Daily 7: Fibonacci Designs

For my Daily 7 I decided to play around with Fibonacci numbers in Geogebra. I made circles with radii of 1,1, 2,3,5,8,13, and 21, which are the first 7 Fibonacci numbers. I played around and rearranged them in some different shapes to try to make some Fibonacci art. Here are a few pictures of what I tried.

Weekly 3: Doing Math

For my third weekly work I thought I would expand on the Chinese Math handout. I worked on the story problems in that handout and thought I would come up with some of my own word problems that follow similar skills for solution. I will also post solutions to each problem.

1. A Corvette travels at a speed of 80 mph, a Malibu travels at a speed of 75 mph. If the Malibu leaves 1 hour before the Corvette, at what point will the Corvette catch up to the Malibu?

It will take the Corvette 15 hours from it's own start time to catch up with the Malibu. They will meet at 1200 miles.

2. Matt has 15 more stickers than Julie, and 3 more than Megan. Megan has 7 more stickers than Bob and Rachel combined. Rachel and Bob have the same amount of stickers.  Rachel has 4 stickers. How many stickers does each person have?

Rachel has 4 stickers
Bob has 4 stickers
Megan has 15 stickers
Matt has 18 stickers
Julie has 3 stickers

3. A large pizza costs $12 and a medium pizza costs $9.  A group of friends want to buy 3 large pizzas and 2 medium pizza and each wants to pay their fair share. There are 8 pieces per pizza and no person gets more than 5 slices and no person gets less than 3 slices.  No one has change so they all pay to the nearest dollar and have two dollars extra. How many friends are paying for the pizzas? and how much did each friend pay?

There are 8 friends paying for pizza. The price of the pizza comes out to $54. Since there are two dollars extra we know that the friends money totaled $56. There are 40 pieces of pizza total and if no one get more than 5 pieces we know there are at least 8 friends. If no person gets less than 3 slices we know there can't be more than 13 friends. The only number that divides evenly into 56 between 8 and 13 is 8 so there has to be 8 friends paying for the pizza.

4. Let's do some more pizza math because pizza always sounds good to me. If two large pizzas a medium and a small cost $32, 1 large, 3 mediums, and 2 smalls cost $40, and 3 mediums and 2 larges cost $42, then how much does each size pizza cost?

A large pizza will cost about $10.36, a medium pizza will cost about $7.09, and a small pizza will cost about $4.18
This can be solved by using a system of equations such as:
2l+1m+2s=42, 1l+3m+2s=40, 2l+3m=42
I like to use matrices to solve systems of equations.

These are just a few examples of my own "story problems" that follow the same style as those given from the Nine Chapters.

Weekly 2: Doing Math:Tesselation

Although it may not look like it. I spent about 2.5 hours constructing this "tessellation".  I'm not the most gifted in the art department, but I took a stab at it.  Two of my biggest problems with the construction were that I didn't have a protractor to make angles equal, and that I didn't start with an equilateral triangle in the center.  Those two errors started to become more apparent as I started to branch out further with the design. Below is a picture on my tessellation spread across 9 sheets of 8.5 by 11 paper.

I started constructing this figure with just an equilateral triangle. As I mentioned I didn't have a protractor so I used the Pythagorean theorem to essentially make two right triangle back to back with a height which was twice the length of the base. From there I wanted to use different shapes to make a more unique construction.  I used barn shaped hexagons which were hexagons with two 90 degree base angles. I simply filled in the rest of the space with whatever other shape would keep the general shape near a triangle. If I were to make this figure into a true tessellation I would make sure to use general shapes that would fill the rest of the figure into a larger equilateral triangle.  By ensuring I have a larger equilateral triangle I would be able to surround it with three more copies of the figure, one on each side, and the pattern could repeat infinitely, by continuously adding copies to each new open side. If there is not a repeatable pattern present, then it is not a tessellation.

Daily 3: Stomachion

After working on the paper version of the stomachion puzzle for about 2 hours and getting no where close to a solution I switched to the online geogebra version.  I was able to finally find a solution with the geogebra version.  A picture of the solution is below.

Nature of Mathematics: Axioms

According to Merriam-Webster an axiom is "a rule or principle that many people accept to be true." The origin of the word comes from the latin word axioma, which means worthy.  In mathematics, axioms are statements which are accepted to be true without proof, because they seem to be evident.  I guess you could say that axioms need no proof, because they are worthy. We do not question them, because we accept them and acknowledge them as true.  One clear example of an axiom is the reflexive axiom used in algebra, which states "a number is equal to itself". This is easily accepted without argument or proof. I could show you that a=a, but by the definition of equality we know that a=a. If a was not equal to itself, then it would not be itself, and we would not be able to say anything for certain about a.

You could argue that axioms while not needing any proof themselves are the basic building blocks for mathematical proofs. In order to prove something, you must use axioms to support your work. Try to prove anything without using something that is already accepted to be true.  You may think that a proof by example would be possible without using axioms. While you may be able to create some sort of proof without a formal axiom used, you will still rely on a universal truth to that is accepted to make your proof true. Let's say that I wanted to prove that 4-3=1.  Maybe I would set out 4 apples on a table.  Then I would take 3 of the apple away and eat them.  This would leave one apple left over. I could claim that I did a proof by example without using any axioms. But would that really be true? The apples could reappear. That would contradict the proof, but we know that 3 apples will not magically appear from nowhere and present themselves on the table. Therefore, one rule that is accepted to be true without any proof is that objects do not spontaneously generate from no where. That statement could be viewed as an axiom in the apple proof.
Axioms are needed for creating and giving validity to proofs.  They are also informally used in everyday life. People do not walk around astonished that aren't turning into dogs.  We accept that we are humans and are not going to change into another animal. No one needs to prove to us that we are humans and are going to stay humans, we just accept this to be true. We further build proofs off of our accepted knowledge.  This relates back to my previous blog about what is math.  Axioms are not just used on paper, and neither is math. Math permeates into everyday life. If nothing is accepted to be true, then there is nothing that can be used to prove anything.

What is Math?

I can not define what math is with a simple statement.  But I will attempt to explain what I view math to be. It is numbers. It is variables. It is arithmetic. It is manipulation. It is also a way of interpreting information.  When there is an issue or problem that needs to be solved; the sorting of information and weighing of options is math.  The issue does not have to be how much water can fit in a fish tank to consist of a mathematical solution. Math is more than the basic manipulation of numbers, but a diverse way of thinking. Mathematical thinking is rational everyday thinking. I believe that mathematical logic often guides decision making. People are not robots without emotions, but when rational decisions must be made I believe the through process of organizing informations and comparing different options is the essence of mathematical thinking.  I very much dislike writing formal proofs, but at the same time mathematical proofs are a great way to form logical arguments and confirm or deny possible truths. Mathematicians are not the only people who like to prove they are right.  To me mathematical thinking is math. Pencil and paper and calculators and graphs do not need to be part of the picture for math to be present. Math permeates our lives, regardless of our feelings for it. 

Top 4 Math Milestones in no particular order (drumroll please......)

1. Descartes' Coordinate Plane: His discovery/creation was a very beneficial representational tool to enhance understanding and development in algebra and geometry.

2. Riemann's Sum (Calculus): I believe that Riemann's Sum is the basic building block for understanding Calculus. 

3. Pythagorean Theorem: Trianlges! I would venture a guess that the Pythagorean Theorem is the most well known mathematical theorem to the general public. 

4. Invention of the Calculator: With the invention of the calculator and other math related technologies, new advancements were able to be developed with less time spent on simple arithmetic.