Timeline: Unit 1: Essentials of Geometry Activities: Resources: 2.5 weeks/12 days 2 weeks/11 days Vocabulary: Undefined terms, Collinear, Perimeter, Coplanar, Line Segment, Between, End Points, Ray, Opposite Rays, Intersection, Postulate, Axiom, Coordinate, Congruent, Midpoint, Bisector, Angle, Acute, Right, Obtuse, Straight, Construction, Perpendicular, Complementary, Supplementary, Adjacent, Linear Pair, Vertical Angles, Polygon, Convex, Concave, Equilateral, Equiangular, Regular New Common Core State Standards: G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. College Readiness: (Range 13-15) Basic Operations and Applications: Perform one-operation computation with whole numbers (Range 13-15) Measurement: Estimate or calculate the length of a line segment based on other lengths given on a geometric figure (Range 13-15) Graphical Representations: Identify the location of a point with a positive coordinate on the number line (Range 16-19) Measurement: Compute the perimeter of polygons when all side lengths are given (Range 16-19) Measurement: Compute the area of rectangles when whole number dimensions are given (Range 16-19) Graphical Representations: Locate points on the number line and in the first quadrant (Range 20-23) Graphical Representations: Comprehend the concept of length on the number line* (Range 20-23) Graphical Representations: Locate points in the coordinate plane (Range 20-23) Properties of Plane Figures: Exhibit knowledge of basic angle properties and special sums of angle measures (e.g., 90, 180, and 360 ) (Range 20-23) Expressions, Equations, &Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities, Add and subtract simple algebraic expressions, Solve routine first-degree equations, Perform straightforward wordto-symbol translations (Range 24-27) Expressions, Equations, &Inequalities :Solve real-world problems using first degree equations (Range 24-27) Graphical Representations: Find the midpoint of a line segment * (Range 28-32) Graphical Representations: Use the distance formula (Range 28-32) Graphical Representations: Match number line graphs with solution sets of linear inequalities NCTM: Algebra: understand the meaning of equivalent forms of expressions, equations, inequalities, and relations; write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases; use symbolic algebra to represent and explain mathematical relationships; Geometry: use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. draw and construct representations of two- and three-dimensional geometric objects using a variety of tools; use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. Concepts and Skills: Know and describe the undefined terms Describe, label, measure and sketch geometric figures (including perimeter, perpendicular and parallel lines). Understand and use equality and congruence of segments and angles. Use properties of angles to solve basic problems using segments and angles including the use of Algebraic equations Understand and use bisectors to solve problems Set up and solve equations comparing angle to complement/ supplement. Use coordinate geometry to find distance, midpoints, and endpoints Classify polygons with sides and angles Solve simple area and perimeter problems (including circles, triangles, and rectangles) Use construction tools to copy angles and segments and construct bisectors Use inequalities to describe geometric figures such as point, line, ray, etc (including domain and range of functions represented as graphs) Strategies: Postulates and Theorems: Ruler Postulate, Protractor Postulate, Segment Addition Postulate, Angle Addition Postulate, Linear Pair Postulate, Vertical Angles Congruence Theorem, Congruent Complements Theorem, Congruent Supplements Theorem, Right Angle Congruence Theorem
analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof. Quality Core: A-1-a. Apply problem-solving skills (e.g., identifying irrelevant or missing information, making conjectures, extracting mathematical meaning, recognizing and performing multiple steps when needed, verifying results in the context of the problem) to the solution of real-world problems A-1- b. Solve single-step and multistep equations and inequalities in one variable B-1- all D-1-a. Identify and model plane figures, including collinear and non-collinear points, lines, segments, rays, and angles using appropriate mathematical symbols D-1-b. Identify vertical, adjacent, complementary, and supplementary angle pairs and use them to solve problems (e.g., solve equations, use in proofs) D-1-d. Use construction techniques, including straightedge and compass, to bisect and trisect segments and to create parallel and perpendicular lines, perpendicular bisectors, and angle bisectors D-2-a. Identify and classify triangles by their sides and angles G-1-b. Apply the midpoint and distance formulas to points and segments to find midpoints, distances, and missing information G-1-c. Use coordinate geometry to solve problems about geometric figures (e.g., segments, triangles, quadrilaterals) Timeline: Unit 2: Logic and Proof Activities: Resources: 2.5 weeks/12 days Starts 12 days from beginning 2.5 weeks/12 days Vocabulary: Conjecture, Inductive Reasoning, Counterexample, Conditional, New State Standards: G.CO.9 Prove theorems about lines and angles. Theorems include: 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. College Readiness: (Range 13-15) Basic Operations and Applications: Solve problems in one or two steps using whole numbers (Range 13-15) Expressions, Equations, and Inequalities: Exhibit knowledge of basic expressions (Range 13-15) Expressions, Equations, and Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals (Range 16-19) Expressions, Equations, and Inequalities: Solve one step equations having integer or decimal answers. (Range 16-19) Expressions, Equations, and Inequalities: Substitute whole numbers for unknown quantities to evaluate expressions (Range 20-23) Expressions, Equations, and Inequalities: Solve routine first-degree equations Concepts and Skills: Use inductive and deductive reasoning Understand geometric relationships in diagrams Proof of geometric relationships Create conditional statements Determine the truth value of conditional and biconditional statements Determine if statements are equivalent Strategies: Postulates and Theorems:
Negation, Equivalent Statements, Biconditional, Deductive Reasoning, Proof, Hypothesis Concusion Theorem Reflexive Symmetric Transitive Substitution Detachment Syllogism Jusification Truth Values Truth Table Contrapositive Converse NCTM: Algebra: understand the meaning of equivalent forms of expressions, equations, inequalities, and relations; write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases; use symbolic algebra to represent and explain mathematical relationships; draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. draw and construct representations of two- and three-dimensional geometric objects using a variety of tools; use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof. Determine if a definition contains all necessary properties Use postulates to prove theorems Use algebraic properties to prove logical arguments Create and use truth tables Identify converse and contrapositive and discuss truth values Quality Core: A-1- b. Solve single-step and multistep equations and inequalities in one variable B-1- all C-1- a. Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems C-1- b. Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions C-1- c. Identify and write conditional and biconditional statements along with the converse, inverse, and contrapositive of a conditional statement; use these statements to form conclusions C-1- e. Read and write different types and formats of proofs including two-column, flowchart, paragraph, and indirect proofs D-1- a. Identify and model plane figures, including collinear and non-collinear points, lines, segments, rays, and angles using appropriate mathematical symbols D-1- b. Identify vertical, adjacent, complementary, and supplementary angle pairs and use them to solve problems (e.g., solve equations, use in proofs)
Timeline: 3 weeks/13 days Starts 24 days from beginning 3 weeks/13 days Vocabulary: Parallel lines, Skew Lines, Transversal, Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Consecutive Interior Angles, Slope Standard Form of a Line Point Slope Form Slope Intercept Form Proportionality Secant line Tangent line Unit 3: Parallel and Perpendicular Lines New State Standards: G.CO.9 Prove theorems about lines and angles. Theorems include: 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. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). College Readiness: Range(13-15) Number Concepts and Properties: Recognize equivalent fractions and fractions in lowest terms Range(20-23) Number Concepts and Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor Range(20-23) Graphical Representations: Exhibit knowledge of slope* Range(20-23) Properties of Plane Figures: Find the measure of an angle using properties of parallel lines Range(28-32) Expressions, Equations, and Inequalities: Write expressions, equations, or inequalities with a single variable for common pre-algebra settings (e.g., rate and distance problems and problems that can be solved by using proportions) Range(24-27) Expressions, Equations, and Inequalities: Solve first-degree inequalities that do not require reversing the inequality sign Range(28-32) Expressions, Equations, and Inequalities: Solve linear inequalities that require reversing the inequality sign Range(28-32) Graphical Representations: Use properties of parallel and perpendicular lines to determine an equation of a line or coordinates of a point NCTM: Algebra: understand the meaning of equivalent forms of expressions, equations, inequalities, and relations; write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases; use symbolic algebra to represent and explain mathematical relationships; draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. draw and construct representations of two- and three-dimensional geometric objects using a variety of tools; use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: Activities: Gates- Parallel and Perpendicular Lines Concepts and Skills: Use properties of parallel and perpendicular lines Prove relationships using angle measures Identify and use angle pair relationships to solve problems Find, compare, and use slopes of lines in a coordinate plane. Find equations of lines to determine parallel, perpendicular or neither. Find intersection of two lines in the coordinate plane Use construction tools to create parallel and perpendicular lines Use proportionality to solve real-world problems Find distance between two parallel lines Find distance from a point to a line Draw secant and tangent line and find equation (Calculus tie-in) Resources: Strategies: Postulates and Theorems: Parallel Postulate, Perpendicular Postulate, Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angle Theorem, Consecutive Interior Angles Theorem,
