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This curriculum is part of the Educational Program of Studies of the Rahway Public Schools.

ACKNOWLEDGMENTS

Christine H. Salcito, Director of Curriculum and Instruction Barbara Pyne, Program Supervisor of Special Education

The Board acknowledges the following who contributed to the preparation of this curriculum.

Subject/Course Title: Date of Board Adoptions:

Special Education June 27, 2013 Algebra I, Geometry & Algebra II Grades 9-12

RAHWAY PUBLIC SCHOOLS CURRICULUM

UNIT OVERVIEW

Content Area: Algebra

Unit Title: Expressions, Equations, and Inequalities

Target Course/Grade Level: Algebra 1/Grade 9

Unit Summary:

• Using the distributive property • Interpret parts of an expression • Solving one, two, and multi-step equations • Solving literal equations • Solving one, two, and multi-step inequalities • Graphing inequalities

Approximate Length of Unit: 8 weeks

Primary interdisciplinary connections: Science, Consumer Science, and Language Arts

LEARNING TARGETS

Unit Understandings Students will understand that…

• You can solve equations and inequalities using the algebraic properties of equality. • Equations can represent real world situations. • Properties are used in solving equations and inequalities. • Real world problems can be solved using equations and inequalities.

Unit Essential Q uestions

• Identify the coefficient, variable, and constant in the expression 4x + 2. • Simplify the expressions: • 2x – 6 +3x • -4(5x – 2) • 3(x – 1) + 2(x + 2) – 6 • Solve varying types of equations. • How can real world situations be represented as equations? • Create an equation for the given situation and solve: Jenna buys 2 sodas and spends $3.10. How much did each soda cost? • Solve a literal equation for a given variable. Example: Solve A = ½bh, for h. • Solve and graph an inequality. • Create an inequality for the given situation and solve: Mike buys a sandwich for $4.50, a soda for $1.89, and wants to buy a snack. He cannot

spend more than $7.00 for his lunch. How much can mike spend on his snack? • Identify the properties used to solve an equation.

Content Area Domain Content Area Cluster Standard

Expressions Seeing Structure in Expressions A-SSE.1, A-SSE.1a

Equations Creating Equations A-CED.1, A-CED.4

Equations Reasoning with Equations and Inequalities A-REI.1

Knowledge and Skills Students will know…

• Vocabulary – o Real numbers o Expressions, coefficient, constant, variable, like terms, simplify, order of operations o Real numbers, positive, negative, integers, irrational numbers, absolute value, o Equations, solutions, evaluate o Algebraic properties of equality, distributive property, substitution property o Literal equations, formulas o Proportion, ratio, probability

• Formulas – o Fahrenheit to Celsius o Simple Interest Formula o Density Formula

Students will be able to…

• use mathematical vocabulary fluently • use appropriate vocabulary to describe expressions • interpret expressions that represent a quantity in terms of its context • interpret parts of an expression, such as terms, factors, and coefficients • use order of operations to simplify expressions • evaluate expressions • solve one, two, and multi-step equations • use the distributive property • create equations and inequalities in one variable and use them to solve problems • explain each step in solving a simple equation from the equality of numbers asserted at the previous step, starting from the assumption that the

original equation has a solution • construct a viable argument to justify a solution • rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • attend to precision • look for and express regularity in repeated reasoning

EVIDENCE OF LEARNING

Assessment What evidence will be collected and deemed acceptable to show that students truly “understand”?

• Unit tests, quizzes, • Open-ended problems that involve written responses • Daily student work • Student/group presentations • Daily Homework

Learning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results?

• Mathematical investigations • Fundraising activity pg. 154 (algebra text). • Study Island assignments • Create real world situations in a word problem format and solve each other’s problems (Language Arts). • Use real-world literal equations and solve them for a variable different than the one given (science formulas: density, Fahrenheit to Celsius, etc.)

RESOURCES Teacher Resources:

• Algebra 1 Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Math websites

Equipment Needed:

• Calculators • Computers • Projectors


UNIT OVERVIEW

Content Area: Algebra 1

Unit Title: Unit Two - Linear Functions


Unit Summary:

• Understand the concept of a function and use function notation • Identify the domain and range of linear functions • Graph linear functions • Relate the domain of a linear function to its graph. • Write a linear function that describes a relationship between two quantities


Primary interdisciplinary connections: Language Arts and Science

LEARNING TARGETS


• Equations and graphs are alternative (and often equivalent) ways of depicting and analyzing data and patterns of change. • Functional relationships can be expressed in a variety of ways: real contexts, graphs, algebraic equations, tables, and words. • Functions can be analyzed using different representations. • A variety of families of functions can be used to model and solve real world problems.


• How can change be best represented mathematically? • How can we use mathematical language to describe change? • How can we use mathematical models to describe change or change over time? • How can patterns, relations, and functions be used as tools to best describe and explain real-life situations? • How are functions and their graphs related? • How can technology be used to investigate properties of linear functions and their graphs?


