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A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Experiment with transformations in the plane. Essential Question: In what ways can congruence be useful? How can undefined terms be used to describe basic geometric figures? AL Cos #1. [G-CO1] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometry• Precise definitions of angles, circle, perpendicular line, and line help.html http://www.geogebra.org/cms/ segment. http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics Skills: Be Able to Show http://www.connected.mcgraw-hill.com • Identify undefined notions such as point, line, and plane. • Identify angles, segments, lines, rays, perpendicular lines, and parallel lines Literacy Standards Writing Standards Practice Standards • Define angles, circles, perpendicular lines, rays, and line RST.9-10.2 WHST.9-10.2 #6 Attend to precision segments using undefined terms and “if-then” and “if and only RST.9-10.4 WHST. 9-10 .3 if” statements. RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Rigid motion, transform, congruence, congruent, corresponding parts, ASA, SAS, SSS, AAS, rotation, reflection, translation, perpendicular bisector, preimage, image, parallel line, angle, center of rotation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Experiment with transformations in the plane. Essential Question: In what ways can congruence be useful? How can transformations be used to describe congruence? AL Cos #2. [G-CO2] 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). Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and/or Literacy Standards Writing Standards Practice Standards software RST.9-10.2 WHST.9-10.2 #6 Attend to precision • Determine the coordinates of the image of a figure when a RST.9-10.4 WHST. 9-10 .3 transformation rule is applied to the preimage RST.9-10.7 WHST. 9-10.4 • Distinguish between transformations that are rigid and those that RST.9-10.10 WHST. 9-10.7 are not. WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Rigid motion, transform, congruence, congruent, corresponding parts, ASA, SAS, SSS, AAS, rotation, reflection, translation, perpendicular bisector, preimage, image, parallel line, angle, center of rotation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Experiment with transformations in the plane. Essential Question: In what ways can congruence be useful? How can transformations be used to analyze geometric figures? AL Cos #3. [G-CO3] Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Describe and illustrate how a rectangle, parallelogram, and an isosceles trapezoid is mapped onto itself using transformations Literacy Standards Writing Standards Practice Standards • Calculate the number of lines of reflection symmetry and the RST.9-10.2 WHST.9-10.2 #7 Look for and make degree of rotational symmetry of any regular polygon RST.9-10.4 WHST. 9-10.3 use of structure RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Rigid motion, transform, congruence, congruent, corresponding parts, ASA, SAS, SSS, AAS, rotation, reflection, translation, perpendicular bisector, preimage, image, parallel line, angle, center of rotation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Experiment with transformations in the plane. Essential Question: In what ways can congruence be useful? What are some ways we can describe simple geometric figures using transformations? AL Cos #4. [G-CO4] Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Define rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines, and line segments, Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #6 Attend to precision RST.9-10.4 WHST. 9-10.3 RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Rigid motion, transform, congruence, congruent, corresponding parts, ASA, SAS, SSS, AAS, rotation, reflection, translation, perpendicular bisector, preimage, image, parallel line, angle,, center of rotation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Experiment with transformations in the plane. Essential Question: In what ways can congruence be useful? How are transformations and congruence related? AL Cos #5. [G-CO5] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Draw a specific transformation when given a geometric figure and a rotation, reflection, or translation. Literacy Standards Writing Standards Practice Standards • Predict and verify the sequence of transformation that will map a RST.9-10.2 WHST.9-10.2 #5 Use appropriate figure onto another. RST.9-10.4 WHST. 9-10.3 tools strategically RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Rigid motion, transform, congruence, congruent, corresponding parts, ASA, SAS, SSS, AAS, rotation, reflection, translation, perpendicular bisector, preimage, image, parallel line, angle, center of rotation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Understand congruence in terms of rigid motions. Essential Question: In what ways can congruence be useful? How is rigid motion and congruence related? AL Cos #6. [G-CO6] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Predict the effect a rigid motion has on figures in the coordinate plane Literacy Standards Writing Standards Practice Standards • Use the fact that rigid motion preserves size and shape to connect RST.9-10.2 WHST.9-10.2 #3 Construct viable the idea of congruency and develop the definition of congruent RST.9-10.4 WHST. 9-10.3 arguments and critique RST.9-10.7 WHST. 9-10.4 the reasoning of others RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Congruence, composition, rigid motions, map, reflection, rotation, reflection, transformation, angle measure, distance A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Understand congruence in terms of rigid