Mathematical Explorers: Building Foundations · 2nd Class · Sorting and Classifying Shapes · Spring Term

Tessellating Patterns

Calculating theoretical probability and conducting simple experiments to determine experimental probability.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Statistics and Probability - S.2.1

About This Topic

Tessellating patterns cover a surface completely with shapes, leaving no gaps or overlaps. In 2nd class, students identify which shapes tessellate, such as equilateral triangles, squares, and rectangles, by arranging them edge to edge. They answer key questions like what it means for a shape to tessellate and how to create repeating patterns, using concrete materials to build understanding.

This topic aligns with the NCCA Sorting and Classifying Shapes unit in Mathematical Explorations: Building Foundations. It develops spatial awareness, pattern recognition, and geometric reasoning, skills that support later work in geometry and design. Students connect tessellations to real-world examples, like floor tiles or honeycombs, fostering appreciation for mathematics in everyday environments.

Active learning benefits this topic greatly because students manipulate physical shapes to test arrangements, gaining intuitive grasp through direct experience. Trial-and-error with cutouts reveals why some shapes fit perfectly while others do not, making abstract concepts concrete and encouraging persistence in problem-solving.

Key Questions

1. What does it mean for a shape to tessellate?
2. How can you use a shape to make a repeating pattern with no gaps or overlaps?
3. Can you create your own tessellating pattern using squares, rectangles, or triangles?

Learning Objectives

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

Before You Start

Identifying and Naming 2D Shapes

Why: Students need to be able to recognize and name basic shapes like squares, rectangles, and triangles before they can explore their properties for tessellation.

Sorting and Classifying Shapes by Attributes

Why: Understanding attributes like sides and angles is foundational for explaining why certain shapes tessellate and others do not.

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.

Watch Out for These Misconceptions

Common Misconception

All shapes tessellate equally well.

What to Teach Instead

Circles and most irregular shapes leave gaps or overlap. Hands-on trials with cutouts let students compare arrangements directly, building evidence-based reasoning. Group sharing highlights patterns in what works.

Common Misconception

Tessellations require identical orientations for every shape.

What to Teach Instead

Rotating or flipping shapes often creates perfect fits, as with triangles. Active manipulation in pairs encourages experimentation, correcting rigid thinking through visible successes.

Common Misconception

Only straight-edged shapes tessellate.

What to Teach Instead

Some curved shapes like certain pentagons can tessellate with modifications. Exploration stations expose students to exceptions, promoting flexible geometric thinking via concrete tests.

Active Learning Ideas

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.

45 min·Small Groups

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.

30 min·Pairs

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.

35 min·Whole Class

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.

20 min·Individual

Real-World Connections

• Architects and interior designers use tessellating patterns when selecting floor tiles, wall coverings, and paving stones to create visually appealing and functional surfaces in homes and public buildings.
• Mosaic artists create intricate designs by fitting together small tesserae, often geometric shapes, to cover surfaces like walls, floors, and sculptures, requiring careful planning of shape placement.
• Engineers consider tessellating principles when designing structures like honeycombs or certain types of solar panels, where efficient packing and stability are crucial.

Assessment Ideas

Quick Check

Provide students with a collection of pre-cut regular polygons (squares, triangles, hexagons). Ask them to select shapes that can tessellate and arrange them on a piece of paper to demonstrate a pattern. Observe which shapes they choose and how they attempt to fit them together.

Exit Ticket

Give each student a card with a picture of a shape. Ask them to write one sentence explaining whether the shape can tessellate and why or why not. For example, 'A square can tessellate because its corners fit together perfectly around a point.'

Discussion Prompt

Present students with two different tessellating patterns, one made with squares and one with triangles. Ask: 'What do you notice about how the shapes fit together at the corners in each pattern?' Guide them to discuss the angles meeting at the vertices.

Frequently Asked Questions

What shapes tessellate for 2nd class?

Equilateral triangles, squares, rectangles, and regular hexagons tessellate reliably at this level. Students discover this by arranging cutouts, noting how edges match perfectly. Introduce rhombi and trapezoids next to expand understanding, always linking back to no-gaps rule.

How can active learning help teach tessellating patterns?

Active approaches like shape manipulation and station rotations make tessellation tangible for 2nd class. Students test fits hands-on, observing gaps firsthand, which deepens retention over diagrams alone. Collaborative designs build communication skills while reinforcing spatial concepts through peer feedback.

Real-world examples of tessellations for primary students?

Point to bathroom tiles, brick walls, soccer balls, and beehives. These connect maths to daily life, motivating exploration. Students sketch classroom examples, then create models, bridging abstract ideas to familiar contexts in the Irish curriculum.

How to assess tessellating patterns understanding?

Observe during activities for edge-matching accuracy and pattern repetition. Use simple rubrics for designs: no gaps, full coverage, creativity in repeats. Exit tickets asking 'Why does this shape tessellate?' reveal reasoning depth effectively.