interpret the structure of and perform operations with polynomials within a geometric context
B - Geometric & Spatial Reasoning
experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
explore the concept of a radian measure and special right triangles
examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena
C - Probabilistic Reasoning
solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions
D - Data & Statistical Reasoning
examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena
communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data
E - Algebraic & Geometric Reasoning
manipulate, prove, and apply trigonometric identities and equations to solve contextual, mathematical problems
communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data
B - Functional & Graphical Reasoning
explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
analyze the behaviors of rational functions to model applicable, mathematical problems
F - Patterning & Algebraic Reasoning
represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications
G - Geometric & Spatial Reasoning
develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle
communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data
B - Functional & Graphical Reasoning
explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
analyze the behaviors of rational functions to model applicable, mathematical problems
C - Patterning & Algebraic Reasoning
represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications
D - Geometric & Spatial Reasoning
develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle
utilize fractions, decimals, percents, and ratios to write and solve a variety of financial problems
B - Functional & Graphical Reasoning
explore and apply functions (i.e., linear, exponential, quadratic, cubic, rational, square root, greatest integer, piecewise) to model and explain real-life phenomena and to solve complex problems in business and financial contexts
C - Patterning & Algebraic Reasoning
explore, evaluate, and rearrange formulas applicable to business and financial contexts
write and solve systems of equations and inequalities in context of financial applications
E - Geometric & Spatial Reasoning
apply properties of polygons, circles, and trigonometry to model and explore real-world applications
F - Data & Statistical Reasoning
collect, analyze, interpret, summarize, and construct displays of data to make predictions within real-world applications
conduct investigative research to solve real-life problems and answer statistical questions involved in business and financial decision-making
make decisions and solve problems using ratios, rates, and percents in a variety of real-world applications
predict potential outcomes by analyzing averages and indices of large data sets through investigations of real-world contexts
B - Patterning and Algebraic Reasoning
develop methods or algorithms to analyze discrete situations
create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money
use vectors and matrices to model and solve real-life situations
make informed decisions and solve problems with a variety of network models in quantitative situations
C - Probabilistic Reasoning
analyze the probability of success or failure in order to make decisions
rationalize decisions based on probabilities and risk of loss and gain of real-life situations
D - Data and Statistical Reasoning
conduct investigative research to solve real-life problems and answer statistical investigative questions involved in business and financial decision-making
E - Functional and Graphical Reasoning
use functions to model problem situations in both discrete and continuous relationships
construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena; use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs
construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model
construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions
B - Geometric & Spatial Reasoning
solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena
C - Patterning & Algebraic Reasoning
create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena
build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations
create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations
D - Numerical Reasoning
investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots
E - Data & Statistical Reasoning
collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems
construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena; use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs
construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model
construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions
B - Geometric & Spatial Reasoning
solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena
C - Patterning & Algebraic Reasoning
create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena
build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations
create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations
D - Numerical Reasoning
investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots
E - Data & Statistical Reasoning
collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems
review function notation, domain/co-domain, identity/associativity, inverse/invertibility connecting to the computer science concept of perfect secrecy (i.e., encryption)
review Python psuedo random number generation, calculating probability distribution and interpreting probability events through application of Caesar's Cypher and other examples of cryptosystems
B - Python Programming
understand how to program and utilize modules and control statements (e.g., loops, conditionals, grouping) in Python
utilize sets, lists, dictionaries, comprehensions, indexing, and tuples in Python
program input and output features to read from and write to files in Python
C - The Complex Field
perform operations of complex number numbers (e.g., absolute value, adding, multiplying) and understand how they produce different transformations
understand how complex numbers connect to the unit circle and represent them in polar form; perform transformations in polar form and utilize Euler's formula/The First Law of Exponentiation to understand these transformations; connect to performing image transformation on a computer graphic program
work with the Galois Field to understand further concepts in perfect secrecy and network coding (i.e., providing efficiency in streaming services)
D - The Vector
connect use of vectors in Galois Field(2) by applying concepts of perfect secrecy, all-or-nothing secret sharing, and programming/solving lights out games
perform vector operations in R(n) including addition, scalar multiplication, and dot product; review concepts of convex and affine combinations; represent and perform these operations using dictionaries and the Vec.py class in Python
find the distance, its unit vector in the same or opposite direction, the projection of a vector onto a given vector or vector space, dot product, inner product, cross product, and angle between two vectors in Euclidean space
solve triangular systems of linear equations using upper-triangular systems, backward substitution, and other algorithms
use the dot product to display the concept of simple authentication schemes and interacting with them, and performing a senator voting record analysis
E - Vector Spaces
determine if a given set of vectors in a vector space is a spanning set for that vector space and if they are linearly independent
define and discuss uses of linear combinations and understand how to solve for coefficients or linear combinations, connect to programmed/solved lights out game
determine if a linear combination is an affine combination and determine if an affine space exists by translating a vector space, represent and affine space as a solution set to a linear system
find whether a vector is a linear combination of a given finite set of vectors in a vector space and provide this linear combination
determine whether a provided subset of a vector space is a subspace and find the dimension of a subspace
define span and what it means for linear combinations to be a span of vectors; connect span to simple authentication schemes; understand the geometric depiction of the span of vectors over R and the geometry of solution sets of homogenous linear equations and systems; understand the geometric interpretations of R^2 and R^3
F - Matrices
define and understand what the null space is by connecting to concepts of homogenous linear systems/matrix equations, and error correcting codes such as linear codes and Hamming's code
factor a given matrix into the product of two elementary matrices, find the adjoint of a matrix and use it to find the inverse of the matrix (understanding the conditions for invertibility) or solve a system of linear equations
generate an augmented coefficient matrix from a system of linear equations
program transformations in 2D geometry using Python and concepts of matrix operations
perform matrix operations including transpose, addition, scaler multiplication, dot product, and multiplication; compute the inner and outer product
program error correcting code concepts such as Hamming's code using matrix operations
review the structure of a matrix and composition of the identity matrix, determine the size, transpose, inverse, rank, and LU-factorization of a matrix; interpret matrices as vectors
G - The Basis and Dimension
determine whether a given set of vectors in a vector space forms a basis for that vector space and recognize standards bases in the vector spaces nth dimensional Euclidean space, the set of all polynomials of degree greater than or equal to n
understand if a linear function is invertible utilizing the concept of dimension and determine if a function is onto or one-to-one; discuss in connection with Kernal-Image Theorem and Rank- Nullity Theorem, demonstrate using checksums
discuss rank theorem and demonstrate its use via the Simple Authentication Schema in computer science
connect the Exchange Lemma to the concept of camera image perspective rendering in a Python lab
define the coordinate representation of a basis and connect to lossy compression in computer science; find a basis for the column or row space of a matrix, and find a basis for and the dimension of the nullspace of a matrix
define what it means for vectors to be linearly dependent and linearly independent and define the Superfluous-Vector Lemma; perform tests of linear dependence
utilize direct sum to add vector spaces and find the basis for the direct sum and understand if two subspaces are complementary
review the minimum spanning forest problem in GF(2) in connection with the Grow and Shrink Algorithms and how to formulate the problem in linear algebra
find the transition matrix from one basis to another (i.e., change of basis)
define and determine the dimension and rank of a basis (and therefore vector space); use it to prove the Morphing Lemma and prove the Superset-Basis Lemma
demonstrate that every vector space has a basis and any finite set of vectors contains a basis for its span (e.g., Subset-Basis Lemma)
H - Gaussian Elimination
solve systems of linear equations and finding the basis for a Null space by use of Gaussian Elimination, Gauss-Jordan Elimination, LU factorization, and Cramer's Rule; show how the simple authentication scheme can be attacked/improved over GF(2) using Guassian Elimination
