Discover Mathematics P5: Term 2, Unit 10, Lessons 1–4
GoalFind the area of a rectangle by covering it with tiles, including tiles that are fractions of a unit.
New words
area · the amount of flat surface a shape coversunit square · a square with side 1 unit; its area is 1 square unittiling · covering a shape with tiles with no gaps and no overlaps
Explanation
To find an area, cover the shape with tiles that have no gaps and no overlaps, then count them. If the tiles are unit squares (1 cm by 1 cm), the area is the number of tiles in square centimetres. A rectangle 4 cm by 3 cm takes 3 rows of 4 tiles = 12 tiles, so its area is 12 square centimetres. When the sides are fractions we use smaller tiles. Half-centimetre tiles are 1/2 cm by 1/2 cm, so each tile has area 1/2 × 1/2 = 1/4 of a square centimetre. A rectangle 2 1/2 cm by 1 1/2 cm needs 5 columns and 3 rows of half-centimetre tiles: 15 tiles. The area is 15 × 1/4 = 15/4 = 3 3/4 square centimetres. This is the same as multiplying the sides: 5/2 × 3/2 = 15/4. So length × width works for fractions too.
Worked examples
A rectangle 4 cm by 3 cm is covered with unit squares: 3 rows of 4 tiles. 3 × 4 = 12 tiles, so the area is 12 square centimetres.
A mat is 3/4 m long and 2/3 m wide. Cut a square metre into 4 columns and 3 rows: 12 equal tiles. The mat covers 3 columns and 2 rows = 6 tiles. 6/12 = 1/2, so its area is 1/2 square metre. Check: 3/4 × 2/3 = 6/12 = 1/2.
A square floor has side 2 1/2 m and is covered with tiles 1/2 m by 1/2 m. Along each side there are 5 tiles, so 5 × 5 = 25 tiles. Each tile is 1/4 square metre, so the area is 25/4 = 6 1/4 square metres. Check: 5/2 × 5/2 = 25/4.
Class activity
Tile a rectangle. On squared paper draw a rectangle 2 cm by 1 1/2 cm. Each square is 1 cm². Draw lines that cut the squares in half so you can count the half-squares, or count whole squares and add the halves. Write the area as a fraction and check by multiplying 2 × 3/2.
Practice
1
Unit squares cover a rectangle 6 cm by 4 cm. How many tiles are needed and what is the area?
4 × 6 = 24 tiles. Area 24 square centimetres.
2
Find the area of a rectangle 3 1/2 cm by 2 cm.
7/2 × 2 = 7 square centimetres.
3
A rug is 1 1/2 m by 1 1/2 m. Tiles of 1/2 m by 1/2 m cover it. How many tiles, and what is the area?
