Lesson 5: Tessellations
Learning Intention
I can create tessellating patterns and explain why some shapes tessellate and others do not.
Success Criteria
- I can define a tessellation (a pattern of shapes that fit together with no gaps or overlaps).
- I can show that angles around a vertex point must add up to 360°.
- I can create a semi-regular tessellation using more than one shape.
🎥 Media Anchor (8 mins)
Video: Tessellations and Repeating Structures
- Which shape combinations tessellate without gaps, and why?
- How can transformation rules help you debug a tessellation error quickly?
Lesson Sequence
1. Investigation: The Floor Tiler (15 mins)
Give small groups a set of plastic polygons (triangles, squares, pentagons, hexagons, octagons).
Challenge: Which ones can tile a floor perfectly? Which ones leave gaps?
Findings: Triangles (Yes), Squares (Yes), Pentagons (No - gap), Hexagons (Yes), Octagons (No - unless you use squares too!).
2. The "Why": Angle Sums (15 mins)
Why do hexagons work but pentagons don't?
Looking at a Vertex point: A full circle is 360°.
- Square (90°): 90 + 90 + 90 + 90 = 360. Fits!
- Hexagon (120°): 120 + 120 + 120 = 360. Fits!
- Pentagon (108°): 108 + 108 + 108 = 324. Gap stays!
3. Escher-Style Art (20 mins)
Demonstrate the "Nibble" technique:
- Start with a square card.
- Cut a shape out of the LEFT side.
- Tape it to the RIGHT side.
- Now the new weird shape will still tessellate!
Students create their own unique tessellating creature.
4. Cultural Connection (10 mins)
Look at Tāniko weaving patterns. They rely on a triangular grid (often diamonds). Discuss how this grid supports different designs compared to a square grid.
Curriculum alignment
- Geometry — Knowledge: - A circle is the path traced out by a point moving in a plane and always a fixed distance (the radius) from a central point. - Angles between parallel lines and a transversal…
- Geometry — Practices: - In similar shapes, corresponding angles are equal and the lengths of corresponding sides are proportional. - Congruent shapes are identical in shape and size.
- Measurement — Practices: - Using Pythagoras’ theorem to:find the length of an unknown side in a right-angled trianglecheck if a triangle has a right anglecalculate the distance between two points in t…
- Measurement — Practices: - Finding:the perimeter of 2D shapesthe circumference of circlesthe area of parallelograms, trapeziums, and kites, relating the formulae used to the formula for a rectangle - …
📋 Teacher Planning Snapshot
Unit-wide intent — Te Aronga o te Wāhanga
Students will engage with this resource to develop geometric thinking through the lens of Māori visual art — exploring symmetry, transformation, tessellation, and spatial reasoning through the mathematical structures embedded in tukutuku panels, kōwhaiwhai rafter patterns, tāniko weaving, and whakairo (carving).
Unit-wide outcomes — Ngā Putanga o te Wāhanga
These describe the whole unit. Assess this lesson against its own success criteria above, not against these.
- ✅ Students can identify and describe geometric properties (symmetry, rotation, reflection, translation) within Māori art forms.
- ✅ Students can design their own pattern using geometric transformations, connecting mathematical precision to cultural meaning.
Differentiation & Inclusion
Scaffold support: Pre-cut shapes that do and do not tessellate, so the question is testable by hand. Extension: prove why a regular pentagon cannot.
ELL / ESOL: Geometry is a highly visual domain — the spatial and pattern-based nature of this content naturally reduces language barriers. Key vocabulary (symmetry, reflection, rotation, translation, tessellation) should be taught using physical models and diagrams before text-based tasks. Students can demonstrate geometric understanding through drawing and construction without requiring English fluency.
Inclusion: Geometric pattern work is inherently accessible and engaging across learning styles — visual, kinaesthetic, and analytical learners all find entry points. Neurodiverse learners often excel at pattern recognition and spatial reasoning. Offer choice in medium: digital tools, grid paper, or physical construction with card. The cultural context provides motivating purpose for students who find abstract geometry disconnected from meaning.
Mātauranga Māori lens: Tessellation asks whether shapes close the plane without gaps — a question you can put to any repeating pattern. Where the pattern is a taonga tradition, use sourced examples and name the source; do not generate a pastiche and call it tukutuku.
Prior knowledge: Students should have foundational understanding of 2D shapes and basic transformation vocabulary. No prior knowledge of Māori art forms required — the unit introduces cultural context alongside mathematical content.
Curriculum alignment
- Te Mātaiaho (2025) · Mathematics and Statistics · Phase 4 (Years 9–10) · Geometry (Practices): “- Transforming 2D shapes in the coordinate plane by translation, reflection about a given line of symmetry, and rotation about a given point by a multiple of 90 degrees”
- NZC (2007) · Mathematics and Statistics · Level 5: “Define and use transformations and describe the invariant properties of figures and objects under these transformations.”