Valley Central School District 944 State Route 17K Montgomery, NY Telephone Number: (845) ext Fax Number: (845)

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1 Valley Central School District 944 State Route 17K Montgomery, NY Telephone Number: (845) ext Fax Number: (845) Geometry R Approved by the Board of Education On November 13, 2017

2 Unit 1 : Essential Geometric Tools and Concepts (12-14 Days) Standards: Focus Standards: G.CO.A.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane. G.CO.D.12 Make, justify and apply formal geometric constructions. Notes: Examples of constructions include but are not limited to: Copy segments and angles. Bisect segments and angles. Construct perpendicular lines including through a point on or off a given line. Construct a line parallel to a given line through a point not on the line. Construct a triangle with given lengths. Construct points of concurrency of a triangle (centroid, circumcenter, incenter, and orthocenter). Constructions of transformations, (see G.CO.A.5) This standard is a fluency recommendation for Geometry. Fluency with the use of construction tools, physical and computational, helps students draft a model of a geometric phenomenon and can lead to conjectures and proofs. See definition for fluency in the Glossary of Verbs Associated with the New York State Math Standards Foundational Standards: 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 7.G.A.2 Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Basic Skills: Naming a line segment, naming an angle. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision.

3 G-CO.D.13 Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle. Progressions in Future Courses: These Geometry Standards will not be used in any future high school course but will have post-high school applications. Key Concepts: Essential Geometric Tools and Concepts Basic Constructions using Circle Properties Topic A: Points Rays and Angles (5-6 Days) Students will use concept of measurements of line segments and angles to begin understanding congruence, adding, subtracting, and angle pairs. U1L1 Points Distances and Segments o Collinearity, Using measurement to understand distance, intro to congruence, adding and subtracting U1L2 Lines Rays and Angles o Lines, Rays, Angles, and Measures of Angles, into to congruence, adding and subtracting U1L3 Types of Angles o Classifying Angles acute, right, obtuse etc through measurement U1L4 Complements and Supplements o Complementary and Supplementary Angles through measurement. Solving algebraically Topic B: Application of Points, Lines, Angles, and Circles (7-8 Days) Students will be able to use the properties of circles to do basic constructions U1L5 Circles and Arcs o Def of Circles, equal radii and understanding how arcs can be used to construct equilateral triangles. U1L6 Constructing a Triangle Given its sides o Construct congruent Triangles through SSS U1L7 Additional Geometric Terminology o Def of midpoint, segment bisector, angle bisector and perpendicular U1L8 More Properties of Lines o Line Axioms and Postulates U1L9 Additional Unit 1 Practice Review

4 Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz on U1L1-L4 Topic A Unit 1 Test

5 Unit 2 : Transformations, Rigid Motions, and Congruence (13-15 Days) Standards: Focus Standards: G.CO.A.2 Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. Note: Instructional strategies may include drawing tools, graph paper, transparencies and software programs. G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments. Notes: Includes point reflections. A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector in addition to Tx,y. The definition of a rotation requires knowing the center (point) and the measure/direction of the angle of rotation. The definition of a reflection requires a line and the knowledge of perpendicular bisectors. Foundational Standards: 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations. 8.G.A.2 Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane. 8.G.A.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision.

6 G.CO.C.10 Prove and apply theorems about the properties triangles. Note: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: Angle Relationships: The sum of the interior angles of a triangle is 180 degrees. The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle. Side Relationships: The length of one side of a triangle is less than the sum of the lengths of the other two sides. In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length. Isosceles Triangles Base angles of an isosceles triangle are congruent. G.CO.B.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. Notes: Description needs to be reproducible. A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector in addition to Tx,y. The definition of a rotation requires knowing the center (point) and the measure/direction

7 of the angle of rotation. The definition of a reflection requires a line and the knowledge of perpendicular bisectors. G.CO.B.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.A.3 Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that map the polygon onto itself. Note: The inclusive definition of a trapezoid will be utilized which states a trapezoid being defined as A quadrilateral with at least one pair of parallel sides. Progressions in Future Courses: These geometry standard will not be used in another high school course but have post high school applications. Key Concepts: Knowing Properties of Basic Transformations Constructing Basic Transformations Congruence Based on Rigid Motions Symmetries of a figure

