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In this
lesson,
we will relate division of whole numbers
to fractions by folding strips of paper into various numbers of parts.
Such activities
will help students
to make sense
out of the relationship between quotients of whole numbers
and fractions.
Suppose a person has three fourths of a cake.
The one whole cake can be represented by this whole bar,
and the three shaded parts of the bar represent
three-fourths of the cake. Next suppose
there are three cakes to be shared equally among four people.
How much cake which person receive?
Now we are going to let the three cakes be represented by this strip of paper
and you can see the strip of the paper has a length of the three whole bars.
We could call this a three bar, this large
strip of paper here,
and we want to
divide is paper equally into four parts.
So, we'll paper fold
this over here
divided this in half and divided in half again.
We'll open this up. We divided the
three into four equal parts, and we draw marks
for each of these parts.
So, we've divided this strip of paper equally into four equal parts
Tape the ends to hold in place.
Each person is going to receive this much cake - four people.
The question is
how much cake is this?
Now we can use these three fourths bar
to show
that's three-fourths right there
three fourths of the cake
this persons is going to get three-fourths each person's going to get three-fourths
of the cake.
So we have divided
three
by four
three
divided by four
and we've seen that it equals
three fourths.
Let's illustrate another quotient.
This time we'll use a strip of paper that has a length of two whole bars.
This could be called a two bar.
Suppose this strip of paper represents two pounds of flour that will be divided into
three equal parts
what will be the weight of each part
We are going to paper fold here
divide this
piece of paper into three equal parts
we'll do that that
and then we'll
draw lines on the creases to mark each part.
We've divided this two-bar, this strip of paper,
that's two bars long
into
three equal parts.
Tape this down.
The question is what is the size of each part?
We'll try a bar with
two parts out of three shaded.
We see that
this is two-thirds
each one of these
parts is two-thirds.
So the weight
of one part of the flower
is two thirds of a pound.
What we've shown
is that two divided by three
equals two thirds.
In this next example we'll also use a two-bar.
This time we want it divided into five equal parts
We are going to have two
divided by five.
Since it is difficult to fold a strip of paper into five equal parts,
we use a different approach.
We know from the two previous examples that when we divide two whole
numbers and get a fraction
having the
two numbers as numerator and denominator.
So, lets just see if two fifths
will work.
Suppose two divided by five is two fifths, then
one of these parts
will be two fifths and another part
will be two fifths
so we just have, hopefully,
five
parts that will all be
two fifths.
We see that's the case one two three four five each part is two fifths.
So, two divided by five
is two fifths.
In this example we used the two fifths bar and worked backward
showing that the two fifths bar could be used to divide
the two bar
into five equal parts.
The strips for these three illustrations were cut from a spool of paper.
In a similar manner we could cut out
bars for
four whole bars
five or six whole bars.
You can carry out paper folding activities.
By listing equations from such activities
students will have no problem
generalizing results.
When one whole number
is divided by another
the resulting fraction
has the first whole number as a numerator,
and the second whole number
as the denominator.
The general statement can be written as
a divided by b
equals a over b
the fraction a over b for
b not equal to zero.
Let's look at another way to make sense
out of the fact that if you divide a whole number a by whole number b
you get the fraction a over b.
We'll take an example here
We know that three divided by four using the sharing concept of division,
means to divide three into four equal parts.
Three divided by four is one of those parts.
We also know
in multiplication of fractions that one fourth
times three
means to divide three
into four equal past and take one of them.
So these mean the same thing.
We know from fractions, multiplying
fractions times whole numbers, that when we multiply
the whole number times the numerator there, so this is just three fourths
so is another way to come out
with the result
dividing two numbers
to get a fraction whose numerator and denominator are two whole numbers.
This relationship
of dividing one whole number by another
and obtaining a fraction
is very convenient.
Let's look at some
examples.
A piece of wood
with a length of 5 feet
is cut into 6 equal pieces what is the length of each piece?
We simply divide five
by six
to obtain
five
sixths
of a foot.
If three chicken pies are shared equally by five by people
what fraction of a pie will each person have?
In this case we divide three by five
and obtain
three-fifths
of a pie
for each person.
If eight pounds of grass seed are divided equally
into five piles
what is the weight
of one of these piles?
In this case we divide eight by five
and get eight fifths,
which equals one and three fifths pounds.
In these video
we saw examples of the sharing concept of division.
When paper strips
were divided into equal numbers of parts
we also generalized the results of the paper folding activity
to conclude
that whole number a
divided by whole number b is equal to the
fraction a over b.
Let's look at the greater quotient game,
from fractionbars.com.
In this game the player clicks the deck
to obtain two fraction playing cards.
The player competes against the computer and the two cards face down are the
computer's cards.
The object is to get
as great a quotient as possible.
This player doesn't have a very good quotient at this point so the player
can click one of the cards to get a new card.
The player has play has a greater quotient. They click GO.
The player with the greater quotient wins the round.
The first player to win three rounds, wins the game.
This player won three rounds and won the game.