Other Materials/Projects

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1 GEOMETRY PACING GUIDE: 1st Nine Weeks UNIT 1: Transformations and Congruence Week Module Standards Learning Target WEEK ONE Goals Expectations Pre-Assessment Proportional Relationships Other Materials/Projects WEEK TWO M1: Tools of Geometry *G-CO.A.1: 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. *G-CO.A.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). *G-CO.C.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 corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Also: G.WCE.4 (+):Know the definition of skew lines; G.WCE.5 (+): Use the distance formula or Pythagorean Theorem to find the length of a segment on the coordinate plane; G.WCE.6 (+):Use the midpoint formula to find the midpoint of a segment on the coordinate plane; G-CO.B.5; G- CO.D.12; G-GPE.B.4 Demonstrate knowledge of the definitions of point, line, plane, line segment, angle, bisectors and describe their characteristics. Copy a line, copy an angle, bisect a line, and bisect an angle. Draw transformations of reflections, rotations, translations, dilations, and vertical/horizontal stretch. Determine the coordinated for the image when a transformation rule is applied. Explain rigid motion/isometry and distinguish between transformations that preserve distance and angle measure (translations, rotations, reflections) and those that do not (dilations, stretches). use inductive reasoning to identify patterns and make conjectures. find counterexamples to disprove conjectures. write and analyze biconditional statements. correctly interpret geometric diagrams by identifying what can and cannot be assumed. identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in proofs. order statements based on logic when constructing my proof. If time, incorporate the use of dynamic software for constructions.

2 WEEK THREE WEEK FOUR M2: Transformations and Symmetry M3: Congruent Figures *G-CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. *G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. *G-MG.A.3(will change to G-MG.A.2): 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). Also: *G-CO.A.1; *G-CO.A.2; *G-CO.A.5; *G- CO.B.6; *G-CO.D.12 *G-CO.A.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. *G-CO.B.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. *G-CO.B.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Also: *G-CO.A.2 Draw transformations of reflections, rotations, translations, dilations, and vertical/horizontal stretch. Determine the coordinated for the image when a transformation rule is applied. Explain rigid motion/isometry and distinguish between transformations that preserve distance and angle measure (translations, rotations, reflections) and those that do not (dilations, stretches). Draw transformations of reflections, rotations, translations, dilations, vertical/horizontal stretch, and combinations of transformations. define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure. define congruent figures as figures that have the same shape and size and state the composition of rigid motions will map one congruent figure onto the other. predict the composition of transformations that will map a figure onto a congruent figure. determine if two figures are congruent by determining if rigid motions will turn one figure onto the other. identify corresponding sides and corresponding angles of congruent triangles. explain that in a pair of congruent triangles, corresponding sides are congruent and corresponding angles are congruent. justify that in congruent triangles distance and angle measure are preserved. If time, incorporate the use of dynamic software for transformations.

3 UNIT 2: Lines, Angles, and Triangles Week Module Standards Learning Target WEEK FIVE WEEK SIX WEEK SEVEN M4: Lines and Angles M4: Lines and Angles M5: Triangle Congruence Criteria *G-CO.C.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 corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. *G-CO.C.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 corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. *G-GPE.B.5: 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). Also: *G-CO.D.12 *G-CO.B.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. *G-CO.B.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in the terms of rigid motion. G.WCE.11(+): Apply congruence theorems of HL and AAS and CPCTC. Also: *G-CO.C.10, *G-SRT.B.5, prove vertical angles are congruent prove and apply challenging theorems about the angles formed by parallel lines and a transversal. construct a line parallel to a given line through a point not on the line. construct perpendicular lines including the perpendicular bisector of a segment. prove points on a perpendicular bisector of a line segment are exactly equidistant from the segment s endpoints. write the equation of a line parallel or perpendicular to a given line that passes through a given point. identify corresponding sides and corresponding angles of congruent triangles. explain that in a pair of congruent triangles, corresponding sides are congruent and corresponding angles are congruent justify that in congruent triangles distance and angle measure are preserved use the definition of congruence, based on rigid motion, to explain the triangle congruence criteria (ASA, SAS SSS, HL, AAS and CPCTC) Other Materials/Projects WCE.G.3(+)

