Post-Doctorate Tutor Puts Math To Work  Calculus and Physics

Mohamed E.

Algebra 1

Algebra 1 requires 75 hours to cover the following 11 topics: 1. Linear Equations 2. Linear Inequalities 3. Graphing Linear Functions 4. Writing Linear Functions 5. Solving Systems of Linear Equations 6. Exponential functions and sequences 7. Polynomial equations and factoring 8. Graphing Quadratic functions 9. Solving Quadratic equations 10. Radical functions and equations 11. Data Analysis and displays Algebra 1 is warming up prelude to the advanced branches of mathematics. It forms the foundation for expressing conceptual idea into literal notation. Algebra 1’s syllabus emphasizes representing equations of functions in graphic formats. That would ingrain visual impressions of mathematical abstract into the mind of the learner. Students will develop confidence in using mathematical symbols to express the thoughts and intentions in a concrete, simple, and universal algebraic formulation. From here, students will part away from laymen by their newly acquired literal depiction of the real world around them. My main emphasis, in teaching Algebra I, is to ensure that my students acquire visual prediction of functions by their power, domain, range, and behavior. That is easy said than done, but the trick lies in the simplicity of lines versus curves, how straight lines have power of one. Curves have greater or lesser powers than one. The domain and range are spotted visually from the horizontal and vertical stretches of the functions.

Algebra 2

Algebra 2 requires 75 hours to cover the following 12 topics: 1. Equations and inequalities ( 5 hours) 2. Linear relations and functions ( 6 hours) 3. Solving system of linear equations ( 5 hours ) 4. Rational functions and relations (6 hours) 5. Matrices ( 5 hours) 6. Polynomials ( 6 hours) 7. Quadratic functions and inequalities ( 6 hours) 8. Exponential and logarithmic functions( 6 hours) 9. Logarithmic functions(5 hours) 10. Arithmetic sequences and series ( 7 hours) 11. Probability and Statistics ( 9 hours) 12. Trigonometry ( 10 hours) In Algebra 2, students will master the fundamental operations on expressions: distribution, commutation, association and identity property. This is a further advent beyond basic operations of addition, subtraction, multiplication, and division. Here, students are introduced to more complex expressions of polynomials, exponentials, logarithms, and trigonometry. After completing Geometry 2, students might still not have connected the greater picture of applying those expressions of real-life problems. For example, polynomials describe relations between various variables in multivariate problems, while exponentials occur in time-changing processes, such as decay and growth over time. While logarithms facilitate reversing exponentials and managing large numbers, trigonometric functions best describe space dimensions and shapes. Beside teaching students the meaning and applications of the rules of geometry, I lean on my engineering training to apply theory to practical real-life problems.

Calculus

Calculus I is a combination of Algebra and Geometry into a powerful tool of differentiation and integration of functions. The main purpose of Calculus is to represent the relationships of assemblies of numbers tied together by some rules of specific functions. Those relationships are Algebraic in nature, yet must represent Geometrical relevance in order to account for variations of numbers governed by the function. Calculus I falls into the following topics: 1. Limits and continuity: • Differentialiblity of functions • Continuity of functions: • The Intermediate Value Theorem. • Asymptotes: vertical, horizontal, and oblique. 2. Differentiation: • Definition of the derivative: rate of change and the slope of a tangent line. • Basic differentiation rules: Power rule, product rule, quotient rule, and chain rule. • Derivatives of trigonometric, exponential, logarithmic, and inverse functions. • Implicit and related rates of transcendental function • Mean Value Theorem: statement and applications. 3. Applications of differentiation: • Maxima, minima, inflection of function behavior • Optimization: maximum and minimum of functions in practical problems. • Related rates: multiple quantities are changing with respect to time. • L'Hôpital's Rule: A method for evaluating limits of indeterminate forms. 4. Integration: • Antiderivatives: indefinite integral of a function. • Definite integrals: the area under a curve. • Fundamental Theorem of Calculus: differentiation and integration. • Integration techniques: substitution and the reverse power rule. • Applications of definite integrals: Calculating areas and volumes. By the end of studying Calculus I, my emphasis focuses on the ability of the learner to grasp the concepts of instances of differentiation versus summative depiction of integration. The learner must acquire confidence in his/her ability to spot the purpose of differentiation as a local variation, while integration as an overall summation. That confidence is gained by solving problems and g

Geometry

Geometry trains learner on visual interpretation of shapes, objects, and lines, areas, and volumes. Combined with Algebra, the study of Geometry will form the pillar of mathematical cognition of the learner. Algebra provides numbers, Geometry puts numbers into forms (shapes). The syllabus of Geometry covers the following topics: 1. Lines and Angles: Line, Line segment and Ray, Angle, Type of angles 2. Parallel Lines: Transversal, Parallel lines 3. Triangles: Properties, Angle sum, Types of Triangles 4. Congruence of triangles: 5. Concurrent lines in Triangles: Median-Centroid, Altitude-Orthocentre, Angular bisector-Incentre. 6. Polygons: Types, Interior angle Sum, Exterior angle sum, Number of diagonals. 7. Quadrilateral: Angle sum properties, Basic properties of Quadrilateral, Types of Quadrilateral Types: Trapezium, Parallelogram, Rectangle, Rhombus, Square 8. Theorems in Quadrilateral: Mid-Point theorem, Intercept theorem 9. Similar Triangles:  Properties of Similar Polygons,  Propertionality theorem  Vertical Angle bisector theorem  Criteria for Similarity of Triangles  Areas and Perimeters  The ratios of similar 10. Right triangle and Pythagorean theorem 11. Proportionality relations  Verifying type of triangle  Right triangle-Trigonometry:  Different systems of measuring angles 12. Trigonometric ratios: T-ratios of (90 – x ) 13. Circles: Length of arc, Area of a sector, Central angle theorem, theorems on Tangents 14. Perimeter and Area: of various triangles, quadrilaterals and Circles 15. Volume and Surface area: of Cuboid, cube, Cylinder, Cone and Sphere During my teaching of Geometry, my emphasis focuses on tying numbers with shapes. That hopeful goal of enhancing the visualization of numbers by their graphic representation transforms the learner onto a conceptual viewer of mathematics, instead of literal receiver of abstract value.

Precalculus

The study of Precalculus constitutes a steep indulgence into the properties of functions. Students will learn how analytical functions are described by their domain, range, limits, continuity, and differentialability. Those properties constitute the essential tools for engineers, scientists, and researchers in the understanding how to analyze and understand relationships between changing variables. In Precalculus, students would start by polynomial function, progress to rational, exponential, logarithmic, and trigonometric functions. The ultimate goal is to learn how to decipher the nature of changing conditions that are governed by particular rules of physics.

SAT Math

Preparing for the SAT exam requires comprehensive review of the basic mathematical disciples: Algebra, Geometry, Trigonometry, and Calculus. My approach in facilitating such overall comprehension of mathematical procedure goes beyond equation, graphs, and calculation. At this pivotal stage of student's transition from high school to college, my emphasis gears towards real-life applications of mathematical logic. Such practical application of mathematics to real-life problems requires clarity of mind and confidence in applying past knowledge on newly encountered challenges. The student has to master the skill of scanning their previously acquired learning, searching for proper fit for the new challenge.

Trigonometry

Trigonometry bridges the gap between Cartesian coordinates and Polar coordinates, thus allowing scientists to implement the most efficient coordinate systems that fits the geometry of a practical problem. Trigonometry exploits the concepts of ratios to tie orthogonal coordinates by angles and arcs, instead of perpendicular vectors. Thus, a basic understanding of one trigonometric function, like tan(x), suffices to solve the remaining five siblings, i.e., sin(x), cos(x), cot(x), sec(x), or csc(x). Together, with the Pythagorean theorem, those six function make the pillars of the imminent branch of trigonometry.