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

Measurement: Metric and American systems

I love the metric system!  So does Bill Nye!











It makes sooooo much sense to me!  All conversions between centimeter and meter and kilometer are related by powers of tens.  And amazingly it's the same with grams and liters, too! For example: 1 kilogram equals 1000 grams and 1 gram equals 1000 mg.  100 centimeters equals 1 meter and 1 centimeter equals 10 millimeters. Should you need a refresher on converting within the metric system, here's a khan academy link:  How to convert within the metric system

Contrast that efficient system with the American system of measurement, and your eyes can begin to glaze over. There are 12 inches to the foot, 3 feet to the yard, 5,280 feet to a mile.  What, you say?  How about 16 ounces to a pound and 2000 pounds to a ton. Then there's the ever confusing volume measurements: 3 teaspoons equals 1 Tablespoon, 16 Tablespoons equal 1 cup, 2 cups equal 1 pint, 2 pints equal 1 quart, 4 quarts equal 1 gallon.  WOW!  How many tablespoons in a gallon?  Do I really NEED to know? (That's 16*2*2*4=256 Tablespoons in a Gallon!)

Here is a fun printable (free!) I found on Teachers Pay Teachers by The Lesson Plan Diva that can help. This is the little mnemonic that I learned a few years ago to help remember those tricky liquid volume conversions--cup, pint, quart, and gallon.  The story takes place in the Land of Gallon where there are four Queens.  Each Queen has a Prince and a Princess.  Each Prince and Princess has two Children (I learned cats, but same idea).  This fun little story helps you to remember 1 Gallon (Land of Gallon) equals 4 Quarts (Queens).  Each Quart has 2 Pints (Prince/Princess), and each Pint has 2 cups (Children)!

Isometries

I think I wanted to talk about Isometries just because I like the word so much.  It's one I've never heard before and it rolls of the tongue so poetically!

I actually feel like Shakespeare when I say it--Oh, I know why--iambic pentameter, another phrase I like the sound of!

Iso means "equal" and Meter means "measure," so it makes sense that isometries would mean equal measure when applied to movement of a shape on a plane.  There are three kind of isometries I learned about: Reflections, Rotations, and Translations.

Reflections (or flips) are exactly what you would think.  A reflected object appears just as if it has been reflected in a mirror.  The reflected object is the exact same distance from the line of reflection as the original, but it appears mirrored.  This concept is a bit difficult to describe with words and deserves a visual example:

In the above image, quadrilateral ABCD is reflected about line x=3.  You can see that A'B'C'D' is the mirror image of the original figure and every point is the exact same distance from from the line of reflection, only in the opposite direction.

Rotations are also know as turns.  The shape is rotated around a single point a certain number of degrees.

In the above image, the quadrilateral ABCD is rotated around point B 180 degrees.  It is as if I picked up and turned the original figure, but kept the shape joined to the grid at point B.  The resulting figure A'B'C'D' is a 180 degree turn from its original location.

Translations are also known as slides.  In a translation, the shape is moved on the plane a certain distance, but the shape appears exactly the same, only in a different place

In this image, quadrilateral ABCD is translated along vector u.  A'B'C'D' is the exact same shape and orientation, it has just been shifted left and a bit down in order to form A'B'C'D'.

Here's a bit more you can do with rotations, reflections, and translations--check out Ms. Pac-man!