make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof. Quality Core: A-1-b. Identify vertical, adjacent, complementary, and supplementary angle pairs and use them to solve problems (e.g., solve equations, use in proofs) A-1-c. Write linear equations in standard form and slope-intercept form when given two points, a point and the slope, or the graph of the equation A-1-d. Recognize the concept of slope as a rate of change and determine the slope when given the equation of a line in standard form or slope-intercept form, the graph of a line, two points, or a verbal description A-1-e. Graph a linear equation using a table of values, x- and y-intercepts, or slope-intercept form B-1- all C-1- a. Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems C-1- d. Use various methods to prove that two lines are parallel or perpendicular (e.g., using coordinates, angle measures) D-1- a. Identify and model plane figures, including collinear and noncollinear points, lines, segments, rays, and angles using appropriate mathematical symbols D-1-b. Identify vertical, adjacent, complementary, and supplementary angle pairs and use them to solve problems (e.g., solve equations, use in proofs) D-1-c. Identify corresponding, same-side interior, same-side exterior, alternate interior, and alternate exterior angle pairs formed by a pair of parallel lines and a transversal and use these special angle pairs to solve problems (e.g., solve equations, use in proofs) D-1-f. Apply properties and theorems of parallel and perpendicular lines to solve problems G-1-a. Use slope to distinguish between and write equations for parallel and perpendicular lines Timeline: 2 weeks/10 days Starts 37 days from beginning 2 weeks/10 days Unit 4: Transformations New State Standards: G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, Activities: Gates-Transformations Concepts and Skills: (Transformations should occur in both a general plane and the coordinate plane) Perform congruence and similarity transformations Resources: Sketchpad Lab- Transformations Strategies: Postulates and Theorems: Translation Theorem, Reflection Theorem,
Vocabulary: Image, Preimage, Isometry, Vector, Reflection, Rotation, Translation, Symmetry, Symmetry of a Transformation Composition, Scalar/Scale Factor, Dilation Magnitude perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. College Readiness: Range(13-15) Number Concepts and Properties: Recognize equivalent fractions and fractions in lowest terms Range (20-23) Basic Operations and Applications: Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off, and computing with a given average NCTM: Algebra: use symbolic algebra to represent and explain mathematical relationships; understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions; interpret representations of functions of two variables judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; use various representations to help understand the effects of simple transformations and their compositions. use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; Apply basic concepts of vectors for translations Reflect a figure in one/two lines Rotate figures about a point Perform compositions of two or more transformations and determine if order matters Create a coordinate rule for a transformation Perform transformations on basic functions Figure out transformations given graphs Identify symmetries of a figure Identify symmetries of a transformation Perform dilations Find and use scale factors (include effect on area) Transformations using matrices (If time allows) Rotation Theorem, Composition Theorem, Reflections in Parallel Lines Theorems, Reflections in Intersecting Lines Theorem
monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; select and use various types of reasoning and methods of proof. Quality Core: B-1- all E-1-a. Determine points or lines of symmetry and apply the properties of symmetry to figures E-1-e. Identify and draw images of transformations and use their properties to solve problems G-1-e. Determine the effect of reflections, rotations, translations, and dilations and their compositions on the coordinate plane Timeline: 3 weeks/14 days Starts 47 days from beginning 3 weeks/13 days Vocabulary: Scalene, Isosceles, Interior Angles, Exterior Angles, Corollary, Congruent Figures, Corresponding Parts, CPCTC, SSS, ASA, SAS, AAS, HL Bases Angles, Legs Vertex Angle Unit 5: Congruent Triangles New State Standards: G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. College Readiness: Range(33-36) Properties of Plane Figures: Draw conclusions based on a set of conditions NCTM: Algebra: use symbolic algebra to represent and explain mathematical relationships; understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions; interpret representations of functions of two variables judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian Activities: Gates- Mystery Triangle Concepts and Skills: Classify a triangle by its angles and sides Use triangle sum and exterior angle sum theorems to solve problems Use isosceles triangle theorems to solve for angles and sides Identify congruent figures Use theorems to prove triangle congruence Use congruence to prove corresponding parts of the figures are congruent Use construction tools to create congruent triangles Perform coordinate proofs of congruent triangles Resources: Strategies: Postulates and Theorems: Triangle Interior Angle Sum Theorem, Triangle Exterior Angle Theorem, Corollary to the Triangle Interior Angle Sum Theorem, Third Angle Theorem, SSS Congruence Postulate, SAS Congruence Postulate, HL Congruence Theorem, ASA Congruence Postulate, AAS Congruence Theorem, Base Angles Theorem, Converse of Base Angles Theorem,