• Vocabulary – o Relation, domain, range, function, correlation, ordered pair o Rise, run, slope, rate of change, x-coordinate, y-coordinate, y-intercept o Slope-intercept form, standard form, and point slope form o Parallel and perpendicular lines

• Formulas –

o Slope intercept form - y = mx + b o Standard form - Ax + By = C o Point slope form


Interpreting Functions Understand the concept of a function and use function notation F-IF.1, F-IF.2, F-IF.4, F-IF.5, F-IF.6

Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.7a, F-IF.8

Building Functions Build a function that models a relationship between two quantities F-BF.1

• How to write linear equations and functions • How to graph linear equations • Whether a relation is a function • The domain and range of a function • How to use slope intercept form, standard form, and point slope form • How to find and identify the slope and y-intercept of a linear function • How to identify and graph horizontal and vertical lines • How to identify parallel and perpendicular lines


• Define relations, functions, domain, range, correlation, slope, and y-intercepts • Calculate the slope of a line • Find the rate of change from a graph • Use slope intercept form, standard form, and point slope form • Write linear equations in all forms and graph them • Graph horizontal and vertical lines • Write linear equations in function notation • Describe and identify parallel and perpendicular lines • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Linear function video at http//www.khanacademy.org/video/basic-linear-function • Study Island assignments • Portfolio Activity pg. 233 • Graphing calculators


• Algebra Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Math websites (Khan Academy).

Equipment Needed:

• Graphing calculators • Projector and computers • Graph paper • Rulers • Tape measures and spheres (portfolio activity)


UNIT OVERVIEW


Unit Title: Unit Three - Systems of Equations and Inequalities


Unit Summary:

• Understand, explain, and solve systems of equations using three methods. • Understand, explain, and solve systems of inequalities.


Primary interdisciplinary connections: Science and Business

LEARNING TARGETS


Creating Equations Create equations that describe numbers or relationships A.CED.2, A-CED.3

Reasoning with Equations and Inequalities Solve systems of equations A-REI.5, A-REI.6,

Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically A-REI.10, A-REI.11. A-REI.12


• A system of equations/inequalities includes two or more equations/inequalities in the same variables • A system of equations can be solved using three methods: graphing, substitution, and elimination • A system of equations may have one, none, or infinite solutions


• Given a system of equations, which method would best be used to solve the system? • Will you get the same solution set if you solve a system using different methods? • How can real world situations be modeled using systems of equations/inequalities?


• Vocabulary – o Systems of equations/inequalities o Classifying systems: Consistent dependent system, consistent independent system, inconsistent system o Graphing method, substitution method, and elimination method

• How to classify systems of equations • How to solve systems of equations/inequalities

Students will be able to… • Classify systems of equations • Graph systems of equations/inequalities • Solve systems of equations using the three methods: graphing, substitution, and elimination • Choose an appropriate method for solving a system of equations • Solve real world problems using systems of equations/inequalities • make sense of problems and persevere in solving them

• reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning



• Unit tests, quizzes • Open-ended problems that involve written responses • Daily student work • Student/group presentations • Daily Homework


• Study Island assignments • Chapter Project pg. 360 (Minimum Cost Maximum Profit) • Portfolio Activity pg. 325 (Salaries)


• Algebra Textbook: Teachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Math websites

Equipment Needed:

• Graph paper • Colored pencils • Graphing calculators • Computers


UNIT OVERVIEW


Unit Title: Unit Four - Exponents and Exponential Functions


Unit Summary: • Understand and use the laws of exponents to simplify and evaluate expressions • Graph exponential functions • Apply exponential functions to real world situations


Primary interdisciplinary connections: Science, history, and economics

LEARNING TARGETS


• The laws of exponents can be used to simplify expressions • Exponential growth and decay can represent real world problems • A comparison can be made between linear and exponential functions • Scientific notation can be used to express large and small numbers


• How do you simplify an expression involving exponents? • How can we apply exponential functions to real world situations? • Why/how can we use scientific notation to represent numbers?


• Vocabulary – o Base, exponent, degree, monomial, coefficient, laws of exponents, product, quotient o Scientific notation o Exponential functions, growth, decay

• How to simplify expressions involving exponents • Scientific notation • How to graph exponential functions • How to describe exponential functions


Interpreting Functions Analyze functions using different representations F-IF.7. F-IF.8

Linear and Exponential Models Construct and compare linear and exponential models and solve problems F-LE.1, F-LE.1a, F-LE.1b, F-

LE.1c

The Real Number System Extend the properties of exponents to rational exponents N-RN.1, N-RN.2

Students will be able to… • Simplify expressions involving exponents • Write numbers in scientific notation • Understand the need for scientific notation • Understand exponential functions and how they are used • Describe if an exponential functions represents growth or decay • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Study Island assignments • Portfolio Activity pg. 395 • Portfolio Activity pg. 415


• Algebra Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Online Websites

Equipment Needed:

• Graphing calculators • Graph paper • Computers


UNIT OVERVIEW


Unit Title: Unit Five - Polynomials and Factoring


Unit Summary:

• Perform operations on polynomials. • Solve problems involving polynomial functions. • Factor polynomials.


Primary interdisciplinary connections: Science

LEARNING TARGETS


Seeing structure in expressions Write expressions in equivalent forms to solve problems A-SSE.3

Arithmetic with Polynomials and Rational

Expressions

Perform arithmetic operations on polynomials A-APR.1

Interpreting Functions Analyze functions using different representations F-IF.7c


• Operations can be performed on polynomials. • Factoring can be used to simplify polynomials. • Equations can be solved by factoring.


• What is a polynomial? • How can factoring be used to simplify polynomials? • What are the different ways to factor polynomials and when are they used? • How can factoring be used to solve equations?


• Vocabulary – o Polynomial, degree, standard form o Polynomial functions o Greatest Common Factor, binomial factor, perfect square trinomial, difference of two squares

• How to perform operations on polynomials • How to factor polynomials • How to find the zeros of a function

Students will be able to… • Perform operations on polynomials • Factor polynomials • Solve equations using factoring • make sense of problems and persevere in solving them • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Study Island assignments • Portfolio Activity pg. 457


• Algebra Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Math websites

Equipment Needed:



UNIT OVERVIEW


Unit Title: Unit Six - Quadratic Functions


Unit Summary:

• Understand and analyze quadratic functions • Solve problems involving quadratic functions • Using the quadratic formula to solve quadratic functions • Complex numbers

Approximate Length of Unit: 3 weeks Primary interdisciplinary connections: Science

LEARNING TARGETS


Seeing structure in expressions Write expressions in equivalent forms to solve problems A-SSE.3a, A-SSE.3b

Reasoning with Equations and Inequalities Solve equations and Inequalities in one variable A-REI.4, A-REI.4a, A-REI.4b

Interpreting Functions Analyze functions using different representations F-IF.7a

The Complex Number System Perform arithmetic operations with complex numbers N-CN.1, N-CN.2

The Complex Number System Use complex numbers and their operations on the complex plane N-CN.7


• Quadratic functions form a parabola when graphed. • Quadratic functions can be solved in a variety of ways. • Quadratic functions are used to represent real world situations.

Unit Essential Q uestions • What is a quadratic function? • How can a quadratic function be solved? • How can you determine the number of real solutions of a quadratic function? • What is a complex number?


• Vocabulary – o Quadratic function, parabolas, axis of symmetry, vertex form, minimum value, maximum value, zeros o Discriminant, imaginary unit, imaginary numbers, complex numbers

• Formulas – o Quadratic Formula

• How to graph quadratic functions • How to solve quadratic functions • How to find the zeros of a function


• Describe the graph of a quadratic function • Determine the vertex and axis of symmetry of a quadratic function • Factor and solve quadratic functions • Use the quadratic formula • Use the discriminant to determine the number of real solutions of a quadratic function • make sense of problems and persevere in solving them • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Study Island assignments • “Rescue at 2000 ft .” Pg. 504-505 activity • Quadratic Function video at http://www.yourteacher.com/algebra2/quadraticfunction.php


• Algebra Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Math websites

Equipment Needed:



UNIT OVERVIEW

Content Area: Geometry

Unit Title: Unit One - Congruence, Proofs, and Constructions

Target Course/Grade Level: Geometry/Grade 10

Unit Summary: • Experiment with transformations in the plane. • Understand congruence in terms of rigid motions. • Prove geometric theorems. • Prove or disprove whether the figure is a certain type of quadrilateral or triangle* • Make geometric constructions.


Primary interdisciplinary connections: Art, Social Studies, Language Arts, and Science

LEARNING TARGETS


• Geometric properties can be used to construct geometric figures. • Geometric relationships provide a means to make sense of a variety of phenomena. • Shape and area can be conserved during mathematical transformations. • Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry. • Coordinate geometry can be used to represent and verify geometric/algebraic relationships.


• Describe the result of applying each rule to a figure in the coordinate plane: o A(x,y) = ( x-6 , y+7 ) o B(x,y) = ( x , y – 14 ) o C(x,y) = ( x + 5 , y) o D(x,y) = ( x , - y ) o E(x,y) = ( - x , y ) o F(x,y) = ( - x , - y )

• Write the transformation that would translate a figure 5 unit to the left and 12 units down • Given four coordinates, prove mathematically which type of quadrilateral is formed. • Determine whether two given triangles are congruent and state which postulate justifies your answer. • Identify the properties of quadrilaterals and the relationships among the properties.

Content Area Domain Content Area Cluster Standard Congruence Experiment with transformations in the plane G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-

CO.5

Congruence Understand congruence in terms of rigid motion G-CO.6, G-CO.7, G-CO.8

Congruence Prove geometric theorems G-CO.9, G-CO.10, G-CO.11

Expressing Geometric Properties With Equations Use coordinates to prove simple geometric theorems algebraically G-GPE.4, G-GPE.5


• Vocabulary – o Point, line, ray, segment, plane, angle, diagonal, endpoint, polygon, center of a polygon, central angle o transformation, translation, reflection, axis of symmetry, rotation o congruent, congruence, corresponding parts, equilateral, equiangular, regular polygon, equidistant o hypothesis, conclusion, conditional, converse, counterexample, bi-conditional, logical chain, proof, theorem o conjecture, theorem, postulate, proof, two-column proof, paragraph proof o vertical angles, adjacent angles, consecutive angles, complementary, supplementary, linear pair o right angel, acute angle, obtuse angle o transversal, alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles o parallel, perpendicular, bisect, perpendicular bisector o CPCTC- Corresponding Parts of Congruent Triangles are Congruent

• Formulas – o distance, midpoint, slope

• Segment Addition and Angle Addition postulates • Overlapping Segments and Overlapping Angles Theorems • Linear Pair property • Vertical Angles Theorem • Transitive, Reflexive and Symmetric Properties • Corresponding Angles Postulate and it’s converse • Alternate Interior, Alternate Exterior, Same-Side Interior Theorems and their converses • Triangle Sum Theorem • Sum of Interior Angles of a Polygon • The Measure of an Interior Angle of a regular polygon • Sum of Exterior Angle of a Polygon • the slopes of parallel and perpendicular lines are opposite reciprocals • the criteria for triangle congruence (ASA, SAS, SSS, and the special case of ASS (HL)) • CPCTC- Corresponding Parts of Congruent Triangles are Congruent


• use mathematical vocabulary fluently • use appropriate vocabulary to describe rotations and reflections • interpret and perform a given sequence of transformations and draw the result • accurately use geometric vocabulary to describe the sequence of transformations that will carry a given figure onto another • use rigid motions to map one figure onto another • use the definition of congruence as a test to see if two figures are congruent • identify the corresponding parts of two triangles • recognize why particular combinations of corresponding parts establish congruence and why others do not • prove theorems about lines and angles (e.g., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles

are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.)

