"Forever January" and some Roman history

In my classroom, one of my homeroom students writes the date up on the whiteboard each morning.  Same place, every day.  This past week, my last block of the day was working, and someone remarked, "We've been in January for a long time!"  None of us took notice, because this is middle school and the occasional non-sequitur is de rigueur.  Then we noticed the student looking curiously up at the date.  Gone was January 20th.  It was now mid-day January 80th!  (Middle-school mischief)   Interesting.  So, I prompted the students for a little extra credit.  Seems we had landed in an alternate universe where it is always January. If we are in the perpetual January world, where our February 1st is Januaryland's January 32nd, then what day on our calendar corresponds to January 80, 2016?  

Would love to see your comments with the date you think is the answer, and a clue about how you got there. (Don't use a calendar.  It's more fun.)

So, then I did a little research about January itself and made up the following problem for my other classes the next day.  You can click on Numa P's name to learn more about the 2nd king of Rome below.

 

Numa Pompilius reigned 715–673 BC was the legendary second king of Rome, who reigned from 715-673 BC, succeeding Romulus. Many of Rome's most important religious and political institutions are attributed to him. In fact, King Pompilius gave January and February their names.  January was named after Janus, the god of beginnings and endings.

There is a folktale that King Pompilius liked the name January so much, he decided not to have any other month names.  The whole year would be called January. So, what we think of as February 1^st would have been called January 32.  If the folktale came true and King Pompilius had his way, think about January 80^th, 2016.

a. What month and day would that be on our calendars?

b. What day of the week is it? 

  

 

Happy New Year - 2016 Fabulous Factoring and a Balrog

During my Christmas break, I began thinking about a good Do Now for the students on their first day back.  What could I do with the number 2016?  Hmmm.  Well, excitingly, it has SO MANY FACTORS!  That would be a great way to get their brains cooking again.  Back in October, my 7th graders explored 504.  We blew out the factor rainbow with a factor "man-oh-war", and proclaimed 504 our favorite number.  It was exhaustingly exciting.  But. Today, 2016 bumped 504 from its pedestal.  This was such a good review of divisibility rules, and dividing, and number patterns, and association.  Once we got into the two digit numbers, the students conjectured whether, for example, 28 would be a factor of 2016.  After all, 2, 4, 7, and 14 were all factors so didn't it make sense?  And they were right.  We're a little stumped why 27 is not a factor of 2016 because 3 and 9 are there.  Someone suggested maybe the larger factor had to be even, but there sat 21 in all its odd glory belying the hypothesis.  We think it might have to do with the fact that 9 is a power of 3 but we're not sure yet, and want to be able to make predictions. 
When you divide 2016 by 4 you get our old friend 504.  That was a huge help when we figured out that if 504 is a factor of 2016, then ALL its factors are also factors of 2016.  Truthfully, took a little convincing on my part, which I demonstrated by showing 36's factors and later on asking for 72's factors.  But, we were rusty from the break.
My 8th graders were worn out by this activity, as they were over a year from factor rainbows, etc.  I told them the sad news that next January, I wouldn't be able to entertain the future 8th graders with a  factoring adventure starring 2017 because, well, 2017 is a factoring dud. "What?!  That's not fair!  Why did we have to have the bad luck to have Jan 2016 in our 8th grade year!!?"  (Did I tell you I love middle school?) I did promise  to inflict 2016 on next year's 8th graders in December just before the break with a "So Long 2016" factoring event. That was some comfort.
So, my 7th graders, who love to factor, loved 2016.  It's not a factor umbrella.  It's not a factor Man-o-War.  It's almost akin to Tolkien's Balrog just before he wraps his sinewy glowing tendrils around Gandalf's leg and drags him into the depths of Moria, but I haven't told them that.  Yet.