Tessellating Patterns

Active learning works because tessellations require spatial reasoning and hands-on trial and error, which concrete materials provide. Students need to physically manipulate shapes to see how angles and edges fit together, making abstract concepts visible. This approach builds confidence as students observe immediate feedback from their arrangements.

• 2nd Class
• Mathematical Explorers: Building Foundations
• 4 activities
• 20 min–45 min

Learning Objectives

1. Identify which regular polygons (squares, equilateral triangles, hexagons) tessellate by arranging them without gaps or overlaps.
2. Explain the properties of shapes that allow them to tessellate, focusing on vertex angles summing to 360 degrees.
3. Create a tessellating pattern using at least two different shapes, demonstrating an understanding of edge-to-edge fitting.
4. Compare and contrast the tessellating abilities of different regular polygons.
5. Classify shapes based on their potential to form tessellating patterns.

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Shape Tessellation Stations

Prepare stations with triangles, squares, rectangles, and hexagons on cardstock. Students cut and arrange shapes to cover paper without gaps. Rotate groups every 10 minutes, then share successful patterns with the class.

Prepare & details

What does it mean for a shape to tessellate?

Facilitation Tip: During Shape Tessellation Stations, circulate with guiding questions like 'What do you notice about how these edges meet?' to prompt reflection.

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

Experiential Learning

⏱ 30 min·Pairs

Pairs: Custom Tessellation Design

Partners select one shape and create a repeating pattern on large paper, rotating and flipping as needed. They trace outlines and colour the design. Pairs present, explaining their choices.

Prepare & details

How can you use a shape to make a repeating pattern with no gaps or overlaps?

Facilitation Tip: For Custom Tessellation Design, provide grid paper with pre-drawn shapes to scaffold students who struggle with freehand drawing.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Experiential Learning

⏱ 35 min·Whole Class

Whole Class: Tessellation Floor Mat

Distribute shape tiles to the class. Students collaborate to cover a floor outline, adjusting placements collectively. Discuss challenges and solutions as a group.

Prepare & details

Can you create your own tessellating pattern using squares, rectangles, or triangles?

Facilitation Tip: When using the Tessellation Floor Mat, stand back to observe peer discussions, listening for geometric vocabulary like 'vertex' or 'edge'.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Experiential Learning

⏱ 20 min·Individual

Individual: Pattern Extension Challenge

Give each student a starter tessellation strip. They extend it across a page, trying different orientations. Collect and display for peer feedback.

Prepare & details

What does it mean for a shape to tessellate?

Facilitation Tip: In the Pattern Extension Challenge, encourage students to rotate their page to see if the pattern still tessellates, reinforcing flexibility in thinking.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Teaching This Topic

Teach tessellations by starting with simple shapes students already know, like squares, before introducing triangles or hexagons. Avoid rushing to abstract rules; let students discover patterns through repeated trials. Research shows that manipulatives and visual models are essential for young learners to grasp spatial relationships. Emphasize that tessellations are about relationships between shapes, not just the shapes themselves.

What to Expect

Successful learning looks like students confidently selecting shapes that tessellate, arranging them without gaps or overlaps, and explaining their choices using geometric terms. They should discuss why certain shapes fit and others do not, showing evidence-based reasoning. Patterns should repeat predictably, demonstrating understanding of edge-to-edge alignment.

Watch Out for These Misconceptions

Common Misconception

During Shape Tessellation Stations, watch for students assuming all shapes tessellate equally. Correction: Direct them to trial different shapes on their station table, comparing gaps when they arrange circles versus squares, then ask, 'What do you observe about the edges that fit perfectly?'

What to Teach Instead

During Custom Tessellation Design, watch for students rotating shapes rigidly. Correction: Remind them to flip or turn pieces, demonstrating with a triangle turned 180 degrees to show how it still fits, then ask, 'How did changing the orientation help?'

Common Misconception

During Tessellation Floor Mat, watch for students believing only squares tessellate. Correction: Place a pentagon cutout nearby and ask, 'Can this shape tessellate? Try arranging it with the squares.' Prompt them to notice angles meeting at vertices.

What to Teach Instead

During Pattern Extension Challenge, watch for students thinking curved shapes cannot tessellate. Correction: Provide a curved pentagon template and ask, 'How can you adjust this shape to fit without gaps?' Encourage them to trim edges if needed.

Assessment Ideas

Quick Check

After Shape Tessellation Stations, collect students’ arrangement sheets and note which shapes they selected and how they fit them together. Ask one student from each group to explain their choice of tessellating shapes.

Exit Ticket

After Custom Tessellation Design, give each student a card with a shape (e.g., octagon, trapezoid). Ask them to write one sentence explaining whether it tessellates and why, using evidence from their design process.

Discussion Prompt

During Tessellation Floor Mat, pause the activity and ask, 'Look at how the triangles and squares meet at the corners. What do you notice about the angles?' Guide students to describe how 360 degrees is formed by the meeting angles.

Extensions & Scaffolding

• Challenge: Ask students to create a tessellating pattern using two different shapes combined, such as a square and a triangle, documenting their process.
• Scaffolding: Provide cut-out shapes with Velcro dots so students can easily rearrange and test fits without frustration.
• Deeper exploration: Introduce the concept of semi-regular tessellations (using multiple regular polygons) and have students research and recreate one example.

Key Vocabulary

tessellate	To tile a surface with one or more geometric shapes, covering it completely without any gaps or overlaps.
vertex	A point where two or more lines or edges meet, forming a corner of a shape.
regular polygon	A polygon where all sides are equal in length and all interior angles are equal in measure.
pattern	A repeating arrangement of shapes or designs.