Learning basic concepts of Mathematics: A Hurricane based approach

1. Learning basic concepts of Mathematics: A Hurricane based approach David Quesada Dept. of Natural Sciences, Mathematics and Computer Science Saint Thomas University, Miami Gardens FL 33054 Lourdes Espana Miami Dade College North Campus Department of Mathematics

2. Annual Meeting AMS, Atlanta January 2006 CNN Courtesy Hurricane science might represent a very interesting subject in order to introduce and learn many of the ideas we discuss in Introductory Algebra courses as well as in Pre-Calculus, Analytic Geometry and Statistics. Based on the Storm tracker facility embedded in the Achieve WeatherBug software, students may access the coordinates for all named storms in the Atlantic basin and perform a research on the concrete shape of the hurricane trajectories and its statistical trends. Based on that information, students are asked to determine an approximate shape for the trajectory and solve a system of linear equations to determine the coefficients of a parabola which better fit the data. A pilot project conducted at Miami Dade College North Campus has given excellent results, and honored students enrolled in the Algebra class were very happy with this experience.  

3. Academic accomplishment in Science and Language: one of the biggest issues for Science and Non Science majors. Student population at STU is entering with a low background in all areas of science, mathematics and grammar. It represents a challenge for all programs offered in campus. Since Weather Studies are too close to our everyday experiences, it might help Students to be engaged in the rigor of the University Life.

4. Occupations requiring higher skill levels are generally the most prestigious and highest paying. Note also that most of these occupations require high skill levels in both language and mathematics, refuting the myth that “if you are good at language you do not have to be good at mathematics,” and vice versa. Do you consider yourself to be either “ math phobic” (fear of mathematics) of “ math loathing” (dislike math )? Many adults harbor fear or loathing of mathematics and, unfortunately, these attitudes are often reinforced by classes that present mathematics as an obscure and sterile subject . Some common misconceptions about Math. 1. Learning mathematics requires special and rare abilities. 2. Math in modern issues is too complex. 3. Math makes you less sensitive. 4. Math makes no allowance for creativity. 5. Math provides exact answers. 6. Math is irrelevant to my life. What is Mathematics after all? The word mathematics is derived from the Greek word Mathematikos, which means “inclined to learn”. Thus, literally speaking, to be mathematical is to be curious, open-minded, and interested in always learning more !!

5. Mathematics as the sum of its branches • Logic – the study of principles of reasoning. • Arithmetic – methods fro operating on numbers • Algebra – methods for working with unknown quantities • Geometry – the study of size and shape • Trigonometry – the study of triangles and their use • Probability – the study of chance • Statistics – methods of analyzing data • Calculus - the study of quantities that change Mathematics also may be viewed as a tool for creating models, or representations that allow us to study real phenomena Bioinformatics Business Management Psychology and Sociology Engineering Mathematical Modeling Biology and Ecology Medicine and Physiology Economics Physics and Chemistry Computer science and Artificial Intelligence

6. www.weatherbug.com Basic ideas and keywords • Hurricane’s science may represent a very interesting topic in order to introduce and learn many of the ideas we discuss in introductory algebra courses as well as in pre-calculus and analytic geometry. • Concepts and definitions as for example: • Domain and Range, Function and Relationship. • Ordered pairs and triplets, system of coordinates. • Trigonometry, planar and spherical. • Trajectories and equations of some planar curves. • Parabola, Trochoids, Spiral curves. Fractal Geometry. • Conics and bodies of revolution. • Probabilities and statistics. Random processes • Numerical analysis. Database analysis. • Computer Modeling. Computational Geometry

7. Most of the Hurricanes follow a parabolic path that can be a motivation for Algebra

9. How to obtain the parabola? • Select three ordered pairs from the hurricane path, one before, one • close to and one after the turning point. • 2. Plug in these points into the equation of the parabola and write the • system of equations in three variables (a,b,c). • 3. Solve the system of equations by subtracting equations in pairs, • resulting in a system of equations in two variables (a,b). • 4. Once the system has been solved and coefficients determined, find • out the vertex and focus of that given parabola. • 5. Compare for different hurricanes if there is any correlation between • the vertex position and the width of the parabola close to the focus with • the intensity of the hurricanes and the moment of the year when it was • formed as a tropical depression.