4 WEEK EIGHT M6: Applications of Triangle Congruence *G-CO.D.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. G-CO.D.13(will drop): Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. *G-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. prove that the results of a construction are valid. use congruence and similarity criteria to solve real-world problems involving triangles. WEEK NINE M7: Properties of Triangles *G-CO.C.10: Prove theorems about triangles. Theorems to 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-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.WCE.18 (+): Find the sum and angle measures of polygons. G-CO.D.13(will drop): Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Also: *G-CO.D.12 prove the sum of the measures of the interior angles of a triangle is equal to 180⁰. prove the base angles of isosceles triangles are congruent. Use theorems about the interior angles of isosceles and equilateral triangles to solve real-world problems. identify that a polygon is a triangle given three side measures (Triangle Inequality Theorem) define inscribed polygons. construct an equilateral triangle inscribed in a circle. construct a square inscribed in a circle. construct a regular hexagon inscribed in a circle. explain the steps to constructing an equilateral triangle, a square, and a regular hexagon inscribed in a circle. generalize conjectures to all similar figures. distinguish among altitudes, angle bisectors, perpendicular bisectors, medians and midsegments in triangles and use their properties to solve problems. distinguish among the centroid, orthocenter, incenter, and circumcenter in a triangle and use the properties of each to solve problems. WCE.G.6(+) G-CO.D.13(+)

5 GEOMETRY PACING GUIDE: 2nd Nine Weeks UNIT 2: Lines, Angles, and Triangles (cont.) Week Lesson Standards Learning Target WEEK ONE M8: Special Segments in Triangles *G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of a quadrilateral inscribed in a circle. *G-CO.C.10: Prove theorems about triangles. Theorems to 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. Also: G.WCE.12(+): Demonstrate understanding of the triangle inequality theorem, the hinge theorem, and the centroid theorem; *G-CO.C.9; *G-CO.D.12; *G-GPE.B.4; *G-GPE.B.5 construct a circumscribed circle of a triangle. prove the segment joining midpoints of two sides of a triangle is parallel to the third side (midsegment) prove the segment joining the midpoints to two sides of a triangle is half the length of the third side. prove the medians of a triangle meet at a point called the centroid. explore additional properties of the centroid theorem. distinguish among altitudes, angle bisectors, perpendicular bisectors, medians and midsegments in triangles and use their properties to solve problems. distinguish among the centroid, orthocenter, incenter, and circumcenter in a triangle and use the properties of each to solve problems. construct special segments in triangles using a compass and a straight edge or patty paper. Other Materials/Projects

6 UNIT 3: Quadrilaterals and Coordinate Proof Week Lesson Standards Learning Target WEEK TWO WEEK THREE M8: Special Segments in Triangles/ M9: Properties of Quadrilaterals M9: Properties of Quadrilaterals *G-CO.C.10: Prove theorems about triangles. Theorems to 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.C.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. WCE.G.5(+): Demonstrate understanding of the hierarchy of quadrilaterals. Also: *G-CO.D.12, *G-GPE.B.4, *G- GPE.B.5 *G-CO.C.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-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Also: *G-CO.C.10, *G-STR.B.5 prove the segment joining midpoints of two sides of a triangle is parallel to the third side (midsegment) define and describe parallelograms. prove the opposite sides of a parallelogram are congruent. prove the opposite angles of a parallelogram are congruent. prove the diagonals of a parallelogram bisect each other. define and describe rectangles, rhombuses, squares, trapezoids and kites. prove rectangles are parallelograms with congruent diagonals. use congruence and similarity criteria for triangles to prove relationships for kites and trapezoids. Other Materials/Projects

7 WEEK FOUR WEEK FIVE M10: Coordinate Proof Using Slope and Distance M10: Coordinate Proof Using Slope and Distance *G-GPE.B.5 (will change to G- GPE.B.3): 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). Also: *G-GPE.B.4 *G-GPE.B.4 (will change to G- GPE.B.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, 3) lies on the circle centered at the origin and containing the point (0, 2). *G-CO.C.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-GPE.B.7 (will change to G- GPE.B.5): Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. (Perimeter only at this time) Also: *G-GPE.B.5 represent the vertices of a figure in the coordinate plane using variables use slope to prove lines are parallel or perpendicular. prove or disprove geometric theorems or definitions in relation to the coordinate plane using slope, distance and midpoint formulas. use coordinate geometry and the distance formula to find the perimeters of polygons.

