Mathematics is taught in both morning lesson blocks and track classes. The track classes extend throughout the year, four times per week, 50 minutes per class. These mixed-grade classes are formed according to competency level. Students use textbooks and do daily homework assignments to work on skills in a logical and continuous progression. The three-week morning lesson blocks, taught by grade level, allow for a more expansive exploration of new mathematical topics, with hands-on work and in-depth research. These morning lessons are one hour/50 minutes in length.

The goals of WHSP mathematics are to awaken in students an awareness and appreciation for the world’s predictable patterns and structures, their harmony and beauty, their order and practicality. In this process, students become aware and reflective of their own thinking process, and ideally they can release old habits of thinking and understanding in order to experience more openness and mobility. They develop rigor in examining assumptions and making decisions, learn to extract pertinent information from complex situations, and develop a repertoire of skills that give confidence when working with the unknown. This can bring them to philosophical reflection through mathematics, as wells as the ability to apply calculation methods to everyday life.

Track Classes

Algebra 1

This introductory course in algebra skills focuses on the ability to calculate and reason symbolically. Topics include variables and equations, applications and problem solving, numbers on a line, the four basic operations on real numbers, transforming equations by these processes, solving problems using these techniques, polynomials and problem solving, factoring polynomials, linear equations and systems, functions, rational and irrational numbers, quadratic functions, the quadratic formula, and joint and combined variation. The students are required to do daily homework and to take quizzes and unit tests.

Geometry

Geometry students study Euclidian geometry with an emphasis on formal logic and proof, visualizing and transforming pictures, and applying geometry to life around us. The students participate in a combination of lecture, group work, and individual practice. They are assessed through class work, homework, chapter tests, a midterm and an end-of-year comprehensive exam. The concepts studied include: the shortest path, distance between points and lines, reflections, types of angles and triangles, parts of triangles, Euler’s line, angles in polygons, triangle theorems, congruency, similarity, tessellations, properties of transversals, types of quadrilaterals and their properties, area, perimeter, surface area, volume, properties of circles, Pythagorean theorem, special right triangles, Fibonacci, and the golden section. Significant attention is placed on problem solving, formal logic and proof, the artistic element of geometry, and the wonder and usefulness of geometry throughout history.

Algebra 2

This course complements and expands the mathematical content and concepts of Algebra 1 and geometry. Students evaluate and simplify basic algebraic expressions, solve linear inequalities and absolute value equations and inequalities, graph and write equations of lines, use laws of exponents, solve equations by factoring, solve polynomial inequalities, simplify rational algebraic expressions, solve equations with fractions and radicals, learn rules and properties of complex numbers, solve equations by completing the square and by the quadratic formula, graph parabolas, solve equations using direct/inverse/joint variation, use synthetic division, extend the laws of exponents to include fractions and irrational numbers, learn and apply basic properties of logarithms.

Pre-Calculus

This course is a continuation of Algebra 2 and a precursor to Calculus. Topics include rectangular coordinates and graphs, linear equations in two variables, transformations of functions, variation, quadratic functions, trinomial factoring, higher degree polynomial functions and their zeros, complex numbers, exponential and logarithmic functions, trigonometric and inverse trigonometric functions and their graphs, laws of sines and cosines, vectors and dot products, trigonometric form of complex numbers, multivariable linear systems of equations, matrices, arithmetic and geometric series, conics, parametric equations, and polar coordinates. The students are required to do daily homework and to take weekly quizzes, unit tests and semester exams.

Calculus

This beginning calculus course meets three times weekly over the year. A prerequisite is credit in a pre-calculus course or familiarity with its content, as determined by the instructor. Topics include libraries of functions, derivatives, shortcuts to differentiation, using the derivative, definite integrals, constructing anti-derivatives, integration, using the definite integral , integration techniques, L’Hopital’s rule, improper integrals, and solutions to differential equations using separation of variables. The students are given daily homework assignments and take unit tests and semester exams.

Morning Lessons

Ninth Grade—Permutations and Combinations

This course explores the basic concepts in counting, probability, and statistics. The students learn how to count arrangements using permutations and combinations, including permutations with repeats, permutations with subsets, arrangements in a circle, standard combination and permutation notation (nCr and nPr), Pascal’s triangle, the Serpinski triangle, the binomial theorem and binomial expansion, mean, median, quartile, range, interquartile range, box and whisker plots, and standard deviation. In addition, students apply their knowledge to deepen their understanding of the impact of technology in studying passwords, password cracking, barcodes, phishing, malware, viruses and the like. They have a direct experience with probability and gaming, which deepens their understanding of the science of chance.

Tenth Grade—Conic Sections

The students begin their study of conic sections through artistic activity and construction, which develops into a more purely numerical representation. They work with the concepts of symmetry, unity, distance, and locus of points to construct the family of conics. Special emphasis is placed on visual representations (form) and how the family of conics can be metamorphosed into each other. The conics are constructed both by pointwise and linewise constructions. Students understand the conics and their relationship to each other within their understanding of infinity and its relationship to lines and circles. They learn the relationship between foci and the directrix and apply this to curvature and eccentricity. Coordinate geometry is used to represent conic sections using formulas and algebra. Students study Dandelin spheres as well as the application of conics to the real world.

Tenth Grade—Trigonometry

The students study the basic concepts of trigonometry, beginning with the definitions in right triangle trig. They create a tool and use it to measure a tall building, tree, etc. in the method of the ancient Greeks. They solve triangles by trigonometry. They learn the six basic trig functions and explore 30-60-90 and 45-45-90 triangles. Upon exploring the unit circle, the students gain facility with quadrantal angles, angles in standard form, and reference angles. Building upon this knowledge, they derive graphs of the sine and cosine functions and explore their application to waves. They learn to solve non-right triangles using the laws of sines and cosines.

Eleventh Grade—Projective Geometry

Projective geometry is approached almost entirely through straightedge constructions, with the resultant 3-dimensionality of the drawings used as a basis for discussions. Topics include point-line duality, Pascal Line and Brianchon Point on a golden ellipse, Pascal Line on a golden hyperbola, Desargues’ Theorem, harmonic four configurations, line and point ellipses based on a pentagram, the Mobius net, a history of geometry, and two of von Baravalle’s exquisite constructions of his Kegelschnitte. Students are required to submit daily pages in a timely manner, to compile these into a morning lesson book, and to take a final exam.

Twelfth Grade—Calculus

The concepts of the calculus are developed from its historical roots, from Archimedes’ use of limits to the formulation of the differential and integral calculus by Newton and Leibniz in the 17th øcentury. We begin by repeating the inclined plane experiment conducted by Galileo in the 16th century as an approach to the idea of instantaneous velocity. There follows the power rule for differentiation, using the method of increments, and the tangent line as the geometrical equivalent of the derivative. Newton’s approach to problems of motion—anti-differentiation—leads to the geometrical interpretation of the integral as the area under a curve. We conclude with a synopsis of the three pillars of mathematics—algebra, geometry, and analysis—and the place of calculus (and science in general) as a model for the physical world. The students are required to turn in daily assignments in a timely manner, to do problems, take a final exam, and submit a morning lesson book.