NCTM Standards alignment:
Geometry: Apply transformations and use symmetry to analyze mathematical situations.

Objective: By interacting with BoxerMath's Tessellation Tool, students understand why equilateral triangles, squares, and regular hexagons tessellate "regularly" in the Euclidean plane.

Introducing the activity: BoxerMath's Tessellation Tool allows the student to build tessellations and other designs by attaching the vertices of various polygons to one another.

Note: BoxerMath's Tessellation Tool will open in a second window. Take a moment to adjust the two windows so that you can work with the applet and also view the text of the activity.

After an introductory exploration period, review the features of BoxerMath's Tessellation Tool that the students have discovered including:

1. use of translate vs. rotate
2. different polygons available
3. how the polygons are moved to the drawing area
4. color options
5. use of reset
6. use of help

Challenge: Ask the students to use triangle(s) and by translating and rotating, cover the plane with no gaps or overlaps. Here are two samples:

1. Have the students compare their results.
   
   
2. How did they make their tessellation? Was it made by using one triangle and both the translate and rotate features? Was it made by using two triangles and the translate feature?

   At this point in the lesson some students may not be able to see beyond the color and/or number of triangles. That is okay. The idea is just to start them thinking about how the two samples are different. With time they should see more, in particular the orientation of the triangles.

3. Ask the students to consider the "different" triangles available in the Tessellation Tool palette of polygons. Ask them, "If you were limited to only one of the triangles, can you make a tessellation? Can you make a tessellation of triangles no matter which of the four triangles is available to you?
   
   
4. Not counting the colors, ask students to discuss or respond to the question: What is similar about these two tessellations? What is different? (Encourage them to use the words rotation and translation in their explanation.)

        

   Students should be able to see that the red/pink tessellation can be rotated to look similar to the grey/yellow tessellation. There are more triangles in the red/pink tessellation but if the same number of triangles were used, it could look the same.

5. Four triangles can be chosen from BoxerMath's Tessellation Tool palette of polygons. Each of them can be rotated to "match" the others. Ask the students to explain the connections between the pairs of triangles shown below.

   Understanding the connection between the pairs of triangles may help students understand how a tessellation can be made with one triangle and the rotation feature.

Using manipulatives: Students' mathematical understanding can be extended if a combination of technology and concrete manipulatives is used. Provide students with activity pattern blocks. If they work with partners or in a group, challenge them to make "different" tessellations using only the equilateral triangles. (Paper activity pattern blocks are available on the Web; see Hand Made Manipulative Instructions by Margo Mankus.)

Revisiting the activity: Now that the students have tessellated with BoxerMath's Tessellation Tool and with activity pattern blocks, have them return to the Tessellation Tool to think about using only one equilateral triangle.

If we know that there are 360 degrees in a circle and if we think of using just one triangle (and the rotation feature) to make our tessellation, how many equilateral triangles do we need to fit together? How many degrees are in the interior measure of the angles that meet?

If students can use the reinforcement of making a tessellation of 6 equilateral triangles, encourage them to do this.

 Formalizing the mathematics:

 Focus students' attention on either of the two units:

   

1. What is the interior measure of the angles of a triangle?
2. Look at the vertex of one triangle that is in a tessellation of triangles. How many triangles, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

 Before returning to BoxerMath's Tessellation Tool, ask the students to predict whether a square will tessellate by answering these questions:

1. What is the interior measure of the angles of a square?
2. Think of the vertex of one square within a tessellation of squares. How many squares, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

After students have tested their prediction, repeat the process with a hexagon. Again ask them to make a prediction by asking:

1. What is the interior measure of the angles of a hexagon?
2. Think of the vertex of one hexagon within a tessellation of hexagons. How many hexagons, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

Now that the students have considered the cases of the equilateral triangle, square, and regular hexagon, ask them to help complete the chart:

NOTE: Numbers indicated in red would not be revealed to students.

Column 1	Column 2	Column 3
name of polygon	degrees of the interior measure of each angle	360 degrees divided by # in Column 2
equilateral triangle	60	360 / 60 = 6
square	90	360 / 90 = 4
regular pentagon	108	360 / 108 = 3.333...
regular hexagon	120	360 / 120 = 3
regular heptagon	128	360 / 128 = 2.8125
regular octagon	135	360 / 135 = 2.666...

 After discussing the numbers needed to complete the table, help students come to the following conclusion:

In a tessellation the polygons used will fit together with their angles arranged around a point with no gaps or overlaps. When using just one polygon (for example, only equilateral triangles), the interior measure of each angle will need to be a factor of 360 degrees (meaning that 360 degrees can be divided evenly by that angle measure). The only regular polygons that qualify are the equilateral triangle, the square, and the regular hexagon.

Assessment:

Ask students to choose any polygon other than a triangle, square or hexagon. Ask them to illustrate and explain why it will not tessellate regularly in the Euclidean plane.

Math Forum Resources

[Information and Lessons]  [Problems of the Week]  [T2T]  [Dr. Math]

What Is a Tessellation?

Definition and illustrated explanation of tessellations.

The Four Types of Symmetry in the Plane

Rotation, translation, reflection, and glide reflection, with illustrations and problems for consideration.

Repeated Reflections of an "R"

Lesson to draw a design with reflectional symmetry and rotational symmetry.

Tessellation Links

Links to other sites on the Web.

Tessellation Tutorials

A series of tutorials that teach students how to tessellate.

Selections from Math Forum Problems of the Week:

MidPoW: Tiling Triangles - posted March 11, 2002

Given the dimensions of a large triangle, find the dimensions of the 25 small triangles that tile it.

MidPoW: A Surfer's Tessellation - posted April 30, 2001 [Pow-teach discussion]

Shane is re-tiling his surf shack. How many of each kind of tile will he need, and what does a big tile cost?

GeoPoW: Transforming a Triangle - posted November 13, 2000

Given two triangles on a coordinate plane, find a pair of transformations that will map the first triangle onto the second triangle.

GeoPoW: Rotating a Triangle - posted April 30, 2001

Given a triangle on a coordinate grid, find the new coordinates of the vertices if you rotate the triangle 90 degrees about the origin.

Selections from Teacher2Teacher Discussions:

Symmetry

Using pattern blocks to teach symmetry operations at a level that is appropriate for secondary level

Tessellations and symmetry - 4th grade

Letters, mirrors, index cards, and software for teaching symmetry.

Selections from Ask Dr. Math Archives:

Symmetry

What is symmetry?

Tessellation

Are there any non-regular convex polygons with more than four sides that can tessellate?

Tessellation Proof

I am looking for the proof of the statement that only three regular polygons tessellate in the Euclidean plane.

Tessellations and Symmetries

How do you make a tessellation with a rotation, reflection, and translation all in one shape?

Types of Tessellations

I am doing a project on tessellations. Can you explain some of the mathematics behind them?

BoxerMath.com offers on-line curricula for Pre-algebra, Algebra I & II, Geometry, and Trigonometry. All courses include practice problems, tests, lesson plans and enrichment activities. BoxerMath.com also correlates course lessons with state and national mathematics standards. A Free Trial is available.