coordinates. understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; use various representations to help understand the effects of simple transformations and their compositions. use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; select and use various types of reasoning and methods of proof. Quality Core: B-1- all C-1- a. Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems C-1- b. Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions C-1- e. Read and write different types and formats of proofs including two-column, flowchart, paragraph, and indirect proofs C-1- f. Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements C-1- g. Use the principle that corresponding parts of congruent triangles are congruent to solve problems D-2-a. Identify and classify triangles by their sides and angles D-2-i. Apply the Angle Sum Theorem for triangles and polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve real-world problems D-2-j. Apply the Isosceles Triangle Theorem and its converse to triangles to solve mathematical and real-world problems G-1-c. Use coordinate geometry to solve problems about geometric figures (e.g., segments, triangles, quadrilaterals) Timeline: 2.5 weeks/12 days Unit 6: Relationships Within Triangles New State Standards: G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are Activities: Concepts and Skills: Use properties of special segments and lines Resources: Sketchpad Lab- Triangle Centers Strategies:
Starts 61 days from beginning 3 weeks/13 days Vocabulary: Midsegment, Perpendicular Bisector, Equidistant, Concurrent, Circumcenter, Incenter, Median, Centriod, Altitude, Orthocenter, Inequalities Inscribed Circumscribed congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. College Readiness: Range (20-23) Basic Operations and Applications: Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off, and computing with a given average Range(24-27) Properties of Plane Figures: Use properties of isosceles triangles * Range(28-32) Expressions, Equations, and Inequalities: Write expressions, equations, and inequalities for common algebra settings Range(28-32) Expressions, Equations, and Inequalities: Solve linear inequalities that require reversing the inequality sign (midsegments, altitudes, medians, angle bisectors, and perpendicular bisector) in triangles to solve problems. Use medians and centroids to find the lengths of segments (2/3 rule) Verify and prove the midsegment theorem Use triangle inequalities to determine if triangles are possible. Find triangle centers in a coordinate plane All right triangles with specified hypotenuse are inscribed in the semicircle Postulates and Theorems: Midsegment Theorem, Perpendicular Bisector Theorem, Angle Bisector Theorem, Triangle Inequality Theorem, Hinge Theorem, a line parallel to one side of a triangle divides the other two proportionally NCTM: Algebra: use symbolic algebra to represent and explain mathematical relationships; understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions; interpret representations of functions of two variables judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; use various representations to help understand the effects of simple transformations and their compositions. use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics;
Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; select and use various types of reasoning and methods of proof. Quality Core: B-1- all C-1- a. Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems C-1- b. Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions C-1- f. Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements C-1- g. Use the principle that corresponding parts of congruent triangles are congruent to solve problems D-2-b. Identify medians, altitudes, perpendicular bisectors, and angle bisectors of triangles and use their properties to solve problems (e.g., find points of concurrency, segment lengths, or angle measures) D-2-c. Apply the Triangle Inequality Theorem to determine if a triangle exists and the order of sides and angles D-2-i. Apply the Angle Sum Theorem for triangles and polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve real-world problems D-2-j. Apply the Isosceles Triangle Theorem and its converse to triangles to solve mathematical and real-world problems Timeline: 2.5 weeks/11 days Starts 73 days from beginning Unit 7: Similarity New State Standards: G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for Activities: Concepts and Skills: Understand similarity in terms of similarity transformations. Use ratios and proportions to solve geometry Resources: Strategies: Postulates and Theorems: AA Similarity Postulate, SSS Similarity Theorem, SAS Similarity Theorem,
