We asked to plot the fractions one-fourth,
three-fourths, six-fourths, and ten-fourths on the
number line. Then write any improper
fractions as mixed numbers. Notice for
all the fractions the denominators are
4.
Because of this, we will cut each length
equal to 1 unit into four equally sized
parts or lengths. Because the length between
0 and 1 is equal to 1 unit, we will cut
this length into four equally sized
parts are lengths and the same thing for
the length between one and two, and the
length between two and three.
By doing this, the lenth between tick
marks is now equal to one-fourth of a
unit. Now we can plot the given
fractions. One-fourth is located one
fourth of the unit to the right of 0, which would
be here.
Three-fourths is located 3/4 of a unit to
the right of zero. Starting at zero and
counting by one fourths we have 0, 1/4,
2/4 and 3/4. we can
also think of this as three copies of
one-fourths. Next we have six fourths. So
let's start at three fourths and count
by fourths: three-fourths, four fourths
which is equal to 1, five-fourths, and six
fourths. Six-fourths is equal to six copies of
one-fourth. And then finally for ten-fourths,
will start at six fourths and count by
one fourths. 6/4, 7/4, 8/4
which is equal to 2, 9/4, and 10/4.
 
Before we express the improper fractions
as mixed numbers. Two of these fractions
can simplify. Fractions simplify if the
numerator denominator share a common
factor other than one. Notice for the
fraction 6/4,  six and four share a
common factor of 2 and for the fraction
10/4,  10 and 4 also share a common
factor of two. When a fraction simplifies
it means it can be expressed using
larger sized pieces or in this case larger
size length. To see how these two
fractions simplify let's cut the length
of each unit now into two equally sized
parts are lenths rather than four. So
here's the length between 0 and 1, which is
equal to 1 unit. Let's cut this into two
equally sized parts or lengths. And the same
thing for the length between one and two
and two and three. Notice by doing this,
now the length between tick marks is equal
to half of a unit. So we have 0, 1/2,
2/2, which is equal to 1,  3/2
4/2 which is equal to 2,  5/2
and 6/2 which is equal to
3.  Now looking at 6/4,  we can see
six fourths is equal to or equivalent to
three halves. Three-halves is the
simplified form of six fourths. More
traditionally to simple 6/4
because six and four share a common
factor of two, we would divide the
numerator and denominator by 2. Notice
how this does give us three halves.
Notice that 10 fourths is equivalent to
five halves. Five-halves is the simplified
form of 10 fourths, or more traditionally
because 10 and 4 share a common factor of 2,
to simplify we divide the numerator and
denominator by 2 which does give us five
halves.
Notice how the fractions of one-fourth
and three-fourths cannot be expressed
with the denominator of 2 because one-fourth
three-fourths do not simplify.
And now will write the improper
fractions as mixed numbers. An improper
fraction is a fraction where the
numerator is greater than or equal to
the denominator, which means six fourths
and 10 fourths are improper fractions
and can be expressed as mixed numbers.
Notice how 6/4 is greater than one. If we
begin at one count by one-fourth, we have
1, 1 and 1/4,  1 and 2/4, which shows that
6/4 is equal to 1 into fourths as a
mixed number.  And looking at ten fourths,
notice how 10 fourths is greater than 2.
If we begin at two and count by 1/4's,
we have 2, 2 and 1/4, 2 and two fourths. You
may also notice that two fourths does
simplify.  Look at the number line, 2/4  for
would be here,  which is equal to
one-half, which means we can also write
the mixed number for 6/4 as
one-and-a-half. We can also write the
mixed number for 10/4 as two and
one-half. Llooking at the partitions equal
to half of a unit, notice that six
fourths, if we  go back to one and count by
one-halves,  we have one and then
one-and-a-half. One and a half is
equal to six fourths and for 10/4,
we begin at two and count by one halves,
we have two,  two and a half which equals
10/4.  I hope you found this helpful.