8 UNIT 4: Similarity Week Lesson Standards Learning Target WEEK SIX WEEK SEVEN M11: Similarity and Transformations M11: Similarity and Transformations *G-SRT.A.1a: 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. *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 corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Also: G-CO.A.1, G-CO.A.2, G- CO.A.5 *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 corresponding pairs of angles and the proportionality of all corresponding 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. Also: *G-SRT.B.4, *G-SRT.B.5 define dilation. perform a dilation with a given center and a scale factor on a figure in the coordinate plane. verify that when a side passes through the center of dilation, the side and its image lie on the same line. verify that corresponding sides of the pre-image and images are parallel and proportional. define similarity as a composition of rigid motions followed by dilations in which angle measure is preserved and side length is proportional. identify corresponding sides and corresponding angles of similar triangles. determine scale factor between two similar figures and use the scale factor to solve problems. demonstrate that corresponding angles are congruent and corresponding sides are proportional in a pair of similar triangles. determine that two figures are similar by verifying that angle measure is preserved and corresponding sides are proportional. Other Materials/Projects

9 WEEK EIGHT WEEK NINE M12: Using Similar Triangles *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. *G-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. *G-GPE.B.6 (will change to G- GPE.B.4): Find the point on a directed line segment between two given points that partitions the segment in a given ratio. WCE.G.7(+): Apply the angle bisector proportionality theorem to triangles. WCE.G.8(+): Apply the geometric mean to right triangles. Also: G-CO.C.10, G-CO.D.12 Tie up loose ends. Review for midterm. Midterm exam. show and explain that when two angle measures are known, the third angle measure is also known. prove a line parallel to one side of a triangle divides the other two proportionally. prove that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. use angle congruence and proportional side relationships to prove similarity in geometric figures. find missing parts of triangles using the Angle Bisector Proportionality Theorem. use triangle similarity theorems such as AA, SSS and SAS to prove two triangles are similar. prove the Pythagorean Theorem using triangle similarity. use properties of similar right triangles to form the definitions of trigonometric ratios for acute angles. WCE.G.7(+) WCE.G.8(+)

10 Geometry MATH PACING GUIDE: 3rd Nine Weeks UNIT 5: Trigonometry Week Module Standards Learning Target WEEK ONE WEEK TWO M13: Trigonometry and Right Triangles M13: Trigonometry and Right Triangles *G-SRT.C.6: 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. *G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. *G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. *G-SRT.C.6: 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. *G-SRT.C.8 (will change to G- SRT.C.8a): Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Also: WCE.G.16(+): Apply the geometric mean to right triangles; WCE.G.17(+): Prove the Pythagorean Theorem using triangle similarity; WCE.G.18(+): Make use if the converse of the Pythagorean theorem, Pythagorean triples, and special right triangles and use them to solve problems; G-SRT.D.9(+), G- GPE.B.7 use properties of similar right triangles to form the definitions of trigonometric ratios for acute angles. calculate sine and cosine ratios for acute angles in a right triangle when two side lengths are given. explain and use the relationship between the sine of an acute angle and the cosine of its complement. use Pythagorean Theorem to solve for unknown side length of a right triangle. use the converse of the Pythagorean Theorem to determine if a triangle is acute, obtuse, or right. list the common Pythagorean triples. solve right triangles including special right triangles by finding the measures of all sides and angles in the triangles. draw right triangles that describe real world problems and label the sides and angles with their given measures. I can solve application problems involving right triangles, including angle of elevation and depression, navigation and surveying. use properties of similar right triangles to form the definitions of trigonometric ratios for acute angles. calculate sine and cosine ratios for acute angles in a right triangle when two side lengths are given. explain and use the relationship between the sine of an acute angle and the cosine of its complement. use Pythagorean Theorem to solve for unknown side length of a right triangle. use the converse of the Pythagorean Theorem to determine if a triangle is acute, obtuse, or right. list the common Pythagorean triples. solve right triangles including special right triangles by finding the measures of all sides and angles in the triangles. draw right triangles that describe real world problems and label the sides and angles with their given measures. I can solve application problems involving right triangles, including angle of elevation and depression, navigation and surveying. Other Materials/Projects

