Course: Geometry Year: Teacher(s): various

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1 Course: Geometry Year: Teacher(s): various Unit 1: Coordinates and Transformations Standards Essential Questions Enduring Understandings G-CO.1. Know 1) How is coordinate Geometric precise definitions geometry used to transformations of angle, circle, find distances and may be explored perpendicular line, area of figures? through the parallel line, and application of line segment, 2) What are concrete materials, based on the transformations? coordinate undefined notions geometry, and the of point, line, 3) What properties use of dynamic distance along a are preserved by geometric software. line, and distance various around a circular transformations? arc. G-CO.2. 4) What properties Represent are common to all transformations in isometries? 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 Approximate Time Frame: 7-8 Weeks Skills Content Vocabulary Use the Pythagorean theorem to solve for a given side of a right triangle. Use the Distance Formula to find the distance between any two points in the coordinate plane and find the perimeter of a given polygon. Explain the relationship between the distance formula and the Pythagorean Theorem. Translate a polygon by a given vector Rotate an object either clockwise or counterclockwise Use both clockwise and counterclockwise rotations to determine the The Pythagorean Theorem and the Distance Formula Vectors and Translations Angles and Rotations Reflections Composition of Transformations Isometries altitude angle base clockwise composition (of transformations) counterclockwise dilation distance distance formula endpoint glide reflection horizontal stretch hypotenuse image isometry isosceles trapezoid isosceles triangle leg line line of symmetry mapping rule midpoint mirror line or reflection line opposite vectors parallel (lines) parallel vectors perimeter perpendicular point

2 not (e.g., translation versus horizontal stretch). G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. measures of angles and reflex angles Applying the slope criteria for perpendicular lines Apply mapping rules to figures in the coordinate plane for rotations of 90, 180 and 270 counterclockwise about the origin. Explain what happens to the coordinates when a point or object is reflected about the x-axis, the y-axis, the line y=x, and the line y = x Use dynamic software (such as Geogebra) to create reflections of objects Explain what the result will be when doing successive reflections in both parallel and intersecting lines and will be able to find the sum of the new angle of rotation in the latter Determine the result of composing two polygon pre-image ray reflection right triangle rise rotation rotational symmetry run segment shear side (of angle) side (of polygon) symmetry transformation translation triangle vector vertical stretch

3 G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 8.G.6. Explain a proof of the Pythagorean Theorem and its converse. 8.G.7. Apply the Pythagorean Theorem to determine the unknown side lengths in right triangles in realworld mathematical rotations and two translations Determine the result of performing glide reflections on an object Define the essential characteristic of an isometry Perform a dilation and a shear on a figure on the coordinate plane Perform a series of isometries on a given figure

4 problems in two and three dimensions. 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Unit 2: Congruence, Construction, and Proof Standards Essential Questions Enduring Understandings G-CO.6. Use 1) What are the Theorems about geometric properties of congruent triangles descriptions of congruent figures? may be proved rigid motions to using transform figures 2) How are theorems transformations and and to predict the and their converses applied to justify effect of a given related to each other? constructions. rigid motion on a given figure; given 3) How can two two figures, use triangles be proved the definition of congruent? congruence in terms of rigid 4) What motions to decide relationships hold if they are among angles formed congruent. by parallel lines and G-CO.7. Use the a transversal? definition of congruence in 5) How can you terms of rigid construct geometric motions to show figures with a that two triangles compass and are congruent if straightedge and how and only if Approximate Time Frame: 6-7 Weeks Skills Content Vocabulary Apply the properties of congruent segments, angles, and circle Understand the distinction between a theorem and its converse. Identify a sequence of isometries that maps one of two congruent figures onto another. Identify parts of congruent polygons. Understand that SAS and ASA are sufficient conditions for establishing that Congruent Figures SAS and ASA Congruence Isosceles Triangles SSS Congruence and HL Theorem Vertical Angles and Parallel Lines Geometric Constructions acute triangle alternate exterior angles alternate interior angles altitude (of triangle) base (of isosceles triangle) base angle (of isosceles triangle) compass conclusion congruent congruence symbol consecutive (sides, angles) construction convex (polygon) angles (formed by transversal) parts dart drawing

