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• • • • • • • • • • • • • • • • • • • • • • • • • • LESSON PLAN • • • • • • • • • • • • • • • • • • • • • • • • • • •

Four suggested activities are outlined here that make use of the puzzle pieces along with the enclosed worksheets. When discussing symmetry in tessellations, there are two details that should be kept in mind. The �rst is that fact that a true tessellation covers the in�nite mathematical plane. The �gures here and any group of puzzle pieces should be considered a �nite portion of an in�nite tessellation. This is an important distinction because a �nite group of pieces cannot perfectly overlap itself under a translation. Another detail is the colors of the individual pieces (tiles). If colors are taken into consideration, a symmetry may not be present that would be present based strictly on the shapes of the tiles. The teacher can decide whether or not to raise the issue of colors.

Activity 1: Translational Symmetry

Objective: Learn and understand the concept of translational symmetry.Materials: One or more In the Ocean puzzles, copies of Worksheet 1, colored pencils (optional).

• Review the definition of a tessellation. (A tessellation is a collection of shapes, called tiles, that �t together without overlaps or gaps to cover the in�nite mathematical plane.)• Go over the definition of translational symmetry. (A tessellation is said to possess translational symmetry if it can be moved by some amount and in some direction such that it remains unchanged (perfectly overlaps itself).)• Hand out copies of Worksheet 1. Have the students mark “T” by each tessellation that possesses translational symmetry. For each of these tessellations, have the students draw an arrow indicating a translation that will cause the tessellation to perfectly overlap itself.(Optional) • For each tessellation that possesses translational symmetry, have the students color the tiles in such a way that the translational symmetry is preserved. (I.e., for example, every green squid would move onto another green squid, not a blue or purple squid.)• Break the class up into groups of 3-5 students, and make sure each group has at least 10 puzzles pieces of each shape.• Ask the students to construct a tessellation possessing translational symmetry that is different from any of those on the worksheet.• Have each group share their tessellation with the class.

Activity 2: Rotational Symmetry

Objective: Learn and understand the concept of rotational symmetry.Materials: One or more In the Ocean puzzles, copies of Worksheet 1, colored pencils (optional).

• Go over the definition of rotational symmetry. (A tessellation is said to possess rotational symmetry if it can be rotated about one or more points such that it remains unchanged (perfectly overlaps itself). If the rotation amount is 1/n of a full revolution, the tessellation is said to possess n-fold rotational symmetry.) • Hand out copies of Worksheet 1. Have the students mark “R” by each tessellation that possesses rotational symmetry. For each of these tessellations, have the students draw a pentagon on each point of 5-fold rotational symmetry and a rectangle on each point of 2-fold rotational symmetry.(Optional) • For each tessellation that possesses rotational symmetry, have the students color the tiles in such a way that the rotational symmetry is preserved. • Break the class up into groups of 3-5 students, and make sure each group has at least 10 puzzles pieces of each shape.• Ask the students to construct two tessellations possessing rotational symmetry, one with 2-fold symmetry and the other with 5-fold symmetry, that are different from any of those on the worksheet.• Have each group share one of their tessellations with the class.

© 2010 Tessellations

In the OceanClassroom Activities

Activity 3: Glide Re�ection Symmetry

Objective: Learn and understand the concept of glide re�ection symmetry.Materials: One or more In the Ocean puzzles, copies of Worksheet 1, colored pencils (optional).

• Go over the definition of glide reflection symmetry. (A tessellation is said to possess glide re�ection symmetry if it can be moved by some amount along some line and re�ected about that line such that it remains unchanged (perfectly overlaps itself). Mirror symmetry is a special case of glide re�ection symmetry in which the glide distance is zero.)• Hand out copies of Worksheet 1. Have the students mark “G” by each tessellation that possesses glide reflection symmetry. For each of these tessellations, have the students draw a dashed line indicating the line about which the tessellation reflects, labeling it “g” or “m” depending on whether it requires a glide along the line or not. For lines marked “g”, have them draw an arrow to indicate an amount of glide that will cause the tessellation to perfectly overlap itself. (The students should ignore the curve in the tails of the rays when considering whether or not a tessellation possesses glide reflection symmetry, as the direction of the curve will always change when a ray is reflected.)(Optional) • For each tessellation that possesses glide reflection or mirror symmetry, have the students color the tiles in such a way that the symmetry is preserved. • Break the class up into groups of 3-5 students, and make sure each group has at least 10 puzzles pieces of each shape.• Ask the students to construct a tessellation possessing glide reflection or mirror symmetry that is different from any of those on the worksheet.• Have each group share their tessellation with the class.

Activity 4: Vertices

One way to characterize and classify tessellations and sets of tiles is through their vertices. A set of tiles can be characterized by the distinct vertices that can be formed with the pieces. A tessellation can be classi�ed by listing the distinct vertices that appear in it. There are 16 distinct vertices using only squid and ray tiles, 13 using only squid and sea turtle tiles, 8 using only sea turtle and ray tiles, and 16 using all three types of tiles. Worksheet 2 describes a method for describing a vertex using letters.

Objective: Learn the role played by vertices in tessellations and how to characterize tessellations by labeling their vertices. Materials: One or more In the Ocean puzzles, copies of Worksheet 2.

• Go over the definition of a vertex. (A vertex is a point at which three or more tiles meet.)• Hand out copies of Worksheet 2. Have the students identify each distinct vertex in the tessellations shown on the worksheet. They should mark each vertex with a large dot and label them V1, V2, etc. Then they should write the letter description of each vertex. Distinct vertices will have different letter descriptions, apart from the starting tile (i.e., two different descriptions describe distinct tessellations unless they are identical under a cyclical permutation). For example, (RT, ST, SH, TH, SH, TH) is the same as (ST, SH, TH, SH, TH, RT), as they just start at different tiles, but it is distinct from (RT, TH, SH, TH, SH, ST). The worksheet describes a way to ensure that a given vertex has only one description, namely by starting with the corner label that would come first alphabetically.• Break the class up into groups of 3-5 students, and make sure each group has at least 10 puzzles pieces of each shape.• Ask the students to construct four distinct vertices that are different from any of those on the worksheet. They should write the letter description of each vertex. • Have each group share their vertices with the class. How many distinct vertices did the class come up with overall?

Worksheet 1

Worksheet 2

ST

SL SR

SH

TT

TL TR

TH

RT

RL RR

RH

The four corners of each tile are labeled above, using S for squid, T for turtle, or R for ray, followed by H for head, L for left, R for right, or T for tail. A vertex can be described using these labels. Pick any tile as the starting point, then write the labels for the tile corners that meet at that vertex, proceeding in a clockwise direction, as shown as right.

Since the description can start with any tile, different descriptions are possible, for example (RT, ST, SH, TH, SH, TH) and (SH, TH, SH, TH, RT, ST). A simple way of making each description unique is to choose the one that would come first if the strings of letters were arranged in alphabetical order. For the vertex at right, for example, the ray tile would be listed first, since it gives the only string of letter that starts with R instead of S or T. (RT, ST, SH, TH, SH, TH)

Mark, label and describe the distinct vertices, V1, V2, etc. in the tessellations below. The top one has four distinct vertices. How many does the bottom one have?

V1: (

V2:

V3:

V4: