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CURRICULUM FOR SPECIAL EDUCATION ALGEBRA A ALGEBRA/GEOMETRY B GEOMETRY C GRADES 9-12 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 Algebra a, Algebra/Geometry B, Geometry C Grades 9-12 June 27, 2013 RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra Unit Title : Unit One - Expressions, Equations, and Inequalities Targe t Course /Grade Le ve l: 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 Le ngth of Unit: 14 weeks Primary interdisciplinary connections: Science and Language Arts LEARNING TARGETS 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 Unit Unde rstandings 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 Esse ntial Q uestions • • • • • • Identify the coefficient, variable, and constant in the expression 4x + 2. Simplify the expressions: o 2x – 6 +3x o -4(5x – 2) o 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. Knowle dge and Skills Students will know… • Vocabulary – o o o o o o o • Students will • • • • • • • • • • • • • • • • • Formulas – o o o Real numbers Expressions, coefficient, constant, variable, like terms, simplify, order of operations Real numbers, positive, negative, integers, irrational numbers, absolute value, Equations, solutions, evaluate Algebraic properties of equality, distributive property, substitution property Literal equations, formulas Proportion, ratio, probability Fahrenheit to Celsius Simple Interest Formula Density Formula 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 Asse ssment 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 Le arning 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 Te ache r Re sources: • • • Algebra 1 T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites Equipme nt Ne e ded: • • • Calculators Computers Projectors RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra 1 Unit Title : Unit One - Linear Functions Targe t Course /Grade Le ve l: Algebra 1/Grade 9 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 Approximate Le ngth of Unit: 8 weeks Primary interdisciplinary connections: Language Arts and Science LEARNING TARGETS Content Area Domain Content Area Cluster Standard 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 Unit Unde rstandings Students will • • • • understand that… 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. Unit Esse ntial Q uestions • • • • • • 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? Knowle dge and Skills Students will know… • Vocabulary – o Relation, domain, range, function, correlation, ordered pair • • • • • • • • • Students will • • • • • • • • • • • • • • • • 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 How to write linear equations and functions How to graph linear equations Whether a relation is a function T he 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 be able to… 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Linear function video at http//www.khanacademy.org/video/basic-linear-function • Study Island assignments • Portfolio Activity pg. 233 • Graphing calculators RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites (Khan Academy). Equipme nt Ne e ded: • • • • • Graphing calculators Projector and computers Graph paper Rulers T ape measures and spheres (portfolio activity) RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra 1 Unit Title : Unit T hree - Systems of Equations and Inequalities Targe t Course /Grade Le ve l: Algebra 1/Grade 9 Unit Summary: • • Understand, explain, and solve systems of equations using three methods. Understand, explain, and solve systems of inequalities. Approximate Le ngth of Unit: 8 weeks Primary interdisciplinary connections: Science and Business LEARNING TARGETS Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A-CED.3 Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the A-REI.5, A-REI.6, A-REI.10 reasoning Unit Unde rstandings Students will understand that… • 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 Unit Esse ntial 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? Knowle dge and Skills Students will know… • 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Study Island assignments • Chapter Project pg. 360 (Minimum Cost Maximum Profit) • Portfolio Activity pg. 325 (Salaries) RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites Equipme nt Ne e ded: • • • • Graph paper Colored pencils Graphing calculators Computers RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/Geometry B Unit Title : Linear Functions Targe t Course /Grade Le ve l: Algebra/Geometry B/Grade 10 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 Approximate Le ngth of Unit: 4 weeks Primary interdisciplinary connections: Language Arts and Science LEARNING TARGETS Content Area Domain Content Area Cluster Standard 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 Unit Unde rstandings Students will • • • • understand that… 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. Unit Esse ntial Q uestions • • • • • • 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? Knowle dge and Skills Students will know… • 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 • How to write linear equations and functions • How to graph linear equations • Whether a relation is a function • T he 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 Students will be able to… • 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Linear function video at http//www.khanacademy.org/video/basic-linear-function • Study Island assignments • Portfolio Activity pg. 233 • Graphing calculators RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites (Khan Academy). Equipme nt Ne e ded: • • • • • Graphing calculators Projector and computers Graph paper Rulers T ape measures and spheres (portfolio activity) RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/ Geometry B Unit Title : Unit T wo - Systems of Equations and Inequalities Targe t Course /Grade Le ve l: Algebra/ Geometry B /Grade 10 Unit Summary: • • Understand, explain, and solve systems of equations using three methods. Understand, explain, and solve systems of inequalities. Approximate Le ngth of Unit: 4 weeks Primary interdisciplinary connections: Science and Business LEARNING TARGETS Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A-CED.3 Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the A-REI.5, A-REI.6, A-REI.10 reasoning Unit Unde rstandings Students will • • • understand that… 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 Unit Esse ntial 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? Knowle dge and Skills Students will know… • 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Study