5 pairs of sides and pairs of angles are congruent. G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G-CO.10. Prove theorems about can you prove they work? two triangles are congruent. Explain transformational proofs of the SAS and ASA congruence theorems. Apply the SAS and ASA Congruence theorems to show that two triangles are congruent. Show that two triangles are congruent and then use this fact to show that pairs of sides or angles are also congruent. Classify triangles by properties of sides and angles Explain the proof of the Isosceles Triangle Theorem and its converse Apply the isosceles triangle theorem Explore relations between medians and altitudes of triangles Explain the role of isosceles triangles in proving the SSS Triangle equilateral triangle Euclidean construction hypothesis included angle included side interior angle of a polygon isosceles triangle kite leg (of isosceles triangle) linear pair of angles median (of triangle) non-convex (polygon) obtuse triangle parallel lines pependicular bisector polygon proof proposition (from Euclid s Elements) right triangle same-side exterior angles same-side interior angles scalene triangle side of a polygon straightedge supplementary angles transversal vertex angle (of isosceles triangle) vertical angles

6 triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the Congruence Theorem Apply the SSS Triangle Congruence Theorem to show that two triangles are congruent and that parts are also congruent. Understand and use the Hypotenuse Leg Congruence Theorem for Right triangles. Identify special pairs of angles (vertical angles, linear pairs, angles formed by two lines and a transversal, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles) Explain proofs of the Vertical Angles Theorem and theorems involving parallel lines and angles formed with a transversal

7 perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Unit 3: Polygons Standards Essential Questions Enduring Understandings G-CO.10. Prove 1) How are polygons Theorems about theorems about classified? polygons are related triangles. to each other in that Theorems include: 2) How can we all are proved based measures of prove properties of on postulates, interior angles of polygons? definitions, a triangle sum to properties and 180 ; base angles 3) How can regular previously proved of isosceles polygons be theorems. triangles are constructed? congruent; the segment joining Apply theorems studied in this investigation Understand the difference between a drawing and a construction and straightedge tools with dynamic geometry software to create Euclidean constructions Write out the steps of a Euclidean Construction and use theorems to prove the validity of such constructions Approximate Time Frame: 7-8 weeks Skills Content Vocabulary Understand and explain the reasoning behind proofs of the Triangle Sum, Quadrilateral Sum and Polygon Sum Theorems Apply the above theorems to find unknown angle measures Sums of Interior Angles of Polygons Inequalities in Triangles Parallel and Perpendicular Lines Regular Polygons Base (of Trapezoid) Bisect each other Centroid (of Triangle) Circumscribed circle Concurrent lines Converse Convex polygon Coordinate proof Diagonal

8 midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G-CO.11. Prove theorems about parallelograms. 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. G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, 4) How does coordinate proof differ from synthetic proof? Prove and apply the AAS Congruence Theorem Understand and explain Proofs of theorems involving inequalities in triangles. Apply theorems involving inequalities in triangles. Apply theorems to prove that two lines are parallel. Construct a line through a point parallel to a given line with compass and straightedge. Given a regular polygon with n sides, find the measures of the interior and exterior angles. Use compass and straightedge tools to construct equilateral triangles, squares, and regular hexagons Find lines of symmetry in a regular polygon Properties of Quadrilaterals Polygons with Coordinates Exterior angle of a polygon Heptagon Hexagon Inscribed polygon Interior angle of a polygon Isosceles Trapezoid Kite Leg (of Trapezoid) Median (of Triangle) Midsegment (of Trapezoid, triangle) Necessary condition Non-adjacent interior angle of a triangle Non-convex polygon Octagon Parallelogram Quadrilateral Rectangle Regular tessellation Rhombus Square Sufficient condition Synthetic proof Tessellation Tiling Transitive Property for > Trapezoid Trichotomy principle

9 string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G-GPE.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, 3) lies on the circle centered at the Understand relationships among special quadrilaterals Prove theorems related to quadrilaterals Derive formulas for the areas of parallelograms, triangles, trapezoids and kites. Prove properties of quadrilaterals and triangles given numerical coordinates for the vertices. Prove properties of quadrilaterals and triangle given variable coordinates for the vertices. Apply properties of quadrilaterals. Find midsegments of trapezoids and triangles Find centroids of triangles.

