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Following areas are covered in 03/53 class:

- Numbers and operations
- Functions and Algebra
- Patterns and relations
- Geometry and Measurement
- Data analysis, Statistics and Probability
- Integration of above areas, open response problems

**Following areas are covered in 47/09 class:**

- Number Sense and Operations
- Patterns, Relations, and Algebra
- Geometry and Measurement
- Data Analysis, Statistics and Probability
- Trigonometry, Calculus, and Discrete Mathematics
- Integration of above areas, open response problems

The emphasis is on mastering skills on which these tests are based. Tests 03/53/47/09 are based on many different math skills. The goal is to master the skills rather than trying to beat the test. Test problems change but skills do not change. I have helped hundreds of teachers pass the test by mastering the skills. Here is a short list of skills. We will focus on many of these skills in the class.

**SKILLS NEEDED TO PASS TEST 03:**

- Number systems, base-10, other bases, ancient number systems.
- Decimal expansion, scientific notation, rounding, grouping, estimation.
- Real numbers, rational numbers, irrational numbers, integers, decimals, percents, fractions, models, number line.
- Graphic, numeric, symbolic representation of numbers.
- Prime, composite numbers, prime factorization, divisibility rules, LCM, GCF.
- Properties: associative, commutative, distributive.
- Non-standard, graphic, pictorial analysis.
- Relationships among addition, multiplication, exponentiation.
- Order of operations, laws of exponents, absolute value, word problems.
- Algebra as generalized arithmetic.
- Variables, functions, equations, inequalities.
- Linear equations, word problems involving fractions, ratios, proportions, percents, real world situations like distance problems, clock problems, solutions concentration problems.
- Functions, input-output, tables, graphs, formulas, mapping diagrams.
- Recognizing patters, sequences, verbal, numeric, pictorial, algebraic, tables, graphs,ratios, rates, real world situations, models.
- Linear equations and functions, graphing, y = mx + b, Ax+By=C, non-linear functions and distinguish between linear and non-linear functions, formulas and their graphs
- Direct and inverse variation, joint variation, models.
- Measurement: customary, metric and nonstandard (ancient) units, unit-conversions, lengths, number lines, perimeters, areas, volumes, change in dimensions, time temperature, rates, average rates, real-world situations.
- Geometry: classify polygons using sides, angles, regular polygons, real-world applications.
- Three-dimensional figures, faces, edges, vertices.
- Translations, rotations, reflections, dilations, similar figures, congruent figures.
- Nets, projections, perspective drawings 3D to 2D conversions.
- Relation between algebra and geometry, supplementary, complementary angles, exterior angles.
- Statistics: measures of central tendency, mean, median, mode, tables, line plots, graphs, Venn diagrams.
- Frequency distributions, percentiles, compare data sets.
- Probability: simple and compound events, independent and dependent events, conditional probability, experimental and theoretical probability, permutations and combinations.
- Integration of knowledge and understanding.

**ADDITIONAL SKILLS NEEDED TO PASS TEST 53:**

- Number theory: properties (e.g. closure, distributive, associative, roots, powers, laws of exponents and proof of theorems (e.g. product of two even numbers is even).
- Operations as related to integers: identity and inverse elements, associative, commutative, distributive properties, area models (represent number operations), origin of algorithms, algebraic concepts.
- Fractions, models, comparisons, order, estimation, mixed numbers.
- Decimals, percents, number line, developing algebra concepts using decimals and percents.
- Collecting, organizing, reporting, analyzing data: line plots, stem-and-leaf plots, box plots, scatter plots, measures of central tendency (e.g. mean, median, mode) and dispersion (e.g. range, standard deviation), frequency distribution, percentile scores, sampling techniques, gathering and organizing data, lines of best fit, interfaces, interpolations, extrapolations from a set of data.
- Probability: geometric probability as a ratio of two areas, permutations, combinations, real world situations.
- Patterns, relations, algebra: numeric, geometric or tabular representations of patterns, sequences, functions (linear, quadratic, exponential, half-life, compound interest), iterative and recursive patterns, domain, range, properties of functions and relations, parent graphs, shifting rules, real world problems.
- Translate problems into algebraic expressions, equations and inequalities, evaluate expression if value of a variable is given, direct and inverse relations, factoring, laws of exponents in algebra, solving equations, simultaneous equations, literal equations.
- Properties of linear relations and functions: vertical line test, difference between a function and a relation. Model rate of change, direct variation, finding equation of a line, parallel and perpendicular lines, systems of equations and inequalities, linear programming, optimization.
- Properties and applications of ratios and proportions: relative and absolute relationships, writing and solving proportions, mixtures, rates, scale factors, scale drawings, perimeters, areas, volumes.
- Properties and applications of quadratic, exponential, polynomial and rational functions and relations: solving quadratic equations by factoring, completion of square, quadratic formula and graphing. Nature of roots, sum and product of roots of a quadratic equation. Real, imaginary and complex numbers, maximum and minimum problems, properties and graphs of exponential functions, exponential growth, decay, compound interest, half-life, rational functions, modeling, real world problems. Joint variation.
- Measurement: unit conversions, length, perimeter, area, volume, mass, capacity, density, time, temperature, angles, rate of change, displacement, velocity, acceleration, similar figures. Effects of measurement errors, percent errors, area density and speed measurement errors.
- Euclidean Geometry: two-column proofs, axioms, postulates as related to lines, angles and planes. Proving triangles congruent and similar, geometric constructions, conjectures and axioms of Euclid.
- Apply Euclid’s concepts to 2D and 3D shapes. Triangles, quadrilaterals, sides, angles, diagonals of polygons, Pythagorean theorem, arcs, circles, obtuse and acute triangles, composite shapes, surface areas and volumes of 3D figures, nets, models.
- Transformational Geometry: Midpoint formula, distance formula, parallel and perpendicular lines, dilations, translations, reflections, glide-reflections, types of symmetry, properties of tessellations, proofs, coordinate geometry.
- Integration of knowledge and understanding.

