Thursday, May 29, 2014

Amicable Numbers- Communicating

What are Amicable Numbers? (Communication Math)

When I first saw this question I really didn't know what it meant. This is not something that I have heard of, or at least remembered about. Therefore, I had to do some digging and figure out what this means. I found a couple different definitions for this.

Amicable numbers are a pair of numbers such that the sum of their proper divisors (not including itself) equals the other number.
     After reading this definition I thought I knew what they were talking about, but as I started to try and come up with some numbers I started having a hard time. Therefore, I had to research some examples before I could start to come up with my own!

(220, 284)- These numbers are said to be amicable numbers.

First we will take 220 and come up with all the proper divisors
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
If we add these numbers together 1+2+4+5+10+11+20+22+44+55+110=284

Now, we can come up with all the proper divisors for 284
1, 2, 4, 71, 142
If we add these numbers together 1+2+4+71+142=220

Therefore, these numbers are amicable numbers, or amicable pairs.

After looking at this example it really seemed to help. But, I couldn't help but wonder how difficult this would be to just pick two numbers. It would take me forever to pick two numbers and see if they were an amicable pair. So, I decided that I would look for another pair and see if there was a pattern.

We will take 2620 and come up with the proper divisors.
1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310
If we add these together 1+2+4+5+10+20+131+262+524+655+1310=2924

Now, we take 2924 and come up with the divisors
1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462
We add these together 1+2+4+17+34+43+68+86+172+731+1462=2620

Therefore, (2620, 2924) are an amicable pair.

I figured out that even after having the numbers of the amicable pair, it was difficult to find all the divisors. That actually took me a long time to do. I had to think about all the rules you learned in high school dealing with the different test.

This was a lot of fun to look up and learn. I had never heard of this before so it was interesting to learn!

Sunday, May 18, 2014

Geometric Tessellation (Doing Math)

Doing Math

In class we worked on the Tessellations on paper and with pattern blocks. I started the above Tessellation in class, but I finished it over the weekend. I added a couple more patterns to the Tessellation and added color. The color really seemed to help. It makes the Tessellation look better and it's easier to see the pieces that are reflected and rotated. 

While creating this pattern I had to start a couple of times. I knew that I needed something that that would repeat, but the repeating needed to be something simple in order to go back and create the colors. This is something that took some time because I kept making it too hard and it was very hard to follow. When it is that hard to follow its very hard to figure out which color you need to use and the shapes keep going together. After thinking about my strategy, I realized that I needed to create simple shapes that don't overlap each other. When the shapes started to overlap it became difficult for an on looker to see the patterns and the tessellation. 

I didn't spend an hour on the above piece so I decided to create another Tessellation...but this time I was going to create a geometric Tessellation. I did this using an app for my iPad called Mandalar. 

This app was very easy to use and would be great to use with students in the classroom. I think this app could really help the students understand shapes and Tessellations. The only problem with this app is that it was really hard to fill in the shapes along the outside of the picture above, that is why you don't see the blue and yellow around the outside.

I have made tessellations from when I was a student in elementary class, but coming back to these really after going through the College of Education, I have a whole different meaning. While creating these tessellations I was thinking about how the students would enjoy this and what they would learn from it. It would be a great way to involve technology into the classroom and have the students work with this app. While creating these tessellation, like stated above, you have to think about what shapes you are going to put next and where the shape would make sense. I feel this would be a great idea for the students to learn and a great problem for the students to work through. 

This is an app that I will save and play along with and try to create even more Tessellations! 

Saturday, May 10, 2014

Who is Euclid?

I spend my hour and a half reading articles upon articles about Euclid. There were so many concepts and ideas placed in these articles that I couldn't figure out what was true and what were rumors. Below is the information that I gathered about Euclid. Most of the facts and ideas seem to be true since I found them on more than one site, but since there are so many rumors out there, some of these might be rumors.

Who is Euclid?
"Father of Geometry"
Not too much is known about Euclid's early life. He was born around 330 B.C. in Alexandria. Euclid's life has been confused with the life of another Euclid, therefore it makes it difficult to believe any of the trustworthy information about this mathematician.

Euclid's Career
He is known as the "Father of Geometry" for a reason. He discovered the gave and really gave it its form. Euclid spent most of his time at the Alexandria library where he did most of his studies and thinking. He started his studies focusing on geometry. " He began developing his theorems and collated it into a colossal treatise called "The Elements"." "The Elements" sold more copies then the Bible and was used numerous times by mathematicians and publishers. There was no end to Euclid's geometry, and he continued to develop many different theories and theorems based on numbers, basic arithmetic, etc. "The Elements" was the first kind of geometry that people of the modern era developed.

"Euclid stated that axioms were statements that were just believed to be true, but he realized that by blindly following statements. there would be no point in devising mathematical theories and formulae." He realized that axioms needed to be proved by proofs. Next, he started to develop logical evidences that would stand against his axioms.

Euclid has a lot of other work as well. He has a wide range of other works actually that are still used and referred to this day. These other works were positions backed with solid proofs.

Death and Legacy
Mankind doesn't know when or why Euclid died. I is said that he might have passed away around 260 B.C.His books, theorems, axioms, and proofs are still used to this day. He left behind a legacy that would be used in mathematics for a numerous amount of years.

Friday, May 9, 2014

What is Math?

This is a question that I ask myself on a regular basis. What is math? Math can be defined in so many ways. I believe that the definition of math depends on the level of mathematician you are. 

When you are first introduced to math you think along the lines of the number line. The younger children think about the numbers and where they would fit on the number line. Also, using manipulatives to try and figure out number sense, which number is bigger than which. 

The older you get the better understand you have of math. Next students start thinking about math as being addition and subtraction. Thinking about number sense and trying to figure out the addition and subtraction facts. Next, comes thinking about multiplication and division. Once you start thinking about these concepts you really start getting an understanding of mathematics. 

Mathematics is a subject that consists so many other subjects, calculus, geometry, trigonometry, algebra, etc. Math is a very important subject because it is the foundation of so many other concepts and ideas. With mathematics you have to show your thinking and explain your reasoning and thinking. This is a great writing tool and many students learn by showing their thinking. As a teacher, you can see so much more in mathematics when the students show and explain their thinking. You can see when and where the students went wrong and you can figure out how to help the students get to the end result.

There are many different concepts and ideas that come to mind when I think about mathematics in general. Dealing with the number system is by far the most interesting. There is so much that you can do with them. for example, Multiplication, Addition, Subtraction, Division, We wouldn't have any type of mathematics without having these numbers.