# Mathematics

**DOVER**** UNION FREE ****SCHOOL DISTRICT**** **

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**New York**** ****State**** **

**Learning Standards**

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**Mathematics**

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**Standard 1: Analysis, Inquiry, and Design**

Students will use mathematical analysis, scientific inquiry, and engineering designs, as appropriate, to pose questions, seek answers, and develop solutions.

**Standard 2: Information Systems**

Students will access, generate, process, and transfer information using appropriate technologies.

**Standard 3: Mathematics**

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

**Standard 4: Science**

Students will understand and apply scientific concepts, principals, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.

**Standard 5: Technology**

Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.

**Standard 6: Interconnectedness: Common Themes**

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

**Standard 7: Interdisciplinary Problem Solving**

Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

** I. ****Integrated Algebra R**

Final Exam No Weighting

45 minutes per day for one year One Credit

** II. ****Course Description**

Integrated Algebra is the first mathematics course in high school. Algebra provides tools and ways of thinking that are necessary for solving problems i a wide variety of disciplines, such as science, business, social sciences, fine arts, and technologty. This course will assist students in developing skills and processes to be applied using a variety of techniques to successfully solve problems in a variety of settings. Problem situations may result in all types of linear equations in one variable, quadratic functions with integral coefficients and roots as well as absolute value and exponential functions. Coordinate geometry will be integratd into the invesigation of these functions allowing students to make connections between their analytical and geometrical representations. Problem situations resulting in systems of equations will also be presented. Alternative solution methods should be given equal value within the strategies used for problem solving. For example, a matrix solution to a system of equations us just as valid as a graphical solution or an algebraic algorithm such as elimination. Measurement within a problem-solving context will include calculating rates using appropriate units and converting within measurement systems. Data analysis including meaures of central tendency and visual representations of data will be studied. An understanding of correlation and causation will be developed and reasonable lines of best fit will be used to make predictions. Students will solve problem situations requiring right triangle trigonometry. Elementary probability theory will be used to determine the probability of events including independent, dependent and mutually exclusive events. **The Integrated Algebra Regents is taken at the conclusion of Integrated Algebra.**

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** III. ****Grade Requirement**

Grade 8-12

** IV. ****Mandatory Prerequisite**

Grade 8: 75% average or above in Math 8 or Math 7A

Grade 9: None

** V. ****Suggested Prerequisite**

None

** VI. ****Comments**

All students graduating must pass the Integrated Algebra Regents exam.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

** I. ****Integrated Algebra 1**

Final Exam No Weighting

45 minutes per day for one year One Credit

** II. ****Course Description**

Integrated Algebra 1A is the first mathematics course of a two year program designed to prepare students for the Integrated Algebra Regents Exam. This course explores concepts including: sets, solving equations in the context of real world problems, two dimensional and three dimensional geometric forms, and right triangle trigonometry. There will be a departmental final exam at the conclusion of this course. Students do not take a Regents exam this year. A scientific or graphing calculator will be used throughout the course.

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** III. ****Grade Requirement**

Grade 9-12

** IV. ****Mandatory Prerequisite**

Math 8

** V. ****Suggested Prerequisite**

None

** VI. ****Comments**

This course is for students who have been identified as needing Academic Intervention Services.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

** I. ****Integrated Algebra 2**

Final Exam No Weighting

45 minutes per day for one year One Credit

** II. ****Course Description**

Integrated Algebra 2 is the second mathematics course of a two year program designed to prepare students for the Integrated Algebra Regents Exam. This course offers a comprehensive review of the concepts covered in Integrated Algebra 1, as well as develops new concepts to complete the curriculum. Additional topics covered include: exponential growth and decay, linear regression, quadratic linear systems and real-life applications of mathematics. At the conclusion of this course students will take the New York State Integrated Algebra Regents Exam. A graphic calculator will be used throughout the course. **The Integrated Algebra Regents is taken at the conclusion of Integrated Algebra 1B.**

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** III. ****Grade Requirement**

Grade 9-12

** IV. ****Mandatory Prerequisite**

Integrated Algebra 1

** V. ****Suggested Prerequisite**

None

** VI. ****Comments**

This course is for students who have been identified as needing Academic Intervention Services.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

**I. Intermediate Algebra **

45 minutes per day for one year One Credit

** II. Course Description**

The purpose of the course is reinforcement and development of algebra skills needed for further study in mathematics. Topics include: Functions, Linear Functions, Quadratic Functions, Exponential Functions, Combing Functions, Solving Equations, Systems of Linear Equations, Factoring and Graphing.

