A 2-session unit for middle school. A tessellation is a pattern of shapes that tiles a surface with no gaps and no overlaps — the honeycomb is the classic example. Students discover which shapes tessellate and why, then use a simple slide-and-tile trick to turn a plain square into an interlocking shape that repeats forever.
A honeycomb tessellates with no gaps. Six hexagons fit around one exactly — the same packing that runs through the Flower of Life.
Level
Intermediate
Grade band
Middle school
Sessions
2 × 45 min
Total time
90 minutes
Overlay
Hexagonal
Learning objectives
By the end of the unit, students will:
Define a tessellation as a tiling with no gaps and no overlaps
Identify the three regular polygons that tessellate and explain why others do not
Connect tessellation to the angles meeting at a point summing to 360 degrees
Use the slide-and-tile method to make an interlocking shape from a square
Repeat the shape into a complete, gap-free pattern
Standards alignment
VA:Cr1.2.7aDevelop criteria to guide making a work of art or design to meet an identified goal.
VA:Cr2.3.7aApply visual organizational strategies to design and produce a work of art, design, or media that clearly communicates information or ideas.
VA:Re8.1.7aInterpret art by analyzing art-making approaches, the characteristics of form and structure, and use of media to identify ideas and mood conveyed.
Materials
Internet-connected device per student to study the hexagonal overlay as a reference
Card stock for cutting a tile template, scissors, and tape
Grid or plain paper, ruler, pencil, and a fine pen
Colored pencils or markers for the finished pattern
Cut-outs of regular triangles, squares, pentagons, and hexagons for the tiling test
Lesson sequence
1
Which shapes tessellate
45 minutes
Warm-up · 5 min
Show a tiled floor, a honeycomb, and a brick wall. Ask what they have in common. The shapes fit together with no gaps. Then ask why we never see floors tiled with circles or regular pentagons — because they leave gaps. The question of which shapes fit is today's puzzle.
(10 min) Tiling test: with cut-out triangles, squares, pentagons, and hexagons, students try to surround a single point. Triangles, squares, and hexagons close up perfectly; regular pentagons leave a stubborn gap.
(8 min) The reason: the angles meeting at a point must add to exactly 360 degrees. Six triangles (6 × 60), four squares (4 × 90), and three hexagons (3 × 120) all reach 360; pentagons (each 108) cannot.
(6 min) Students sketch each of the three regular tessellations and label the angle sum at a meeting point.
(2 min) They predict whether a mix of shapes might tile, setting up the extension.
Angles must total 360° at a meeting point. Triangles, squares, and hexagons reach it exactly; regular pentagons always leave a gap.
Reflection · 10 min
Why do the angles at a meeting point have to add to 360 degrees?
Which regular shapes tessellate, and which one surprised you by failing?
Where do you see tessellations in everyday life?
2
Building an interlocking tile
45 minutes
Warm-up · 5 min
Show a drawing where lizards or birds interlock with no gaps, in the style of M. C. Escher. Ask how a shape can be a creature and still tile perfectly. The trick is that whatever is cut from one side is added to the opposite side, so the area never changes. They are about to do it.
Main activity · 30 min
(8 min) Students cut a small square from card stock. They cut a bump out of the left edge and tape it onto the right edge in the same position — a slide, not a flip.
(4 min) They do the same from top to bottom: a piece off the top, taped to the bottom. The tile now has wiggly edges but tiles exactly, because every cut is matched by an addition.
(14 min) Using the template, students trace the tile across the page, sliding it over and down so each copy locks into the last with no gaps. They add a little face or detail so each tile reads as a shape.
(4 min) They color the pattern so neighbouring tiles contrast and the interlock is easy to see.
The slide-and-tile trick: a bump cut from the left edge, added to the right. Area stays the same, so the new shape tiles perfectly.
Reflection · 10 min
Why does sliding a cut to the opposite side keep the tile able to tessellate?
Did your pattern have any gaps, and if so, where did the method go wrong?
What did your interlocking shape end up looking like?
