# William A.

## Tutor

### Largo, FL 33773

Will travel 10 miles

\$44 per hour.

4.6 20 ratings

### Let a Professor help you; I care that you succeed in Math and Science

I am an applied Physicist with 6 years of experience teaching and tutoring undergraduate physics lecture and laboratory, and more than 20 years in aerospace systems and electronics engineering, all with computer support. I am dedicated to student success, particularly with "at-risk" students. My teaching includes general physics step-wise methods of solution and unifying concepts based on physical principles and expressed by theory and mathematics. I have many years of experience as an eng... [more]

Algebra 1

Algebra is our introduction to that branch of mathematics that substitutes letters for numbers. An algebraic equation represents a balance scale; what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors, etc. Here in Algebra 1, we will work on the basics of algebraic concepts including; ? Slope of a straight line ? Exponents and Exponential Decay ? Fraction equations ? Functions ? Linear Equations and Inequalities ? Parabolas and Quadratic Equations ? Percent, Ratios and Proportions Also included are multiplying polynomials, the Distributive property for simplifying monomial expressions, the Symmetric Property of Equality (If a = b then b = a), how to solve Word Problems and problems involving distance, rate (speed) and time.

Algebra 2

Algebra is our introduction to the branch of mathematics that substitutes letters for numbers. An algebraic equation represents a balance scale; what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors, etc. Here in Algebra 2, we will work on these concepts and; ? Properties of functions and the algebra of functions, matrices, and systems of equations. Linear Equations and Inequalities, quadratic, exponential, logarithmic, polynomial, and rational functions and the analytic skills for solving them. ? Radical expressions and exponents. ? Binomial theorem. ? Graphing. ? Graph parabolas, ellipses and hyperbolas given the equations. ? In-depth graphing of systems of linear equations and inequalities, quadratics functions and equations. ? Probability and statistics using standard deviation measures to compare the scatter of sets of real-world data. ? Permutations and combinations. ? Mathematical modeling of real life applications.

Biology

Biology is the science of life, living and non-?living? organisms, and living matter, in all its forms and phenomena, including its structure, function, growth, origin, evolution, behavior and distribution. It includes Botany, the study of plants; Zoology, the study of animals; and Microbiology, the study of microorganisms and all their subdivisions. Having a Ph.D. Biologist as my parent has been invaluable in understanding Biology and its principles. Some of my first jobs were as a junior Field Technician doing Biological research for the University of Illinois. My own PhD work, while in microscopy, was heavily influenced by the cells, cancers and microorganisms I was imaging, in addition to the training I received at Illinois in cellular mechanics. Major topics include; the philosophy of science, scientific method, chemical organization and constituents of life, cell biology, molecular genetics, molecular evolution and biodiversity. An introduction to higher levels of biological organization from the organism to the ecosystem is presented. Additional topics include; organismal structure and physiology, behavior, population ecology, community ecology, ecosystem ecology and environmental biology. This biology material involves the scientific study of living organisms and considers the interactions among the vast number of organisms that inhabit planet Earth. Herein, it presents the basic form and function of these organisms, from cells to organ systems, from simple viruses to complex humans. More detailed specific topics include; basic chemistry, biomolecules; bioenergetics, kinetics, enzyme catalysis, cellular respiration and metabolism, photosynthesis, nutrients and tropism, DNA structure and replication, protein synthesis, regulation of gene expression, genetic engineering, and cancer. Also available are; gas exchange, translocation mechanisms, circulation, hormones, excitable cells, nerve cell function, muscle cell function, regulation and homeostasis. Cell topics include; differentiation, metamorphosis and regeneration, embryonic development, cell cycles, mitosis and meiosis, Mendelian inheritance, gene expression and gene interactions. Macro-principles include; human inheritance, mechanisms of evolution, natural selection, adaptation, population genetics, speciation, sexuality. Discussed in environmental biology are; the biosphere, communities and community succession, energy and productivity, mineral cycles, community structure, population dynamics, population growth and reproductive strategies, population regulation, human populations and human intrusion in ecosystems.

Calculus

Here in calculus, we examine the key concepts of the limit, the derivative, and continuity, as well as their main applications in graphing and optimizing functions. From there, we will go on to explore the fundamental theorem of calculus, which leads to the concept of integration and integrals, and eventually study polynomial approximations and series. Logarithmic, exponential, and other transcendental functions are included. My experience with differential and integral calculus covers undergraduate Calculus I, II and III, and extends well into their advanced applications, some of which I inevitably used in my doctoral research. Throughout my collegiate education, I have always come across some idea or technique associated with integral or differential calculus and used many of the techniques covered herein in my everyday employment in the aerospace industry. In this study of calculus, we want to start by fostering an intuitive understanding of the limiting process (including one-sided limits) based upon functions, estimating limits from graphs or tables of data, and calculating limits using algebra. We will also understand asymptotes and unbounded behavior in terms of graphics and describe asymptotic behavior in terms of limits involving infinity. An intuitive understanding of continuity as a property of functions in terms of limits, and the Intermediate Value and Extreme Value theorem are also anticipated. The concept of the derivative is presented graphically, numerically and analytically and interpreted as an instantaneous rate of change. The derivative is identified as being the slope of a curve at a point, a tangent line to a curve, and a local linear approximation, in addition to being the instantaneous rate of change as the limit of the average rate of change. The derivative is also shown to be a function, and the corresponding characteristics of the graphs of ? and ??, the relationship between the increasing and decreasing behavior of ? and the sign of ??, the 1st derivative test, Rolle?s theorem and the mean value theorem with their geometric interpretations are examined. Second derivatives, the corresponding characteristics of the graphs of ?, ?? and ???, and the relationship between the concavity of ? and the sign of ?? are also demonstrated. Techniques are presented for computation of derivatives of basic functions, including power, exponential, logarithmic, and trigonometric functions, as well as differentiation rules for sums, products and quotients of functions, the chain rule and implicit differentiation. Applications of derivatives including analysis of curves/sketching, the notions of monotonicity and concavity, optimization, modeling rates of change, and interpretation of the derivative as a rate of change are presented as well as L?Hospital?s Rule for determining limits and convergence of improper integrals and series. Basic properties of definite integrals (including additivity and linearity) with interpretations, the definite integral as a limit of Riemann sums, and the definite integral of the rate of change of a quantity over an interval are presented. Use of the Fundamental Theorem of Calculus to unite differential and integral calculus, evaluate definite integrals and represent particular antiderivatives is demonstrated. Techniques of antidifferentiation, including antiderivatives following directly from derivatives of basic functions, antiderivatives by substitution of variables parts, simple partial fractions, improper integrals, indefinite integration, and calculating volumes through solids of revolution are shown. Applications of antidifferentiation, including the finding of specific antiderivatives using initial conditions, motion along a line, solving separable differential equations, use in modeling, as well as Riemann and trapezoidal sums to numerically approximate definite integrals are presented. The concept of series as a sequence of partial sums and convergence is defined in terms of the limit of partial sums. Series of constants forming a geometric series (with applications), the harmonic series, terms of series as areas of rectangles, their relationship to improper integrals, and the ratio test for convergence and divergence are shown. Taylor polynomial approximation with graphical demonstration of convergence, Maclaurin series for various functions, the general Taylor series centered at x = a, and power series functions are defined. The natural logarithmic function, inverse functions, exponential functions their differentiation and integration, differential equations for growth and decay and other transcendental functions are included as necessary.

Chemistry

Chemistry is the branch of physical science concerned with the composition, structure, properties, and reactions of matter, and with the transformations that they undergo, especially of atomic and molecular systems. The topics studied herein include atomic theory, stoichiometry, chemical bonding, thermochemistry, periodicity, solution chemistry and selected topics in descriptive chemistry. Much of my chemical background was learned on-the-job (OJT) while operating and designing semiconductor processes. My academic instruction included undergraduate atomic theory, solid state/thermodynamics, radioactivity/nuclear science, and high school college prep chemistry. Specific material presented herein includes: 1)Matter and Measurement a. SI units, scientific notation b. linear equations, graphing, ratio and proportion c. significant digits, dimensional analysis 2) Atoms, Molecules and Ions a. mass/atomic number; mass/atomic number; isotopes/half-lives/nuclear particles; particle/mass charge; families/groups; series/periods chemical/physical properties b. naming and writing chemical formulas c. molar mass- measurements of gas density, freezing and boiling-points d. percentage composition-- Mole fractions; molar and molal solutions 3) Chemical Reactions a. reaction types -- synthesis, decomposition, single and double, precipitation replacement, Oxidation number, oxidation-reduction b. acid/base theories: Arrhenius, Br?nsted-Lowry; amphoterism c. electrochemistry i. Faraday?s laws ii. standard half-cell potentials; Nernst equation iii. prediction of the direction of redox reactions iv. strong/weak electrolytes, dissociation/ionization (pH, pOH) 4) Stoichiometry a. net ionic equations; presence of ionic and molecular species b. balancing equations c. molar mass and volume relations d. limiting reagent/reactant, percent yield; titration e. balanced equations express conservation of energy and matter 5) Energy and Chemical Reactions - Thermodynamics a. conservation of energy; State functions b. First law: change in enthalpy; heat of formation; heat of reaction; Hess?s law; heats of vaporization and fusion; calorimetry c. Second law: entropy; free energy of formation; free energy of reaction; free energy vs. enthalpy and entropy changes d. relationship of change in free energy to equilibrium constants and electrode potentials 6) The Structure of Atoms a. historical/quantum models b. periodic Trends i. atomic/nuclear radii, electronegativity, shielding effect ii. electron energy levels: atomic spectra, quantum numbers, atomic orbitals iii. electron configurations/oxidation numbers; ionization energies c. periodic chart 7) Chemical Bonding a. bonding Theories b. types: ionic, covalent, metallic, hydrogen bonding, van der Waals (including London dispersion forces), binding forces c. polarity of bonds, electronegativities d. electron dot diagrams e. resonance forms f. organic bonding models g. relationships to states, structure, and properties of matter 8) Molecular models and Molecular Structure a. valence bond: hybridization of orbitals, resonance, sigma and pi bonds b. valence shell electron pair repulsion (VSEPR) c. molecular structure and geometric shape i. structural isomerism and coordination complexes ii. dipole moments of molecules 9) Carbon-Select Topics a. organic chemistry b. hydrocarbons and functional groups c. relationship to biochemistry 10) States of Matter a. gases i. pressure, temperature, and volume ii. gas laws - Ideal, Dalton?s and Graham?s; Equation of state iii. vapor pressure; partial pressures b. kinetic molecular theory i. Avogadro?s hypothesis and the mole concept ii. dependence of kinetic energy on temperature c. intermolecular forces and liquids i. molar heats of fusion and vaporization ii. specific heat capacity iii. changes of state, including critical and triple points d. solids - structure of solids; lattice energies e. solutions and their behavior i. solution concentrations ii. Raoult?s law, colligative properties (nonvolatile solutes); osmosis 11) Chemical Kinetics: The Rates of Chemical Reactions a. catalysis, reaction rates and kinetics, order b. temperature dependence c. activation energy d. rate-determining step and mechanism relationships 12) Reactivity Principles a. equilibrium; simultaneous equilibria; Chatelier?s principle b. equilibrium constants for gaseous reactions: Kp, Kc c. aqueous equilibria, acid/base theory; constants for acids & bases; pK, pH d. solubility product constants and precipitation e. dissolution of slightly soluble compounds f. ion effects; buffers; hydrolysis g. Entropy and Free Energy; Electron Transfer Reactions 13) Chemistry of the Main Group Elements and of the Transition Elements a. families/groups; series/periods b. trends/patterns on chart: horizontal, vertical and diagonal 14) Nuclear Reactions a. nuclear equations b. nuclear chemistry

Microsoft Excel

General Computer

Tailored to the individual student?s needs and requirements, General Computer instruction is designed to address basic computer skills necessary for the use of today?s tools. Material covered includes basic hardware features, specific items such as memory and storage, operating systems and modes, networks and wireless, basic internet, browsers and search engines and introductory computer training not covered under other subject sections.

Geometry

Geometry describes mathematically the properties, measurement, and relationships of points, segments, lines, angles, surfaces, triangles, polygons, circles and solids. Through their introduction, the student develops an understanding of the theorems and postulates that form the foundation of geometry. The concepts of Euclidean geometry are stressed through definitions and deductive proofs. Emphasis is placed on the description and use of inductive, deductive, and intuitive reasoning skills. Geometry is studied as a mathematical system through the deductive development of relationships in the plane and space developed intuitively in previous years. Powers of abstract reasoning, spatial visualization and logical reasoning patterns are improved through this study. The focus is on comparisons between these objects concerning surface areas, volumes, congruency, similarity, transformations, and coordinate geometry. The structures studied include congruent segments and angles, circle chords, secants and tangent segments, parallel and perpendicular lines, angle measure in triangles, direct and indirect triangle congruence and similarity, proofs, solids of revolution, logic, similar triangles, transformations, the Pythagorean Theorem, geometric constructions, coordinate geometry, and surface area and volume of solids. This includes the study of transformations and right triangle trigonometry. Inductive and deductive thinking skills are used in problem solving situations, and applications to the real world are stressed. Also emphasized is writing proofs to solve (prove) properties of geometric figures. Algebra I skills are used throughout.

Microsoft Word

Physics

Physics (Physics / General Physics) is the branch of science that deals with matter, energy, motion, and force. More specifically, the science of matter and energy, and of their interactions, grouped in traditional fields such as acoustics, optics, mechanics (statics and dynamics), thermodynamics, and electromagnetism, as well as in modern extensions including atomic and nuclear physics, cryogenics, solid-state physics, particle physics, and plasma physics. This material provides a systematic introduction to the main principles of physics and emphasizes the development of conceptual understanding and problem-solving ability using algebra and trigonometry, and calculus when requested. In any course of instruction that addresses physics, the student is often asked to think and make decisions in ways and situations that they might not have done so previously. This is accomplished by using applied mathematics in accordance with a set of rules, laws and ?physical? principles. Providing sufficiently deep insight and coaching successful performance is often left as the responsibility of the physics tutor. In order to hopefully simplify physics instruction, homework and exam questions, many students and instructors categorize physics problems into one of two classes; the conceptual and the solution-oriented. The solution-oriented approach, when necessary, proceeds in a structured, step-wise fashion, i.e., a problem statement, a physical diagram, identified ?knowns? and unknowns, selection of governing equations and principles, etc. Step-wise ?solution-type? methods are easily taught (and even memorized), and key features are knowing when to use which tool, specific approach or equation. For a conceptual-type problem, the method of solution can often be an ?acquired taste,? akin to solving proofs in mathematics, requiring a study of similar situations, and even ?reflective? thought. To meet our student?s needs, this material is an algebra and trigonometry-based study of classical mechanics, [linear and rotational kinematics and dynamics (including work, energy, impulse, momentum, and collisions)], fluids, heat, thermodynamics, periodic motion, wave motion, vibrations and sound, electricity, electrostatics, electric fields, Gauss' law, capacitance, current, resistance, magnetic forces and fields, electromagnetic induction, DC and AC circuits, electromagnetic waves, mirrors, lenses, geometrical optics, and modern physics. Additional topics include vector algebra, motion in a plane, internal energy, gravity, angular momentum, conservation laws, measurement concepts, heat and electrical circuits, relativity, atomic structure, the nucleus, fundamental particles and nuclear physics. When requested, calculus-based topics include dynamics of single and many-particle systems, circular and rigid body motion, elasticity, and the laws of thermodynamics with applications to ideal gases, thermodynamic processes, electromagnetism and optics. Modern physics, based on quantum theory, includes atomic, nuclear, particle, and solid-state studies. Physics also embraces many other applied fields such as geophysics and meteorology.

Microsoft PowerPoint

Prealgebra

Algebra is that branch of mathematics that substitutes letters for numbers. An algebraic equation represents a balance scale; what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors, etc. Pre-algebra explores mathematical concepts that are foundational for success in algebra including the fundamentals of arithmetic, algebraic expressions, positive and negative integers, rational numbers, equations, decimals, fractions, ratios, proportions, percents, area, volume, and probability. Students will develop and expand problem solving skills (creatively and analytically) in order to solve word problems and continue to practice using math skills to solve word problems and think creatively and analytically. Pre-algebra introduces beginning topics in number theory and algebra, including common divisors and multiples, primes and prime factorizations, basic equations, expressions and inequalities, ratios, Squares, Cubes, and Higher Exponents, Multiples and Divisibility Tests, Primes and Prime Factorization, Least Common Multiple, Greatest Common Divisor, Advanced Equations and Word Problems, Speed and Rates, Percent Increase and Decrease, Angles and Parallel Lines, Angles and Polygons, Perimeter and Area, Circles, Pythagorean Theorem, Special Triangles and Quadrilaterals, Tables, Graphs, Charts and Counting as Arithmetic. Pre-algebra includes square roots, a thorough exploration of geometric tools and strategies, an introduction to topics in discrete mathematics and statistics, and a discussion of general problem-solving strategies. After completing pre-algebra, students will be able to: ? evaluate numerical and algebraic expressions using order of operations; ? use basic operations (addition, subtraction, multiplication, division, and absolute value) with integers and rational numbers; ? convert within metric and other customary measuring systems; ? apply properties to find missing angle measures in triangles; ? find area of circles, rectangles, triangles, and trapezoids; ? find perimeter of polygons and circumference of circles; ? find volume and surface areas of rectangular and other geometric solids; ? locate ordered pairs on a coordinate plane; ? graph inequalities and linear equations on a coordinate plane; and ? solve single-operation equations containing whole numbers, integers, and rational numbers. Successful completion of this material prepares students for success in Algebra 1.

Precalculus

The main goal of Pre-calculus is for students to gain a deep understanding of the fundamental concepts and relationships of functions. Students will expand their knowledge of quadratic, exponential, and logarithmic functions to include power, polynomial, rational, piece-wise, and trigonometric functions. Students will investigate and explore mathematical ideas, develop multiple strategies for analyzing complex situations, make connections between representations, and provide support in solving problems. Students will analyze various representations of functions, sequences, and series. Discrete topics include the Principles of Mathematical Induction and the Binomial Theorem. Students will analyze bi-variate data and data distributions and apply mathematical skills to make meaningful connections with real-world situations. Pre-calculus applies modeling, and problem-solving skills to the study of trigonometric and circular functions, identities and inverses, and the study of polar coordinates and complex numbers. Vectors in two and three dimensions are studied and applied. Problem simulations are explored in multiple representations?algebraic, graphic, and numeric. Quadratic relations are represented in polar, rectangular, and parametric forms. The concept of the limit is applied to rational functions and to discrete functions such as infinite sequences and series. The formal definition of limit is applied to proofs of the continuity of functions and provides a bridge to the calculus. Instructional material includes a review of algebraic concepts and skills: the real and complex number systems, polynomials, algebraic fractions, exponents and radicals, linear and quadratic equations, inequalities, rectangular coordinate systems, lines and circles; Linear and quadratic equations and inequalities; graphs of equations, including lines, circles, parabolas; composition, inverses of functions; transformations of graphs; linear and quadratic models; equations and inequalities involving polynomials and rational functions; exponentials and logarithms with applications; trigonometric functions and inverse trigonometric functions: definitions, graphs, identities; real and complex zeros of polynomials; polar coordinates; DeMoivre's Theorem; conic sections; solutions of systems of equations by substitution and elimination; systems of inequalities; arithmetic sequences and geometric series. A summary of specific tasks includes: A. Compute with matrices and use matrices to solve problems. B. Analyze the behavior of sequences and series. C. Analyze and solve problems using polynomial functions. D. Model and graph functions and transformations of functions. E. Analyze the behavior of functions. F. Solve problems using trigonometry. G. Graph curves using polar and parametric equations. H. Solve problems involving the geometric properties of conic sections. I. Compute probabilities for discrete distributions and use sampling distributions to calculate approximate probabilities. J. Analyze bi-variate data using linear regression methods. Further emphasis is given to exponential form of complex numbers, geometric representation of complex numbers, roots of unity, applications of complex numbers to geometry, two-dimensional and three-dimensional vectors and matrices, determinants, dot and cross product, applications of vectors and matrices to geometry; probability, circular functions, and two- and three-dimensional vectors. The concepts of limit, derivatives, and power series are introduced. Trigonometry concepts such as Law of Sines and Cosines will be introduced. Students will then begin analytic geometry and calculus concepts such as limits, derivatives, and integrals.

Trigonometry

Trigonometry is the branch of mathematics that studies the properties of the sides and angles of plane or spherical triangles, the trigonometric functions, the calculations based on them and their applications. This course focuses entirely on plane trigonometry. Specifically addressed are; Measures of angles, Coordinate in a Plane, Applications of Special angles, The four solutions (cases) of an oblique triangle, the Laws of Sines, Laws of Cosines, graphical representations of the trigonometric functions, graphs of sin x and cos x, graphs of tan x, cot x, sec x, csc x; Fundamental Identities, Proving Trigonometric Identities, Inverse Trigonometric Functions, Trigonometric Ratios in Right Triangles, Trigonometric Functions on the Unit Circle, Linear and Angular Velocity, Translations of Sine and Cosine Functions, Modeling Real World Data with Sinusoidal Functions, Trigonometric Inverses and Their Graphs, Sum and Difference Identities, Double-Angle and Half-Angle Identities, Solving Trigonometric Equations, Normal Form of a Linear Equation Trigonometric functions have applications in the physical and life sciences. The six trigonometric functions which are defined in terms of ratios are used routinely in calculations made by surveyors and navigators. Procedures using trigonometric tables and those using a calculator are included as needed to solve problems. Triangle solution problems, trigonometric identities, and trigonometric equations require knowledge of elementary algebra.

Astronomy

There are certain subjects, like Astronomy, that study not only ?the world?, actually ?The Universe?, in which we live, they also invoke spectacular imaginations to students of all levels. I feel very much the same sort of awe; from the books and toy telescopes that I received as a child, to the telescopes and space programs I work on today. Having worked on missions such as the NEAR-Shoemaker Asteroid Rendezvous for The Johns Hopkins University and the Mars Observer for RCA, I feel that my perspective is somewhat uniquely qualified by my own personal experiences and achievements. Being a physicist certainly helps as well. I am happy to share with my students the following (academic) topics; The Field of Astronomy, Units and Measurements, Day and Night; Rotation of the Earth; The Seasons; Phases of the Moon; Eclipses, Constellations; Planet Motions; The Scientific Method, Earth-centered and Sun-centered Models of the Solar System; Brahe and Galileo's observations; Kepler's Laws, Newton's Laws of motion; Gravitation, Orbital Mechanics, Light, Blackbody Radiation, The Big Bang, Formation of Elements, Different Kinds of Radiation, The Physics of Space Plasmas, Discovery of the Galaxy and the Vastness of Space ? Galaxies and Expansion of the Universe, Age and Origin of the Solar System - Discovery of the Solar System, Clues from Meteorites, Clues from Comets, Methods of Observational Astronomy - Telescopes, Spectroscopy and Stars, Measuring Distances to Stars, star charts and nebulae, Observatories and Radio Telescopes, The Sun - The Electromagnetic Spectrum, The Sun?s Structure, Nuclear Fusion, Life on Earth, Planets of the Solar System - The Jovian Planets, The Terrestrial Planets, The Earth in Space - Introduction to Earth, Comparing Earth to other Terrestrial Planets, The Search for Extrasolar Planets - First Discoveries, Are there other Earth-like planets?, Search methods, Mars Observation and Missions; Atmosphere and Geology, Characteristics of Stars; Star Birth; The Lengths of Star Lives, The Main Sequence; Star Death, The Milky Way; Black Holes; Astrophysics and Cosmology, Dark Energy and Matter, Predictions for the Future of our Universe; Are We Alone in the Universe?

Statistics

Statistics has to do with the collecting and analyzing of numerical data in large quantities. Topics include sampling and experimental design, sampling errors, descriptive statistics, basic probability (random variables, expected values, normal and binomial distributions); binomial and normal distributions, estimation, hypothesis testing (including Type I and Type II errors) and confidence intervals for means, proportions, and regression parameters; descriptive methods in correlation, and simple linear regression; analysis of variance, Student's t-test, percentiles, and z-scores. Important principles include; 1. Exploratory analysis of data making use of graphical and numerical techniques to study patterns and departures from patterns. 2. The realization that data must be collected according to a well developed plan if valid information on a conjecture is to be obtained. 3. The realization that probability is a valuable tool for anticipating what the distribution of data should look like under a given model. 4. The fact that models and data interact in statistical work. Models are used to draw conclusions from data. Inference from data is a process of selecting a reasonable model, including a statement in probability language, of how confident one can be about the selection.

SAT Math

GRE

Physical Science

The definition of what constitutes Physical Science may change from program to program; often consisting of just (some) chemistry and (some) physics. I prefer a broader definition such as; "Physical Science includes astronomy, atmospheric science, chemistry, geology, physical geography and physics." Significant themes include the Scientific Method, Scientific Measurements and Tools of Science. An introduction to Astronomy is treated as a broad survey of modern astronomy examining the solar and stellar systems, an overview of the structure and motion of comets, asteroids, the planets and their natural satellites, and an examination of our present understanding of the nature, origin and evolution of the sun, stars, galaxies, and special objects. Opportunities include learning about lenses and mirrors, construction and use of telescopes, how to make measurements, and how to read star charts and locate objects in the heavens. Atmospheric Science is an introductory examination of the Earth's weather and climate. This segment covers a broad range of topics including the origin, composition, and structure of the atmosphere; the formation of clouds and precipitation; the formation of organized weather systems; weather prediction; air pollution; and climates. The foundations of Chemistry segment addresses matter, its composition and the changes it undergoes. This segment also examines the influence of chemistry on society through studies on topical subject areas in chemistry such as energy, environmental and health issues. This material provides a broad background in general chemical principles including Chemical vs. Physical changes, compounds, formula naming, and fundamental concepts in inorganic chemistry. Atoms, atomic theory and structure, organization of the Periodic Table, and chemical bonding are also presented as well as chemical reactions and their balancing chemical equations. Stoichiometry, gas laws, solutions, equilibria, redox, acid-base theory, pH scale, fundamental concepts in organic chemistry, and nuclear chemistry are also topics. The focus of the introduction to Geology segment is on the physical composition of the Earth and the dynamic processes that affect it. Topics covered include plate tectonics, mountain building, volcanoes, earthquakes, glaciers, rivers, minerals, and rocks. Physical Geography is an introduction to the geographical features of the Earth's natural environment. This segment examines the physical, chemical and biological processes that shape these features and control their spatial distribution; the dependence of human society on the natural environment; and the ways in which humans intentionally and unintentionally modify the natural environment. An algebra/trigonometry-based study of Physics includes matter and energy, force and motion, kinematics, Newton's Laws, Universal Gravitation, simple machines, dynamics, momentum, conservation and transformation of energy. Rotational motion, mechanics, and forces in fluids such as Bernoulli's Principle and Archimedes' Principle are presented. Fundamental concepts of the Kinetic Theory dealing with molecular motion and energy, gas laws, stresses and strain in materials and the phases of matter are addressed. Topics include measurement, work, power, heat and heat transfer - conduction, convection, and radiation. Also considered are thermodynamics, waves, sound, light, mirrors and lenses, color, fundamental concepts of static and dynamic electrical charges and their characteristic behavior. Electricity and magnetism, circuits, magnets, computers, and Nuclear Reactions, Decay and Half-Life, Fission and Fusion Reactions, and Issues with Nuclear Energy are covered. With an understanding of this material, students should be able to demonstrate an understanding of the physical environment and be able to apply the scientific principles to observations experienced. The role of the student here is to develop inquiry and problem solving skills within the context of scientific investigation. Students will apply what they learn to everyday situations by conducting investigations and formulating and testing their own hypotheses.

ASVAB

Probability

The probability of an event is a measure of the chance that the event occurs; namely the relative possibility that an event will occur, as expressed by the ratio of the number of actual occurrences to the total number of possible occurrences. This instruction provides an elementary introduction to probability with applications. I?ve learned about probability in various places including; on the job training (OJT) at Johns Hopkins Applied Physics Laboratory, Six Sigma training in Fortune Five aerospace, and collegiate Thermodynamics/ Solid State Physics instruction. Quantifying such a complex and illusive subject is no small success for the mathematics and, a point of great satisfaction for me. Topics in this instruction include: the theory of probability and its applications; basic probability models (coin and dice tosses, cards, etc.); Venn diagrams, tree diagrams, odds, combinatorics (permutations, nPk and combinations, nCk); the axioms of probability, conditional probability and independence of events; sample spaces; random variables; the Strong and Weak Laws of Large Numbers, discrete and continuous random variables/vectors and probability distributions; Pascal's triangle, expected values, mean, standard deviation, Gaussian distributions, standardized test scores, statistical estimation and testing; Poisson and related distributions; joint, marginal, and conditional densities, moment generating function; binomial, univariate, and bivariate normal distributions; confidence intervals, correlation, and limit theorems; and introduction to linear regression. Associated material; Probability? Sample Spaces and the Algebra of Sets, The Probability Function, Combinatorial Probability, Axiomatic probability, a priori probability, Bayes' theorem, likelihood function, posterior probability distribution. Random Variables? Hypergeometric probabilities, the variance, joint densities, combining random variables, further properties of the mean and variance, order statistics, random sampling and sampling distributions. Special Distributions? The Geometric distribution, the Negative Binomial distribution and the Gamma distribution. The Normal Distribution? Inferences about a population mean; Normal, t, ?2 and F distributions. This course should also impart to the student the important idea that real phenomena can be modeled stochastically using random variables/vectors and their distributions. These modeling aspects, can be imparted through computer simulations, real experiments, and the use of historical data, and should make the course very useful to students in the physical, engineering, biological and social sciences. Calculus-level proofs of important results are presented or outlined.

PSAT

ACT Math

The ACT Mathematics Test is a 60-question, 60-minute test designed to measure the mathematical skills that students have typically acquired by the end of 11th grade. The test presents multiple-choice questions that require one to use reasoning skills to solve practical problems in mathematics. You aren't required to know complex formulas and perform extensive computation, but you do need knowledge of basic formulas and computational skills to answer the problems. All of the problems can be solved without a calculator. I took this test during High School and found the mathematics, while challenging, to be quite tractable. Preparing for this test should put you in quite good shape. Note: unless otherwise stated on the test, one should assume that: 1. Figures accompanying questions are not drawn to scale. 2. Geometric figures exist in a plane. 3. When given in a question, ?line? refers to a straight line. 4. When given in a question, ?average? refers to the arithmetic mean. The format of the ACT Math Test is straightforward; ACT simply lumps all of the problems into one big list of math questions. There are two types of questions: basic problems and word problems. Word problems tend to be more difficult than basic problems simply because they require the additional step of translating the words into a numerical problem that you can solve. Of course, a basic problem on a complex topic will still likely be more difficult than a word problem on a very easy topic. Content that could appear on the ACT Mathematics Test. In the Mathematics Test, three sub-scores are based on six content areas as follows: Pre-Algebra/Elementary Algebra ? Pre-Algebra (23%). basic operations using whole numbers, multiples & primes, decimals, fractions, and integers; divisibility, remainders and place value; square roots and approximations; the concept of exponents; scientific notation; factors; ratio, proportion, and percent; number problems, linear equations in one variable; absolute value and ordering numbers by value; elementary counting techniques and simple probability; series, data collection, representation, and interpretation; mean, median, & mode and understanding simple descriptive statistics. ? Elementary Algebra (17%). properties of exponents and roots, evaluation of algebraic expressions through substitution, using variables to express functional relationships, understanding algebraic operations, writing expressions & equations, simplifying algebraic expressions, multiplying binomials and the solution of quadratic equations by factoring. Intermediate Algebra/Coordinate Geometry ? Intermediate Algebra (15%). Relationship between sides of an equation, an understanding of the quadratic formula, rational and radical expressions, logarithms, absolute value equations and inequalities, sequences and patterns, systems of equations, quadratic inequalities, functions, modeling, matrices, roots of polynomials, and complex numbers ? Coordinate Geometry (15%). graphing and the relations between equations and graphs, number lines & inequalities; the (x,y) coordinate plane including points, lines, polynomials, circles, and other curves; graphing inequalities; slope; parallel and perpendicular lines; distance; midpoints; and conics. Plane Geometry/Trigonometry ? Plane Geometry (23%). the properties and relations of plane figures, including angles and relations among perpendicular and parallel lines; properties of circles, triangles, rectangles, parallelograms, trapezoids and polygons; transformations; the concept of proof and proof techniques; volume; and applications of geometry to three dimensions. ? Trigonometry (7%). understanding trigonometric relations in right triangles; values and properties of trigonometric functions (SOHCAHTOA); solving triangles; graphing trigonometric functions; modeling using trigonometric functions; use of trigonometric identities; and solving trigonometric equations. Calculators may be used for any problem on the test, but could be more harm than help for some questions. You are responsible for knowing if your calculator model is permitted. If the test staff finds that you are using a prohibited calculator or are using a calculator on any test other than the Mathematics Test, you will be dismissed and your answer document will not be scored. In terms of time allotted per question, the Math Test ranks the highest among the subject tests, averaging one minute per question. The drawback is that the Math Test is relatively difficult for most people. It tests learned knowledge, not just intuitive knowledge.

ACT Science

Discrete Math

Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. Often called ?Everything except calculus,? the term ?discrete mathematics? is therefore used in contrast with ?continuous mathematics,? which is the branch of mathematics dealing with objects that can vary smoothly. From my 25 years in the aerospace electronics industry, I have been fortunate to have used the following discrete mathematics techniques in my career, namely: Having worked extensively with the design and production of microcircuits, I?ve used boolean algebra (a ?Discrete Algebra?) and logic extensively for understanding digital gates and programming. My work with large databases acquainted me with relational algebras and I?ve used linear algebra (matrices and arrays) in situations such as image processing and mechanics. As a Reliability Engineer for complex space electronic systems, I developed an in-depth understanding of Combinatorics (permutation, combination and partitioning) and calculation of probabilities in order to produce failure rates, Mean-Time Between Failures (MTBF) and to conduct Failure Modes, Effects and Criticality Analyses (FMECA). As a Systems Engineer, I?ve used Program Evaluation and Review Technique (PERT) and Operations Management Theory, including Gantt chart analyses and Linear Programming, to model many cost and schedule relations and process dynamics for Highly-Capable/Highly-Reliable aerospace programs. This required a sophisticated understanding and implementation of Graph and Network Theory. The largest cost account that I defined and managed was \$13.4m in Labor and Material. As a Technical Entrepreneur, I?ve performed Strategic Marketing Analyses which requires a visceral understanding of Utility Theory in order to evaluate quantitative measures of the relative economic satisfaction from the consumption of various goods and services. I am more than happy to share these practical applications of Discrete Mathematics, and the theory behind them, with my students.

Biostatistics

Biostatistics is a powerful and productive application of statistics to biological systems. Statistics has to do with the collecting and analyzing of numerical data in large quantities. Biology is the science of life, including its structure, function, growth, origin, evolution, behavior and distribution and most biological data has to deal with large numbers. Applications encountered can be anything from the number of Escherichia coli in a set of water samples to deer populations grazing along power lines. I had a job once with the University of Illinois as a Field Technician on an Environmental Impact Study. The work entailed collecting and processing soil samples that contained ground mites and then counting the various microscopic mite distributions to see if there was any difference between our test and control sites. What we were looking for was any manifestation of statistical differences between the test and control population distributions. You may also wish to see my WyzAnt certifications for statistics (and probability) and biology along with their subject descriptions and my general profile for more information.

Linear Algebra

Linear algebra is the branch of mathematics that is concerned with mathematical structures such as matrices and determinants. This includes the theory of systems of linear equations, vector spaces, and their transformation properties. The structures are called ?closed? under the operations of addition and scalar multiplication because using these two operations on these structures simply results in a similar structure. Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, analytic geometry, as well as many other problems in mathematics, physics, and engineering. I have had extensive experience using, teaching and training in linear algebra throughout my career in areas ranging from electrical engineering, computer science and practical mathematics applications, to advanced mechanics, numerical methods and group theory. I?ve taught basic linear algebra in order to deal with the geometries for classical mechanics and linear system techniques used for Kirchhoff?s Equations in circuit analysis. My doctorate involved the digital acquisition and fast Fourier transformation of large video arrays involved in image processing. My graduate studies at The Johns Hopkins University included readings and lessons out of Arfken?s, Mathematical Methods for Physicists, and Goldstein?s Classical Mechanics. My teaching, coursework and research with linear algebra has included, but is not limited to; Vector Spaces. Vectors in 2-space and 3-space - Geometry of vectors; Dot Product & Cross Product; Inner products and spaces, orthogonal projection, Cauchy-Schwarz and triangle inequalities. Euclidean n-Space; Linear Transformations;Subspaces - vector spaces that live inside of other vector spaces; Subspaces of Rn;Linear Independence (or dependence); Basis and Dimension; Change of Basis; Orthonormal Basis - the Gram-Schmidt process; Least Squares, QR-Decomposition; Linear combinations of vectors - the column space; The Nullspace of A: Solving Ax = 0;The Rank and the Row Reduced Form; Affine subspaces and the set of solutions to Ax=b; Dimensions of the Four Subspaces Solving systems of linear equations and matrices. Vectors and Linear Equations The Idea of Elimination - Elimination Using Matrices; reduced row-echelon form. Basic theorems on the existence and uniqueness of solutions. Matrix notation; Rules for Matrix Operations; Matrix Arithmetic & Operations; transpose and trace of a matrix. Special Matrices - Diagonal, Triangular and Symmetric matrices;Inverse Matrices and Elementary Matrices; Finding Inverse Matrices; Transposes and Permutations Augmented matrices and row operations Gaussian Elimination and Gauss-Jordan Method LU-Decompositions - a way of ?factoring? certain kinds of matrices. A = LU; The lower triangular L holds all the forward elimination steps, and U is the matrix for back substitution. Linear transformations, their algebra and geometry. The Idea of a Linear Transformation; The Matrix for a Linear Transformation; Diagonalization and the Pseudoinverse; Representing linear transformations from Rn to Rm by m by n matrices. Rotations, reflections, projections. Composition of linear transformations and matrix multiplication. Injections, surjections, isomorphisms. Inverses. Representing linear transformations in different coordinate systems Determinants. Inverses, pivots and volumes in n-dimensional space; Permutations and Cofactors; Using Row Reduction to Find Determinants; Cramer's Rule, Inverses, and Volumes Eigenvalues and Eigenvectors. Applications to Differential Equations; Symmetric Matrices; diagonalizing a symmetric matrix. Positive Definite Matrices; Similar Matrices; Singular Value Decomposition (SVD); the diagonalization of any matrix. Complex Vectors and Matrices. Complex Numbers. Hermitian and Unitary Matrices; The Fourier matrix F and the Fast Fourier Transform (multiplying quickly by F and F-1) Applications. Matrices in Engineering ? differential equations replaced by matrix equations; Graphs and Networks?leading to the edge-node matrix for Kirchhoff?s Laws; Linear Programming?minimization of the cost; Fourier Series?linear algebra for functions and digital signal processing; Matrices in Statistics and Probability; Computer Graphics ? matrices move and rotate and compress images. The inertia tensor.

Electrical Engineering