beginning the lesson print off examples and non examples of tessellations. Non
examples should include images that are almost, but not quite, tessellations as
well as images that are clearly not tessellations. There should be one
example and one non-example for every group of four students in the class.
This can include artwork (MC Escher), patterns using polygons (one
polygon and two) as well as tessellations and images that have irregular
tessellations. This variety will allow students to see that tessellations
can come in multiple forms and will help avoid misconceptions. If you choose
to, you may use the PowerPoint below for your class.
Tessellation or Not? PowerPoint
Note that there is not
always a consensus about what a true tessellation is (ex: some allow for slight gaps in between images, such as
grout between shower tiles); thus, allow for students to defend their positions.
In general, the answers are:
- Image 1-
no, due to overlap
- Image 2-
- Image 3-
- Image 4-
- Image 5-
no, due to gaps
- Image 6-
no, due to overlap (tail of orange cat)
- Image 7-
- Image 8-
yes, if you allow for tessellations to divide surfaces that are not planar
- Image 9-
- Image 10-
- Image 11- no, due to the air and water (they have no shape, and can therefore be considered a gap in the drawing)
(End users have
permission to print out or download these images for use in presenting this
lesson plan. Users may not reproduce the art on other websites without the
express prior permission of Seth Bareiss, copyright holder, at
your class by having students play one game neXtu on Calculation Nation.
This will help ensure that they stay on task during the main lesson as
well as give them an opportunity to examine the geometry and strategy used in
the game without specifically focusing on it. While students are playing the game, circulate the room,
and ask students questions such as:
- How are you choosing
where to put your next tile?
- Which type of tile do
you like using the best? Why?
- Why do you think the
game creators designed the game board in this way?
as an opportunity to assess students informally by using Class Notes. Once students have completed one game, have select students
share out strategies they used for winning or playing the game. Below is a sample of the neXtu game board.
Optional: If you are able to complete the game on a smartboard as a
class, this will allow your students to verbalize strategies as well as use
math vocabulary. It would be beneficial for students to describe to you
where they want the next piece placed on the board. This will encourage
the use of precise math language. During this time have students give
explanations as to why they want to place a piece in a particular location.
5-10 minutes of game play, ask students to examine the game board. Ask
them what they notice about the game board. Students should notice that:
- The game board is made
up of three different regular polygons (triangles, squares and hexagons)
- All the shapes are
connected to make a larger design
- The larger design is
- Shapes are adjacent to
each other (there are no gaps in between the polygons and no overlap of
- It looks as if the design
could go on
students describe the game board to you, listen and accept all their responses
but only record the responses on the board or overhead that specifically apply
to a tessellation. Note that you may have to revisit terms,
such as “regular” and “adjacent.” A tessellation is a tiling of one or more
figure(s) that fit together without any overlap or gaps; furthermore, each vertex
is made up of the same configuration. If students leave out any part of the
definition for a tessellation, use guided questions to draw their attention to
that aspect of the game board (ex: “What could I do to make this game board
Once all parts of the definition have been stated, tell students that they have
just described a tessellation. Write this on the board on top of the list you
recorded. Once you have reviewed the definition for tessellation, ask
students to complete a think-pair-share so they can reflect on where else they
may have seen tessellations in nature (ex: beehives, soccer balls, etc.).
have developed a definition of a tessellation, break students into groups of
four and provide each group with two images, an example and a non-example of a
tessellation. The group should work together to determine which image is a
tessellation and give a detailed explanation as to their answer. Be sure to
have students also describe why the non-example is not a tessellation, as to
clear up any misconceptions. Give students approximately three to five minutes
to develop their response. During this time, you will want to touch base
with each group to ensure that they have correctly identified their image as an
example of a tessellation or a non example. Once all groups are ready, have
a few speakers from each group show their image and share their group’s
response with the class. After each group responds, give other students
an opportunity to ask the group a question or add on to their explanation.
neXtu Tessellations Activity Sheet
neXtu Tessellations Answer Key
As a wrap up, have students
return to their computer screens and play the game a second time. Ask
them to notice how the tessellated game board impacts playing and winning the
game. Have students complete the neXtu Tessellations Activity Sheet while
they play the game. Once all students have finished playing the
game one time through, begin a class discussion on the importance of the design
of the game board.
- It is important that
all the shapes connect because you can capture points from the other player
- It is important that
all the shapes connect because you can increase your own points by placing your
own tiles adjacent to one another.
- If none of the shapes
on the game board were adjacent you would not be able to win the game.
- If some of the shapes
were adjacent you may have fewer ways to capture another player’s points or to
increase your own.
- It provides for
diversity in the strategy (ex: a triangle has no influence over a hexagon and
1. What do
you notice about the game board?
[Answers Vary. Sample answers: the game board is made up of regular
triangles, hexagons, and squares; all the shapes fit together to make a pattern
that repeats; triangles aren’t adjacent to hexagons; no shape is adjacent to
its own shape.]
2. How do
the shapes fit together?
[Answers Vary. Sample answers: the shapes are side by side; shapes don’t
overlap and there aren’t any gaps; each vertex in the game board is made up of
one hexagon, one triangle and two squares.]
3. What did
the game designers do to the individual shapes to make them fit together?
[Answers will vary. Sample
answer: transformations were performed to each shape to make it fit in with the
pieces around it. Students may use words like flip or reflect, turn or
rotate, slide or translate to describe what is happening.]
4. What do
you see in this image that tells you it is not a tessellation?
[Answers may vary. Sample
answers: there isn’t a repeating pattern; there are gaps between the figures; the
figures are overlapping.]
5. Would it
be possible to win this game if the game board wasn’t a tessellation?
[Answers will vary.
Sample 1: If all the
triangles, squares and hexagons were separated you wouldn’t be able to capture
other points or gain points for having several of your tiles in the same place.
Each player would have the exact same number of points because we each
start with the same number of points.
Sample 2: Yes, because shapes
can be side by side without tessellating.]
- How did you guide your
class discussion to help students develop a definition of a tessellation?
- Describe the observations
you made regarding your students use of vocabulary related to geometric
- Describe the benefits
and/or detriments you observed from the game based focus of this lesson. How would
you alter this lesson to ensure maximum engagement from all students?
- Develop a definition
for a tessellation.
- Analyze the importance
of the tessellated game board in playing and winning the game neXtu.
- Explore the use of
geometry in a computer game.
NCTM Standards and Expectations
- Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.
- Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.
- Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.
- Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.
- Build and draw geometric objects.
- Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.
Common Core State Standards – Mathematics
Grade 4, Algebraic Thinking
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.