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The School District of Palm Beach County GEOMETRY REGULAR Unit A: Essentials of Geometry 2015 - 2016 Topic & Suggested Pacing Standards G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties with Equations G-MG Geometry – Modeling Geometry G-GMD Geometry – Geometry Measurement & Dimension Student Target Core Mathematics Florida Standards August 18 - August 28 • name and sketch geometric figures Identify Points, Lines, and Planes • use segment posulates to identify congruent segments Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, Use Segments and Congruence • find lengths of segments in reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting MAFS.912.G-CO.4.12 the coordinate plane a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line Use Midpoint and Distance Calculator Neutral • copy and bisect segments segment; and constructing a line parallel to a given line through a point not on the line. Formulas and angles using construction 1.4 tools Measure and Classify Angles Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance • use special angle MAFS.912.G-GPE.2.7 formula. relationships to find angle Calculator Neutral Measure and Classify Angles 1.2, 1.3, 1.4 measures Constructions Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a • classify polygons MAFS.912.G-MG.1.1 human torso as a cylinder). • name, measure, and classify Calculator Neutral Describe Angle Pair Relationships angles 1.6 MAFS.912.G-CO.1.1 Calculator: Neutral 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 Classify Polygons FSQ Unit A Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Larson Geometry 1.1 1.2 1.3 1.4 & Constructions 1.5 1.6 The School District of Palm Beach County GEOMETRY REGULAR Unit B: Reasoning and Proof 2015 - 2016 G-CO Geometry - Congruence Topic & Suggested Pacing Standards Mathematics Florida Standards • write definitions as conditional statements • use postulates involving Analyze Conditional Statements points, lines, and planes • use algebraic properties in Use Postulates and Diagrams logical arguments • write proofs using geometric Reason Using Properties from theorems Algebra • use properties of special pairs of angles Prove Statements about Segments and Angles Core August 31 - September 9 MAFS.912.G-CO.1.1 Calculator: Neutral 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 MAFS.912.G-CO.3.9 Calculator: Neutral 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 Prove Angle Pair Relationships FSQ Unit B Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Student Target Larson Geometry 2.2 2.4 2.5 2.6 2.7 The School District of Palm Beach County GEOMETRY REGULAR Unit C: Parallel and Perpendicular Lines 2015 - 2016 Topic & Suggested Pacing Standards Mathematics Florida Standards 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 MAFS.912.G-CO.3.9 Calculator: Neutral 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 MAFS.912.G-CO.4.12 a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line Calculator: Neutral 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 USA Unit A, B, C Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Student Target • identify angle pairs formed by three intersecting lines • use angles formed by parallel Identify Pairs of Lines and Angles lines and transversals • use angle relationships to Use Parallel Lines & Transversals prove that lines are parallel • find and compare slopes of Prove Lines are Parallel lines • find equations of lines Find and Use Slopes of Lines • find the distance between a point and a line Write and Graph Equations of • construct parallel and Lines perpendicular lines Core September 10 - September 29 MAFS.912.G-CO.1.1 Calculator: Neutral MAFS.912.G-GPE.2.5 Calculator: Neutral G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties with Equations Prove Theorems About Perpendicular Lines Prove Theorems About Perpendicular Lines (Constructions) Larson Geometry 3.1 3.2 3.3 3.4 3.5 3.6 & Constructions The School District of Palm Beach County GEOMETRY REGULAR Unit D: Congruent Triangles 2015 - 2016 G-CO Geometry - Congruence Topic & Suggested Pacing Standards Mathematics Florida Standards MAFS.912.G-CO.1.2 Calculator: Neutral MAFS.912.G-CO.1.5 Calculator: Neutral MAFS.912.G-CO.2.6 Calculator: Neutral 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 October 5 - October 22 Apply Triangle Sum Properties Apply Congruence and Triangles Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph Relate Transformations & paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure Congruence onto another. 