 ## Math Program

The mathematics program includes the study of quantities, of figures and of relationships between quantities and figures. This study is marked by intuition, analysis, logical rigor, elegance and simplicity. Mathematics has a dual nature, which is reflected in our program. As a language, it has elegance, reflected most clearly in the study of patterns. It is also used to express quantitative relations in other disciplines, which demands proficiency in mathematical skills and problem solving. We believe all students are capable of appreciating the beauty of mathematics and of solving sophisticated problems. Therefore, we require all students to complete the entire mathematics curriculum.

At Trinity School, mathematics instruction uses lecture, demonstration, exploration and discussion. During the foundational years of prealgebra and algebra, the teacher is more likely to lecture on the topics. The students do not have the foundations yet to lead the discussions but should be encouraged to ask probing questions. As the students continue, they will engage in more discussion and begin to take the lead in exploring the topics. By senior year, the students should be able to carry the discussion and arrive at meaningful conclusions. The objective is that our students know how mathematics works and are proficient in its use and can apply it in scientific settings.

Math and Logic (sixth grade) - In this course, students begin the transition from the concrete, arithmetic thought of elementary school years to more general algebraic thought of the middle school years. This transition starts with the study of fractions as numbers and continues with performing arithmetic operations on them and understanding their equivalent representations as decimals, percents and points on the number line. Students will also grow in their ability to formulate questions, solve word problems, and express the logic of their solutions through investigations and logic puzzles.

Prealgebra (seventh grade) - This class prepares the student for the study of algebra, and also introduces some basic geometry.Students learn to work with variables and with expressions involving variables. These skills are then used to help formulate and solve more challenging problems (story problems). The students are also introduced to solving equations using fractions and decimals. This course also includes some material on ratios, proportions and percents.

The prealgebra course includes geometry. This segment includes the properties of points, lines and planes, a study of the properties of parallel lines, congruence of triangles and measurements of elementary geometric figures such as rectangles, triangles, parallelograms and circles. If there is time, students learn to compute volumes and surface areas of three-dimensional geometric objects, such as prisms, cylinders and spheres.

The geometry segment also includes some work with constructions using straightedge and compass. This particular section begins with student-generated experimentation. Students are not told how to do the constructions, at least at the beginning, but they are simply given the rules. Once students have demonstrated competence with whole-number arithmetic, they are allowed to use calculators in class and on tests. Usually this occurs toward the beginning of the second semester.

Algebra (eighth grade) - This course is the heart of the curriculum. Students need to be competent in algebra in order to complete the rest of the mathematics curriculum successfully. At the end of the year an algebra diagnostic is given to each student.

In algebra the students begin to experience a three-pronged approach to mathematics, namely, the symbolic, the graphical and the numerical approaches. Most of the emphasis is on the symbolic. However, in this course the students begin to see the relationship between the symbolic and the graphical approaches.

As part of the symbolic approach, students learn to compute with mathematical expressions that include one or more variables. They solve linear and quadratic equations, as well as systems of linear equations. They multiply polynomials, and factor quadratic polynomials. They learn to work with fractions made up of polynomials, and to solve fractional equations. They are also introduced to equations involving square roots and to using radicals to solve equations.

As part of the graphical approach, students learn to work with the real-number line and the coordinate plane. They graph numbers and equations, both linear and quadratic. They learn to work their way around the triangle of equation, graph and slope. They also learn to find solutions graphically.

Geometry (first semester of ninth grade) - The geometry course covers only one semester, in order to make room for the three semesters of precalculus. Some of the material normally covered in geometry is found elsewhere in the program.

The emphasis in this course is on developing geometric intuition and a familiarity with the basic facts about spatial relations that are used in mathematics and science. But there is some emphasis on proofs. It is in this course that students first begin to reason formally about mathematical facts.

The students learn to use parallel lines, congruent and similar triangles and quadrilaterals, right triangles and the Pythagorean Theorem, and circles. If there is time, the students also review area and volume of figures.

Precalculus (second semester of ninth grade and both semesters of tenth grade) - This is primarily a course on the theory of functions. It prepares the student for calculus by a study of the functions used in calculus. It also includes material needed for physics. One of the key concepts is the relationship between an equation and its graph. Graphing calculators (TI-83’s) are used extensively.

The first semester is focused on the Algebra II skills. The students study higher degree polynomials and rational functions. They also learn how transformations of a function alter the corresponding graph of the function. Exponential and logarithmic functions are presented at the end of the semester.

The second semester begins with a study of trigonometry, which lasts about twelve weeks. When this is completed, students begin a study of matrices and systems of linear equations. The students are also introduced to linear programming.

Near the beginning of the third semester, students begin using a set of notes on vector spaces and linear transformations. The material on vector spaces ties in well with their physics course. This material takes about eight weeks.

Students then study conic sections. With about six weeks left in the semester, students finish precalculus by studying topics in discrete mathematics. They study sequences, series, probability and statistics.

Calculus (eleventh grade) - Calculus is taught in the junior year, and continues into the first semester in the senior year. Derivatives are covered in the first semester, and integrals in the second. In the third semester the emphasis has gradually moved toward three-dimensional calculus.

The first semester begins with a study of functions used in calculus. Students study the concept of the derivative, regarded as a rate of change, the slope of a tangent, and the limit of slopes of secants. Following this, they learn to compute derivatives. The semester ends with a group project on the applications of the derivative.

A similar sequence is followed for the study of integrals. Integrals are presented as areas that can be approximated by counting squares, and sometimes calculated by the use of geometry. Then the concept of an area function is introduced. This leads to a motivation of the Fundamental Theorem of Calculus. This motivation is so effective that often students guess this theorem before we come to it in the text. Finally, students are at least exposed to the notion that the integral is actually a limit of the sums of areas of rectangles.

The study of the concept is followed by a study of various methods of integration, including u- substitution, trigonometric substitution, partial fractions and integration by parts. Since many integrals encountered in the real world cannot be solved by closed- form methods, we then study methods of approximation. Students learn to compute with right and left sums, midpoint sums, trapezoidal sums and Simpson’s method. Finally, there is a unit on applications of the integral. Again, this concludes with a group project. The students are introduced to differential equations in the second semester. They use slope fields to visualize the multiple solutions to a differential equation. They learn to solve first-order differential equations and apply these rules in basic applications like the simple harmonic oscillator.

Calculus and linear algebra (twelfth grade) - Senior year mathematics begins with the study of multivariable calculus in two dimensions. Partial derivatives, gradients, the double integral, line integrals and Green’s Theorem in the plane comprised most of the first third of the year. Linear algebra is studied next, organized around the solution of the matrix linear equations Ax=b and around the eigenvalue/eigenvector problem. The last third of the year is an introduction to mathematical modeling. Differential equations and linear algebra, especially eigenvalues are the primary mathematical tools. Applications studied are in the area of biology, physics and economics. MATLAB is used extensively for its graphing capabilities and as a tool in calculus, differential equations and linear algebra. Students continue to write programs in all parts of the course, as programming, mathematics and science are increasingly integrated.