## Mathematics

### From developmental math to calculus and beyond, Smarthinking delivers.

Using our proprietary virtual whiteboard for live, real-time collaboration, tutors guide students through solving math problems. Smarthinking’s tutors do not solve the problem for students. Instead, tutors use a problem-solving based teaching style to help students learn underlying math concepts. In this way, students learn to handle future assignments on their own.

Except where noted, mathematics and statistics tutors are available 24/7 during the standard school year. See the Hours of Service schedule for additional details.

**Mathematics: Topics Covered**

**Basic Math:** Operations on Whole Numbers, Addition and Subtraction Using Fractional Notation, Addition and Subtraction Using Decimal Notation, Multiplication and Division Using Fractional Notation, Multiplication and Division Using Decimal Notation, Ratio and Proportion, Percent Notation, The Real Number System, Dimensional analysis, Number systems other than base 10, Order of Operations, Exponents, Set Operations, Venn diagrams, Logic; Truth tables, Conditionals and Bi-conditionals.

**Algebra:** Linear equations; Applications, Absolute value - Inequalities; Linear, Absolute value, Non linear - Functions / Graphs; Function notation, Linear, Polynomials, Translations / Shifting / reflecting graphs, Composition, Inverse functions - Roots of polynomials; Quadratic, Factoring, Complete square, Quadratic formula, Complex Numbers, Higher Degree, Factoring special cases, Synthetic division - Exponentials and Logs; Graphs, Properties, Solving equations - Systems of linear equations; 2 equations 2 unknowns, Matrix algebra, Gaussian elimination, Inverses, Determinates - Conics - Mathematics of finance; Simple/compound interest, Annuities, Amortization - Linear programming; Geometric approach, simplex method.

**Liberal Arts Math: **Voting: Voting Methods, Fairness Criteria, Weighted Voting; Fair Division: Divider-Chooser, Lone-Divider, Lone-Chooser, Last-Diminisher, Sealed Bids, Markers; Apportionment; Consumer Mathematics: Interest, Ordinary Annuities, Fixed Installment Loans, Amortization, Average Daily Balance; Brief Survey of Graph Theory: Introductory Euler and Hamilton Circuits and Paths, Fleury's Algorithm, Nearest-Neighbor and Repetitive Nearest Neighbor Algorithms, Cheapest Link Algorithm, Minimal Spanning Trees, Kruskal's Algorithm; Introductory Scheduling: Digraphs, Priority Lists; Introductory Counting: Fundamental Counting Principal, Permutations, Combinations; Basic Probability; Mean; Median; Mode; Frequency Distributions; Percentile Rank

**Geometry:** Basic Definitions; Points, Lines, Rays, Angles, Triangles - Angles, Types, Measurements - Congruent Triangles - Planes and Parallel Lines - Circles, Polygons, Quadrilaterals - Coordinate Geometry, Midpoint, Slope, Distance Formula - Theorems, Postulates, and Proofs.

**Trigonometry:** Trigonometric functions; Angles and their measure, Right triangle trigonometry, Functions of any angle, Applications - Graphs of trigonometric functions; Inverse trig functions - Analytic Trigonometry, Solving equations, Verifying identities, Formulas -Law of Sines, Law of Cosines - Complex numbers - Vectors - Analytic Geometry; Conics, Polar equations.

**Calculus (Single Variable):** Limits; definition, limit theorems, L'Hopital's Rule - Continuity; intermediate value theorem - Derivatives; chain rule, implicit differentiation - Applications of Derivatives; Analysis of Graphs, mean value theorem, max/min, related rates - Integrals; definition, definite integrals, fundamental theorem of calculus - Integration techniques; polynomials, exponential and logs, trigonometric, substitution, trig substitution, parts, partial fractions - Applications of Integrals; length of curves, work, volume, surface area - Parametric equations - Polar coordinates - Sequences and Series; Convergence test, Power series, Taylor series.

**Statistics:** Descriptive Statistics, Data Analysis (Graphic Representations, Measures of Central Tendency, Dispersion, Position, Regression and Correlation); Probability (Combinatorics, Random Variables, Probability Distributions for Discrete and Continuous Random Variables; Inferential Statistics (Sampling and Sampling Distributions, Central Limit Theorem, Confidence Intervals, Hypothesis Testing, Inference Concerning Correlation and Regression); Analysis of Variance (Categorical Data Analysis; Chi-square; Contingency Tables; Homogeneity tests; Decision Theory); Process and Quality Control (Control Charts)

**Math en Español:** Bilingual Math tutoring is available in all of the content areas described under Mathematics and Statistics.