**ADDITIONAL SKILLS NEEDED TO PASS TEST 47:**

- Place value with bases 2,3, 5,8, 16 etc., multiplication as repeated addition, Mayan, Mesopotamian, Egyptian number systems.
- Integers, fractions, percents, ratios, proportions, identity, inverse, signed numbers, area models for multiplication, algorithms for multiplication and division of integers and rational numbers.
- Percent increase and decrease, average rate of change, approximation, reasonableness of answers.
- Real numbers, rational numbers, complex numbers.
- Functions and relations, linear, quadratic, cubic, absolute value, exponential, infinite series and sequences, Fibonacci, arithmetic and geometric sequences.
- Equations, inequalities and models. Functions, domain , range, continuity, piecewise-defined functions, addition, subtraction, composition of functions, transformations, f(x), f(x+k), f(x)+k etc.
- Linear equations and applications, inequalities, graphing, complex roots of quadratic equations, population growth and decay, half-life, rational functions, rational roots theorem, absolute value functions, poles and zeros, factor theorem, remainder theorem.
- Measurement: Indirect measurement, percent error, area, density, speed.
- Proving geometry theorems, Greek, Hindu, Chinese systems of geometry.
- Triangles, quadrilaterals, polygons and their properties, special triangles, arcs, angles, unit circle, radians, degrees.
- Prisms, pyramids, cylinders, cones, perspective drawings, nets.
- Data analysis: graphs, charts, tables, linear regression, interpolation.
- Probability: permutations and combinations, simple and compound events.
- Trigonometric functions and identities, degrees, radians, the law of sines, the law of cosines, graphs of trigonometric functions and their inverses, amplitude, period, phase shift, using trigonometry for real world problems.
- Calculus: limits, rate of change, derivatives, maxima, minima, points of inflections, concavity, integration, area under a curve, development of calculus.
- Discrete Mathematics: linear programming, matrices, sets, iteration, finite differences, graphs, trees.
- Integration of knowledge and understanding.

**ADDITIONAL SKILLS NEEDED TO PASS TEST 09:**

- Real and Complex numbers, vectors, trigonometric applications, proofs.
- Number theory: Euclid’s algorithm, congruence classes, modular arithmetic, Mersenne primes, perfect numbers, Fermat’s last theorem, fundamental theorem of arithmetic.
- Conjectures, geometric, numeric and tabular, Pascal’s triangle, the binomial theorem, finite and infinite sequences.
- Linear and Abstract Algebra: Matrices, determinants, geometric and algebraic properties of vectors, matrix representing linear transformations, groups, rings, fields.
- Quadratic maxima and minima problems, inverse and joint variation problems, zeros and of polynomials, asymptotes, radical, rational, step functions.
- Exponential and logarithmic functions, properties, graphs, decibel scale, Richter scale, inverse relations between exponential and logarithmic functions to solve problems.
- Geometry: proofs, constructions, circles, polygons.
- Rectangular and polar coordinates, conic sections, hyperbolic and elliptic geometries (non-Euclidian)
- Statistics: range, standard deviation, inter-quartile range, percentile scores, variability.
- Methods of gathering data: random sampling, avoidance of bias, simulations, random variables, probability distributions ( e.g. uniform, normal, binomial).
- Limits, continuity. Trigonometric equations, geometric series, discontinuous functions, secant line, average rate of functions and related problems.
- Differential Calculus: tangent to a curve, properties of derivatives, differentiability, product rule, chain rule, transcendental functions, velocity, acceleration, optimization, related rates, verifying that a function is a solution of a differential equation.
- Integral calculus: area under a curve, Riemann sum, fundamental theorem of calculus, techniques of integration, area, work, volume, arc length, displacement, velocity, solving differential equations by separation of variables.
- Discrete/ finite mathematics: sets, permutations, combinations, finite differences, mathematics of finance (compound interest, annuities, amortization), recursive problems, linear programming, finite graphs and trees.
- Integration of knowledge and understanding.