4.9 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on Prove Triangles Congruent by SSS a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are Prove Triangles Congruent by SAS congruent. & HL 4.3, 4.9 MAFS.912.G-CO.2.7 Calculator: Neutral 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. MAFS.912.G-CO.2.8 Calculator: Neutral 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 Triangles Congruent by ASA and AAS Use Congruent Triangles 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 MAFS.912.G-CO.3.10 congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; Calculator: Neutral the medians of a triangle meet at a point. 4.1., 4.4, 4.5, 4.6, 4.7, 4.8 FSQ Unit D Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Use Isosceles and Equilateral Triangles Perform Congruence Transformations Student Target • 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 Core Larson Geometry 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 The School District of Palm Beach County GEOMETRY REGULAR Unit E: Relationships within Triangles 2015 - 2016 Topic & Suggested Pacing Standards Mathematics Florida Standards MAFS.912.G-CO.3.9 Calculator: Neutral 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 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 Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Student Target • use properties of midsegments and write coordinate proofs Midsegment Theorem and • use perpendicular bisectors Coordinate Proof to solve problems • use angle bisectors to find Use Perpendicular Bisectors distance relationships • use medians and altitudes of Use Angle Bisectors of Triangles triangles • find possbile side lengths of Use Medians and Altitudes triangles • use inequalities to make Use Inequalities in a Triangle comparisons in two triangles Core October 26 – November 6 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 MAFS.912.G-CO.3.10 congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; Calculator: Neutral the medians of a triangle meet at a point. 5.1, 5.3, 5.4, 5.6 MAFS.912.G-GPE.2.4 Calculator: Neutral G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties with Equations Inequalities in Two Triangles and Indirect Proof Larson Geometry 5.1 5.2 5.3 5.4 5.6 The School District of Palm Beach County GEOMETRY REGULAR Unit F: Similarity 2015 - 2016 Standards Mathematics Florida Standards MAFS.912.G-C.1.1 Calculator: Neutral Prove that all circles are similar. 6.2 MAFS.912.G-CO.1.2 Calculator: Neutral 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. MAFS.912.G-GPE.2.4 Calculator: Neutral 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 MAFS.912.G-GPE.2.6 Calculator: Neutral 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 MAFS.912.G-MG.1.3 Calculator: Neutral MAFS.912.G-SRT.1.1 Calculator: Neutral 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 MAFS.912.G-SRT.1.2 Calculator: Neutral 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. MAFS.912.G-SRT.1.3 Calculator: Neutral Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 6.3 MAFS.912.G-SRT.2.4 Calculator: Neutral 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. MAFS.912.G-SRT.2.5 Calculator: Neutral 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 Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties with Equations G-MG Geometry - Modeling Geometry Topic & Suggested Pacing Student Target Core November 9 – November 24 • use proportions to identify similar polygons Larson Use Similar Polygons • identify similarity Geometry transformations called 6.1 Relate Transformations and dilations 6.2 Similarity • explain what it means for two figures to be similar 6.3 Prove Triangles Similar by AA • use the AA Similarity 6.4 Posulate to show two triangles 6.5 Prove Triangles Similar by SSS and similar SAS • use the SSS and SAS Similarity 6.6 Theorems to show to triangle 6.6 Use Proportionality Theorems similar • use proprotions with a Extension Perform Similarity triangle or parallel lines Transformations • perform dilations Blender • find a point on a directed line Supplmental Partition Segments segment that partitions the segment in a given ratio Lessons OCG 5-3 OCG 8-3 The School District of Palm Beach County GEOMETRY REGULAR Unit G: Right Triangles and Trigonometry 2015 - 2016 Topic & Suggested Pacing Standards Mathematics Florida Standards MAFS.912.G-MG.1.1 Calculator: Neutral MAFS.912.G-SRT.2.4 Calculator: Neutral MAFS.912.G-SRT.2.5 Calculator: Neutral MAFS.912.G-SRT.3.6 Calculator: Neutral MAFS.912.G-SRT.3.7 Calculator: Neutral MAFS.912.G-SRT.3.8 Calculator: Neutral Student Target Core • use the Pythagorean Theoerm to find the side Larson lengths