11 WEEK THREE M14: Trigonometry with All Triangles G-SRT.D.9 (+): Derive the formula A = ½ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.D.10 (+): Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.D.11 (+): Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right and non-right triangles (e.g., surveying problems, resultant forces) **All will change to G-SRT.C.8b: Know and use the Law of Sines and Law of Cosines to solve problems in real life situations. Recognize when it is appropriate to use each. derive the formula, A = ½ absin(c), for the area of a triangle. calculate the area of a triangle using the formula A=1/2absin(C) derive the Law of Sines and Law of Cosines. use the Law of Sines and Law of Cosines to solve real world problems. I can distinguish between situations that require the Law of Sines or the Law of Cosines. I can represent real world problems with diagrams of right and nonright triangles and use them to solve for unknown side lengths and angle measures. G-SRT.D.9 G-SDT.D.11 WEEK FOUR M14: Trigonometry with All Triangles G-SRT.D.9 (+): Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.D.10 (+): Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.D.11 (+): Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right and non-right triangles (e.g., surveying problems, resultant forces) **All will change to G-SRT.C.8b: Know and use the Law of Sines and Law of Cosines to solve problems in real life situations. Recognize when it is appropriate to use each. derive the formula, A = ½absin(C), for the area of a triangle. calculate the area of a triangle using the formula A= ½absin(C) derive the Law of Sines and Law of Cosines. use the Law of Sines and Law of Cosines to solve real world problems. I can distinguish between situations that require the Law of Sines or the Law of Cosines. I can represent real world problems with diagrams of right and nonright triangles and use them to solve for unknown side lengths and angle measures. G-SRT.D.9 G-SDT.D.11

12 UNIT 6: Properties of circles Week Module Standards Learning Target WEEK FIVE WEEK SIX M15: Angles and Segments in Circles M15: Angles and Segments in Circles *G-C.A.1 (will change to recognize ): Prove that all circles are similar. *G-C.A.2: 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.) Also: G-C.A.4(+), G-C.D.13 *G-C.A.2: 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.) *G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Also: G.WCE.22(+): Explore segment relationships in circles; G.WCE.20(+): Revisit area formulas and composite area problems describe the relationship between two secants, a secant and a tangent or two tangents in relation to the intercepted circle. verify that inscribed angles on a diameter are right angles. verify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. construct the inscribed circle whose center is the point of intersection of the angle bisectors (incenter). prove that the opposite angles in an inscribed quadrilateral are supplementary. construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors (circumcenter). find the lengths of segments formed by lines that intersect circles. describe the relationship between two secants, a secant and a tangent or two tangents in relation to the intercepted circle. verify that inscribed angles on a diameter are right angles. verify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. construct the inscribed circle whose center is the point of intersection of the angle bisectors (incenter). prove that the opposite angles in an inscribed quadrilateral are supplementary. construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors (circumcenter). I can find the lengths of segments formed by lines that intersect circles. use similarity to calculate the length of an arc. define the radian measure of an angle as the ratio of arc length to its radius, and calculate a radian measure when given an arc length and its radius. convert degrees to radians using the constant of proportionality (2π x angle measure/360 ). use similarity to derive the formula for the area of a sector. Other Materials/Projects

13 WEEK SEVEN WEEK EIGHT M16: Arc Length and Sector Area M17: Equations of Circles and Parabolas *G-GMD.A.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-C.A.1 (will change to recognize ): Prove that all circles are similar. G-C.B.5 (will drop): 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. Also: WCE.G.21(+): Use the similarity ratio between two solids to fine the volume; WCE.G.8(+): Find the lateral area and surface area of threedimensional figures; *G- MG.A.1 *G-GPE.A.1 (wording will change): 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. G-GPE.A.2 (BOOK ONLY) Also: *G-GPE.B.4, define π _as the ratio of a circle s circumference to its diameter. use algebra to demonstrate that because π _is the ratio of a circle s circumference to its diameter that the formula for a circle s circumference must be C = π d. find the surface area of prisms, cylinders, pyramids, spheres and cones. describe the difference between lateral and surface area. explain the meaning of the symbols in the formulas. develop formulas to calculate the volumes of 3-D figures including spheres, cones, prisms, and pyramids. use the similarity ratio between two solids to find the volume use the Pythagoren Theorem to locate points on a circle, the center of a circle, or the radius of a circle. derive the equation of a circle given the center and the radius of the circle. given the equation, identify the center and the radius of a circle and graph the circle. WEEK NINE Make-Up This week will be for catching up on any missed days due to testing, holidays, or snow.

