Crossing Borders/Breaking Boundaries V
Looking East, Looking West: Europe and Arabia, 1450-1750
July 18-25, 2005
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Lesson Title: Geometry in Islamic Design

Name: Christina Kim and Heather Huntington

Discipline: Math - Geometry HSA

School: Eleanor Roosevelt High School, Greenbelt, MD

Grade Level: 9-12

Time Period: 45 minutes

Standards:

  • Indicator 1.1.1
    The student will recognize, describe, and/or extend patterns and functional relationships that are expressed numerically, algebraically, and/or geometrically.
  • Indicator 2.1.1
    The student will analyze the properties of geometric figures.
  • Indicator 2.1.3
    The student will use transformations to move figures, create designs, and/or demonstrate geometric properties.
  • Indicator 2.1.4
    The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology.

Specific Objectives:
The students will learn how to construct a polygon in order to locate symmetries and translate polygons. To encourage the students to make mathematical connections to the real world, they will learn about the geometries exhibited in Islamic designs.

Vocabulary/Concepts:

  • Islam - the religious system revealed to Muhammad that entails the complete acceptance of the wisdom of Allah (God). The word “Islam” is derived from the Arabic word for “peace.”
  • Geometric - containing lines, angles, circles, triangles, squares, and other similar shapes.
  • Tessellation - a covering of a plane with congruent copies of the same region with no holes or overlaps.
  • Polygon - a closed, many-sided figure that has sides that are all line segments.
    • Triangle - three-sided polygon
    • Square - four-sided polygon
  • regular polygon - a polygon whose sides are all the same length and whose angles all have equal measure.
  • congruency - when two shapes are the exact same size.
  • similarity - when two polygons have corresponding angles that are congruent, as well as proportional sides.
  • transformation - a mapping of one space onto another or onto itself.
    • translation - the composite of two reflections over parallel lines or a slide.
    • rotation - the composite of two reflections over intersecting lines.
    • symmetry - a figure has symmetry if there is an isometry that maps the figure onto itself. Students should be aware of two types of symmetry: point and line.
      • point - point symmetry occurs if there is a rotation of 180 degrees or less that maps the figure onto itself.
      • line (or reflectional) - this occurs if there is a reflection that maps the figure itself.
    • reflection - a reflection in line r is a transformation such that if a point A is on line r, then the image of A is itself and if point B is not on line r, then its image B’ is the point such that r is the perpendicular bisector of line segment BB”.
  • Alhambra - a citadel and palace on a hill overlooking Granada, Spain. Built by Moorish kings in the 12th and 13th centuries, the Alhambra is the finest example of Moorish architecture in Spain.

Materials/Resources (For Both Students and Teacher):

General Resource (For Both Students and Teacher)

http://www.salaam.co.uk/themeofthemonth/march02_index.php?l=3


PowerPoint Resources (For Teacher)

The Elements of Art. J. Paul Getty Museum Education Department: A Guide to Building a Visual Arts Lesson.

Islamic Geometrical Patterns for the Teaching of Mathematics Symmetry. S. Jan Abas, School of Information, University of Wales, Bangor, UK.

http://www.crescentmoonweb.co.uk/gallery/nigel/islamic04.htm

http://www.hellam.net/twinsite/ispat2.gif

http://www.thegrid.org.uk/hertsmathsyear2000/mecss/islamic.html

http://www.anthonyslayter-ralph.com/main.html

Classroom Materials (For Students)

  • Blank paper
  • Islamic graph paper
  • Colored pencils
  • Compass
  • Straight edge
  • Tracing paper
  • PowerPoint Presentation, “Geometry in Islamic Design”
  • Islamic graph paper printed on a transparency
  • Overhead projector
  • Worksheet, “Geometry in Islamic Design - Warm Up”
  • Worksheet, “Creating Islamic Designs”
  • (Optional: one prize)

Lesson Abstract:
Students begin the lesson with a warm-up which investigates the symmetries of shapes. To conclude the warm-up the teacher will show a short PowerPoint presentation on Islamic Design as an attempt to evoke a connection. This presentation will be revisited at the very end of class.
As part of the body of the lesson, the students will be separated into groups and then challenged to recreate Islamic graph paper with a compass and ruler. This construction will later be modeled for the entire class to see. After the students discover how to create this grid paper, they will use the original complete grid paper given to them and tracing paper to create polygons on the tracing paper using a ruler. They will be expected to know the symmetries and regularity of the shapes they create and one student from each group will reconstruct their polygon on the overhead and share it with the class.

