These pages are dedicated to the teachers in I-MATH. In this course, the teachers each create a project for their students. The following is a project completed by one of our I-MATH teachers.

We started our Geometry unit by beginning with points, lines, and planes. The students studied different types of lines and angles learned how to construct congruent segments and angles, using compasses and straight edges, and discovered how to bisect line segments, a necessary step for the mask project development.

I introduced them to polygons using geoboards and rubber bands. In small groups, they created regular and non-regular polygons, and explored angle relationships. From there, it was natural to move on to symmetry, line and rotational, and then to tangrams, assembling different polygons together to make various shapes and pictures.  Using tangrams was a brain opener for some they were challenged to think more spatially than usual.  Many initially commented how difficult tangram puzzles were, but I could see an improvement in their ability to problem solve by the second day.

We then moved on to transformations; translations, rotations, and reflections, and finally tessellations and patterns. I brought in a book, "The Life and Works of the Escher" by Miranda Fellows, and we pulled the web sites listed below. The students really enjoyed creating transformations and tessellations, and they were fascinated by Escher's work. Many had very interesting comments about seeing goodness and darkness in his work, and there were several conversations as to his work being "art" or math". Since his work was done during the WWII era, and the students were studying this era in Social Studies, it was easy for me to tie the two disciplines together.

Here are a few links to some fascinating websites about M.C. Escher:

http://www.mcescher.com/

http://www.nga.gov/collection/gallery/ggescher/ggescher-main1.html

http://www.iproject.com/escher/escher100.html

And here's another interesting site, with lots of information on tessellations and symmetry:

http://www.camosun.bc.ca/~jbritton/jbsymteslk.htm

All along, the students worked on mini-projects. I gave them hand-held mirrors to study their own face for symmetry, and they had to draw what they saw.  The students had the option of taking pictures of things around campus that had symmetry, or could draw them, and then describe the line or rotational symmetry they saw.

At the same time, some students cut out magazine pictures that had symmetry or pictures and figures that showed one or more forms of transformation or tessellation. Others created their own designs.

For tessellations, we used pattern blocks in small groups to make tessellations, and then they all drew and colored their own tessellations. I had a couple of "lab days" where the students pulled the IMath websites (chapters 3-8) as well as the topics on angles, triangles, polygons, and tessellations.

I also incorporated a software program "Cosmic Geometry" where they could create their own polygon robots and different tessellations. The students were very excited to design on the computer; they really enjoyed experimenting with the tools of design. Some of their work is quite creative! I am hopeful that they will be able to use Sketchpad someday to elaborate on what they have done for this unit.

This introduction to Geometry lasted a couple of weeks, and all the while I was preparing them for designing their mask. The Mask activity was to create a mask that incorporated all the geometry concepts they had learned up to then, and to demonstrate this knowledge in a creative mask form. For demonstration purposes, I had done a couple of samples that represented both A and B grades, and left them up in the room for reference, along with the rubric we created together.

We talked about the rubric guide and what was needed in order to achieve the highest level of points. There needed to be polygons, line segments, and angles all clearly marked and identified; line and rotational symmetry, including turn and degrees; bisectors marked; all three forms of transformations; and at least one tessellation in the mask. I gave them a copy of the rubric two weeks before the due date, and assisted and answered questions along the way.

Most of their questions were centered around, "Is this good enough?", to which I asked, "Does this tell me everything you know about geometry?" They were concerned with being more neat than displaying all the lines of symmetry and compass bisecting marks. Just from a glance, I could quickly tell who really "got it" and who didn't. Even noting the sum of degrees in various polygons alerted me to who knew the interior sums. Some students actually went back to the Angle web site chapter to figure out the correct degrees, and how to bisect a line segment.

Some students created a cat mask, or Halloween mask, or other common mask, while others were more creative. I had several students comment that they want to draw it on the computer, using the geometry tools, and one wants to create his 3-dimensionally, now that we have moved onto Solids in the unit. This mask project can be extended many ways!

Overall, the comments afterwards were very favorable. Most students said they like being able to be creative with the masks. Some saw no limit; others limited themselves by being too concerned with neatness. One student said he liked to "color and draw even if it was math because it didn't feel like math homework." Another commented on the freedom it allowed her to express her geometry knowledge.

The timing of this Geometry unit was perfect since I was learning this same material at the same time in our I-MATH lessons. It made the unit so much more exciting and fun for me as a teacher! 

You need a pencil! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Name:______________________________________

You need to complete the following activities today in the lab:

Go to Tessellations Station in Mighty Math Cosmic Geometry

Purpose: to learn about translation, reflection, rotation, and tessellation while strengthening your visual skills in math and art.

Objectives:

Explore Geometry concepts of translation, reflection, and rotation

Enhance understanding of symmetry

Exercise spatial, visual, and artistic creativity while exploring patterns in math and design

Discover connections between art and mathematics

Listen carefully to the directions and take a tour of the controls so you know what tools to use. Click on the screen when you are ready to experiment.

Create and design your own tessellation. Choose shapes, pull and stretch them, fill them with color, and tessellate them to create interlocking patterns. Print the design, or make a copy of it on the back side of this paper.

Go to www.punahou.edu/acad/sanders/geometrypages/GPLinesList.html

Click on Topic 3 (Polygons) and browse through. Look at the names and designs of polygons. At the bottom, click on Topic 4.

Topic 4 (Symmetry). Look at the pictures; notice Line Symmetry and Rotational Symmetry. At the bottom, click on Topic 5.

Topic 5 (Construction) Could you construct one with compass if it was on a test? At the bottom, click on topic 6.

Topic 6 (Angles of a Polygon) What is the sum of the interior angles in a triangle? In a square? In a pentagon? In a hexagon?

Topic 7 (Tessellations) check the sites out. What are the only 3 regular polygons that fit together perfectly to form tessellations?

Topic 8 (Creating Tessellations) Practice designing and creating your own. Visit the web site at page bottom and see "What is a Tessellation?"

Reflection:

write a page in Math journal about today's lab experience

Turn in this paper with answers to all the questions, your tessellation, and reflection.

When you are done, you may go back to any site and experiment. Have fun!!

Mighty Math Cosmic Geometry, ßœHarcourt Brace, Grades 6-8, supplemental computer program.