## Mathematics

Aligned with the National Council of Teachers of Mathematics’ (NCTM) Curriculum Focal Points and Curriculum and Evaluation Standards, the Academy’s math program is guided by the principle that "people who understand and can use mathematics in a variety of environments will be successful in college and in life." The Academy’s program is inspired by NCTM’s challenge to the notion "that mathematics is only for the select few. On the contrary … all students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding." The Academy uses Bridges to Mathematics in the Lower School.

### List of 10 items.

• #### Early Childhood (PK-K)

Young children learn by exploring. The approach of the Early Childhood mathematics program is hands-on and experiential, encouraging discovery and application of mathematical concepts to everyday life. Children develop number concepts through counting skills, numeral recognition, and one-to-one correspondence. They learn to sort and classify and to identify, create, and extend patterns. By the end of Kindergarten, most students can solve simple problems, are familiar with the concept of time, and have been introduced to value through manipulating coins. They are also familiar with the concepts of addition and subtraction.

The Academy's math program is developmental and recognizes that children are at different places on a continuum. Throughout the early years, children are exposed to activities designed to explore essential mathematical building concepts. Everyday Math and Primary Concepts are used through the middle of Grade 1, while Techniques of Problem Solving (TOPS), Minute Math, and manipulatives (Cuisenaire rods, Unifix cubes, base 10 blocks, geoboards, etc.) are used throughout the entire Lower School Math program.

In Grade 1, children master numbers through 100 and develop familiarity with numbers through 1000. They begin to understand place value are able to write three-digit numbers. The concept of fact families and the reciprocity of addition and subtraction are introduced. They begin to identify geometric shapes, congruence, and symmetry. By the end of Grade 1, most students are able to translate words into math by practicing solutions to word problems.

Children develop competency in manipulating numbers, including working with number sense, place-value concepts, fractions, money with decimals, measurement operations, and computation. Mental computation is introduced, along with estimation, developing simple thinking strategies for basic facts and studies of patterns and relationships. In Grade 2, instructional practices include use of manipulatives, cooperative work, learning to frame good questions, writing about mathematics problems and solutions, and beginning to use calculators and computers as tools in mathematics.

In Grade 3, students build on their counting skills, counting forward and backward and adding whole numbers. They begin to understand and practice multiplicative reasoning, the understanding that certain situations require multiplication or division as appropriate operations. They solidify counting patterns and place value to read and write whole numbers through 100,000. Students begin multiplication and division, develop addition and subtraction multi-digit procedures, and explore properties of addition and subtraction. They develop accuracy with basic operations and relations.

Three central mathematical themes become increasingly important throughout upper elementary student development: multiplicative reasoning, equivalence, and computational fluency. After focusing on multiplicative reasoning in Grade 3, equivalence is introduced. Equivalence is the manipulation of numbers and figures of equal values in different forms; for example, a fraction can also be represented as a decimal and inches can be represented as feet. Within this context, students learn the properties of operations and work with fractions and decimals. They also organize and interpret data, learn to use appropriate units and tools for measurement, and demonstrate a consistent understanding of patterns, relations, and functions.

Computational fluency with whole numbers is a major focus in Grade 5. Students practice and experiment with basic number combinations and develop the ability to formulate efficient methods of computing. Problem solving in new environments begins the process of "transfer," the capacity for a student to apply a set of rules or concepts to a new situation. The classroom is a place where students begin to take intellectual risks as they try new approaches and strategies. Students solidify their understanding and practice with whole numbers and decimals, addition and subtraction, multiplication and division, measurement, and integers.

Students are given an opportunity to participate in the MATHCOUNTS program.

Grade 6 students master the basic skills related to fractions, whole numbers, decimals, and percents, and recognize the equalities associated with these different numerical representations. Students also focus on introductory algebra and geometry skills with a heavy focus on mathematical vocabulary.

Grade 7 math seeks to develop and strengthen students’ calculative skills in basic operations with whole numbers, integers, positive and negative fractions, and decimals, as well as solving simple equations, working with ratios, proportions, and the interrelationships among fractions, decimals, and percents.

Pre-Algebra includes solving multi-step equations and inequalities, computing and applying statistical data and probability, solving and graphing linear equations, and applying basic geometric principles in plane figures.

Honors Pre-Algebra is an intensified course that addresses the same topics of pre-algebra and geometry but in greater depth and complexity.

Grade 8 students develop and use critical thinking skills while incorporating technology to solve everyday problems and situations through math. Students will understand, appreciate, and ultimately express with clarity the power of algebra in the development of effective problem solving, as well as its role in providing a firm foundation for success in all subsequent higher level mathematics courses.

In Algebra and Honors Algebra, working with linear relationships and corresponding representations in graphs, tables, and equations, factoring, systems of equations, and word problems is critical. Students will be required to demonstrate proficiency in their ability to accurately and effectively communicate in algebraic language orally and in writing.
• #### Grades 9 - 12

ALGEBRA COURSES

Algebra I establishes the vocabulary and symbolism of algebra and includes evaluating expressions, properties of real numbers, rational and irrational numbers, square roots, function theory, solving and graphing linear equations and systems, solving and graphing linear inequalities and systems, applying exponent properties, scientific notation, simplifying polynomial expressions, solving polynomial equations, basic factoring, solving and graphing quadratic functions, exponential growth and decay, and word problems. Students are introduced to matrices, probability, data analysis, and simplifying and solving rational expressions and equations.

