The Classroom Tile Pattern
A classroom floor is covered with square tiles arranged in a repeating pattern. Each tile is a square of side 25 centimetres. The pattern along one wall uses three shapes: a large square, a triangle, and a small square, repeating in that order. The caretaker notices that the diagonal of each square tile, measured corner to corner, is longer than the side. A learner claims that if you double the side of a square tile, the diagonal also doubles. The room is 5 metres long and 4 metres wide.
1. How many tiles fit along the length and along the width of the room? How many tiles cover the floor in total?
Read the worked answer
Convert the room to centimetres: length 5 m = 500 cm, width 4 m = 400 cm.
Along the length: 500 ÷ 25 = 20 tiles. Along the width: 400 ÷ 25 = 16 tiles.
Total tiles = 20 × 16 = 320 tiles.
Each tile is a square so the tiles fit exactly with no cutting needed.
2. A learner says doubling the side of a square doubles its diagonal. Use the relationship between the side and diagonal of a square to judge this claim.
Read the worked answer
For a square of side s, the diagonal d satisfies d² = s² + s² = 2s², so d = s√2.
The diagonal is proportional to the side. If the side doubles from s to 2s, the new diagonal is 2s√2 = 2 × (s√2), which is exactly double the old diagonal.
So the learner is correct: doubling the side doubles the diagonal. For example, a 25 cm tile has diagonal 25√2 ≈ 35.4 cm, and a 50 cm tile has diagonal 50√2 ≈ 70.7 cm, which is twice as long.
3. The repeating pattern goes large square, triangle, small square, large square, triangle, small square, and so on. If the pattern starts with a large square at position 1, what shape is at position 50? Explain your reasoning.
Read the worked answer
The pattern repeats every 3 tiles: positions 1, 4, 7, … are large squares; positions 2, 5, 8, … are triangles; positions 3, 6, 9, … are small squares.
Divide 50 by 3: 50 = 3 × 16 + 2. So position 50 is the 2nd shape in the 17th repeat of the cycle.
The 2nd shape in the cycle is the triangle. So position 50 is a triangle.