Mathematics · Class 5 · Term 1: Foundations of Number and Geometry · Term 1

Introduction to Tessellations

Students will explore how regular and irregular polygons can tile a plane without gaps or overlaps, creating tessellations.

• CBSE Learning Outcomes
• NCERT: G-2.2

About This Topic

Tessellations cover a plane completely with shapes that fit together without gaps or overlaps. In Class 5, students discover which regular polygons, such as equilateral triangles, squares, and regular hexagons, can tessellate. They learn the key reason lies in the interior angles: at each vertex, angles from surrounding polygons must sum exactly to 360 degrees. For example, six equilateral triangles meet at a point since each has 60-degree angles.

This topic aligns with CBSE geometry standards, fostering spatial awareness and pattern recognition skills essential for higher maths. Students also examine tessellations in Indian art like Rangoli designs and kolam patterns, as well as nature, such as honeycomb structures created by bees. These connections make the concept culturally relevant and engaging.

Active learning shines here because students manipulate shapes physically to test arrangements, observe failures, and refine designs. Hands-on trials reveal why certain polygons work while others leave gaps, turning abstract angle rules into intuitive understanding. Collaborative pattern creation builds persistence and creativity.

Key Questions

1. Explain why only certain regular polygons can tessellate a plane.
2. Analyze examples of tessellations in art and nature.
3. Design a new tessellation pattern using a combination of different shapes.

Learning Objectives

• Classify polygons as regular or irregular based on side and angle properties.
• Explain the condition for tessellation using the sum of interior angles at a vertex.
• Analyze given patterns to identify whether they are tessellations and justify the reasoning.
• Design a tessellation using a combination of at least two different polygons.
• Create a tessellation pattern that demonstrates no gaps or overlaps.

Before You Start

Identifying Polygons

Why: Students need to be able to recognize and name basic polygons like triangles, squares, and hexagons before exploring their properties for tessellations.

Measuring Angles

Why: Understanding how to measure interior angles is crucial for determining if shapes can fit together at a vertex to sum to 360 degrees.

Key Vocabulary

Tessellation	A pattern made of shapes that fit together perfectly without any gaps or overlaps, covering a flat surface.
Polygon	A closed shape made of straight line segments, such as a triangle, square, or hexagon.
Regular Polygon	A polygon where all sides are equal in length and all interior angles are equal in measure.
Irregular Polygon	A polygon where sides or angles are not all equal.
Vertex	A corner point where two or more line segments or edges meet.
Interior Angle	The angle formed inside a polygon at one of its vertices.

Watch Out for These Misconceptions

Common Misconception

All regular polygons can tessellate a plane.

What to Teach Instead

Only equilateral triangles, squares, and regular hexagons tessellate because their interior angles divide 360 degrees evenly at a vertex. Pentagons and heptagons leave gaps or cause overlaps. Hands-on sorting activities let students test shapes directly and discover the angle rule through trial and error.

Common Misconception

Irregular shapes cannot form tessellations.

What to Teach Instead

Many irregular polygons tessellate if edges match perfectly, as in bathroom tiles or giraffe skin patterns. Peer sharing of successful designs corrects this by showing real examples. Group challenges encourage experimentation with modified shapes.

Common Misconception

Tessellations require identical shapes only.

What to Teach Instead

Combinations of different shapes can tessellate if they fit without gaps, like in Escher art. Collaborative mural building helps students see how varied polygons interlock, building confidence in complex patterns.

Active Learning Ideas

Pairs Tiling Challenge: Regular Polygons

Provide cut-out equilateral triangles, squares, pentagons, and hexagons. Pairs arrange them around a central shape to cover paper without gaps. They record which succeed and measure angles at vertices with protractors. Discuss findings as a class.

30 min·Pairs

Small Groups: Rangoli Tessellation Hunt

Groups search classroom for tessellated patterns, like floor tiles or printed fabrics, then sketch and classify shapes used. Extend by photographing Indian art examples from books. Present observations, noting regular versus irregular polygons.

25 min·Small Groups

Whole Class: Collaborative Mural Tessellation

Distribute grid paper and shape templates. Each student adds a repeating pattern section using triangles and squares. Connect pieces into a large mural. Reflect on how individual contributions fit seamlessly.

45 min·Whole Class

Individual: Custom Shape Design

Students draw an irregular polygon that tessellates, like an arrowhead shape. Trace and cut multiples to tile a page. Test rotations and flips, then colour for a decorative pattern inspired by kolam.

20 min·Individual

Real-World Connections

• Architects and interior designers use tessellations to create visually appealing and functional floor tiles, wall coverings, and ceiling designs, ensuring efficient use of space.
• Artists, like M.C. Escher, have famously used tessellations in their artwork to create optical illusions and intricate patterns, demonstrating the aesthetic potential of repeating shapes.
• Beekeepers observe the hexagonal tessellations in honeycomb structures, which are nature's efficient way of storing honey and maximizing space within the hive.

Assessment Ideas

Quick Check

Provide students with cut-out shapes of equilateral triangles, squares, and regular pentagons. Ask them to arrange these shapes around a single point on their desks. Observe which shapes successfully tessellate (sum of angles at the vertex is 360 degrees) and which leave gaps or overlap. Ask: 'Which shapes worked? Why do you think they worked?'

Exit Ticket

Give each student a printed image of a pattern. Ask them to write 'Yes' if it is a tessellation and 'No' if it is not. Below their answer, they must write one sentence explaining their choice, referring to gaps or overlaps.

Discussion Prompt

Show students examples of Indian art like Rangoli or Kolam patterns. Ask: 'Can you identify any tessellating shapes in these designs? How do these patterns relate to the mathematical concept of tessellations we have learned?' Encourage students to share their observations.

Frequently Asked Questions

Which regular polygons tessellate the plane?

Equilateral triangles, squares, and regular hexagons tessellate because the angles around a point sum to 360 degrees: six 60-degree angles for triangles, four 90-degree for squares, three 120-degree for hexagons. Regular pentagons fail as five 108-degree angles total 540 degrees, causing overlaps. Students verify this by arranging cut-outs.

How do tessellations appear in Indian art and nature?

Rangoli and kolam patterns use tessellated dots and shapes for intricate floor designs. In nature, beehives form hexagonal tessellations for efficiency, and pineapple skins show triangular patterns. Exploring these builds cultural pride and shows practical geometry applications.

How can active learning help teach tessellations?

Physical manipulation of cut-out shapes lets students test arrangements, feel gaps, and adjust intuitively before angle calculations. Group rotations through tiling stations promote discussion of successes and failures. Designing personal patterns reinforces rotation and reflection symmetries, making rules memorable through discovery.

Why do some polygons fail to tessellate?

Failure occurs when interior angles do not sum to 360 degrees at vertices, leading to gaps or overlaps. For instance, regular pentagons have 108-degree angles, so five exceed 360 degrees. Classroom trials with paper shapes reveal this concretely, aiding retention over rote memorisation.