Area with Fractions
Find the area of a rectangle with fractional side lengths by tiling with unit-fraction squares; show that the area equals the product of the side lengths
What a learner can do afterwards
- Tile a 2/3 × 3/4 rectangle with 1/12 squares and count to find area = 6/12 = 1/2
- Explain why the number of unit-fraction tiles equals the product of the two fractions
- Calculate the area of a rectangle with sides 1 1/2 and 2/3 using fraction multiplication
1 · Read
You already know how to multiply fractions. Today you will use that skill to find the area of a rectangle when the length and width are both fractions, like a rug that is 2/3 meter long and 3/4 meter wide.
A rectangle is 2/3 m long and 3/4 m wide. Line up 2 of the 3 columns and 3 of the 4 rows. That covers 6 of the 12 tiles, so the area is 6/12 square meters, which is the same as 1/2 square meter.
A garden bed is 1 1/2 m long and 2/3 m wide. First write 1 1/2 as 3/2. Then multiply: 3/2 x 2/3 = 6/6 = 1. The bed's area is 1 square meter, the same answer tiling would give.
Turn mixed numbers into improper fractions before you multiply. Then estimate first: since 2/3 and 3/4 are both less than 1, the area has to be smaller than 3/4. That helps you catch mistakes.
Area with fraction sides follows the same rule as always, length times width, and tiling with tiny fraction squares shows exactly why it works.
More to read
Key ideas: area with mixed number side lengths
Finding the area of a rectangle with fraction sides.
You can do it in two ways, and they always give the same answer.
Way 1: tile it with little squares. Say a rectangle is 2/3 of a unit long and 3/4 of a unit wide. Imagine a big unit square split into 12 equal small squares, 4 across and 3 down. Each small square has an area of 1/12. The rectangle covers 3 of the 4 columns and 2 of the 3 rows, so it covers 3 times 2, or 6 small squares. Its area is 6 times 1/12, which is 6/12, or 1/2 square unit.
Way 2: just multiply the side lengths. 2/3 times 3/4 is 6/12, which is 1/2.
Why do the tile count and the multiplication match? The denominators tell you how to split the unit square. The 12 tiles come from 3 times 4. The number of tiles you count is the numerators multiplied: 2 times 3. So counting tiles is secretly the same as multiplying fractions.
It also works with mixed numbers. A garden that is 1 1/2 yards by 2/3 yard has area 3/2 times 2/3, which is 6/6, or 1 whole square yard.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
15 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.