Tessellations

M. C. Escher was an artist who used strong mathematical understanding in his art.  He is know for many optical illusions such as

Hand with Reflecting Globe Escher, M. C. (Maurits Cornelis), 1898-1972 1935

and

Concave/Convex Escher, M. C. (Maurits Cornelis), 1898-1972 1955

Escher is also know for his tessellations.  Tessellations are images that are created when a single shape is repeated over and over again, completely covering the paper (or plane) with no gaps between the shapes.  A "fun" question to ask students in early geometry is, "Does this shape tessellate?"  Meaning, can you arrange this shape over and over again in the same plane (you can turn it) and cover the whole plane with no empty spaces?

Here is a simple video showing some of Escher's tessellations

Geogebratube has a fun interactive to play with a tessellation
and here's a fun tessellation project for middle school aged kids

GeoGebra

Have you ever felt like there should just be a way to work with geometry and algebra, but on a computer? Well there is a great one, so let me introduce you to: 

GeoGebra is a super useful online tool for drawing those pesky shapes that just won't come out right on paper.  I have used it to analyze triangles; to construct parallel lines cut by a transversal, and to analyze the relationship of the resulting angles; to examine the interior and exterior angles of a polygon; and to construct repeating congruent polygons.  But there are so many other uses!  Not only can you construct your own geometric figures, you can also access Geogebratube to utilize other people's creations.  

Check out the following:

Tangram Puzzle Cat (Geometry for the Elementary Set)

Circumference of a Circle (Geometry for Middle School)

The Unit Circle (High School Concepts)

There are so many uses for Geogebra: things I haven't even imagined yet!  What a wonderful, FREE tool to have at our fingertips.  I plan on using lots of Geogebra in my future career as an educator!

Check out Spirograph, just for fun!

How many degrees in a shape?

Even if it's been ages since high school geometry, I imagine that most people still remember how many degrees there are in a triangle.  (Just in case you forgot--it's 180 degrees.)


Remember?  There you go!

Now, how many degrees are there in square?--That's 360 degrees.

Now, what about a pentagon?  I don't know if I ever even learned that one in high school!  It's easy, though, to determine how many degrees there are in any convex polygon.  One method is to use a process called triangulation, which is essentially drawing triangles inside the polygon, all originating from one point. If you draw a line segment from one point to every other point in the triangle (except the two points closest because there is already a line there), you will draw triangles.  Since every triangle's interior angles sum to 180 degrees, you can count the number of triangles you have just drawn in the polygon and then multiply that number by 180 degrees to get the total number of degrees in the polygon. Check out this method from Khan academy

Back to that first assumption.  How do we know there are 180 degrees in a triangle?  Is this something that the math teachers know and that we everyday non-mathy people just can't understand?  The answer is a resounding NO! There's an easy way to prove that there are 180 degrees in a triangle, just click on the Geogebra image below to play with an interactive geogebra tube and prove it to yourself!

"Triangle.Equlilateral.png" by en:user:herbee is licensed by CC under the GFDL

"GeoGebra" by gengebra.com