Surface Area, Nets and the Pythagorean Theorem

We had a tough problem in class today, and I thought I'd share it.

The Scorpion and the Cricket

Inside a rectangular room measuring 30 feet in length and 12 feet in width and height, a scorpion is at a point on the middle of one of the end walls, 1 foot from the ceiling as shown in point A, and a cricket is on the center of the opposite wall, 1 foot from the floor, as shown at point B.  What is the shortest distance that the scorpion must crawl to reach the cricket, which remains stationary?  Of course, the scorpion never drops or falls, but crawls fairly.

Once I figured out that the scorpion was on the wall on the right of the box, 1 foot from the ceiling, and the cricket was on the wall on the left of the box, 1 foot from the floor, I figured I had it all figured out.  I labeled the box with the dimensions that were listed in the initial problem, had the scorpion crawl down the right wall, across the floor, and up the left wall.

To me, clearly, the scorpion crawled 11 feet down the right wall, 30 feet across the bottom panel, and 1 ft up the left wall.  11ft+30ft+1 ft= 41 ft.  So I figured the scorpion crawled 41 feet.  When I got to class, the other students were discussing the phrase from the problem: "the shortest distance."  When I really started to think about this problem, I realized that there MIGHT be a shorter distance that the scorpion could travel.  I know that if the scorpion could fly, the shortest distance would be a straight, downward trending line through the air of the box, directly to the cricket (the shortest distance between two points is a straight line).   That length of that line would be the absolute shortest way the scorpion could travel.  But since the scorpion can not fly, it has to crawl.  My instructor suggested that we explore nets to inspire an alternate path for the scorpion.    

In the above image, you can see that the first net that a drew did not help me.  Because I drew the cricket and the scorpion in such a way that they appeared in a straight line in relationship to each other, I could not figure out a way to draw the triangle I needed, so I abandoned this net.  Net #2 allowed me to draw a diagonal line that the scorpion could travel, but because of where I positioned the side flaps on my diagram, it made the cricket and the scorpion even farther apart.  In this diagram, I have one leg measuring 47 feet, and the other leg measuring 7 feet, making the hypotenuse measure 47.5 feet.  This net created a diagonal line for the scorpion to travel, but the distance was longer than my initial distance.  I saw that I needed to make my triangle's "legs" shorter in order to shorten the length of the hypotenuse.  Net #3 solved the problem for me!

What a fun problem!  The shortest distance the scorpion could travel is 40ft!

Circle Geometry--Discovering Pi

Okay, I believe you, π is a number.  It may not be a number as I am used to working with numbers, but it represents an irrational number that starts with the digits 3.1415 and then continues on indefinitely.  Even more important than knowing the exact value of π is recognizing that π is a constant that turns up when investigating the geometry of a circle.

Any circular shape has a diameter, or distance across.  That diameter is consistently related to its circumference.  No matter how large or small you make a circle, if you measure the circumference of the circle and then measure its diameter, you will find that two measurements are related.  In fact, the Diameter of a circle times π equals the circumference of the circle.  Neat, huh! 

Here's a really great video about the history of π

Perhaps if you celebrate pi day, you might like to try cutting π this year!

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!

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