Tessellations

1
Tessellations

• The Fusion of Math and Art

2
Introduction
A constructivist lesson is rooted in an authentic
context that draws upon the students' real-world
experience. Such a lesson is especially
beneficial for teaching abstract mathematical
concepts, which are designed to teach the
geometric concept of tessellations.
3
Introduction
Rather than presenting the definition of
tessellations directly to the class, these
lessons are structured to facilitate the
students' discovery of the concept through a
series of teacher-guided constructivist
activities. Students construct their
understanding of tessellations by using Web-based
resources, participating in multimedia
activities, and applying their knowledge to a
real-world problem.
4
Age
This curriculum is focused in the Middle school
to Junior High school level, specifically for an
eighth grade level.
5
Length of Time
The curriculum planed for tessellations will be a
two week period.

• The first five days will be done in class. After
  this period the students have the rest of the two
  weeks to finish their assignment and
  presentations will follow after the two week
  period.

6
New Jersey Standardsin Math and Art
7
New Jersey StandardsMath Standards
Standard 4.2 Geometry and Measurement All
students will develop spatial sense and the
ability to use geometric properties,
relationships, and measurement to model,describe
and analyze phenomena.

8
Math Standards to be Addressed in This Lesson
Building upon knowledge and skills gained in
preceding grades, by the end of Grade 8,
students will
A. Geometric Properties
3. Understand and apply properties of polygons.
Quadrilaterals, including squares, rectangles,
parallelograms, trapezoids, rhombi
Regular polygons
Sum of measures of interior angles of a polygon

Which polygons can be used alone to generate
a tessellation and why
9
New Jersey StandardsArt Standards
Standard 1.2 All Students will utilize those
skills, media, methods and technologies
appropriate to each art form in the creation of
visual art.
10
Art Standards to be Addressed in This Lesson
Building upon knowledge and skills gained in
preceding grades, by the end of Grade 8, students
will

• D. Visual Art
• Incorporate various art elements and principles
  in the creation of works of art.
• Explore various media, technologies and processes
  in the production of two and three dimensional
  art.
• Identify form, function, craftsmanship, and
  originality when creating a work of art.
• Identify careers and lifelong opportunities for
  making art.

