Going Beyond Basic Geometry Using Symmetry and Tesselations
Using Bloom's Taxonomy you can help advanced students go beyond basic geometry.
By Erin Bailey
By the time students reach the upper elementary grades, there seems to be a division between those who “get” geometry and those who do not. As teachers, we know that students start to utilize their abstract thinking skills more in the fourth through sixth grades. However, we know that not all students develop skills according to a predetermined schedule. There may be some students who are ready to go beyond basic geometry, and some who aren't. Challenging students who have more than a basic understanding of any topic is, well . . . challenging. But every student deserves to learn something new every day, even if their classmates may not be ready.
Incorporating Bloom's Taxonomy
A good way to meet the needs of all students is to use Bloom’s Taxonomy. It is one way to encourage students to reach a deeper level of understanding on any topic. For students who gasp new concepts easily, have achieved mastery of old ones, or crave a challenge, you can focus on the upper three levels: analysis, evaluation, and creation.
Here are two ideas that target the higher end of Bloom’s Taxonomy. The first center involves symmetry, and the second has students explore the properties of polygons using tessellations.
Activities to Identify Symmetry
Symmetry is an important concept for finding mathematical patterns and making scientific observations. Using
tools in Microsoft Word, it is easy to build a worksheet displaying three-dimensional figures. Ask students to alter the pictures you have made to make them symmetrical either by adding or subtracting blocks. To control the challenge level, you can decide how many lines of symmetry must be included. If you don’t have time to create the needed pictures, use photocopies of the examples of non-symmetry in the student textbook for the same purpose. Another idea is to provide building blocks, drinking straws, or even Legos and have students construct a three-dimensional structure that has more than one line of symmetry. As a follow up, students can look in books, online, or in magazines for symmetrical buildings and other symmetrical structures (even faces!). Have them note the lines of symmetry on a photocopy.
Discovering How Tesselations Work
While most of your students are busy exploring the properties of polygons, ask your advanced students to discover how tessellations work. There are only three regular geometric shapes that tessellate. By "regular," I mean that they are figures in which all sides and internal angles are equal. These are the equilateral triangle, square, and hexagon. After explaining what a tessellation is, provide students with a page of geometric shapes or tiles and have them find which shapes can be used to fill the area of a piece of paper with no gaps or overlaps. If you have access to computers with Microsoft Word loaded onto them, have students go to the “Shapes” category under the “Insert” menu. Using this, they can experiment with creating basic tessellations. However, the objective should be for them to discover that geometric tessellations require the interior angles to add up to 360 degrees. Be sure to ask them to explain why some shapes work and others don’t. Once they have mastered the mathematical principals, they can move onto more complicated tessellations based on congruent shapes like the ones created by M.C. Escher. What follows are more lessons that go beyond basic geometry.
Geometry Lessons and Activties:
Using the book "A Cloak for the Dreamer" by Aileen Freedman, students create designs with pattern blocks. The focus of the lesson is on students discovering which regular polygons can be used, which will not work, and the properties that make it so.
After watching a Quick Time video, students explore how wallpaper uses both regular and irregular shapes. This lesson builds nicely off of “Tiling Tessellations” or can be a substitute for more advanced students.
This is a good introduction to seeing symmetry in a three-dimensional object. The lesson requires a large chunk of time (two hours), so you will have to plan accordingly.
This requires Internet access as well as an introduction to chip carving. Students study rosettes to discover radial symmetry and then create their own rosettes on paper. Although it requires some advance legwork, the lesson can be integrated into art, math, and social studies.