** III. Grade Requirement**

High School (generally grades 10-12)

** IV. Mandatory Prerequisite**

Passing Grade on the Integrated Algebra Regents Exam or class

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** V. Suggested Prerequisite**

None

** VI. Comments**

None

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

**I. ****Geometry**

Final Exam No Weighting

45 minutes per day for one year One Credit

**II. ****Course Description**

Geometry is intended to be the second couse in mathematics for high school students. Within this course, students will have the opportunity to make conjectures about geometric situations and prove in a variety of ways, both formal and informal, that their conclusion follows logically from their hypothesis. This course is meant to employ an integrated approach to the study of geometric relationships. Integrating synthetic, transformational, and coordinate approaches to geometry, students will justify geometric relationships and properties of geometric figures. Congruence and similarity of triangles will be established using appropriate theorems. Transformations including rotations, reflections, translations, and glide reflections and coordinate geometry will be used to establish and verify geometric relationships. A major emphasis of this course is to allow students to investigate geometric situations. Properties of triangles, quadrilaterals, and circles should receive particular attention. It is intended that students will use the traditional tools of compass and straightedge as well as dynamic geometry software that models these tools more efficiently and accurately, to assit in these investigations. Geometry is meant to lead students to an understanding that reasoning and proof are fundamental aspects of mathematics and something that sets it apart from other sciences. ** The Geometry Regents is taken at the conclusion of Geometry.**

**III. ****Grade Requirement**

Grade 9-12

**IV. ****Mandatory Prerequisite**

This course is restricted to those students who successfully completed Integrated Algebra.

**V. ****Suggested Prerequisite**

None

**VI. ****Comments**

This course is strongly recommended for all college bound students.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

**I. Algebra II and Trigonometry**

Algebra 2 and Trigonometry Regents Exam No Weighting

45 minutes per day for one year One Credit

**II. ****Course Description**

Algebra 2 and Trigonometry is the final course of the three units of credit required for a Regents diploma. This course is a continuation and extension of the two courses that preceded it. While developing the algebraic techniquyes that will be required of those students that continue their study of mathematics, this course is also intended to continue developing alternative solution strategies and algorithms. For example, technology can provide to many students the means to address a problem situation to which they might not otherwise have access. Within this course, the number system will be extended to include imaginary and complex numbers. The families of functions to be studied will include polynomial, absolute value, radical, trigonometric, exponential, and logarithmic functions. Problem situations involving direct and indrect variation will br solved. Problems resulting in systems of equations will be solved graphically and algebraically. Algebraic techniques will be developed to facilitate rewriting mathematical expressions into multiple equivalent forms. Data analysis will be extended to include measures of dispersion and the analysis of regression that model functions studied throughout this course. Associated correlation coefficients will be determined, using technology tools and interpreted as a measure of strength of the relationship. Arithmetic and geometric sequences will be expressed in multiple forms, and arithmetic and geometric serices will be evaluated. Binomial experiments will provide the basis for the study of probability theory and the normal probability distribution will be analyzed and used as an approximation for these binomial experiments. Right triangle trigonometry will be expanded to include the investigation of circular functions. Problem situations requiring the use of trigonometric equa5tions and identities will also be investigated.

**III. ****Grade Requirement**

Grade 10-12

**IV. ****Mandatory Prerequisite**

This course is restricted to those students who successfully completed Integrated Algebra and Geometry

**V. ****Suggested Prerequisite**

None

**VI. ****Comments**

This course is strongly recommended for all college bound students.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

**I. Applied ****Geometry **

Final Exam No Weighting

45 minutes per day for one year One Credit

**II. ****Course Description**

This course is for students who hae successfully completed the objectives for Integrated Algebra. Applied Geometry is designed to enhance the understanding of geometric concept, terminology, and applications. Topics include Euclidean and coordinate geometry, similarity, triangle relationships, geometry of quadrilaterals, transformations, locus, and geometry of the circle.

**III. ****Grade Requirement**

Grade 10-12

**IV. ****Mandatory Prerequisite**

This course is restricted to those students who successfully completed Integrated Algebra.

**V. ****Suggested Prerequisite**

None

**VI. ****Comments**

Students completing this course will receive a Math credit.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

** I. ****Business Mathematics**

Local Exam No weighting

One Year One Credit

** II. ****Course Description**

Business Mathematics will help students develop the specific skills required to solve a wide variety of mathematical problems commonly found in everyday business situations and in their personal business situations. Topics covered include personal financial management: business operations, office applications, and banking and finance for personal and business use. A student completing the course in business mathematics will be better prepared to handle the computational skills of an entry-level business position as wll as his/her own personal financial management requirements.

** III. ****Grade Requirement**

Grades 9-12

** IV. ****Mandatory Prerequisite**

None

**V. Suggested Prerequisite**

None

** VI. ****Comments**

Business Mathematics is cross-referenced with the Business Department.

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

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**I. Calculus C #119**

45 minutes per day for one year 1.25 Weighting

** II. Course Description**

This course is the first of a three-semester sequence developing

calculus for the student majoring in engineering, mathematics,

or the sciences. Topics include the derivative, limits, continuity,

differentiability, the definite integral, the Fundamental Theorem of

Calculus, techniques of differentiation (including for transcendental

functions), applications of differentiation, mathematical modeling

and computer applications.