motions. Essential Question: In what ways can congruence be useful? How can transformations be used to preserve criteria for congruence? AL Cos #7. [G-CO7] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Based on idea of rigid motion, show that two triangles are congruent if and only if their corresponding sides and Literacy Standards Writing Standards Practice Standards corresponding angles are congruent WHST.9-10.2 #3 Construct viable WHST. 9-10.3 arguments and critique WHST. 9-10.4 the reasoning of others WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Congruence, composition, rigid motions, map, reflection, rotation, reflection, transformation, angle measure, distance, figure, corresponding sides, corresponding angles, triangle A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Understand congruence in terms of rigid motions. Essential Question: In what ways can congruence be useful? How can rigid motion be used to develop the criteria for triangle congruence? AL Cos #8. [G-CO8] Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Based on idea of rigid motion, develop the triangle congruence criteria, ASA, SSS, and SAS. Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #2 Reason abstractly RST.9-10.4 WHST. 9-10.3 and quantitatively RST.9-10.7 WHST. 9-10.4 #3 Construct viable RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 Key Vocabulary Congruence, composition, rigid motions, map, reflection, rotation, reflection, transformation, distance, figure, corresponding sides, corresponding angles, triangle, SAS, ASA, SSS A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Prove geometric theorems Essential Question: In what ways can congruence be useful? How can proofs of geometric ideas be useful when studying geometric figures? AL Cos #9. [G-CO9] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Prove vertical angles are congruent • Prove when a transversal crosses parallel lines, alternate interior Literacy Standards Writing Standards Practice Standards angles are congruent and corresponding angles are congruent RST.9-10.2 WHST.9-10.2 #2 Reason abstractly • Prove points on a perpendicular bisector are equidistant from the RST.9-10.4 WHST. 9-10 .3 and quantitatively segment’s endpoints RST.9-10.7 WHST. 9-10.4 #3 Construct viable RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 Key Vocabulary Theorem, linear pair, vertical angles, alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles, perpendicular bisector, supplementary angles, complementary angles, equidistant, congruent, adjacent, consecutive/non-consecutive, reflection, Law of Syllogism, Law of Detachment. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Prove geometric theorems Essential Question: In what ways can congruence be useful? How can proofs of geometric ideas be useful when studying geometric figures? AL Cos #10. [G-CO10] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Prove vertical angles are congruent Literacy Standards Writing Standards Practice Standards • Prove base angles of an isosceles triangle are congruent RST.9-10.2 WHST.9-10.2 #2 Reason abstractly • Prove the segment joining midpoints of two sides of a triangle is RST.9-10.4 WHST. 9-10 .3 and quantitatively parallel to the third side and half the length RST.9-10.7 WHST. 9-10 .4 #3 Construct viable • Prove the medians of a triangle meet at a point RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 Key Vocabulary Theorem, linear pair, vertical angles, alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles, perpendicular bisector, supplementary angles, complementary angles, equidistant, congruent, adjacent, Consecutive/non-consecutive, reflection, Law of Syllogism, Law of Detachment. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Prove geometric theorems Essential Question: In what ways can congruence be useful? How can proofs of geometric ideas be useful when studying geometric figures? AL Cos #11. [G-CO11] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show In relation to parallelograms • Prove opposite sides are congruent Literacy Standards Writing Standards Practice Standards • Prove opposite angles are congruent RST.9-10.2 WHST.9-10.2 #2 Reason abstractly • Prove the diagonals of a parallelogram bisect each other, and RST.9-10.4 WHST. 9-10 .3 and quantitatively conversely, rectangles are parallelograms with congruent RST.9-10.7 WHST. 9-10 .4 #3 Construct viable diagonals. RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 Key Vocabulary Quadrilateral, parallelogram, rectangle diagonals, distance formula, ,midpoint formula, slope, bisector, congruence properties A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Make geometric constructions Essential Question: In what ways can congruence be useful? How can constructions of geometric ideas be useful when studying geometric figures? AL Cos #12. [G-CO12] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Copy a segment Literacy Standards Writing Standards Practice Standards • Copy an angle RST.9-10.2 WHST.9-10.2 #5 Use appropriate • Bisect a segment RST.9-10.4 WHST. 9-10 .3 tools strategically • Bisect an angle RST.9-10.7 WHST. 9-10.4 #6 Attend to precision • Construct perpendicular lines including perpendicular bisectors RST.9-10.10 WHST. 9-10.7 • Construct a line parallel to a given line through a point not on the WHST. 9-10.9 line WHST. 9-10.10 Key Vocabulary Segment, angle, perpendicular lines, bisector, parallel lines, perpendicular lines, bisect, formal construction, informal construction, compass, straightedge A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Congruence To successfully complete the Geometry Course, the learner will Make geometric constructions Essential Question: In what ways can congruence be useful? How can constructions of geometric ideas be useful when studying geometric figures? AL Cos #13. [G-CO13] Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • That proving and applying congruence provides a basis for http://www.dynamicgeometry.com/General_Resources/Links.html modeling situations geometrically http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Construct an equilateral triangle inscribed in a circle • Construct a square inscribed in a circle Literacy Standards Writing Standards Practice Standards • Construct a regular hexagon inscribed in a circle RST.9-10.2 WHST.9-10.2 #5 Use appropriate RST.9-10.4 WHST. 9-10.3 tools strategically RST.9-10.7 WHST. 9-10.4 #6 Attend to precision RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Construction, equilateral triangle, square, regular hexagon, inscribe, circle A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Understand similarity in terms of similarity transformations Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #14. [G-SRT1] Verify experimentally the properties of dilations given by a center and a scale factor a. [G-SRT1a] 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. [G-SRT1b] The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Given a center and scale factor, verify experimentally that when dilating a figure, a segment of the preimage that does not pass Literacy Standards Writing Standards Practice Standards through the center of the dilation is parallel to its image when the RST.9-10.2 WHST.9-10.2 #5 Use appropriate dilation is preformed RST.9-10.4 WHST. 9-10.3 tools strategically • Given center and scale factor, verify experimentally that when RST.9-10.7 WHST. 9-10.4 #8 Look for and performing dilations of a line segment, the segment that becomes RST.9-10.10 WHST. 9-10 .7 express regularity in the image is longer or shorter based on the scale factor WHST. 9-10.9 repeated reasoning WHST. 9-10.10 Key Vocabulary Dilation, center, scale factor, image, slope, parallel, corresponding sides, pre-image, distance, ratio, segment A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Understand similarity in terms of similarity transformations Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #15. [G-SRT2] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use the idea of dilation transformations to develop the definition of similarity Writing Standards Practice Standards • Given two figures, determine whether they are similar and explain Literacy Standards RST.9-10.2 WHST.9-10.2 #3 Construct viable their similarity based on the equality of corresponding angles and RST.9-10.4 WHST. 9-10 .3 arguments and critique the proportionality of corresponding sides RST.9-10.7 WHST. 9-10.4 the reasoning of others RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Similarity, composition, rigid motion, dilation, angle measure, side length, proportional, corresponding sides, corresponding angles, SSS similarity, SAS similarity. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Understand similarity in terms of similarity transformations Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #16. [G-SRT3] Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use the properties of similarity transformations to develop the criteria for proving similar triangles. AA Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #3 Construct viable RST.9-10.4 WHST. 9-10.3 arguments and critique RST.9-10.7 WHST. 9-10.4 the reasoning of others RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Similarity transformation, angle measure, similar, AA A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Prove theorems involving similarity Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #17. [G-SRT4] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use AA, SAS, SSS similarity theorems to prove triangles are similar Writing Standards Practice Standards • Prove a line parallel to one side of a triangle divides the other two Literacy Standards RST.9-10.2 WHST.9-10.2 #3 Construct viable proportionally and its converse RST.9-10.4 WHST. 9-10 .3 arguments and critique • Prove the Pythagorean Theorem using similarity RST.9-10.7 WHST. 9-10.4 the reasoning of others RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Proof, corresponding angles, similarity, segment addition, parallel, intersect, Pythagorean Theorem, AA, SAS, SSS similarity. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Prove theorems involving similarity Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #18. [G-SRT5] Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Prove two triangles are congruent using similarity theorems • Prove that geometric figures, other than triangles, are congruent Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #1 Make sense of RST.9-10.4 WHST. 9-10.3 problems and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Congruence, side length, angle measure, proportional, corresponding sides, triangle congruence, triangle similarity, AA, SSS, SAS similarity. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Define trigonometric ratios and solve problems involving right triangles Essential Question: How might the features of one figure be useful when solving problems about a similar figure? AL Cos #19. [G-SRT6] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Using a corresponding angle of similar triangles, show that the relationships of the side ratios are the same which leads to the Literacy Standards Writing Standards Practice Standards definition of trigonometric ratios for acute angles. RST.9-10.2 WHST.9-10.2 #2 Reason abstractly RST.9-10.4 WHST. 9-10.3 and quantitatively RST.9-10.7 WHST. 9-10.4 #7 Look for and make RST.9-10.10 WHST. 9-10.7 use of structure WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Similarity, rigid motion, dilation, angle measure, proportional, right triangle, line segment, parallel, leg, hypotenuse, AA similarity, corresponding sides, tangent, sine, cosine, acute angle, ratio, trigonometry, constant A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Define trigonometric ratios and solve problems involving right triangles Essential Question: How might the features of one figure be useful when solving problems about a similar figure? How can sine and cosine be used to solve problems involving triangles? AL Cos #20. [G-SRT7] Explain and use the relationship between the sine and cosine of complementary angles. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Explore the sine of an acute angle and the cosine of its complement and determine their relationship Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #2 Reason abstractly RST.9-10.4 WHST. 9-10.3 and quantitatively RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Complementary angles, acute angle, sine ratio, cosine ratio, right triangle A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Define trigonometric ratios and solve problems involving right triangles Essential Question: How might the features of one figure be useful when solving problems about a similar figure? How can sine and cosine be used to solve problems involving triangles? AL Cos #21. [G-SRT8] Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Apply trigonometric ratios and the Pythagorean Theorem to solve application problems involving right triangles Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #1 Make sense of RST.9-10.4 WHST. 9-10.3 problems and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 #4 Model with WHST. 9-10.9 mathematics WHST. 9-10.10 Key Vocabulary Sine ratio, cosine ratio, tangent ratio, right triangle, inverse trigonometric ratio, acute angle, Pythagorean Theorem, side, angle, triangle A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Apply trigonometry to general triangles Essential Question: How might the features of one figure be useful when solving problems about a similar figure? How can sine and cosine be used to solve problems involving triangles? AL Cos #22. [G-SRT10] (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Using trigonometry and the relationship among sides and angles of any to prove the Law of Sines Literacy Standards Writing Standards Practice Standards • Using trigonometry and the relationship among the sides and RST.9-10.2 WHST.9-10.2 #1 Make sense of angles of any triangle, use the Pythagorean Theorem to prove the RST.9-10.4 WHST. 9-10 .3 problems and persevere Law of Sines RST.9-10.7 WHST. 9-10 .4 in solving them • Use the Law of Sines to Solve Problems RST.9-10.10 WHST. 9-10 .7 #2 Reason abstractly • Use the Law of Cosines to solve problems WHST. 9-10.9 and quantitatively WHST. 9-10.10 #7 Look for and make use of structure Key Vocabulary Law of Sines, Law of Cosines, altitude, right triangle, side, Pythagorean Theorem. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Similarity, Right Triangles, and Trigonometry To successfully complete the Geometry Course, the learner will Apply trigonometry to general triangles Essential Question: How might the features of one figure be useful when solving problems about a similar figure? How can sine and cosine be used to solve problems involving triangles? AL Cos #23. [G-SRT11] (+) 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). Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Dilations, similarity, and the properties of similar triangles allow http://www.dynamicgeometry.com/General_Resources/Links.html for the application of trigonometric ratios to real-world situations http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right triangles Literacy Standards Writing Standards Practice Standards • Understand and apply the Law of Sines and law of Cosines to RST.9-10.2 WHST.9-10.2 #1 Make sense of find unknown measure in non-right triangles. RST.9-10.4 WHST. 9-10.3 problems and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 #4 Model with WHST. 9-10.9 Mathematics WHST. 9-10.10 Key Vocabulary Law of Sines, Law of Cosines, right triangle, triangle inequality A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Circles To successfully complete the Geometry Course, the learner will Understand and apply theorems about circles Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos #24. [G-C1] Prove that all circles are similar. Learning Targets Concepts: Need to Know • The properties of polygons, lines, and angles can be used to understand circles; the properties of circles can be used to solve problems involving polygons, lines, and angles. Skills: Be Able to Show • Using the fact that the ratio of the circumference to diameter is the same for all circles, prove that all circles are similar Digital Resources: http://www.webmath.com/ http://www.coolmath.com/0-geometrymath-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Literacy Standards RST.9-10.2 RST.9-10.4 RST.9-10.7 RST.9-10.10 Key Vocabulary Circle, similar figures, rigid motion, dilation, angle measure, pre-image, image central angle Writing Standards WHST.9-10.2 WHST. 9-10.3 WHST. 9-10.4 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Practice Standards #3 Construct viable arguments and critique the reasoning of others A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Circles To successfully complete the Geometry Course, the learner will Understand and apply theorems about circles Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos #25. [G-C2] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • The properties of polygons, lines, and