# Mathematics

At Manlius Pebble Hill, mathematics classes are composed of multi-age, multi-grade student groups. That approach allows every student to complete a required three-year sequence of college preparatory mathematics within a framework of flexibility that empowers students to progress at their own pace. Whenever possible, we utilize a five-point approach to presenting material: numerically, algebraically, graphically, verbally (descriptively) and concretely (through an activity or with a picture). This makes math approachable to students with a range of abilities and interests, and enables teachers to emphasize rational and critical thinking skills.

MPH math teachers blend the best of traditional pedagogy with proven contemporary teaching practices, including frequent collaborative projects and open-ended investigative activities. Faculty members encourage students to take intellectual risks by raising questions and formulating conjectures using mathematical arguments. Interactive computer software, graphing calculators, and the Calculator-Based Lab (CBL) are used in courses whenever possible. As part of the School’s commitment to written communication, students are required to express mathematical concepts in clear, coherent prose in their math courses.

Most students complete four years of math in the Upper School, and learn to apply mathematical thinking in many areas of their academic study. Many students also pursue a variety of elective courses, including advanced calculus-based mathematics, statistics, and independent studies with faculty. Many students enjoy participating in math so much that they choose to spend their free time sharpening the skills they have acquired. Our math league team is a popular extracurricular activity and consistently places first among similarly sized schools in Onondaga County!

completed Algebra II/Trigonometry AC. This course is for students who embrace challenges, function independently and enjoy delving into how and why mathematical concepts work. Pre-calculus AC builds on the skills developed in the accelerated Upper School mathematics sequence. It places a strong emphasis on problem solving. Sound manipulative algebra skills are necessary. Students analyze the relationships between numeric, algebraic and graphic representations of linear, quadratic, exponential, logarithmic, polynomial, rational, and trigonometric functions, along with the special characteristics of each function. The graphing calculator, Calculator Based Laboratory (CBL), various probes and programs, and computer software and applications provide a variety of ways to explore and create mathematics. Algebraic proofs are discussed to provide a greater understanding and appreciation of our mathematical system in preparation for Advanced Placement and college level math courses. One to three essays, lab reports or projects are completed each quarter to expand the student’s ability to communicate mathematical knowledge.

**Courses for Graduation Credit**

**Algebra 1 ** (*Prerequisite: successful completion of Math 8*)

In this course students review traditional topics of algebra: solving equations and inequalities, linear functions and graphing, and rational numbers. New topics include systems of linear functions and inequalities, operations with polynomials, quadratic equations, and irrational numbers. The course pays special attention to algebraic manipulation skills, communication of ideas, and the use of the graphing calculator.

**Algebra 1 AC **(*Prerequisite: teacher recommendation required*)

This accelerated (AC) course is offered to 7^{th}, 8^{th}, and 9^{th} grade students recommended by their teacher. This course is for students who embrace challenges, function independently, and enjoy delving into how and why mathematical concepts work. Students pursue traditional topics of algebra: solving equations and inequalities, linear functions and graphing, systems of linear functions and inequalities, operations with polynomials, quadratic equations, rational and irrational numbers, and logic. The course devotes special attention to problem solving skills, written communication of ideas, developing the relationship between algebraic models and graphs, and the use of the graphing calculator.

**Geometry** (*Prerequisite: successful completion of Algebra*)

The second course in this mathematics sequence introduces the student to geometric concepts. Students examine topics in plane geometry using algebra as a foundation for each unit. Euclidean geometry is introduced as an axiomatic mathematical model founded on postulates. Theorems and definitions are used to justify equations for solving problems focused on segments, angles, triangles, parallel lines, quadrilaterals, and circles. Activities are used to explore the properties of geometric shapes using hands- on explorations, including constructions with the compass and straight edge.

**Geometry AC **(*Prerequisite: successful completion of Algebra 1AC)*

The second course in the accelerated (AC) mathematics sequence is offered to students who have successfully completed Algebra 1AC. This course introduces Euclidean geometry as an axiomatic mathematical model founded on postulates, and students experience its development through the proof, exploration of theorems and properties, and applications of algebra. Students focus on creating two-column proofs for triangles, parallel lines, quadrilaterals, and circles. Activities using the compass and straight edge are used to explore the properties of geometric shapes.

**Algebra 2/Trigonometry ** (*Prerequisite: successful completion of Geometry*)

This course stresses algebraic manipulations, problem solving, exploring rational, radical, and quadratic equations. Students continue their study of algebraic structures, including the real number system. The course begins the development of function theory. Algebraic manipulations involving whole number, integral and fractional exponents are examined. Trigonometric functions are introduced from the viewpoint of the unit circle and students explore their graphs and applications. The graphing calculator is used to explore and solve equations, to check solutions, to discover properties of functions, and to simplify calculations.

**Algebra 2/Trigonometry AC **(*Prerequisite: successful completion of Geometry AC*)

This course stresses algebraic techniques, problem solving, and exploring rational, radical, and quadratic equations. Students continue their study of algebraic structures, including the real and complex number systems. The course introduces the theory of functions. Trigonometric functions are introduced from the viewpoint of the unit circle, then explored through graphs and applications. Exponential and logarithmic functions are introduced. The graphing calculator is used to solve and check equations, and to discover the properties of all the functions studied.

**Advanced Courses**

*Meet 3 credit graduation requirement.*** **

**Pre-Calculus **(*Prerequisite: successful completion of Algebra 2/Trigonometry*)

This course is for those students who would like further practice with algebraic manipulations and the study of functions. Topics include a review of algebraic manipulations, linear and quadratic equations and inequalities, characteristics of functions, and manipulations with linear, quadratic and higher degree polynomial functions, rational, exponential, and logarithmic functions. The unit circle, right triangles, graphs, and applications of trigonometry are also studied. The calculator plays an integral role in discovering mathematical concepts.

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**Pre-Calculus AC** (*Prerequisite: successful completion of Algebra 2/Trigonometry AC*)

Pre-calculus AC builds on the skills developed in the accelerated Upper School mathematics sequence. It places a strong emphasis on problem solving. Sound manipulative algebra skills are necessary. Students analyze the relationships between numeric, algebraic and graphic representations of linear, quadratic, exponential, logarithmic, polynomial, rational, and trigonometric functions, along with the special characteristics of each function. The graphing calculator, Calculator Based Laboratory (CBL), various probes, programs, computer software, and applications provide a variety of ways to explore and create mathematics. Algebraic proofs are discussed to provide a greater understanding and appreciation of our mathematical system in preparation for Advanced Placement and college level math courses.

**Advanced Placement Calculus AB **(*Prerequisite: successful completion of Pre-Calculus AC*)

Calculus allows us to analyze the behaviors of functions by relating limits to differentiation and integration. Using derivatives to describe rates of change of one variable with respect to another or using definite integrals to describe the net change in one variable over an interval of another, enables students to understand change in a variety of contexts. The relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus is a central idea in AP Calculus AB. Using definitions and theorems to build arguments and justify conclusions are a major emphasis. This course includes comprehensive preparation for the AP examination.

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**Advanced Placement Calculus BC **(*Prerequisite: successful completion of AP Calculus AB)*

The second year of calculus covers topics unique to the Advanced Placement Calculus BC curriculum and numerous applications of calculus. Topics include vector and parametric functions and their derivatives, polar coordinates, rigorous definitions of limits, advanced integration techniques with improper integrals, and an extensive treatment of infinite sequences and series. Using definitions and theorems to build arguments and justify conclusions are a major emphasis of the AP course. The course includes a thorough preparation for the AP Calculus BC exam, including a demanding review of Calculus AB from an advanced viewpoint.

**Advanced Placement Statistics ** (*Prerequisite: successful completion of Algebra 2/Trigonometry)*

AP Statistics focuses on the analysis of data with an emphasis on observing patterns in data and the departures from those patterns. Students produce models of data using regression analysis, probability, and simulation in order to anticipate and predict patterns beyond the measured data. They observe the normal distribution and learn how to mathematically describe variations from the norm. Students study the process of sampling and sampling distributions to produce a confidence interval and to make an inference about a population based on the sample. The binomial and normal distributions provide good models for inference. Students use several tests of significance to make inferences, including the “z,” “t,” and Chi-Square tests. The course includes a thorough preparation for the AP Statistics exam.

**Elective Courses**

__First Semester__

**Explorations in Math **(*Prerequisite: successful completion of Algebra 2/Trigonometry*)

Students will dive into a variety of mathematical topics typically not covered in core classes. Topics include, but are not limited to, paradoxes, proof by induction, properties of infinity, and mathematical fluency. Additional topics may be added based on the mathematical interests of the students. At the end of the course, students are expected to research a topic of interest to present to the class.

**Financial Algebra **(*Prerequisite: successful completion of Algebra 2/Trigonometry*) * *

In this introductory course to personal finance and decision making, students apply what they have learned about functions to understand income taxes, credit and debt, loans, banking practices, starting a business, car and home ownership, personal budgets, retirement planning, and the stock market. This course is designed to provide students a strong foundation in financial problem solving that will enable them to make informed decisions regarding matters of money and finance in their daily lives.

**Probability & Statistics **(*Prerequisite: successful completion of Algebra 2/Trigonometry)*

This course offers a hands-on introduction to the study of statistics and probability. This course aims to give students an understanding of the main ideas of statistics, as well as useful skills for working with data and evaluating the results of studies. Topics include exploratory data analysis, experimental design, basic probability, and methods for statistical inference. Practical examples based on reliable data are used throughout the course. Students will plan and conduct experiments or surveys and analyze the resulting data.

__Second Semester__

**Accounting **(*Prerequisite: successful completion of Algebra 2/Trigonometry)*

This course introduces students to the basics of financial accounting. Students learn the rules for tracking debit and credit as well as the structure and preparation of a General Journal and a General Ledger. The content of the course includes the preparation of a worksheet from which the students write a business’s financial statements. Students study cash controls, the maintenance of a checking account, and various special journals to make the recording of repetitive transactions more efficient. Students prepare year-end adjustments, write the financial statements of a corporation, close the books at the end of a fiscal period and study payroll accounting.** **

**Explorations in Math **(*Prerequisite: successful completion of Algebra 2/Trigonometry)*

Students will dive into a variety of mathematical topics typically not covered in core classes. Topics include, but are not limited to, paradoxes, proof by induction, properties of infinity, and mathematical fluency. Additional topics may be added based on the mathematical interests of the students. At the end of the course, students are expected to research a topic of interest to present to the class.

**Innovative Math with Coding **(*Prerequisite: successful completion of Algebra 1)*

The TI-Innovator Rover helps introduce students to coding and robotics. The simple programming language is built into the TI-84+ graphing calculator and makes it easy to program the system, run it, and trouble shoot to correct or fine-tune performance. With the TI-Innovator Rover, students will roll over roadblocks to learning by experiencing – not just seeing – math. The physical representation creates an entry point to problem solving that connects math, coding, and movement. Students will learn basic coding and use their algebra and geometry skills to solve various challenges.