Lesson 5 Workshop: Testing Tessellations
Supported route
This workshop separates the local vertex test from the larger repeat, so learners can explain why a trial succeeds or fails. The open-inquiry version of Lesson 5 is available when learners are ready to select and justify their own tessellation method.
Learning target
Ākonga will construct a gap-free, overlap-free repeat and use the angles meeting at one vertex as evidence for whether the selected regular polygon can tessellate by itself.
Prepare the vertex laboratory
Cut sets of equilateral triangles, squares, regular pentagons and regular hexagons in four colours. Add a 360° circle mat, protractors, glue sticks and one recording card per pair.
Local test before full pattern
Fit corners around one point, record each interior angle, and check whether the total is exactly 360°. Passing this local test is necessary; learners must still extend the arrangement to verify that it repeats without a gap or overlap.
Tessellation workshop · 60 minutes
1. Sort the examples · 7 minutes
Pairs sort six small diagrams into gap, overlap or tessellation. They mark one vertex in each diagram so the discussion stays tied to visible evidence.
2. Model triangle and pentagon tests · 13 minutes
Arrange six 60° triangle angles around a point to make 360°. Contrast this with regular pentagon angles of 108°: no whole number of identical angles makes one complete turn.
3. Guided test cards · 16 minutes
Each pair tests squares and regular hexagons, records a repeated-addition equation, and extends the successful vertex arrangement in at least three directions.
Checkpoint E: the equation, physical arrangement and written conclusion must agree.
4. Independent repeat · 16 minutes
Choose one successful regular polygon, trace a six-repeat patch, and colour corresponding copies of the motif. Label the translation used to move between two adjacent repeats.
5. Exit explanation · 8 minutes
Complete: “My shape can tessellate by itself because ___ angles of ___° meet to make ___°, and the extended patch shows ___.”
Teacher checkpoint notes
Reduce measurement load: provide the four interior-angle values and ask the learner to select and test the correct repeated-addition equation.
Keep claims bounded: the vertex sum establishes a local condition; require the physical patch before accepting that the arrangement extends as a tessellation.
Ready to progress: the learner links a 360° vertex equation to a repeat that is demonstrably free of gaps and overlaps.
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.”