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Surprise! Leap Year Doesn't Occur Every Four Years

We live by and teach our students that there are exactly 365 days in a year. In reality, the earth turns approximately 365 and a quarter times (six extra hours) on its axis by the time it has completed a full year's orbit around the sun, which means that every so often the calendar has to catch up.  Since those six extra hours add up to 24 hours over the course of four years (4 × 6 = 24), our calendar includes a leap year every fourth year.  (It's similar to receiving a free ice cream cone after getting your frequent buyers card stamped the fourth time.)  That is the reason the month of February has 29 days instead of 28 for a total of 366 days in the year. 

This year of 2020 is a leap year, but why is the word "leap" used?  Believe it or not, it has to do with patterns. (All math is based on patterns!)  Typically, a calendar date that is on, say, a Monday one year will fall on a Tuesday the next year; then Wednesday the year after that, and so on. However every fourth year, thanks to the extra day in February, we "leap" over Thursday and that same calendar date lands on a Friday instead.  (For example, in 2016, Christmas was on a Sunday, but because 2020 is a leap year, this year, Christmas will be on Friday, not Thursday.)

Believe it or not, there is a mathematical formula for figuring out leap years. (Don’t you love it?) It goes like this: A leap year is any year whose date is exactly divisible by four except for those years that are divisible by 100, not 400. (No, I didn’t make this up!) So years that are evenly divided by 100 are not leap years; however, if the years are also evenly divisible by 400, they are leap years. BUT this is only a temporary fix! This will work for about 3,300 years, at which point we will be a day off - again!

For example, 1600 and 2000 were leap years, but 1700, 1800, and 1900 were not. In the same way, 2100, 2200, 2300, 2500, 2600, 2700, 2900, and 3000 will not be leap years, but 2400 and 2800 will be. Therefore, in a period of two thousand years, we will have 485 leap years. By this rule, the average number of days per year will be 365 + 1/4 − 1/100 + 1/400 = 365.2425, which is 365 days, 5 hours, 49 minutes, and 12 seconds.

So why does this formula have to be so difficult? Because, in reality, the exact number of days in a solar year is slightly less than 365.25 (365.242374, to be exact), so the algorithm is designed so that a leap year is omitted every so often to account for underestimating the length of the earth's orbit.

Unfortunately, there's an exception to the "divide by 4" rule.  (You knew there would be).  For some time, astronomers have been able to more precisely estimate the earth's orbit. In reality, that number is roughly 365.2422 days, or 365 days, 5 hours, 48 minutes and 46 seconds, just a smidgen under the 365.25 days previously discussed.  By comparing the numbers, we see that the number above is off by 26 seconds. To make up for this, a rule states there can only be 97 leap years over the span of 400 years, not 100 as you may think. [Source: U.S. Navy Astronomical Center] One way to remember the rule is this:  Years that occur at the turn of centuries such as 1900 and 2000 must be evenly divisible by 400. This is why 1900 wasn't a leap year but the year 2000 was.

Does A Circle Have Sides?

Believe it or not, this was a question asked by a primary teacher.  I guess I shouldn't be surprised, but in retrospect, I was stunned. Therefore, I decided this topic would make a great blog post.

The answer is not as easy as it may seem. A circle could have one curved side depending on the definition of "side!"  It could have two sides - inside and outside; however this is mathematically irrelevant. Could a circle have infinite sides? Yes, if each side were very tiny. Finally, a circle could have no sides if a side is defined as a straight line. So which definition should a teacher use?

By definition a circle is a perfectly round 2-dimensional shape that has all of its points the same distance from the center. If asked then how many sides does it have, the question itself simply does not apply if "sides" has the same meaning as in a rectangle or square.

I believe the word "side" should be restricted to polygons (two dimensional shapes). A good but straight forward definition of a polygon is a many sided shape.  A side is formed when two lines meet at a polygon vertex. Using this definition then allows us to say:

1) A circle is not a polygon.

2) A circle has no sides.

One way a primary teacher can help students learn some of the correct terminology of a circle is to use concrete ways.  For instance,  the perimeter of a circle is called the circumference.  It is the line that forms the outside edge of a circle or any closed curve. If you have a circle rug in your classroom, ask the students is to come and sit on the circumference of the circle. If you use this often, they will know, but better yet understand circumference.

For older students, you might want to try drawing a circle by putting a pin in a board. Then put a loop of string around the pin, and insert a pencil into the loop. Keeping the string stretched, the students can draw a circle!

And just because I knew you wanted to know, when we divide the circumference by the diameter we get 3.141592654... which is the number π (Pi)!  How cool is that?

If you are studying circles in your classroom, you might like this resource. It is a set of two circle crossword puzzles that feature 18 terms associated with circles. The words showcased in both puzzles are arc, area, chord, circle, circumference, degrees, diameter, equidistant, perimeter, pi, radii, radius, secant, semicircle, tangent and two.. It is a great way for students to review vocabulary.

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Heart Rebus Puzzles - Pictures that Represent the Word "Heart"

Hearts and Valentines Rebus Puzzles
 Only $4.75

Do you have any idea what a rebus puzzle is? Essentially they are little pictures, often made with letters and words, which cryptically represent a word, phrase or saying. My college students love to solve rebus puzzles like the one on your right. Can you figure out what the picture represents? The clue I will give you is that it has to do with an expression that contains the word "heart". (See the answer at the end of this post.)

Since it is close to Valentine's Day, and college students don't have Valentine's parties, I decided to create several rebus puzzles that represent familiar expressions that contain the word "heart". (e.g. "From the Bottom of My Heart" or "Cross My Heart") Each illustration uses a picture or symbol to represent a word or phrase. The students must use logic and reasoning skills to solve the 24 rebuses. (Yes, I ended up with 24!)

If you purchase this resource, all you have to do is copy the 12 pages of illustrations (two per page) using a color copier. If you do not have access to a color copier, you can enhance the hearts by hand coloring them or have a student help you color. (My grandkids are great at coloring!)

Free Resource
Each class period during the month of February, I put up two heart illustrations as a focus activity. However, you could place one or more up at one time or all of them up at the same time. As my college students enter the room, they try to figure out what heart expressions the two pictures represent. Sometimes they solve the puzzle immediately; other times it takes them a while.  But no matter how long it takes, my students find that it is fun and engaging, in addition to being a very challenging Valentine's Day activity!

If you aren't sure that you wish to purchase the full resource, download the free one! It contains four rebus puzzles ready for you to copy.

*The answer to the above rebus is "A heart full of love."  Did you get it?

Two Day Teachers Pay Teachers Sale

Teachers Pay Teachers is having a two day sale on Tuesday, February 4th and Wednesday, February 5th!

During these two days, many TPT sellers will offer sale discounts anywhere from 5% to 20% off. Everything in my store is on sale as well.  At checkout, you can receive another 5% discount by TPT if the special code of FEBSALE is entered. 

Free Resource for Grades 5-8
So here is your chance to purchase all of the products on your Teachers Pay Teachers wish list, and you don't even have to worry about shipping since the items are digital and available for download immediately after purchase. So grab that on-line shopping cart and rush on over to Teachers Pay Teachers to stock up.

By the way, when you stop by my store, be sure and download one of my free items that uses number tiles. Just click under the resource cover on your right.

Want to win a $100 TpT Gift Certificate? Three will be given away. Check out the details at: a Rafflecopter giveaway