The School District of Palm Beach County GEOMETRY HONORS Unit A: Essentials of Geometry

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1 MAFS.912.G-CO.1.1 MAFS.912.G-CO.4.12 MAFS.912.G-GPE.2.7 MAFS.912.G-MG.1.1 Unit A: Essentials of 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. 1.1, 1.2, 1.4, 1.5 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. 1.4 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 1.2, 1.3, 1.4 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 1.6 August 18 - August 28 Identify Points, Lines, and Planes Use Segments and Congruence Use Midpoint and Distance Formulas Measure and Classify Angles Measure and Classify Angles Constructions Describe Angle Pair Relationships Classify Polygons G-CO - Congruence G-GPE Expressing Geometric Properties with Equations G-MG Modeling G-GMD Measurement & Dimension name and sketch geometric figures use segment posulates to identify congruent segments find lengths of segments in the coordinate plane copy and bisect segments and angles using construction tools use special angle relationships to find angle measures classify polygons name, measure, and classify angles & Constructions FSQ Unit A

2 MAFS.912.G-CO.1.1 MAFS.912.G-CO.3.9 Unit B: Reasoning and Proof G-CO - Congruence 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. 2.2 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. 2.2, 2.4, 2.5, 2.6, 2.7 August 31 - September 9 Analyze Conditional Statements Use Postulates and Diagrams Reason Using Properties from Algebra Prove Statements about Segments and Angles write definitions as conditional statements use postulates involving points, lines, and planes use algebraic properties in logical arguments write proofs using geometric theorems use properties of special pairs of angles Prove Angle Pair Relationships FSQ Unit B

3 Unit C: Parallel and Perpendicular Lines G-CO - Congruence G-GPE Expressing Geometric Properties with Equations MAFS.912.G-CO.1.1 MAFS.912.G-CO.3.9 MAFS.912.G-CO.4.12 MAFS.912.G-GPE.2.5 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. 3.1 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. 3.1, 3.2, 3.3, 3.6 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. 3.6 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). 3.4, 3.5 September 10 - September 29 Identify Pairs of Lines and Angles Use Parallel Lines & Transversals Prove Lines are Parallel Find and Use Slopes of Lines Write and Graph Equations of Lines Prove Theorems About Perpendicular Lines Prove Theorems About Perpendicular Lines (Constructions) identify angle pairs formed by three intersecting lines use angles formed by parallel lines and transversals use angle relationships to prove that lines are parallel find and compare slopes of lines find equations of lines find the distance between a point and a line construct parallel and perpendicular lines & Constructions USA Unit A, B, C

4 MAFS.912.G-CO.1.2 MAFS.912.G-CO.1.5 MAFS.912.G-CO.2.6 MAFS.912.G-CO.2.7 MAFS.912.G-CO.2.8 MAFS.912.G-CO.3.10 Unit D: Congruent Triangles Mathematics Florida 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). 4.3, 4.9 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. 4.9 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. 4.3, 4.9 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. 4.2, 4.3 *Not included on district assessment FSQ Unit D. Standard will be assessed on USA Unit D, E, F. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse Leg) follow from the definition of congruence in terms of rigid motions. 4.4, 4.5, 4.6, 4.7 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. 4.1., 4.4, 4.5, 4.6, 4.7, 4.8 FSQ Unit D October 5 - October 22 Apply Triangle Sum Properties Apply Congruence and Triangles Relate Transformations & Congruence Prove Triangles Congruent by SSS Prove Triangles Congruent by SAS & HL Prove Triangles Congruent by ASA and AAS Use Congruent Triangles Use Isosceles and Equilateral Triangles Perform Congruence Transformations G-CO - Congruence classify triangles and find measures of their angles identify congruent figures use transformations t show congruence use the sides lengths to prove triangles congruent use sides and angles to prove congruence prove congruence through ASA and AAS use congruent triangles to prove corresponding parts congruent use theorems about isosceles and equilateral triangles create an image congruent to a given triangle

5 MAFS.912.G-CO.3.9 MAFS.912.G-CO.3.10 Unit E: Relationships within Triangles Mathematics Florida 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. 5.2 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. 5.1, 5.3, 5.4, 5.6 October 26 November 6 Midsegment Theorem and Coordinate Proof Use Perpendicular Bisectors Use Angle Bisectors of Triangles Use Medians and Altitudes Use Inequalities in a Triangle G-CO - Congruence G-GPE Expressing Geometric Properties with Equations use properties of midsegments and write coordinate proofs use perpendicular bisectors to solve problems use angle bisectors to find distance relationships use medians and altitudes of triangles find possbile side lengths of triangles use inequalities to make comparisons in two triangles MAFS.912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2). 5.1 FSQ Unit E Inequalities in Two Triangles and Indirect Proof

