Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to nctm.org/illuminations. Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on nctm.org.

Geometry in Computer Games? An Exploration of Tessellations Used in neXtu

• Lesson
3-5
1

In this lesson students will develop a definition for tessellations.  They will also analyze the importance of the tessellated game board in playing and winning the game NeXtu on the Calculation Nation website.

Prior to 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.

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- yes
• Image 3- yes
• Image 4- yes
• Image 5- no, due to gaps
• Image 6- no, due to overlap (tail of orange cat)
• Image 7- yes
• Image 8- yes, if you allow for tessellations to divide surfaces that are not planar
• Image 9- yes
• Image 10- yes
• 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 sethnesstessellations.org).

Opening Activity

Begin 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?

Use this 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.

Main Activity

After 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 repeated.
• Shapes are adjacent to each other (there are no gaps in between the polygons and no overlap of polygons)
• It looks as if the design could go on forever

As 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 bigger?”). 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.).

Once students 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.

Closing Activity

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 vice versa).

Assessment Options

1. After completing the lesson have students draw in their journals an example and a non example of a tessellation.  Ask them to explain why each picture is an example or a non example of a tessellation.  For students that need more challenge, give them two specific shapes and ask them to tessellate those shapes.  One example could be an octagon and a square.  Ask students to decide whether the shapes can be tessellated and explain their reasoning as to why or why not.
2. Give students several images, some tessellations and some not, and ask them to describe why the image is or is not a tessellation.

Extensions

1. Students should visit the Tessellation Creator and explore which shapes or combinations of shapes can make tessellations.  You may need to give them direction in this area so ask specifically if a triangle can tessellate and what steps they may need to take to make the triangle tessellate.
2. Encourage students to investigate the possibility of shapes that are not regular polygons being able to tessellate.
3. Students should visit the Tessellation Creator to create a different game board.  Have students choose a different set of shapes to create a viable NeXtu game board.  Students may need to play the game again to gain an understanding of how the game is played in order to create a different game board. Students can choose to play their game board using a spinner (to determine the value of their piece) and markers or colored pencils.

Questions for Students

Main Activity

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?

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.]

Teacher Reflection

• 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 concepts.
• 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?

Tessellation Creator

3-5, 6-8
Use this applet to create patterns to cover the screen using regular polygons.

Learning Objectives

Students will:

• 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.