Transformational Geometry

Essential content for elementary teachers

Understand symmetry and identify line symmetry and rotational symmetry in geometric shapes and objects.
Understand transformations of geometric shapes.
1. Congruence
  1. Translation (slide)
  2. Reflection (flip)
  3. Rotation (turn)
  4. Glide reflection (glide)
2. Similarity: dilation (enlarge and shrink)
3. Topological: distortion
Define, understand, and develop tessellations by hand and on the computer; understand which shapes will tessellate and why.

Essential content for students grades K-3

Apply transformations and use symmetry to analyze mathematical situations.12
1. Identify lines of symmetry in nature, in regular polygons and other two-dimensional geometric shapes; make shapes that have lines of symmetry and fold shapes to find lines of symmetry.
2. Identify slides, flips, and turns.
3. Investigate tessellations with regular polygons (square, equilateral triangle, hexagon) and other polygons (quadrilaterals-rectangle, parallelogram, rhombus, kite, and trapezoid).

Essential content for students grades 4-5

Maintain and expand on concepts introduced in primary grades.
Apply transformations and use symmetry to analyze mathematical situations.13
1. Recognize line and rotational symmetry in two- and three-dimensional geometric shapes and objects.
2. Use slides, flips, and turns to understand position and congruence of geometric shapes.
3. Define and create tessellations of simple polygonal shapes; determine regular polygons that can be tessellated; and develop a tessellation by hand and on the computer.

See Appendix A for a list of sample state standards for mathematics.

Instructional activities and lesson plans for transformational geometry

The brief activities and lesson plans listed here (by grade level) are described more extensively in Instructional Activities and Lesson Plans, where they are listed alphabetically.

Activity	Description
Creating Tessellations grades 1-5	Have students use index cards to create a shape that will tessellate the plane.
Line Symmetry grades K-3	Have students show lines of symmetry using pattern blocks, a pattern block applet, or paper folding.
Simply Symmetrical grades 1-5	Have students explore the classroom for symmetrical designs with both reflection and rotation symmetry. Have students draw the missing half of an object using reflection symmetry. Have students explore symmetry in block letters.
Geo-Dot Paper grades 3-5	Have students draw different quadrilaterals on dot paper and then compare their shapes with a partner. Explore properties of shapes, congruent and similar shapes, and translations and reflections.
Regular Polygons grades 4-5	Have students create regular polygons with flexible straws or K'NEX™ materials and explore which of the regular polygons will tessellate.
Pentomino Pieces grades 3-5	Have students use pentomino pieces to explore reflection and rotation symmetry. Use paper folding, mirrors, or Miras™.
Creating Pentominoes grades 4-5	Have students work together, using square tiles, to find the 12 different pentomino shapes and record their results on graph or square dot paper. Have students use transformations to verify the uniqueness of shapes.

Lesson Plan	Description
Tiling the Plane grades 2-3	Students use pattern blocks and triangle paper to develop an understanding of tessellations while reviewing names of geometric shapes.
Rep Tiles grade 3	Students develop a deeper understanding of similarity and how perimeter changes as a result of increase in size. They create "rep tiles" using four pattern blocks ("rep-4 tiles").
Rep Tiles grades 4-5	Similar to the grade 3 lesson above, with added element of how area changes as a result of an increase in size.
Reflections grade 5	Students discover properties of reflections and glide reflections and review names of shapes and the concepts of congruence and perpendicular lines. Use pattern blocks, Miras™, and protractors.
Tiling the Plane grade 5	Similar to the grades 2-3 lesson above, and addressing more complex tessellations.

National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, p. 396.
Ibid.