Area with Fractional Sides: Tiling and Multiplying | Grade 5 Math | LevelKite

Area is the amount of flat space a shape covers. You can find it by covering the shape with equal square tiles that leave no gaps and do not overlap. Covering a shape this way is called tiling.

The usual tile is a unit square. It is a square whose sides are each $1$ unit long. The area of one unit square is $1$ square unit. Unit squares work well when the side lengths are whole numbers.

Now think about a rectangle that is $\frac{3}{5}$ unit long and $\frac{2}{5}$ unit wide. Not even one unit square fits inside it, so you need smaller tiles. In this lesson you will tile rectangles like this one, count the tiles, and see why multiplying the side lengths gives the same area.

A unit fraction is a fraction whose numerator, the top number, is $1$ . One example is $\frac{1}{5}$ . You can make a square tile whose side length is a unit fraction.

Start with a unit square. Cut its length into $5$ equal parts and its width into $5$ equal parts. Now the unit square holds $5$ rows of $5$ small squares. That is $5 \times 5 = 25$ tiles, and each tile has sides $\frac{1}{5}$ unit long.

The $25$ tiles are all the same size, and together they fill $1$ square unit. So one tile is $\frac{1}{25}$ of a square unit. This matches a multiplication you already know: $\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}$ .

The figure below shows this unit square. Count $5$ rows and $5$ columns, which makes $25$ tiles. The figure also marks two bands, or wide stripes. One band covers $3$ of the $5$ rows, and the other covers $2$ of the $5$ columns. The bands cross in a rectangle that is $3$ tiles down and $2$ tiles across. Its sides are $\frac{3}{5}$ unit and $\frac{2}{5}$ unit long.

The unit square holds

25

square tiles, and the two bands cross in a rectangle with

3

rows of

2

tiles.

Worked example

Tile a rectangle and count the tiles

A rectangle is $\frac{3}{5}$ unit long and $\frac{2}{5}$ unit wide. Tile it with squares whose sides are $\frac{1}{5}$ unit long. What is the area of one tile? What is the area of the rectangle?

Find the area of one tile. A unit square holds 5 rows of 5 of these tiles, so 25 equal tiles fill 1 square unit. Each tile is one twenty-fifth of a square unit.
$\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}$
See how many tiles fit along each side. Three fifths is 3 copies of one fifth, so 3 tiles fit along the length. Two fifths is 2 copies of one fifth, so 2 tiles fit along the width.
$\frac{3}{5} = 3 \times \frac{1}{5} \quad \text{and} \quad \frac{2}{5} = 2 \times \frac{1}{5}$
Count the tiles in the rectangle. It has 3 rows with 2 tiles in each row. These are the tiles where the two bands cross in the figure.
$3 \times 2 = 6$
Add up the area of the tiles. The rectangle is made of 6 tiles, and each tile is one twenty-fifth of a square unit. So the area is six twenty-fifths of a square unit.
$6 \times \frac{1}{25} = \frac{6}{25}$

Answer

One tile has an area of $\frac{1}{25}$ square unit. The rectangle has an area of $\frac{6}{25}$ square unit.

You can also work the other way. A product is the answer to a multiplication. When two fractions are each less than $1$ , you can show their product as the area of a rectangle inside a unit square. Each fraction becomes one side length of the rectangle.

When the two fractions have different denominators, or bottom numbers, the tiles change shape. For $\frac{2}{3} \times \frac{4}{5}$ , you cut the unit square into $3$ equal rows and $5$ equal columns. Each tile is then $\frac{1}{5}$ unit wide and $\frac{1}{3}$ unit tall. It is a small rectangle, not a square. That is fine. The tiles are still equal, so you can still count them.

The figure below shows this unit square. One band covers $2$ of the $3$ rows, and the other covers $4$ of the $5$ columns. The bands cross in a rectangle with sides $\frac{2}{3}$ unit and $\frac{4}{5}$ unit long. The area of that rectangle is the product.

The band over

2

of the

3

rows and the band over

4

of the

5

columns cross in

8

of the

15

equal tiles.

Worked example

Show a product as an area

Show $\frac{2}{3} \times \frac{4}{5}$ as the area of a rectangle inside a unit square. Then use the rectangle to find the product.