8 Topic A: Basic Transformations (7-8 Days) Students will be able to identify, construct and state properties of basic rigid motions U2L1 Transformations o Identify and Create Basic Transformation U2L2 Rotations o Construct Rotations, and on the coordinate plane U2L3 Reflections o Construct and on the coordinate plane U2L4 Isosceles Triangles o Def and properties, perpendicular bisectors (construct) U2L5 Translations o Def and properties, on the coordinate plane, construction Topic B: Congruence using Basic Transformations etc (6-7 Days) Students will be able to apply the properties of basic rigid motions to triangle congruence proofs U2L6 Congruence and Rigid Motion o sequence of rigid motions to map one unto the other U2L7 Basic Rigid Motion Proofs o transformation proofs U2L8 Congruence Reasoning with Triangles o transformation proofs using vertices U2L9 Symmetries of a Figure Review Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 2 Test

9 Unit 3: Euclidean Triangle Proof (15-19 Days) Standards: Focus Standards: G-CO.A.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane. G-CO.B.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.B.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions. G-CO.C.9 Prove and apply theorems about lines and angles. Note: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: Vertical angles are congruent If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. The points on a perpendicular bisector are equidistant from the endpoints of the line segment. Foundational Standards: 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations. 8.G.A.2 Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane. 8.G.A.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. 8.G.A.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

10 G-CO.C.10 Prove and apply theorems about the properties triangles. Note: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: Angle Relationships: The sum of the interior angles of a triangle is 180 degrees. The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle. Side Relationships: The length of one side of a triangle is less than the sum of the lengths of the other two sides. In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length. Isosceles Triangles Base angles of an isosceles triangle are congruent. Progressions in Future Courses: These Geometry Standards will not be used in any future high school course but will have post-high school applications. Key Concepts: Learning the Basics of Proof Congruent Triangle Proofs CPCTC

11 Topic A: Drawing Inferences and Basic Axioms (2-3 Days) Students will be able to be able to draw conclusions from given information U3L1 Drawing Inferences from Givens Segment/Angle/Perpendicular Bisectors Perpendicular Median and Altitude of a triangle U3L2 Axioms of Equality Partition, Addition, Subtraction, and Substitution Properties Topic B: Congruent Triangle Proofs (6-7 Days) Students will be able to congruent triangle proofs by SSS, SAS, ASA U3L3 Triangle Congruence Theorems SAS,ASA,SSS U3L4 CPCTC CPCTC U3L5 Proofs with Partitioning Partition, Addition, Subtraction, and Substitution Properties Complements and Supplements of Equals are Equal Review Topic C: Parallel line proofs (2-3 Days) Students will be able to do proofs involving parallel lines U3L6 Parallel Properties Review Corresponding, Alternate Interior/Exterior, Same Side Interior U3L7 More Work with Parallel Lines Exterior Angle Theorem Topic D: Congruent Triangle Proofs (5-6 Days) Students will be able to prove triangles using AAS and HL U3L8 A.A.S. and Isosceles Triangles AAS Properties of Isosceles Triangles(Revisit) applied to AAS U3L9 Hypotenuse-Leg HL Properties of Angle Bisectors applied to HL Additional Triangle Proof

12 Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 3 Test

13 Unit 4: Constructions (10-11 Days) Standards: Focus Standards: G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL (hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions. G-CO.D.12 Make, justify and apply formal geometric constructions. Notes: Examples of constructions include but are not limited to: Copy segments and angles. Bisect segments and angles. Construct perpendicular lines including through a point on or off a given line. Construct a line parallel to a given line through a point not on the line. Construct a triangle with given lengths. Construct points of concurrency of a triangle (centroid, circumcenter, incenter, and orthocenter). Constructions of transformations, (see G.CO.A.5) This standard is a fluency recommendation for Geometry. Fluency with the use of construction tools, physical and computational, helps students draft a model of a geometric phenomenon and can lead to conjectures and proofs. See definition for fluency in the Glossary of Verbs Associated with the New York State Math Standards G-CO.D.13 Make and justify the constructions for inscribing an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Foundational Standards: 7.G.A.2 Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Progressions in Future Courses: These Geometry Standards will not be used in any future high school course but will have post-high school applications. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.

