Numbers and Their BFFs

Amicable numbers. First off, what are they? Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number (thank you Wiki). But what does that exactly mean? Let's look at an example:

To show that two numbers are amicable you must first find their proper divisors. So first you need to know what a proper divisor is: a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3 (again, thank you Wiki). Let's work with 220 and 284.

Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
Proper divisors of 284: 1, 2, 4, 71, 142

If we add together all the proper divisors of 220 we obtain 284.
If we add together all the proper divisors of 284 we obtain 220.

Weeeeeird. In short, that is what amicable numbers, or more commonly amicable pairs, are.

This site gives a nice little brief history on amicable numbers. Briefly discuss famous mathematicians who found amicable pairs, Pythagoras being the first to find 220 and 284 as an amicable pair, and several other devoting their time to finding others. The page even offers a way to discover other amicable pairs. Fancy.

Who knew that numbers had BFFs. I guess everyone deserves them.


FUN FACTS

• There are some amicable pairs (m, n), in which the sum of digits of m and n is equal [4]. For example, consider amicable pair ( 69615, 87633), 
• Sum of digits of 69615 = 6+9+6+1+5 = 27 
• Sum of digits of 87633 = 8+7+6+3+3 = 27 
• There is no amicable pair in which one of the two numbers is a square

(http://www.shyamsundergupta.com/amicable.htm)

Tessellations!

Growing up my sister and I would always color in these books with our colored pencils at my grandma's house. Little did I know that what I was actually coloring were tessellations. I wish I could remember the names of the books but apparently I have been a fan of tessellations for quite some time now.

In MTH 221 we also work with tessellations but to a different extent. In MTH 221, we took a square, cut piece(s) out of it, then reattached them somewhere else on the square. This topic was covering the different types of tessellations so we created two different shapes. My first shape was created using slides. From this, my shape resembled the Teenage Mutant Ninja Turtles. My second shape was created using rotations. This shape resembled distorted bats, so I colored them Halloween colors. I wish I had my past school stuff with me so I could post a picture, hopefully I come back to this with an update.

For MTH 495 I first started using the pattern blocks to create tessellations. I later moved to the isometric dot paper. I think overall I probably spent 3-4 drawing the page. Obviously there were distractions but I watched two movies while doing the drawing. Below of pictures of the beginning and the in progress. I'll updated with the completed colored version when it's finished.

I also really enjoyed this website that allows you to create patterns online. I wouldn't be able to spend as long on a site versus doing a tessellation with pen and pencil but it's still an excellent resource and a good time killer. 

MathMagicLand

Math really is involved in everything we do. This video was actually quite interesting to watch. A simplified cartoon that is actually full of so much information. Who knew that Disney was not only creative but educational? Music, art, nature, chess, sports, and the list continues. I would love to use this as a tie in in my classroom one day.

Pythagoras is most likely one of the name that will come out of your mouth when asked about famous mathematicians. And rightfully so. Simply searching Pythagoras and his theorem generates lists upon lists of his history and his work.

The video above gives a nice little visual to the Pythagorean Theorem. This model has a single triangle in the middle with the squares built off each leg and the hypotenuse of the triangle. I'm sure you know Pythagoras's Theorem, but as a refresher:

a² + b² = c²

Where c is the longest side and a and b are the other two sides

In the water demo, the water beings in the squares built of the smaller legs. Each of these two areas are completely full. The board is the rotated to reveal that all the water that was contained in the two smaller squares, completely fills the larger square, the hypotenuse. Pretty cool, right?

I wish it gave details onto how this model was created but I love how great of a visual it is. Granted, when showing this to kids they would already need to have the background knowledge but what a great anchor visual for them to have.

Also, another great activity is a proof of the Pythagorean Theorem using jelly beans! Cool, right? Not only does this activity allow students to work hands on with the Theorem but it also allows them to get a better understanding as to why and how the theorem works. Similar to the water demonstration, the Jelly Bean activity has students section off the squares created off each leg of the triangle. The students then slide the Jelly Beans into the largest square, not removing any of the Jelly Beans. Students are able to visual see the theorem in action, which is what I love about hands on activities. When I first learned this theorem, I never really understood why it worked. Through my college courses I have most definitely learned more about the workings of this theorem and it really is a great one that deserves to be fully understood.

You can check out the Jelly Bean activity here.

Math Is...

Reasoning, understanding, explaining, and counting are all words that come to mind when I think of math. Math is not simply computing numbers to receive an answer or inputting numbers into an equation to generate another number. Math is using numbers to receive a better concept of the world around you. Math is involved in everything we use, even if you don't think it is, there is a way math is involved. I think one of the greatest things is that math is universal. While each country has different was of teaching math, the concepts are all the same.

As far as milestones or Top 5, there are so many. There's famous people, theorems, inventions, fields, concepts, etc. For sure I would say that Euclid is important. He built the foundation for modern mathematics by introducing a set of axioms; I mean there's an entire class at Grand Valley dedicated to him, must be pretty important.

Along with Euclid, you have Pythagoras and Fibonacci; there's Euler and Newton, each contributing their own piece to the world of mathematics.

Then there are calculators. Huge milestone in the world of mathematics. Calculators allowed for quicker computation and a simple way to check answers. Calculators themselves have progressed so much that no one could have imagined the effectiveness of them when they were first created.

Math is also several different fields. We have algebra (my favorite), calculus, geometry, non-Euclidean geometry, differential equations, and the list goes on. What's interesting though is that each field builds upon and integrates different fields.

Math is patterns, logical reasoning, relationships, explanations, and application. Math is far more than numbers and computation.