Mathematics

Department Overview:

A mathematics education should provide students with the necessary mathematics and problem-solving skills to model and apply mathematics to the real world and to a variety of disciplines.  The mathematics department has high expectations for achievement for every student.  The goal of the department is to enable all students to reach high standards and full potential in a supportive, academically focused environment.  We seek to engage students in meaningful math that stimulates curiosity and enjoyment while balancing conceptual understanding, procedural fluency, and application of mathematics.   The math program is designed to present math as a cohesive whole while emphasizing problem solving, reasoning, and modeling.  Throughout our curriculum we strive to incorporate the eight mathematical Practice Standards outlined in the Massachusetts Frameworks:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the arguments of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Daily homework is an integral part of the learning process.  Assessment may include homework, notebook checks, class work, quizzes, tests, portfolios, projects or essays. In addition, students should be prepared to express their understanding of mathematical concepts and to describe in writing how they arrived at solutions.  All students are expected to come to class daily with a scientific calculator.  The Math Department offers a calculator loan program for students on free or reduced-price lunch plans. Students should see their teachers for more information.

Accelerating in Mathematics

The purpose of accelerating in mathematics is to be able to take one of the  two AP math courses offered at Amherst Regional High School and we recognize that some students may wish to accelerate their math program in order to achieve this.  This option is demanding and should only be attempted by those students who are truly fascinated by mathematics, are highly independent, and who have a deep commitment to hard work.   Students may only accelerate in their 9th or 10th  grade year by taking Geometry H and Algebra 2H in the same year.  This is the only acceleration offered.   There is no summer acceleration in math.

Prerequisites for Mathematics Courses:

In an effort to ensure that students have the necessary background and readiness skills to be successful, most courses in mathematics have both course and grade prerequisites.  It is important for students to be aware of these requirements so that they select the appropriate courses to match their interests and prior mathematical preparation.  Because mathematics is so sequential in nature, most courses require that students earn a final grade of C- or better to continue on to the next course in the sequence.   Students who earn a grade of a D during the regular year course have two options to prepare for the next course.  The first option is to register for an appropriate summer school course at ARHS.  Students must earn a C- or better in the summer school course to move into the next course.  These students will earn two general credits during summer school.  Students not meeting these standards are expected to repeat the course during the next academic year.  The second option is to take the math elective before or concurrently with the next math course.  All grade prerequisites must be satisfied prior to beginning the next course in the sequence or the elective must be taken.  Students who earn a failing grade must repeat the course during the next academic year.

ARHS Graduation Requirement:

Eight (8) credits in mathematics are required for graduation. These can be earned only after the student has completed the eighth grade.  We recommend that every student complete the study of Algebra 1 and Geometry during their four years in order to prepare for the mathematics MCAS test.  Students must pass MCAS mathematics in order to be eligible to earn a diploma in the state of Massachusetts.  Additionally, students who do not score in the categories of “proficient” or “advanced” on the mathematics MCAS will have an Educational Proficiency Plan implemented.  These students must enroll in and pass the equivalent of a full year’s math course in each of their eleventh and twelfth grade years for a total of 8 credits in order to earn a diploma.  Students who plan to further their education should also plan to complete at least through Algebra 2H or Introduction to Pre-calculus and earn grades of C- or better, to be prepared for the SAT or ACT.  

Massachusetts State University Admissions Requirement:

For students entering Massachusetts state universities and UMass campuses, four years of math are required including Algebra 1, Geometry and Algebra 2.  Students must be enrolled in a math course in their senior year.

Grouping:

Honors classes are designed for students who enjoy mathematics and are motivated to develop their mathematical thinking.  Students should expect the work in an Honors course to have a high level of abstraction and sophistication. The work will not be just harder problems; rather, it will be work at a different cognitive level. The teacher will help students to learn at this level and will consider it a major task of the course to provide students with extensive opportunities to improve their abilities in this area. 

Ninth grade students who are interested in moving from Math 8 into Geometry Honors must take the Math Elective 2H before Geometry Honors.  All other students must have their current teacher recommend them as well as with their parent/guardian approval. Students who move from college prep to honors will need to spend additional time and effort in order to bridge the gap of skills and content acquisition. Students will need to take the Math Elective 2H if moving to Geometry H.    The department head or a designated math teacher will help students by carefully outlining expectations and offering guidance.

Course Sequences:

These are representative of the path most students follow but do not represent every pathway possible.  The Math Electives 2H is for students transitioning into the honors sequence or struggling in the honors sequence and the Math Elective is meant for students that need extra support.

Sequences for Rising 9th Graders and Rising 10th graders:

Students taking Integrated Math 1 in 9th Grade:

Course Descriptions:

Concepts in Algebra (8211)

Credits: 4

Prerequisite:  Teacher recommendation or permission of department head

This course is designed for 11th and 12th grade students that need more time to build upon and extend their foundation on the Algebra 1 and Geometry standards.  It is appropriate for students needing review in fundamental skills as well as the content in algebra and geometry.  Students review operations on positive and negative numbers, order of operations, proportional reasoning (fractions, decimals, percents and ratios), scientific notation, and operations on exponents while also exploring and practicing key algebra and geometry concepts such as graphing linear equations and systems of linear equations, slope, solving one and two step equations, area/perimeter/volume of a variety of shapes, angle relationships, Pythagorean theorem, and circles Possible Next Courses:  Geometry CP or Algebra 2 CP

Integrated Math 1: Algebra, Geometry, Probability and Statistics (351C and 361)

Credits: 6 (¾ of the year)

Prerequisite: Teacher recommendation

This course is for students who need to develop a stronger background and greater confidence in mathematics, and stronger abstract reasoning abilities before attempting Algebra 1. Units examine and connect topics in Algebra, Geometry, and Probability and Statistics.  Students will also investigate a number of long-term non-routine problems. Students will represent a linear function with tables, rules, graphs and contexts.  Students will create one representation from another.  Algebra topics include the ideas of variables, expressions, linear equations and their graphs.  Geometry topics include similarity, and right triangle trigonometry. Students work on a variety of problems involving chance occurrences in the Probability and Statistics unit. Students will solve contextual word problems using multiple strategies.    Possible Next Courses:  Concepts in Algebra, Algebra 1

Math Elective 2 (364)

Credits: 2 

Prerequisite: Teacher recommendation or permission of department head

Students are required to take this course if they received a D in either Algebra 1 or Geometry and did not attend summer school.  The results of standardized local assessments, statewide (MCAS) testing, and teacher recommendation are used as a basis for placing students in this course.   This course is designed to target key areas outlined in the Massachusetts State Frameworks to give students the support necessary to be successful in Geometry or Algebra 2 through extra time on concepts, revisiting prior concepts, and pre-teaching.   This course is taken before or concurrently with Geometry or before Algebra 2.

Algebra 1 (312)

Credits: 8  (Update - 6 credits - 3/4 year course)

Prerequisite: A minimum grade of C- in Math 8, Integrated Math 1 or permission of department head

This course will focus on teaching students to model real-world situations through the development of fundamental algebraic concepts necessary for further study of mathematics.  Students are expected to be competent in operations with fractions, decimals, and positive and negative numbers.  Students will investigate different types of expressions, equations, and inequalities.  Students will represent linear functions, exponential functions and quadratic functions with a graph, table, rule and context.  Students will create one representation from another.   Students will solve contextual word problems and non-routine problems using multiple strategies. Some other areas of study include systems of equations and expressions involving exponents.  Students will learn to graph a variety of functions and systems of equations on the coordinate plane including the use of graphing calculators and computer software.  Possible Next Courses:  Geometry or Geometry Honors 

Algebra 1 Honors Option (3121)  

Students should expect the additional work to have a high level of abstraction and sophistication. The work will not be just harder problems; rather, it will be work at a different cognitive level.  Students will be assessed at this higher level of abstraction.  Possible Next Courses:  Geometry or Geometry Honors

Math Elective 2H (365)

Credits: 2 (1 quarter class)

Prerequisite: Teacher recommendation or permission of department head

This course is designed to give students the foundation necessary for success in Geometry H.   Math 8 students must take this before entering Geometry H.  This course focuses on the remaining Concepts in Algebra 1 not taught in Math 8.  It should also be considered for Algebra 8 students that need more time with the concepts developed in eighth grade.  This course will focus on key areas such as graphical and algebraic reasoning to solve a variety of optimization problems, quadratic and exponential functions.   This course is taken before the Geometry or Geometry Honors.

Math Competency Portfolio (2601)

Credits: 8 

Prerequisite:  Teacher recommendation or permission of department head

This yearlong course is open to 10th and 11th graders who have a history of not passing the Math MCAS or who have never taken the Math MCAS in grades K-8. Students compile a portfolio of work to demonstrate that they possess the knowledge and skills normally tested on the Math MCAS. There are 60 MA standards that students must show competency on. As a result, the portfolio may take more than one year to complete.  Students engaged in compiling a portfolio also still have to SIT for the regular MCAS test.  A student whose competency portfolio demonstrates a sufficient level of achievement may be awarded a Competency Determination, which makes a student eligible for graduation.  Massachusetts requires that all students submitting MCAS Competency portfolios have attendance rates of at least 95% during the school year prior to and during the year of portfolio submission.  

