Pre-Calculus

MATH ELECTIVE

This course offers a comprehensive study of pre-calculus mathematics with a Waldorf-inspired approach, integrating experiential learning, creativity, and an appreciation for the elegance and practicality of mathematics. It provides students with a strong foundation in mathematical concepts and prepares them for advanced mathematics.

First Semester

Weeks 1–2: Introduction to Functions

• Topics: Definition of functions, domain and range, evaluating functions.

• Activities:

  • Explore function notation f(x)f(x)f(x), and identify domains/ranges.

  • Introduce linear and quadratic functions.

  • Talmudic Application: Calculate the halachic hours (sha'ot zmaniyot) for a given day length.

  • Assessment: Short quiz on function notation and basic graphing.

Weeks 3–4: Linear and Quadratic Functions

• Topics: Linear functions (slope-intercept form, point-slope form), quadratic functions (standard form, vertex form).

• Activities:

  • Graph linear and quadratic functions and find roots, intercepts, and vertices.

  • Solve quadratic equations by factoring, completing the square, and using the quadratic formula.

  • Finance Application: Model cost-revenue functions to find break-even points.

  • Assessment: Problem set involving graphing and solving quadratic functions.

Weeks 5–6: Transformations of Functions

• Topics: Translations, reflections, scalings of linear and quadratic functions.

• Activities:

  • Explore transformations: vertical/horizontal shifts, stretching/compressing, and reflecting graphs.

  • Talmudic Application: Adjust halachic time calculations for seasonal changes (e.g., adjusting sunrise and sunset).

  • Assessment: Practice problems on transformations.

Weeks 7–8: Polynomial Functions

• Topics: Polynomial functions, end behavior, finding zeros, synthetic division.

• Activities:

  • Graph polynomial functions, and analyze their behavior based on the degree and leading coefficient.

  • Find and factor roots of polynomial functions using synthetic division and the Rational Root Theorem.

  • Assessment: Problem set on polynomial functions.

Weeks 9–10: Introduction to Trigonometry

• Topics: Unit circle, radian measure, sine and cosine functions.

• Activities:

  • Explore the unit circle, and convert between degrees and radians.

  • Learn to calculate sine and cosine values for key angles.

  • Talmudic Application: Use the unit circle to calculate lunar cycles and molad (new moon) for Jewish months.

  • Assessment: Short quiz on the unit circle and radian measure.

Weeks 11–12: Trigonometric Functions and Graphs

• Topics: Graphing sine and cosine functions, amplitude, period, and phase shifts.

• Activities:

  • Graph transformations of sine and cosine functions, and adjust amplitude, period, and phase shifts.

  • Finance Application: Use sinusoidal functions to model periodic business or market trends.

  • Assessment: Problem set on graphing trigonometric functions.

Weeks 13–14: Trigonometric Identities

• Topics: Pythagorean identity, sum and difference identities, double angle formulas.

• Activities:

  • Prove and apply trigonometric identities.

  • Solve trigonometric equations using identities.

  • Assessment: Midterm exam covering functions, polynomials, and basic trigonometry.

Weeks 15–16: Exponential Functions

• Topics: Exponential growth and decay, solving exponential equations.

• Activities:

  • Explore exponential functions and applications such as population growth, decay, and half-life.

  • Finance Application: Model compound interest and growth using exponential functions.

  • Assessment: Problem set on exponential functions and their applications.

Weeks 17–18: Logarithmic Functions

• Topics: Logarithmic properties, solving logarithmic equations.

• Activities:

  • Explore logarithms as inverses of exponential functions.

  • Solve logarithmic equations and apply logarithmic properties.

  • Finance Application: Use logarithms to solve for time in compound interest formulas.

  • Assessment: Practice problems on solving logarithmic equations.

Weeks 19–20: Review and Final Exam

• Activities:

  • Comprehensive review of all topics covered in the first semester.

  • Assessment: First-semester final exam.

Second Semester

Weeks 21–22: Introduction to Complex Numbers

• Topics: Complex numbers, imaginary unit iii, adding, subtracting, and multiplying complex numbers.

• Activities:

  • Explore the algebra of complex numbers and solve equations with imaginary solutions.

  • Talmudic Application: Apply complex numbers to model abstract halachic structures.

  • Assessment: Short quiz on basic operations with complex numbers.

Weeks 23–24: Complex Numbers in Polar Form

• Topics: Converting between rectangular and polar form, modulus, and argument of complex numbers.

• Activities:

  • Convert complex numbers from rectangular to polar form.

  • Explore modulus and argument and their geometric interpretations.

  • Assessment: Problem set on polar form of complex numbers.

Weeks 25–26: Sequences and Series – Arithmetic

• Topics: Arithmetic sequences and series, sum of an arithmetic series.

• Activities:

  • Derive the formulas for the nth term and the sum of an arithmetic sequence.

  • Talmudic Application: Apply arithmetic sequences to the Omer count or other time-based Jewish cycles.

  • Assessment: Problem set on arithmetic sequences and series.

Weeks 27–28: Sequences and Series – Geometric

• Topics: Geometric sequences and series, sum of an infinite geometric series.

• Activities:

  • Explore the formulas for geometric sequences and series, including applications in finance.

  • Finance Application: Use geometric series to calculate the present value of an annuity.

  • Assessment: Practice problems on geometric sequences and series.

Weeks 29–30: Vectors – Basic Operations

• Topics: Vector addition, scalar multiplication, magnitude, and direction.

• Activities:

  • Perform vector operations and explore the geometric interpretation of vectors.

  • Talmudic Application: Use vectors to model directional problems in halachic structures (e.g., eruv boundaries).

  • Assessment: Problem set on vector operations.

Weeks 31–32: Vectors – Dot Product and Cross Product

• Topics: Dot product, cross product, applications of vectors in geometry.

• Activities:

  • Calculate the dot product and cross product of vectors.

  • Apply vectors to solve geometric problems.

  • Assessment: Problem set on the dot product and cross product.

Weeks 33–34: Matrices and Systems of Equations

• Topics: Matrix operations, solving systems of equations using matrices, Gaussian elimination.

• Activities:

  • Perform matrix operations and solve systems of linear equations using Gaussian elimination and matrix inverses.

  • Finance Application: Apply matrix methods to optimize financial portfolios.

  • Assessment: Problem set on matrices and systems of equations.

Weeks 35–36: Limits and Continuity

• Topics: Definition of limits, calculating limits, continuity.

• Activities:

  • Explore limits and continuity of functions.

  • Talmudic Application: Apply the concept of limits to halachic boundary problems (e.g., determining forbidden actions as limits are approached).

  • Assessment: Practice problems on limits and continuity.

Weeks 37–38: Review and Begin Culminating Project

• Activities:

  • Review all topics from the second semester.

  • Begin independent or group projects applying pre-calculus concepts to a real-world problem.

Weeks 39–40: Culminating Project and Final Exam

• Activities:

  • Complete the final project and present findings.

  • Assessment: Second-semester final exam and culminating project presentations.