10 origin and containing the point (0, 2). Unit 4: Similarity Standards Essential Questions Enduring Understandings G-SRT.A.1.A A 1) How is the image The concept of dilation takes a under a dilation similarity enables line not passing related to its preimage? us to explore through the center geometric of the dilation to a relationships. parallel line, and 2) What are leaves a line similarity passing through transformations? the center unchanged. 3) How is G-SRT.A.1.A The congruence a special dilation of a line case of similarity? 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 the meaning of similarity for triangles as the equality of all Approximate Time Frame: 4-5 weeks Skills Content Vocabulary Understand that dilations map lines not passing through the center of dilation to a parallel line and lines passing through the center of dilation remain unchanged. Construct dilations of figures using a compass and straightedge when the center of dilation is on the figure, in the interior of the figure, or in the exterior of the figure. Identify the scale factors of images that have been dilated. Draw the image of a given image that has undergone a dilation. Determine if two given figures are Properties of Dilations Similar Figures Proving Similar Triangles Parallel Lines in Triangles Similarity in Right Triangles Special Right Triangles adjacent (leg) angle of descent angle of depression angle of elevation altitude center of dilation composition (of transformations) dilation directed line Segment division (of a segment) geometric mean hypotenuse indirect measurement leg (of right triangle) opposite (leg) mean proportional partitioning proportion right triangle scale factor similar figures similarity transformation simplified square root

11 pairs of angles and the proportionality of all pairs of sides. G-C.A.1 Prove that all circles are similar G-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G-SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. the result of a dilation and if so, determine the center of dilation. Understand that figures are similar if their angles are congruent and all pairs of sides are in proportion. Use similarity transformations to prove that two polygons or circles are similar. Understand that congruence is a special case of similarity. Determine if two figures are similar or not using similarity transformations Prove AA Similarity, SAS Similarity, and SSS Similarity theorems. Use similarity transformations and similarity theorems to prove two or more triangles similar. special right triangles 30 o -60 o -90 o and 45 o -45 o -90 o

12 G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Prove that all equilateral triangles are similar. Prove the Side Splitting Theorem and its converse. Use basic construction tools to divide a line segment into n congruent parts. Decompose figures and prove that triangles that make up the whole are similar. Prove the Right Triangle Similarity Theorem and use it to prove the Pythagorean Theorem. Solve problems involving proportions in right triangles. Determine missing side lengths in a special right triangle. Determine acute angle measures in a given right triangle with side lengths that determine a

13 special right triangle. Unit 5: Circles and other Conics Standards Essential Questions Enduring Understandings G-CO.9. Prove 1) How are the Most properties of theorems about equations of conic circles may be lines and angles. sections related to derived from the Theorems include: their locus definition of a circle vertical angles are definitions? as the locus of congruent; when a points at a given transversal 2) What relations distance from a crosses parallel among angles, given point. lines, alternate chords, and tangents interior angles are to circles can be congruent and proved? angles are 3) How are the congruent; points lengths of arcs and on a areas of sectors perpendicular related to central bisector of a line angles in circles? segment are exactly those 4) What are the equidistant from properties of the segment s inscribed and endpoints. circumscribed G-C.2. Identify triangles and and describe inscribed relationships quadrilaterals? among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed Approximate Time Frame: 5-6 Weeks Skills Content Vocabulary Define the term locus of points. Use the Pythagorean theorem to derive the formula for circles centered at origin. Complete the square to find the center and radius of a circle given by an equation in the form: x 2 + ax + y 2 + by + c = 0 Construct a perpendicular bisector Explain why the perpendicular bisector of any chord in a circle must include the center Use constructions to find the center of a circle given two chords Circumscribe a circle about a triangle Derive an equation that relates the measure of the Circles in the Coordinate Plane Radii and Chords Central Angles, Arcs and Sectors Tangents to Circles Angle Bisectors Inscribed Angles and Cyclic Quadrilaterals Parabolas Arc arc length center central angle chord circle circumscribed circle (of a triangle) completing the square conic section cyclic quadrilateral diameter directrix (of conic section) equidistant focus (of conic section) inscribed angle (of a circle) inscribed quadrilateral (of a circle) inscribed circle (of a triangle) intercepted arc locus of points major arc minor arc parabola point of tangency radius

