/28 days. Plus we’ll add weeks in as well. Shouldn’t weeks fit nicely into months and years?
Students will quickly discover that we earthlings have been cursed with a horrible number. 365 is very nearly prime. In fact, we call it a semiprime (another great investigation opportunity!) since it only has one set of factors besides than 1 and 365.
Our only options are either 5 months with 73 days each, or 73 months with 5 days each. Neither one is that great. Let’s discuss the tradeoffs with those options, class!
What if we tried 10 months? 11 months? 13 months? 9 months?
This is such a great example of a fuzzy problem. Even our optimal outcome is not totally clear.
What Other Folks Have Done…
We have (yet another) opportunity here for exploring. What have other cultures done about dividing 365 into nice months?
Students might discover solar vs lunar vs solar-lunar calendars. Encourage them to stay with solar (unless you’d like to venture into a whole different problem!). Eventually, they’ll note that many cultures have tried intercalary months, which are little chunks of time that aren’t actually in a month at all. They are often used as a time for celebration.
We kinda do this with the period of time between Christmas and New Years Day. Those days almost exist outside of the usual calendar, but other systems have made this official. Then one could have 10 months of 36 days, plus a 5-day festival time. On a leap-year, that festival could extend to 6-days. So many possibilities!
Some starting points:
So, after I went on this whole journey, here’s what I’d ask from students:
- Create a nice set of months with an equal number of days in each month.
- You may create an intercalary period outside of these months.
- Create a system of weeks that will fit into those months and into the year evenly.
- Research the origin of the names of the months and weekdays, then create your own names yours.
- Explain the tradeoffs you had to make to get this system to work.
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