Tessellations: Repeating ShapesActivities & Teaching Strategies
Students learn best when they move from abstract rules to concrete evidence. Tessellations rely on precise angle sums and spatial reasoning, which are clearer when students manipulate shapes physically rather than just observing them. The hands-on approach in these activities builds intuition that textbooks alone cannot provide.
Learning Objectives
- 1Identify the specific angle measurements at the vertices of regular polygons that allow them to tessellate.
- 2Design a tessellating pattern using a single irregular quadrilateral.
- 3Analyze how M.C. Escher used transformations like translation, rotation, and reflection to create complex tessellations.
- 4Create an original tessellating artwork inspired by M.C. Escher's style.
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Pairs: Shape Fitting Trials
Pairs cut paper shapes like triangles, squares, pentagons, and hexagons. They arrange pieces to cover a square mat without gaps or overlaps, rotating or flipping as needed. Pairs note which shapes succeed and sketch their findings.
Prepare & details
Explain the mathematical principles that allow shapes to tessellate.
Facilitation Tip: During Shape Fitting Trials, circulate with a timer to keep pairs focused on testing each shape for gaps or overlaps before moving to the next one.
Setup: Groups at tables with problem materials
Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric
Small Groups: Escher Tile Design
Groups select a simple animal outline and modify it to tessellate by adjusting edges for perfect fits. They cut multiples, arrange into a repeating pattern, and add colour. Groups present their tile to the class.
Prepare & details
Design a tessellating pattern using a single irregular shape.
Facilitation Tip: For Escher Tile Design, provide only safety scissors and pre-cut cardstock to avoid wasted time and encourage precision in edge modification.
Setup: Groups at tables with problem materials
Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric
Whole Class: Tessellation Mural
Each student creates one tessellating tile based on a class-chosen shape. Tiles combine on a large wall display to form a mural. Students discuss how individual pieces contribute to the whole.
Prepare & details
Analyze how M.C. Escher transformed simple tessellations into complex artistic compositions.
Facilitation Tip: Before the Tessellation Mural starts, assign roles (cutter, gluer, designer) so each student contributes meaningfully to the collaborative piece.
Setup: Groups at tables with problem materials
Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric
Individual: Irregular Shape Creator
Students draw and cut an irregular shape that tessellates, using reflection or rotation. They tile a page and describe their design process in a short label.
Prepare & details
Explain the mathematical principles that allow shapes to tessellate.
Setup: Groups at tables with problem materials
Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric
Teaching This Topic
Teach tessellations by letting students discover the 360-degree rule through trial and error, then formalize it afterward. Avoid front-loading too much vocabulary; let the activities reveal the concepts naturally. Research shows that students retain geometric reasoning better when they first experience the frustration of mismatched shapes, then find the satisfying fit that works.
What to Expect
By the end of these lessons, students will confidently identify tessellating shapes, modify irregular ones to fit, and create purposeful patterns that mirror both mathematical and artistic goals. They will articulate why certain angles work and how artists use tessellations in creative ways.
These activities are a starting point. A full mission is the experience.
- Complete facilitation script with teacher dialogue
- Printable student materials, ready for class
- Differentiation strategies for every learner
Watch Out for These Misconceptions
Common MisconceptionDuring Shape Fitting Trials, watch for students who force irregular shapes together without considering angles.
What to Teach Instead
Prompt them to measure the angles with a protractor and test how many fit around a central point before arranging them on the table.
Common MisconceptionDuring Escher Tile Design, watch for students who assume only regular polygons can be used.
What to Teach Instead
Remind them to cut irregular edges and test the new shape’s fit by tracing and sliding it along the tile’s perimeter.
Common MisconceptionDuring Tessellation Mural, watch for students who create repetitive patterns without transformation.
What to Teach Instead
Ask them to consider how Escher used color and scale changes to turn simple tiles into complex figures.
Assessment Ideas
After Shape Fitting Trials, provide students with pre-cut regular polygons and ask them to arrange them around a central point, recording which shapes sum to 360 degrees at the vertex. Ask: ‘Which shapes fit together perfectly here?’
During Irregular Shape Creator, give students a worksheet with a simple irregular quadrilateral. Ask them to draw one modification to the shape that would allow it to tessellate. On the back, they write one sentence explaining why their modified shape will tessellate.
After Escher Tile Design, students display their artwork in pairs and use a checklist: ‘Does the pattern repeat without gaps or overlaps?’ ‘Are there recognizable figures within the tessellation?’ ‘Did the artist use color effectively?’ Partners provide one specific positive comment.
Extensions & Scaffolding
- Challenge students who finish early to design a tessellation using only a scalene triangle, requiring them to rotate and flip the shape to cover the plane.
- Scaffolding: Provide a template with a 60-degree angle guide for students who struggle to visualize the 360-degree sum during Shape Fitting Trials.
- Deeper: Invite students to research and present on tessellations in architecture, such as Islamic geometric patterns or M.C. Escher’s architectural illusions.
Key Vocabulary
| Tessellation | A pattern made of shapes that fit together perfectly without any gaps or overlaps. |
| Vertex | A point where two or more lines or edges meet; a corner of a shape. |
| Irregular Polygon | A polygon where not all sides are equal in length and not all angles are equal in measure. |
| Translation | Moving a shape from one place to another without rotating or flipping it; a slide. |
| Reflection | Creating a mirror image of a shape by flipping it across a line; a flip. |
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