Cut the unit square to match the denominators. Make 3 equal rows for the thirds and 5 equal columns for the fifths. That gives 3 rows of 5 tiles, so 15 equal tiles fill 1 square unit. Each tile is one fifteenth of a square unit.
$\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}$
Mark the two side lengths. A band over 2 of the 3 rows is two thirds of a unit tall. A band over 4 of the 5 columns is four fifths of a unit wide. The rectangle where the bands cross has both of these side lengths.
Count the tiles in that rectangle. It has 2 rows with 4 tiles in each row, as the figure shows.
$2 \times 4 = 8$
Write the area. The rectangle is made of 8 tiles, and each tile is one fifteenth of a square unit. This area is the product. It matches what you get when you multiply the numerators and multiply the denominators.
$\frac{2}{3} \times \frac{4}{5} = 8 \times \frac{1}{15} = \frac{8}{15}$

Answer

The product $\frac{2}{3} \times \frac{4}{5}$ is the area of a rectangle with sides $\frac{2}{3}$ unit and $\frac{4}{5}$ unit long. That area is $\frac{8}{15}$ square unit.

Key concept

Multiply the side lengths to find the area

The area of a rectangle is its length times its width. This is true when the side lengths are fractions too, and tiling shows why. Cut the unit square into rows and columns that match the two denominators. Then:

multiplying the numerators counts the tiles in the rectangle
multiplying the denominators counts the tiles in the whole unit square, which tells you the size of one tile

Look back at the first rectangle: $\frac{3}{5} \times \frac{2}{5} = \frac{3 \times 2}{5 \times 5} = \frac{6}{25}$ . It has $3 \times 2 = 6$ tiles, and the unit square has $5 \times 5 = 25$ tiles. So counting the tiles and multiplying the side lengths give the same area.

\text{Area} = \text{length} \times \text{width}

Worked example

Multiply, then check with square tiles

A rectangular poster is $\frac{3}{4}$ meter long and $\frac{1}{2}$ meter wide. Find its area by multiplying the side lengths. Then check your answer by tiling the poster with square tiles.

Multiply the length by the width. Multiply the numerators, then multiply the denominators. The sides are measured in meters, so the area is three eighths of a square meter.
$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$
Choose a square tile that fits both sides. One half equals two fourths, so a square with sides one fourth of a meter long fits both. Then 3 tiles fit along the length and 2 tiles fit along the width.
$\frac{3}{4} = 3 \times \frac{1}{4} \quad \text{and} \quad \frac{1}{2} = \frac{2}{4} = 2 \times \frac{1}{4}$
Find the area of one tile. A square meter holds 4 rows of 4 of these tiles, so each tile is one sixteenth of a square meter.
$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
Count the tiles and add up their area. The poster holds 3 rows of 2 tiles, which is 6 tiles. Each tile is one sixteenth of a square meter.
$6 \times \frac{1}{16} = \frac{6}{16}$
Compare the two results. Divide the numerator and the denominator of six sixteenths by 2, and you get three eighths. The two fractions name the same amount, so counting tiles and multiplying agree.
$\frac{6}{16} = \frac{6 \div 2}{16 \div 2} = \frac{3}{8}$

Answer

The area of the poster is $\frac{3}{8}$ square meter. Multiplying the side lengths and counting the tiles give the same area.

Counting tiles and multiplying give the same area

Rectangle	Counting tiles	Multiplying side lengths	Area
$\frac{3}{5}$ unit by $\frac{2}{5}$ unit	$6$ tiles, each $\frac{1}{25}$ square unit: $6 \times \frac{1}{25} = \frac{6}{25}$	$\frac{3}{5} \times \frac{2}{5} = \frac{6}{25}$	$\frac{6}{25}$ square unit
$\frac{2}{3}$ unit by $\frac{4}{5}$ unit	$8$ tiles, each $\frac{1}{15}$ square unit: $8 \times \frac{1}{15} = \frac{8}{15}$	$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$	$\frac{8}{15}$ square unit
$\frac{3}{4}$ meter by $\frac{1}{2}$ meter	$6$ tiles, each $\frac{1}{16}$ square meter: $6 \times \frac{1}{16} = \frac{6}{16} = \frac{3}{8}$	$\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$	$\frac{3}{8}$ square meter

Here is the big idea. You can tile a rectangle that has fractional side lengths with equal tiles cut from a unit square. To find its area, count the tiles and multiply by the area of one tile, or multiply the two side lengths. Both ways give the same area. Multiplying the numerators counts the tiles, and multiplying the denominators tells you the size of each tile. Always state the area in square units.

Next, you will find the area of rectangles whose side lengths are mixed numbers, such as $2\frac{1}{2}$ units.