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Determining how many times
one amount fits into another
or how many times bigger
one amount is than another
may involve whole numbers or fractions. For both types of numbers the
basic idea division
is the same
in forty years teacher math courses for future teachers
there were always students
who were surprised to find
that division of fractions had the same meaning
as division of whole numbers.
In this video
division of fractions
will be connected to division of whole numbers
in two ways.
If a cook has two-thirds of a cup of cocoa,
represented by the shaded amount of the top bar,
and a batch of chocolate cookies requires
one sixth cup of cocoa,
represented by the lower bar,
how many batches of cookies can be made?
Now we can see that the shaded amount of the top bar
is four times greater than the shaded amount of the lower bar,
or that the shaded amount of the lower bar fits into the shaded amount of the top bar
one, two, three,
four times.
The idea of goes into
is one way division of whole numbers is taught.
For example how many times does three go into (or divide into) fifteen?
We write
fifteen
divided by three
equals five.
Similarly we can use the operation of division
in this example
two-thirds
divided by
one sixth
equals four.
In both of these examples
we can ask how many times does one amount fit into another,
and in both
with whole numbers and fractions
we can answer the question by division.
This is one connection from whole numbers to fractions.
The concept of division stays the same
Let's consider another example.
If one-half teaspoon baking soda is available,
as represented by the top bar,
and one twelfth teaspoon is needed for each pan of biscuits,
represented by the lower bar,
how many pairs of biscuits can be made?
Well we can see that the
shaded amount of the lower bar fits into the shaded amount of the top bar
one, two, three, four, five,
six times.
So we can write one half
divided by
one twelfth
equals six.
So, six pans of biscuits
can be made.
Let's look at these special cases involving division of fractions.
One third divides into one three times.
One fourth into one
for times.
One fifth, five times and one sixth into a whole bar six times.
Now one thing students might notice is that these numbers are
reciprocals of each other
So another thing they might notice is that
one divided by one-third could be written as one
times three.
One divided by one fourth equals one times four.
One divided by one fifth equals one times five.
One divided by one sixth equals one times six.
So to divide the first number by the second,
we can multiply the first number
times the reciprocal of the second.
First number divided by the second, multiply the
first number times the reciprocal of the second.
Students can see this relationship
visually from these special cases.
Students can now use this invert and multiply process on the results for the bars
that we showed earlier
in this video.
We saw that one sixth
fits into two thirds, four times.
Now we can use the invert and multiple rule.
Two-thirds divided by
one sixth,
will be two thirds
times the reciprocal of one sixth,
is twelve over three,
this is equal to four.
Notice what happens when we replace the two thirds bar
by the equivalent
four sixths bar.
We can see that one sixth fits in here four times in fact this is just a matter
of dividing
whole numbers, dividing four by one, so
two thirds divided by one six
equals
four sixths
divided by one sixth.
Now we've got the same size parts,
both cases this is four divided by one
which is four.
So, to divide two fractions replace by fractions having the same denominator,
and then divide the numerators.
Doesn't this sound too good to be true?
You can divide fractions by dividing whole numbers.
Once you get the common denominators.
This provides another connection
from whole numbers to fractions.
Let's make it one more example
of dividing fractions by getting common denominators.
Here we have two thirds
divided by one-fifth.
Two-thirds divided by one fifth.
Now we can see that
one-fifth divides in once, twice, three times, a little bit more than three times.
So, we've got two-thirds
divided by one fifth.
Now we'll get common denominators.
The common denominator is fifteen.
So we multiply three by five
to get fifteen.
Five times two is ten.
Three times five is fifteen and three times one is three.
Now we have parts of the same size.
If these were bars, both bars would have fifteen parts. We have ten parts shaded, three parts shaded.
So, we merely divide
ten by three which
is three and one-third.
So two-thirds divided by one fifth
is three and one-third.
Now let's use the
invert the divisor method.
We've got two thirds,
divided by one fifth.
Can be two thirds times five.
Ten over three
equals
three and one-third.
Let's label these methods.
Common denominator
Down here I'll call this "Invert".
The invert method
is quicker than first getting common denominators
but there's something nice about the common denominator method.
In this case we have
a common denominator of fifteen
the ten parts and three parts
are the same size so we can divide by whole numbers.
The division concept can be seen
But the invert method
does not give a sense that one amount has been divided by another.
It is easy to see
why students using the invert method do not have
a conceptual understanding
for division fractions.
In this video
We saw two ways in which division of whole numbers is related to
division of fractions.
One very important connection
is that the concept of division
is the same for division of whole numbers as for division of fractions
regardless
of the type of numbers being used
we're determining how many times greater
one number is
than another.
Then we looked at two ways
to divide one fraction by another.
Either invert and multiply,
or get common denominators for the two fractions
and divide one numerator by the other.
According to Eugene Smith
in his "History Mathematics"
getting common denominators
to divide one fraction by another
was a method used to divide fractions many years before invert and multiply
became popular.
The Equations Game
on fractionbars.com can be played with either
Fraction Bars or Fraction Playing Cards.
The object is to drag the bars
into the ten blanks, to form all four equations.
This game can be a challenge.
Moving the bars involves computation and trial and error.
It always be possible to form all four equations.
The player has formed one equation
The equation involving division.
The player has one more equation to form.
The player has formed all four equations and will click
GO to check the results.
All four equations are correct as shown by the checkmarks.
The player scored ten points for these four equations.
There are three rounds of ten fractions each and the player wins the game with
a total of at least twenty one points.