understand how Threshold Secret Sharing works in conjunction with Gaussian Elimination through a programming lab in Python
understand how factoring integers can be performed using Euclid's algorithm and utilizing prime set factors in Python
use elementary row operations to create matrices in row-echelon and reduced row-echelon form
I - Orthogonalization
define and perform QR factorization of a matrix to compute solutions to the matrix equation Ax=b; use to perform the application of least squares to find the line or curve of best fit (linear/quadratic) to approximate data in the industrial espionage problem/sensor node problem/machine learning problem
find an orthogonal basis for a given basis/subspace/inner product space by applying the Gram- Schmidt orthonormalization process
given the solution space of a homogenous system of linear equations, find an orthonormal basis
determine if two given vectors/sets of vectors(complements)/subspaces are orthogonal, parallel, or neither; find the orthogonal component of a given subspace
use orthogonalization to find the closest point in the span of many vectors, compute a basis/subset basis, direct sums of complements
J - Special Bases
utilize compression by suppression to find the closest k-sparse vector coordinate representation in terms of an orthogonal basis
understand how images and sounds can be represented as wavelets and the bases representation of wavelets as well as wavelet transformation, implementation, and decomposition, perform Python lab on using wavelets to perform file compression
define and demonstrate the Fourier transform connecting how a sound is stored as a sequence of amplitude samples and how the Fast Fourier Transform Algorithm is utilized/derived/coded
K - The Eigenvalue/Eigenvector
connect use of the determinant and eigenvectors to code functionality of Google's PageRank search engine in Python
discuss how Markov chains work to model various concepts such as population movement, dance patterns, literary documents, and Google's search engine PageRank
utilize eigenvalues/vectors and single value decomposition to program face recognition software (Eigenfaces)
determine an orthogonal matrix that diagonalizes a given matrix
find a nonsingular matrix(D) for a given matrix (if one exists) such that D^-1AD is diagonal; find a basis for the domain of a linear transformation such that the matrix of the linear transformation relative to the basis is diagonal
find the determinant, minors, and cofactors of a given matrix and use the determinant to find whether a given matrix is singular/non-singular; use determinant properties to characterize eigenvalues
find the eigenvalues of a given symmetric matrix and find the dimension of the corresponding eigenspace
find the characteristic equation and eigenvalues/corresponding eigenvectors of a given matrix and determine if the matrix is diagonizable/symmetric/orthogonal
verify the eigenvalue/eigenvector of a given matrix while understanding the geometric interpretation and coordinate representation; connect to the Internet Worm case of 1988; perform eigen theorem proofs
L - Linear Programming
perform a Python lab to explore concepts of linear programming
explore samples of linear programming cases including the diet problem, the vertices of polyhedra (polyhedral combinatorics), the simplex algorithm, game theory, nonzero-sum games, and compressed sensing for MRI imaging
perform a machine learning lab on a large set of health care data that incorporates concepts of linear programming
utilize exact and approximate calculations to quantify real-world phenomena and solve problems
B - Patterning & Algebraic Reasoning
construct expressions, equations, and inequalities, and use them to represent and solve problems by choosing appropriate procedures and interpreting solutions in context
C - Functional & Graphical Reasoning
define, build, and interpret functions that arise in various contexts by applying knowledge of the characteristics of the different families of functions, and analyze the effects of parameters
D - Geometric & Spatial Reasoning
reason deductively and inductively about figures and their properties and make sense of geometric situations using measurements in real-world contexts
E - Data & Statistical Reasoning
make sense of and reason about variation in data using graphs, tables, and probability models to solve problems and draw appropriate conclusions from solutions
learn to evaluate matrices and apply their properties to solve engineering problems; calculate determinants of matrices; express systems of linear equations in matrix equation form; use Gaussian elimination to compute solution sets of linear systems
investigate functions of two and three independent variables to model engineering systems; compute limits of scalar and vector-valued functions; identify, interpret and graph level curves of multivariate functions; calculate regions of continuity of such functions
apply knowledge of mathematics, science, and engineering design to solve problems; determine the equations of lines and surfaces using vectors and 3D graphing; apply dot and cross products of vectors to express equations of planes, parallelism, perpendicularity, angles; describe the role of vectors in engineering applications, such as modeling the velocity of moving objects or static forces on structures and objects
use visual and written communication to express basic design elements in the appropriate mathematics notation; demonstrate fundamentals of technical sketching using computer- generated visuals by using the appropriate mathematics scale; present a technical design, using computer-generated model, for an assigned design project utilizing the appropriate scientific units (US standards and SI units)