• construct proofs using a variety of methods: Two-column, paragraph, flowchart • use distance, slope and midpoint formulas then use the information to solve geometric problems • calculate slopes of lines and use the information to determine whether two lines are parallel, perpendicular or neither • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Mathematical investigations • Construct and analyze geometric figures • Make conjectures from geometric figures and data and then prove or disprove them • Work with tools of geometry • Use geometric properties to solve real-world problems • Design a piece of art that illustrates reflectional and/or rotational symmetry. Describe the symmetries in detail. • Create a translation or rotation tessellation. (examples by M.C. Esher)


• Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott • Movie: Flatland (starring Martin Sheen) • Geometers Sketchpad • Visual aids (suggestions) e.g.

o Dominos for logical chains o AngLegs for triangle congruence

Equipment Needed:

• Rulers/straight edge • Protractors • Compasses • Patty paper • Graph paper • Geometers Sketchpad


UNIT OVERVIEW


Unit Title: Unit Two - Similarity, Right Triangles and Trigonometry


Unit Summary: • Understand similarity in terms of similarity transformations. • Prove theorems involving similarity. • Define trigonometric ratios and solve problems involving right triangles • Apply geometric concepts in modeling situations.


Primary interdisciplinary connections: Art, Social Studies, Language Arts, and Science

LEARNING TARGETS


Similarity, Right Triangles & Trigonometry Understand similarity in terms of similarity transformations G-SRT.1, G-SRT.2, G-SRT.3

Similarity, Right Triangles & Trigonometry Prove theorems involving similarity G-SRT.4, G-SRT.5

Similarity, Right Triangles & Trigonometry Define trigonometric ratios and solve problems involving right

triangles

G-SRT.6, G-SRT.7, G-SRT.8

Expressing Geometric Properties With

Equations

Use coordinates to prove simple geometric theorems algebraically G.GPE.6, G.GPE.7


• Geometric properties can be used to establish similarity. • Comparing similar figures is useful when there is a need for indirect measurement. • Smaller scale models can be created by using similarity • Geometric relationships provide a means to make sense of a variety of phenomena. • Measurements can be used to describe, compare, and make sense of phenomena.

Unit Essential Q uestions • Show how to find the measure of a tree using indirect measurement. • What properties do all triangles share? How are triangles classified? • How are similarity and congruence established? Why is this important? • Prove that two given triangles are similar. • Given a side and an angle of a right triangle, use the trigonometric ratios to find the remaining angles and sides of the triangle.


• Vocabulary – o Dilation, center of dilation, contraction, expansion, scale factor, similar, proportionate, corresponding parts o Isosceles triangle, vertex angle, base angle, base and legs of an isosceles triangle o Corollary

o Altitude, base and height of a parallelogram, trapezoid and triangle o Legs of a trapezoid o Apothem

• Formulas – o Trigonometric ratios: SohCahToa o Area of a triangle, parallelogram, trapezoid and a regular polygon

• Triangle similarity theorems and postulate (ASA, SAS, SSS, the special case of ASS (HL)and AA) • Side-splitt ing theorem • Proportional Altitudes, Medians, Angle Bisectors and Segments Theorems • Polygon similarity Postulate • Pythagorean Theorem and it’s converse • Pythagorean Inequalities • Pythagorean triples • 45-45-90 Triangle Theorem • 30-60-90 Triangle Theorem • Isosceles Triangle Theorem and it’s converse • Triangle Midsegment Theorem • Triangle Inequality Theorem


• develop a hypothesis based on observations • make connections between the definition of similarity and the attributes of two given figures • set up and use appropriate ratios and proportions • recognize why particular combinations of corresponding parts establish similarity and why others do not • construct a proof using one of a variety of methods • use information given in verbal or pictorial form about geometric figures to set up a proportion that accurately models the situation • use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. • apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost;

working with typographic grid systems based on ratios). • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Mathematical investigations • Construct and analyze geometric figures • Make conjectures from geometric figures and data and then prove or disprove them • Work with tools of geometry • Use geometric properties to solve real-world problems • (HRW) Text, Pg.542 Portfolio Activity – Techniques for Indirect Measurement

Use proportions and at least two of the methods listed below to find the dimensions of a building or other structure at your school or in your neighborhood.

Measure the shadow of the building and the shadow of a person or object with a known height Use a mirror to create similar triangles Take a photograph of the building with a person or object of known height standing in front of it . Measure the building and person in the

photograph. • (HRW) Text, Pg.552 Chapter Eight Project – Indirect Measurement

o Build a scale model of your school and possibly the area around it


• Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott • Movie: Flatland (starring Martin Sheen) • Geometers Sketchpad • Visual aids (suggestions) e.g.

o Examples of scale models Equipment Needed:

• Tape measure • Ruler • Protractor • Poster board • Scissors • Tape • Geometers Sketchpad


UNIT OVERVIEW


Unit Title: Unit Three - Extending to Three Dimensions


Unit Summary: • Explain volume formulas and use them to solve problems. • Visualize the relation between two‐dimensional and three‐dimensional objects. • Apply geometric concepts in modeling situations.


