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.

What is the area of this rectangle? Notice, its side lengths are fractional (1/2 inch and 2/3 inch). Let’s extend its sides and draw a square inch around it. Surely the area of our rectangle is less than a half square inch. But how much is the area exactly?

To solve this problem, let’s draw a grid inside our square inch: Now it is easy to see that the area of the colored rectangle is exactly 2/6 or 1/3 of the square inch. (Why? Because the square inch is divided into 6 equal parts, and our rectangle covers two of them). Notice that we get the same result (1/3 square inch) if we multiply the side lengths, using fraction multiplication: 2 3 in. ×  1 2 in.  =  2 6  in2 =  1 3  in2

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 Area (from the picture):   m2

b. Side lengths: in.  and  in. Area (from the picture):   in2

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): m ×  m  =	b. Side lengths: in.  and  in. Area (by multiplication): in. ×  in.  =
c. Side lengths: m  and  m Area (by multiplication): m ×  m  =	d.  Side lengths: km  and  km Area (by multiplication): km ×  km  =

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

a.   1 4 m  ×  1 2 m  =  1 8 m2	b.   1 2 in.  ×  4 6 in.  =  4 12 in2
c.   3 4 ft  ×  2 7 ft  =	d.   3 5 km  ×  5 6 km  =

The area of this rectangle can be found by multiplication: 3 4 m  ×  1 3 m  =  1 4 m2; however, we want to verify this using a visual method. For that reason, let’s sketch a unit square around the rectangle and tile it.

We need to extend the sides of the rectangle to draw the square. The 1/3-meter side simply needs to be three times as long to make it 1 meter. Then, divide the side that is 3/4 meters long into three equal parts— each part is 1/4 m long. Then extend that side by another 1/4 meter.

Lastly, draw the entire square. Draw gridlines to show the tiles within the square meter: one side is divided into 3 equal parts, and the other to 4 equal parts. We get 12 tiles. Now it is easy to see that the area of the colored rectangle is 3 tiles out of 12, or 3/12 of a square meter. That simplifies to 1/4 of a square meter.

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.

a. 1/3 m 1/3 m     Area: m ×  m  =	b. 1/5 m 1/3 m
c. 1/5 m 1/2 m	d. 1/4 m 1/4 m

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.

a. 3/4 m 1/2 m	b. 2/5 m 3/4 m
c. 2/3 m 2/3 m	d. 3/5 m 1/2 m
e. 3/4 m 3/4 m	f. 5/6 m 1/2 m

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).

a.  ×    =	b.  ×    =
c.  ×    =	d.  ×    =

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 7/8-mile sides,   or a rectangle that is 1/4 mile by 3 miles?

This lesson is taken from Maria Miller's book Math Mammoth Fractions 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

Math Mammoth Fractions 2

A self-teaching 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.

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