** III. Grade Requirement**

High School (Grade 12)

** IV. Mandatory Prerequisite**

For Seniors:Placement determined by Compass Testing done in Winter of Junior Year.

OR

Algebra II/Trig Regents Score of 65 or Better AND Pre-Calculus Grade of C (70)

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** V. Suggested Prerequisite**

None

** VI. Comments**

None

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

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**I. Statistics C #120**

45 minutes per day for one year 1.25 Weighting

** II. Course Description**

Satisfies the mathematics requirement of the Associate in Arts degree

program. Basic statistical procedures are developed. Topics include

descriptive statistics; probability; probability distributions; hypothesis

testing; confidence intervals; correlation and regression

** III. Grade Requirement**

High School (Grades 11 & 12)

** IV. Mandatory Prerequisite**

For Seniors: Determined by Compass Testing done in Winter of Junior Year

OR

Integrated Algebra Regents Score of 75 or better in the previous 2 years

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** V. Suggested Prerequisite**

None

** VI. Comments**

None

**MATHEMATICS EDUCATION COURSE DESCRIPTION**

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**I. Pre-Calculus C #121**

45 minutes per day for one year 1.25 Weighting

** II. Course Description**

This course is intended primarily for students planning to take

calculus. Topics include a review of the fundamental operations;

polynomial, rational, trigonometric, exponential, logarithmic, and

inverse functions; modeling and data analysis.

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** III. Grade Requirement**

High School (Grades 11 & 12)

** IV. Mandatory Prerequisite**

For Seniors: Determined by Compass Testing done in Winter of Junior Year.

OR

Algebra II/Trig Regents Score of 65 or Better

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** V. Suggested Prerequisite**

None

** VI. Comments**

None

**Common Core Curriculum **

**Common Core High School Algebra I**

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the

middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra I

than has generally been offered. The critical areas, called units, deepen and extend understanding of linear and

exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a

linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical

Practice Standards apply throughout each course and, together with the content standards, prescribe that students

experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of

problem situations.

**Critical Area 1:** By the end of eighth grade, students have learned to solve linear equations in one variable and have

applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now,

students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and

translating between various forms of linear equations and inequalities, and using them to solve problems. They master

the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and

solution of simple exponential equations.

**Critical Area 2:** In earlier grades, students define, evaluate, and compare functions, and use them to model

relationships between quantities. In this unit, students will learn function notation and develop the concepts of

domain and range. They explore many examples of functions, including sequences; they interpret functions given

graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations

of various representations. Students build on and informally extend their understanding of integer exponents

to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing

between additive and multiplicative change. Students explore systems of equations and inequalities, and they find

and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as

exponential functions.

**Critical Area 3:** This unit builds upon students’ prior experiences with data, providing students with more

formal means of assessing how a model fits data. Students use regression techniques to describe approximately

linear relationships between quantities. They use graphical representations and knowledge of the context to make

judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the

goodness of fit.

**Critical Area 4:** In this unit, students build on their knowledge from unit 2, where they extended the laws of exponents

to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure

in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of

equations involving quadratic expressions.

**Critical Area 5:** In this unit, students consider quadratic functions, comparing the key characteristics of quadratic

functions to those of linear and exponential functions. They select from among these functions to model phenomena.

Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions.

In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function.

Students expand their experience with functions to include more specialized functions—absolute value, step, and

those that are piecewise-defined.

**Common Core Geometry**

The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences

from the middle grades. Students explore more complex geometric situations and deepen their explanations of

geometric relationships, moving towards formal mathematical arguments. Important differences exist between this

Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized

early in this course. The Mathematical Practice Standards apply throughout each course and, together

with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject

that makes use of their ability to make sense of problem situations. The critical areas, organized into six units are as

follows.

**Critical Area 1**: In previous grades, students were asked to draw triangles based on given measurements. They also

have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions

about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria,

based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation

for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about

triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain

why they work.

**Critical Area 2:** Students apply their earlier experience with dilations and proportional reasoning to build a formal

understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply

similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles

and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures

of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first

course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many

triangles.

**Critical Area 3:** Students’ experience with two-dimensional and three-dimensional objects is extended to include

informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of

two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object

about a line.

**Critical Area 4:** Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a

rectangular coordinate system to verify geometric relationships, including properties of special triangles and

quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course.

Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.

**Critical Area 5:** In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a

radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths

and angle measures. They study relationships among segments on chords, secants, and tangents as an application of

similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when

given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate

plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to

determine intersections between lines and circles or parabolas and between two circles.

**Critical Area 6:** Building on probability concepts that began in the middle grades, students use the languages of set

theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound

events, attending to mutually exclusive events, independent events, and conditional probability. Students should make

use of geometric probability models wherever possible. They use probability to make informed decisions.