Mastering Mathematical Thinking: 4th Class · 4th Class · Shape, Space, and Symmetry · Summer Term

Tessellations

Investigating how certain shapes can tile a plane without gaps or overlaps.

• NCCA Curriculum Specifications
• NCCA: Primary - Shape and Space
• NCCA: Primary - Symmetry

About This Topic

Tessellations cover a plane completely with shapes that fit without gaps or overlaps. In 4th class, students test regular polygons to see which ones work: equilateral triangles, squares, and regular hexagons succeed because their interior angles add up to 360 degrees at each vertex. They examine why shapes like regular pentagons fail, creating spaces, and explore combinations of shapes that tessellate together. This hands-on analysis answers key questions from the NCCA Shape and Space strand.

Tessellations connect symmetry and spatial reasoning across the Summer Term unit. Students spot tessellations in everyday items, such as floor tiles, honeycombs, or Islamic art, and create their own patterns using rotations and translations. These activities build skills in recognizing geometric properties, predicting fits, and describing patterns verbally, which support broader mathematical thinking.

Active learning suits tessellations perfectly. When students cut, arrange, and rearrange physical shapes on tables or floors, they experience angle relationships directly. Group trials encourage debate over successes and failures, while designing personal patterns reinforces rules through creative application, turning geometric theory into intuitive understanding.

Key Questions

1. Why do some shapes tessellate perfectly while others leave gaps?
2. Analyze the properties of shapes that allow them to tessellate.
3. Construct a tessellating pattern using a regular polygon.

Learning Objectives

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

Before You Start

Identifying and Classifying Polygons

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

Measuring and Comparing Angles

Why: Understanding interior angles and their measures is fundamental to determining if shapes can fit together without gaps.

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.

Watch Out for These Misconceptions

Common Misconception

All regular polygons tessellate if made small enough.

What to Teach Instead

Size does not affect tessellation; interior angles must sum to 360 degrees at vertices. Hands-on sorting and fitting activities let students test various sizes directly, revealing the angle rule through repeated trials and peer comparisons.

Common Misconception

Only squares tessellate perfectly.

What to Teach Instead

Triangles and hexagons also tessellate alone, and combinations work too. Group explorations with multiple shapes build collections of examples, helping students revise narrow views via shared evidence and discussion.

Common Misconception

Tessellations require straight edges only.

What to Teach Instead

Curved shapes can tessellate if edges match precisely. Manipulating puzzle pieces with curves in pairs shows matching is key, shifting focus from straight lines through tactile experimentation.

Active Learning Ideas

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.

45 min·Small Groups

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.

30 min·Pairs

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.

35 min·Whole Class

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.

25 min·Individual

Real-World Connections

• Architects and designers use tessellations in tiling floors and walls, creating visually appealing and practical surfaces in buildings like the National Museum of Ireland. The repeating patterns add aesthetic value and structural integrity.
• Mathematicians study tessellations to understand geometric principles, which have influenced art movements like Cubism and are found in the intricate patterns of Islamic geometric art, seen in historical mosques and palaces.

Assessment Ideas

Exit Ticket

Provide students with cut-out shapes of equilateral triangles, squares, and regular hexagons. Ask them to select one shape and demonstrate how it tessellates by arranging it on grid paper. On the back, they should write one sentence explaining why their chosen shape works.

Discussion Prompt

Show students images of a regular pentagon pattern with gaps and a regular hexagon tessellation. Ask: 'What is the key difference in the angles at the points where the shapes meet? How does this difference affect whether the shapes can cover the entire surface?'

Quick Check

Observe students as they work in small groups to create a tessellating pattern using at least two different regular polygons. Note which groups successfully achieve a gap-free pattern and which students can articulate the angle sum rule (360 degrees) at the vertices.

Frequently Asked Questions

What shapes tessellate in 4th class maths?

Equilateral triangles, squares, and regular hexagons tessellate alone because their 60, 90, and 120-degree angles fit six, four, or three around a point to total 360 degrees. Combinations like triangles and hexagons also work. Students discover this by testing NCCA-specified regular polygons, linking to symmetry in patterns.

How can active learning help teach tessellations?

Active methods like cutting shapes, arranging them on tables, and adjusting fits give direct feedback on gaps or overlaps. Small group rotations build collaboration as students debate angle fits, while creating floor designs scales concepts spatially. These approaches make abstract angle sums concrete, boost engagement, and help articulate rules, aligning with NCCA emphasis on practical exploration.

Why do some shapes tessellate and others not?

Tessellation depends on interior angles dividing evenly into 360 degrees. For example, a square's 90 degrees fits four times; a pentagon's 108 degrees leaves a gap. Classroom trials with compasses and protractors confirm this, connecting to unit key questions on properties.

How to connect tessellations to real life?

Point out tessellations in tiles, bricks, soccer balls, or nature like pineapple skins. Students hunt examples in school or photos, then replicate with shapes. This links NCCA Shape and Space to the environment, deepening recognition of symmetry in architecture and art.