We're asked to plot the mixed numbers
two and one-fourth and negative one and
three-fourths on the number line. Then
write the mixed numbers as improper
fractions. The mixed number positive two
and one-fourth is located two and
one-fourth units to the right of 0 and
the mixed number negative one and
three-fourths is located one and
three-fourths of a unit to the left of
zero. Let's first plot two and one-fourth
on the number line. From zero move right
two units and then from here we're going to
move one-fourth of the unit to the right
of two because the lenth between two and
three is equal to 1 unit, let's cut the
length between two and three into four
equally sized parts or pieces. By doing
this now the lengths between tick marks
between two and three are equal to
one-fourth of the unit. So from 2 we move
right  one-fourth of the unit to the
mixed number two and one-fourth. Now let's
plot negative one and three-fourths on
the number line. Starting at zero, move
left one unit and now we need to move left
three fourths of a unit. Because the length
between negative 1 negative 2 is equal
to 1 unit, let's cut this length into four
equally sized parts or pieces. The length
between tick marks between negative 1
negative 2 are now equal to one-fourth
of the unit. So here we're at negative 1
negative 1 1/4,  negative 1 2/4,
and negative one and three-fourths. Now
let's write both of these mixed numbers
as improper fractions. To do this using
the number line, we want to cut each unit
into four equally sized parts or pieces,
which means we'll cut the length between 0
and 1 into four equally sized parts or
pieces as well as a length between one and
two.
To the left of zero we will cut the length
between 0 negative 1 into four equally
sized parts or pieces as well as a length
between negative 2 negative 3. Now
starting at zero we can determine how
many copies of one fourths it takes to
reach two and one-fourth. Starting at 0 we
have one-fourth, two-fourths
three-fourths, four fourths which is
equal to 1, 5 fourths, 6 fourths, seven-fourths,
eight fourths which is equal to 2, and
finally nine fourths.  Two and one-fourth
is equal to nine fourths or nine copies
of 1/4.  And now starting at zero moving
left, we will count by negative one
fourths. We have negative one-fourth,
negative two fourths, negative
three-fourths, negative four fourths
which equals negative 1, negative 5
fourths, negative 6 fourths, and finally negative
seven-fourths. Negative one and
three-fourths equals negative
seven-fourths, or we can say negative one
and three-fourths equals seven copies of
negative one-fourth. The more traditional
algorithm to convert a mixed number to
an improper fraction is: step one, the
denominator remains the same, and then
the numerator of the improper fraction
is the denominator of the mixed number
times the whole number plus the
numerator. Notice four times two equals 8
plus 1 which equals nine, giving us the
improper fraction of nine fourths. For a negative
mixed number, we first make the improper
fraction negative. The denominator
remains the same, and the numerator is 4
times 1 plus 3 which is equal to seven.
So when the mixed number is negative
we do need to be careful about the sign. I
hope you found this helpful.
