Sample Lesson Plans on Tessellations
Objective: By interacting with Boxer Math’s Tessellation Tool, students
understand why equilateral triangles, squares, and regular hexagons tessellate
"regularly" in the Euclidean plane.
Introducing the activity: Boxer Math’s Tessellation Tool allows the student to
build tessellations and other designs by attaching the vertices of various polygons to
Note: Boxer Math’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 Boxer Math’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:
A. Have the students compare their results.
B. 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.
C. 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?
D. 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.
E. 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
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.
Revisiting the activity: Now that the students have tessellated with Boxer Math’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 Boxer Math’s Tessellation Tool, ask the students to
predict whether a square will tessellate by answering these questions:
4. What is the interior measure of the angles of a square?
5. Think of the vertex of one square within a tessellation of squares. How
many squares, in all, are touching at that vertex?
6. 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:
7. What is the interior measure of the angles of a hexagon?
8. Think of the vertex of one hexagon within a tessellation of hexagons.
How many hexagons, in all, are touching at that vertex?
9. 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
degrees of the interior 360 degrees divided
name of polygon
measure of each angle 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
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
By: Suzanne Alejandre
Definition: A tessellation is a pattern made up of one or more shapes, completely
covering a surface without any gaps or overlaps.
Before you start: Read in you book about tessellations. All quadrilaterals can tile a plane
by themselves. The pentagon and the octagon can not. Also, it would be to your benefit to
purchase a template with geometric figures on it or to make you own out of cardboard.
Note: Follow the following procedure exactly. Any deviation must be approved by me
ahead of time. Your grade will greatly be affected by your ability to follow these
Assignment: You must do a total of eight(8) different tessellations.
Choose a Format:
Option 1: Use poster board, draw a 2 inch boarder around it and divide it into 8 regions in
any way you wish, but make the regions big enough to see the full pattern of the
Option 2: Use letter size white paper, draw a 1/2 inch boarder around each and draw one
tessellation per page. Staple them together like a book. Include a title page and a table of
I. Do five(5) tessellations using only one figure.
1) Regular hexagon
4) One of parallelogram, rhombus, OR rectangle
5) One of trapezoid OR Kite
II. Earn 5 points by doing any of the above in a clever manner (arrange the shapes
creatively) OR use different sizes in one of your tessellations.
III. Do two(2) tessellations using more than one type of shape.
(1) One must use two or more different shapes AND
(2) One must use three or more different shapes.
Your pattern can comprise of any combination of shapes (no circles allowed).
IV. Do one tessellation using a no special convex quadrilateral OR a nonconvex
quadrilateral OR create your own shape.
V. Add color and design.
By: Diane Coates