Tiling the Plane

· Activity Name: Tiling the Plane with Math Activity Tiles

· Objectives:

The students will explore the properties of given polygons and basic tessellations. Some facts to be learned: the angle sum at each point will add up to 360 degrees; squares, equilateral triangles, and hexagons are the only regular shapes that tessellate by themselves; some shapes can be paired up or put in triples to tessellate; some shapes tessellate by sliding while other tessellate by rotating.

· EALR/Standards:

1.2 understand and apply concepts and procedures from measurement.

1.3 understand and apply concepts and procedures from geometric sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

· Materials:

MAT shapes (Only the regular shaped ones are necessary)

· Teacher Notes

o Prerequisites for the learner:

Definition of “regular” shapes

Ability to find interior angles

o Teacher hints for the activity:

This activity works on almost any schedule. Allow the students a few extra minutes at the beginning of the lesson to gather all of the shapes, and to make designs if they have never used the manipulative.

Working on large tables or the floor is best for this activity. When the students find a tiling that works, have them keep a sample of it to show the class during discussion time.

o Introductory questions:

What is a tiling?

Where have you seen tilings used in the “real world?”

Can tilings have gaps or overlaps?

o Wrap-up questions:

Which shapes tiled by themselves?

Which shapes tiled in pairs?

Which shapes tiled in triples?

Did any shapes tile with four or more different shapes?

Is there a pattern in which shapes tile?

Can you tell if you have all the possible tiling patterns?

Are there any patterns that you thought would work, but could not build?

Are there any tiling groups that can make two designs with the same shapes?

o Solutions:

Introductory questions: A covering of the plane.

Sidewalks, brick walls, floorings, etc.

No.

Wrap-up questions: Answers can be found on the table on the back page.

o Assessment suggestions:

Verify that students participated and understand the concepts by collecting their chart papers. A simple homework activity is to design five different “bathroom floor” tile patterns based on the lesson. Points can be given for number of shapes used, explanation of why a certain tile is “best,” or mathematics that show why a drawing will work.

· The Activity:

Have the students divide into groups of four and then divide the MAT tiles between the groups. (These work best on large tables or carpeted floors) Give at least 5 minutes of time for the students to experiment with the tiles. Ask them to make a list of all the given shapes and as much information as they can find about each one. (Their lists should include side lengths, angle measures, number of vertices and edges, etc.)

Once the groups have completed their lists ask the students to use just one shape and tile the plane. They should attempt this with all of the shapes and record those that work. After finding all singular tiling shapes, ask the students to find all the pairs of shapes that tile the pane, followed by sets of three, four, five, etc.

The groups should record their answers on some type of chart so that they can look for patterns and make some conjectures. The ultimate goal is for a group to justify why they know that they have found all of the possible combinations and why there can be no others. (A hint can be “look at the measures of the angles that meet at a point.”)

· Assessment material:

The following table can be created by the students as they work through this project, or used as a quick reference for correcting their work.

· Extensions:

Ask the students if they can find all of the different tiling in “real world” places. Allow them to bring in pictures or sketches of the tilings along with the address of where they found it.

Polygon Name (Greek)		Number of sides	Interior angle sum	Interior Angle	Division into 360
			(N-2)180	(N-2)180/N	360 (Interior angle)

Trigon (Triangle)		3	180	60	6
Tetragon (Quadrilateral)		4	360	90	4
Pentagon		5	540	108	3.333333333
Hexagon		6	720	120	3
Heptagon		7	900	128.5714286	2.8
Octagon		8	1080	135	2.666666667
Enneagon		9	1260	140	2.571428571
Decagon		10	1440	144	2.5
Hendecagon		11	1620	147.2727273	2.444444444
Dodecagon		12	1800	150	2.4

Polygons that Tile alone
Trigon	3.3.3.3.3.3	6 at a vertex		6*60	360
Tetragon	4.4.4.4	4 at a vertex		4*90	360
Hexagon	6.6.6	3 at a vertex		3*120	360

Polygons that tile in pairs
Trigon/tetragon	3.3.3.4.4 or 3.3.4.3.4	3 trigon, 2 tetragon		360+290	360
Tetragon/octagon	4.8.8	1 tetragon, 2 octagon		90+2*135	360
Trigon/dodecagon	3.12.12	1 trigon, 2 dodecagon		60+2*150	360
Trigon/hexagon	3.6.3.6	2 trigon, 2 hexagon		260+2120	360
**Pentagon/Decagon	5.5.10	2 pentagon, 1 decagon		2*108 +144	360

Polygons that tile in triples
Trigon/tetragon/hexagon	3.4.6.4	1 trigon, 2 tetragon, 1 hexagon		60+2*90+120	360
Tetragon/hexagon/dodecagon	4.6.12	1 tetragon, 1 hexagon, 1 dodecagon		90+120+150	360
**Trigon, tetragon, dodecagon	3.3.4.12	2 trigon, 1 tetragon, 1 dodecagon		2*60+90+150	360

** This does not give a continuous tiling