in right triangles Geometry Apply the Pythagorean Theorem • use the converse of the 7.1 Pythagorean Theorem to 7.2 Use the Converse of the determine if a triangle is a right Pythagorean Theorem triangle 7.3 • use properties of the altitude 7.4 Use Similar Right Triangles of a right triangle 7.5 • use the relationships among Special Right Triangles the sides in special right 7.6 triangles 7.6 Apply the Tangent Ratio • use the tangent ratio for Additional indirect measurment Apply the Sine and Cosine Ratios • use the sine and cosine ratios Lesson AL 2 • explain the relationships 3 Trig Ratios of Complementary between the sine and cosine of Angles complementary angles 7.7 • use the inverse tangent, sine, Solve Right Triangles and cosine ratios January 5 – January 20 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 FSQ Unit G Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education G-GPE Geometry – Expressing Geometric Properties with Equations G-MG Geometry - Modeling Geometry - The School District of Palm Beach County GEOMETRY REGULAR Unit H: Quadrilateral 2015 - 2016 G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties with Equations G-MG Geometry - Modeling Geometry Topic & Suggested Pacing Standards Student Target Core • find angle measures in Larson polygons Geometry • find angle and side measures 8.1 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: Find Angle Measures in Polygons in parallelograms MAFS.912.G-CO.3.11 opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and 8.2 • know the diagonal properties Calculator: Neutral conversely, rectangles are parallelograms with congruent diagonals. Use Properties of Parallelograms in parallelograms 8.3 8.2, 8.3, 8.4, 8.6 • use properties to identify 8.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure Sides and Angles of parallelgrams MAFS.912.G-GPE.2.4 defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the 8.5 Parallelograms • use properties of rhombuses, Calculator: Neutral circle centered at the origin and containing the point (0, 2). rectangles, and squares 8.6 8.5 Show that a Quadrilateral is a • use the properties of Parallelogram trapezoids and kites Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance Blender MAFS.912.G-GPE.2.7 • identify special quadrilaterals formula. Supplemental Calculator: Neutral Properties of Rhombuses, Rec8.4 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. Lessons tangles, and Squares Mathematics Florida Standards January 21 – February 5 MAFS.912.G-MG.1.1 Calculator: Neutral 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. MAFS.912.G-SRT.2.5 Calculator: Neutral 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. FSQ Unit H Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Special Parallelograms Use Properties of Trapezoids and Kites Identify Special Quadrilaterals 8.2 Part 2 OCG 4-2 & 3 8.4 Part 2 OCG 4-5 The School District of Palm Beach County GEOMETRY REGULAR Unit I: Properties of Transformations 2015 - 2016 Topic & Suggested Pacing Standards MAFS.912.G-CO.1.3 Calculator: Neutral MAFS.912.G-CO.1.4 Calculator: Neutral MAFS.912.G-CO.1.5 Calculator: Neutral MAFS.912.G-SRT.1.1 Calculator: Neutral MAFS.912.G-SRT.1.2 Calculator: Neutral Student Target • use a vector to translate a figure • reflect a figure in any given Represent transformations in the plane using, e.g., transparencies and geometry software; describe Translate Figures and Use Vectors line transformations as functions that take points in the plane as inputs and give other points as outputs. Compare • rotate figures about a point transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Perform Reflections • carry a polygon onto itself 9.3, 9.5, 9.7 through rotations and Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it Perform Rotations reflections onto itself. • perform combinations of two 9.6 Rotations & Reflections - Carry a or more transformations Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel Polygom onto Itself • identify line and rotational lines, and line segments. symmetries of a figure 9.1, 9.3, 9.4 Apply Compositions of • use drawing tools to draw Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph Transformations dilations paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Identify Symmetry 9.1, 9.3, 9.4, 9.5, 9.6, 9.7 Identify and Perform Dilations 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. Mathematics Florida Standards MAFS.912.G-CO.1.2 Calculator: Neutral G-CO Geometry - Congruence February 8 – February 22 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 USA Unit G, H, I Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education Core Larson Geometry 9.1 9.3 9.4 9.5 9.6 9.7 Blender Supplemental Lessons Supplement Rotations & Reflections The School District of Palm Beach County GEOMETRY REGULAR Unit J: Properties of Circles 2015 - 2016 Standards Mathematics Florida Standards MAFS.912.G-C.1.2 Calculator: Neutral MAFS.912.G-C.1.3 Calculator: Neutral MAFS.912.G-C.2.5 Calculator: Neutral MAFS.912.G-CO.1.1 Calculator: Neutral MAFS.912.G-CO.1.2 Calculator: Neutral 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). 