angles can be used to http://www.dynamicgeometry.com/General_Resources/Links.html understand circles; the properties of circles can be used to solve http://www.analyzemath.com/geometry.html problems involving polygons, lines, and angles. http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use definitions, properties and theorems to describe relationships Literacy Standards Writing Standards Practice Standards among inscribed angles, radii, and chords. RST.9-10.2 WHST.9-10.2 #1 Make sense of • Understand that inscribed angles on a diameter are right angles RST.9-10.4 WHST. 9-10 .3 problems and persevere • Understand that the radius of a circle is perpendicular to the RST.9-10.7 WHST. 9-10 .4 in solving them tangent where the radius intersects the circle. RST.9-10.10 WHST. 9-10.7 #6 Attend to precision WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Central angle, inscribed angle, circumscribed angle, diameter, radius, chord, tangent, circle, intersect, endpoints, right angle, perpendicular A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Circles To successfully complete the Geometry Course, the learner will Understand and apply theorems about circles Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos #26. [G-C3] Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • The properties of polygons, lines, and angles can be used to http://www.dynamicgeometry.com/General_Resources/Links.html understand circles; the properties of circles can be used to solve http://www.analyzemath.com/geometry.html problems involving polygons, lines, and angles. http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Construct inscribed circles of a triangle Literacy Standards Writing Standards Practice Standards • Construct circumscribed circles of a triangle RST.9-10.2 WHST.9-10.2 #5 Use appropriate • Using definitions, properties, and theorems, prove properties of RST.9-10.4 WHST. 9-10 .3 tools strategically angles for a quadrilateral inscribed in a circle RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Inscribed, circumscribed, angle bisector, perpendicular bisector, construction, compass, straightedge, intersection, incenter, circumcenter, quadrilateral, arc, inscribed angle, Arc Addition Postulate, equation, opposite angles, supplementary A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Circles To successfully complete the Geometry Course, the learner will Understand and apply theorems about circles Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos #27. [G-C4] (+) Construct a tangent line from a point outside a given circle to the circle. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • The properties of polygons, lines, and angles can be used to http://www.dynamicgeometry.com/General_Resources/Links.html understand circles; the properties of circles can be used to solve http://www.analyzemath.com/geometry.html problems involving polygons, lines, and angles. http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Construct a tangent line from a point outside the circle to the Literacy Standards Writing Standards Practice Standards given circle. RST.9-10.2 WHST.9-10.2 #5 Use appropriate RST.9-10.4 WHST. 9-10.3 tools strategically RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Tangent line, circle, perpendicular, radius, endpoint, midpoint, line segment, point A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Circles To successfully complete the Geometry Course, the learner will Understand and apply theorems about circles Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos #28. [G-C5] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • The properties of polygons, lines, and angles can be used to http://www.dynamicgeometry.com/General_Resources/Links.html understand circles; the properties of circles can be used to solve http://www.analyzemath.com/geometry.html problems involving polygons, lines, and angles. http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use similarity to derive the fact that the length of the arc Literacy Standards Writing Standards Practice Standards intercepted by an angle is proportional to the radius, RST.9-10.2 WHST.9-10.2 #6 Attend to precision identifying the constant of proportionality as the radian RST.9-10.4 WHST. 9-10 .3 #7 Look for and make measure of the angle. RST.9-10.7 WHST. 9-10 .4 use of structure • Find the arc length of the circle RST.9-10.10 WHST. 9-10 .7 • Using similarity, derive the formula for the area of a sector WHST. 9-10.9 • Find the area of a sector of a circle WHST. 9-10.10 Key Vocabulary Similarity, rigid motion, dilation, angle measure, length, proportional, arc, constant or proportionality, radian, angle, area, circle, sector, formula, intercepted arc A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Expressing Geometric Properties with Equations To successfully complete the Geometry Course, the learner will Translate between the geometric description and the equation for a conic section. Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems? AL Cos # 29. [G-GPE1] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Algebra can be used to efficiently and effectively describe and http://www.dynamicgeometry.com/General_Resources/Links.html apply geometric properties http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use the Pythagorean Theorem to derive the equation of a circle given the center and the radius Literacy Standards Writing Standards Practice Standards • Given the equation of a circle complete the square to find the RST.9-10.2 WHST.9-10.2 #2 Reason abstractly center and radius of a circle RST.9-10.4 WHST. 9-10.3 and quantitatively RST.9-10.7 WHST. 9-10.4 #3 Construct viable RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 #7 Look for and make use of structure Key Vocabulary Distance formula, Pythagorean theorem, difference, coordinates, radius, circle, hypotenuse, equation, center, complete the square, quadratic equation, conic equation, standard form, general form A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Expressing Geometric Properties with Equations To successfully complete the Geometry Course, the learner will Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Essential Question: How can algebra be useful when expressing geometric properties? AL Cos #30. [G-GPE4] Use coordinates to prove simple geometric theorems algebraically. 