Geometry CP (316B)

Credits: 4

Prerequisite: A minimum grade of C- in Algebra 1 or permission of department head

Geometry is a language intensive course in which students will investigate relationships and formulate and test hypotheses.  Students are required to interpret what they read, develop a working knowledge of new vocabulary.  and write observations, conjectures and logical supporting arguments.  In this course, students will build an understanding of similarity based on dilations and proportional reasoning. Students will investigate geometry theorems about triangles, parallel lines, and circles, and prove them by writing deductive proofs and coordinate proofs. Some additional concepts students will investigate are transformations, congruence, similarity, right triangle trigonometry, volume and surface area of three-dimensional figures and coordinate geometry.  Students will also compute and interpret probabilities of compound events and conditional probabilities. Geometric relationships will be investigated through the use of geometric constructions, physical models, and computer simulations.  Possible Next Courses:  Algebra 2H or Algebra 2

Geometry Honors (318B)

Credits: 4

Prerequisite: A minimum grade of C- in Algebra 8, Algebra 8 and Math Elective 2H, or taking Math Elective 2H concurrently or permission of department head

Students should expect the work in this course to have a high level of abstraction and sophistication. Geometry is a language intensive course.   The focus of this course is to develop deductive reasoning skills that allow students to construct logical mathematical arguments, and to improve each student’s ability to visualize and understand abstract concepts.  Students are required to interpret what they read, to learn and use new vocabulary, and to write observations, conjectures and supporting arguments.  Students will develop the concept of formal proof.  Some of the topics students will study include similarity, transformations, congruence, volume and surface area of three-dimensional figures, coordinate geometry, circles, right triangle trigonometry, the Law of Sines and the Law of Cosines.  Geometric relationships will be investigated through the use of geometric constructions, physical models, and computer simulations.  Possible Next Courses:  Algebra 2H or Algebra 2

Algebra 2 CP (322B)

Credits: 4

Prerequisite: A minimum of grade of C- in Geometry or Geometry/Honors or permission of department head

This course takes an in-depth look at functions, including direct and inverse variations, linear, quadratic, higher degree polynomials, and rational functions.  Students explore these functions graphically, numerically, and analytically and enhance their ability to work with the symbols of algebra in solving practical, real-world problems.  Students will also use systems of equations and inequalities to solve problems.  Possible Next Courses:  Introduction to Precalculus, Introduction to Statistics 1 & 2 or AP Statistics

Algebra 2 Honors (324B)

Credits: 4

Prerequisite: A minimum of grade of C- in Geometry Honors or by permission of department head

This course is a continuation and intensification of algebraic topics begun in previous courses. Students will expand their understanding of functions to include polynomial, exponential, and logarithmic functions. Students will continue representing functions with a graph, table, rule and context.  Students will create one representation from another.   Students will solve contextual word problems and non-routine problems using multiple strategies.  It is intended for students who have a strong commitment to the study of mathematics.  Major topics include sequences and series, polynomials, exponential functions, logarithms, trigonometric functions, transformations of parent graphs, systems of inequalities, solving nonlinear systems of equations, and their applications.   Students will rigorously explore the course content by using graphical, numerical, and analytical methods.

Possible Next Courses:  Pre-Calculus Honors, Introduction to Statistics 1 & 2, or AP Statistics

Introduction to Pre-Calculus (335B)

Credits: 4

Prerequisite: A minimum of grade of C- in Algebra 2 CP or by permission of department head

This course continues the exploration of many topics introduced in Algebra 2 and then students will study some of the concepts from Pre-calculus.  Topics of study include inverses and logarithms, solving systems of equations and inequalities, graphing in three-dimensions, trigonometric functions, and polynomial and rational functions. Students will rigorously explore the course content by using graphical, numerical, and analytical methods. Possible Next Courses:  Pre-Calculus Honors, Introduction to Statistics 1 & 2, or AP Statistics

Pre-Calculus Honors (337B)

Credits: 4 (Update - 6 credits - 3/4 year course)

Prerequisite: A minimum of grade of C- in Algebra 2 Honors or by permission of department head

This course is designed for students with a strong interest in the STEM fields. The course is composed of selected topics needed for entry into a Calculus course.  Students will develop their understanding of general function concepts such as function operations, composition and inverses, and transformations. Building on their knowledge of function behavior, students will investigate the graphical characteristics of rational and polynomial functions. They will expand their knowledge of trigonometric functions, focused on function graphs, equation solving, identities, and modeling. Students will learn how to express complex numbers in rectangular and polar forms and will investigate the graphs. Some of the other topics included are exponential and logarithmic functions, vectors, sequences, limits, conics and parametric functions, and matrices.   Possible Next Courses:  AB Calculus, BC Calculus, or Introduction to Statistics 1 & 2 or AP Statistics

Introduction to Statistics 1 (357a) 

Introduction to Statistics 1 Honors (357b)

Credits: 2

Prerequisite: A minimum grade of C- in Algebra 2 CP or Honors or permission of department head. This course is appropriate to take at any point after Algebra 2. 

Did you ever wonder what you can really know from a set of data or statistics about, say, medical testing or global warming? This course begins with an in-depth study of variability, in particular categorizing and quantifying different sources of variability in a data set. Topics include: Measurement variation, natural variation, production variation, sample variation, data and probability distributions, and measures of central tendency. These concepts are then applied to the analysis of bivariate data sets: correlation, residuals and least-squares lines, and linear model fitting. Emphasis is placed on assessing the predictive value of the models. This course challenges students to analyze real data of current events and confront the assumptions, power and limits of statistical analysis. The course makes use of statistical analysis software. Students analyze data, prepare reports and make presentations of their findings throughout the course.  Possible Next Course:  Introduction to Statistics 2

Introduction to Statistics 2 (358a)

Introduction to Statistics 2 Honors (358b)

Credits: 2

Prerequisite: A minimum grade of C- in Introduction to Statistics 1 or permission of department head

The topics in this course will build upon the concepts in Introduction to Statistics 1.  This course will focus on developing a deeper understanding of sample variation and the question of what can be inferred about a population from a sample.  Inferences about proportions and means are explored through polling theory, investigation of racial and gender profiling, and scientific measurement.  Related topics include sampling protocols and bias, Central Limit Theorem, confidence levels and intervals and margin of error.  The course extends these ideas and techniques to comparison of groups and populations based on samples to theoretical models using Chi Squared statistics.  This course continues to challenge students to analyze real data and confront the assumptions, power and limits of statistical analysis. The course makes extensive use of statistical analysis software. Students analyze data, prepare reports and make presentations of their findings throughout the course.  Possible Next Course:  Introduction to Precalculus

Statistics/AP (381A)

Credits: 4

Prerequisite: A minimum grade of C- in Algebra 2 Honors, a B+ in Algebra 2 CP or permission of department head. This course is appropriate to take at any point in a student’s high school math sequence after Algebra 2.  

AP Statistics prepares students to take the College Board Advanced Placement Exam.  Statistics is a full-year, rigorous, entry-level course that introduces students to the major concepts and tools for collecting, analyzing and drawing conclusions from data.  Students will explore data, plan and conduct studies, explore random phenomena using probability and simulations and study statistical inference.   Based on a syllabus set by the College Board, topics include:  Univariate data analysis, Bivariate data analysis including correlation, residuals, least-square lines, and linear model fitting, sampling protocols and bias, Central Limit Theorem, confidence levels and intervals and margin of error.  Emphasis is placed on assessing the predictive power of the models.  The ethical guidelines for statistics will be discussed throughout the course.   The course makes extensive use of graphing calculators and statistical analysis software. Students analyze data, prepare reports and make presentations of their findings throughout the course.  Students will engage regularly in solving AP Statistics Exam problems released by the College Board.    Possible Next Course:  Introduction to Precalculus, Precalculus Honors, AB Calculus or BC Calculus (successfully completed the pre-requisites)

Calculus AB/AP (341C)

Credits: 6 (¾ of the year)

Prerequisite: A minimum grade of B Precalculus Honors or permission of department head

A model rocket is launched straight upward at a velocity of 40 feet per second.  How fast is it traveling 3.5 seconds later?  What is the sum of infinitely many infinitely small quantities?  Calculus investigates these ideas.  Calculus AB prepares students to take the AB level of the College Board Advanced Placement Exam.  Based on a syllabus set by the College Board, this course contains topics taught in first-semester college calculus courses as well as some of the topics taught in the second semester. Students who are successful in this course will be prepared to take a second-semester college calculus course. Students who elect this course need a strong background in algebra, geometry and trigonometry.  The ability to think abstractly is very important.  Graphing calculators are used in class and on homework.  It is expected that students have access to a graphing calculator outside of class.  Possible Next Courses:   Introduction to Statistics 1 & 2, AP Statistics, College Class

Calculus BC/AP (342A)

Credits: 8

Prerequisite: A minimum grade of B+ in Pre-Calculus Honors or permission of department head

The BC level calculus course is an extension of the AB level course (see description above). Therefore, this yearlong course will move at a much faster pace.   It prepares students to take the BC level College Board Advanced Placement Exam.  BC calculus is an intensive course equivalent to most first-year college calculus courses.    Students who are successful in this course will be prepared to take a multivariable calculus course (usually the third semester of a college calculus sequence).  Because the AP exams are given in May, this extended syllabus (set by The College Board) must be completed by the end of April.  Students who elect this course need a strong background in the additional topics of polar and parametric equations and their graphs, logarithms, and finite and infinite series.  It is expected that students have access to a graphing calculator outside of class.  Possible Next Courses:   Introduction to Statistics 1 & 2, AP Statistics, College Class