Primary interdisciplinary connections: Art, Social Studies, Language Arts, Science and Business

LEARNING TARGETS


Geometric Measurement & Dimension Explain volume formulas and use them to solve problems G-GMD.1, G-GMD.2, G-GMD.3

Geometric Measurement & Dimension Visualize relationships between two-dimensional and three-dimensional objects G-GMD.4

Modeling With Geometry Apply geometric concepts in modeling situations G-MG.1, G-MG.2, G-MG.3

Expressing Geometric Properties With Equations Use coordinates to prove simple geometric theorems algebraically G-GPE.7


• Three dimensional figures can be represented in two dimensions by using the orthographic projections • Three dimensional figures can be drawn/created by referring to the orthographic projections of the figure. • Geometric properties can be used to construct geometric figures. • Geometric relationships provide a means to make sense of a variety of phenomena. • Coordinate geometry can be used to represent and verify geometric/algebraic relationships.


• Given a choice of two box designs with different dimensions, which of the two is better from the manufacturer’s point of view. Justify your answer.

• Give examples of when it is better to maximize the surface area to volume ratio and when it is better to minimize the surface area to volume ratio. Explain what makes it better. (e.g. Tums vs. Tylenol, Plants in hot climates vs. cold climates)

• Calculate the volume, surface area or specific dimensions of a variety of polyhedral.


• Vocabulary – o Orthographic projection, isometric drawing, parallel planes o Polyhedron, faces, edges, vertices, dihedral angle, cross section, o Prism, base, height, slant height, lateral height, lateral edge, lateral face, right prism, oblique prism, altitude o Cylinder, pyramid, cone, sphere o Surface area, volume, density o Surface area to volume ratio

• Formulas – o Area of a regular polygon o Volume of prism, cylinder, pyramid, cone, sphere o Surface Area of a right prism, right cylinder, right pyramid and right cone o Diagonal of a right rectangular prism o Distance Formula in Three Dimensions

• Cavalieri’s Principle

Students will be able to… • Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. • Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. • Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. • Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by

rotations of two-dimensional objects. • Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). • Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost;

working with typographic grid systems based on ratios). • Make connections between two-dimensional figures such as rectangles, squares, circles, and triangles and three-dimensional figures such as

cylinders, spheres, pyramids and cones. • Connect experiences with this standard as it related to the two-dimensional shapes studied in Unit 2 to three-dimensional shapes. • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Mathematical investigations • Construct and analyze geometric figures • Make conjectures from geometric figures and data and then prove or disprove them • Work with tools of geometry • Use geometric properties to solve real-world problems • (HRW) Text, Pg.485 Portfolio Activity Creating Solids of Revolution • (HRW) Alternative Assessment Chapter 7 Form A – Product Packaging • “Building a Castle” – Find the volume and surface area of a ‘castle’ built with a variety of children’s blocks


• Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Geometers Sketchpad • Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott • Movie: Flatland (starring Martin Sheen)

• Visual aids (suggestions) e.g. o Unit blocks, children’s blocks o Deck of cards to illustrate Cavalieri’s Principle o Set of Geometric Solids, everyday solids o Nets of solids o Real life examples of solids o Assortment of packaging options o Cutouts made from Honeycomb balls to identify three-dimensional objects generated by rotations of two-dimensional objects.

Equipment Needed:

• Isometric Dot paper • Graph paper • Rulers • Calculators • Student set of Geometric Solids • Nets of solids • Poster board • Scissors • Tape/Glue • Geometers Sketchpad


UNIT OVERVIEW


Unit Title: Unit Four - Circles With and Without Coordinates


Unit Summary:

• Understand and apply theorems about circles. • Find arc lengths and areas of sectors of circles. • Translate between the geometric description and the equation for a conic section. • Use coordinates to prove simple geometric theorems algebraically. • Apply geometric concepts in modeling situations.


Primary interdisciplinary connections: Art, Social Studies, Language Arts, Science

LEARNING TARGETS


• Circles have unique properties and applications which are different from those of other geometric figures. • Measures of line segments and angles associated with a circle are found by using the properties of a circle • Geometric relationships provide a means to make sense of a variety of phenomena. • Coordinate geometry can be used to represent and verify geometric/algebraic relationships.

Unit Essential Q uestions • What is a circle? • Given the center and the radius, find the equation of the circle in the coordinate plane or given the equation of a circle in center-radius form,

state the center and the radius of the circle. • How do you find the measure of an arc?