14 Key Concepts: Basic Constructions and their applications Constructing Points of Concurrencies of a triangle Constructing Inscribed or Circumscribed Polygons Topic A: Constructions Students will be able to construct parallel, perpendicular lines, angle bisectors, inscribed and circumscribed polygons and circles U4L1 Introduction to Constructions o Tools of Construction o Construct isosceles and congruent triangles given segments U4L2 Constructing Angles and Parallel Lines o Copy Angles to Construct Parallel Lines U4L3 Constructing Perpendicular Lines o Perpendicular bisector, through a point off and on the line U4L4 The Circumscribed Circle o Constructing the Circumcenter U4L5 Bisecting an Angle o Construct Angle Bisector and Proof of why it works U4L6 The Inscribed Circle of a Triangle o Constructing the Incenter, and Incircle U4L7 Inscribed Regular Polygons o Constructing an inscribed Triangle, Square and Hexagon Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 4 Test

15 Unit 5: The Tools of Coordinate Geometry (16-19 days) Standards: Focus Standards: G.GPE. B.5 On the coordinate plane: a) Explore the proof for the relationship between slopes of parallel and perpendicular lines; b) Determine if lines are parallel, perpendicular, or neither, based on their slopes; and c) Apply properties of parallel and perpendicular lines to solve geometric problems. Foundational Standards: 8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. e.g., compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equations y=mx+b for a line intercepting the vertical axis at b. Progressions in Future Courses: These geometry standards will not be used in another high school course but have post-high school applications. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning. Key Concepts: Review of the slope formula and the criteria for parallelism Exploration of the slope criteria for perpendicularity Review of Pythagorean Theorem (no proof of it yet) Derivation and use of the distance formula Derivation and use of the midpoint formula Equations of lines in slope-intercept, y = mx + b form Equations of lines in point-slope, y 2 - y 1 = m(x 2 - x 1 ) form Equations of horizontal and vertical lines Transformations in the Coordinate Plane

16 Topic A: Applications of Slope Formula (4-5 days) Students will be able to use the slope formula to determine if lines are parallel or perpendicular or neither Students will be able to write/graph lines using slope/y-int form U5L1 Slope and Parallelism o Slope Formula U5L2 Slope and Perpendicularity o Slope Formula U5L3 Equations of Lines o Write equation in slope/int form and graph using point and parallel or perpendicular line Topic B: Special Lines; Distance Formula (Pythagorean Theorem) and Midpoint Formula (7-8 days) Students will be able to Use Slope and Point to Write Equation of a Line Students will be able to Use Pythagorean Theorem; Simplify in Simplest Radical Form Students will be able to Use/Apply Distance Formula Students will be able to Use/Apply Midpoint Formula U5L4 The Point-Slope Form of a Line o Graphing o Writing Equations U5L5 Horizontal and Vertical Lines o Connection to parallel and perpendicularity U5L6 The Pythagorean Theorem (Review) o Simplifying Radicals o Solving for missing sides U5L7 The Distance Formula o On and off coordinate plane U5L8 The Midpoint Formula o On and off coordinate plane Topic C: Transformations in the Coordinate Plane (5-6 days) Students will be able to Perform Rotations, Reflections and Translations on the Coordinate Plane U5L9 Rotations in the Coordinate Plane o Rules for rotation o Rotation about the origin and a point not on origin U5L10 Reflections in the Coordinate Plane o Reflections over axes o Reflections over special lines in coordinate plane U5L11 Translations in the Coordinate Plane o Recognize a sequence of transformations

