Perfect Tile Patterns

Overview

Today students investigate tessellations using regular polygons, then connect what they observe to a key interior-angle rule: the angles meeting at a vertex must add to exactly 360° for a perfect fit with no gaps or overlaps. This builds classification and geometric reasoning about 2D shapes.

Learning intentions

Students will be able to:

• define a tessellation as a pattern of identical shapes with zero gaps and zero overlaps
• test regular polygons (equilateral triangles, squares, regular pentagons, regular hexagons) to determine whether they tessellate
• explain, using interior angles, why certain regular polygons can tile around a vertex but others cannot

Success criteria

Students can:

• describe what “tessellate” means and identify examples
• correctly record which polygon tiles and which does not, using evidence from their tests
• explain the role of interior angles at a vertex, using the idea that they must total 360° for a perfect tiling

Curriculum links

• MA3-2DS-01: Students investigate and classify two-dimensional shapes (including triangles and quadrilaterals) based on properties, and use geometric reasoning about 2D shape properties.
• MA3-2DS-01: Students identify and determine geometric properties by measuring/considering angle sizes to support classification.
• MA3-GM-03 (supporting): Students measure and construct angles and identify relationships between angles formed by intersecting lines.
• MA3-GM-02 (supporting): Students select appropriate units/devices to measure lengths and angles where needed to justify angle reasoning.

Lesson structure (total minutes)

1. 0–10 min · Introduction & Hook. Teacher shows a beehive honeycomb or bathroom tiling image and explains tessellation using student-friendly language (identical shapes, zero gaps, zero overlaps). Students discuss in pairs: “What makes this pattern keep fitting forever?” and share one idea.

2. 10–30 min · Hands-on Investigation. Teacher places students in pairs, gives each pair a set of plastic pattern blocks or paper cutouts of the regular polygons (equilateral triangles, squares, regular pentagons, regular hexagons), and provides a “Tiling Test” recording sheet with columns: Tiles? (Yes/No), Evidence (gap/overlap), and “How many at a point?” Students work on one polygon type at a time to cover a small flat area (or within a drawn outline), trying different arrangements until they either achieve a perfect local tiling or clearly see a gap/overlap.

3. 30–45 min · Mathematical Proof (Angle Rule). Teacher regroups students and guides a comparison: triangles and squares typically succeed; pentagons often fail; hexagons typically succeed. Teacher directs attention to a single vertex where tiles meet and states the rule: for shapes to fit perfectly around one point, the interior angles at that vertex must add to 360° (a full turn). Students turn and talk: “Why would the rule predict failure for the pentagon?” Teacher listens for mentions of “leftover angle” or “doesn’t reach 360°”.

4. 45–55 min · Guided Practice (Square Works). Teacher writes: square interior angle = 90°. Students model “four squares around a point”: 90° × 4 = 360°, so they tile perfectly. Students complete a short calculation in books for the square and then draw a small “vertex diagram” showing four angles meeting at a point.

5. 55–60 min · Reflection & Exit Check. Teacher gives a quick written prompt: “Use the 360° vertex rule to explain why a regular pentagon leaves a gap when three meet.” Students write 3–4 sentences. Teacher circulates for immediate feedback and collects for checking.

Resources

• Image of tessellations: honeycomb or bathroom tile pattern (printed or on screen)
• Pattern blocks or sturdy paper cutouts: equilateral triangles, squares, regular pentagons, regular hexagons
• “Tiling Test” recording sheet (Tiles? evidence; number meeting at a point)
• Workbook pages with space for vertex diagrams and angle calculations
• Rulers and protractors (only if students need to verify angles)
• Coloured pencils for marking gaps/overlaps
• Class board/markers for demonstrating the 360° rule

Assessment

• Formative: observation during the tiling task—can students identify gaps/overlaps and record evidence clearly?
• Formative: during discussion, listen for accurate use of “vertex” and the “360° around a point” idea
• Exit check: written explanation using the pentagon interior angle (108°) and the leftover angle concept

Differentiation

• Support: provide sentence starters for the final explanation (e.g., “The interior angle of a regular pentagon is… When three meet, the total is… That leaves…”).
• Support: offer a model “vertex diagram” frame (circle/point with spokes) for students who need structure.
• Extension (for fast finishers): ask them to test whether 4 or 5 pentagons would overshoot/undershoot 360° and describe what that would look like.
• EAL/SEN: reduce cognitive load by limiting to one polygon per round at first; keep recording sheet simple and highlight key terms (tessellation, vertex, interior angle).

(Required) Teacher notes for the angle reasoning

• Square: interior angle = 90°, so four meet exactly: 90° × 4 = 360°.
• Regular pentagon: interior angle = 108°, so three meet: 108° × 3 = 324°, leaving 360° − 324° = 36° gap (hence no perfect tessellation around a single vertex).