2 weeks/10 days Vocabulary: Ratio, Proportion, Geometric Mean, Scale Factor, Similar, Dilation, Reduction, Enlargement, AA, SSS, SAS triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. College Readiness: Range(20-23) Number Concepts and Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor. Range (20-23) Basic Operations and Applications: Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off, and computing with a given average Range(24-27) Basic Operations & Applications: Solve multistep arithmetic problems that involve planning or converting units of measure (e.g., feet per second to miles per hour) Range(33-36) Measurement: Use scale factors to determine the magnitude of a size change NCTM: Algebra: use symbolic algebra to represent and explain mathematical relationships; understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions; interpret representations of functions of two variables judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; use various representations to help understand the effects of simple transformations and their compositions. use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; problems. Use geometric means (include geometric construction) Use guess and check method to find arithmetic and geometric means Use indirect measurement and similarity to solve problems Prove triangles similar using postulates and theorems Identify relationships within similar right triangles when an altitude to a hypotenuse is made. Use proportions with triangles and parallel lines to solve problems. Perform dilations graphically and algebraically Triangle Proportionality Theorem, Parallel Transversal Proportionality Theorem, Angle Bisector of a Triangle Proportionality Theorem, Altitude to Hypotenuse Theorem, Side-Splitter Theorem
solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving. Reasoning and Proof: recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; select and use various types of reasoning and methods of proof. Quality Core: B-1- all C-1- a. Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems C-1- h. Use several methods, including AA, SAS, and SSS, to prove that two triangles are similar, corresponding sides are proportional, and corresponding angles are congruent. D-2-d. Solve problems involving the relationships formed when the altitude to the hypotenuse of a right triangle is drawn. E-1-c. Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g., finding the height of a tree using the shadow of the tree and the height and shadow of a person) E-1-d. Use the definition of similarity to establish the congruence of angles, proportionality of sides, and scale factor of two similar polygons E-1-f. Apply relationships between perimeters of similar figures, areas of similar figures, and volumes of similar figures, in terms of scale factor, to solve mathematical and real-world problems E-1-g. Determine the geometric mean between two numbers and use it to solve problems (e.g., find the lengths of segments in right triangles) Timeline: 3 weeks/15 days Starts 84 days from beginning 3 weeks/15 days Vocabulary: Pythagorean Triple, Pythagorean Theorem, Converse of Pythagorean Theorem, Trigonometric Ratio, Tangent, Unit 8: Right Triangles and Trigonometry New State Standards: G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Activities: Concepts and Skills: Use the Pythagorean Theorem and its converse to solve real world problems Classify a triangle as acute, obtuse, or right based on its sides Use special relationships in right triangles (30-60-90 and 45-45-90) to solve for missing sides Use trigonometric ratios to solve for missing information in right triangles Use trigonometric ratios to solve for missing information in any Resources: Strategies: Postulates and Theorems: Pythagorean Theorem, Converse to the Pythagorean Theorem, Altitude to Hypotenuse Theorem, Geometric Mean Theorem, 45-45-90 Theorem, 30-60-90 Theorem, Law of Sines, Law of Cosines
Sine, Cosine, Inverse Tangent, Inverse Sine, Inverse Cosine, Angle of Elevation, Angle of Depression, Law of Sines, Law of Cosines G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.9 (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). triangles Construct Unit Circle using special right triangles Use trigonometric ratios to solve real world problems College Readiness: Range (20-23) Basic Operations and Applications: Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off, and computing with a given average Range(24-27)Properties of Plane Figures: Recognize Pythagorean triples* Range(24-27) Functions: Express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given side lengths Range(28-32) Basic Operations & Applications: Solve word problems containing several rates, proportions, or percentages Range(28-32) Properties of Plane Figures: Apply properties of 30-60 -90, 45-45 -90, similar, and congruent triangles Range(28-32) Properties of Plane Figures: Use the Pythagorean theorem Range(28-32) Functions: Apply basic trigonometric ratios to solve right-triangle problems Range (33-36) Functions: Exhibit knowledge of unit circle trigonometry NCTM: Algebra: use symbolic algebra to represent and explain mathematical relationships; judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. draw reasonable conclusions about a situation being modeled. Geometry: analyze properties and determine attributes of two- and three-dimensional objects; explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations; investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates. use trigonometric relationships to determine lengths and angle measures.. use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. use geometric models to gain insights into, and answer questions in, other areas of mathematics; Measurement: make decisions about units and scales that are appropriate for problem situations involving measurement. analyze precision, accuracy, and approximate error in measurement situations; use unit analysis to check measurement computations. Problem Solving: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems;