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.

### They’ll Discover…

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:

### The Task

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.

I turned the whole thing into two Byrdseed.TV projects: a math-focused one and an older one that emphasizes the social studies aspects.