• Vocabulary – o Circle, arc, radii, diameter, semicircle, minor arc, major arc. chord, tangent, point of tangency, tangent segment, perpendicular,

perpendicular bisector, secant o Central angle, arc length, arc measure, intercepted arc, inscribed angle

• Formulas – o Equation of a circle o Circumference, Area o Central angle, intercepted arc o Length of an Arc o Pythagorean Theorem o Distance


Circles Understand and apply theorems about circles G-C.1, G-C.2, G-C.3

Circles Find arc lengths G-C.5

Expressing Geometric Properties With Equations Translate between the geometric description and the equation for a conic section G-GPE.1

• Chords and Arcs theorem and it’s converse • Tangent theorem and it’s converse • Radius and Chord Theorem • Inscribed Angle theorem, Right Angle Corollary, Arc Intercept Corollary


• prove that all circles are similar • identify and describe relationships among inscribed angles, radii, and chords • derive the equation of a circle • connect experiences from Unit 2 and Unit 3 with two-dimensional and three-dimensional shapes to circles • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning





• Construct and analyze geometric figures • Make conjectures from geometric figures and data and then prove or disprove them • Work with tools of geometry • Use geometric properties to solve real-world problems


• Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments • Teacher developed worksheets and activities • Geometers Sketchpad • Visual aids (suggestions) e.g.

o Geometer o Compass (for drawing circles) o Compass (for directions)

Equipment Needed:

• Compass (for drawing circles) • Compass (for directions) • Straight edge • Graph paper • Geometers Sketchpad


UNIT OVERVIEW


Unit Title: Unit One - Expressions, Equations, and Inequalities

Target Course/Grade Level: Algebra II/11th Grade

Unit Summary: Unit 1 expands on students’ understandings and skills related to expressions, equations, and inequalities.

Students will develop the answers to the following essential questions:

• How do variables help you model real-world situations? • How can you use the properties of real numbers to simplify algebraic equations? • How do you solve an equation or an inequality?


Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies

LEARNING TARGETS


• some patterns can be represented using diagrams, words, numbers, or algebraic expressions • variables can represent variable quantities in real world situations and in patterns • the set of real numbers has subsets related in particular ways • the properties of real numbers are relationships that are true for all real numbers (except, in one case, zero) • some mathematical phrases and real-world quantities can be represented using algebraic expressions • the properties that apply to real numbers also apply to variables that represent them • properties of numbers and equality and inverse operations can be used to solve an equation by finding increasingly simpler equations that have

the same solution as the original equation • important properties of equality include reflexive, symmetric, transitive, substitution, addition, subtraction, multiplication and division • just as properties of equality can be used to solve equations, properties of inequality can be used to solve inequalities • an absolute value quantity is nonnegative • since opposites have the same absolute value, an absolute value equation can have two solutions • absolute value inequalities can be written as compound inequalities without absolute value signs


• How do variables help you model real-world situations? • How can you use the properties of real numbers to simplify algebraic equations? • How do you solve an equation or an inequality?


Seeing Structure in Expressions Interpret the structure of expressions A-SSE.1, A-SSE.1.a

Seeing Structure in Expressions Write expressions in equivalent forms to solve problems A-SSE.3,

Creating Equations Create equations that describe numbers or relationships A-CED.1, A-CED.4


• Vocabulary – o absolute value o algebraic expression o compound inequality o like terms o literal equations o term o variable o Properties of Equality o Properties of Inequality


• use variables to represent unknown quantities in real-world situations • apply properties of real numbers to simplify algebraic expressions • apply the Properties of Equality to solve an equation • apply the Properties of Inequality to solve an inequality • find all of the values of a variable that make an equation or inequality true





• Individualized practice and instruction using Khanacademy.org videos and exercises • Study Island assignments • Cooperative learning opportunities


• Algebra II Math Textbook: Teachers’ Edition & accompanying resources, e.g. Lesson presentations, practice worksheets, assessments, writing assignments

• Teacher developed worksheets and activities Equipment Needed:

• Graphing calculators • Internet access for all students


UNIT OVERVIEW


Unit Title: Unit Two - Functions, Equations, and Graphs


Unit Summary: Unit 2 expands on students’ understandings and skills related to functions, equations, and graphs.

Students will develop the answers to the following essential questions: • Does it matter which form of a linear equation you use? • How do you use transformations to help graph absolute value functions? • How can you model data with a linear function?



LEARNING TARGETS


• a pairing of items from two sets is special if each item from one set pairs exactly with one item from the second set • a relation is a set of pairs of input and output values • relations can be represented with ordered pairs, mapping diagrams, tables of values, and graphs • some quantities are in a relationship where the ratio of corresponding values is constant • When moving from any point on a non-vertical line in the coordinate plane to any other point on the line, the ratio of the vertical change to the

horizontal change is constant. That constant ratio describes the slope of the line. • slope can be calculated by finding the ratio of the difference in the y-coordinates to the difference in the x-coordinates for any two points on the

line • the slopes of two lines in the same plane indicate how the lines are related • Linear functions can be represented by the slope-intercept, point-slope, or standard form. One version can be transformed to another as needed. • Sometimes it is possible to model data from a real-world situation with a linear equation. The equation can then be used to draw conclusions

about the situation. • the equation of a trend line, or line of best fit , can be used to model data that cluster in a linear pattern • a scatter plot can be used to determine the strength of the relationship, or correlation between data sets • there are sets of functions called families, in which each function is a transformation of a special function called the parent • a parent function is the simplest form of a set of functions that form a family. Each function in the family is a transformation of the parent

function • the simplest example of an absolute value function is f(x) = x • the values of a, b, and k, in the form y = ax – h + k determine how the parent function y = x can be transformed

Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A-CED.2

Interpreting Functions Understand the concept of a function and use function notation F-IF.1, F-IF.2

Interpreting Functions Interpret functions that arise in applications in terms of the context F-IF.6

Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8,

Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials A-APR.1

Building Functions Build a function that models a relationship between two quantities F-BF.1, F-BF.1b. F-BF.1c

Building Functions Build new functions from existing functions F-BF.4

• the graph of the linear inequality contains all points on one side of a line and may or may not include the points on the line • arithmetic operations can be formed on functions • some functions have inverses


• Does it matter which form of a linear equation you use? • How do you use transformations to help graph absolute value functions? • How can you model data with a linear function? • How do you find the inverse of a function?


