Algebra

Students will use vectors and matrices to organize and describe problem situations.

Represent situations and solve problems using vectors in areas such as transportation, computer graphics, and the physics of force and motion.

Represent geometric transformations and solve problems using matrices in fields such as computer animations.

Students will use a variety of network models to organize data in quantitative situations, make informed decisions, and solve problems

Solve problems represented by a vertex-edge graph, and find critical paths, Euler paths, and minimal spanning trees.

Construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures.

Investigate the scheduling of projects using PERT.

Consider problems that can be resolved by coloring graphs

Students will create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money.

Use exponential functions to model change in a variety of financial situations

Determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments.

Students will analyze and evaluate the mathematics behind various methods of voting and selection

Evaluate various voting and selection processes to determine an appropriate method for a given situation

Apply various ranking algorithms to determine an appropriate method for a given situation.

Students will demonstrate knowledge of both the definition and the graphical interpretation of limit of values of functions

Use theorems and algebraic concepts in evaluating the limits of sums, products, quotients, and composition of functions

Verify and estimate limits using graphical calculators

Students will demonstrate knowledge of both the definition and graphical interpretation of continuity of a function.

Evaluate limits of functions and apply properties of limits, including one-sided limits.

Estimate limits from graphs or tables of data.

Describe asymptotic behavior in terms of limits involving infinity.

Apply the definition of continuity to a function at a point and determine if a function is continuous over an interval.

Students will demonstrate knowledge of differentiation using algebraic functions.

Use differentiation and algebraic manipulations to sketch, by hand, graphs of functions.

Identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing

Use differentiation and algebraic manipulations to solve optimization (maximum - minimum problems) in a variety of pure and applied contexts.

Students will explore and use historical methods for expressing and solving equations.

Express the geometrical algebra found in historical works (such as the Elements of Euclid) in modern algebraic notation.

Solve systems of linear and nonlinear equations using Diophantus' method.

Translate into modern notation problems appearing in ancient and medieval texts that involve linear, quadratic, or cubic equations and solve them.

Use Cardanos cubic formula and Khayyams geometric construction to find a solution to a cubic equation.

Students will explore abstract algebra and group-theoretic concepts.

Add, subtract, and multiply two quaternions

Explore matrix products other than the Cayley product (including Lie and Jordan) by determining whether these products are associative or commutative.

Identify whether a given set with a binary operation is a group.

Students will use and apply number theoretic concepts.

Find the first four perfect numbers using Euclid's formula.

Prove statements concerning figurate numbers using both graphical (as in the manner of the Greeks) and algebraic methods.

Solve simple linear congruences of the form ax = b mod m.

Use Fermat's Little Theorem and Euler's Theorem to simplify expressions of the form a k mod m.

Use Gauss Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form x 2 = p mod q.

Discover that the real primes that can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.

Students will use the algebraic techniques of Fermat, Barrow, and Newton to determine tangents to quadratic curves.

Students will investigate the relationship between points, lines, and planes in three-dimensions.

Represent equations of lines in space using vectors

Express analytic geometry of three dimensions (equations of planes, parallelism, perpendicularity, angles) in terms of the dot product and cross product of vectors.

Students will recognize and apply properties of matrices.

Find the determinant of 2-by-2 and 3-by-3 matrices

Represent a 3-by-3 system of linear equations as a matrix and solve the system in multiple ways: the inverse matrix, row operations, and Cramer's Rule

Apply properties of similar and orthogonal matrices to prove statements about matrices.

Find and apply the eigenvectors and eigenvalues of a 3-by-3 matrix.

Students will explore functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0.

Evaluate such functions at a point in the plane

Graph the level curves of such functions

Determine points or regions of discontinuity of such functions.

Students will use basic functions to solve and model problems related to stock transactions, banking and credit, employment and taxes, rent and mortgages, retirement planning, and other related finance applications.

Apply linear, quadratic, and cubic functions.

Apply rational and square root functions.

Apply greatest integer and piecewise functions.

Apply exponential and logarithmic functions.

Students will understand the characteristics of these functions as they relate to financial situations.

Understand domain and range when limited to a problem situation.

Understand and apply limits as end behavior of modeling functions.

Students will use formulas to investigate investments in banking and retirement planning.

Apply simple and compound interest formulas.

Apply future and present value formulas.

Students will understand and use matrices to represent data and solve banking and retirement planning problems.