Mathematics · Class 2 · Shapes and Space · Term 1

Tessellations and Tiling

Exploring how shapes fit together without gaps or overlaps to create patterns.

About This Topic

Tessellations cover a surface completely with shapes that fit together without gaps or overlaps. In Class 2, students examine how equilateral triangles, squares, and regular hexagons tile planes perfectly, while shapes like circles or regular pentagons leave spaces or overlap. They answer questions such as which shapes tile a floor without gaps, why some shapes fail on their own, and how to design patterns using squares and triangles. This builds spatial awareness through simple experiments with everyday examples like floor tiles or honeycombs.

Within the CBSE Mathematics curriculum's Shapes and Space unit for Term 1, tessellations connect basic geometry to pattern-making and problem-solving. Students develop skills in observing angles and edges, which prepare them for advanced topics like symmetry and transformations in later grades. Regular practice helps children recognise tessellations in their surroundings, such as brick walls or patchwork quilts.

Active learning suits this topic well because children learn best by handling shapes directly. Cutting paper polygons, arranging them on desks or floors, and adjusting for fits or gaps lets students test ideas quickly. Group sharing of designs encourages explanation of choices, corrects errors through peer feedback, and sparks creativity in pattern invention. This method makes abstract geometry concrete and enjoyable.

Key Questions

1. Which shapes can tile a floor without leaving any gaps?
2. Explain why some shapes cannot tessellate on their own.
3. Design a simple tessellation pattern using a square and a triangle.

Learning Objectives

• Identify shapes that can tessellate a plane without gaps or overlaps.
• Explain why certain regular polygons (e.g., pentagons, heptagons) cannot tessellate a plane on their own.
• Design a simple tessellation pattern using two different regular polygons.
• Demonstrate how to create a tessellation by arranging and repeating a single shape.

Before You Start

Identifying Basic Shapes

Why: Students need to be able to recognise and name common 2D shapes like squares, triangles, and circles before they can explore how they fit together.

Understanding Sides and Corners

Why: Knowledge of the number of sides and corners (vertices) of shapes is helpful for understanding how they connect to form tessellations.

Key Vocabulary

Tessellation	A pattern made of shapes that fit together perfectly with no gaps or overlaps, covering a flat surface.
Tile	To cover a surface with shapes that fit together without any spaces in between.
Gap	An empty space or opening between two things.
Overlap	To lie over or on top of something else, creating a space where two things cover each other.

Watch Out for These Misconceptions

Common Misconception

All shapes can tessellate if turned or flipped.

What to Teach Instead

Shapes tessellate only if their angles add up to 360 degrees at each vertex. Hands-on trials with cutouts show circles leaving gaps and pentagons overlapping, while peer discussions help students compare results and refine their understanding of edge-matching.

Common Misconception

Only squares tessellate.

What to Teach Instead

Many polygons like equilateral triangles and hexagons also tessellate. Activity rotations where groups test different shapes reveal successes beyond squares, building confidence as students discover patterns through trial and shared observations.

Common Misconception

Tessellations must use identical shapes only.

What to Teach Instead

Combinations of shapes can tessellate too. Collaborative mural activities demonstrate this, as groups experiment with mixes like squares and triangles, learning through visible failures and successes in real-time arrangements.

Active Learning Ideas

Shape Sorting: Tiling Challenge

Give students cutouts of triangles, squares, rectangles, circles, and hexagons. Instruct them to arrange shapes on A4 paper to cover it fully without gaps or overlaps. Have them record which shapes succeed and why others fail, then share findings.

30 min·Small Groups

Pattern Creation: Square-Triangle Designs

Provide grid paper and coloured squares and equilateral triangles. Students create repeating border patterns or floors by fitting shapes edge-to-edge. Encourage them to colour and label their designs, explaining the tiling rule.

25 min·Pairs

Floor Simulation: Classroom Tiles

Use cardboard shapes to mimic floor tiling on a marked classroom area. Groups place shapes to cover the space, noting adjustments needed. Conclude with a class vote on the best seamless design.

40 min·Small Groups

Mural Building: Group Tessellation

On a large chart paper, students add tessellating shapes one group at a time to build a class mural. Each group uses one shape type first, then combines with others. Display and discuss the final pattern.

45 min·Whole Class

Real-World Connections

• Architects and interior designers use tessellations when planning floor tiles, wall coverings, and mosaic art to create visually appealing and functional surfaces.
• Bricklayers use the principle of tessellation when building walls, ensuring bricks fit together tightly to create strong structures without gaps.

Assessment Ideas

Quick Check

Provide students with cut-out shapes (squares, triangles, circles, pentagons). Ask them to select shapes that can tile a surface without gaps and place them on their desks to demonstrate. Observe which shapes they choose and how they arrange them.

Exit Ticket

Give students a worksheet with a grid. Ask them to draw a tessellation pattern using only squares and triangles. They should then write one sentence explaining why their chosen shapes fit together without gaps.

Discussion Prompt

Show students images of different tiled surfaces (e.g., a honeycomb, a brick wall, a tiled floor). Ask: 'Which of these are tessellations? How do you know? What shapes are used, and why do they fit together so well?'

Frequently Asked Questions

Which shapes tessellate for Class 2 students?

Equilateral triangles, squares, and regular hexagons tessellate because their angles fit exactly around a point. Rectangles work along straight lines but need careful alignment. Circles and pentagons do not, as they create gaps or overlaps. Use physical models to let students verify this through direct experience.

How to teach why some shapes cannot tessellate?

Demonstrate with cutouts around a central shape; show how pentagon angles exceed 360 degrees, causing overlaps. Students measure angles with paper protractors or compare visually. Group challenges to force fits highlight the issue, leading to discussions on angle sums and edge matches.

How can active learning help students understand tessellations?

Active methods like manipulating cutouts or tiling floors engage spatial senses fully. Children test hypotheses immediately, see gaps form, and adjust shapes, which cements the concept better than diagrams. Pair work fosters explanation skills, while class murals build collaboration and pride in collective designs, ensuring lasting retention.

Ideas for tessellation patterns using squares and triangles?

Create borders by alternating squares and triangles along edges, or fill areas with rows of triangles framed by squares. Students draw on grid paper first, then cut and assemble. This practice answers unit questions directly and extends to cultural patterns like kolam designs, linking maths to art.