14 Geometry MATH PACING GUIDE: 4th Nine Weeks UNIT 7: Measurement and Modeling in Two and Three Dimensions Week Lesson Standards Learning Target WEEK ONE WEEK TWO M18: Volume Formulas M18: Volume Formulas/ M19: Visualizing Solids *G-GMD.A.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-MG.A.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Also: *G-GMD.A.2, G-GMD.A.3, *G-MG.A.1 *G-GMD.A.2(+): Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G-GMD.B.4 (will drop): Identify the shapes of two dimensional cross sections of three-dimensional objects, and identify three dimensional objects generated by rotations of two dimensional objects. *G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Also: G-GMD.A.3, G-MG.A.1, G- MG.A.2 state that if two solid figures have the same total height and their crosssectional area are identical at every level, then the figures have the same volume. compute the volume of a prism, pyramid, cylinder, cone, and composite solid. state that if two solid figures have the same total height and their crosssectional area are identical at every level, then the figures have the same volume. compute the volume of a prism, pyramid, cylinder, cone, sphere, and composite solid. identify the shapes of two dimensional cross sections of threedimensional objects. compute the surface area of a prism and cylinder. Other Materials/Projects

15 WEEK THREE WEEK FOUR WEEK FIVE M19: Visualizing Solids/ M20: Modeling and Problem Solving M20: Modeling and Problem Solving *G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G-GMD.A.3 (will drop): 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). Also: G-GPE.B.7, G-MG.A.2 *G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Also: G-MG.A.3 G.WCE.7(+) Understand and apply the rules of logic as they relate to proving geometric theorems. G.WCE.8(+) Investigate forms of non-euclidean geometry. G.WCE.9(+) Understand and apply the rules of logic as they relate to proving geometric theorems. G.WCE.10(+) Construct truth tables to determine the truth value of logical statements. compute the surface area of a prism, pyramid, cylinder, cone, sphere, and composite solid. apply concepts of proportionality and scale factor to solve problems. use properties and measurements to describe objects. use properties and measurements to describe objects. solve problems with given contraints. find counterexamples to disprove conjectures. write and analyze biconditional statements. write the inverse, converse, and contrapositive of a conditional statement. order statements based on logic when constructing my proof. research and explain the other types of geometry besides Euclidean such as spherical, hyperbolic, and elliptical. identify, write, and analyze the truth value of conditional statements. apply the Law of Detachment and Law of Syllogism in logical reasoning. follow logical steps to write a simple indirect proof. construct truth tables to determine the truth value of logical statements. Review concepts from throughout the year with comprehensive learning tasks.

16 WEEK SIX WEEK SEVEN WEEK EIGHT G.WCE.13 Construct the points of concurrency within a triangle and solve problems using the properties of the centroid, orthocenter, incenter, and circumcenter. Review concepts from throughout the year with comprehensive learning tasks. G.C.4(+) Construct a tangent line from a point outside a given circle to the circle. G.GPE.2(+) Derive the equation of a parabola given a focus and directrix. G.GPE.3(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Review concepts from throughout the year with comprehensive learning tasks. Review concepts from throughout the year with comprehensive learning tasks. distinguish among altitudes, angle bisectors, perpendicular bisectors, medians and midsegments in triangles and use their properties to solve problems. distinguish among the centroid, orthocenter, incenter, and circumcenter in a triangle and use the properties of each to solve problems. construct special segments in triangles using a compass and a straight edge or patty paper. use points of concurrency to construct and make conjectures about the Euler Line. construct a tangent line from a point outside a given circle to the circle. derive the equation of a parabola given a focus and directrix. derive the equation of an ellipse given the foci, noting that the sum of the distances from the foci to any fixed point on the ellipse is constant. apply concepts of geometry to solve problems explain properties and geometric relationships mathematically WEEK NINE Final assessments and yearending activities