Tessellating
Concept: Tessellating
Time: 2+ hour lesson
Definition: A tessellation is a tiling, made up of the repeated use of polygons
and other curved figures to completely fill a plane without gaps or overlapping.
Minnesota Graduation Standard:
The student shall demonstrate an understanding of characteristics of the physical
world and the ability to create a model to illustrate a concept.
Assessment: Tessellate a plane using regular polygons.
Creates own irregular polygon to use in tessellating another plane.
One Rich, Multi-Step Problem to Solve:
How can you make an irregular shape tessellate?
Materials:
Pattern blocks
Pencils and sharpener
Markers/Crayons/Colored pencils
Lined index cards cut into squares (uncut cards could be used, but squares leave
room for more creative designs)
Scissors
Tape
Legal sized paper or larger (one per student, for their final project)
Printer paper
Magazines
Optional: Pictures of M.C. Escher’s work
Naturally occurring tessellations-honeycomb, snakeskin, sand dollars and flowers
are the easiest to find (check online under natural tessellations)
Directions to Students as They Start Activity on Own:
1. Divide the students into pairs or small groups. Give each group one color/shape
of the pattern blocks. Have the students use their blocks to completely cover
an index card (leaving no spaces and not overlapping). (15 minutes)
2. As the teacher, wander around and help during this process.
3. As a class, talk about what patterns like those they created, the students
have seen or can find in the classroom.
4. Share the pictures of the naturally occurring tessellations or other man-made
ones that have been found in magazines.
5. Tell the students that the patterns they created and are looking at are called
tessellations.
6. Pass out the magazines and have the students look for tessellations. Let
each group share with the class what they have found. Identify regular and irregular
shapes. (10 minutes)
7. Share M.C. Escher art examples.
8. Teach the students how to do the “nibble” technique. (25-30 minutes)
a. Hand out scissors, pencils, tape, and square index cards to each student.
b. Demonstrate cutting the “nibble.”
Start on one corner and cut across to an adjacent corner (adjacent means one
that “shares” a side)
Remind the students that the shape must retain the same area as the original.
There is NO TRIMMING and don’t overlap!!
Allow the students time to cut. Teachers: walk around and check to make sure
the cuts were to adjacent corners, if not give them a new card and explain it
again.
c. Demonstrate moving the “nibble.”
Show the students how each of the three movements works.
Have the students each select the movement they want, and then tape down their
“nibble.”
Remember NO overlapping!
Teacher needs to assist during this step.
d. Create a second nibble. Use the same movement selected for the first one.
9. Share an example made by the teacher. Brainstorm with the class about the
object (This is the shape you have…what could this be?) It may not look
like something, use your imagination, and ask for suggestions. (5-10 minutes)
10. Have the students look at their own shape and come up with an idea of what
it might be.
11. Pass out the large sheets of paper.
12. Show the students how to trace their tile. Make sure to be careful in lining
it up and realize that you may need to “fudge.” (Tracing will take
at least 15 minutes)
13. Have the students trace their irregular shape to completely cover their
sheet of paper.
14. Add color and design to transform the shapes into an object. Some helpful
suggestions: (This will vary on the student…some will take 10 minutes and
others will not finish)
Use contrasting colors like a checkerboard
Color each row differently
Less detail tends to be easier
Picking just a few colors (instead of every one in the box!)
15. Let the students know that they can take it home, so they don’t have
to rush.
Whole Group Presentations:
If the students are interested, allow five minutes or so for sharing of their
artwork. Have them show their shape, where they cut and then the final product
they have created. This would be great to do during snack time!
Questions to Help Students Think:
What other objects could this be? (have them turn their shapes)
Why did you cut there? How would your shape be different if you had cut on a
different side?
How could your shape be different if you had chosen another movement?
(note to teacher: if the student really dislikes their shape remove the tape
and try a different movement)
What movement did you use? (Reflection, rotation, or slide)
Variations: Provide equilateral triangles, hexagons, or other regular polygons
that tessellate for use in addition to squares (or as a further activity).
Use combinations of regular shapes.
Cut out your shapes to make a puzzle.
Teacher Resources:
Software: TesselMania!, by Kevin Lee, MECC Software, 1994, 1996.
Look at the local library for books with M.C.
Escher’s artwork.
Authors: Angie Petit, Sherri Peterson
3rd and 4th Grade – 50 minutes
Concept: Learning about problem solving through trial and error, spatial
reasoning, and geometry
Minnesota Graduation Standard:
1.. Describe and compare three-dimensional geometric figures existing in the
physical world
2.. Extend or create geometric patterns to solve problems
Assessment: Students will be assessed upon whether or not they were actively
involved in building and analyzing the design of free-standing paper towers,
whether or not they were able to recognize geometric shapes in their tower designs,
if they were to measure accurately, and if they were able to explain similarities
and differences between real towers and their own paper towers.
Multi-Step Problem to Solve: The challenge is to build the tallest, free-standing
tower possible from a single piece of paper and 30 cm of masking tape
Materials Needed:
¸ Standard 8.5 x 11 inch paper
*3 pieces per group – 1 white for official tower and 2 blue (or other color)
for practice towers
*extra white paper if time permits for extension activity
¸ Masking tape
*60 cm. per group – 30 cm. for practice and 30 cm. for official tower
¸ Meter sticks
¸ Scissors
Directions to Students: (5 minutes)
Before beginning the lesson and the challenge, brainstorm with students they
have seen or visited. What do towers have in common? How are they constructed?
(leave the list up for the students to see and refer to)
Rules for building the tower:
1. Your final tower may only contain those materials supplied by the teacher
2. You will have thirty minutes for official design and construction
3. Your tower must not be attached at the base and may not lean against any
other surface
4. Your tower must stand on its own for 10 seconds or longer
5. The height will be measured from the base to the highest point
6. You can have your tower “officially” measured as may times as possible
within the 30-minute time limit and you can keep adding to it as time permits
Scoring plan: (write on white board)
Over 50 cm = Good
Over 80 cm = Outstanding
Over 100 cm = Spectacular
Over 150 cm = A masterpiece of Engineering and design
Group Work (40 minutes)
Tower building time will be broken up into two sections: practice time and official
time. Each will last 20 minutes. Break the students into partners or small groups.
Each group will be given two pieces of paper, scissors, 30 cm of masking tape,
and a meter stick.
During practice time, brainstorm with the students and ask, “What possible
shapes and designs could be used?” Encourage a wide variety of tower designs
in order to find the one with the greatest potential. Be willing to help with
design ideas.
Before official time begins, make sure to collect extra paper and the tape that
is left over from practice time. You may want to take some time to brainstorm
again about strategies that worked and didn’t work before the official
time begins.
When official time does begin, each group receives a single sheet of paper and
another 30 cm of tape. While construction is taking place, make sure to circulate
around the room and measure the towers when asked.
Whole Group Presentations (5 minutes)
Finish the activity with a discussion of the activity, including a presentation
of each group’s towers. Ask what they did differently the second time,
what they enjoyed, and what they would like to know about towers.
Questions for Discussion
1. Which designs worked? Which didn’t? How do you explain these results?
2. What geometric shapes do you see? Why do you suppose that certain shapes
work better than others?
3. How important were measurements in this activity?
4. How do your completed measures compare with the actual towers we discussed
at the beginning of this activity?
5. What sort of training do you think a person would need if someone wanted
to design and build real towers?
Extension:
Given a section of the newspaper, how tall can you make
a tower?