6 MAFS.912.G-C.1.1 MAFS.912.G-CO.1.2 MAFS.912.G-GPE.2.4 MAFS.912.G-GPE.2.6 MAFS.912.G-MG.1.3 MAFS.912.G-SRT.1.1 MAFS.912.G-SRT.1.2 MAFS.912.G-SRT.1.3 MAFS.912.G-SRT.2.4 MAFS.912.G-SRT.2.5 Prove that all circles are similar. 6.2 Unit F: Similarity Mathematics Florida 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). 6.6 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2). 6.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 6.5, 6.6 Extension 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). 6.5 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. 6.2, 6.6 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. 6.2 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 6.3 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. 6.4, 6.5 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.1, 6.5 USA Unit E, F November 9 November 24 Use Similar Polygons Relate Transformations and Similarity Prove Triangles Similar by AA Prove Triangles Similar by SSS and SAS Use Proportionality Theorems Perform Similarity Transformations Partition Segments G-CO - Congruence G-GPE Expressing Geometric Properties with Equations G-MG - Modeling use proportions to identify similar polygons identify similarity transformations called dilations explain what it means for two figures to be similar use the AA Similarity Posulate to show two triangles similar use the SSS and SAS Similarity Theorems to show to triangle similar use proprotions with a triangle or parallel lines perform dilations find a point on a directed line segment that partitions the segment in a given ratio Extension Blender Supplmental Lessons OCG 5-3 OGC 8-3

7 MAFS.912.G-MG.1.1 MAFS.912.G-SRT.2.4 MAFS.912.G-SRT.2.5 MAFS.912.G-SRT.3.6 MAFS.912.G-SRT.3.7 MAFS.912.G-SRT.3.8 MAFS.912.G-SRT.4.10 MAFS.912.G-SRT.4.11 Unit G: Right Triangles and Trigonometry 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). 7.3 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. 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. 7.1, 7.3 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 7.3 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. 7.4, 7.5 Explain and use the relationship between the sine and cosine of complementary angles. 7.6 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 7.1, 7.2, 7.5, 7.6, 7.7 Prove the Laws of Sines and Cosines and use them to solve problems. 7.7 Extension *Not included on district assessment for this unit. 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). 7.7 Extension *Not included on district assessment for this unit. FSQ Unit G January 5 January 20 Apply the Pythagorean Theorem Use the Converse of the Pythagorean Theorem Use Similar Right Triangles Special Right Triangles Apply the Tangent Ratio Apply the Sine and Cosine Ratios Trig Ratios of Complementary Angles Solve Right Triangles Law of Sines and Cosines G-GPE Expressing Geometric Properties with Equations G-MG - Modeling use the Pythagorean Theoerm to find the side lengths in right triangles use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle use properties of the altitude of a right triangle use the relationships among the sides in special right triangles use the tangent ratio for indirect measurment use the sine and cosine ratios explain the relationships between the sine and cosine of complementary angles use the inverse tangent, sine, and cosine ratios use trigonometry with acute and obtuse triangles Additional Lesson AL Extension

8 MAFS.912.G-CO.3.11 MAFS.912.G-GPE.2.4 MAFS.912.G-GPE.2.7 MAFS.912.G-MG.1.1 MAFS.912.G-SRT.2.5 Unit H: Quadrilateral Mathematics Florida Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 8.2, 8.3, 8.4, 8.6 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 8.4 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 8.1 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 8.2, 8.3, 8.4, 8.5 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. January 21 February 5 Find Angle Measures in Polygons Use Properties of Parallelograms Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure Sides and Angles of Parallelograms defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2). 8.5 Show that a Quadrilateral is a Parallelogram Properties of Rhombuses, Rectangles, and Squares Special Parallelograms Use Properties of Trapezoids and Kites Identify Special Quadrilaterals G-CO - Congruence G-GPE Expressing Geometric Properties with Equations G-MG - Modeling find angle measures in polygons find angle and side measures in parallelograms know the diagonal properties in parallelograms use properties to identify parallelgrams use properties of rhombuses, rectangles, and squares use the properties of trapezoids and kites identify special quadrilaterals Blender Supplemental Lessons 8.2 Part 2 OCG 4-2 & Part 2 OCG 4-5 FSQ Unit H

