Integrated Math III

In this course, students generalize their understanding of quadratic functions to include cubic and other polynomials. Students gain a solid experience of fundamental concepts involving these higher-level polynomials, including factorization, division, solving polynomial equations, and essential polynomial theorems and their applications.

Knowledge of inequalities and their solutions is essential when studying college-level mathematics. In this course, students build on existing knowledge of functions to solve various types of inequalities, including quadratic, polynomial, rational, and two-variable nonlinear inequalities.

In this course, students further develop the knowledge gained in Integrated Math II to carry out deep explorations of trigonometric functions. They will learn how to apply the law of sines and cosines, derive and apply trigonometric identities, solve trigonometric equations, and explore inverse trigonometric functions.

At this level, students experience their first taste of alternatives to the Cartesian coordinate system, exploring the basics of parametric and polar coordinates. They will learn how to convert to and from Cartesian coordinates and plot simple curves in these coordinate systems.

Students explore advanced mathematical objects such as vectors and matrices at this level. Students will master vector addition, scalar multiplication and learn about different types of vector products. They will learn to apply various operations to matrices and explore determinants. In addition, they will use these ideas to solve geometric problems involving length, angle, area, and volume. Students will also gain a concrete understanding of linear transformations in the plane and relate these operations to their current knowledge of transformations.

Students take their knowledge of complex numbers to new depths by exploring complex numbers in polar form, De Moivre’s theorem, Euler’s theorem, and the fundamental theorem of algebra. They will also explore how operations on complex numbers can be interpreted as transformations of vectors in the complex plane.

Students complete their understanding of the four conic sections to include ellipses and hyperbolas. In addition, students achieve mastery of radical and rational functions, including sketching their graphs and describing properties.

Finally, students will explore advanced concepts in probability and statistics, including conditional probability, discrete random variables, and the normal distribution.

• Explore finite geometric series, including computing the sum of a geometric series, and use them to model financial problems.
• Dive deep into polynomials, including factoring, dividing, solving polynomial equations, the factor, remainder, and rational root theorems, graphing polynomials, and describing their closure properties.
• Simplify and manipulate rational expressions, and solve rational equations.
• Thoroughly analyze properties of reciprocal and rational functions, including exploring their end-behavior, roots, vertical asymptotes, and holes. They will describe the behavior of rational functions using limits and describe closure properties of rational functions.
• Study radical expressions and functions, including simplifying square root expressions, rationalizing denominators, and graphing radical functions.
• Solve various inequalities, including quadratic, polynomial, non-polynomial, two-variable, and rational inequalities.
• Extend their knowledge of conic sections to include ellipses and hyperbolas. This includes describing their equations and working with properties such as foci, directrices, and eccentricity.
• Delve into trigonometry with general triangles, including the law of sines, the law of cosines, areas of general triangles, and modeling and solving problems using these methods.
• Understand inverse trigonometric functions, including their graphs, domain, and range, and evaluate expressions involving these functions.
• Master trigonometric identities, including the Pythagorean identities, cofunction identities, sum and difference, and double-angle formulas.
• Solve trigonometric equations, including describing their general solutions, cases with transformed functions, and quadratic trigonometric equations.
• Develop a solid foundation of parametric equations, including graphing curves, finding intersections, and working with parametric representations of some types of conic sections.
• Understanding polar coordinates and equations, converting between Cartesian and polar coordinates, and sketching and analyzing simple polar curves.
• Explore complex numbers in depth, including how operations with complex numbers manifest in the complex plane, the polar form of a complex number, De Moivre's theorem, Euler's formula, the roots of a complex number, the fundamental theorem of algebra, and extending polynomial identities.
• Gain proficiency in vector operations, such as addition, scalar multiplication, dot product, and cross product.
• Learn the basics of matrices, including addition, subtraction, scalar multiplication, matrix multiplication, and determinants, and apply these concepts to solve systems of equations and perform linear transformations in the plane.
• Deepen their understanding of probability and statistics, including regression models, conditional probability, discrete random variables such as the binomial and geometric distribution, and the normal distribution.