• Vocabulary – o correlation o direct variation o domain o function o linear equation o range o relation o slope o slope-intercept form of a linear equation o point-slope form of a linear equation o standard form of a linear equation o inverse of a function


• identify different forms of linear equations • determine which from of a linear equation is most easily found with the given information • convert between various forms of linear equations • identify the different kinds of transformations • determine whether a transformation changes the location or shape of the graph or both • make a scatter plot of linear data • determine the correlation of linear data • perform operations on functions • find the inverse of a function





• Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities






UNIT OVERVIEW


Unit Title: Unit Three - Linear Systems



Students will develop the answers to the following essential questions: • How can you solve systems of equations using three methods? • How can you solve systems of inequalities? • Do you understand that when solving a system you may have one solution, no solutions, or infinite solutions?



LEARNING TARGETS


• a system of equations is solved by finding a set of values that replace the variables in the equations and make each equation true • a point of intersection (x, y) of the graphs of the functions f and g is a solution of the system y = f(x), y = g(x) • a system of equations can be solved by writing equivalent systems until the value of one variable is clear, then substituting to find the value(s) of

the other variable • if the equations of two systems are equivalent, then a solution of the system that is easier to solve is also a solution of the more difficult system • Graphing is usually the most appropriate method to solve a system of inequalities • the solution set is the set of all points that are the solutions of each inequality in the system • Some real-world problems involve multiple linear relationships. Linear programming accounts for all these linear relationships and gives the

solution to the problem. • the feasible region contains all the points that satisfy all the constraints

Unit Essential Q uestions • Given a system of equations, which method would best be used to solve the system? • Will you get the same solution set if you solve a system using different methods? • How can real world situations be modeled using systems of equations/inequalities?

Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A-CED.2, A-CED.3

Reasoning with Equations and Inequalities Solve systems of equations A-REI.5, A-REI.6,

Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically A-REI.10, A.REI.11, A.REI.12


• Vocabulary – o Systems of equations/inequalities o Classifying systems: Consistent dependent system, consistent independent system, inconsistent system o Graphing method, substitution method, and elimination method

• How to classify systems of equations • How to solve systems of equations/inequalities


• Classify systems of equations • Graph systems of equations/inequalities • Solve systems of equations using the three methods: graphing, substitution, and elimination • Choose an appropriate method for solving a system of equations • Solve real world problems using systems of equations/inequalities • make sense of problems and persevere in solving them • reason abstractly and quantitatively • construct viable arguments and critique the reasoning of others • model with mathematics • use appropriate tools strategically • attend to precision • look for and make use of structure • look for and express regularity in repeated reasoning











UNIT OVERVIEW


Unit Title: Unit Four - Matrices


Unit Summary: Unit 4 expands on students’ understandings and skills related to data representations.

Students will develop the answers to the following essential questions: • How can data be represented in a matrix? • How can you model real world situations using matrices?

Approximate Length of Unit: 1 week

Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Social Studies

LEARNING TARGETS


• A matrix is another way to represent data. • Operations can be performed on matrices • Real world data can be represented in a matrix

Unit Essential Q uestions • How can data be represented in a matrix? • How can you model real world situations using matrices?


• Vocabulary – o Matrices o dimensions o entry


• represent data in a matrix • perform operations on matrices

Content Area Domain Content Area Cluster Standard Vector and Matrix Quantities Perform operations on matrices and use matrices in applications N-VM.6, N-VM.7, N-VM.8











UNIT OVERVIEW


Unit Title: Unit Five - Quadratic Functions and Equations



Students will develop the answers to the following essential questions: • What are the advantages of a quadratic equation in vertex form? In standard form? • How is any quadratic function related to the parent quadratic function y = x2 ? • How are the real solutions of a quadratic equation related to the graph of the related quadratic function?



LEARNING TARGETS


• the graph of any quadratic function(parabola) is a transformation of the parent function, y = x2 • the vertex form of a quadratic function is f(x) = a( x – h )2 + k, where a ≠ 0 • the axis of symmetry is a line that divides the parabola into two mirror images • the vertex of the parabola is (h,k), the intersection of the parabola and its axis of symmetry • the standard form of a quadratic function is is f(x) = ax2 + bx + c, where a ≠ 0 • for any quadratic function f(x) = ax2 + bx + c, the values of a, b, and c provide key information about ifs graph • three non-collinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function • many quadratic trinomials (ax2 + bx + c) can be factored into products of two binomials • The Distributive Property or FOIL method can be used to multiply two binomials. FOIL can be used in reverse to factor • the zeros of a quadratic function y = ax2 + bx + c can be found by solving the related quadratic equation • some quadratic equations in standard form can be solved by factoring the quadratic expression and using the Zero-Product Property • the real solutions of a quadratic equation show the zeros of the related quadratic function and the x-intercepts of its graph • completing a perfect square trinomial allows the completed trinomial to be factored as the square of a binomial • An equation that contains a perfect square can be solved by finding square roots. The simplest of this type of equation has the form ax2=c • a quadratic equation ax2 + bx + c = 0 can be solved by a formula that gives values of x in terms of a, b, and c • every quadratic equation has complex number solutions (that sometimes are real numbers) • The imaginary unit is the complex number whose square is -1. So i2 = -1 and i = √-1

