General Description

In this course students begin to master the craft of mathematical problem solving and to develop an understanding of the power of abstraction in mathematics. Algebra focuses on functions, their properties, and their use both in mathematics as well as in other disciplines. Students will learn about linear functions and their properties (e.g., slope and intercept), functions of two or more variables, and, finally, quadratic equations. Students will also become proficient in the use of graphing calculators. In addition, they will create algebraic models for real-world situations. By the conclusion of the course all students will be capable of explaining and describing properties of linear and quadratic equations and where we find them in the world. Finally, students will improve their mathematical vocabulary.

Essential Questions

 �     What is algebra?
 �     In what ways can algebra be used to model situations in the world?
 �     Why do we study algebra? Why do mathematicians study algebra?
 �     How does algebra connect with other disciplines?
 �     How is mathematics used to present information in the media?

Units of Study/Activities and Projects:

 �     Introduction to Algebra
   

       �            Investigation of slope and intercept using graphing calculators
       �            Description and interpretations of situations presented in two dimensional graphs

 �     Linear Functions

       �     solving systems of linear equations
       �     modeling Distance=Rate x Time problems algebraically

 �     Quadratic Equations

�     solving by factoring
�     using logic symbolism
�     Projects using graphing calculator
�     Journal writing & Mathematical vocabulary

Learning Outcomes

As a result of taking algebra students will:

       �     master the art of translating verbal expressions into algebraic equations.
       �     be able to write an equation as a verbal sentence.
       �     understand the use of variables to represent unspecified amounts or numbers.
       �     solve algebraic problems using and not using a graphing calculator.
       �     understand how algebra can be summoned to solve problems in other disciplines in the sciences and
             the humanities

Forms of Assessment

       �     Group problem solving
       �     Project presentations
       �     Class participation in problem solving activities
       �     Homework assignments
       �     Formal tests.

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General Description

This mathematics course focuses on two- and three-dimensional Geometry.
As students encounter new concepts in geometry, basic concepts in algebra are reinforced forming an integrated approach. The first term of this course begins with Euclidean concepts using the compass, straightedge, and protractor. The course examines theoretical forms as well as shapes and structures in the real world (The Brooklyn Bridge and a cathedral are among the field trips we take.) Students learn to use numbers and symbols to communicate mathematical ideas. Measurement in both British and Metric units is practiced. Theorems will be learned and applied to problems (e.g. the opposite-angle and Pythagorean theorems). In the second term of this course students will learn about conic sections and how to perform geometric transformations. Real-world applications, hands-on activities, and self discovery are the primary methods use in this class to prepare students for study of Trigonometry and the world around them.

Essential Questions

�           What is geometry?
�           How do I use a compass, straightedge, and protractor to display knowledge of Euclidean plane
            geometry?
�           How do I measure and calculate surface area and volume of theoretical and real-world objects?
�           What are the important theorems in geometry and how do I use them in proofs and reasoning?
�           What are conic sections and how do I graph them?

Units of Study/ Activities & Projects

            �           Algebra review
            �           points, lines, rays
            �           congruence
            �           angles (use of a protractor and compass), angle theorems perpendicular, bisecting segments
                        and angles
            �           triangles: right, isosceles, equilateral, Pythagoras� theorem
            �           measuring length
            �           calculating perimeter and area
            �           quadrilaterals, circles
            �           volume
            �           solids of the above dimensions: cubes, pyramids, spheres, cones, conic sections
            �           the coordinate plane
            �           transformations (Eischer�s mathematically based designs)

Learning Outcomes

 As a result of Geometry class, students will gain:

            �           knowledge of Geometry according to New York State standards (see attached)
            �           knowledge of Euclidean plane geometry
            �           increased linguistic knowledge and performance skill. With an emphasis on communicating
                        concepts of Geometry in speech and writing.
            �           ability to to solve complex problems and tasks in mathematics (in groups and individually). 

Forms of Assessment

            �           completion and competency in group and individual activities
            �           glossary and journal responses
            �           individual projects; two- and three-dimensional
            �           class participation and group participation
            �           performance of skills using tools (i.e. straight edge, compass and protractor)

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Advanced
Algebra and
Trigonometry

General Description

This mathematics class is divided into two sections.  In the first section, basic concepts in Algebra, Geometry, and Trigonometry are reinforced and new, more advanced applications of these disciplines are explored.  This is done in order to increase and strengthen the mathematical foundation needed for Calculus.  In addition, the concept of a function is introduced and some basic functions are examined.  In the second section, higher mathematical functions are examined and manipulated, more complex number systems are applied, and calculus principles introduced.  Real world applications, hands-on activities, and self-discovery are the primary tools used in this class to prepare students mathematically for Calculus and the world.

EssentialQuestions

�         What is Pre-calculus?
�         What is a function?
�         How are these functions used in our world?
�         What are the characteristics of different functions?
�         How do I graph different functions?
�         What are the different number systems and when/how do I use each?
�         What is the rate of change?