After a discussion of the shapes presented on the overhead, the students will complete a worksheet which will assess their understanding of the lesson. After collection of the worksheet, the PowerPoint presentation will be revisited to encourage a more in-depth discussion of the geometry in Islamic Design.

Lesson Development:

Motivation/Warm Up (10 minutes):

  • [2.1.1] Students will be given a “Geometry in Islamic Design - Warm Up” worksheet and will be asked to draw lines of symmetry through four shapes. If the shape has rotational symmetry, students draw a point in the center of the shape.
  • Students will be shown a PowerPoint presentation with various Islamic tile patterns. They will be asked the following questions:
    • [2.1.1] Do you notice any geometric figures? If so, what are the properties of these figures?
    • [1.1.1] Do you see any patterns? If so, describe them.
    • What is unique about the designs?
    • Compare and contrast the designs you see in the slides with the patterns in clothing people are wearing in our classroom today

Challenge/Guided Practice (8 minutes):

  • The teacher will separate the class into groups of four and give each student two compasses, straight edge, paper, and the Seven Circle Islamic graph paper.
  • [2.1.4] The teacher will explain that the first group to copy the graph paper onto white paper by constructing it, only using a compass, will come up to the overhead and explain it to the class (and possibly earn a teacher predetermined prize).
  • The teacher will walk around the class to assist the students.
  • The student will model the drawing of Islamic graph paper on the overhead or board. As the student draws, the teacher should explain that this design often lays the groundwork for Islamic patterns.

Modeling (2 minutes):

  • The teacher will put an Islamic graph paper transparency on the board. The teacher will demonstrate how to locate and dot several intersections. Intersections will be connected in order to form an equilateral triangle.

Independent Practice (13 minutes):

  • Each person in the group will be given a piece of tracing paper and asked to place it on top of the Seven Circle Islamic graph paper.
    • [2.1.4] Each person in the group will be required to construct polygons on the tracing paper by connecting intersections from the Seven Circle Islamic graph paper with a straight edge. The teacher will explain that in each group, each member should find a unique shape. The teacher should ask the class to think about line and point symmetry, regularity, and similarity for each shape.
    • [2.1.3] After three minutes, the teacher will allow each group to choose one person from the group to construct their shape on the overhead. Their team will be asked to comment on symmetry lines, rotational symmetry, whether the shape is regular, and whether it is similar to other shapes constructed by other teams. Each team should attempt to introduce a new polygon on the overhead.
    • As each group presents, the teacher will comment on the Islamic symbolism of different types of shapes:
      • Star - symbolizes equal radiation in all directions from a central point.
      • Circle - emphasis on Islam’s one God, symbolizing also, the role of Mecca, which is the center of Islam toward which all Muslims face in prayer. Also regarded as a symbol of eternity, without being and without end, and is not only considered the perfect expression of justice-equality in all directions in a finite domain--but also the most beautiful parent of all polygons, both containing and underlying them.
      • Triangle - the triangle, by tradition, is symbolic of human consciousness and the principle of harmony.
    • [2.1.3] Students will return to their groups and align their own polygon with the graph paper again, and create an exact copy of their shape on the tracing paper in another position. The teacher will then ask, “Who can tell me what type of transformation this is?” (ANSWER: translation)

Assessment (9 minutes):

  • Students will be given a blank “Creating Islamic Designs” worksheet and colored pencils.
    • [2.1.3 & 2.1.4] They will create an Islamic design, using colored pencils, that includes all three transformations. They should color the entire grid. Students should explain why their design represents an Islamic pattern. They should also explain where and how the transformations occur.
    • The teacher will give the students five minutes to individually complete the worksheet.
    • The teacher will collect the worksheet for formal assessment.

Closure/Summary (3 minutes):

  • The teacher will return to the original PowerPoint presentation and discuss the symmetries in the tiles in more depth.

Lesson Extensions:

  • Geometer’s Sketchpad
    • Explore the pre-made lesson on Islamic Art from www.keypress.com
      • Introduce the idea of regular and irregular tessellations.
      • Demonstrate translations, reflections, and rotations using the pre-made presentation.
  • Area and Ration Connection
    • Have students count the number of triangles in different shapes to compare the ratio of triangles from one to another.
  • Concept of Infinity
    • Have students discuss how far the patterns could go in each direction to lead into a discussion of the topic of “infinity.”
  • Algorithm
    • Using Islamic graph paper, have students write an algorithm for a design based on coloring in the different segments.
  • Transformations as a way to earn money
    • Explore the work of Scott Kim at his web site www.scottkim.com
    • Have students come up with their own logo to show rotations, reflections, or translations.

Sponsored by
the Center for Renaissance & Baroque Studies
and the Maryland State Department of Education