Lesson Guide
Explanation
Shapes tessellate when copies of them cover a surface with no gaps and no overlaps — floor tiles, brickwork, honeycomb. Turning and flipping a tile is allowed, which is how triangles fit: every second one is upside down. A single gap anywhere, however small it looks, means the shape does not tessellate, and circles never do because curved edges always leave gaps.
Your Turn
Do circles tessellate? Check: no — curved edges leave gaps.
Watch Out
Overlapping is not allowed either. Covering the page by letting tiles sit on top of each other is not a tessellation.
Explanation
The test is the angles where the corners meet: they must add to exactly three hundred and sixty degrees with nothing left over. Four squares of ninety make it, and six equilateral triangles of sixty. Three regular pentagons make three hundred and twenty-four, leaving a gap of thirty-six, which is far too small for a fourth pentagon's hundred and eight — so they do not tessellate. A regular hexagon's angle is a hundred and twenty and three of them make three hundred and sixty, so hexagons do.
Your Turn
A regular hexagon has angles of 120°. How many meet at a point? Check: 360 ÷ 120 = 3.
Watch Out
The angles must make 360 exactly. 324 is not close enough — a gap is a gap.
Explanation
Every triangle tessellates and so does every quadrilateral, however irregular. The reason is the angle sums: a triangle's three angles make a hundred and eighty, so putting two of each corner at a point gives three hundred and sixty; a quadrilateral's four angles make three hundred and sixty already, so one of each corner at a point is exactly right. That is why a question asking which shape does not tessellate is nearly always pointing at a regular pentagon or a circle.
Your Turn
Does an irregular four-sided shape tessellate? Check: yes — its angles add to 360.
Watch Out
Irregular does not mean it fails. The angles still add to 180 or 360, so it still fits.
Explanation
On a scale drawing, going from the drawing to the real thing you multiply by the scale, and coming back you divide. With one grid unit standing for two metres, a room drawn four units by three is really eight metres by six; a wall fourteen metres long is drawn seven units. On a map where a centimetre stands for fifty metres, a path six centimetres long is really three hundred metres. Write the scale out as a sentence first, and check at the end that the real thing came out bigger than the drawing.
Your Turn
On a map, 1 cm stands for 50 m. A path measures 6 cm. How long is it really? Check: 300 m.
Watch Out
The real thing is bigger. If your answer is smaller than the drawing, you divided when you should have multiplied.
Explanation
One trap is worth naming. A room drawn four units by three is twelve square units on the paper, and at one unit for two metres it is really eight metres by six — an area of forty-eight square metres, not twenty-four. Doubling both sides quadruples the area. So convert both lengths to real measurements first, then multiply. Quiz and exercises now.
Your Turn
A room is drawn 4 units by 3 units at 1 unit = 2 m. What is its real area? Check: 8 × 6 = 48 square metres.
Watch Out
Do not scale an area by the scale. Convert the two lengths first, then multiply them.
📌 Key Points — this lesson in a nutshell
- Shapes tessellate when copies cover a surface with no gaps and no overlaps.
- They fit round a point only if the angles there add to exactly 360°. Squares (4 × 90) and equilateral triangles (6 × 60) do; regular pentagons (108°) do not.
- Every triangle and every quadrilateral tessellates, even irregular ones.
- On a scale drawing, real = drawing × scale, and drawing = real ÷ scale. Write the scale as "1 unit stands for ___" before you start.
Step 1: Introduction
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