For educators who have access to LogoWriter software, this is another possibility for introducing or reinforcing the idea of tessellations with students. LogoWriter is also an excellent tool when working with:

the Cartesian plane
degrees (circles)
direction (left, right, positive, negative)
polygons
patterns

and as an introduction to an "object-oriented" computer language.

Objectives: [NCTM Standards Grades 6-8: Geometry, Communication, Reasoning and Proof]

1. To use the software, LogoWriter, to tessellate in the plane.
2. To learn how to make a repeating pattern of triangles, hexagons and squares.

Materials

1. Computer
2. LogoWriter software

Procedure:

[It is assumed that the basics of LogoWriter have already been introduced.]

1. Instruct the students to make a new page.
2. Instruct the students to type the following procedures in the "flipside":

[NOTE: Instead of just giving these procedures to the students, an introductory activity could be presented in which the students generate the procedures to produce a variety of polygons.]

to triangle :side
repeat 3[fd :side rt 120]
end

to hexagon :side
repeat 6[fd :side rt 60]
end

to square :side
repeat 4[fd :side rt 90]
end

3. Challenge the students to discover a procedure that will result in tessellations of:

squares
triangles
hexagons

Sample Procedures

1. Triangular Tessellation

to tessTri
pu setpos [-242 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-212 -18.5] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-182 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-152 -18.5] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-122 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-92 -18.5] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-62 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-32 -18.5] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-2 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [28 -18.5] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [58 0] pd
repeat 5[triangle 35 fd 37 triangle 35]
pu setpos [-203 -39] setc 5 pd fill
pu setpos [-193 -20] setc 3 pd fill
pu setpos [-202 -3] setc 2 pd fill
pu setpos [-223 -1] setc 6 pd fill
pu setpos [-234 -21] setc 4 pd fill
pu setpos [-222 -39] setc 15 pd fill
pu setpos [68 -95] setc 2 pd fill
pu setpos [67 -57] setc 3 pd fill
pu setpos [67 -19] setc 4 pd fill
pu setpos [66 17] setc 5 pd fill
pu setpos [67 52] setc 6 pd fill
pu setpos [67 90] setc 15 pd fill
end

2. Hexagonal Tessellation

to tessHex
pu setpos [170 0] pd hexagon 30
rt 180 bk 30 hexagon 30
repeat 6[pu rt 90 fd 51.5 lt 90 pd hexagon 30]
pu setpos [196 -16] pd hexagon 30
repeat 6[pu rt 90 fd 51.5 lt 90 pd hexagon 30]
pu rt 180 setpos [-165 46] pd hexagon 30
repeat 6[pu rt 90 fd 51.5 lt 90 pd hexagon 30]
pu setpos [-64 12] setc 14 pd fill
pu setpos [39 12] setc 3 pd fill
pu setpos [145 12] setc 7 pd fill
pu setpos [-166 11] setc 4 pd fill
end

3. There are many procedures which could be written to create tessellations.
Please send me your favorites!