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COURSE CONTENT MATHEMATICS GEOMETRY CCRS QUALITY CORE EVIDENCE OF STUDENT ATTAINMENT CONTENT STANDARDS RESOURCES FIRST SIX WEEKS 1 C.1.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.1] Students: Given undefined notions of point, line, distance along a line, and distance around a circular arc, 12 D.1.a 12. Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.12] Develop precise definitions of angle, circle, perpendicular line, parallel line, and line segment, Identify examples and nonexamples of angles, circles, perpendicular lines, parallel lines, and line segments. Glencoe Geometry Text, p.17, 30, 39, 40, 55, 207, 222, 245, 266, 267, 273, 275, 323, 334, 494, 670-671, 718 Students: Glencoe Geometry Text, Ch. 1, p.5-39 and Ch. 10, p.694-714 Make and justify formal geometric constructions with a variety of tools and methods (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) including the following: 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, Compare and contrast different methods for doing the same construction, and identify 1 COURSE CONTENT MATHEMATICS GEOMETRY geometric properties that justify steps in the constructions. 13 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO.13] 2 G.1.e 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.2] Glencoe Geometry Text, p.740 Students: Use tools (e.g., compass, straight edge, geometry software) and geometric relationships to construct regular polygons inscribed in circles, Explain and justify the sequence of steps taken to complete the construction. Students: Given a variety of transformations (translations, rotations, reflections, and dilations), Represent the transformations in the plane using a variety of methods (e.g., technology, transparencies, semitransparent mirrors (MIRAs), patty paper, compass), Describe transformations as functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g., f(x,y) → f(x+a, y+b)], Glencoe Geometry Text, p. 296-302, 511-517, 623-631, 632-638, 639, 640-646, 650, 651659, 674-681 2 COURSE CONTENT MATHEMATICS GEOMETRY 3 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO.3] Students: Given a collection of figures that include rectangles, parallelograms, trapezoids, or regular polygons, 4 C.1.a E.1.e 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO.4] Compare transformations that preserve distance and angle to those that do not. Identify which figures that have rotations or reflections that carry the figure onto itself, Perform and communicate rotations and reflections that map the object to itself, Distinguish these transformations from those which do not carry the object back to itself, Describe the relationship of these findings to symmetry. Glencoe Geometry Text, p.623-631, 632-638, 640-646, 650, 651-659 Students: Glencoe Geometry Text, p.663-669 Use geometric terminology (angles, circles, perpendicular lines, parallel lines, and line segments) to describe the series of steps necessary to produce a rotation, reflection, or translation, Use these descriptions to communicate precise definitions of rotation, reflection, and translation. 3 COURSE CONTENT MATHEMATICS GEOMETRY 5 E.1.a E.1.e 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.5] Students: Given a geometric figure, 9 C.1.a C.1.e D.1.c C.1.b D.1.b 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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [G-CO.9] Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or geometry software, Describe and justify the sequence of transformations that will carry a given figure onto another. Glencoe Geometry Text, p. 144-150, 151-159, 180-186, 207-214, 324-333 Students: Glencoe Geometry Text, p.294-295, 623-631, 632-638, 639, 640-646, 650, 651-659 Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology, Explain the requirements of a mathematical proof, Present a complete mathematical proof of geometry theorems including the following: 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, Critique proposed proofs made by others. SECOND SIX WEEKS 4 COURSE CONTENT MATHEMATICS GEOMETRY 6 E.1.a 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.6] Students: Given geometric descriptions of rigid motions, Glencoe Geometry Text, p. 294-295, 296-302, 623-631, 632-638, 640-646, 651-659, 682683 Predict the effect of the rigid motion on a given figure, Produce the image of a figure under the transformation, Compare and contrast the predictions to the actual transformation. Given two figures, 7 C.1.g E.1.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.7] Determine if a sequence of rotations, reflections, and translations will carry the first to the second, and if so justify their congruence by the definition of congruence in terms of rigid motions. Students: Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), Glencoe Geometry Text, p. 255-263, 294-295, 296-302, 623-631, 632-638, 640-646, 651659, 682-683 Verify that corresponding sides and corresponding angles are congruent. Given two triangles that have the same side lengths and angle measures, Find a sequence of rigid motions that will map one onto 5 COURSE CONTENT MATHEMATICS GEOMETRY the other. 8 10 C.1.f C.1.e D.2.b D.2.j 8. Explain how the criteria for triangle congruence, angle-side-angle (ASA), sideangle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO.8] Students: 10. Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, 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, and the medians of a triangle meet at a point. [G-CO.10] Students: Glencoe Geometry Text, p. 296-302, 682-683 Use rigid motions and the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof to establish that the usual triangle congruence criteria make sense and can then be used to prove other theorems. Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology, Explain the requirements of a mathematical proof, Present a complete mathematical proof of geometry theorems about triangles, including the following: measures of interior angles of a triangle sum to 180o; 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, Critique proposed proofs made by others. Glencoe Geometry Text, p. 246-254, 255-263, 264-272, 275-282, 285-293, 303-309, 324333, 335-343, 344-351, 355-362, 371-380, 490-499, 546 6 COURSE CONTENT MATHEMATICS GEOMETRY 11 C.1.e C.1.i D.2.g 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.11] 14 14. Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT.1] 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.1.a] Glencoe Geometry Text, p. 403-411, 413-421, 423-429, 431-438 Students: Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology, Explain the requirements of a mathematical proof, Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals, Critique proposed proofs made by others. Students: Given a center of dilation, a scale factor, and a polygonal image, b.The dilation of a line segment is longer or shorter in the ratio given by the scale factor. [G-SRT.1.b] Glencoe Geometry Text, p. 672-673, 674-681 Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex, Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line, Find the ratio of the length of the line segment from the 7 COURSE CONTENT MATHEMATICS GEOMETRY 15 C.1.h E.1.c E.1.d 15. 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.2] center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor, Conjecture a generalization of these results for all dilations. Students: Given two figures, Glencoe Geometry Text, p. 469-477, 478-487, 511-517, 682-683 Determine if they are similar by demonstrating whether one figure can be obtained from the other through a dilation and a combination of translations, reflections, and rotations. Given a triangle, Produce a similar triangle through a dilation and a combination of translations, rotations, and reflections, Demonstrate that a dilation and a combination of translations, reflections, and rotations maintain the measure of each angle in the triangles and all corresponding pairs of sides of the triangles are proportional. THIRD SIX WEEKS 16 C.1.h 16. Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT.3 Students: Given two triangles, Glencoe Geometry Text, p. 478-487, 511-517, 682-683 Explain why if the measures of 8 COURSE CONTENT MATHEMATICS GEOMETRY 17 D.2.e 17. Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT.4] two angles from one triangle are equal to the measures of two angles from another triangle, then measures of the third angles must be equal to each other, Use this established fact and the properties of a similarity transformation to demonstrate that the Angle-Angle (AA) criterion for similar triangles is sufficient. Students: Given a triangle and a line parallel to one of the sides, Glencoe Geometry Text, p. 478-487, 490-499, 501-508, 537-545 Prove the other two sides are divided proportionally by using AA, similarity properties, previously proven theorems and properties of equality (Table 4). Given a triangle with two of the sides divided proportionally, Prove the line dividing the sides is parallel to the third side of the triangle. Given a right triangle, Use similar triangles and properties of equality (Table 4) to prove the Pythagorean Theorem. 9 COURSE CONTENT MATHEMATICS GEOMETRY 18 D.2.d E.1.e E.1.g 18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [G-SRT.5] Students: Given a contextual situation involving triangles, Glencoe Geometry Text, p. 255-263, 264-272, 273, 275-282, 283-284, 478-487, 490-499, 501-508, 511-517, 537-545 Determine solutions to problems by applying congruence and similarity criteria for triangles to assist in solving the problem, Justify solutions and critique the solutions of others. Given a geometric figure, 19 H.1.b 19. 