17 Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz on U5L1-L3 Topic A Quiz on U5L4-L8 Topic B Quiz on U5L9-L11 Topic C Unit 5 Test

18 Unit 6: Quadrilaterals and Polygons (11-13 Days) Standards: Focus Standards: G.CO.C.11 Prove and apply theorems about parallelograms. Notes: Include multi-step proofs and algebraic problems built upon these concepts. Based on the inclusive definition of a trapezoid (specifically a quadrilateral with at least one pair of parallel sides), a parallelogram is a trapezoid. Examples theorems include but are not limited to: A diagonal divides a parallelogram into two congruent triangles. Opposite sides of a parallelogram are congruent. The diagonals of parallelogram bisect each other. If the diagonals of quadrilateral bisect each other, then quadrilateral is a parallelogram. If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle. Additional theorems covered allow for proving a given quadrilateral a particular parallelogram (rhombus, rectangle, square) based on given properties. G.GPE. B.4 On the coordinate plane, algebraically prove geometric theorems and properties. Notes: Examples include but not limited to: Given points and/or characteristics, prove or disprove a polygon is a specified quadrilateral or triangle based on its properties. Given a point that lies on a circle with a given center, prove or disprove that a specified point lies on the same circle. This standard is a fluency recommendation for Geometry. Fluency with the use of coordinates to establish geometric results, calculate Foundational Standards: 6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices. Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

19 length and angle, and use geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields. See definition for fluency in the Glossary of Verbs Associated with the New York State Math Standards. Progressions in Future Courses: These geometry standards will not be used in another high school course but have post-high school applications. Key Concepts: Review of parallel line properties Proving Theorems About Trapezoids and Parallelograms Opposite sides are congruent Opposite angles are congruent Diagonals bisect each other Congruent diagonals lead to rectangles Perpendicular diagonals lead to rhombi Topic A: Properties of Parallelograms and Trapezoids; Midsegment of Triangle (4-5 days) The students will build on their prior knowledge of quadrilaterals to explore the properties of parallelograms and trapezoids. The students will discover the midsegment of a triangle. U6L1 Trapezoids and Parallelograms Properties of Quadrilaterals Definitions of Trapezoids and Parallelograms U6L2 Properties of Parallelograms U6L3 What Makes a Parallelogram Properties of Parallelogram Euclidean Proof U6L4 The Midpoints of a Triangle Midpoints on the coordinate plane Discovering the midsegment of a triangle and it s properties Topic B: Properties of Special Quadrilaterals and Coordinate Proof (7-8 days) Students will be able to apply properties of parallelograms to prove rectangles, rhombi, and squares on the coordinate plane. U6L5 Rectangles o Properties of a Rectangle o Coordinate proof and algebraic applications U6L6 Rhombi o Properties of a Rhombus o Coordinate proof and algebraic applications U6L7 Squares o Properties of a Square o Coordinate proof and algebraic applications U6L8 Additional Quadrilateral Practice o Mixed Review of all quadrilateral properties and coordinate proof Review (2 Days)

20 Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 6 Test

21 Unit 7: Dilations and Similarity (15-18 Days) Standards: Focus Standards: G.CO.A.2 Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. Note : Instructional strategies may include drawing tools, graph paper, transparencies and software programs. G.SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor. G.SRT. A.1a Verify experimentally that 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. G.SRT. A.1b Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT. A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Note : Description needs to be reproducible. The center and scale factor of the dilation must always Foundational Standards: 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

22 be specified with dilation. A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector in addition to Tx,y. The definition of a rotation requires knowing the center (point) and the measure/direction of the angle of rotation. The definition of a reflection requires a line and the knowledge of perpendicular bisectors. 7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. G.SRT. A.3 Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar. G.SRT. B.4 Prove and apply similarity theorems about triangles. Notes : Include multi-step proofs and algebraic problems built upon these concepts. Examples theorems include but are not limited to: If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally (and conversely). The length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse. The centroid of the triangle divides each median in the ratio 2:1. G.SRT. B.5 Use congruence and similarity criteria for triangles to: a. Solve problems algebraically and geometrically. b. Prove relationships in geometric figures. Notes : ASA, SAS, SSS, AAS, and Hypotenuse-Leg (HL) theorems are valid criteria for triangle

23 congruence. AA~, SAS~, and SSS~ are valid criteria for triangle similarity. This standard is a fluency recommendation for Geometry. Fluency with the triangle congruence and similarity criteria will help students throughout their investigations of triangles, quadrilaterals, circles, parallelism, and trigonometric ratios. These criteria are necessary tools in many geometric modeling tasks. See definition for fluency in the Glossary of Verbs Associated with the New York State Math Standards G.GPE. B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Note : Midpoint formula is a derivative of this standard. Progressions in Future Courses: These geometry standards will not be used in another high school course but have post-high school applications. Key Concepts: Non-rigid motions and dilations in the Euclidean and Cartesian planes Properties of dilations Similarity Transformation The AA Criteria for Similarity and Dilations The Triangle Proportionality Theorem - Proof and Applications Partitioning a Line Segment in the Coordinate Plane S.A.S. and S.S.S. Criteria for Similarity Similarity proofs and results : o The Pythagorean Theorem o The Medians of a Triangle o Similarity and Right Triangles

24 Topic A: Intro to Dilations and Similarity (5-6 Days) Students study and prove the properties of dilations. U7L1 Dilation Constructions U7L2 Dilation in Coordinate Plane The Two Primary Properties of Dilations U7L3 Dilation and Angles Preserving Angle Measure U7L4 Similarity Definition of Similarity Similarity Transformations Topic B: Similarity and Dilations (3-4 Days) Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. U7L5 Similarity Criteria o AA, SSS, SAS Similarity U7L6 Reasoning with Similarity o Similarity Proofs U7L7 More Similarity Reasoning o Means and Extremes o Mean Proportional Topic C: Proportionality and Similarity (7-8 Days) Students divide a line segment into equal pieces by the Side Splitter and Dilation Methods. Students examine how an altitude drawn from the vertex of a right triangle to the hypotenuse creates two similar sub-triangles. Students work with adding, subtracting, multiplying, and dividing radical expressions. Finally, students prove the Pythagorean Theorem using similarity. The students study the ratio formed by the centroid. U7L8 Side Splitter Theorem U7L9 Partitioning a Line Segment o Geometrically o Algebraically U7L10 Medians of a Triangle o Centroid o Ratio formed by the Centroid U7L11 Right Triangle and Similarity o Use the Mean Proportion to Find Measures U7L12 Proving the Pythagorean Theorem (+)

25 Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 7 Test

26 Unit 8: Right Triangle Trigonometry (11-13 Days) Standards: Focus Standards: Foundational Standards: G.GPE. B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Note : Midpoint formula is a derivative of this standard. G.SRT. C.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT. C.8 Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems. Note : Special right triangles refer to the and triangles. G.SRT. D.9 Justify and apply the formula A= 1/2 ab sin (C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 8.G.B.6 Understand a proof of the Pythagorean Theorem and its converse. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Progressions in Future Courses: F-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. F-TF.C.8 Prove the Pythagorean identity. Key Concepts: The definition of the trigonometric ratios with similar right triangles Finding the area of a non-right triangle using A=1/2 ab sin (C) The sine and cosine of complementary angles Trigonometry and the calculator Using trigonometry to find missing sides and angles of right triangles Applications of right triangle trigonometry

27 Topic A: Intro to Right Triangle Trig and the Calculator (6-7 Days) Students link their understanding of similarity and relationships within similar right triangles formally to trigonometry. In addition to the terms sine, cosine, and tangent, students study the relationship between sine and cosine. U8L1 Similar Right Triangles o Equal rations among similar right triangles U8L2 The Trigonometric Ratios o Sine, cosine and tangent ratios U8L3 Trigonometry and the Calculator (2 Days) o Solving proportion and comparing ratios o Recognizing ratios of complementary angles are congruent (cofunctions) o Using Inverse Function to solve for missing angles Topic B: Missing Sides and Angles and Applications (5-6 Days) Students will show how to apply the trigonometric ratios to solve right triangle problems and real world applications. U8L4 Solving for Missing Sides of a Right Triangle U8L5 Trigonometric Applications o Finding sides and angles o Angle of elevation and depression U8L6 More Trigonometric Applications o Including polygons o Multi-step real world applications o Area of oblique triangles Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 8 Test

28 Unit 9: Circles With and Without Coordinates (18-21 Days) Standards: Focus Standards: G.C.A.1 Prove that all circles are similar. G.C.A.2a Identify, describe and apply relationships between the angles and their intercepted arcs of a circle G.C.A.2b Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle. G.CO.A.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane. G.CO.A.2 Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. Note: Instructional strategies may include drawing tools, graph paper, transparencies and software programs. G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another. Notes: Instructional strategies may include graph paper, tracing paper, and geometry software. A point Foundational Standards: G.CO.D.12 Make, justify and apply formal geometric constructions. G.CO.D.13 Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle. G.SRT. B.5 Use congruence and similarity criteria for triangles to: a. Solve problems algebraically and geometrically. b. Prove relationships in geometric figures. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

29 reflection or glide reflection could be used here. Students can investigate compositions of transformations and the single transformation that is equivalent G.GPE. A.1 1a. Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the equation of the circle. 1b. Graph circles given their equation. Note for 1a. Finding the center and radius could involve completing the square. The completing the square expectation for Geometry follows Algebra I: leading coefficients will be 1 (after possible removal of GCF) and the coefficients of the linear terms will be even. Note for 1b. For circles being graphed, the center will be an ordered pair of integers and the radius an integer. Students need to be able to graph circles in Algebra II with respect to standard A-REI.C.7, solving quadratic and linear systems algebraically and graphically. Progressions in Future Courses: A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line and the circle. G.SRT. B.5 Use congruence and similarity criteria for triangles to: a. Solve problems algebraically and geometrically. b. Prove relationships in geometric figures. Notes : ASA, SAS, SSS, AAS, and Hypotenuse-Leg (HL) theorems are valid criteria for triangle G.GMD. A.1 Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

30 Key Concepts: The locus definition of a circle and similarity of circles Angles, radii, and chord relationships : o Central, inscribed, and circumscribed angle relationships o Tangent relationships o Cyclic quadrilaterals o Euclidean circle proof The circumference and area formulas for a circle Arc length and the area of a sector Radian measure of angels Equations of Circles Including Completing the Square Circles Problems in the Coordinate Plane (such as writing the equation of the tangent at a point on the circle) Topic A: Intro to Circles-Angles and Arcs (4-5 Days) Students will be able to Identify the relationships between the diameters of a circle and other chords of the circle. Students will explore the relationship between inscribed angles and central angles and their intercepted arcs Students will be able to recognize and use different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure. Students will be able to find the measure of a central angle that is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Students will be able to use the inscribed angle theorem to find the measures of unknown angles. Students will be able to prove relationships between inscribed angles and central angles. U9L1 Circle Terminology o Definition of a circle o Parts of the circle o Major and minor arcs o Central Angles U9L2 Inscribed Angles o Inscribed angles formed by chords o Exterior Angle Theorem and the connection to inscribed angles U9L3 More Work with Inscribed Angles o Inscribed angles intercepting same arc o Angles inscribed in a semi-circle Topic B: Circles: Segments and Angles (8-9 Days) Students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even two tangent lines), meeting a point inside or outside of a circle. They establish the secant angle theorems and tangent-secant angle theorems. By drawing auxiliary lines, students also

31 notice similar triangles and thereby discover relationships between lengths of line segments appearing in these diagrams. U9L4 Intersecting Chords o Intersecting chords and arcs o Intersecting chords and angles U9L5 Tangents to a Circle o Tangent / Radius relationship o Tangents drawn from an external point U9L6 Tangents Secants and their Angles U9L7 Tangent and Secant Proofs and Practice U9L8 Tangent and Secant Lengths Topic C: Equation of a Circle and Tangent Lines (6-7 Days) Students write the equation for a circle in center-radius form using the Pythagorean theorem or the distance formula. Students write the equation of a circle given the center and radius. Students identify the center and radius of a circle given the equation. Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in x and y is the equation for a circle. Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point. U9L9 Equations of Circles o Centers on and off the origin U9L10 Placing Circles in Standard Form o Completing the square U9L11 Constructing Tangents o Constructing from point on circle o Constructing from point off circle U9L12 Equations of Tangent Lines o Writing equations graphically and algebraically Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 9 Test

32 Unit 10: The Geometry of Three Dimensions (15-17 Days) Standards: Focus Standards: G.GMD. A.1 Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. G.GMD. B.4 Identify the shapes of plane sections of three dimensional objects, and identify three dimensional objects generated by rotations of two-dimensional objects. Note : Plane sections are not limited to being parallel or perpendicular to the base. G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. G.MG.A.2 Apply concepts of density based on area and volume of geometric figures in modeling situations. G.MG.A.3 Apply geometric methods to solve design problems. Note : Applications could include designing an object or structure to satisfy constraints such as area, volume, mass and cost. G.GPE. B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. Note: This standard is a fluency recommendation for Geometry. Fluency with the use of coordinates to establish geometric results, calculate length and angle, and use geometric representations Foundational Standards: 7.G.A. 3 Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the base. 7.G.B.4 Apply the formulas for the area and circumference of a circle to solve problems. 7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two and three-dimensional objects composed of triangles, trapezoids, parallelograms, cubes, and right rectangular prisms. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.C.9 Solve problems, mathematical and real world, which use the formulas for the volume of cones, cylinders, and spheres. Standards for Mathematical Practice: CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

33 as a modeling tool are some of the most valuable tools in mathematics and related fields. See definition for fluency in the Glossary of Verbs Associated with the New York State Math Standards. Progressions in Future Courses: These geometry standards will not be used in another high school course but have post-high school applications. Key Concepts: Area and perimeter with and without coordinates 3-Dimensional Figures and Their Relationship to 2-D Shapes Volume Formulas and Cavalieri's Principle Modeling with Geometry Topic A: Modeling and Measurement in Two Dimensions (7-8 Days) The student will be able to calculate perimeter, circumference and area of circles on/off the coordinate plane. The student will be able to calculate perimeter and area of irregular polygons The student will be able to calculate the area of a circle s sector, length of a circle s arc The student will conclude that all circles are similar The student will understand radian measure U10L1 Perimeter On and of the coordinate plane U10L2 The Circumference of a Circle Relationship to Perimeter of a Polygon U10L3 The Area of Polygons U10L4 The Area of a Circle Parts of Circles Shaded Regions U10L5 Sectors of Circles Area Arc Length U10L6 Radian Measures of Angles All Circles are Similar Radian Rotation Definition/Formula

34 Topic B: Include the title as well as the length of time (8-9 Days) The students will be able to provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. The students will be able to identify the shapes of plane sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. The students will be able to use geometric shapes, their measures, and their properties to describe objects. The students will be able to apply concepts of density based on area and volume of geometric figures in modeling situations. The students will be able to apply geometric methods to solve design problems. U10L7 Solids and Their Cross Sections Introduction to Prisms, Cylinders, Pyramids and Cones U10L8 Volume of Prisms and Cylinders Calculate Volume of Right Prisms and Right Cylinders Solve Modeling Problems U10L9 The Volume of Pyramids and Cones Develop the Formula for the Volume of Pyramids and Cones Solve Modeling Problems U10L10 Spheres Calculate Volume Solve Modeling Problems U10L11 The Volume of Truncated Cones Applications and Modeling Problems Review (2 Days) Evidence of Learning/Assessments: Do Now Activities Collected Do Now Quizzes Homework Assignments Quiz Unit 10 Test