11
New Jersey StandardsArt StandardsStandard
1.2All Students will utilize those skills,
media, methods and technologies appropriate to
each art form in the creation of visual art.
In these lessons students will learn how to
identify and solve design problems in space,
structures, and objects for the home and
workplace.
12
Day 1 Exploring Geometric Shapes For the first
day, several learning centers will be set up
around the classroom to help students enhance
their understanding of geometric shapes.
13
Center 1 Getting to know your shapes.
There will be a center where students can
experiment with actual geometric shapes (little
plastic ones). They will be encouraged to make
patterns and try fitting all different types of
shapes together.
1
14
Center 2 What else can you do? In Center 1,
students were able to explore a variety of
Shapes, but the shapes were all of the same
size. (i.e. all squares where the same size, so
were all the triangles, etc.)
This station will be set up with a number of
geoboards.Geoboards are boards that have a bunch
of pegs on them (blue dots). If you wrap rubber
bands around these pegs, you can create all types
of shapes of different sizes. Another way to
explore patterns and the relationships between
shapes!
2
15
Center 3 Explore The Concept of Repeating
Patternsin Shapes, also Known as Tessellations.
By now the students have gained some experience
with fitting geometric shapes together. With
technology, they will be able to explore a wider
variety of shapes and manipulate them in all
sorts of ways.
In the Computer Resource Center, students will be
able surf through an array of quality, pre-select
ed websites aimed to expand their overall
knowledge of tessellations. To avoid
students just surfing the web while they
should be learning, I would require that they
print out two ORIGINAL tessellations, which will
be used later. Watch...
3
16
Center 4 Its Time to Reflect!
A key part of constructivism is that you allow
the students a chance to reflect on the learning
experiences they have had. Allot time to write in
journals at the end of class.
Now that those gears are turning, put the
thoughts to words. Encourage students to record
anything new that they learned. If you see
dumfounded looks, ask students to pull out the
tessellations they did online and write about
why they chose to do it the way they did, etc.
Its a good idea to have your students leave
their journals with you It will help construct
Day 2s lesson. You can build off of the
conclusions your students have already drawn.
4
17
CONNECTIONS ARE MADE!!!!
18
DAY 1s GOAL - ACCOMPLISHED!!
Lets move on
19
Day 2 How Come Some Shapes Fit Together and
Others Do not? Learn the Math Behind the Mayhem!
Again, the internet serves as an important tool.
Students will be provided with pre-selected sites
and objective sheets at the beginning of class.
Today, the objective is to find out why all this
works!
20
Students want to discover relationships like
these
And YES, students will be asked to find a
mathematical Formula! This is the general
formula to find the measure of an interior angle
of a regular n-gon is
21
Objective sheets will encourage the students to
take notice to things like
What is a Vertex?
Where have we seen a vertex before? How many
degrees in a circle? Is there anything you can
tell about these four equal quadrants?
22
What is the Relationship Between Verticesand
Tessellations?
Students will be asked to use the tessellations
they made yesterday. They will go to a vertex and
write what number-side shapes are around each
vertex.
Like this
23
Reflection Time!
Again, make time at the end of the period for
students to reflect on the lesson. Students have
already determined what shapes are around the
vertices of their own tessellations now
encourage them to find the measure of the
interior angles of those shapes and draw
conclusions on why tessellations work.
24
Day 3 exploring the world of art and culture
What do two-dimensional tessellations look like?
Where in art can they be found?
25
Exploring M.C. Escher on Tessellations
This exploration of the world of art and culture,
including the works of M.C. Escher, provides
students with an opportunity to identify and
create original tessellations. Using the Internet
and handouts, students will explore the artistic
representations of tessellations and
transformations and enhance their understanding
of the underlying mathematical principles.
26
Research
Students will explore the works of M.C. Escher
and other artist on the Internet sites provided,
or in other outside information, to identify
similarities and differences in the patterns he
created. The students will be given samples of
Escher's work and facilitate a discussion about
the nature of patterns for the next session.
Students will take notes in their journals on
their findings.
27
Day 3
Complete
28
Day 4 Students Findings
This period will be used for the students
research from the day before. The students will
share their findings in the class discussion.
29
Discussion
Students can work in small groups to discuss
their ideas about Escher's patterns. In groups,
students can categorize the different kinds of
tessellations they find using their knowledge of
symmetry, similarity, and congruency.
30
Reflection time!
The students will also assess the knowledge they
learned in this lesson and previous lesson at
this time. The students are to use the tools
provided by the teacher to create their own
original tessellation. Can you come up with other
ways to tessellate a pattern?
31
Day 4
Complete
32
Day 5 It's Time to Tessellate!
By day five most students should have a good
understanding of tessellations both mathematical
and artistic.
The students will be in groups no larger than
five students.
33
Lets tessellate
The class will be present with the following
problem
You are an interior designer that has been
selected to do Tommy Tessellations bathroom.
Your client likes his bathroom very much but he
wants new tiling for his floors.
The floors dimensions are 60X97 feet. He wants a
uniform pattern that will tie in together and
look completely uniform when each tile is place.
(He wants no real distinction between tiles.) He
wants his tiles to be geometric shapes only in
any size. Your job is to make a sample of the
floor tiling before you start construction.
34
Lets tessellate!
The time period for this project will be a week
after the problem is presented.
They will use all the information that they
collected from the previous lesson from the notes
in their journal and any additional information
provided by the teacher
35
Use your imagination!
At the end of the week each group will present
their sample tiles. The students can present
their tiles in any way they wish.
The students can used any material to complete
the project
36
Questions to Consider on Presentation
Think about the following questions as you
proceed and refer to the principles of design.

• Can you identify the pattern unit in your
  tessellation?
• What art materials did you use?
• What expressive qualities, feelings or moods does
  the artwork display?
• Did you please your customer?

37
Student References

• http//www.mathartfun.com/shopsite_sc/store/html/i
  ndex.html
• http//mathforum.org/sum95/suzanne/tess.intro.html
  
• http//www.sasked.gov.sk.ca/docs/midlmath/escher.h
  tml
• http//www.scienceu.com/geometry/articles/tiling/
• http//www.ScienceU.com/geometry/articles/tiling/w
  allpaper.html
• http//www.ScienceU.com/geometry/handson/trigrp/tr
  igrp.cgi?group5on
• http//www.ScienceU.com/geometry/articles/tritile/

38
Grading Rubric
Scale points 2 and 4 are not explicitly
defined. These will be used to score students
who exceed one criteria, but dont quite meet the
next.
39
Who said math and art were not friends