# Geometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute

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1 Geometry Cluster: Experiment with transformations in the plane. G.CO.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.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.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. G.CO.1 Cluster: Experiment with transformations in the plane. Knowledge: Describe the terms point, line, and distance along a line in a plane Knowledge: Define perpendicular lines, parallel lines, line segments, and angles Knowledge: Define circle and the distance around a circular arc G.CO.2 Cluster: Experiment with transformations in the plane. Knowledge: Describe the different types of transformations including translations, reflections, rotations and dilations Knowledge: Describe transformations as functions that take points in the coordinate plane as inputs and give other points as outputs Reasoning: Write functions to represent transformations Reasoning: Compare transformations that preserve distance and angle to those that do not (e.g., translation vs. horizontal stretch) 78

2 G.CO.3 Cluster: Experiment with transformations in the plane. Reasoning: Describe rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself. G.CO.4 Cluster: Experiment with transformations in the plane. Knowledge: Recall definitions of angles, circles, perpendicular and parallel lines and line segments Knowledge: Define rotations, reflections, and translations. G.CO.5 Cluster: Experiment with transformations in the plane. Knowledge: Given a geometric figure and a rotation, reflection or translation, draw the transformed figure Demonstration: Draw a transformed figure and specify the sequence of transformations that were used to carry the given figure onto the other 79

3 Cluster: Understand congruence in terms of rigid motions. 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-CO.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.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.6 Cluster: Understand congruence in terms of rigid motions. Reasoning: Determine if two figures are congruent using the definition of congruence in terms of rigid motions Demonstration: Use geometric descriptions of rigid motions to transform figures Demonstration: Use geometric descriptions of rigid motions to predict the effect of a given motion on a given figure G-CO.7 Cluster: Understand congruence in terms of rigid motions. Knowledge: Identify corresponding angles and sides of two triangles Knowledge: Identify corresponding pairs of angles and sides of congruent triangles after rigid motions Reasoning: Justify congruency of two triangles using transformations Demonstration: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if corresponding pairs of sides and corresponding pairs of angles are congruent Demonstration: Use the definition of congruence in terms of rigid motions to show that if the corresponding pairs of sides and corresponding pairs of angles of two triangles are congruent, then the two triangles are congruent G-CO.8 Cluster: Understand congruence in terms of rigid motions. Knowledge: Informally use rigid motions to take angles to angles and segments to segments (from 8th grade) Knowledge: Formally use dynamic geometry software or straightedge and compass to take angles to angles and segments to segments Reasoning: Explain how the criteria for triangle congruence (ASA, SAS, SSS) follows from the definition of congruence in terms of rigid motions 80

4 Cluster: Prove geometric theorems. 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 corresponding 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 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.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.9 Cluster: Prove geometric theorems. Knowledge: Identify and use properties of vertical angles, parallel lines with transversals, all angle relationships, corresponding angles, alternate interior angles, perpendicular bisector, equidistant from endpoint Demonstration: Prove vertical angles are congruent Demonstration: Prove corresponding angles are congruent when two parallel lines are cut by a transversal and converse Demonstration: Prove alternate interior angles are congruent when two parallel lines are cut by a transversal and converse Demonstration: Prove points on a perpendicular bisector of a line segment are exactly equidistant from the segments endpoint G.CO.10 Cluster: Prove geometric theorems. Knowledge: Identify the hypothesis and conclusion of a theorem Reasoning: Analyze components of a theorem Demonstration: Prove theorems about triangles Demonstration: Design an argument to prove theorems about triangles 81

5 G.CO.11 Cluster: Prove geometric theorems. Knowledge: Classify types of quadrilaterals Knowledge: Explain theorems for parallelograms and relate a figure Demonstration: Use the principle that corresponding parts of congruent triangles are congruent to solve problems Demonstration: Use properties of special quadrilaterals in a proof Cluster: Make geometric constructions. 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 perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.12 Cluster: Make geometric constructions. Knowledge: Explain the construction of geometric figures using a variety of tools and methods Reasoning: Apply the definitions, properties, and theorems about line segments, rays, and angles to support geometric constructions Reasoning: Apply properties and theorems about parallel and perpendicular lines to support constructions Demonstration: Perform geometric constructions using a variety of tools and methods, including: 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.13 Cluster: Make geometric constructions. Demonstration: Construct an equilateral triangle, square, and regular hexagon inscribed in a circle 82

6 Cluster: Understand similarity in terms of similarity transformations. G.SRT.1: 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. G.SRT.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.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.1 Cluster: Understand similarity in terms of similarity transformations. Knowledge: Define image, pre-image, scale factor, center, and similar figures as they relate to transformations Knowledge: Identify a dilation stating its scale factor and center Knowledge: Explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image Demonstration: Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor Demonstration: Verify experimentally that a dilated image is similar to its pre-image by showing congruent corresponding angles and proportional sides Demonstration: Verify experimentally that a dilation takes a line not passing through the center of the dilation to a parallel line by showing the lines are parallel Demonstration: Verify experimentally that dilation leaves a line passing through the center of the dilation unchanged, by showing that it is the same line G.SRT.2 Cluster: Understand similarity in terms of similarity transformations. Reasoning: Given two figures, decide if they are similar by using the definition of similarity in terms of similarity transformations 83