Courses in Algebra II serve as a natural extension of topics covered in Algebra I. The content and pace of the course are rigorous and require students to develop higher-order thinking skills in preparation for precalculus. Topics include polynomial and rational expressions and functions, systems of equations and inequalities, functions, radicals, irrational numbers, complex numbers, synthetic substitution, graphing polynomial functions, composition of functions, inverses, exponential and logarithmic functions, and curve fitting.

Honors Algebra II covers all topics, however, more difficult problems are explored with the expectation that students are highly proficient in Algebra I topics, can work at a very fast pace, will complete extensive assignments, and require minimal extra help from the instructor. Additional topics include advanced problem-solving along with an in-depth examination of functions.

GEOMETRY COURSES

Geometry courses require mastery of the concepts of algebra including quadratics and radical expressions. Students explore both Euclidean and solid geometries with a particular emphasis on plane geometry. Topics of study include an introduction to logic and proofs, triangles, special quadrilaterals, polygons, perimeter and area of figures, surface area and volume of solids, similar shapes (ratio and proportion), circles, and trigonometry. Applications of these topics are incorporated into the lessons and assignments.

Honors Geometry studies additional topics which may include: indirect proofs, sequences, pattern recognition, vectors, volumes of revolution, equations of lines in three space and planes. However, important differences lie in the pacing and emphasis of the course. Additionally, an emphasis is placed on independent learning and higher-level thinking skills. Students are routinely expected to successfully tackle the more challenging problems in planar and solid geometry.

PRECALCULUS & TRIG COURSES

Students in Precalculus/Trig are expected to work at a rigorous pace and to spend a significant amount of time on homework assignments and related activities. Precalculus/Trig topics include conic sections, binomial theorem, sequences and series, counting principles, and introductory probability concepts, compositions of functions, inverse functions, exponential and logarithmic functions. Trigonometry is explored with the emphasis on the circular functions. Students will work extensively on graphing, identities, solutions of right and oblique triangles, and inverse functions and their graphs. Students also study proofs, parametric functions, and complex numbers.

Honors Precalculus/Trig covers all of the topics covered in Precalculus/Trig as well as additional topics in vectors, complex numbers, graphing techniques, end-behavior models, applications, inequalities, parametric and polar equations, conic sections, partial fractions, combinatorics, probability, and sequences and series. Additionally, an emphasis is placed on independent learning and higher-level thinking skills.

CALCULUS COURSES

In Calculus and AP Calculus AB* courses, the theory of calculus, understanding why and how techniques work and when to use them, is a central focus each time a new topic is presented. The course follows closely, but is not limited to, the topics and techniques specified by The College Board. The differential calculus topics include limits, continuity, curve sketching, derivatives of polynomial functions, exponential/ logarithmic functions, trigonometric and inverse trigonometric functions, and optimization and related rates applications. The integral calculus topics include Riemann sums, the Fundamental Theorem of Calculus, methods of integration, area under a curve, volumes of revolution, differential equations, slope fields, and applications (such as exponential growth and decay).

AP Calculus BC* is a challenging course that continues from where Calculus left off. Students need to have already mastered differentiation and basic integration. The course will review some of the concepts covered in AP Calculus AB but at a higher level. The course follows closely but is not limited to the topics and techniques specified by The College Board. Topics include L’Hopital’s Rule, advanced methods of integration, improper integrals, the calculus of polar functions, infinite sequences and series, Taylor and power series, vector functions, polar calculus, and first-order differential equations and slope fields. Applications will focus on area accumulation, volume, surface area, applied differential equations, growth models, approximation techniques, work, fluid force, center of mass, and business applications.

Multivariable Calculus continues from where AP Calculus BC ended. Specific topics include: three-dimensional coordinate systems, lines, planes, and quadric surfaces; vector-valued functions, parametric equations, and curves in two- and three-dimensional space; arc length and curvature; differential calculus of functions of more than one variable with limits, continuity, partial derivatives differentials, tangent planes, the chain rule, directional derivatives and gradients; maximizing and Lagrange multipliers; integral change of variables; multiple integration in various coordinate systems; line integrals and surface integrals; curl and divergence; The Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theorem and the Divergence Theorem.

STATISTICS COURSES

Students in Statistics and AP Statistics* learn mathematical skepticism and rigorously controlled experimental design and analysis. Topics include counting principles, probability, sampling techniques, exploratory data analysis, probability, probability distributions, normal distributions, and hypothesis testing. Students also engage in a rich and varied experience with applied mathematical concepts including data analysis and interpretation, methods of data collection, and planning and conducting studies. Major topics include descriptive statistics, probability, normal, Chi-Square and t-distributions, confidence intervals and tests of significance. Data analysis requires the use of statistical graphing calculators and modern statistical software.

*Advanced Placement and honors-level courses cover material that is found in regular classes but at a much faster pace, depth, and breadth.

### List of 7 members.

• #### Shane Mills

Senior School Mathematics Teacher
Grove City College - B.S.
Saint Vincent College - M.S.
• #### John Basinger

Senior School Mathematics Teacher
Penn State - B.A
West Virginia University - M.S.
• #### Seamus Coons

Senior School Mathematics Teacher
State University of Cortland - B.S.
• #### Christy Fairman

Senior School Mathematics Teacher
Westminster College - B.A.
College of William and Mary - M.S.
• #### Matthew Michaels

Senior School Mathematics Teacher
University of Pittsburgh - B.S.
University of Pittsburgh - M.A.
• #### Frederick Moreno

Middle School Mathematics Teacher
Duquesne University - B.S.
Duquesne University - M.Ed.
• #### Regina Ricci

Middle School Mathematics Teacher
Duquesne University - B.S.

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