10.1 *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 MAFS.912.G-CO.4.12 a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line Calculator: Neutral segment; and constructing a line parallel to a given line through a point not on the line. 10.4 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Calculator: Neutral 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 MAFS.912.G-GPE.1.1 find the center and radius of a circle given by an equation. Calculator: Neutral 10.7 FSQ Unit J Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education G-C Geometry - Circles G-CO Geometry - Congruence G-GPE Geometry – Expressing Geometric Properties Topic & Suggested Pacing Student Target Core • use properties of a tangent to a circle Larson • use angle measures to find Geometry Use Properties of Tangents arc measures 10.1 • use relationships of arcs and 10.2 Find Arc Measures chords in a circle • use inscribed angles of circles 10.3 Apply Properties of Chords • find the measures of angles 10.4 & interior and exterior a circle Use Inscribed Angles and • construct equilateral triangle, Constructions Polygons regular hexagon, square in a circle Use Inscribed Angles and • find segment lengths in Polygons(Constructions) circles • construct a circle that Apply Other Angle Relationships circumscribes a triangle Blender in Circles • write equations of circles in Supplemental the coordinate plane Lessons Find Segment Lengths in Circles • explain how the Pythagorean Theorem can be used to derive 10.6 Part 2 Constructing Circumscribed the equation of a circle OCG 7-2, 7Circles February 29 – March 16 10.5 10.6 10.7 3, 7-6 Write and Graph Equations of Circles The School District of Palm Beach County GEOMETRY REGULAR Unit K: Measurement of Figures and Solids 2015 - 2016 Topic & Suggested Pacing Standards Mathematics Florida Standards Student Target Core • find arc lengths and other measures Larson • find the radian measure of an Geometry Circumference and Arc Length angle • find the areas of circles and Measure Angles in Radians sectors 11.1 Extension • fine the areas of regular Areas of Circles and Sectors polygons inscribed in circles • identify solids Areas of Regular Polygons • find the volume of prisms and cylinders Explore Solids • use density to solve problems • find the volume of pyramids Volume of Prisms and Cylinders and cones • describe solids produced by 11.6 Extension Density rotating two-dimensional firgures around an axis in space Volume of Pyramids and Cones • find the surface area and volume of spheres Solids of Revolution • use properties of similar solids Surface Area and Volume of Blender Spheres Supplemental Lessons Similar Solids March 28 – April 11 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 MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Calculator: Neutral 11.3 MAFS.912.G-C.2.5 Calculator: Neutral Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, MAFS.912.G-GMD.1.1 pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Calculator: Neutral 11.1, 11.7 MAFS.912.G-GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Calculator: Neutral 11.6, 11.7, 11.8, 11.9 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional MAFS.912.G-GMD.2.4 objects generated by rotations of two-dimensional objects. Calculator: Neutral 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 MAFS.912.G-MG.1.2 cubic foot). Calculator: Neutral 11.6 Extension MAFS.912.G-MG.1.3 Calculator: Neutral 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). 11.7 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. MAFS.912.G-SRT.3.8 Calculator: Neutral Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 11.3 *Not included on district assessment for this unit. Standard will be assessed elsewhere as appropriate. USA Unit J, K Copyright © 2015 by School Board of Palm Beach County, Department of Secondary Education G-C Geometry - Circles G-CO Geometry - Congruence G-GMD Geometry - Geometry Measurement & Dimension G-MG Geometry - Modeling Geometry G-SRT Geometry - G-SRT Geometry - Similarity, 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 OCA2 8-1