7 G.SRT.3 Cluster: Understand similarity in terms of similarity transformations. Knowledge: Recall the properties of similarity transformations Reasoning: Establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general case of any two similar triangles Cluster: Prove theorems involving similarity. G.SRT.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.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.4 Cluster: Prove theorems involving similarity. Knowledge: Recall postulates, theorems, and definitions to prove theorems about triangles Demonstration: Prove theorems involving similarity about triangles G.SRT.5 Cluster: Prove theorems involving similarity. Knowledge: Recall congruence and similarity criteria for triangles Demonstration: Use congruency and similarity theorems for triangles to solve problems Demonstration: Use congruency and similarity theorems for triangles to prove relationships in geometric figures Cluster: Define trigonometric ratios and solve problems involving right triangles. G.SRT.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.7: Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 84

8 G.SRT.6 Cluster: Define trigonometric ratios and solve problems involving right triangles. Knowledge: Name the sides of right triangles as related to an acute angle Knowledge: Recognize that if two right triangles have a pair of acute, congruent angles, that the triangles are similar Reasoning: Compare common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometry ratios G.SRT.7 Cluster: Define trigonometric ratios and solve problems involving right triangles. Reasoning: Explain how the sine and cosine of complementary angles are related to each other Demonstration: Use the relationship between the sine and cosine of complementary angles G.SRT.8 Cluster: Define trigonometric ratios and solve problems involving right triangles. Knowledge: Recognize which methods could be used to solve right triangles in applied problems Knowledge: Solve for an unknown angle or side of a right triangle using sine, cosine, and tangent Reasoning: Apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems Cluster: Apply trigonometry to general triangles. G.SRT.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.10: (+) Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11: (+) 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). G.SRT.9 Cluster: Apply trigonometry to general triangles. Knowledge: Recall right triangle trigonometry to solve mathematical problems Reasoning: 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 85

9 G.SRT.10 Cluster: Apply trigonometry to general triangles. Reasoning: Prove the Law of Sines Reasoning: Prove the Law of Cosines Reasoning: Recognize when the Law of Sines or Law of Cosines can be applied to a problem and solve problems in context using them Demonstration: Use the Laws of Sines and Cosines to find missing angles or side length measurements G.SRT.11 Cluster: Apply trigonometry to general triangles. Reasoning: Determine from given measurements in right and non-right triangles whether it is appropriate to use the Law of Sines or Cosines Reasoning: 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) Cluster: Understand and apply theorems about circles. G.C.1: Prove that all circles are similar. G.C.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.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.4: (+) Construct a tangent line from a point outside a given circle to the circle. G.C.1 Cluster: Understand and apply theorems about circles. Knowledge: Recognize when figures are similar Reasoning: Compare the ratio of the circumference of a circle to the diameter of the circle Reasoning: Discuss, develop, and justify the ratio of the circumference of a circle to the diameter of the circle for several circles 86

10 G.C.2 Cluster: Understand and apply theorems about circles. Knowledge: Identify inscribed angles, radii, chords, central angles, circumscribed angles, diameter, tangent Knowledge: Recognize that inscribed angles on a diameter are right angles Knowledge: Recognize that the radius of a circle is perpendicular to the radius at the point of tangency Reasoning: Examine the relationship between central, inscribed, and circumscribed angles by applying theorems about their measures G.C.3 Cluster: Understand and apply theorems about circles. Knowledge: Define inscribed and circumscribed circles of a triangle Knowledge: Recall midpoint and bisector definitions Knowledge: Define a point of concurrency Demonstration: Prove properties of angles for a quadrilateral inscribed in a circle Demonstration: Construct inscribed circles of a triangle Demonstration: Construct circumscribed circles of a triangle G.C.4 Cluster: Understand and apply theorems about circles. Knowledge: Define tangent, radius, perpendicular bisector, and midpoint Knowledge: Identify the center of the circle Reasoning: Synthesize the theory that applies to circles that tangents drawn from a common external point are congruent Reasoning: Synthesize the theory that applies to circles that a radius is perpendicular to a tangent at the point of tangency Demonstration: Construct the perpendicular bisector of the line segment between the center, C to the outside point, P Demonstration: Construct arcs on circle C from the midpoint Q, having length of CQ Demonstration: Construct a tangent line 87

