Calendar of Best Fit

All my life I've operated under the assumption that a leap year occurs once every four years as our way to account for calendar discrepancies... a way of handling the actual length of a year: 365.25 days.

Vicky Rauch enlightened me a little today. In reality, the length of the year is closer to 365.2421897 days, and that's if you are measuring the mean distance between equinoxes, which is probably a good idea so that seasons stay intact. With this figure in mind, it has been calculated that we only need to have 97 leap years every 400 years! So, that's what we do. We, in fact, skip the leap year at the turn of the century 3 out of every 4 times. So, 2000 WAS a leap year, but 2100, 2200, and 2300 will not be. Cool.

But, here's a question, because surely even this system will not be entirely accurate: How long will it take before the calendar will be off by a full day, despite our leap year alterations? And for all you scientists out there, the tropical year (measurement based upon equinoxes) varies slightly from the sidereal year (measurement based upon earth's orbit in relation to fixed stars). How would the calendar change if it was based on the sidereal year (365.256363004 days)? And dare I even mention that the tropical year varies slowly as the years progress? The calendar is such an easy thing to take for granted!

Some fascinating classroom discussion and fabulous mathematics work might just ensue on Wednesday. Happy Leap Day!

Just for Fun: Problem 2

How do you find the center of the circumscribed sphere of any triangular pyramid (not necessarily regular)?

For Free or Not for Free?

For free or not for free: that is the question; 

Whether 'tis nobler in the mind to covet
The wealth and fame of outrageous fortunes,
Or to take arms against the seas of ignorance,
And by benevolence, end them? To donate: to cede;
No more; 

Honestly, I can see both sides:

• Providing original curricular resources openly and freely creates an atmosphere of collegiality and solidarity among teachers. Ideas are given more room to grow, although they are potentially less developed (which can often be a good thing). Plus, ideas can be more widely spread, since there is no cost involved.
• Offering original resources on a fee basis limits their influence to those that are willing to pay the price, but rewards the author for his time and creative genius. This incentive has the capacity to encourage greater care in the production, and can lead to higher quality and more thorough resources.

On a personal level, I believe in the benefits of benevolence, but I also obsess about perfection. After I have put together a lesson, activity, or unit for my students, I try it out. Sometimes it's great, sometimes, not so much. But then in the hours of afterthought and redesign, I try to address the quirks: design away the flaws, fill in the gaps, remove the bumps, and polish it up with some serious rationalizations. I have been known to spend an additional 17 hours on this revision process for a single lesson.

And you must be thinking, "Who has this kind of time?" 

I do. I suppose it's time for me to be transparent: I've been without a classroom since June. I've been embarrassed to admit it - afraid of a loss of credibility and upset by my role as victim of the down-turned economy. Nevertheless, here I am, hoping for a new position in September, and filling the time with lots of intense self-reflection and curricular revision. Sometimes I'm empowered to share my work freely, but lately I feel validated in asking a small fair price for my time and ideas.

And in the spirit of sharing, I'd like to open a forum for you and I to share some original resources. What is the best thing you created for your students? Link it up below, free or not. We'll let the submissions determine the mood of the masses. Add a couple things if you like, but please: 

• only post links to actual resources and not your general website or blog. 
• only middle/high school math products, like algebra, geometry, trig, calculus, stats, etc.
• free or cost items are both welcome. If you would like a nice recommendation for a marketplace to host your items, click here to join the TeachersPayTeachers community. You can give your things away or name your own price. It's a lovely community, and they could use some more good secondary math products.
• in the URL field, put the location of the actual product, and in the Name field, write a short description (subject and topic are good to know!)

Just for Fun: Problem 1

Because math is fun, and sometimes I like to work on interesting problems. 

AND because I see so many UNinteresting problems.

Just For Fun, Problem #1: