# Mathematics

## High School

The goal of the Mathematics Department is to build mathematical competence in its students. This is achieved by providing course offerings that develop a deep understanding of mathematics and actively engage all students in meaningful mathematics, discussing mathematical ideas, and applying mathematics to real-world problems. The habits of problem solving, communicating, reasoning and proof, making connections, and using representations and mathematical models are emphasized in each of the courses.

## Courses

- Foundations Algebra I (CP -3121F/3122F)
- Algebra I (CP -3121/3122), (H -312111/312212)
- Algebra I (Acc) -313111/313212
- Foundation of Geometry (CP) -3221F/3222F
- Geometry (CP -3221/3222)
- Geometry (H -3131/3132)
- Geometry (Acc) -3141/3142
- Foundations of Algebra II -3321F/3322F
- Algebra II (CP -3321/3322), (H -3231/3232)
- Algebra II (Acc) -32231/32232

## Foundations Algebra I (CP -3121F/3122F)

**(2) Semester Course ** 2.5 credits/semester

**Open to: ** Grade 9

**Prerequisites:** Grade 8 teacher recommendation

This course is designed to support the needs of students who benefit from a individualized environment or have foundational gaps in their knowledge base. It is structured to meet individual needs in terms of pacing, rigor, and scope. Ample scaffolding of topics and review will be incorporated to support mastery of the course content. Students will benefit from frequent and varied instruction/assessment so students will receive consistent feedback on their progress. Curriculum will focus on the same big ideas with reduced depth and complexity as College Prep Algebra I. In this course, students will: 1) Solve one variable equations, systems, and inequalities; (2) Examine linear, exponential, and quadratic functions graphically and algebraically (3) Perform operations on polynomials, including addition, subtraction, multiplication, and factoring; (4) Graph and analyze the graphs of linear, exponential, and quadratic functions and (5) Extend their understanding of the laws of exponents and apply them to simplify problems.

**The course objectives are taught each year according to each student’s instructional level.**

Curriculum Units and Learning Outcomes:

#####
Pre-Algebra and Solving One Variable Equations

Solve Advanced One Variable Equations and Graphing Lines

Solving Systems of Equations

Solving and Graphing Inequalities and Special Equations

Exponential Expressions and Functions

Polynomials and Factoring

Graphing Parabolas and Solving Quadratic Equations

Functions

Statistics

## Algebra I (CP -3121/3122), (H -312111/312212)

**(2) Semester Course ** 2.5 credits/semester

**Open to: ** Grade 9

**Prerequisites: ** Grade 8 Math or Honors Grade 8 Math / teacher recommendation

The main areas of concentration of College Prep Algebra I are to 1) deepen and extend understanding of linear and exponential relationships; (2) contrast linear and exponential relationships with each other and engage in methods for analyzing, solving, and using quadratic functions; (3) extend the laws of exponents to square and cube roots; and (4) apply linear models to data that exhibit a linear trend. Students will also summarize, interpret and represent one or two variable data distribution. All topics will be covered with rigor to ensure mathematical success at the college level. College prep students will benefit from frequent and varied instruction and assessment so students will receive consistent feedback on their progress. High expectations are coupled with many vehicles for success to ensure all students can reach their potential. Honors Algebra I students should possess a strong and consistent work ethic as well as an ability to learn independently. While some review of prerequisite topics will occur, the class depth and challenge of the class content requires students enter the course with a firm mathematical foundation from middle school.

(CP) Curriculum Units and Learning Outcomes:

#####
Evaluate Expressions, Solve Equations, Graph Linear Equations

Write linear equations, solve and graph inequalities

Solving Linear Systems, Graphing Piecewise Functions

Exponents and Exponential Functions

Polynomials and Factoring

Functions

Quadratics, Graphing and Solving

Statistics

(Honors) Curriculum Units and Learning Outcomes:

#####
Prerequisite Review/ Linear Functions

Inequalities

Linear Systems

Exponents and Exponential Functions

Polynomials and Methods of Factoring

Advanced Factoring, Quadratic Graphing, and Linear Regression

Solving Quadratics

Topics in Statistics

## Algebra I (Acc) -313111/313212

**(2) Semester Course **2.5 credits/semester

**Open to:** Grade 9

**Prerequisites: ** H Algebra I / teacher recommendation

The main areas of concentration of Honors Accelerated Algebra I are to 1) deepen and extend understanding of linear and exponential relationships; (2) contrast linear and exponential relationships with each other and engage in methods for analyzing, solving, and using quadratic functions; (3) extend the laws of exponents to square and cube roots; and (4) apply linear models to data that exhibit a linear trend. Students will also summarize, interpret and represent one or two variable data distribution. In addition, students simplify and solve rational expressions and equations using various operations. Lastly, students will explore some of the important aspects of functions that they will further develop in Algebra II including but not limited to composition, inverse, and operations.

It is important to understand that Accelerated Algebra I requires not only solid mathematical prerequisite skills, but also a strong and consistent work ethic. Mastery of all middle school math topics is necessary for success because in place of review and remediation are the extra topics covered necessary for Calculus preparation.

Curriculum Units and Learning Outcomes:

#####
Evaluate Expressions, Solve Equations, Graph Linear Equations

Write linear equations, solving and graphing inequalities, systems of equations

Linear Programming and Rules of Exponents

Polynomials

Quadratics

Radicals and Rationals

Statistics

## Foundation of Geometry (CP) -3221F/3222F

**2) Semester Course** 2.5 credits/semester

**Open to: ** Grade 10

**Prerequisites:** 9th grade teacher recommendation

This course is designed to support the needs of students who benefit from a more supported environment or have foundational gaps in their knowledge base. Students are expected to take notes, create study guides, build models and use manipulatives that can be utilized during assessments. Curriculum will focus on the same big ideas with reduced depth and complexity as College Prep Geometry. In addition, there will be an emphasis on MCAS prep throughout the year. Exams, pacing, and instruction are adapted and modified to meet the needs of the students in the class. In this course, students will cover topics of congruence, transformations, similarity, constructions, algebraic proofs, modeling, classifications of polygons, introductory trigonometry, measurement (area, volume distance, etc.), and probability.

**The course objectives are taught each year according to each student’s instructional level.**

Curriculum Units and Learning Outcomes:

#####
Points, Lines, Planes and Angles

Logic and Reasoning

Parallel / Perpendicular Lines and an Introduction to Triangles

Congruent Triangles

Similarity

Right Triangles and Trigonometry

Circles and Area and Volume of Two / Three Dimensional Figures

## Geometry (CP -3221/3222)

**(2) Semester Course** 2.5 credits/semester

**Open to:** Grade 10

**Prerequisites:** Algebra I, Teacher Recommendation

The fundamental purpose of the high school Geometry is to formalize and extend students’ geometric experiences in the areas of congruence, transformations, similarity, constructions, theorem and algebraic proofs, circle theory and application, coordinate geometry, modeling, introduction trigonometry, measurement, and probability. Upon completion of this course, the student will be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; and apply transformations to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems; understand measurable attributes of objects and the units, systems, and processes of measurement; and apply appropriate techniques, tools, and formulas to determine measurements.

Honors Geometry will include the derivation and application of trigonometric formulas and employ much algebra to solve geometric problems. Note: Geometry and Algebra II may be taken concurrently.

Curriculum Units and Learning Outcomes:

#####
Tools of Geometry and Logic and Reasoning

Parallel and Perpendicular Lines andCongruent Triangles

Special Segments in Triangles

Quadrilaterals; Ratio and Proportions and Similarity; Probabilty and Measurement

Right Triangles and Trigonometry

Transformations and Circles

Extending Surface Area and Volume and Areas of Polygons

## Geometry (H -3131/3132)

**(2) Semester Course** 2.5 credits/semester

**Open to:** Grade 9 or 10

**Prerequisites:** Algebra I / teacher recommendation

The fundamental purpose of the high school Geometry is to formalize and extend students’ geometric experiences in the areas of congruence, transformations, similarity, constructions, theorem and algebraic proofs, circle theory and application, coordinate geometry, modeling, introduction trigonometry, measurement, and probability. Upon completion of this course, the student will be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; and apply transformations to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems; understand measurable attributes of objects and the units, systems, and processes of measurement; and apply appropriate techniques, tools, and formulas to determine measurements.

Honors Geometry will include the derivation and application of trigonometric formulas and employ much algebra to solve geometric problems. Honors Geometry students require a mastery of Algebra I content (including quadratics). Students in honors should also be able to work independently and solve rigorous problems.

Note: Geometry and Algebra II may be taken concurrently.

Curriculum Units and Learning Outcomes:

#####
Tools of Geometry and Logic and Reasoning

Parallel/Perpendicular Lines and Congruent Triangles

Proportions/Similarity and Probabilty

Right Triangles, Trigonometry and Areas of Polygons

Transformations and Circles

Extending Surface Area and Volume

## Geometry (Acc) -3141/3142

**(2) Semester Course **2.5 credits/semester

**Open to: ** Grades 9-10

**Prerequisites: ** Grade 9 Accelerated Algebra I or Grade 8 Advanced Algebra / teacher recommendation

Accelerated Honors Geometry focuses on six critical areas: (1) establish criteria for congruence of triangles based on rigid motions; (2) establish criteria for similarity of triangles based on dilations and proportional reasoning; (3) informally develop explanations of circumference, area, and volume formulas; (4) apply the Pythagorean Theorem to the coordinate plane; (5) prove basic geometric theorems; and (6) extend work with probability. A rigorous study of triangle trigonometry, evaluating with the unit circle, conic sections, introduction to graphing sinusoids, and an in-depth study of circles and their properties are explored. It is important to understand that Accelerated Geometry requires not only solid mathematical prerequisite skills, but also a strong and consistent work ethic. Self-motivation and the ability to work independently are also required.

Note: Geometry and Algebra II may be taken concurrently.

Curriculum Units and Learning Outcomes:

##### Points, Lines and Planes, Logic, and Parallel Lines

Triangles and their Relationships

Quadrilaterals/Polygons and Proportions/Similarity

Probability

Right Triangle Trigonometry and Transformational Geometry

Circles and Conic Sections

3D Solids and Intro to PreCalculus

## Foundations of Algebra II -3321F/3322F

**(2) Semester Course** 2.5 credits/semester

**Open to: ** Grade 11

**Prerequisites: **10th grade teacher recommendation

This course is designed to support the needs of students who benefit from a more supported environment or have foundational gaps in their knowledge base. Students will receive more in-class time spent on instruction and student practice for various topics, a significantly smaller class size and access to a lead teacher and support staff during class. Curriculum will focus on the same big ideas with reduced depth and complexity as College Prep Algebra II. Students will have more frequent formative assessment feedback opportunities than college preparatory level. Students will demonstrate their understanding of curriculum taught on assessments with support of their own graphing calculator (or class notes, as applicable) and test items are scaffolded when multiple understandings are required within the same question (for example, separating one question into two or more questions each assessing only one or two discrete skills). In this course, students will: (1) investigate and understand the characteristics of quadratic graphics; (2) use various methods for solving quadratics; (3) simplify exponential expressions and (4) perform operations on, as well as graph, polynomials and rational functions.

**The course objectives are taught each year according to each student’s instructional level.**

Curriculum Units and Learning Outcomes:

#####
Solve One Variable Equations

Functions, Quadratic Functions, Exponential Functions

Matrices

Quadratic Equations, Quadratic Formula, Complex Numbers

Polynomials, Zeros, Complex Numbers

Right Triangle Trigonometry

Composition of Functions, Inverse Functions

Logarithm and Exponential Functions

## Algebra II (CP -3321/3322), (H -3231/3232)

**(2) Semester Course** 2.5 credits/semester

**Open to: ** Grade 11

**Prerequisites: ** Algebra I / teacher recommendation

In second year course, students will explore algebra in symbolic and graphical contexts. Upon completion of this course, the student will investigate and understand the characteristics of quadratic graphics and use various methods for solving quadratics.hey will simplify exponential expressions and perform operations on, as well as graph, polynomials functions. Right triangle trigonometry, as well as, exponential and logarithmic functions are also discussed. Students will understand meanings of operations and how they relate to one another; compute fluently and make reasonable estimates; understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships and organize and display relevant data. Honors Algebra will also include topics in piecewise and composite functions; series and simplifying rational expressions.

Note: Geometry and Algebra II may be taken concurrently.

(CP) Curriculum Units and Learning Outcomes:

#####
Right Triangle Trigonometry

Relations and Family of Functions

Polynomials and Graphing Quadratics

Radicals

Solving Quadratics and Factoring

Exponential and Inverse Functions

Logarithms

Directed Angles

(Honors) Curriculum Units and Learning Outcomes:

#####
Quadratic Functions

Relations and Functions

Polynomials

Rational Exponents

Exponential and Logarithmic Funcitons

Rational Functions

## Algebra II (Acc) -32231/32232

**(2) Semester Course** 2.5 credits/semester

**Open to: ** Grades 10-11

**Prerequisites: ** Accelerated Geometry / teacher recommendation

This course extends the completed content in Accelerated Geometry and Algebra I by deepening students’ understanding polynomial, rational/radical, exponential, logarithmic, trigonometric, and composite functions using a graphic and symbolic approach. Students will further their study of trigonometry to include identities, inverses and equations. In addition, matrices, piecewise functions, and the complex number system operations, and limits are explored.

It is important to note that all topics traditionally covered in precalculus will be covered in Accelerated Algebra II ensuring students are prepared for Calculus following this course.

Curriculum Units and Learning Outcomes:

#####
Rate of Change, Functions, Matrices, and Systems

Quadratics, Polynomials, Radicals, and Rationals

Exponentials, Logarithms, and Limits

Right Triangle Trigonometry

Trigonometric Identities

- Pre-Calculus (CP -3421/3422), (H -3331/3332)
- Math for Career & College Readiness (CP) -3361/3362
- Advanced Quantitative Reasoning (CP) -3491/3492
- Calculus (H) -3441/3442
- Advanced Placement Calculus AB (AP) -3461/3462
- Advanced Placement Calculus BC (AP) -3471/3472
- Advanced Placement Statistics (AP) -3501/3502
- General Math
- ELE Math -3001ELE/3002ELE
- Mathematical Modeling
- Sports Analysis
- Statistics (H)

## Pre-Calculus (CP -3421/3422), (H -3331/3332)

**(2) Semester Course**

**Open to: ** Grade 11-12 2.5 credits/semester

**Prerequisites: ** Algebra II and Geometry / teacher recommendation

This advanced course will thoroughly cover functions and trigonometry in preparing students for future studies in calculus. This course is approached from both graphic and symbolic perspectives. The student will explore polynomial, rational, exponential, logarithmic, and trigonometric functions in great depth. This course requires that students compute fluently, work independently, have mastered prerequisite algebra skills and are well-versed in efficient use of graphic calculators.

(CP) Curriculum Units and Learning Outcomes:

#####
Applications of Trigonometry

Intro to Trigonometry Vocabulary

Circular Trigonometry

Sinusoidal Models

Analytic Trigonometry

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Matrices

(Honors) Curriculum Units and Learning Outcomes:

#####
Trigonometric Functions and Vectors

Trigonometric Graphs

Trigonometric Identitites

Trigonometric Equations

Polar and Parametric

Rational Functions

Advanced Functions

Rational Exponents and Radical Functions

Exponential and Logarithmic Functions

Conics (Time Permitting)

Sequence, Series, Probability (time permitting)

Limits, Intro to Calculus

## Math for Career & College Readiness (CP) -3361/3362

**(2) Semester Course **2.5 credits/semester

**Open to: ** Grade 12

**Prerequisites:** Algebra I / teacher recommendation

This course is designed to support the needs of students who benefit from a more supported environment or have foundational gaps in their knowledge base. Students will receive more in-class time spent on instruction and student practice for various topics, a significantly smaller class size and access to a lead teacher and support staff during class. This course is intended for grade 12 students. Placement in this course is based on the recommendation of classroom teacher and guidance counselor. Students are required to participate in a combination of in-class activities and online, self-paced studies. Class activities will include project-based and real-world applications.

## Advanced Quantitative Reasoning (CP) -3491/3492

**(2) Semester Course** 2.5 credits/semester

**Open to: ** Grade 12

**Prerequisites:** Algebra II and Geometry / teacher recommendation

This course is designed as a mathematics course alternative to pre-calculus. Throughout Quantitative Reasoning, students are encouraged to continue their study of mathematical ideas in the context of real-world problems and decision-making through the analysis of information, modeling change, and mathematical relationships. Topics include analysis of quantitative data, modeling with a variety of functions, applying concepts of vectors and matrices, advanced algebra, and applications of trigonometry.

Curriculum Units and Learning Outcomes:

#####
Fractions

Matrices

Geometry

Right Triangle Trig

Rational Functions

Vectors

Statistics

## Calculus (H) -3441/3442

**2) Semester Course** 2.5 credits/semester

**Open to: ** Grade 12

**Prerequisites: ** Acc. Algebra II or H Algebra II & H Pre-Calculus recommended / teacher recommendation

Key topics in this course include: analysis of graphs, limits of functions, asymptotic and unbounded behavior, continuity, the concept of a derivative, the derivative at a point, the derivative as a function, second derivatives, applications of derivatives, and the computation of derivatives, interpretations and properties of definite integrals, applications of integrals, the Fundamental Theorem of Calculus, techniques of antidifferentiation, applications of antidifferentiation, and numerical approximations to definite integrals. Honor Calculus is intended for those students who will take calculus in college. Students enrolled in Honors Calculus show interest in algebra-based mathematics and have mastered simplifying algebraic expressions, factoring, and graphing polynomial, rational and trigonometric functions.

Students are expected to work questions involving basic trigonometry, factoring and graphing without having to rely on the assistance of a graphing calculator.

Curriculum Units and Learning Outcomes:

#####
Prerequisites and Limits

Derivative Concept, Rules and Introductory Applications

Advanced Derivatives / Applications of a Derivative

Optimization

The Antiderivative and Basics of Integration

Methods of Integration

Topics in Calculus

## Advanced Placement Calculus AB (AP) -3461/3462

**(2) Semester Course **2.5 credits/semester

**Open to: ** Grades 11-12

**Prerequisites: ** Acc. Algebra II / teacher recommendation

This course will prepare students to take the Advanced Placement Calculus AB Examination in May, which may allow them to be awarded credit or a course waiver in college. The material covered in this course is the Calculus AB curriculum published by The College Board. The basic ideas of differential and integral calculus are developed. Work is assigned for the student to do during the summer before taking the course.

Curriculum Units and Learning Outcomes:

#####
We follow the AP College Board curriculum

## Advanced Placement Calculus BC (AP) -3471/3472

**(2) Semester Course** 2.5 credits/semester

**Open to:** Grade 11-12

**Prerequisites:** AP Calculus AB / teacher recommendation

This high-level course will prepare students to take the Advanced Placement Calculus BC Examination in May, which may allow them to be awarded credit or a course waiver in college. The material covered in this course is the Calculus BC curriculum published by The College Board. It is more extensive than the AB course. The student contemplating a career in engineering or mathematics will find this course an essential preparation. The basic ideas of differential and integral calculus will be developed. Work is assigned for the student to do during the summer before taking the course.

Curriculum Units and Learning Outcomes:

#####
We follow the AP College Board curriculum

## Advanced Placement Statistics (AP) -3501/3502

**(2) Semester Course ** 2.5 credits/semester

**Open to: ** Grades 11-12

**Prerequisites: ** Algebra II / teacher recommendation

The purpose of this statistics courses is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. In this full year course, students learn to organize data, interpret data, study elementary probability theory, and use random variables with binomial, normal, and standard distributions. The student later concentrates on inferential statistics. The student learns sampling principles and how to apply various tests to determine validity of the samples and implications to the entire population. Hypothesis testing, including t-tests, chi-square tests, and analysis of variance will also be covered. The course concludes with a section on nonparametric statistics.

Curriculum Units and Learning Outcomes:

#####
We follow the AP College Board curriculum

## General Math

**(2) Semester Course** 2.5 credits/semester

**Open to: **Grades 9-12

**Prerequisites:** Teacher recommendation

The general math curriculum enables students to develop skills and work toward progression of their math skills. The course objectives are taught each year according to each student’s instructional level.

## ELE Math -3001ELE/3002ELE

**(2) Semester Course** 2.5 credits/semester

**Open to: ** Foundational ELs and SLIFE

**Prerequisites: ** None

ELE Math is designed to provide the language of the content area of mathematics to foundational and SLIFE students in the English Learner Education program. Through this course, students will work toward progression of their English language and literacy proficiencies in the domains of listening, speaking, reading, and writing within the mathematics content area. Advancement in usage of linguistic complexity, language conventions, and academic vocabulary will be the focus of this course for English learners.

**Content for this course is based on WIDA standards and tailored to the needs of the students who are assigned.**

* EL = English Learner *SLIFE = Students with Limited or Interrupted Formal Education

## Mathematical Modeling

Semester Course

Open to: Grades 11-12

2.5 credits

Prerequisites: Algebra II

In this course, students are introduced to the cornerstones of mathematical modeling: problem identification / formulation/ investigation, development and refinement of model, experimentation, and interpretation of results. This course fosters both logical and creative thinking through a project-based curriculum. Projects are chosen based on student interests.

## Sports Analysis

Semester Course

Open to: Grades 11-12

2.5 credits

Prerequisites: Algebra II

Sports analysis immerses students in the analysis of sports through a mathematical lens. Students will use their statistical toolbox to examine sports performance, records, and strategy. Contemporary methods of data organization and sorting will be employed to teach students computational efficiency. This course focuses on math application and inquiry. Students will engage in project based learning based on their interests.

## Statistics (H)

Two Semester Course

Open to: Grade 12

2.5 credits per semester

Prerequisites: Algebra I / teacher recommendation

In this course, students are introduced to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students learn methods for gathering and summarizing data, computing measures of central tendency and variation, and making inferences about a population. Students also study elementary probability theory and use random variables with binomial and normal distributions. At the end of the semester, students design, implement, and present a statistically-based research project.

This second part of this course will concentrate on inferential statistics. The student will learn sampling principles and how to apply various tests to determine validity of the samples and implications to the entire population. Hypothesis testing, including t-tests, chi-square tests, and analysis of variance will also be covered. The course will conclude with a section on nonparametric statistics.

## Contact Us

### Kenneth (John) Astill

**Titles:**Long-Term Substitute Mathematics Teacher

**Locations:**Hopkinton High School

**Email:**

### Jennifer Fairbanks

**Titles:**Math Teacher, Boys Cross Country Coach

**Locations:**Hopkinton High School

**Email:**

### Michael Greco

**Titles:**Math Teacher, Varsity Girls Basketball Coach

**Locations:**Hopkinton High School

**Email:**