High School Math with Mr. Starzynski

^{Whether you're a fan of math or find it challenging, Mr. Starzynski's lessons will help you see and understand math through the Waldorf approach.}

<sup>Our math courses provide an academic year-long math track designed to enhance your ability for logical thinking by engaging in and honing math skills. In this way, you may advance your learning,  increase your appreciation for, and improve your overall understanding of the beauty of mathematics.

Additionally, each course features a 4 week Math Main Lesson block, allowing the student to immerse themselves in a new topic thoroughly. This deep dive ensures proper introduction and exploration of the topic, engaging head, heart, and hands through historical context, artistic expression, and problem-solving activities.

Course access for 18 months.</sup>

Here's a glimpse into what each course provides:

• Algebra I:

  This course focuses on the fundamental structure of the real number system and the application of algebraic and combinatoric reasoning. Students will develop mastery in solving, graphing, and modeling linear equations. The curriculum integrates analytic geometry, focusing on the relationship between algebraic functions and their graphical representations on the Cartesian plane.

  Key areas of study include:

  • Linear Functions: Mastery of multiple techniques for solving linear equations.
  • Quadratic Equations: Solving second-degree equations through factoring.
  • Geometry: The study of angles, area and volume.
  • Analytic Geometry: The study of slope, intercepts, and coordinate transformations.
  • Combinatorics and Probability: An exploration of counting principles, including permutations and combinations, to determine the likelihood of outcomes in complex events.
  • Discrete Mathematics: Application of logical reasoning and mathematical theory to problem-solving.

• Geometry: 

  This course provides a rigorous investigation into Euclidean geometry, deductive reasoning, and the mathematical properties of two- and three-dimensional figures. Students develop a foundational understanding of geometric logic through formal mathematical proofs, postulates, and definitions.

  Core Areas of Study:

  • Geometric Congruence and Similarity: Analysis of triangles, quadrilaterals, and polygons; application of ratios and proportions to geometric figures.
  • Mensuration and Solid Geometry: Calculation of area and volume for complex three-dimensional figures, including cones, spheres, and polyhedra.
  • Right Triangle Trigonometry: Application of the Pythagorean Theorem and basic trigonometric ratios (sine, cosine, tangent).
  • Analytic Geometry: Integration of algebraic functions with geometric coordinates. Systems of equations.
  • Constructions and Logic: Use of compass and straightedge for systematic geometric constructions; exploration of incommensurable magnitudes and irrational numbers.
  • Advanced Algebraic Transitions: Simplification of complex expressions and the solution of multi-step linear and quadratic equations as they apply to geometric modeling.

• Algebra II: 

  This course expands upon the concepts of Algebra I, focusing on the study of advanced functions, systems of equations, and the complex number system. Students develop proficiency in analyzing, graphing, and solving non-linear equations and exploring the relationships between algebraic and geometric structures.

  Core Areas of Study:

  • Polynomial Functions: Analyzing graphs, determining roots, and performing operations on higher-degree polynomials.
  • Complex Numbers: Introduction to the imaginary unit i and performing operations within the complex number system.
  • Exponential and Logarithmic Functions: Modeling growth and decay and solving equations involving logarithms and exponents.
  • Advanced Trigonometry: Comprehensive study of the unit circle, trigonometric ratios, and the law of sines/cosines (including the ambiguous case).
  • Systems of Equations and Inequalities: Solving multi-variable systems using algebraic and graphical methods in 2D Cartesian coordinates.
  • Synthetic and Projective Geometry: A study of geometric invariance and transformations. This includes the application of theorems by Pappus, Desargues, and Pascal to explore perspective and coordinate projections.
  • Analytical Modeling: The use of coordinate geometry to solve complex equations and simplify intricate algebraic expressions.

• Pre-Calculus:

  This course serves as a comprehensive bridge between advanced algebra and calculus, emphasizing the theoretical and practical application of functions, limits, and data analysis. Students develop a rigorous understanding of the properties of functions and the historical evolution of mathematical theory.

  Core Areas of Study:

  • Advanced Function Analysis: Systematic study of rational expressions, radicals, and the Fundamental Theorem of Algebra.
  • Analytic Geometry and Transcendental Functions: Advanced investigation of logarithmic and trigonometric functions, focusing on their properties, graphs, and applications in modeling real-world phenomena.
  • Statistics and Quantitative Data Management: Analysis, organization, and graphical representation of large data sets. Students utilize statistical methods to interpret data trends and present findings through project-based assessments.
  • The Theory of Limits and Continuity: Introduction to the concept of a limit to resolve the "division by zero" paradox. Students apply limit properties to define continuity and transition from average to instantaneous rates of change.
  • Differential Calculus Foundations: Derivation of the derivative for power functions using the formal definition of a limit. Applications include modeling kinematics (speed) and economic fluctuations.
  • Historical Development of Mathematical Theory: A technical survey of mathematical progress from antiquity to the modern era. Topics include the discrete vs. continuous debate, the development of infinite series, the formalization of irrational numbers, and the chronological discovery of the Calculus.

Mr. Starzynski