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
Ngā Whāinga Ako — Learning Intentions
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).
Ngā Paearu Angitū — Success Criteria
- ✅ 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: Provide grid templates and partially completed pattern examples for entry-level construction tasks. Allow students to trace and analyse existing patterns before creating their own. Extend capable students by asking them to calculate the mathematical properties of their design (angles, lines of symmetry, ratio of repeat unit to total pattern) and explain the transformation rules in formal mathematical language.
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: Māori art is mathematics made visible. Tukutuku panels encode precise geometric grids requiring exact calculation of spacing, proportion, and symmetry. Kōwhaiwhai patterns use translational symmetry along the length of rafter beams — a sophisticated application of repeating geometric units. Tāniko weaving requires mental rotation and spatial reasoning to maintain pattern integrity across diagonal threads. Whakairo (carving) encodes symbolic meaning within geometric forms. Far from being decorative, these art forms represent generations of mathematical knowledge encoded in cultural practice. Teaching geometry through this lens shows ākonga that mathematics belongs to all cultures — and that Māori ancestors were sophisticated mathematicians.
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
- Geometry and Measurement — Shape: Apply the properties of symmetry, including line and rotational symmetry, to identify, describe, and create patterns and shapes.
- Geometry and Measurement — Transformation: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).