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.6] Establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles. Students: Given a collection of right triangles, Glencoe Geometry Text, p. 558-566, 567, 568-577, 578 Construct similar right triangles of various sizes for each right triangle given, Compare the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles, Communicate observations made about changes (or no change) to such ratios as the length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of 10 COURSE CONTENT MATHEMATICS GEOMETRY 20 H.1.b 20. Explain and use the relationship between the sine and cosine of complementary angles. [G-SRT.7] similar triangles, remains the same, Summarize these observations by defining the six trigonometric ratios. Students: Given a right triangle, Glencoe Geometry Text, p. 568-577 Explain why the two smallest angles must be complements, Compare the side ratios of opposite/hypotenuse and adjacent/hypotenuse for each of these angles and discuss conclusions. Given a contextual situation involving right triangles, 21 D.2.e H.1.b H.1.c H.1.a 21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. [G-SRT.8] Compare solutions to the situation using the sine of the given angle and the cosine of its complement. Students: Given a contextual situation involving right triangles, Glencoe Geometry Text, p. 547-555, 568-577, 580-587, 588-597 Create a drawing to model the situation, Find the missing sides and/or angles using trigonometric ratios, Find the missing sides using the Pythagorean Theorem, Use the above information to interpret results in the context 11 COURSE CONTENT MATHEMATICS GEOMETRY of the situation. 22 D.2.f 22. (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [G-SRT.10] Students: Given any triangle, 23 23. (+) 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.11] Derive the Law of Sines and the Law of Cosines. Use the Law of Sines or the Law of Cosines to find unknown lengths of sides and measures of angles. Students: Given a contextual situation, Glencoe Geometry Text, p. 588-597 Glencoe Geometry Text, p. 588-597, 598 Choose the appropriate law and apply it to determine the measures of unknown quantities. FOURTH SIX WEEKS 34 G.1.c 34. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. [G-GMD.5] Students: Given the vertices of a regular polygon. Glencoe Geometry Text, p. 807-817 Find the area. Find the perimeter. Find the area of the inscribed or circumscribed polygon. Find the perimeter of the inscribed or circumscribed polygon. Given the apothem, perimeter, area, or measurement of a side, 12 COURSE CONTENT MATHEMATICS GEOMETRY 35 D.4.a F.2.a F.2.b E.1.f E.1.h 35. Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [G-GMD1] Find the area. Find the perimeter. Find the area of the inscribed or circumscribed polygon. Find the perimeter of the inscribed or circumscribed polygon. Students: Given a circle, Glencoe Geometry Text, p. 697-705, 798-804, 863-870, 873-879, 880-887 Use repeated reasoning from multiple examples of the ratio of circle circumference to the diameter, to informally conjecture that the circumference of any circle is a little more than three times the diameter, Divide the circle into an equal number of sectors, and rearrange the sectors to form a shape that is approaching a parallelogram, Make conjectures about the area and perimeter of the new shape as the number of sectors becomes larger, and relate those conjectures to the original circle. Given a cylinder, Explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the 13 COURSE CONTENT MATHEMATICS GEOMETRY cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same crosssectional area at every level, then they have the same volume. Given a pyramid or cone, 36 D.4.a F.2.a F.2.c 36. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. [G-GMD3] Students: Given a contextual situation that requires finding the volume of a cylinder, pyramid, cone, or sphere as part of its solution, 37 D.4.a F.2.a F.2.c 37. Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama) [G-GMD.6] Explain that the shapes could be divided into cross-sections, and the area of the crosssections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone. Glencoe Geometry Text, p. 67-74, 863-870, 873-879, 880-887 Use an appropriate shape or 2D drawing to model the situation, Solve using the appropriate formula, Justify and explain the solution and solution path in the context of the given situation. Students: Given similar figures, Glencoe Geometry Text, p. 896-902 14 COURSE CONTENT MATHEMATICS GEOMETRY 38 D.4.b 38. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. [G-GMD4] Determine the relationship between surface areas and volumes of similar figures. Students: Given 3-D objects, Glencoe Geometry Text, p. 647-648, 839-844 Conjecture about the characteristics of geometric shapes formed if a crosssection of a 3-D shape is taken, Take 2-D cross-sections at different angles of cut, Explain the shape formed by taking 2-D cross-sections, Compare and contrast the figures formed when the angle of the cut changes. Given 2-D objects, 39 D.1.a E.1.h 39. 