Tessellations

Active learning lets students feel the fit of shapes as they test, fail, and revise. When students rotate through stations with hands-on pieces, they experience the angle rule in real time rather than memorize it abstractly. The tactile work turns abstract geometry into something they can see and adjust immediately.

• 4th Class
• Mastering Mathematical Thinking: 4th Class
• 4 activities
• 25 min–45 min

Learning Objectives

1. Analyze the properties of regular polygons that enable them to tessellate a plane.
2. Compare and contrast the tessellating abilities of different regular polygons.
3. Create a tessellating pattern using at least two different regular polygons.
4. Explain why certain combinations of angles at a vertex are necessary for a tessellation.
5. Identify examples of tessellations in architectural designs and natural structures.

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Polygon Tessellation Tests

Prepare stations with cutouts of equilateral triangles, squares, hexagons, and pentagons. Groups test each shape by arranging them around a point and extending to cover paper, noting gaps or overlaps in journals. Rotate every 10 minutes and share findings with the class.

Prepare & details

Why do some shapes tessellate perfectly while others leave gaps?

Facilitation Tip: In Personal Tessellation Design, prompt students to outline their final pattern in marker so the tessellating units stand out clearly.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 30 min·Pairs

Pairs Challenge: Mixed Shape Tessellations

Partners receive assorted regular polygons and try combining two or more types to cover a worksheet without gaps. They sketch successful patterns and explain the angle sums verbally. Switch partners midway to compare strategies.

Prepare & details

Analyze the properties of shapes that allow them to tessellate.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 35 min·Whole Class

Whole Class: Collaborative Floor Tessellation

Project a large grid on the floor with tape. Class works together to fill it using shape cutouts, adjusting as needed. Discuss properties that made it work and photograph the final design.

Prepare & details

Construct a tessellating pattern using a regular polygon.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Individual

Individual: Personal Tessellation Design

Each student selects a tessellating shape, colors it, and repeats it to fill A4 paper, adding symmetry elements. They label angles and present one unique feature to peers.

Prepare & details

Why do some shapes tessellate perfectly while others leave gaps?

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teachers should avoid telling students the angle rule up front; instead, let the materials reveal it through repeated trials. Move between groups asking guiding questions like 'Where are the vertices meeting?' and 'Can you rotate that piece to close the gap?' to steer thinking without giving answers. Research shows that self-discovery of the 360-degree rule sticks better than direct instruction.

What to Expect

Successful learning looks like students confidently selecting, rotating, and matching shapes to cover paper without gaps or overlaps. They should explain why only certain regular polygons work by pointing to the angles at a vertex. Groups should also create mixed-shape patterns and defend their choices with angle sums.

Watch Out for These Misconceptions

Common Misconception

During Station Rotation: watch for students who believe scaling a regular pentagon down will make it tessellate.

What to Teach Instead

Hand them a set of pentagons in three sizes and ask them to test each one on grid paper, then compare vertex gaps with the hexagon set to see that size does not fix the angle problem.

Common Misconception

During Pairs Challenge: watch for students who assume only squares can tessellate when mixed with other shapes.

What to Teach Instead

Have them rotate through mixed sets and compare their mixed patterns to the single-shape stations, asking them to list all successful combinations and explain why those angles fit.

Common Misconception

During Collaborative Floor Tessellation: watch for students who think tessellations must have straight edges.

What to Teach Instead

Place curved puzzle pieces at the station and ask groups to match edges precisely, then compare straight-edge and curved-edge tessellations side by side.

Assessment Ideas

Exit Ticket

After Station Rotation, provide cut-out shapes of equilateral triangles, squares, and regular hexagons. Ask students to select one shape, arrange it on grid paper to demonstrate tessellation, and write one sentence explaining why their chosen shape works.

Discussion Prompt

During Collaborative Floor Tessellation, show images of a regular pentagon pattern with gaps and a regular hexagon tessellation. Ask students to point to the vertices and compare the angles, then explain how the angle difference affects coverage.

Quick Check

During Pairs Challenge, observe which groups create gap-free mixed-shape patterns and which students can articulate the 360-degree angle sum rule at the vertices while defending their designs.

Extensions & Scaffolding

• Challenge: Ask students to design a single tile that combines two regular polygons to tessellate, then cut and test their composite shape.
• Scaffolding: Provide hexagon and triangle templates with pre-marked vertices so students can focus on matching angles.
• Deeper exploration: Introduce semi-regular tessellations using three regular polygons and challenge students to find and document all eight possible combinations.

Key Vocabulary

Tessellation	An arrangement of shapes that fits perfectly together without any gaps or overlaps, covering a flat surface.
Vertex	A point where two or more lines or edges meet; in tessellations, it is where the corners of shapes join.
Interior Angle	The angle inside a polygon, measured at each vertex.
Translation	A transformation that moves every point of a figure the same distance in the same direction, creating a repeating pattern.
Rotation	A transformation that turns a figure around a fixed point, often used to create repeating elements in a tessellation.