Pattern Power: The Art and Math of Tessellations

Materials Needed

• Grid/graph paper
• Stiff cardstock or index cards (to make shape templates)
• Colored construction paper
• Scissors
• Pencils and rulers
• Glue sticks
• Colored markers or pencils
• A set of plastic or cardboard shape cutouts (equilateral triangles, squares, rectangles, regular pentagons, and regular hexagons)

Learning Objectives

By the end of this lesson, you will be able to:

• Define what a tessellation is in your own words.
• Determine and test whether a specific shape (like a triangle, square, rectangle, or pentagon) can tessellate a flat surface.
• Explain the mathematical rule of tessellation (shapes must fit together around a vertex to equal exactly 360 degrees without gaps or overlaps).
• Design and construct a colorful tessellation art piece using triangles, squares, or rectangles.

Success Criteria

• Your final design has absolutely no gaps and no overlaps between the shapes.
• Your shapes cover the designated surface area completely.
• You can explain why your chosen shape tessellates, using the term "vertex" correctly.

Introduction (Hook & Objectives)

Imagine you are a video game designer tasked with building a brick wall, a honeycomb in a beehive, or a tiled bathroom floor in a virtual world. If you leave even a tiny gap between the tiles, the game character might fall through the world! If you overlap them, the graphics will glitch.

To make these designs perfect, you need to use tessellations. A tessellation is a pattern made of one or more geometric shapes that fit together perfectly to cover a flat surface without leaving any gaps and without overlapping. Today, we are going to become pattern detectives. We will test different shapes to see which ones have the "magic powers" to tessellate, and then we will create our own piece of mathematical art!

Body: Instruction & Practice (I Do, We Do, You Do)

1. I Do: The Math Secret of Tessellations

First, let's look at the secret math behind why some shapes tessellate and others don't. It all comes down to angles and vertices.

A vertex (plural: vertices) is a corner where two or more lines meet. When we put shapes together to tile a surface, their corners meet at a single point.

The 360-Degree Rule: For shapes to fit together perfectly around a single point (vertex) without gaps or overlaps, the angles meeting at that point must add up to exactly 360 degrees (a full circle).

• Let's look at a square: Every corner of a square is a right angle (90 degrees). If we put four squares together corner-to-corner, the math is:
  90° + 90° + 90° + 90° = 360°.
  Because it equals exactly 360°, squares tessellate perfectly!

2. We Do: The Shape Investigation

Now, let's explore together! Grab your pre-cut shapes (triangles, squares, rectangles, pentagons, and hexagons). Let's test them on a flat table to see which ones can tile the space.

Step-by-step Investigation:

1. Test the Rectangles: Place four rectangles together so their corners meet at one point. Do they fit? Yes! Because like squares, their corners are 90 degrees.
2. Test the Equilateral Triangles: Place the corners of your triangles together around a single point. How many does it take to make a flat circle? (Hint: It takes 6!). Each angle of an equilateral triangle is 60 degrees.
   60° × 6 = 360°. Triangles tessellate!
3. Test the Regular Pentagons (5-sided shapes): Try to place pentagons corner-to-corner. What happens? Each corner of a regular pentagon is 108 degrees. If we put three together:
   108° + 108° + 108° = 324° (leaves a gap).
   If we try to add a fourth:
   324° + 108° = 432° (they overlap).
   Conclusion: Regular pentagons cannot tessellate on their own!

Fill out this Shape Detective Table together:

3. You Do: Create Your Tessellation Masterpiece!

Now it's your turn to be the artist! You will create a colorful tessellation mosaic.

Your Mission:

1. Choose your shape(s): Decide if you want to use triangles, squares, rectangles, or a combination of them.
2. Create your tiles: Use a ruler to draw your chosen shape on different colored construction papers, then carefully cut them out. You will need about 15–20 of them (all identical in size).
   Tip: You can use grid paper to make sure your squares or rectangles are perfectly equal!
3. Plan your layout: Before gluing anything down, arrange your shapes on a blank sheet of paper to make sure they fit perfectly without any gaps or overlaps. Try to make an interesting color pattern!
4. Secure your design: Carefully glue your shapes down onto the backing paper. Make sure the edges touch exactly.
5. Add Details: Once glued, use markers or colored pencils to turn your geometric shapes into something creative (e.g., transform your triangles into a flock of birds, your squares into a futuristic city wall, or your rectangles into a colorful brick path).

Conclusion & Reflection

Great job! Let's review what we discovered today:

• What are the two golden rules of tessellation? (Answer: No gaps, no overlaps!)
• Why do squares tessellate while regular pentagons do not? (Answer: The corner angles of squares meet to make exactly 360 degrees, while pentagons leave a gap or overlap because their angles don't add up to 360 degrees).

Show and Tell: Show off your final artwork! Point out the vertices where your shapes meet and explain the color pattern you chose to use.

Assessment

Formative Assessment: During the "We Do" activity, the student will be observed testing the shapes and filling out the Shape Detective Table correctly.

Summative Assessment: The final tessellation project will be assessed based on the Success Criteria:

• Accuracy (40%): Are there zero gaps and zero overlaps in the pattern?
• Completion (40%): Is the designated area fully covered by the tessellating shapes?
• Explanation (20%): Can the student explain how the shapes fit together at the vertex?

Differentiation Options

For Learners Needing Support (Scaffolding):

• Use pre-cut foam tile stickers instead of cutting out paper shapes with scissors. This keeps the focus entirely on the spatial arrangement and eliminates frustration with cutting.
• Stick to a single shape type (only squares or only rectangles) on a piece of grid paper to help guide alignment.

For Advanced Learners (Extensions):

• The Hexagon Challenge: Try to tessellate regular hexagons. Calculate the angle of a regular hexagon (120 degrees) and explain mathematically why they work (120° × 3 = 360°).
• The Escher "Nibble-and-Tape" Technique: Start with a cardstock square. Cut a wavy shape out of the left side, slide it directly across to the right side, and tape it down. Cut another wavy shape out of the top, slide it down to the bottom, and tape it. Use this new "crazy" shape as a template to trace and cut out shapes that will still tessellate perfectly!