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.1] Conjecture about the characteristics of geometric shapes formed from rotating a 2-D shape about a line, Rotate the object about given lines, Explain the 3-D objects formed if the 2-D object is rotated about a line. Students: Given a real-world object, Glencoe Geometry Text, throughout the text; examples: p. 13, 75-77, 393-401, 818-824, 854-86 Select an appropriate geometric shape to model the 15 COURSE CONTENT MATHEMATICS GEOMETRY 40 40. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2] Students: Given a contextual situation involving density, 41 41. 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-MG3] object, Provide a description of the object through the measures and properties of the geometric shape which is modeling the object, Explain and justify the model which was selected. Model the situation by creating an average per unit of area or unit of volume, Generate questions raised by the model and defend answers they produce to the generated questions (e.g., should population density be given per square mile or per acre? What insights might one yield over the other?), Explain and justify the model in terms of the original context. Students: Given a contextual situation involving design problems, Glencoe Geometry Text, p. 797, 863-870, 873-879 Glencoe Geometry Text, various pages(see p. T34 for a list) Create a geometric method to model the situation and solve the problem, Explain and justify the model which was created to solve the problem. 16 COURSE CONTENT MATHEMATICS GEOMETRY FIFTH SIX WEEKS 24 24. Prove that all circles are similar. [G-C.1] Students: Given a collection of circles, 25 D.3.a D.3.b D.3.c 25. 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.2] Show that in each case there exists a transformation that consists of a dilation and a combination of rigid motions that will take one of the circles to any of the others, Verify that the ratio of the circumference of the circles created through dilations is equal to the ratio of the radii of the circles, which is the same as the scale factor of the dilation, Explain with logical reasoning how a circle is fully defined by a single parameter "r" so the only non-translational changes that can be made is alteration of "r", which changes the size and not the shape and therefore the circles are similar. Students: Given circles with two points on the circle, Glencoe Geometry Text, p. 697-705 Glencoe Geometry Text, p. 697-705, 706714, 715-722, 723-730, 732-739 Compare the measures of the angles (with and without technology) formed by creating radii to the given points, creating chords from a third point on the circle to the given points, and creating tangents from a third point outside the circle to the given points, and conjecture about possible relationships among 17 COURSE CONTENT MATHEMATICS GEOMETRY the angles, Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle is one half the central angle cutting off the same arc, and the circumscribed angle cutting off that arc is supplementary to the central angle relating all three). Given circles with chords from a point on the circle to the endpoints of a diameter, Find the measure of the angles (with and without technology), conjecture about and explain possible relationships, Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle on a diameter is a right angle). Given a circle with a tangent and radius intersecting at a point on the circle, Find the measure of the angle at the intersection point (with and without technology), conjecture about and explain possible relationships, Use logical reasoning to justify (or deny) the conjectures (in particular justify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 18 COURSE CONTENT MATHEMATICS GEOMETRY 26 D.2.b D.3.d 26. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3] Students: Given a triangle, Glencoe Geometry Text, p. 723-730, 740 Use tools (e.g., compass, straight edge, geometry software) to construct inscribed and circumscribed circles, Explain and justify the sequence of steps taken to complete the construction. Given a quadrilateral inscribed in a circle, 27 D.3.c 27. (+) Construct a tangent line from a point outside a given circle to the circle. [GC4] Students: Given a circle and an external point, 28 F.1.d F.1.e 28. 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-C5] Conjecture possible relationships among the angles through the use of inscribed angles use logical reasoning to justify (or deny) the conjectures (in particular, justify that diagonally opposite angles of a quadrilateral are supplementary). Use tools to construct a tangent line to the circle from the point. Students: Given an arc intercepted by an angle, Glencoe Geometry Text, p. 732-739 Glencoe Geometry Text, p. 706-714, 798-804 Use dilations to create arcs intercepted by the same central 19 COURSE CONTENT MATHEMATICS GEOMETRY 29 G.1.d D.1.e 29. 