Tessellations using Transformations

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SWBAT create and tessellate figures on the coordinate plane using transformations.

Big Idea

Students will explore tessellations and create their own design.

Do Now

5 minutes

As students walk in the room, they are given a small sheet of paper with a grid on it. I also project the Do Now on the Smartboard. Students are instructed to draw a design on the grid and are given an example to look at. The design students create should be very simple. They are instructed to use straight segments only and to make sure the endpoints of each segment drawn are located on an intersection point of the horizontal and vertical lines in the grid. The last step is to shade in the design.  This design will be used later in the lesson to create a template for tessellating. 


12 minutes

I begin the Mini-Lesson by showing the students pictures of tessellations I have taken.  We go over the definition of the term “tessellation.” I ask the students, “How do tessellations relate to transformations?”  Based on their knowledge of transformations, students are able to figure out that tessellations are shapes that are transformed across a plane. We use the pictures to identify the transformations used to tessellate the shapes. The picture from the synagogue in Budapest is a translation tessellation, the picture from the museum in Madrid is a rotation or reflection tessellation and the picture from the church in Rome is a combination of transformations.


I then show students a design of tessellating hexagons. Inside the hexagons, I have put a picture of the Chesire Cat from Alice’s Adventures in Wonderland to connect to previous lessons in the unit. The picture is an example of a translation tessellation. To show students how the shape tessellates, I have used Smart Notebook to infinitely clone the shape in order to tessellate it. I call on a student to continue the tessellation by translating the blue hexagon.


We next look at how to create the tessellation template. Students use the design they created in the Do Now. To create a translation tessellation template, students will translate the top figure -6 units vertically and the left figure 6 units horizontally on the grid. Press the play button on the Smart Notebook page to show how the shapes translate. 


18 minutes

There are two ways the tessellation template can be used. Depending on the level of the students, either method can be used. One way is to cut the template out and trace it on the “Transformation Tessellations Grid.” It is helpful to have the students glue their template to an index card and then cut it out again to make it sturdier. The template boxes and the grid boxes are the same size, which make it easy for the students to use in order to trace.


The second way is to have students identify the coordinates of the vertices of the template and then plot them on the grid. I give these students rulers to connect the points of the translated shapes. They then translate each point on the grid to create the tessellation.


After the students have tessellated their shape at least five times on the grid, they label each image and write the rules for the translation that mapped the preimage to each image.


As students work, I circulate around the room and check their translations are accurate and help them make any adjustments if necessary. At the end of the activity, I choose students to show their designs to the rest of the class. 


5 minutes

In the summary, we look at an MC Escher tessellation called “Reptiles” from 1943. This work can be found on the website www.mcescher.com  in the “Back in Holland 1941-1954” section of the picture gallery. I ask the students, “What type of transformation can be used to create the tessellation?”  We then discuss how to create a rotation tessellation. This will be used in the next lesson for the performance task.