At a glance. One lesson of about 60 minutes. Interior angles of regular polygons, the 360-degree vertex condition, regular and semi-regular tessellations.
Calculator: no calculator. Every interior angle in the questions is a whole number of degrees, so no calculator is needed. The figure rounds the 7-sided and 11-sided angles to one decimal place. The work is in the vertex condition.
What students will learn
Know
- A tessellation has no gaps and no overlaps, so the angles at a vertex add to 360°.
- A regular n-gon has interior angle (n − 2) × 180 ÷ n; only triangles, squares and hexagons tile on their own.
- Semi-regular tessellations use two or more kinds of regular polygon with the same arrangement at each vertex.
Understand
- The polygon's size does not change its angles, so it does not change whether it tiles.
- Angles adding to 360° at one vertex is necessary; the same arrangement must also continue across the plane.
Do
- Find an interior angle of a regular polygon.
- Decide whether a regular polygon, or a mix, fills a vertex exactly.
- Name regular and semi-regular tessellations by their vertex arrangement.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: interior angles and vertex sums (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find other arrangements that reach 360, and to test them.
- Criterion C. Communicating: explaining why a polygon does or does not tile using the angle at the vertex.
- Criterion D. Applying mathematics in real-life contexts: floors, tiles and mosaics. This lesson prepares the unit's own tessellation task.
The one thing to do with the figure
Before moving the slider, ask for the number. Ask how many pentagons fit round a point and what is left, then show it.
On the second figure, ask: “This one adds to 360. Is that enough to prove it tiles the whole floor?” It is needed, and it works for these, but the arrangement must repeat at every vertex.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2 (angles at a point). |
|---|---|
| 10 min | Hook, prediction for pentagons, then the first figure. |
| 10 min | Key idea; the second figure, mixing polygons. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 180°; 360°; it can. |
|---|---|
| Worked examples | No (144°, 2.5); 60 + 150 + 150 = 360; 15 sides; 2 squares and 3 triangles. |
| Faded example | No: 140°, and 360 ÷ 140 is not whole. |
| 1. Which regular polygon tiles | B. 120° fits 3 times; 108°, 135° and 144° do not divide 360. |
| 2. Interior angle | 160. 2880 ÷ 18. |
| 3. Angle left at a vertex | 132. 360 − 228. |
| 4. Sides from exterior angle | 5. 360 ÷ 72. |
| 5. Whole copies that fit | 2. Angle 162°; 360 ÷ 162 ≈ 2.2. |
| 6. Fits exactly | A. B, C and D add to 288, 324 and 258. |
| 7. Missing polygon | 12. Angle 150°; exterior 30°. |
| 8. Triangles to fill the gap | 2. 120 ÷ 60. |
| 9. Why it tiles | A. The test is angles summing to 360°. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used an angle that does not divide 360. | Compute each angle: 108, 120, 135, 144. |
| 20 or 180 in question 2 | Gave the exterior angle, or left out the minus 2. | Check with a hexagon: 120, not 180 or 60. |
| 228 or 12 in question 3 | Gave the angles already used, or took one angle from the other. | Draw a point and mark the two angles that are already there. |
| 108 or 288 in question 4 | Gave the interior angle, or subtracted from 360. | Exterior angles of a polygon always add to 360. |
| 3 or 18 in question 5 | Rounded up, or used the number of sides as the angle. | Three copies need 486°: more than a full turn. |
| B, C or D in question 6 | Did not add the angles. | Add the angles in each option on a separate line. |
| 150 or 30 in question 7 | Gave the angle, or the exterior angle, not the number of sides. | Find the angle, then convert it to a number of sides. |
| 6 or 4 in question 8 | Ignored the hexagons, or used only one. | Take both hexagons' 120° from 360 before dividing. |
Questions worth asking
- Why does the size of the polygon not matter? Listen for “the angles stay the same however big it is”.
- Which regular polygons have an interior angle that divides 360? 3, 4 and 6 sides only: the three regular tessellations.
- Does the order of the polygons round the vertex matter? Listen for “yes”: 3.6.3.6 and 3.3.6.6 use the same polygons and add to 360, but only the first continues across the plane.
Differentiation
A way in for anyone stuck
- Cut out paper polygons with equal sides and fit them round a point.
- Make a table of n, interior angle and 360 ÷ angle for n = 3 to 12.
- Give the interior angles in a list, and ask which sums reach 360.
Stretch
- The arrangement 5.5.10 adds to 108 + 108 + 144 = 360, but it does not tile the plane. Why might the angle test be necessary but not sufficient? (The ring closes at one vertex but cannot be repeated at every vertex.)
- Every triangle and every quadrilateral tessellates. Use the angle sums 180 and 360 to explain how the corners meet. (Rotate copies about the midpoint of a side so that each angle of the shape meets round a point.)
Students learning in English as an additional language
Tessellation, vertex and interior angle are new words. Pair each with a picture: a tiled floor for tessellation, a corner for vertex, and the angle inside a corner for interior angle.
| Word | Plain meaning | Example |
|---|---|---|
| tessellation | a pattern of shapes that covers a flat surface with no gaps and no overlaps | a tiled floor |
| vertex | a corner where edges meet | the corner of a tile |
| interior angle | the angle inside a polygon at a corner | 120° in a regular hexagon |
| regular polygon | a polygon with all sides and all angles equal | a square |
| semi-regular | two or more regular polygons with the same arrangement at every vertex | 4.8.8 |
Sentence starters for explaining aloud or in writing:
- Each interior angle is …, and 360 ÷ … is …
- The angles at the vertex add to …, so there is a gap / overlap / no gap of …
- It tessellates / does not tessellate because …
Two things not to say
“It looks like it fits.” A gap of 36° looks small on a drawing. Add the angles.
“Shapes with more sides tile better.” More sides means a bigger angle, and fewer copies fit.
“Only regular shapes tessellate.” Every triangle and every quadrilateral tessellates.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Interior angleFind the interior angle of a regular 12-sided polygon.
150°. - Earlier work, enlargementEnlarge (2, −3) by scale factor 5 from the origin.
(10, −15).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- VertexA regular hexagon and two squares meet at a vertex. What angle is left?
360 − 120 − 90 − 90 = 60°. - SidesA regular polygon has an exterior angle of 40°. How many sides?
9.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why no regular polygon with more than 6 sides tiles on its own.
Its angle is more than 120° and less than 180°, so 360 ÷ angle is between 2 and 3, never a whole number. - Earlier work, rotationsTurn (3, −1) through 90° anticlockwise about the origin.
(1, 3).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to the slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Paper polygons with equal sides: fit copies round a point and look for the gap.