Units of Study/Activities & Projects

�         Algebra/Geometry/Trigonometry review
�         Area Studies: working with variables, multiplying binomials, least common denominator (L.C.D.),       real numbers, right triangles
�         Number game boards, trigonometry angle relationship poster, L.C.D. game, student step by step        instructions for binomial multiplying, journal entries, native language & English glossary of words
      and definitions

�         Introduction to Functions
�         Area Studies: what is a function?, domain and range, factoring, working with graphs, imaginary
      and complex numbers, & constant and linear functions.
�         Graph scrapbook of functions, expanded number game board, creating students' own imaginary
      number problems, journal entries, native language & English glossary of words and definitions

�         Basic Functions

�         Area Studies: constant functions, linear functions, cubing functions, quadratic functions, & rate of
      change
�         Graph scrapbook of variations of these four functions, self discovery and analysis of movements of
      graphs of functions, real-world applications of these functions and their graphs, journal entries,
      native language & English glossary of words and definitions

�         Advanced Functions & Basic Calculus
�         Area Studies: polynomial functions, logarithmic functions, exponential functions, basic
      calculus
�         graph scrapbook of variations of these complex functions, real-world applications of these functions       and their graphs, journal entries, native language & English glossary of words and definitions, & (in  
      the near future) using graphing calculators and/or computer software to examine and manipulate
      these functions and their graphs.

Learning Outcomes

As a result of Pre-calculus class, students will gain:

• knowledge of pre-calculus material according to state standards.

• awareness of multiple representations of functions.

• knowledge of various number systems, their importance, and derivation 

• increased linguistic knowledge, with emphasis on the ability to communicate math ideas and to explain intricate concepts in speech and writing

• ability to work in groups to solve complex problems and to complete difficult tasks

Forms  of Assessment

• group and individual activities completion and competency
• glossary and journal responses
• quizzes to assess knowledge
• participation and role in group and in class

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General Description

This math class begins by combining previous mathematics courses and applying the material learned in the solving of complex real-life problems.  Then the class begins to analyze functions.  Students examine functions in terms of different limits and derivatives.  Different methods of differentiation and integration and the concept of continuity are also studied.  All of these areas can be applied to subjects outside of calculus such as in physics and economics.

Essential Questions

•     What is a limit?

•          When is a function continuous?

•          What is a rate of change?

•          How can we find the rate of change of a non-linear function?

•          What are some short-cuts that we can use to find the derivative?

•          How can we apply the derivative to our lives?

•          Does differentiability imply continuity?

•          How can we find the height, velocity, or acceleration of a free falling object?

•     How' is the rate of change used in economics?

•         What is integration and how is it applied?

Units of Study/Projects

Introduction

�         Area studies: Pre-Calculus review: intervals, inequalities, inequalities and absolute value, exponents and radicals, operations with exponents, finding the domain, special product and factorization techniques, complex fractions, rationalization.

�         "9 types of intervals" activity, "inequalities" group and individual activity. math in economics activity, "prove the 7 properties" activity, "find the domain" activity, " special products and factorization" poster activity, journal entries.

Functions, Gral2hs and Limits

�         Area studies: distance formula, n-dd-point formula, graphs of equations, break-even points, slope, inverse, introduction to limits, properties of limits, one-sided limits, continuity.

�         "Football distance" and "estimating sales" activities, "economics break-even point" activity, "using the slope int he real world" activity, graphs of basic functions, �properties of limits" and "operations with limits" activities, continuity reading, "determining continuity" activity, journal entries and quiz.

Differentiation:  Part One

• Area studies: derivative and the slope, rate of change-secant line and tangent line, definition of the slope of a graph, definition of a derivative, differentiability, continuity, differentiation rules.
  
• Rate of change questioning, "slope activity", "definition of the derivative application", "differentiability and continuity " reading and questions, "differentiation rules, proofs and applications", journal entries, quiz.

Differentiation:  Part Two

�         Area studies: average rate of change, instantaneous rate of change, position functions, velocity functions, more differentiation rules, marginal cost/revenue/profit.

�         Rate of change-velocity and economics activity. "Create your own 2 rate of changes", activity, "summary of differentiation rules" poster activity.

Differentiation:  Part Three

• Area studies: higher order derivatives, acceleration, implicit differentiation, related rates.
  
• "Higher order derivative " activity, acceleration word problems, related rates word problems, oral quiz

Learning Outcomes

As a result of studying calculus, students will gain:

•          Knowledge of limits and derivatives and using them to find rates of change

•      Increased linguistic knowledge, with emphasis on the ability to communicate math ideas and 
       concepts through writing and oral presentation.

•          Ability to work in groups to solve more complex mathematical problems

•           Knowledge of integration

Forms of Assessment

•          Group and individual activities

•          Journal entries and responses 

•          Written and oral quizzes

•          Participation and role in group work

•          Group discussion panel evaluation

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