11 Cluster: Find arc lengths and areas of sectors of circles. 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.C.5 Cluster: Find arc lengths and areas of sectors of circles. Knowledge: Recall how to find the area and circumference of a circle Knowledge: Explain that 1 = /180 radians Knowledge: Determine the constant of proportionality (scale factor) Reasoning: Justify the radii of any two circles (r 1 and r 2 ) and the arc lengths (s 1 and s 2 ) determined by congruent central angles are proportional Reasoning: Verify that the constant of a proportion is the same as the radian measure, Θ, of the given central angle. Conclude s = r Θ Cluster: Translate between the geometric description and the equation for a conic section. G.GPE.1: 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.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. G.GPE.1 Cluster: Translate between the geometric description and the equation for a conic section. Knowledge: Define a circle Knowledge: Use Pythagorean Theorem Knowledge: Complete the square of a quadratic equation Reasoning: Derive equation of a circle using the Pythagorean Theorem, given coordinates of the center and length of the radius Reasoning: Determine the center and radius by completing the square 88

12 G.GPE.2 Cluster: Translate between the geometric description and the equation for a conic section. Knowledge: Define a parabola including the relationship of the focus and the equation of the directrix to the parabolic shape Reasoning: Derive the equation of a parabola given the focus and directrix G.GPE.3 Cluster: Translate between the geometric description and the equation for a conic section. Reasoning: Derive the equations of ellipses given the foci, using the fact that the sum of distances from the foci is constant Reasoning: Derive the equations of hyperbolas given the foci, using the fact that the difference of distances from the foci is constant Demonstration: Use definitions of conic sections and the Pythagorean Theorem to derive equations Cluster: Use coordinates to prove simple geometric theorems algebraically. 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, the square root of 3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.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). G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.GPE.4 Cluster: Use coordinates to prove simple geometric theorems algebraically. Knowledge: Recall previous understandings of coordinate geometry including distance, midpoint, and slope formulas, equation of a line, and definitions of parallel and perpendicular lines Reasoning: Use coordinates to prove simple geometric theorems algebraically Reasoning: Derive the equation of a line through 2 points using similar right triangles Reasoning: Derive simple proofs involving circles 89

13 G.GPE.5 Cluster: Use coordinates to prove simple geometric theorems algebraically. Knowledge: Recognize that slopes of parallel lines are equal Knowledge: Recognize that slopes of perpendicular lines are opposite reciprocals Reasoning: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems Demonstration: Find the equation of a line parallel to a given line that passes through a given point Demonstration: Find the equation of a line perpendicular to a given line that passes through a given point G.GPE.6 Cluster: Use coordinates to prove simple geometric theorems algebraically. Knowledge: Recall the definition of ratio Knowledge: Recall previous understandings of coordinate geometry Demonstration: Given a line segment (including those with positive and negative slopes) and a ratio, find the point on the segment that partitions the segment into the given ratio G.GPE.7 Cluster: Use coordinates to prove simple geometric theorems algebraically. Knowledge: Use the coordinates of the vertices of a polygon to find the necessary dimensions for finding the perimeter Knowledge: Use the coordinates of the vertices of a triangle to find the necessary dimensions (base, height) for finding the area Knowledge: Use the coordinates of the vertices of a rectangle to find the necessary dimensions (base, height) for finding the area Reasoning: Formulate a model of figures in contextual problems to compute area and/or perimeter 90

14 Cluster: Explain volume formulas and use them to solve problems. 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.2: (+) Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.1 G.GMD.2 Cluster: Explain volume formulas and use them to solve problems. Reasoning: Give an informal argument for the formulas for the circumference and area of a circle Reasoning: Give an informal argument for the formulas for the volume of a cylinder, pyramid, and cone Demonstration: Use dissection arguments, Cavalieri s principle, and informal limit arguments Cluster: Explain volume formulas and use them to solve problems. Reasoning: Give an informal argument using Cavalieri s principle for the formula for the volume of a sphere and other solid figures G.GMD.3 Cluster: Explain volume formulas and use them to solve problems. Knowledge: Utilize the appropriate formula for volume, depending on the figure Reasoning: Use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems Cluster: Visualize relationships between two-dimensional and three-dimensional objects. G.GMD.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.. G.GMD.4 Cluster: Visualize relationships between two-dimensional and three-dimensional objects. Reasoning: Relate the shapes of two-dimensional cross-sections to their three-dimensional objects Demonstration: Discover three-dimensional objects generated by rotations of two-dimensional objects Demonstration: Use strategies to help visualize relationships between two--dimensional and three dimensional objects 91

15 Cluster: Apply geometric concepts in modeling situations. G.MG.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.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). G.MG.1 Cluster: Apply geometric concepts in modeling situations. Reasoning: Given a real world object, classify the object as a known geometric shape use this to solve problems in context Reasoning: Focus on situations well modeled by trigonometric ratios for acute angles Demonstration: Use measures and properties of geometric shapes to describe real world objects G.MG.2 Cluster: Apply geometric concepts in modeling situations. Knowledge: Define density Reasoning: Apply concepts of density based on area and volume to model real-life situations G.MG.3 Cluster: Apply geometric concepts in modeling situations. Knowledge: Describe a typographical grid system Reasoning: 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) 92

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