Lesson 2: Symmetry Investigations
Learning Intention
I can identify lines of symmetry and rotational symmetry in 2D shapes and patterns.
Success Criteria
- I can draw lines of symmetry where a shape matches exactly if folded.
- I can determine the order of rotational symmetry (how many times a shape fits onto itself in 360°).
- I can create a simple motif with exactly 2 lines of symmetry.
🎥 Media Anchor (8 mins)
Video: Symmetry in Geometric Patterns
- How do you test whether a line is a true axis of symmetry?
- Where do you see symmetry choices influencing visual balance in Māori patterns?
Lesson Sequence
1. Warm Up: Mirror Mirror (10 mins)
Use handheld mirrors. Place them on letters of the alphabet (A, B, C...). Which letters have a line of symmetry? Which have two?
2. Concept Builder: Line vs Rotation (20 mins)
Line Symmetry (Reflectional): "The folding line."
Rotational Symmetry: "The spinning match."
Activity: Show a square. How many fold lines? (4). If I spin it, how many times does it look like a square? (4). Order of rotation = 4.
Contrast: Show a rectangle. Lines = 2. Order = 2.
Contrast: Show a Kōwhaiwhai spiral (koru). Does it fold? (Usually no). Does it spin? (Often order 2).
3. Creator Task: The Symmetry Designer (20 mins)
Brief: On grid paper, design a "tile" that meets these specific rules:
- Must have at least 1 line of symmetry (vertical or horizontal).
- Must use only straight lines (for easy drawing).
- Must NOT be just a square or cross (be creative!).
Challenge: A neighbour tries to find the order of rotational symmetry for your design.
4. Reflection (10 mins)
Why do cultural patterns (like in churches, marae, mosques) use so much symmetry? (Balance, beauty, order, storytelling).
Curriculum alignment
- Geometry — Practices: - Representing and constructing 3D shapes, including rectangular and triangular prisms and pyramids, from nets and plan views drawings - Transforming 2D shapes in the coordina…
- Algebra — Practices: - The gradient m of a straight line can be determined with the formula m = rise—run = Δy—Δx . - A vertical line has an infinite gradient. This cannot be expressed in the formu…
- Algebra — Practices: - Multiplying or dividing by a negative number reverses an inequality. - The constant rate of change of a linear graph is the vertical change (how far it goes up or down) divi…
- Geometry — Knowledge: - A set of points in a plane can be transformed by translation, reflection about a line, and rotation about a fixed point.
- Statistics — Practices: - For relationship investigations, drawing an eyeballed line or curve of best fit to predict possible y-values (the response variable) for given x-values (the explanatory vari…
📋 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: A symmetry checker (mirror or tracing paper) so ākonga TEST rather than guess. Extension: find a pattern with rotational but not reflective symmetry.
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: Symmetry in kōwhaiwhai and tāniko is exact enough to analyse, and the exactness is worth noticing. But 'has reflective symmetry' is a mathematical description; what the pattern MEANS belongs to those who hold it, and that meaning is not recoverable from the geometry.
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.”