**Available Upon Request via Prescheduled Sessions:**

**Multivariable Calculus** (i.e. Calculus III): Vectors: dot product, cross product, lines, planes; Vector-Valued Functions: limits, derivatives, curves, tangents, curvature; Partial Derivatives: chain rule, directional derivatives, gradient, Lagrange multipliers; Multiple Integrals: surface area, polar & cylindrical coordinates, moments and center of gravity; Vector Calculus: vector fields, line integrals, Green's theorem, surface integrals, Stokes Theorem, Divergence Theorem.

**Differential Equations** (i.e. Introductory Differential Equations): Solution of First-order ODE's: analytical, graphical and numerical methods; Linear ODE's; Undetermined Coefficients and Variation of Parameters; Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance; Complex Numbers and Exponentials; Fourier Series, Periodic Solutions; Delta Functions, Convolution, and Laplace Transform Methods; Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

**Linear Algebra** (Sophomore Level) - Linear Equations: row reduction, vector and matrix equations, linear independence, linear transformations; Matrix Algebra: matrix operations, inverse, subspaces, dimension, rank; Determinants: properties of determinants, Cramer's rule; Vector Spaces: subspaces, null, column & kernel spaces, bases; Eigenvalues: eigenvalues & eigenvectors, characteristic equation, diagonalization, linear transformations, complex eigenvalues, discrete dynamical systems; Orthogonality: inner product, Gram-Schmidt process, least-squares, inner product spaces, Symmetric Matrices: quadratic forms; Optimization: Simplex method.

**Discrete Mathematics** - Logic and proofs: propositional logic & equivalences, proof methods, and strategies; Discrete structures: sets, set operations, functions, sequences, and series; Algorithms: integers, primes, gcd, matrices, induction, recursion; Counting: pigeonhole principal, permutations, combinations, linear recurrence relations, generating functions; Discrete probability: Baye's theorem, expected value, variance; Graphs: representation, isomorphism, connectivity, Euler & Hamiltonian paths, trees; Boolean Algebra: functions, logic gates, minimization of circuits.

**Advanced Statistics:
**Basic Review: Box plots, histograms, bar charts, pie charts, counting principles; descriptive statistics, mean, median, mode, five-number summary, standard deviation, range, IQR, Probability distributions.

Estimation Theory: Estimates by method of moments, their properties; Maximum likelihood estimates & their properties, Fisher information, Rao-Cramer inequality, efficient estimates; Bayes estimates, prior and posterior distributions, conjugate priors; Sufficient and jointly sufficient statistics, Neyman-Fisher factorization criterion, Rao-Blackwell theorem; Estimates for parameters of normal distribution, their properties; Chi-square, Fisher and Student distributions; Sampling distributions; Confidence intervals (For sampling distribution and for parameters of normal distribution).

Hypotheses Testing: Testing simple hypotheses, Bayes decision rules, types of error, most powerful tests, likelihood ratio tests, randomized tests; Composite hypotheses, power function, monotone likelihood ratio and uniformly most powerful tests; t-tests and F-tests; Goodness-of-fit tests, chi-square tests, tests of independence and homogeneity, Kolmogorov-Smirnov test, Effect Sizes; Two independent samples, paired sample t-tests; Test for equality of variance.

Regression and Classification: Simple linear regression, least-squares fit, statistical inference in simple linear regression, confidence intervals, prediction intervals; Classification problem, boosting algorithm; Multiple linear regression; Correlation; Normal probability plots and other assumption checking techniques; Effect Sizes; Logistic regression; Correlation and regression techniques for quantitative and qualitative data analysis; nominal scales, interactions; other related multivariate methods.

ANOVA: Basic One-Way, repeated measures, mixed model, factorial, randomized block ANOVA, ANCOVA; Effect Sizes; Preplanned comparisons; Post-hoc analysis/comparisons: Bonferroni, Tukey, LSD, Dunnett's.

Non-parametric Statistics: Kruskal Wallis; Sign Test; Wilcoxin Signed-Rank; Wilcoxin Rank Sum Test; Other tests.