The School District of Palm Beach County GEOMETRY HONORS Sections 1 & 2: Introduction to Geometry, Transformations, & Constructions

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1 MAFS.912.G CO.1.1 MAFS.912.G CO.1.2 MAFS.912.G CO.1.4 MAFS.912.G CO.1.5 MAFS.912.G CO.4.12 Calculator: Neutral The School District of Palm Beach County Sections 1 & 2: Geometry, Transformations, & Constructions Mathematics Florida 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. 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). Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. August 16 August 30 Students will defining and representing points, lines, line segments, planes, rays, and angles, as the building blocks of Geometry. Basics of Geometry defining and representing points, lines, line segments, planes, rays, and angles, as the building blocks of Geometry. Midpoint and midpoint and distance, and applications on a coordinate plane by Distance in the either finding midpoint coordinates, endpoint coordinates or length of Coordinate Plane segments. use the commutative and associative properties to identify Partitioning a Line equivalent expressions. Students will determine which properties Segment (distribute, associative, and commutative) have been used when writing equivalent expressions. Parallel and finding the point on a directed line segment between two given Perpendicular Lines Transformations Examining and Using Translations Examining and Using Dilations Examining and Using Rotations Examining and Using Reflections points that partitions the segment in a given ratio. finding the point on a directed line segment between two given points that partitions the segment in a given ratio. identifying parallel and perpendicular lines, and writing equations of lines parallel or perpendicular to another line. introducing rigid and non rigid transformations, translation, reflection, rotation, and dilation. performing translations of points and line segments on a coordinate performing dilations of points and line segments on a coordinate performing dilations of points and line segments on a coordinate performing rotation of points and line segments on a coordinate performing reflection of points and line segments on a coordinate Basic Constructions constructions of line segments MAFS.912.G GPE.2.5 Calculator: Neutral Prove the slope criteria for parallel and perpendicular lines and use 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). MAFS.912.G GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MAFS.912.G GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. FSQ Sections 1 & 2 1 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

2 Section 3: Angles MAFS.912. G CO.1.1 MAFS.912. G CO.1.2 MAFS.912. G CO.1.4 Mathematics Florida 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. 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. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. August 31 September 16 Angles Angle Pairs Special Types of Angle Pairs Formed by Transversals and Non Parallel Lines Angle Pairs Formed by Transversals and Parallel Lines Perpendicular Transversals Students will... use the precise definitions of angles, circles, perpendicular lines, parallel lines, and line segments, basing the definitions on the undefined notions of point, line, distance along a line, and distance around a circular arc. represent transformations in the 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. use definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. prove theorems about lines. prove theorems about angles. use theorems about lines to solve problems. use theorems about angles to solve problems. identify the result of a formal geometric construction. determine the steps of a formal geometric construction Angle Preserving Transformations MAFS.912.G CO.1.5 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. Construction of Angles, Perpendicular and Parallel Lines MAFS.912. G CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. 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. MAFS.912. G CO.4.12 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. FSQ Section 3 2 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

3 Sections 4 & 5: MAFS.912. G CO.1.2 MAFS.912. G CO.1.3 MAFS.912. G CO.1.4 MAFS.912. G CO.1.5 MAFS.912.G CO.2.6 MAFS.912.G CO.2.7 MAFS.912. G CO.3.10 MAFS.912.G MG.1.1 MAFS.912. G SRT.1.1 Mathematics Florida September 19 October 17 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 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. 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. 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. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180 ; triangle inequality theorem; 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. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 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. Angles of Translation of Reflection of Rotation of Dilation of Compositions of Transformations of Symmetries of Regular Congruence and Similarity of Students will distinguish between a rectangle, parallelogram, trapezoid, or regular polygon. describe the rotations and reflections a rectangle, parallelogram, trapezoid, or regular polygon carries onto itself. apply two or more transformations to a given figure to draw a transformed figure. specify a sequence of transformations that will carry a figure onto another. provide descriptions of rigid motions and explain how each preserves distance and angle. be able to predict the effect of a given rigid motion on a given figure. prove theorems about triangles using deductive reasoning (such as the law of syllogism). prove a theorem about triangles such as measures of interior angles of a triangle sum to 180. use geometric shapes to describe objects. use the measures of geometric shapes to describe objects. use the properties of geometric shapes to describe objects. explain the properties of dilations given by a center and a scale factor. perform dilations given by a center and a scale factor on figures in a verify that 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. verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. explain similarity in terms of similarity transformations where angle measure is preserved and side length changes proportionally. determine if two figures are similar, including triangles MAFS.912.G SRT.1.2 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. USA Sections of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

4 Sections 6 & 7: Triangles MAFS.912.G CO.2.8 MAFS.912.G CO.3.10 MAFS.912.G SRT.1.2 MAFS.912.G SRT.1.3 MAFS.912.G SRT.2.4 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse Leg) follow from the definition of congruence in terms of rigid motions. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180 ; triangle inequaliy theorem; 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. 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. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems about triangles. Mathematics Florida October 24 November 10 Triangles Area and Perimeter on the Coordinate Plane Triangle Congruence SSS and SAS Triangle Congruence ASA and AAS Using Triangle Congruency to Find Missing Variables Triangle Similarity Triangle Mid Segment Theorem Students will distinguish between inscribed and circumscribed circles of a triangle. prove properties of angles for a quadrilateral inscribed in a circle, such as opposite angles in an inscribed quadrilateral are supplementary. recognize triangle congruence (ASA, SAS, SSS) in terms of rigid motions that preserve distance (S) and angle (A). show how preserving correlating distances (S) and angles (A) between two triangles results in congruence. prove theorems about triangles using deductive reasoning prove a theorem about triangles such as measures of interior angles of a triangle sum to 180. use coordinates to compute perimeters of polygons and areas of triangles and rectangles. explain similarity in terms of similarity transformations where angle measure is preserved and side length changes proportionally. determine if two figures are similar, including triangles. explain why, if two angle measures are known, the third angle is also known using the properties of similarity transformations. prove theorems about triangles. apply the concepts of congruence and similarity criteria to solve problems involving triangles. apply the concepts of congruence and similarity criteria to prove relationships in geometric figures MAFS.912.G SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Triangle Inequalities Triangle Proofs MAFS.912.G GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Inscribed and Circumscribed Circles of Triangle Medians in Triangles MAFS.912. G C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. FSQ Sections 6 & 7 4 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

5 Section 8: Right Triangles MAFS.912. G CO.2.8 MAFS.912. G GPE.2.4 MAFS.912.G GPE.2.7 MAFS.912.G.SRT.2.4 MAFS.912.G SRT.2.5 MAFS.912.G SRT.3.6 MAFS.912.G SRT.3.7 Mathematics Florida Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse Leg) follow from the definition of congruence in terms of rigid motions. Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 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. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 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. Explain and use the relationship between the sine and cosine of complementary angles November 14 November 22 The Pythagorean Theorem The Converse of the Pythagorean Theorem Right Triangle Congruency Special Right Triangles Special Right Triangles Right Triangle Similarity Trigonometry Students will... recognize triangle congruence (ASA, SAS, SSS) in terms of rigid motions that preserve distance (S) and angle (A). show how preserving correlating distances (S) and angles (A) between two triangles results in congruence. use the Pythagorean Theorem to determine if the point (a, b) lies on a circle centered at the origin and containing the point (x, y). use coordinates to compute perimeters of polygons and areas of triangles and rectangles. prove theorems about triangles, such as a line parallel to one side of a triangle divides the other two proportionally, and conversely. prove theorems about triangles, such as using triangle similarity to prove the Pythagorean Theorem. apply the concepts of congruence and similarity criteria to solve problems involving triangles. apply the concepts of congruence and similarity criteria to prove relationships in geometric figures. explain by angle angle similarity of two right triangles that side ratios are properties of the angles in the triangle. use similarity to define trigonometric ratios (tangent, sine, and cosine) for acute angles in right triangles. determine cosine and sine rations for acute angles in right triangles given the lengths of two sides. explain the relationship between sine and cosine of complementary angles and construct a diagram to illustrate the relationship. express the Pythagorean Theorem as a2 + b2 + c2 and use it to find the unknown length of a right triangle side. use trigonometric ratios and the Pythagorean Theorem to solve realworld application problems MAFS.912.G SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. USA Sections of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

6 Sections 9 & 10: Quadrilaterals MAFS.912.G CO.1.1 MAFS.912. G CO.3.11 MAFS.912. G CO.4.12 MAFS.912.G GPE.2.4 Mathematics Florida 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. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Use coordinates to prove simple geometric theorems algebraically. January 9 January 27 Quadrilaterals Parallelograms Rectangles and Squares Rhombi Kites Trapezoids Mid segment of Trapezoids Students will... prove theorems about parallelograms using deductive reasoning prove theorems about parallelograms, such as the diagonals of a parallelogram bisect each other create geometric constructions such as 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. identify the appropriate algebraic method to prove or disprove simple geometric theorems given a set of coordinates. use slope to determine if lines in a polygon are parallel. determine if two lines are parallel by examining their slopes. determine if two lines are perpendicular by examining their slopes. use coordinates to compute perimeters of polygons and areas of triangles and rectangles. use the properties of geometric shapes to describe objects MAFS.912.G.GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Quadrilaterals in the Coordinate Plane Parts 1 & 2 MAFS.912.G GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Constructions of Quadrilaterals MAFS.912.G MG.1.1 Use geometric shapes, their measures, and their properties to describe objects. FSQ Sections 9 & 10 6 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

7 Section 11: Properties of N-Gons MAFS.912. G SRT.2.5 MAFS.912. G GPE.2.7 MAFS.912.G MG.1.1 MAFS.912.G MG.1.3 Mathematics Florida Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use geometric shapes, their measures, and their properties to describe objects. Apply geometric methods to solve design problems. January 30 February 10 Inroduction to N gons Angles of N gons Segments in Regular N gons Area of N gons Coordinate Geometry Students will... apply the concepts of congruence and similarity criteria to solve problems involving triangles. apply the concepts of congruence and similarity criteria to prove relationships in geometric figures. use coordinates to compute perimeters of polygons and areas of triangles and rectangles. use geometric shapes to describe objects Students will explain the relationship between the circumference and area of a circle. use the measures of geometric shapes to describe objects. use the properties of geometric shapes to describe objects. apply geometric methods to solve design problems FSQ Section 11 7 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

8 Sections 12 & 13: Circles MAFS.912. G C.1.1 MAFS.912. G C.1.2 MAFS.912.G C.1.3 MAFS.912.G C.2.5 MAFS.912.G GPE.1.1 Mathematics Florida Identify and describe relationships among inscribed angles, radii, and chords Identify and describe relationships among inscribed angles, radii, and chords Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. February 13 February 28 Circumference of a Circle Area of a Circle Sectors of a Circle Circles in the Coordinate Plane Circle Transformations supplementary. Radians and Degrees Arcs and Inscribed Angles Inscribed Constructing Inscribed in a Circle Tangent Lines, Secants and Chords Circumscribed Angles and Beyond Constructing Inscribed and Circumscribed Circles of Triangles Students will prove similarity among all circles by demonstrating that the pre image of a dilation central to the circle is equal to the image in terms of the measure of the central angles. define central angle, inscribed angle, circumscribed angle, diameter, radius, and chord. explain the relationship between central, inscribed, and circumscribed angles explain that inscribed angles on a diameter are right angles explain that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. distinguish between inscribed and circumscribed circles of a triangle. prove properties of angles for a quadrilateral inscribed in a circle, such as opposite angles in an inscribed quadrilateral are explain similarity in terms of similarity transformations where angle measure is preserved and side length changes proportionally. derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius. define the radian measure of the angle as the constant of proportionality. derive the formula for the area of a sector. derive the equation of a circle of radius (0,0) by applying the Pythagorean Theorem to the right angle triangle formed by extending the radius as the hypotenuse from the circle s center to a point on the circle (x, y). determine the center of a circle given the equation of the circle. complete the square to find the center and radius of a circle given by an equation. use the Pythagorean Theorem to determine if the point (a, b) lies on a circle centered at the origin and containing the point (x, y) MAFS.912.G GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. USA Sections of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

9 Section 14: Three Dimensional Geometry MAFS.912. G GMD.1.1 MAFS.912. G GMD.1.3 Mathematics Florida Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. March 6 March 31 Geometry Nets and Three Dimensional Figures Cavalieri s Principle for Area Cavalieri s Principle for Volume Volume of Prisms and Cylinders Students will define the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. explain the relationship between the circumference and area of a circle. inscribe a polygon to determine its area. calculate the base area for a prism, cylinder, cone, and pyramid. determine the volume for a prism, cylinder, cone, and pyramid. explain the conceptual relationships among the volume formulas of prisms, cylinders, cones, and pyramids. use dissection arguments, Cavalieri s principle, and informal limit arguments MAFS.912.G GMD.2.4 Identify the shapes of two dimensional cross sections of three dimensional objects, and identify three dimensional objects generated by rotations of two dimensional objects. Surface Area of Prisms and Cylinders MAFS.912.G SRT.1.2 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. Volume of Pyramids and Cones Surface Area of Pyramids and Cones Spheres Similar Shapes Cross Sections and Plane Rotations FSQ Section 14 9 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

10 MAFS.912. G MG.1.1 MAFS.912. G MG.1.2 The School District of Palm Beach County Section 15: Additional Modeling with Geometry Mathematics Florida Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). April 3 April 10 Density Minimizing and Maximizing Angles of Elevation and Depression Students will... use geometric shapes to describe objects Students will explain the relationship between the circumference and area of a circle. use the measures of geometric shapes to describe objects. use the properties of geometric shapes to describe objects. apply concepts of density based on area in modeling situations. apply concepts of density based on volume in modeling situations. apply geometric methods to solve design problems MAFS.912.G MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios) Topographic Grid System Based on Ratios Areas in Real World Contexts Volume in Real World Contexts USA Sections 14 & of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016

11 Post Assessment Content MAFS.912. G GMD.1.2 MAFS.912.G GPE.1.2 MAFS.912.G GPE.1.3 MAFS.912.G SRT.4.10 MAFS.912.G SRT.4.11 MAFS.912.G C.1.4 Mathematics Florida May 8 May 26 Post Assessment Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. Derive the equation of a parabola given a focus and directrix. Derive the equations of ellipses and hyperbolas given the foci and directrices. Prove the Laws of Sines and Cosines and use them to solve problems. 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). Construct a tangent line from a point outside a given circle to the circle Students will... use Cavalieri s Principle when finding the volume of a sphere. derive the equation of ellipses and hyperbolas given a focus and directrx derive the equation of an ellipse. prove the Law of Sines and Cosines apply the Law of Sines and Cosines construct a tangent line from a point outside a given circle to the circle. Supplement and Larson 11 of 11 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 27, 2016