Island assignments • Chapter Project pg. 360 (Minimum Cost Maximum Profit) • Portfolio Activity pg. 325 (Salaries) RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites Equipme nt Ne e ded: • • • • Graph paper Colored pencils Graphing calculators Computers RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/ Geometry B Unit Title : Unit T hree - Exponents and Exponential Functions Targe t Course /Grade Le ve l: Algebra/ Geometry B/Grade 10 Unit Summary: • • • Understand and use the laws of exponents to simplify and evaluate expressions Graph exponential functions Apply exponential functions to real world situations Approximate Le ngth of Unit: 6 weeks Primary interdisciplinary connections: Science, history, and economics LEARNING TARGETS Content Area Domain Content Area Cluster Standard 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- The Real Number System Extend the properties of exponents to rational exponents LE.1c Unit Unde rstandings Students will • • • • understand that… T he 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 Unit Esse ntial Q uestions • • • 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? Knowle dge and Skills Students will know… • 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 N-RN.1, N-RN.2 • • Students will • • • • • • • • • • • • How to graph exponential functions How to describe exponential functions 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Study Island assignments • Portfolio Activity pg. 395 • Portfolio Activity pg. 415 RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Online Websites Equipme nt Ne e ded: • • • Graphing calculators Graph paper Computers RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/ Geometry B Unit Title : Unit Four - Polynomials and Factoring Targe t Course /Grade Le ve l: Algebra/ Geometry B /Grade 10 Unit Summary: • • • Perform operations on polynomials. Solve problems involving polynomial functions. Factor polynomials. Approximate Le ngth of Unit: 8 weeks Primary interdisciplinary connections: Science LEARNING TARGETS Content Area Domain Content Area Cluster Standard Seeing structure in expressions Write expressions in equivalent forms to solve problems A-SSE.3 Arithmetic with Polynomials and Rational Perform arithmetic operations on polynomials A-APR.1 Analyze functions using different representations F-IF.7c Expressions Interpreting Functions Unit Unde rstandings Students will • • • understand that… Operations can be performed on polynomials. Factoring can be used to simplify polynomials. Equations can be solved by factoring. Unit Esse ntial Q uestions • • • • 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? Knowle dge and Skills Students will know… • 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Study Island assignments • Portfolio Activity pg. 457 RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites Equipme nt Ne e ded: • • • Graphing calculators Graph paper Computers RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/ Geometry B Unit Title : Unit Five - Quadratic Functions Targe t Course /Grade Le ve l: Algebra/ Geometry B / Grade 10 Unit Summary: • • • • Understand and analyze quadratic functions Solve problems involving quadratic functions Using the quadratic formula to solve quadratic functions Complex numbers Approximate Le ngth of Unit: 5 weeks Primary interdisciplinary connections: Science LEARNING TARGETS Content Area Domain Content Area Cluster Standard 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 Unit Unde rstandings Students will • • • understand that… 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 Esse ntial 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? Knowle dge and Skills Students will know… • Vocabulary – o Quadratic function, parabolas, axis of symmetry, vertex form, minimum value, maximum value, zeros o Discriminate, imaginary unit, imaginary numbers, complex numbers • • • • Students will • • • • • • • • • • • • Formulas – o Quadratic Formula How to graph quadratic functions How to solve quadratic functions How to find the zeros of a function be able to… 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 discriminate 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Study Island assignments • “ Rescue at 2000 ft.” Pg. 504-505 activity • Quadratic Function video at http://www.yourteacher.com/algebra2/quadraticfunction.php RESOURCES Te ache r Re sources: • • • Algebra T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher developed worksheets and activities Math websites Equipme nt Ne e ded: • • • Graphing calculators Graph paper Computers RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Algebra/ Geometry B Unit Title : Unit Six - Congruence, Proofs, and Constructions Targe t Course /Grade Le ve l: Algebra/Geometry B/ Grade 10 Unit Summary: • • • Experiment with transformations in the plane. Understand congruence in terms of rigid motions. Prove geometric theorems. Approximate Le ngth of Unit: 4 weeks Primary interdisciplinary connections: Art, Social Studies, Language Arts, and Science LEARNING TARGETS 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 Unit Unde rstandings Students will • • • • understand that… 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. Unit Esse ntial Q uestions • • • 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 units to the left and 12 units down Identify the properties of quadrilaterals and the relationships among the properties. Knowle dge and Skills Students will know… • Vocabulary – o Point, line, ray, segment, plane, and angle o o o o o o o o • • • • • • • • Students will • • • • • • • • • • • • • • • • • transformation, translation, reflection, axis of symmetry, rotation congruent, congruence, corresponding parts hypothesis, conclusion, conditional, converse, counterexample, bi-conditional, logical chain, proof, theorem conjecture, theorem, postulate, proof, two-column proof, paragraph proof vertical angles, adjacent angles, consecutive angles, complementary, supplementary, linear pair right angel, acute angle, obtuse angle transversal, alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles parallel, perpendicular, bisect, perpendicular bisector Segment Addition and Angle Addition postulates Overlapping Segments and Overlapping Angles T heorems Linear Pair property Vertical Angles T heorem T ransitive, Reflexive and Symmetric Properties Corresponding Angles Postulate and it’s converse Alternate Interior, Alternate Exterior, Same-Side Interior T heorems and their converses T riangle Sum T heorem be able to… 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 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: T wo-column, paragraph, flowchart 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • 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 te ssellation. (examples by M.C. Esher) RESOURCES Te ache r Re sources: • • • • • • Geometry T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher 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 Equipme nt Ne e ded: • • • • • • Rulers/straight edge Protractors Compasses Patty paper Graph paper Geometers Sketchpad RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Geometry Unit Title : Unit One - Congruence, Proofs, and Constructions Targe t Course /Grade Le ve l: Geometry/Grade 11 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. Approximate Le ngth of Unit: 16 weeks Primary interdisciplinary connections: Art, Social Studies, Language Arts, and Science LEARNING TARGETS 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, GCO.