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). Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Algebra can be used to efficiently and effectively describe and http://www.dynamicgeometry.com/General_Resources/Links.html apply geometric properties http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use coordinate geometry to prove geometric theorems algebraically Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #2 Reason abstractly RST.9-10.4 WHST. 9-10.3 and quantitatively RST.9-10.7 WHST. 9-10.4 #3 Construct viable RST.9-10.10 WHST. 9-10.7 arguments and critique WHST. 9-10.9 the reasoning of others WHST. 9-10.10 #7 Look for and make use of structure Key Vocabulary Distance formula, slope formula. A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Expressing Geometric Properties with Equations To successfully complete the Geometry Course, the learner will Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Essential Question: How can algebra be useful when expressing geometric properties? AL Cos #31. [G-GPE5] 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). Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Algebra can be used to efficiently and effectively describe and http://www.dynamicgeometry.com/General_Resources/Links.html apply geometric properties http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Using slope, prove lines are parallel or perpendicular • Find equations of lines based on certain slope criteria such as Literacy Standards Writing Standards Practice Standards finding the equation of a line that is parallel to or perpendicular RST.9-10.2 WHST.9-10.2 #3 Construct viable to a given line that passes through a given point RST.9-10.4 WHST. 9-10.3 arguments and critique RST.9-10.7 WHST. 9-10.4 the reasoning of others RST.9-10.10 WHST. 9-10.7 #8 Look for and WHST. 9-10.9 express regularity in WHST. 9-10.10 repeated reasoning Key Vocabulary Slope, parallel, perpendicular, product, line, linear equation, slope-intercept form, point-slope form A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Expressing Geometric Properties with Equations To successfully complete the Geometry Course, the learner will Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Essential Question: How can algebra be useful when expressing geometric properties? AL Cos #32. [G-GPE6] Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Algebra can be used to efficiently and effectively describe and http://www.dynamicgeometry.com/General_Resources/Links.html apply geometric properties http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Given two points, find the point of the line segment between the two point that divides the segment into a given ratio Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #7 Look for and make RST.9-10.4 WHST. 9-10.3 use of structure RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Directed line segment, endpoint, ratio A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Expressing Geometric Properties with Equations To successfully complete the Geometry Course, the learner will Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Essential Question: How can algebra be useful when expressing geometric properties? AL Cos #33. [G-GPE7] Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Algebra can be used to efficiently and effectively describe and http://www.dynamicgeometry.com/General_Resources/Links.html apply geometric properties http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use coordinate geometry and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #1 Make sense of RST.9-10.4 WHST. 9-10.3 standards and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Coordinate plane, coordinates, distance formula, perimeter, polygon, area, triangle, rectangle A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Geometric Measure and Dimension To successfully complete the Geometry Course, the learner will Explain volume formulas and use them to solve problems. Essential Question: How can two dimensional figures be used to understand three dimensional objects? AL Cos #34. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. [Alabama] Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Two-dimensional figures can be stacked to create three http://www.dynamicgeometry.com/General_Resources/Links.html dimensional objects and generate volume formulas http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Given the coordinates of the vertices of a polygon, determine the areas and perimeters of regular polygons Literacy Standards Writing Standards Practice Standards • Given other characteristics of a polygon, determine the areas and RST.9-10.2 WHST.9-10.2 #3Constrcut viable perimeters of regular polygons RST.9-10.4 WHST. 9-10 .3 arguments and critique • Find the area and perimeters of the inscribed or circumscribed RST.9-10.7 WHST. 9-10 .4 the reasoning of others polygons RST.9-10.10 WHST. 9-10.7 #7 Look for and make WHST. 9-10.9 use of structure WHST. 9-10.10 Key Vocabulary Pi, circle, circumference, diameter, dissection, equivalent, ratio, area, regular, polygon, perimeter, side, apothem, radius, base, prism, cylinder, pyramid, cone, volume, substitute, height A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Geometric Measure and Dimension To successfully complete the Geometry Course, the learner will Explain volume formulas and use them to solve problems. Essential Question: How can two dimensional figures be used to understand three dimensional objects? AL Cos #35. [G-GMD1] 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. Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Two-dimensional figures can be stacked to create three http://www.dynamicgeometry.com/General_Resources/Links.html dimensional objects and generate volume formulas http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Using Cavalieri’s principle, provide informal arguments to develop the formulas for the volume of spheres and other solid Literacy Standards Writing Standards Practice Standards objects RST.9-10.2 WHST.9-10.2 #2 Reason abstractly • Explain the formulas for the circumference of a circle and the RST.9-10.4 WHST. 9-10 .3 and quantitatively area of a circle by determining the meaning of each term or factor RST.9-10.7 WHST. 9-10 .4 #3Construct viable • Explain the formulas for the volume of a cylinder, pyramid, and RST.9-10.10 WHST. 9-10 .7 arguments and critique cone by determining the meaning of each term or factor WHST. 9-10.9 the reasoning of others WHST. 9-10.10 #7 Look for and make use of structure Key Vocabulary Height, base area, volume, Cavalieri’s Principle, cross sectional area, Pi, circle, circumference, diameter, dissection, equivalent, ratio, area, regular, polygon, perimeter, side, apothem, radius, base, prism, cylinder, pyramid, cone, volume, substitute, height A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Geometric Measure and Dimension To successfully complete the Geometry Course, the learner will Explain volume formulas and use them to solve problems. Essential Question: How can two dimensional figures be used to understand three dimensional objects? AL Cos #36. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3] Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Two-dimensional figures can be stacked to create three http://www.dynamicgeometry.com/General_Resources/Links.html dimensional objects and generate volume formulas http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Solve problems using volume formulas for cylinders, pyramids, cones, and spheres Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #4 Model with RST.9-10.4 WHST. 9-10.3 mathematics RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Volume, cylinder, pyramid, cone , sphere A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Geometric Measure and Dimension To successfully complete the Geometry Course, the learner will Visualize relationships between two-dimensional and three-dimensional objects. Essential Question: How can two dimensional figures be used to understand three dimensional objects? AL Cos #37. Determine the relationship between surface areas of similar figures and volumes of similar figures. [Alabama] Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • The formulas for surface area and volume of geometric figures http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics Skills: Be Able to Show http://www.connected.mcgraw-hill.com • Students should be able to determine the relationship between the surface area and volume of similar figures. Literacy Standards RST.9-10.2 RST.9-10.4 RST.9-10.7 RST.9-10.10 Key Vocabulary Formulas for surface area of geometric figures, formulas for the volume of similar figures Writing Standards WHST.9-10.2 WHST. 9-10.3 WHST. 9-10.4 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Practice Standards #4 Model with mathematics #7 Look for and make use of structure A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Modeling with Geometry To successfully complete the Geometry Course, the learner will Apply geometric concepts in modeling situations Essential Question: In what ways can geometric figures be used to understand real world situations? AL Cos #38. [G-GMD4] Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Learning Targets Concepts: Need to Know • Two-dimensional figures can be stacked to create three dimensional objects and generate volume formulas Digital Resources: http://www.webmath.com/ http://www.coolmath.com/0-geometrymath-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Given a three-dimensional object, identify the shape made when the object is cut into cross-sections • When rotating a two-dimensional figures, such as a square, know Literacy Standards the three dimensional figure that is generated, such as a cylinder. RST.9-10.2 Understand that the cross section of a solid is an intersection of a RST.9-10.4 RST.9-10.7 plane and a solid. RST.9-10.10 Key Vocabulary Cross-section, rotate Writing Standards WHST.9-10.2 WHST. 9-10.3 WHST. 9-10.4 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Practice Standards #4 Model with mathematics #7 Look for and make use of structure A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Modeling with Geometry To successfully complete the Geometry Course, the learner will Apply geometric concepts in modeling situations Essential Question: In what ways can geometric figures be used to understand real world situations? AL Cos #39. [G-MG1] Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Geometric definitions, properties, and theorems allow one to http://www.dynamicgeometry.com/General_Resources/Links.html describe, model, and analyze solutions in the real world. http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use geometric shapes, their measures, and their properties to describe objects Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #4 Model with RST.9-10.4 WHST. 9-10.3 mathematics RST.9-10.7 WHST. 9-10.4 RST.9-10.10 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Key Vocabulary Circumference, area, perimeter, volume A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Modeling with Geometry To successfully complete the Geometry Course, the learner will Apply geometric concepts in modeling situations Essential Question: In what ways can geometric figures be used to understand real world situations? AL Cos #40. [G-MG2] 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).* Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Geometric definitions, properties, and theorems allow one to http://www.dynamicgeometry.com/General_Resources/Links.html describe, model, and analyze solutions in the real world. http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Use the concept of density when referring to situations involving area and volume models such as persons per square mile. Literacy Standards Writing Standards Practice Standards RST.9-10.2 WHST.9-10.2 #1 Make sense of RST.9-10.4 WHST. 9-10.3 problems and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 #4 Model with WHST. 9-10.9 mathematics WHST. 9-10.10 Key Vocabulary Unit of measure, area, perimeter, volume, convert, density, composite figures A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Modeling with Geometry To successfully complete the Geometry Course, the learner will Apply geometric concepts in modeling situations Essential Question: In what ways can geometric figures be used to understand real world situations? AL Cos #41. [G-MG3] 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).* Digital Resources: Learning Targets http://www.webmath.com/ http://www.coolmath.com/0-geometryConcepts: Need to Know math-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ • Geometric definitions, properties, and theorems allow one to http://www.dynamicgeometry.com/General_Resources/Links.html describe, model, and analyze solutions in the real world. http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Skills: Be Able to Show • Solve design problems by designing an object or a structure that satisfies certain constraints such as minimizing cost or working Literacy Standards Writing Standards Practice Standards with a grid system based on ratios. RST.9-10.2 WHST.9-10.2 #1 Make sense of RST.9-10.4 WHST. 9-10.3 problems and persevere RST.9-10.7 WHST. 9-10.4 in solving them RST.9-10.10 WHST. 9-10.7 #4 Model with WHST. 9-10.9 mathematics WHST. 9-10.10 Key Vocabulary Geometric model, graph, equation, table, formula A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Using Probability to Make Decisions To successfully complete the Geometry Course, the learner will Cluster: Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules) Essential Question: How is probability used to make informed decisions about uncertain events? AL Cos #42 (+) [S-MD6] Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Learning Targets Concepts: Need to Know • The rules of probability can lead to more valid and reliable predications about the likelihood of an event occurring. Skills: Be Able to Show • Use random processes to generate probabilities to make fair decisions. Digital Resources: http://www.webmath.com/ http://www.coolmath.com/0-geometrymath-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Literacy Standards Writing Standards Practice Standards RST.9-10.2 RST.9-10.4 RST.9-10.7 RST.9-10.10 WHST.9-10.2 WHST. 9-10.3 WHST. 9-10.4 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 #1 Make sense of problems and persevere in solving them #2 Reason abstractly and quantitatively. #3 Construct viable arguments and critique the reasoning of others. #4 Model with mathematics. #5 Use appropriate tools strategically. #7 Look for and make use of structure. Key Vocabulary Random number generator, simulation A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator Geometry – Using Probability to Make Decisions To successfully complete the Geometry Course, the learner will Use probability to evaluate outcomes of decisions Essential Question: How is probability used to make informed decisions about uncertain events? AL Cos #43. [S-MD7] (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Learning Targets Concepts: Need to Know • The rules of probability can lead to more valid and reliable predictions about the likelihood of an event occurring. Skills: Be Able to Show • Students should be able to analyze decisions and strategies using probability concepts. • Students should be able to analyze whether or not the best decision was made. • Students should be able to analyze the available strategies, recommend a strategy, and defend their choice of strategies. Digital Resources: http://www.webmath.com/ http://www.coolmath.com/0-geometrymath-art.html http://www.onlinemathlearning.com/geometryhelp.html http://www.geogebra.org/cms/ http://www.dynamicgeometry.com/General_Resources/Links.html http://www.analyzemath.com/geometry.html http://library.thinkquest.org/2647/geometry/geometry.htm http://etc.usf.edu/clipart/galleries/722-mathematics http://www.connected.mcgraw-hill.com Literacy Standards RST.9-10.2 RST.9-10.4 RST.9-10.7 RST.9-10.10 Key Vocabulary Sample space, probability, event Writing Standards WHST.9-10.2 WHST. 9-10.3 WHST. 9-10.4 WHST. 9-10.7 WHST. 9-10.9 WHST. 9-10.10 Practice Standards #3Construct viable arguments and critique the arguments of others #4 Model with mathematics #5 Use appropriate tools strategically A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator A COMMUNITY CONNECTED BY EDUCATION Athens City Schools Math Standards Navigator
