M. C. Escher and the world of Geometry
Curricular Area: Mathematics/Geometry
Grade level: 9th-10th
Time Frame: One week
Goals
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Students will show understanding in identifying and understanding tessellations and
transformations i.e. translations, reflection, rotation and glide reflection.
They will use works of M.C. Escher to represent tessellations.
Students will use the Internet to research M.C. Escher and one of his works.
Objectives
1. Given 10 minutes, students will classify 11 transformations based on coordinate plane images
with a 90% accuracy.
2. Given 20 minutes and a teacher demonstration, students will accurately analyze transformations
in a three works by M.C. Escher with a 90% accuracy.
3. In partners, students will create a presentation that represents geometry in an M.C. Escher work
in one week.
§111.34. Geometry
(b) 2.A: The student is expected to use constructions to explore attributes of geometric figures and to
make conjectures about geometric relationships
(B) 5.C: The student is expected to use properties of transformations and their compositions to make
connections between mathematics and the real world, such as tessellations
(b) 10.A: The student is expected to use congruence transformations to make conjectures and justify
properties of geometric figures including figures represented on a coordinate plane
(b) 11.A: The student is expected to use and extend similarity properties and transformations to explore
and justify conjectures about geometric figures
Context/Modifications
 Students already have been introduced to how geometry is present in architecture and artwork.
 Students already know how to physically represent transformations
Key Terms
Transformation-A change in the position, size or shape of a figure. The most common transformations
include translation, rotation, reflection and enlargement.
Translation-a transformation in which every point of a geometric figure is moved the same distance in
the same direction.
Reflection-(in a plane) the replacement of each point on one side of a line by the point symmetrically
placed on the other side of the line; a mirror image of a figure
Rotation-an operation that rotates a geometric figure about a fixed point.
Glide Reflection-is a composite transformation which is a translation followed by a reflection in line
parallel to the direction of translation.
Tessellation-is a pattern of shapes repeated to fill a plane without any overlapping of shapes and
without any gaps; also called tiling.
Materials
 Printed examples of M.C. Escher's works
 Pencils, Pens, Paper
 Tiles, Cutouts of shapes
 Internet
 White board/Overhead Projector
 Makeshift coordinate plane, either on board or paper
Anticipatory Focus
 Have students research M.C. Escher before they come to class to see example of his works and
have them select a favorite piece of his artworks.
 Introduce the concept of transformations and how they are represented in artwork and
architecture.
 Show students particular websites they may use to research M.C. Escher and where they can
find useful information about him
 Show students examples of past group presentations
Procedures
1. Teacher will reintroduce transformations to class and have students properly define each form
of transformations based on prior knowledge. (15 minutes)
2. Teacher will have students use tiles and makeshift coordinate planes to represent each form of
transformation. (25 minutes)
3. Teacher will hand out worksheet to each student and will do one example with them (10
minutes)
4. Teacher will allows students time to complete worksheets alone or with partners while assisting
those who need help (20 minutes)
5. Teacher will ask students about where they see transformations or tessellations (5 minutes)
6. Have students share the information they researched on M.C. Escher (5-10 minutes)
7. Introduce Escher assignment to students and go over requirements for project. (5-10 minutes)
8. Students will be assigned their partners they will work with (3 minutes)
9. Show students each M.C. Escher artwork (10 minutes)
10. Students will identify any transformations in each artwork. (5 minutes)
11. Allow students time at the end of class to meet with partners and work on collaborating ideas
for project (10 minutes)
Setting Expectations
 Students will participate in class discussions
 The teacher will make sure that each student can properly represent transformations on their
own graph
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The teacher will assist students with accessing information on M.C. Escher
Students will provide a reasonable amount of information on M.C. Escher
Students will respect each others choice of artwork to use
Students will have a complete project (in some form) to present to class at the end of the week
Input
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Visual display of M.C. Escher artwork
Visual display of transformations
Visual display of websites for students through showing on computer
Modeling
 The teacher will explain the use of transformations and tessellations in architecture (around the
room or in pictures) and tessellations (in artwork).
 The teacher will show examples of past presentations for students to see how their presentation
should be organized.
 The teacher will show how to properly search for information about M.C. Escher and his works.
Check for Understanding
 The teacher will make sure that the students can represent transformations properly.
 The teacher will circulate throughout the classroom while students are working on solving the
transformations using the tiles and makeshift coordinate plan or the worksheet.
 The teacher will have students use white board to visual show their understanding about
transformations and tessellations.
Guided Practice
 The teacher will guide the students while working on the initial problems for transformations.
 The teacher will allow students to use white board to solve any problems they may have, with
his/her help.
 The teacher will monitor students while working on problems and while collaborating with their
partners.
 The teacher will guide students in identifying transformations and tessellations in M.C. Escher's
works.
Reteach
 The teacher will have students redefine transformations and re-identify transformations on
examples.
 The teacher will go over requirements for the presentations
Independent Practice
 Students will work with tiles on their own to show their own representations of transformations.
 Students will work on given problems with their tiles.
Mastery check
 Teacher will have students represent a select problem for their classmates as a way to check
answers on worksheet.
 The teacher will walk throughout the class to make sure a student doesn't have questions on
how to create a presentation.
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A teacher will have certain checkpoints for students to reach each day.
Technology Uses
1. Internet searches
2. Overhead project/White board/ Promethean board
Websites
http://filipspagnoli.files.wordpress.com/2008/06/mc-escher-waterfall.jpg
http://marbsonline.com/wordpress/wp-content/uploads/2009/02/mcesher_day__night.jpg
http://library.thinkquest.org/16661/gallery/escher/8.html
http://library.thinkquest.org/16661/gallery/escher/12.html
Extensions
 Students may further research on transformations and tessellations in artwork
Closure
 Students will discuss their final presentations with their classmates
 Students will discuss their newfound information on M.C. Escher
 Teacher will ask students about how when they are ever in a new place, they should look around
and see if transformations are all around them in the buildings they see or the artwork they see.
Reflective Critique
 The teacher should make time available for students collaborate with him/her
 Students should be able to grasp their understanding of the concept at the end of the week
Evaluation/Assessment
 Final presentation