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 Use coordinates to prove simple geometric theorems algebraically G-GPE.4, G-GPE.5 Equations Unit Unde rstandings Students will understand that… • 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. Unit Esse ntial Q uestions • Describe o o o the result of applying each rule to a figure in the coordinate plane: A(x,y) = ( x-6 , y+7 ) B(x,y) = ( x , y – 14 ) 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. Knowle dge and Skills Students will know… • 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 T riangles are Congruent • Formulas – o distance, midpoint, slope • Segment Addition and Angle Addition postulates • Overlapping Segments and Overlapping Angles T heorems • Linear Pair property • Vertical Angles T heorem • T ransitive, Reflexive and Symmetric Properties • Corresponding Angles Postulate and it’s converse • Alternate Interior, Alternate Exterior, Same-Side Interior T heorems and their converses • T riangle Sum T heorem • Sum of Interior Angles of a Polygon • T he 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 T riangles are Congruent Students will be able to… • 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: T wo-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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • 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 te ssellation. (examples by M.C. Esher) RESOURCES Te ache r Re sources: • • • • • • Geometry T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher 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 Equipme nt Ne e ded: • • • • • • Rulers/straight edge Protractors Compasses Patty paper Graph paper Geometers Sketchpad RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Geometry Unit Title : Unit T wo - Similarity, Right T riangles and T rigonometry Targe t Course /Grade Le ve l: Geometry/Grade 11 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. Approximate Le ngth of Unit: 8 weeks Primary interdisciplinary connections: Art, Social Studies, Language Arts, and Science LEARNING TARGETS Content Area Domain Content Area Cluster Standard 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 Unit Unde rstandings Students will understand that… • 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 Esse ntial 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. Knowle dge and Skills Students will know… • 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 T rigonometric ratios: SohCahT oa o Area of a triangle, parallelogram, trapezoid and a regular polygon • T riangle similarity theorems and postulate (ASA, SAS, SSS, the special case of ASS (HL)and AA) • Side-splitting theorem • Proportional Altitudes, Medians, Angle Bisectors and Segments T heorems • Polygon similarity Postulate • Pythagorean T heorem and it’s converse • Pythagorean Inequalities • Pythagorean triples • 45-45-90 T riangle T heorem • 30-60-90 T riangle T heorem • Isosceles T riangle T heorem and it’s converse • T riangle Mid-segment T heorem • T riangle Inequality T heorem Students will be able to… • 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • 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) T ext, Pg.542 Portfolio Activity – T echniques 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 T ake 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) T ext, Pg.552 Chapter Eight Project – Indirect Measurement o Build a scale model of your school and possibly the area around it RESOURCES Te ache r Re sources: • • • • • • Geometry T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher 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 Equipme nt Ne e ded: • • • • • • • T ape measure Ruler Protractor Poster board Scissors T ape Geometers Sketchpad RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW Conte nt Are a: Geometry Unit Title : Unit T hree - Extending to T hree Dimensions Targe t Course /Grade Le ve l: Geometry/Grade 11 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. Approximate Le ngth of Unit: 8 weeks Primary interdisciplinary connections: Art, Social Studies, Language Arts, Science and Business LEARNING TARGETS Content Area Domain Content Area Cluster Standard Geometric Measurement & Dimension Explain volume formulas and use them to solve problems G-GMD.1, G-GMD.2, GGMD.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 Unit Unde rstandings Students will understand that… • T hree dimensional figures can be represented in two dimensions by using the orthographic projections • T hree 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. Unit Esse ntial Q uestions • • • 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. T ums vs. T ylenol, Plants in hot climates vs. cold climates) Calculate the volume, surface area or specific dimensions of a variety of polyhedral. Knowle dge and Skills Students will know… • 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 T hree 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 EVIDENCE OF LEARNING Asse ssment 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 Le arning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results? • 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) T ext, 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 RESOURCES Te ache r Re sources: • • • • • • Geometry T extbook: T eachers’ Edition & accompanying resources, e.g. T ransparencies, practice worksheets, assessments, writing assignments T eacher 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 twodimensional objects. Equipme nt Ne e ded: • • • • • • • • • • Isometric Dot paper Graph paper Rulers Calculators Student set of Geometric Solids Nets of solids Poster board Scissors T ape/Glue Geometers Sketchpad