1. Problem Solving Project

1. 1. How Mathematical Problem Solving Can Lead to Excellence, Personal Development and Integrity Outside of a Mathematics Classroom

The philosophical understanding of the world, its general laws and basic scientific concepts is not possible without mathematics. Mathematical problem solving leads to excellence, personal development and integrity outside of a mathematics classroom. It helps to analyze, compare and classify objects, establish cause-and-effect relations and patterns, build logical chains of reasoning and develop algorithmic thinking.

Mathematics allows us to develop some important mental faculties: analytical, deductive (the ability to generalize), critical, forward-looking (the ability to forecast, to think several steps ahead) capacity. The mathematical problem solving improves the abstract thinking, the ability to concentrate; it also trains memory and enhances the speed of thought.

Mathematical thinking helps lawyers, like a good chess players, to build complex combinations of variants for protection in court, or to invent clever ways to interact with the legal framework and come up with all sorts of artful and nontrivial solutions. Mathematical problem solving can be used in different areas of life, for example in private life. If we want to renew our flat we have to calculate how much paint, wallpaper, tile, adhesives and cement to buy. When we cut out and sew clothes the result depends on the accuracy of measurement.

Mathematical calculation can be used everywhere: in the car, which you drive, in your computer or a portable device. All the buildings do not collapse under their own weight due to the fact that all the data necessary for the construction were calculated in advance by formulas. Medical and health care also exists because mathematics is used, firstly in the design of medical devices, and secondly, for the analysis of data on the effectiveness of a treatment. Even the weather forecast is not completed without the use of mathematical models.

1. 2. The Mathematics Behind the Joke

1. 2. 1. This joke is based on the fact that mathematical expressions, like expressions in natural language, can have several meanings. As in other jokes based on wordplay, humor obtained at the expense of an ambiguous meaning, in this case, the expression 10 in the binary system is equal to the decimal number 2:

– Why do mathematicians always confuse Halloween with Christmas? – Because 31 Oct = 25 DecHumor is in the similarity of abbreviations in the English language in October / octal number system and December / decimal number system, and coincidence that these two representations are equal (318 = 3 * 8 + 1 = 25 = 2510). (Mathgarage. WordPress. com)

1. 2. 2. This well-known mathematical riddle about a fly is very popular:

Two trains, 200 km apart, are moving toward each other at the speed of 50 km/hour each. A fly takes off from one train flying straight toward the other at the speed of 75 km/hour. Having reached the other train, the fly bounces off it and flies back to the first train. The fly repeats the trip until the trains collide and the bug is squashed. What distance has the fly traveled until its death?

There is a complicated way counting a sequence or simply knowing that if the fly is flying for 2 hours still at the same speed of 75 km/h then it flies a distance of 150 km. (BrainDen. com)

1. 2. 3. Statement c. All students have a computer.

Inductive reasoning: Sam has a computer, his neighbor has a computer, all of the students in the group have a computer, all students at the University have a computer. Therefore all students have a computer (“ Inductive Reasoning Examples”).

Deductive reasoning: All students have a computer. Sam is a student, therefore he has a computer (“ Deductive Reasoning Examples”).

2. Geometry Project

Maurits Cornelis Escher (1898 -1972) – Dutch graphic artist known primarily for his conceptual lithographs, woodcuts and metal engraving, in which he masterfully examined aspects of plastic concepts of infinity and symmetry as well as the characteristics of the psychological perception of complex three-dimensional objects (M. C. Esher).

The mathematicians were his enthusiastic fans, they saw in his work an original visual interpretation of certain mathematical laws. It’s very interesting with regard to the fact that Escher had no special mathematical education.

He drew ideas for his work from mathematical articles which described the mosaic plane partition, plane projection of three-dimensional shapes and non-Euclidean geometry. He was fascinated by all sorts of paradoxes, including ” impossible figures”. The most interesting of Escher’s ideas are all kinds of plane partitions and the logic of three-dimensional space. The regular partition of the plane, called ” mosaic” is a set of closed figures that cover a plane without crossing figures and gaps between them. Typically, simple polygons are used for mosaic composition, e. g., squares or rectangles. But Escher was interested in all kinds of mosaics – regular, which form repeating patterns, and irregular (non-recurring patterns), and has also introduced his own view, which he called ” Metamorphosis”, where figures change and interact with each other, and sometimes change the plane itself.

The following 6 ways of a plane partition with parallelograms, rectangles, squares, triangles, diamonds (with an angle of 60 degrees), hexagons (with an angle of 120) formed the basis of all his mosaic works (see fig. 1).

Mathematicians have proven that only three regular polygons fit the regular partition of a plane: triangle, square and hexagon (see fig. 2).

## Regular partition by birds Reptiles Evolution1

Cycle

Maurits Cornelis Escher was fascinated by rectilinear geometric shapes – polyhedrons. Polyhedrons are major figures in many of his works and appear as auxiliary members in even greater number of works. The woodcut ” Four Regular Solids” (see fig. 3) shows the intersection of main regular polyhedrons located on the same axis of symmetry, besides it polyhedrons look translucent and it’s possible to see the rest through any of them (Impossible World). Fig. 3. Four Regular Solids

3. Statistics Project

3. 1. The statement that Saint Leo University is the largest of three Catholic institutions in Florida is true (see table 1) and is supported by statistical data of 2015 (Collegestats. org).

## The largest Catholic institutions in Florida

3. 2. Saint Leo University is one of the highest providers of higher education online.

The ranking of the best online bachelor’s degree programs 2015 is available on the website USNews. com. It’s based on such factors as graduation rates, indebtedness of new graduates and academic and career support services offered to students.

The ranking indicators for the institutions with highest, lowest rank and Saint Leo University are set out in the table below (see table 2):

## Comparison of Online Bachelor’s Program Providers

Saint Leo University has rank #50 with the overall score 79. The lowest published rank is #214 with the overall score 44 (University of Wisconsin—River Falls ), for the college with the highest rank (Pennsylvania State University—World Campus ) the overall score is 100.

The following statistical diagram (see fig. 4) allows comparing the institutions providing online bachelor’s programs by categories and proves the statement that Saint Leo University is one of the highest providers of higher education online.

3. 3. Saint Leo University is ranked as the third largest among private universities and colleges in Florida.

The intersection of sets of the largest institutions in Florida (Collegestats. org) and private colleges in Florida (StateOfFlorida. com) gives us the following selection of the largest private colleges in Florida (see table 3):

## The largest private colleges in Florida

The results can be visualized in the following diagram (see fig. 5):

## Works Cited

BrainDen. com. Web. 18 June 2015.

Collegestats. org. Web. 20 June 2015.

Impossible World. Web. 18 June 2015. < http://im-possible. info/english/articles/

escher_math/escher_math. html>.

“ Deductive Reasoning Examples.” YourDictionary. LoveToKnow, Corp. 2015. Web. 18 June 2015.

“ Inductive Reasoning Examples.” YourDictionary. LoveToKnow, Corp. 2015. Web. 18 June 2015.

Mathgarage. WordPress. com. Web. 18 June 2015. < https://mathgarage. wordpress. com/

2012/10/30/why-do-mathematicians-always-confuse-halloween-and-christmas/>.

M. C. Esher. MCEsher. com. Web. 18 June 2015. < http://www. mcescher. com/>.

StateOfFlorida. com. Web. 20 June 2015. < http://www. stateofflorida. com/

colleges-in-florida. aspx >.

USNews. com. Web. 20 June 2015. < http://www. usnews. com/education/online-education/bachelors/rankings >.