9 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-SRT.1.1 MAFS.912.G-SRT.1.2 Unit I: Properties of Transformations Mathematics Florida 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). 9.3, 9.5, 9.7 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 9.6 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 9.1, 9.3, 9.4 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. 9.1, 9.3, 9.4, 9.5, 9.6, 9.7 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. 9.7 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. 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. 9.7 February 8 February 22 Translate Figures and Use Vectors Perform Reflections Perform Rotations Rotations & Reflections - Carry a Polygom onto Itself Apply Compositions of Transformations Identify Symmetry Identify and Perform Dilations G-CO - Congruence use a vector to translate a figure reflect a figure in any given line rotate figures about a point carry a polygon onto itself through rotations and reflections perform combinations of two or more transformations identify line and rotational symmetries of a figure use drawing tools to draw dilations Blender Supplemental Lessons Supplement Rotations & Reflections USA Unit G, H, I

10 MAFS.912.G-C.1.2 MAFS.912.G-C.1.3 MAFS.912.G-C.2.5 MAFS.912.G-CO.1.1 MAFS.912.G-CO.1.2 MAFS.912.G-CO.4.12 MAFS.912.G-CO.4.13 MAFS.912.G-GPE.1.1 MAFS.912.G-GPE.1.2 Unit J: Properties of Circles Mathematics Florida Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 10.1, 10.3, 10.4, 10.5, 10.6 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 10.3, 10.4, 10.6 Part 2 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. 10.4, 10.5 *Not included on district assessment FSQ Unit J. Standard will be assessed on USA Unit J, K. 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. 10.1, 10.2 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) *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. 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 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 10.4, 10.6 Part 2 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 Derive the equations of ellipses and hyperbolas given the foci and directrices Part 2 AL *Honors only. Not included on district assessment for this unit. FSQ Unit J February 29 March 16 Use Properties of Tangents Find Arc Measures Apply Properties of Chords Use Inscribed Angles and Polygons Use Inscribed Angles and Polygons(Constructions) Apply Other Angle Relationships in Circles Find Segment Lengths in Circles Constructing Circumscribed Circles Write and Graph Equations of Circles Equation of a Parabola G-C - Circles G-CO - Congruence G-GPE Expressing Geometric Properties use properties of a tangent to a circle use angle measures to find arc measures use relationships of arcs and chords in a circle use inscribed angles of circles find the measures of angles interior and exterior a circle construct equilateral triangle, regular hexagon, square in a circle find segment lengths in circles construct a circle that circumscribes a triangle write equations of circles in the coordinate plane explain how the Pythagorean Theorem can be used to derive the equation of a circle write the equation of a parabola given a focus and directrx & Constructions AL Blender Supplemental Lessons 10.6 Part 2 OCG 7-2, 7-3, 7-6

11 MAFS.912.G-C.2.5 MAFS.912.G-CO.4.13 MAFS.912.G-GMD.1.1 MAFS.912.G-GMD.1.3 MAFS.912.G-GMD.2.4 MAFS.912.G-MG.1.2 MAFS.912.G-MG.1.3 MAFS.912.G-SRT.3.8 Unit K: Measurement of Figures and Solids Mathematics Florida 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. 11.1, 11.1 Extension, 11.2 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle 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. 11.1, 11.7 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 11.6, 11.7, 11.8, 11.9 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. 11.5, 11.7 Extension Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot) Extension 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) *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. March 28 April 11 Circumference and Arc Length Measure Angles in Radians Areas of Circles and Sectors Areas of Regular Polygons Explore Solids Volume of Prisms and Cylinders Density Volume of Pyramids and Cones Solids of Revolution Surface Area and Volume of Spheres Similar Solids G-C - Circles G-CO - Congruence G-GMD - Measurement & Dimension G-MG - Modeling G-SRT - G-SRT - find arc lengths and other measures find the radian measure of an angle find the areas of circles and sectors fine the areas of regular polygons inscribed in circles identify solids find the volume of prisms and cylinders use density to solve problems find the volume of pyramids and cones describe solids produced by rotating two-dimensional firgures around an axis in space find the surface area and volume of spheres use properties of similar solids Extension Extension Blender Supplemental Lessons OCA2 8-1 USA Unit J, K

12 Unit L: Post Assessment Content MAFS.912.G-GPE.1.3 Derive the equations of ellipses and hyperbolas given the foci and directrices. Supplement Post Assessment derive the equation of ellipses and hyperbolas given a focus and directrx Supplement