Content Area Domain Content Area Cluster Standard The Complex Number System Perform arithmetic operations with complex numbers N.CN.1, N.CN.2

The Complex Number System Use complex numbers in polynomial identities and equations N.CN.7, N.CN.8

Arithmetic with Polynomials and Rational Expressions Understand the relationship between zeros and factors of polynomials A.APR.3

Creating Equations Create equations that describe numbers or relationships A.CED.1,A.CED.2,A.CED.3

Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4, F.IF.5

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.8, F.IF.9

Building functions Build a function that models a relationship between two quantities F.BF.1

Unit Essential Q uestions • What is a quadratic function? • What are the advantages of a quadratic equation in vertex form? In standard form? • How is any quadratic function related to the parent quadratic function y = x2 ? • How can a quadratic function be solved? • How are the real solutions of a quadratic equation related to the graph of the related quadratic function? • What is a complex number?


• Vocabulary – o axis of symmetry o complex number o discriminant o greatest common factor o imaginary number o parabola o quadratic function o standard form o vertex form o zero of a function o Quadratic formula

• Formulas – o Quadratic formula o Vertex form


• Describe the graph of a quadratic function • identify the vertex, line of symmetry, maximum or minimum, domain, range, and translations of a quadratic function • graph quadratic functions with and without graphing calculators • Factor and solve quadratic functions • Use the quadratic formula • use quadratic functions as models • identify the x-intercepts of the graphs of related quadratic functions • Use the discriminant to determine the number of real solution of a quadratic function






RESOURCES

Teacher Resources:





UNIT OVERVIEW


Unit Title: Unit Six - Polynomials and Polynomial Functions



Students will develop the answers to the following essential questions: • What does the degree of a polynomial tell you about its related polynomial function? • For a polynomial function, how are the factors, zeros, and x-intercepts related? • For a polynomial equation, how are factors and roots related?



LEARNING TARGETS


• The algebraic form of a polynomial function gives information about its graph. Its graph gives information about its algebraic form. • the shape and end behavior of the graph of a polynomial is determined by the degree of the polynomial and by the sign of the leading coefficient • knowing the zeros of a polynomial function gives information about its graph • A polynomial of degree n has n linear factors. The graph of the related function crosses the x-axis an even or odd number of t imes depending on

whether n ix even or odd • ( x – a ) is a linear factor if and only if a is a zero • If ( x – a ) is a factor of a polynomial, then the polynomial has value 0 when x = a. If a is a real number, then the graph of the polynomial has

(a,0) as an intercept • polynomials can be divided using steps that are similar to the long division steps that are used to divide whole numbers • ( x – a ) is a linear factor if and only if a is a root of the related polynomial equation • the degree of a polynomial equation tells how many roots the equation has • To expand the power of a binomial in general, first multiply as needed. Then write the polynomial in standard form • The behavior of the graphs of polynomial functions of different degrees can suggest which will best fit a particular real-world data set


• What does the degree of a polynomial tell you about its related polynomial function? • For a polynomial function, how are the factors, zeros, and x-intercepts related? • For a polynomial equation, how are factors and roots related?

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A-SSE.1.a, A-SSE.2

Arithmetic with Polynomials and Rational Expressions Understand the relationship between zeros and factors of polynomials A.APR.2, A.APR.3

Arithmetic with Polynomials and Rational Expressions Rewrite rational expressions A.APR.6

Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4, F.IF.5

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.8, F.IF.9


• Vocabulary – o Degree, interval, increasing, decreasing, end behavior o multiplicity o polynomial function o relative maximum, relative minimum o standard form of a polynomial function o synthetic division/ long division o turning point


• write a polynomial function given a polynomial equation • identify the degree of a polynomial equation • identify the highest power of a polynomial function • write a polynomial given its factors or zeros • identify the zeros of a polynomial function by finding the x-intercepts of its graph • factor a polynomial equation • apply the Zero-Product Property











UNIT OVERVIEW

Content Area: Algebra, Functions

Unit Title: Unit Seven - Rational Functions


Unit Summary: Unit 6 expands on students’ understandings and skills related to radical functions and rational exponents.

Students will develop the answers to the following essential questions: • What is a rational function? • Describe the graphs of rational functions (domain, asymptotes, holes in the graph). • How can you simplify rational expressions?



LEARNING TARGETS


• Real world problems can be solved using inverse, joint, and combined variation. • Rational functions can be graphed and will identify/describe the features of their graphs. • Rational expressions can be simplified.

Unit Essential Q uestions • What is a rational function? • Describe the graphs of rational functions (domain, asymptotes, holes in the graph). • How can you simplify rational expressions?


• Vocabulary – o Inverse, joint, and combined variation o Rational functions o Domain, asymptotes, holes o Rational expressions

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A-SSE.2

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A-REI.2

Arithmetic with Polynomials and Rational

Expressions

Rewrite rational expressions A-APR.6

Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8

Students will be able to… • Identify inverse, joint, and combined variation • solve real-world problems involving inverse, joint, and combined variation • identify and evaluate rational functions • graph rational functions and describe their important features • simplify rational expressions






RESOURCES

Teacher Resources:




Documents

CCUURRRRIICCUULLUUMM FFOORR SSPPEECCIIAALL ......• Use real-world literal equations and solve them for a variab le different than the one given (science formulas: density, Fahrenheit