Activity One
Objective
In this activity students will use a geometry computer program to create a tessellation using a specific quadrilateral as the fundamental domain, and then analyze the symmetrical properties of the tessellation to determine if these properties match the properties of the quadrilateral from which they were built.
Procedures
Before this lesson, students will have had some time to get familiar with the geometry program. Since my school does not have Geometer's Sketchpad, I will be using a program that can be downloaded for free called Geogebra. Even though the students will have had some time working with Geogebra before this lesson, I will provide an instruction sheet to help them construct their tessellations.
Figure 1.9
Each student will be assigned a different parallelogram (rectangle, square, rhombus, or a typical parallelogram, that is not any of the more special types). This will be the fundamental domain they are going to create for their tessellation. This is illustrated in Figure 1.9 with a rectangle labeled with the number one. On each side of the quadrilateral, they will then construct a square (using the side length as the length of each side of the corresponding square). This is shown above in the areas labeled with a two. Then, using the side lengths of the squares, create another quadrilateral to fill in the remaining area of the fundamental domain, as shown by the areas labeled with a three.
After they are finished constructing the students will get with the other students that used the same quadrilateral as the center of their fundamental domain and discuss the following questions.
What symmetries define the quadrilateral that you started with?
What do you notice about fundamental domain that was created?
What are the symmetries of the entire picture you have created?
Can you make any connections to the symmetries of the quadrilateral you started with to the figure that you created?
Does this pattern that you created show up in your everyday life? Where might you see this pattern?
The students will discuss and write down their ideas about the answers to these questions. Then the students will form groups in which everyone started with a different quadrilateral. The students will share their patterns and share their ideas, and then discuss if there is a conjecture that they can come up with that explains the symmetry of these kinds of patterns based on the quadrilateral that they were created from.
As an extension to this activity, have students start with any arbitrary quadrilateral. Then have the students find the midpoints of the sides and rotate by 180o around those points. Then have the students use the same questions to examine this tessellation. In this case the students will find that some of the tessellations they have created will not yield four-fold rotational symmetry, as the parallelograms did.
Assessment
Students will be assessed on the different components of their work in class. Each student will be responsible for creating a pattern (even if they worked in partners on the computer for this part of the activity). Each student will also need to have written down his/her ideas about the questions they discussed in their groups. For homework, the students will look for these kinds of patterns in the real world. They can take pictures of places in their house or around the town or in the school to find examples. They can also use the internet to find pictures of these kinds of patterns.
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