Leapin' Lizards - Leap Year is Mathematical!

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 2012 is a leap year.

But why is the word "leap" used?  Believe it or not, it has to do with 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 2010, Christmas was on a Sunday, but because 2012 is a leap year, this year, Christmas will be on Tuesday, not Monday.)

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.  

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.

Here is something fascinating for those whose birthday falls on February 29^th.  Over the course of their lives, these people will enjoy 75% fewer birthdays than the rest of us.  Does that also mean they are 75% younger, too?  So if your birthday is Leap Year Day, Happy, Happy Birthday to You!

Put a LID on It!

There are so many things we consider to be trash, when in reality, they are treasures for the classroom. One that I often use is plastic lids from things like peanut canisters, Pringles, coffee cans, margarine tubs, etc.  These lids can be made into stencils to use when completing a picture graph.

Students must first of all understand what a picture graph is.  A pictorial or picture graph uses pictures to represent numerical facts. Sometimes it is referred to as a representational graph. Each symbol or picture used on the graph represents a unit decided by the student or teacher. Each symbol could represent one, two, or whatever number you want.  This type of graph is used when the data being gathered is small or approximate figures are being used, and you want to make simple comparisons.

Here is what you do to make ready-made picture graph stencils.

• Choose the size of lid that you want and turn it over. Then trace a pattern on the plastic lid.  Make sure you are using the bottom of the lid so the rim does not interfere when the children use it to trace. 
• To make the stencil, cut out the pattern using an Exacto knife. You might choose to do zoo animals:  a zebra, a lion, a bear, an elephant or a giraffe. 
• Have a large sheet of paper ready with a question on it such as: “What is your favorite zoo animal?”
• The students then select the stencil (picture) that is their favorite animal and trace it in the correct row on the graph.

Below is a sample of this type of graph. It is entitled, What is Your Favorite Season?  A leaf is used for fall; a snowflake represents winter; a flower denotes spring, and the sun is for summer. Notice at the bottom of the graph that each tracing will represent one student.

You could craft stencils for modes of transportation, geometric shapes, pets, weather, etc. The list is infinite.  But what if you don't want to or don't have time to make all of those stencils? Then save the strips that are left when you punch out shapes using a die press. They are instant stencils!

If you are interested in additional graphing ideas, check out the resource entitled: Graphing Without Paper or Pencil. 

Also I am featuring a freebie that goes along with Trash to Treasure Ideas.  It is entitled: Milk Lid Math.  This four page handout contains numerous math activities that utilize this free manipulative. 

When Dividing, Zero Is No Hero

Have you ever wondered why we can't divide by zero?  I remembering asking that long ago in a math class, and the teacher's response was, "Because we just can't!"  I just love it when things are so clearly explained to me. So instead of a rote answer, let's investigate the question step-by-step.

The first question we we need to anwer is what does a does division mean?  Let's use the example problem on the right.

1)   The 6 inside the box means we have six items such as balls.  (dividend)

2)   The number 2 outside the box (divisor) tells us we want to put or separate the six balls into two groups.

3)   The question is, “How many balls will be in each group?”

4)   The answer is, “Three balls will be in each of the two groups.”  (quotient)

                                      

Using the sequence above, let's look at another problem, only this time let's divide by zero.

1)  The 6 inside the box means we have six items like balls.  (dividend)

2)  The number 0 outside the box (divisor) tells us we want to put or separate the balls into groups into no groups.
3)  The question is, “How many balls can we put into no groups?”

4)  The answer is, “If there are no groups, we cannot put the balls into a group.”

5)  Therefore, we cannot divide by zero because we will always have zero groups (or nothing) in which to put things.  You can’t put something into nothing.

Let’s look at dividing by zero a different way.  We know that division is the inverse (opposite) of multiplication; so………..

1)   In the problem 12 ÷ 3 = 4.  This means we can divide 12 into three equal groups with four in each group.

2)  Accordingly, 4 × 3 = 12.  Four groups with three in each group equals 12 things.

So returning to our problem of six divided by zero..... 

1)   If 6 ÷ 0 = 0.......

2)  Then 0 × 0 should equal 6, but it doesn’t; it equals 0. So in this situation, we cannot divide by zero and get the answer of six.

We also know that multiplication is repeated addition; so in the first problem of 12 ÷ 3, if we add three groups of 4 together, we should get a sum of 12.  4 + 4 + 4 = 12 

As a result, in the second example of 6 ÷ 0, if six zeros are added together, we should get the answer of 6.   0 + 0 + 0 + 0 + 0 + 0 = 0    However we don’t. We get 0 as the answer; so, again our answer is wrong.

It is apparent that how many groups of zero we have is not important because they will never add up to equal the right answer. We could have as many as one billion groups of zero, and the sum would still equal zero. So, it doesn't make sense to divide by zero since there will never be a good answer.  As a result, in the Algebraic world, we say that when we divide by zero, the answer is undefined. I guess that is the same as saying, "You can't divide by zero," but now at least you know why.

A Plateful of Ideas

When I taught the primary grades in a Title I school, I often found homework was seldom returned. I knew I had to come up with an idea that would be unique; something the parents would recognize as homework; something the kids would want to complete.  Thus a Plateful of Ideas was created. 

What this entailed was using paper plates on which the assignment was to be completed.  I bought about 300 at the local dollar store.  The children wrote their name on the back of the plate, and I would put the assignment on the front.  (Since I didn't want to write it 25 times, I would copy it, then glue it to the center of the plate).  Some sample assignments were:

1. Find pictures of things that are the color blue and paste them on your plate.
2. Find words that start with the letter "S" and glue them on your plate.
3. Find things that come in pairs or twos.  Paste the pictures on your plate. You may also draw items that come in twos.
4. Write as many ways as you can to add and get the answer of ten.
5. Write at least eight different three digit numbers on your plate.
6. Find pictures or draw pictures of at least six vegetables.
7. Around the rim of the paper plate, write the numbers from 1-25.
8. Around the rim of the paper plate, write all the alphabet letters as capitals.
9. On the plate, draw your family, including your pets.

The next day, those children who brought their plate back with the assignment completed would receive a small reward for doing so such as a sticker, a small box of raisins, a new eraser, a new pencil, etc. (I love the Oriental Trading Company for this!)  I would place the reward on their plate which, of course, would bring a big smile.  No plate = no reward!  As you can imagine, few plates were left at home, and few assignments were incomplete. I then displayed the plates in the classroom and used them during the day for sharing or reteaching, or just praising a child.  Since the child's name was on the reverse side, I could hang them up on the hall bulletin board as well.

I did not do this assignment every day, but at least once a week, the children would have a Plateful of Ideas assignment to complete.  Parents liked it because when they saw the paper plate, they knew their child had homework.  Children were encouraged by their parents to complete the assignment; so, besides the physical reward, they were given the much needed parental encouragement to do homework.  I found it to be a win-win situation for everyone.

So if your plate isn't too full right now, I hope you will give this idea a try. 

I have just posted a math freebie in honor of the Irish.  It's entitled: March Fraction Words and is a word/riddle activity that requires the student to practice identifying fractional parts.