10. Hurricanes move following a trochoidal trajectory even though the envelope in most of the cases becomes a parabola. For students, the connection between Trochoides and physical situations we meet every day may result interesting. Trochoid describe a family of curves. Trochoid is defined as the trace of a point fixed on a circle that rolls along a line. Sometimes the name trochoid is used to mean hypotrochoid and epitrochoid. (curve traced by rolling circle on another circle) More generally, trochoid is any curve that is the locus of a point fixed to a curve A, while A rolls on another curve B without slipping. • If the point lies on the rolling circle, it is a cycloid. • If the point lies outside the rolling circle, it is sometimes called prolate (extended) cycloid. • If the point lies inside the rolling circle, it is curtate (contracted) cycloid.

11. Uniform, Constant Electric and Magnetic Fields Uniform, Constant Electric Field The center of gyration will move in the direction of the y-axis with constant velocity w = cE/B. It will start from a position on the x-axis, since in the moving system we must assign an intial velocity -w along the y-axis so the particle is intially at rest in the space system. This determines the radius of gyration r = (cE/B)/ωc = mc2E/qB2. We now know that the center of gyration will move on the line x = r with constant velocity w. Since the velocity of the point in contact with the y-axis will always be -w with respect to the center of gyration, its instantaneous velocity in the space system will be zero. That is, the circle of gyration will roll without slipping on the y-axis. We know that any point on the circumference of the circle will trace out a cycloid. The cycloid will be traced on the other side of the y-axis. If the initial conditions are different, many possibilities result, and the path may be a hypocycloid or a prolate cycloid, making little loops, or just waving back and forth Motion in a uniform, constant electric field is motion with constant acceleration a = (q/m)E, which is in the direction of the field for a positive charge q and in the opposite direction for a negative charge. Choose the x-axis in the direction of the electric field, and the y-axis so that the xy-plane contains both the electric field and the initial velocity Uniform, Constant Magnetic Field

12. Spiral patterns in hurricanes

13. Connecting Meteorology and Astronomy Logarithmic Spirals Isabel and M51

14. Entry level student’s research projects The mathematics of weather phenomena: From water droplets to fractal snowflakes. Goals: To introduce student to the fascinating world of geometry and its applications. Nature is full of patterns, and how we recognize them helps us to understand the laws governing nature. Some of these patterns are geometrical in nature and comprise a wide set of phenomena: water droplets, rainbows, cloud shapes, lightning, runoff paths, snowflakes, hurricanes, crystals, and many others. This activity allows students to identify many of these patterns and link those with physical laws. Keywords: Patterns, geometry, circle, rectangle, sphere, cylinder, Euclid, branching processes, trees, fractals, arrangement, solids, crystals, liquid crystals, quasi-crystals, order, disorder, surface tension, gravity, wetting, seeds or crystallization center, packaging. Weather forecasting and statistics: The exciting world of uncertainties and central tendencies. Goals: In this activity, students will learn the basics of statistics and the reason why nature sometime prefer random variables or unpredictable processes? Why, after watching TV forecasting sometimes, weather results in something completely different? The answer for this question lies in the statistical character of the weather behavior. What are the differences between weather and climate? Which one of these two is a short term behavior which one a long term behavior? Why can we not predict weather for more than three days with high accuracy? Keywords: Mean, mode, median, standard deviation, variance, hypothesis testing, probability, random variable, sampling, data, correlation, regression, curve fitting, mining, forecasting, guessing, uncertainty, interval of confidence, outdoor temperature, pressure and humidity.

15. Statistics and the Frequency of Hurricane Landfalls in Continental USA