14 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. G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G-C.5. 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. G-GPE.1. Derive the equation of a circle of given center and radius central angle, the length of the radius, and the length of the intercepted arc (arc length = ) Calculate arc length given the radius of the circle and the measure of the central angle. Derive an equation that relates the area of a sector, the length of the radius, and the measure of the central angle (sector area ). Explain why a tangent to a circle is perpendicular to the radius drawn to the point of tangency. Explain and apply the Tangent Segments Theorem Use coordinate geometry to solve problems involving circles and tangents Explain why the bisector of an angle is the locus of points equidistant from secant sector semicircle tangent tangent segment vertex (of parabola)

15 using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G-GPE.2. Derive the equation of a parabola given a focus and directrix. G-GPE.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, 3) lies on the circle centered at the origin and containing the point (0, 2). the sides of the angle. Inscribe a circle in a triangle Know and apply the Inscribed Angle Theorem and its corollaries Derive the standard form equations for the parabola with a horizontal directrix and the parabola with a vertical directrix. Given the equation of a parabola, determine both the focus and the directrix. Given two of the following, determine the equation of the parabola: vertex, focus, or directrix Solve real world applications involving parabolas. Unit 6: Three Dimensional Geometry Standards Essential Questions Enduring Understandings Approximate Time Frame: 5-6 Weeks Skills Content Vocabulary

16 6G4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. 7G6 Solve realworld and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 8G9 Know the formulas for the volumes of cones, cylinders and spheres and use them to solve realworld and mathematical problems. 1) What kinds of solid geometric figures are there? 2) What are some of the properties of these solids? 3) How can we find surface areas and volumes for these solids? 4) Where do the formulas come from? 5) How can we use these formulas to solve problems we might face in the real world? 6) Is there only one kind of geometry, the Euclidean Geometry with which we are familiar? Formulas for solid figures relate to each other in meaningful ways, just as area formulas for plane figures connect to each other. Construct models of prisms, pyramids, regular and semiregular polyhedra from concrete materials. Use strategies to count the numbers of vertices, edges and faces, and determine a relationship between these numbers (Euler s Formula). Examine a polyhedron, cylinder, or cone to identify the surfaces that make up the surface area, and compute the surface area using formulas for plane figures. Draw nets for solid figures, build models and calculate the surface areas of the shapes. Understand Cavallieri s principle and how it is applied to finding the volumes three- Polygons and Polyhedra Nets and Surface Area Volume Cross Sections and Solids of Revolution Spheres Geometry on the Sphere Size and Shape in the Real World altitude Archimedean solid axis of rotation CAD (computer aided design) circumscribed (prism or pyramid) concave cone convex cross-section cube cylinder density dodecahedron edge (of polyhedron) Equator face (of polyhedron) frustum (of cone or pyramid) geodesic great circle height (of prism, pyramid, cylinder or cone) hemisphere hexahedron inscribed (prism or pyramid) icosahedron isometric drawing lateral surface latitude longitude meridian net non-convex

17 G-GMD.1. 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. G-GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G-GMD.4. Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects G-MG.1. Use geometric shapes, their measures, and their dimensional figures. Find the volumes of prisms, cylinders, pyramids and cones. Visualize the cross section of a three dimensional figure. Identify the solid figure formed by rotation a twodimensional figure about an axis. Locate points in space with threedimensional rectangular coordinates. Explore informally various ways to demonstrate the surface area and volume of spheres. Use the formulas for surface area and volume in applications involving spheres. Use spherical coordinates to locate a point on any sphere; longitude and latitude to locate a point on Earth. octahedron Platonic solid polyhedron Prime Meridian prism pyramid projection regular polyhedron semi-regular polyhedron Schläfli symbols Schlegel diagram semi-regular polyhedron slant height solids generated by rotations of plane figures sphere spherical coordinates spherical excess spherical triangle surface area surface of revolution tesseract tetrahedron three-dimensional figure trilateration vertex (of polyhedron) volume

18 properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). * G-MG.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. Recognize that the shortest path between two points on a sphere lies on a great circle. Understand how the sum of its interior angles is related to the area of a spherical triangle. Recognize that any projection of a spherical surface onto a plane involves some distortion of size or shape. Understand how the properties of spheres are used in GPS. State a problem clearly and plan a step-by-step approach to solving the problem.