Fraction Multiplication and Area
This fifth grade lesson explores the area of a rectangle with fractional side lengths. We tile the rectangle with unit rectangles, and show that the area is the same as would be found by multiplying the side lengths. We multiply fractional side lengths to find areas of rectangles.



1. Each picture shows some kind of square unit, and a colored
rectangle. Figure out the side lengths
and the area of the rectangle from the picture.
a. Side lengths: m and m

b. Side lengths: in. and in.

2. Again, figure out the side lengths of the colored rectangle from the picture. Then multiply the side
lengths to find its area.
Check that the area you get by multiplying is the same as what you can see
from the picture.
a. Side lengths: m and m Area (by multiplication):

b. Side lengths: in. and in. Area (by multiplication):


c. Side lengths: m and m Area (by multiplication):

d. Side lengths: km and km Area (by multiplication):

3. Shade a rectangle inside the square so that its area can be found by the fraction multiplication.










4. Extend the sides of the rectangle so you get a square meter (unit square).
Draw gridlines into the
square as in the example above. Write a multiplication for the area
of the colored rectangle. Verify
that the area you get by multiplying is the same as what you can see
in the picture.





5. Extend the sides of the
rectangle so you get a square meter (unit square). Draw gridlines into the
square as in the example above. Write a multiplication for the area
of the colored rectangle. Verify
that the area you get by multiplying is the same as what you can see
in the picture.








6. In the pictures below, the outer square is one square unit. Write a multiplication for the area of
the
colored rectangle. This time, we are not using meters or inches, just “units”
and “square units,” and
you do not have to include those in the multiplication
(simply write the fractions without any units).





7. a. Draw a 1 in. by 1 in. square. What is its area?
b. Draw a rectangle with 3/4 in.
and 5/8 in. sides
inside the square you drew so that
the two
sides
of the rectangle touch the sides of the square.
See the illustration below (not to
scale).
c. Find the area of your rectangle.
8. a. Draw a square centimeter.
b. Draw the rectangle with 3/10 cm and
7/10 cm sides
inside the square centimeter so that
the two
sides
of the rectangle touch the sides of the square.
c. Calculate the area of the rectangle in square
centimeters
using both fractions and decimals
(calculate it two times).
Using fractions:
Using decimals:
9. a. Find the area of a rectangular suburb that is 3 km by 1/2 km.
b. A village lies inside a 5/8 mile by 3/4 mile
rectangle. Find its
area.
10. The pony ride costs $3.50 per kilometer. You go for 3 3/5 km.
a. Calculate the total
price using fractions.
Use 3 ½ for the price per kilometer.
b. Calculate the total
price using decimals.
Use 3.6 for the kilometers.
11. a. A stamp measures 7/8 in. by 3/4 in.
Amanda
puts 6 of them onto an
envelope, side by side.
Find the total area these
stamps occupy.
b. The envelope is 8 in. by 5 in.
About what
part of the envelope do the six stamps cover?
Which has a larger area, a square with 7/8mile sides, or a rectangle that is 1/4 mile by 3 miles?

This lesson is taken from my book Math Mammoth Fractions 2. It ties in with the Common Core Standard 5.NF.4.b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Math Mammoth Fractions 2
A selfteaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.
Download ($5.75). Also available as a printed copy.