 ## Senior High Math

The approach to teaching mathematics varies greatly from school to school and from country to country. At St. Ambrose we believe in a classical approach, striving to develop mathematical intuition and problem-solving skills in our students. Successful students approach mathematical problems as they would an investigation or puzzle. They ask questions, try out different approaches, and reflect and revisit their solutions. We do not want to limit ourselves to a cookbook-style approach in which students follow a prescribed set of steps by rote without a feeling for how the method works and why. For St. Ambrose math students, how one reaches a solution is as important as the solution itself.

We believe that problem solving is central to the teaching and learning of mathematics. Problem-solving involves logical reasoning; it is important to explore the reasons why step two follows step one; but successful students also understand the process of modifying, adapting, and combining mathematical tools to find new ways to reach a solution.

Our philosophy is to place students in the proper math sequence – one that provides enough challenge to reach their potential without needlessly pushing them beyond their means. Each student that comes to SAA is given our own in-house math placement exam in order to determine the best possible starting point. Our in-house curriculum goes as high as Calculus II and advanced students have the opportunity to take their own off-campus courses if our curriculum is exceeded. Business Math is offered as an elective for Juniors and Seniors.

## Algebra I

Algebra is a gateway to all higher mathematics. Students will learn the skills and concepts that will help them think logically, problem-solve, recognize relationships and patterns, and represent verbal situations as equations.

Topics covered in the course include:

• Relations and functions
• Linear equations and functions, linear inequalities
• Absolute value equations and inequalities
• Systems of linear equations and inequalities
• Operations with polynomials
• Factoring polynomials
• Rational expressions and equations

A more detailed list of topics is provided here.

## Geometry

Geometry introduces the nature of proofs and the use of deductive and inductive reasoning. A successful competition of Geometry provides a foundation for the more sophisticated reasoning required in the mathematical analysis of higher math courses.

While Jacobs’ Geometry: Seeing, Doing, and Understanding is the primary textbook, students may also read excerpts from Euclid’s Elements at the teacher’s discretion. Students will also continue to review and solidify their understanding of Algebra and will work to improve their Algebraic abilities.

Topics covered in the course include:

• Introduction to geometry
• Inductive and deductive reasoning
• Lines and angles
• Exploring congruence and similarity
• Parallel Lines
• Inequalities
• The right triangle
• Circles

A more detailed list of topics is provided here.

## Algebra II

Algebra II expands on the topics of Algebra I in both depth and complexity. Students are introduced to the concept of functions and families of functions, including quadratic, exponential, and power functions. Students will also explore conic sections including ellipses and hyperbolas: their interpretation and application. The exploration of functions segues into an introduction of logarithms and the application of logarithms to STEM subjects.

The course covers the solution of systems of linear equations involving two or more variables and provides a foundation for future exploration of Linear Algebra. Students will be introduced to matrices and the use of the discriminant for evaluation systems of equations.

The course concludes with an introduction to sequences and series and the exploration of higher-order functions and their relationship to complex numbers and the complex plane.

Topics covered in the course include:

• Understanding Functions and Relations
• Linear Functions for Modelling
• Systems of Linear Equations: Matrix Mathematics
• Quadratic Functions and Complex Numbers
• Exponential and Logarithmic Functions
• Rational Algebraic Expressions
• Irrational and Radical Algebraic Expressions
• Quadratic Relationships and Systems of Equations
• Higher-order Functions and their Applications
• Trigonometric and Circular Functions

A more detailed list of topics is provided here.

## Pre-Calculus

Precalculus provides the culmination of several mathematical topics previously explored in Algebra II and provides an introduction to the Calculus for students who will be taking that course in the Senior or Junior year. The course begins with an exploration of functions and transformations, helping students to see functions as objects with properties that can be explored and analyzed.

The course goes deeply into trigonometric identities and application of sinusoids to real-world problems. Students will built upon their knowledge of plane geometry in constructing trigonometric proofs and relationships and apply their understanding to STEM problems.

The second half of the course explores more properties of functions including a deeper dive into logarithmic functions and logistic equations and their applications.

Precalculus also will further the student’s understanding of combinatorics, probabilities and the functions of a random variable with explorations involving the application of the binomial theorem and regression analysis.

The course concludes with a review of analytical geometry and conic sections and a more detailed appreciation of complex numbers and their relationship to trigonometry with De Moivre’s Theorem.

Topics covered in the course include:

• Functions and Mathematical Modelling
• Dilations, Transformations, and Inverses of Functions
• Sinusoids and the Application of Circular Functions
• Trigonometric Identities, Relations, and Proofs
• Combining Sinusoids with Sum and Product Properties
• Triangle Trigonometry with Vectors
• Properties of Elementary Functions
• Properties of Random Variables
• Probability and Binomial Theorem
• Polar Coordinates and Complex Numbers

A more detailed list of topics is provided here.

## Calculus

Calculus is an advanced-level course designed for the motivated Junior and Senior student. The course is divided into three main topic areas: Differentiation, Integration, and Differential Equations. In the Differentiation section, the course introduces limits, differentiation rules, curve sketching, relative rates, and optimization problems. Integration covers Riemann sums, areas under a curve, volumes of revolution, and volumes of cylindrical shells. The Differential Equation section provides the foundation for a study of diffy-q’s and their application in STEM fields.

The SAA Calculus curriculum can provide a foundation for students who wish to receive AP credits through the successful competition of the Calculus AB exam.

Topics covered in the course include:

• Rules of Differentiation
• Related Rates
• Maxima and minima and curve sketching
• Optimization Problems
• L’Hopital’s Rule
• Linear Approximations
• Riemann Sums and the Trapezoidal Rule
• The Antiderivative and the Fundamental Theorem of Calculus
• Applications of Accumulation Functions
• Areas under a Curve
• Volumes of Revolution
• Differential Equations
• AP Calculus Preparation: AB Exam

A more detailed list of topics is provided here.

## Calculus II

Calculus II builds upon the foundation set in the Calculus I course and is designed for the advanced and motivated Senior. The course provides a review of Calculus I topics from limits through derivatives and Integration. It expands on integration by providing additional instruction in integration techniques including integration by parts, partial fractions, and trigonometric substitution.

The course provides instruction in the application of Calculus to infinite series and sums through the use of the power series and the Taylor series. Students will learn how to apply divergence tests to infinite series and will also learn how to derive fundamental identities such as Euler’s Formulas using Calculus techniques. The course will also cover the application of Calculus to parametric equations and polar curves.

The SAA Calculus II curriculum can provide a foundation for students who wish to receive AP credits through the successful competition of the Calculus BC exam.

Topics covered in the course include:

• Review of Derivatives
• Review of Integration
• Integration Techniques: Partial Fractions, Substitutions, Integration by Parts
• Sequences and Series: Convergence Tests
• Working with the Taylor Series
• Deriving Trigonometric and Exponential Identities: Euler’s Formula
• Applying Calculus to Parametric Equations
• Applying Calculus to Polar Curves
• AP Calculus Preparation: BC Exam

A more detailed list of topics is provided here.