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Once Upon a Time,

            you had a teacher that said to you that

"Mathematics is about you everywhere."

It's true.

There is rotation or reflection

 symmetry in the wings of a butterfly,

 

in the petals of a daisy

and many tree leaves.

There is the triangle shape of the top of our CLA building.   

There is the randomness of color in the coat of an alley cat kitty

Even stop signs display the geometric shape of an octagon.

                                      Numbers   

are about you everywhere.

They denote what time it is.

They are the essence of all money 

                  transactions.

They define the  gigabyte  storage space on the hard drive of your computer.

        Even when you drive your car, you are subtracting the number of gallons of gasoline you use from what was the total number of gallons you had at the gas station

                            as well as adding to your

 stress level.

Every day

subtracts one more day from the total number of days left in this

                        year.     .  

The following analogy is given to show you the quintessence of our place value number system and the

    power     of the different algorithms that you take for granted every day. 

And what is an algorithm you ask?

Definition:  Algorithm:

  An algorithm is a rule, process, or method of performing a mathematical operation.  All of the algorithms we have ever used are based on the basic properties of the real number system. They are just abbreviated ways (short-cuts) to get an answer to a mathematical problem in a quicker manner than going through all the theoretical steps that make the result valid. 

We especially call your attention below to the Cashier's Algorithm and how simple it is to use it to do all of your subtractions.

You use the Cashier's algorithm all the time and are rarely aware of it.

                  You use it every time you get groceries,

                                           buy shoes, or

 pay for anything.

Because the modus operandi of the algorithm works in any place value number system, we're going to go over it with you in base five to let us concentrate on the process of subtraction rather than wanting just to get the answer.

Ready, Set, Go!

You ask the students to imagine (pretend) that they are going to take a vacation on Hand Island.  You explain that this place is called "Hand Island" because the people there consider a group to be the number of fingers they have on one hand. They count as follows:

      0, 1, 2, 3, 4, five by which they mean one hand's worth or five things. They use place value to write their numbers so that one zero, "10 " stands for the quantity of one group of five objects, any five objects, rocks, books, candies, etc.

 While vacationing your mother asks if you would like to have your allowance so that you can go to the Kandy Store and buy a piece of taffy.  By all means you say, "YES!"

Your allowance for this week is forty-two cents, so your mother gives you the least number of coins that totals that amount (she wants you to be able to walk to the candy store and not have your pockets so heavy that she has to drive you).

You get one-quarter, three nickels and two pennies. You check and agree that the six coins add up to forty-two cents.

    1Quarter + 3Nickels + 2Pennies

Your mother cautions you to be careful walking to the store.

         But you don't walk,

              you run to the store.

You get to the store and find that your taffy costs nine cents.

On Hand Island nine cents is written 14_five. 

You reach your hand into your pocket and pull out all of your money and remember that

You only have Two pennies.

What do you do?

You give the cashier two nickels

and

wait for the cashier to return your change.

You expect the Cashier to keep one nickel, give five pennies back to you for your second nickel, wait for you to count out and hand four pennies back at which time you expect the cashier to give you the taffy. 

After all, this is how you write out your pencil and paper subtractions in school, right!

But,

The Cashier takes both nickels and hands you the taffy and one cent in change.

T h a t' s    a l l !

As you walk out of the store, you mentally go over the arithmetic of the two options and see them pictorially in your head as follows:

Option 1.

 The Give Two nickels, Get Back Five pennies, Return Four Pennies and Keep One–penny Scenario.

+

=

+

+

Option 2.

The Give Two Nickels, Get One Penny and the Candy Back Scenario.

+

=

+

Which is the simpler procedure?

               Which is the simpler diagram?

v

You realize that the process of Option 2 is not only easier, but it also takes less time and it can be applied to any subtraction problem in any base. 

You're so excited; you can't wait to share your Cashier's  Algorithm with your classmates.                   

But how do you explain the process?

How do you put your thoughts into words that your classmates will understand?

You think for a while and carefully analyze the steps that were just taken. 

You realize that the process is nothing more than an application of the commutative, associative and distributive properties of the real number system you've been taught in school.  You see how these properties together with place value allow you to subtract the same way in any base.

You are so anxious to share your experience with the class that you can hardly wait for the next day's

    "Show and Tell ."    

You tell the class how you ran  to the store with the following coins:

	1Q +3N + 2P = forty-two cents.
	Taffy cost nine cents, so you need to pay the cashier the equivalent of one nickel and four pennies, or 14 ¢.

You said how set back you were for a moment because you had only

two (not four) pennies!

No mind, you also had two nickels.

You tell the class about how the cashier took your two nickels and handed you the taffy and one penny in change.

You tell them how you then realized that this is how subtraction is done in the real world every day. 

         You tell them that if they used what you called your Cashier's Algorithm for all of their subtractions, then they would never have to re-group (borrow) or exchange again.

You say that you could hardly wait to show them how the Algorithm works so they too can save time writing out their own subtractions if they wanted to.

Recap:

	The cashier takes your two nickels, subtracts four pennies from the value of one of the nickels  and gives you back one penny.
	Two pennies plus one penny is three pennies.
	Subtracting the second nickel, you have one quarter, one nickel, three pennies in your pocket and the candy in your hand.

Walla !   That is the

                       end

                           of your story.