B - Differentiation In Engineering
evaluate and apply partial differentiation of multivariable functions with two or more independent variables; compute the first and second partial derivatives of a function; use the general chain rule to determine the partial derivatives of composite functions; compute and apply the gradient of multivariable functions; solve engineering optimization problems by applying partial differentiation or Lagrange multipliers; utilize partial derivatives in developing the appropriate system balances (e.g., mass balance) in engineering problems
C - Multidimensional Integration in Engineering Systems
apply the techniques of double and triple integration to multivariable scalar- and vector-valued functions; manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates; evaluate and apply line integrals that are independent of path; apply properties of integrals to calculate and represent area, volume, or mass; use integrals of vectors to define and apply the gradient, divergence, or the curl e. Interpret the theorems of Green, Stokes, or Gauss and apply them to the study of real-world phenomena
communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data
B - Functional & Graphical Reasoning
explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
analyze the behaviors of rational functions to model applicable, mathematical problems
analyze the behaviors of rational and piecewise functions to model contextual mathematical problems
utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions
C - Patterning & Algebraic Reasoning
represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications
demonstrate how geometric sequences and series apply to mathematical models in real-life situations
D - Geometric & Spatial Reasoning
develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle
analyze the behaviors of conic sections and polar equations to model contextual, mathematical problems
E - Algebraic & Geometric Reasoning
manipulate, prove, and apply trigonometric identities and equations to solve contextual, mathematical problems
represent and model vector quantities to solve problems in contextual situations
interpret the structure of and perform operations with polynomials within a geometric context
B - Geometric & Spatial Reasoning
experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
explore the concept of a radian measure and special right triangles
examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena
C - Probabilistic Reasoning
solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions
D - Data & Statistical Reasoning
examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena
interpret the structure of and perform operations with polynomials within a geometric context
B - Geometric & Spatial Reasoning
experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
explore the concept of a radian measure and special right triangles
examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena
C - Probabilistic Reasoning
solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions
D - Data & Statistical Reasoning
examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena
A - Abstract Reasoning & Deterministic Decision-Making
solve contextual, mathematical problems involving linear programming and use the mathematics as a model to make decisions about real-life phenomena
solve contextual, mathematical problems involving optimal locations and use the mathematics as a model to make decisions about real-life phenomena
solve contextual, mathematical problems involving optimal paths and use the mathematics as a model to make decisions about real-life phenomena
B - Abstract Reasoning & Probabilistic Decision-Making
solve contextual, mathematical problems with normal distributions to make appropriate decisions
solve contextual, mathematical problems using other distributions (e.g., binomial, geometric, and Poisson) as well as simulations to make appropriate decisions
use simulations to make appropriate decisions
using quantitative reasoning, determine fair methods to reflect the wishes of a larger population with representatives
C - Probabilistic Reasoning
use probabilistic models to make appropriate decisions
express spatial and functional relationships with vectors and matrices, functions, and analytic geometry in three dimensions, and use these relationships to solve contextual, mathematical problems
B - Abstract & Quantitative Reasoning
define, describe, and represent the differentiation of functions of two independent variables and differential vectors to solve contextual, mathematical problems and to explain real-life phenomena
C - Abstract & Quantitative Reasoning
interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain real-life phenomena
interpret, represent, and communicate logical arguments to explain reasoning and justify thinking when solving problems and to include real-life phenomena
apply methods of proof to prove or disprove mathematical statements; explain reasoning and justify thinking through mathematical induction when formulating mathematical arguments
B - Abstract and Quantitative Reasoning
use sets to describe relationships and equivalence when solving contextual, mathematical problems used to explain real-life phenomena
C - Numerical Reasoning
apply properties of numbers to uncover mathematical patterns
explore and solve problems using properties of integer divisibility
explore and apply mathematical congruences in real-life phenomena
explore and apply mathematical theorems and conjectures related to prime numbers in real-life phenomena
