Wednesday, May 30, 2012

"Sum" More Quick Tricks

Sometimes, my students think, I am a magician who pulls answers out of a hat.  Over the years, I have learned that mathematicians are ingenious people who are always looking for quick and easy ways to do things.  Maybe that's why we now have graphing calculators and computer programs to figure taxes.

I have a friend who teaches math on the college level in North Carolina.  In fact, we have been friends since 6th grade, but that's another story.  When she read my posting of May 10th, she shared another trick for quickly finding a sum.  Her trick has to do with a sequence that begins with any number, with any number of terms as long as they are separated by the same amount.  For instance, the series below is a six number sequence with a difference of two between each number.
Here is what you do to quickly to find the sum.  Add the first and last terms.  5 + 15 = 20.  Now multiply by the number of terms which in this case is 6.  20 x 6 = 120   Finally, divide by 2.  So, mentally this is what it would look like.


Now, how many of you went back to add up 5 + 7 + 9 + 11 + 13 + 15?  Did you get the answer of  60?  Isn't it amazing!?!  Maybe math teachers are magicians after all!

   

Wednesday, May 23, 2012

The Left Angle Mystery

Geometry is probably my favorite part of math to teach because it is so visual.  Plus the subject lends itself to doing many hands-on activities, even with older students.  When our unit on points, lines and angles is finished, it is time for the unit test.  Almost every year I ask the following question:  What is a left angle?   Much to my chagrin, here are some of the responses I have received over the years.

  1)      A left angle is the opposite of a right angle.

  2)      On a clock, 3:00 o'clock is a right angle, but 9:00 o'clock is a left angle.


              3)      A left angle is when the base ray is pointing left instead of right.

    4)      A left angle is 1/2 of a straight angle, like when it is cut into two pieces, only it is the part on the left, not the part on the right.

5)      A left angle is 1/4 of a circle, but just certain parts. Here is what I mean.


Now you know why math teachers, at times, want to pull their hair out!  Just to set the record straight, in case any of my students are reading this, there is no such thing as a left angle!  No matter which way the base ray is pointing, any angle that contains 90is called a right angle.



If you would like  some different ways to teach angles, you might look at the resource entitled,  Angles: Hands-on Activities.

Wednesday, May 16, 2012

Recycled Butterflies

My two granddaughters are in preschool and of course, everything is new and exciting to them.  They came home one day with egg carton caterpillars.  I know most of us have made one of these in our lifetime, but to these two little girls, they were the best craft ever!

They told me that Mrs. Madison was raising butterflies in their classroom, and soon they would hatch.  Anticipation and excitement reigned until the day they came out of school telling everyone that one of the butterflies had hatched.  However, much to their chagrin, Mrs. Madison was going to let it go.  They just couldn't understand why or how their teacher could do that!

But, here is the good part!  The girls got to make a cocoon out of a toilet paper cylinder.  They covered it by gluing on white cotton balls.  Then the made a butterfly out of tissue paper and a small plastic bag tie.  They put the butterfly inside the cocoon and then pretended to have the butterfly hatch!  This was done over and over and over until the cocoon was no more.  Luckily, I was able to get pictures before both were literally destroyed!

Now, what does all of this have to do with math?  Well, based on a blog , Flapjack Products, I contenplated all the ways to use recycled products to make items for the classroom.  Thus Trash to Treasure was created.  It is 34 pages of clever art ideas, fun and engaging mini-lessons as well as cute and easy-to-construct crafts all made from recycled or common, everyday items. 
Trash to Treasure

Find out more than 14 ways to use milk lids for math.  Did you know that you can practice math facts using clear plastic containers?  Learn how to take two plastic plates and turn them into angle makers.  How about using two plastic beverage lids to make card holders for kindergartners or for those whose hands are disabled?  Discover ten ways to use carpet squares as well as nine ways to use old calendars.  How about playing hop scotch on old carpet squares? Were you aware that butter tubs can become an indoor recess game to practice addition or multiplication facts?  These are just a few of the fun and exciting activities that use recycled items found in this 34 page resource entitled Trash to Treasure.

Because these numerous activities vary in difficulty and complexity, they are appropriate for any  PreK - 3rd classroom, and the visual and/or kinesthetic learners will love them.  You can download a free eight page version at Trash to Treasure FREE  or check out the thumbnails for the entire resource by clicking under the cover page on the right.

Thursday, May 10, 2012

"Sum" Trick

In my last posting, I talked about how you would need 55 dots to make a Ten Black Dots book.  Normally, to find that answer, you would add the numbers together.

But did you know there is an easier way?  Take 10 and divide it by 2.  That equals 5.  Multiply 10 x 5 and you get 50 then add in the 5 which equals 55.  Too confusing?  Well let's look at it in groups that equal 10.

As illustrated above, 10 is by itself so it is 10.  Then if we group the numbers so that each group equals ten, we have four additional sets.  So all together, we have five groups of ten with five left over which equals 55.        5 x 10 = 50 + 5 = 55

This will work for every sequence of consecutive numbers which begins with one and contains an even set.  In other words, sets that contain 2, 4, 6, 8, 10, 12... numbers.  Merely divide the largest number by 2; multiply the largest number by the quotient, and then add the quotient.

Example:  14, 13, 12, 11, 10,  9,  8,  7,  6,  5,  4,  3,  2, 1

This will also work for an odd numbered sequence like 11 but the formula or quick trick for finding the sum is a little different.  As seen below, we again divide 11 by 2, which 5.5 or rounded up equals 6. Again we group sets of two that equal 11. There are five groups plus 11 by itself so that makes a total of six groups.


Since there are no numbers left by themselves, simply multiply 11 by 6 (the rounded up quotient) to get the sum which is 66.

I love to write a series of consecutive numbers which begin with one on the board, and have the students find the answer using their calculators while I do the math in my head.  Of course, they are amazed and swear that I have memorized the answer.  I then ask me to give me a series (not off the wall or so large that it would take forever to use the calculator) and again I quickly give them the answer.  I then teach them that math trick. 

Students love "tricks" like this, but I always burst their bubble by telling them mathematicians are astute people.  That's why they are always looking for faster, quicker, and smarter ways to do math!