ACT Math,
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.
Calculus,
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 ƒ and ƒ’, the relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’, 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 ƒ, ƒ’ and ƒ’’, and the relationship between the concavity of ƒ and the sign of ƒ’ 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.
Electrical Engineering,
Electrical Engineering
Electrical engineering deals with the practical application of the theory of electric and magnetic forces and their effects on the design and construction of machinery, especially the design and application of circuitry and equipment for power generation and distribution, machine control, and communications.
Having worked in aerospace electronics for over 20 years, I have the following industry experience that is readily transferable to my students:
Engineering assignments in component test and evaluation, environmental testing, power systems and fiber optics gyroscope testing and instrumentation; Radio frequency (RF) and microwave experience includes device applications engineering and radio operator training; analog modulation, including: double-sideband modulation, amplitude modulation, single-sideband modulation, frequency modulation, and phase modulation; implementations of transmitters and receivers, plane wave propagation in free space, reflections, standing waves and antennas; the design, manufacture and test of inertial guidance electronics for aerospace including ring laser gyroscopes, digital signal processors, components (resistors, voltage and current sources, capacitors, inductors, and operational amplifiers) and their mathematical models, computer and paper analysis of circuits with transient and steady state signals, and printed wiring boards and assemblies; computer electronics design, manufacture and test, performance evaluation of computer processors and Reduced Instruction Set Computer (RISC) microcontrollers, datapath components and control units, instruction set architecture, hardware & software interface, interfacing hardware components (such as Analog-to-Digital Converters (ADC)/Digital-to-Analog Converters (DAC), sensors, transducers, etc.), memory and input/output (I/O) circuitry.
Industry experience in semiconductors includes process and design engineering, production, device testing, laboratory evaluations (construction analysis, Destructive Physical Analysis (DPA) and device Failure Analysis (FA)), training engineers in device physics including characteristics of large and small signal models, thermal characteristics, models, analysis, and design of circuits using semiconductor diodes and transistors (field-effect and bipolar); experience with integrated circuits includes microprocessor chip sets and Application Specific Integrated Circuits (ASIC) design, process engineering, production, packaging, testing, evaluation, and application; Very Large Scale Integration (VLSI) design, linear circuit applications, gain, frequency response, gain stages and output stages, internal circuitry of op-amps, the application and interfacing of integrated circuits logic families and integrated circuits for use in analog applications.
Participation in industry committees for electronics reliability and field environments, presentation of electrical engineering papers on Dynamic Random Access Memories (DRAMs) and Charge Injection Device cameras and membership in professional organizations including; the Institute of Electrical and Electronics Engineers (IEEE), the National Association for Radio, Telecommunications and Electromagnetics (NARTE) and the American Radio Relay League (ARRL).
Formal electrical engineering instruction and laboratory work not necessarily addressed on-the-job includes;
Kirchhoff' s laws and Thevenin's Theorem, analysis of circuits with sinusoidal signals, superposition, phasors, complex frequency and frequency response, alternating current (AC) steady state analysis, AC power, mutual inductance, equivalent circuits, nonlinear elements, series and parallel resonance and Bode plots; feedback of linear small-signal amplifiers, low and high frequency analysis of differential amplifiers, current sources, gain stages and output stages, internal circuitry of op-amps, op-amp configurations, op-amp stability and compensation, feedback control systems modeling and transfer functions, stability, root locus, continuous and discrete-time systems/signal analysis; performance of radio systems in the presence of white gaussian noise, electricity and magnetism, Maxwell’s equations, plane wave propagation in free space, reflections and standing waves; number systems, Boolean algebra, functions, minimization, computer design including datapath components and control unit, machine and assembly language programming; the Laplace transform, Fourier transform and series, and their application to the systems analysis, and the Nyquist and Shannon theorems.
Physics,
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.
Prealgebra,
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,
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.
Probability,
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.