#### Happy Leap Year!

Do you know why we have Leap Years?

It’s because the time it takes for the Earth to travel around the Sun takes longer than 365 days – 365 days, 5 hours, 48 minutes and 46 seconds, to be exact. Last week, our blog explored Patient Problem Solving, encouraging teachers to remove the unnecessary “givens” when it comes to solving a math problem, and instead, letting students think through and build problem solving steps on their own. Leap Day is a great opportunity to put this into practice, by going through an exercise to understand why we have Leap Years. Having your students come up with their own rules for how to account for the difference between solar and calendar years is a great way to appreciate the wonderful math behind Leap Day!

##### Part I: The Only Given

Start with this fact: **The exact time for the Earth to travel around the Sun, is 365 days, 5 hours, 48 minutes, and 46 seconds.**

Since our calendar year is only 365 days long, what happens to those 5 extra hours, 48 minutes, and 46 seconds? After 4 years? How about 100 years?

After 4 years, the extra time would put the calendar year behind the solar year (the time it takes for the Earth to travel around the Sun) by about a day. After a century, the difference would be 25 days!

##### Part II: Arriving at Every Four Years

How do we make up for that extra time in the solar year?

Without any calendar adjustment, the difference between the calendar and solar years will only grow larger. Soon enough, our months and seasons won’t sync up! As the general “every 4 years” Leap Year rule implies, the creators of our calendar decided to add an extra day to our calendar every four years. 5 hours, 48 minutes, and 46 seconds is almost 6 hours, and every 4 years, that adds up to 24 hours, or 1 day.

##### Part III. That Extra 11 Minutes, 14 Seconds

Here comes the lesser-known part: what happens to the 11 minutes, 14 seconds that we add per year to round out 5 hours, 48 minutes, and 46 seconds to 6 hours? That’s an extra 44 minutes, 56 seconds every 4 years!

Conclusion: every 4 years is not good enough. After 128 years, the the calendar would gain an entire extra day.

##### Part IV. What now?

At this point, students should recognize that the calendar needs to be shortened somehow to get rid of the annual excess of 11 minutes, 14 seconds.

The creators of our calendar fixed this problem by adding an exception: every year divisible by 100 is NOT a leap year unless the year is also divisible by 400. This rule eliminates three leap years per century. For example, there will be 8 years between leap years 2096 and 2104. The year 2100, while divisible by 100, is NOT divisible by 400. With this rule, it would take 3,300 years for the calendar and solar year to diverge by a day.

What kinds of solutions did your students come up with? Share with us on Twitter! @Knowre.

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