Sometimes the most interesting explorations are staring us right in the face. This math calendar project began with a series of realizations:
- The days-per-month are a mess. As a grown-up, I still cannot remember how many days are in a given month.
- Weeks do not fit into months evenly (30 ÷ 7 or 30 ÷ 7 are both messes).
- Weeks don’t fit into years evenly (no, 365 ÷ 52 ≠ 52)
- Since the calendar is human-created, why doesn’t it work out nicer?
- Many months have number-related roots that don’t line up with their month number. Oct means 8, yet it’s the 10th month. Dec means 10, yet it’s the 12th month. See September and November as well.
These questions led to more questions:
- So, why do some months have a number root (Sept, Oct, etc) and some don’t? Where did the other month names come from?
- Where the heck did the names of the weekdays come from? I get Sunday and Mo(o)nday, but what’s a Wednesday?
- But why are there seven days in a week?
- How have other cultures/civilizations handled all of this?
Eventually, all of this turned into a math calendar project about factoring with loads of cross-disciplinary options.
Any Calendar’s Problem
The core problem is that, because of the earth’s movement, we’re stuck with 365 days in a year. For some reason (another great investigation opportunity for students) we have ended up with 12 months. I’d point out to students that this is a factoring problem.
Ask students to calculate 365 ÷ 12 to see the problem. 12 isn’t a factor of 365, so, we end up alternating between 30-day months and 31-day months. Plus weird February (another great investigation opportunity for students – why doesn’t February fit in?).
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 the weeks fit nicely into months and years? Currently, neither months nor years break evenly into weeks.
So this is all a set up. Students are going to run into a delightful wall here.
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 beyond itself and 1. In this case: it’s 5 and 73.
So our options are either 5 months with 73 days each or 73 months with 5 days each. Neither one is 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 People 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 an end-of-year 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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