Thursday, April 24, 2014

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 need to answer 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.

If you would like a resource about this very topic click here:  Why We Can't Divide By Zero

Wednesday, April 16, 2014

He IS Risen...Just as He Said!

For me and my family, next week is Holy week.  Next Thursday we celebrate the Lord’s Supper together at our church because it will be Maundy Thursday which may be unfamiliar to some of you.

Maundy Thursday always falls on the Thursday (when else?) following Palm Sunday which was the triumphal entrance of Jesus into the city of Jerusalem. Maundy is from the Latin "mandatum" or command (John 13:34, "Love one another as I have loved you.") and reminds us of the institution of the Lord's Supper (Communion). The Last Supper commemorates when Jesus shared the Passover meal with His twelve disciples previous to His betrayal, trial, and crucifixion as described in the gospels of Matthew, Mark, Luke, and John.  During the meal, Jesus gave the disciples bread and wine which symbolized His body and His blood.  Christians partake of communion to replicate Jesus' giving of the bread and wine and to remember His death on the cross. While Palm Sunday is usually more joyful, Maundy Thursday services are more somber since it marks the betrayal of Jesus by Judas Iscariot.

For most schools here in Kansas, there is no school on Friday because it is Good Friday.  Good Friday is always two days before Easter and commemorates the crucifixion of Jesus Christ and His cruel death at Calvary. Christian theology teaches that Christ's death is the perfect atonement for sins, and as a result, the crucifix, or cross, is one of the fundamental symbols of Christianity.

The end of Holy week is Easter, when our family gathers at church to celebrate Jesus Christ's resurrection from the dead.  The New Testament teaches that the resurrection of Jesus is the foundation of Christianity.  It establishes Jesus as the powerful Son of God and as a living Savior because He conquered death through His resurrection. It is a day of joyous celebration when many hymns as well as arrangements of special music and songs of the faith are sung.  I would like to share one of my favorites.

Christ Arose
Lyrics and Music by Robert Lowry (1826–1899)

Verse 1
Low in the grave He lay,
Jesus my Savior,
Waiting the coming day,
Jesus my Lord!

Up from the grave He arose,
With a mighty triumph o’er His foes,
He arose a Victor from the dark domain,
And He lives forever, with His saints to reign.
He arose! He arose!
Hallelujah! Christ arose!

Verse 2
Vainly they watch His bed,
Jesus my Savior;
Vainly they seal the dead,
Jesus my Lord!

Verse 3
Death cannot keep its prey,
Jesus my Savior;
He tore the bars away,
Jesus my Lord!


Thursday, April 10, 2014

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.   You might also like Milk Lid Math.  This four page handout contains numerous math activities that utilize a free manipulative. 

Thursday, April 3, 2014

Happy Earth Day

Happy Earth Day Everyone! 

Earth Day is observed each year on April 22nd.  The purpose of the day is to encourage awareness of and appreciation for the earth's environment. It is usually celebrated with outdoor shows, where individuals or groups perform acts of service to the earth. Typical ways of observing Earth Day include planting trees, picking up roadside trash, and conducting various programs for recycling and conservation.

Symbols used by people to describe Earth Day include: an image or drawing of planet earth, a tree, a flower or leaves depicting growth or the recycling symbol. Colors used for Earth Day include natural colors such as green, brown, or blue.  The universal recycling symbol as seen on your left is internationally recognized and used to designate recyclable materials.  It is composed of three mutually chasing arrows that form a Mobius strip which, in math, is an unending single-sided looped surface.  (And you wondered how I would get math in this article!?!)  This symbol is found on products like plastics, paper, metals and other materials that can be recycled. It is also seen, in a variety of styles, on recycling containers, at recycling centers, or anywhere there is an emphasis on the smart use of materials and products.

Free Handout

Inspired by Earth Day, Trash to Treasure is an eight page FREE handout. It features clever ideas, fun and engaging mini-lessons in addition to cute and easy-to-construct crafts, all made from recycled or common, everyday items.  Discover how to take old, discarded materials and make them into new, useful, inexpensive products or tools for your classroom. To download the free version, just click under the cover page on your right.