Mathematics

# How many Pieces Complete the Jigsaw Puzzle

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How many pieces complete the jigsaw puzzle?  Three, ten, twenty-five, 100, 500, 1000,...more?

Some children seem to enjoy the success of putting together jigsaw puzzles.

This is developed by encouraging these children to piece together puzzles with the fewest pieces first.

Could it be that developing the ability to work with fractions is just such a progression, too?

Clearly this is the case.

"How many pieces are in this puzzle?"

"Teacher, there are 25 pieces in this puzzle."

"Good, Johnny.  Let's place the number, '25,' on the bottom of the fraction in this exercise.  The total number of pieces is called the 'denominator' of the fraction."

"Here are two piles of puzzle pieces.  How many are in each?"

"Teacher, one pile has seven pieces.  The other pile has two pieces."

"Good, Johnny.  Put the two piles together.  How many pieces are in the new pile?"

"Teacher, there are nine pieces in the new pile."

"Good, Johnny.  Now let's write this activity as an addition equation, using fractions.  We called the bottom number, the 'denominator,' which means the 'name' (or family or language or operating system) of the fraction equation/sentence.  The top number of every fraction is called, the 'numerator,' which means the portion (or the number or the total of each) of the smaller piles."

"Teacher, do you mean that the bottom number is like using the English language to speak with other people, who, also, know how to speak English?"

"That is exactly right, Johnny.  Do you remember how sad Corazon's Mommy looked last week, when she visited our class?"

"Yes, Ma'am.  Why did she look so sad?"

"Corazon's family moved here from Venezuela last year, and even though Corazon now knows English pretty well, her Mommy is still having trouble learning English."

"You and I can talk to each other, Teacher, because we both speak English well."

"Very good, Johnny.  We can add these two piles together because they both 'speak' the 'language' of '25.'"

"They speak '25,' Teacher.  I don't understand."

"Maybe it would be better, if I show you, Johnny.

7          2              9

_    +    _       =     _

25        25            25

"You see, Johnny, every fraction is in the same denominator (or family or language of math) on the bottom.  When the bottom number is the same, then all we have to do is to add the top numbers in order to answer the equation."

"Teacher, twenty-five is a kind of big number for me.  Is there another way to learn fractions?"

"Of course, there is, Johnny.  There are three children in your family, right?"

"Yes, Ma'am."

"If you want to go out to play in the yard with your brother, you have to find him in order to take him with you to respectfully ask your mom or dad, right?"

"Yes, Ma'am.  That is the way that we get permission in our family."

"What fraction of the children in your family are asking to go out to play in this case?"

"Well, Teacher, we have three children. Would that be the...denominator?"

"The numerator would be one for each fraction, meaning that one child plus one child equals two children or

1           1             2

__   +    __     =   __

3           3             3

"Is that right, Teacher?"

"It certainly is, Johnny.  Three is the total number of children in your family.  This is the denominator.  One child joined one child to equal two children or two-thirds of the children in your family, who ask to go out to play in the yard."

The dialog in this story seems very dated, but it seemed necessary in order to help the flow of the concepts in logical progression.

The approach is merely one way of stating the ideas to be taught, but it is hoped that the reader will find a sufficient number of starting-points in order to create a learning environment that will work the students, being taught.

This presentation is intended for introducing the most base-line beginners to the addition of simple fractions, (which would most-commonly be the early elementary grades.)  Subtraction is easy enough to see, using the concepts written herein, but the presentation the multiplication and the division of simple fractions (as well as any conceptualization of compound fractions) would understandably be more beneficial at a later date.

To review, the visual (as well as preferably audible and tactile) engagement of fractions needs to show the total group of blocks, puzzle pieces, children, balls, or marshmallows, etc. on the bottom of each fraction, repeatedly being called, the "denominator."

The portions being added together would be placed on the top of each fraction, and in turn would be repeatedly called the "numerator."

Repeat, repeat, repeat, repeat,...since as we all know repetition is the best teacher for long-term memory.

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