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-GPE1] angle with radii of various sizes (including using dynamic geometry software), and use the ratios of the arc lengths and radii to make conjectures regarding possible relationship between the arc length and the radius, Justify the conjecture for the formula for any arc length (i.e., since 2πr is the circumference of the whole circle, a piece of the circle is reduced by the ratio of the arc angle to a full angle (360)), Find the ratio of the arc length to the radius of each intercepted arc and use the ratio to name the angle calling this the radian measure of the angle by extending the definition of one radian as the angle which intercepts an arc of the same length as the radius, Develop the formula for the area of a sector by interpreting a circle as a complete revolution and a sector as a fractional part of a revolution. Students: Given the center (h,k) and radius (r) of a circle, Glencoe Geometry Text, p. 757-763 Explain and justify that every point on the circle is a combination of a horizontal and vertical shift from the center with a length equal to the radius, Create a right triangle from the 20 COURSE CONTENT MATHEMATICS GEOMETRY center of a circle to a general point on the circle, and show that the legs of the right triangle are the absolute values of x-h and y-k, and the hypotenuse is r, then apply Pythagorean theorem to show that r2 = (x - h)2 + (y - k)2. Given an equation of a circle in general form, Complete the square to rewrite the equation in the form r2 = (x - h)2 + (y - k)2 and determine the center and radius. SIXTH SIX WEEKS 30 G.1.b G.1.c 30. Use coordinates to prove simple geometric theorems algebraically. [G-GPE4] 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 point (0,2). Students: Given coordinates and geometric theorems and statements defined on a coordinate system, 31 C.1.d D.1.f G.1.a G.1.c 31. 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-GPE5] Use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others. Students: Given a line, Glencoe Geometry Text, p. 303-309, 403-411, 413-421, 423-429, 430-438, 439-448, 757763 Glencoe Geometry Text, p. 187, 188-196, 198-205, 206, 488-489 Create lines parallel to the given line and compare the slopes of parallel lines by examining the rise/run ratio of each line, Create lines perpendicular to 21 COURSE CONTENT MATHEMATICS GEOMETRY the given line by rotating the line 90 degrees and compare the slopes by examining the rise/run ratio of each line, Use understandings of similar triangles and logical reasoning to prove that parallel lines have equal slopes and the slopes of perpendicular lines are negative reciprocals. Given a geometric problem involving parallel or perpendicular lines, 32 G.1.c 32. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6] Apply the appropriate slope criteria to solve the problem and justify the solution including finding equations of lines parallel or perpendicular to a given line. Students: Given two points and a ratio that partitions the segment between the points, Glencoe Geometry Text, p. 25-35, 490-499, 600-608, 674-681, 757-763 Construct a circle using one of the given points as the center and the distance between the points as the radius, Construct a dilation of the circle using the given ratio as the scale factor and find the intersection between the dilation and the equation of the line passing through the given points, Justify and explain the reasons for each step in the process of finding a point that partitions a 22 COURSE CONTENT MATHEMATICS GEOMETRY segment in a given ratio. 33 F.2.a G.1.c 33. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [G-GPE7] Students: Given a contextual situation that requires the perimeter and/or area of a polygon as part of its solution, 42 F.1.c 42. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6] 43 F.1.c 43. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7] Find the solution to the situation through the use of coordinates and the distance formula as appropriate, through modeling the situation in a Cartesian coordinate system and explain and justify the solution. Students: Given a contextual situation in which a decision needs to be made, Glencoe Geometry Text, p. P8-P9, 939-946 Use a random probability selection model to produce unbiased decisions. Students: Given a contextual situation in which a decision needs to be made, Glencoe Geometry Text, p. 56-64, 779-786 Glencoe Geometry Text, p. P8-P9, 931-937, 947-953 Use probability concepts to analyze, justify, and make objective decisions. 23