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Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 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. Congruence Experiment with transformations in the plane. Geometry G-CO.1 Evidence of Student Attainment Students: Given undefined notions of point, line, distance along a line, and distance around a circular arc, Develop precise definitions of angle, circle, perpendicular line, parallel line, and line segment, Identify examples and non-examples of angles, circles, perpendicular lines, parallel lines, and line segments. 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). Congruence Experiment with transformations in the plane. Geometry G-CO.2 Franklin County Schools 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 Teacher Vocabulary Knowledge Skills Students know: Students are able to: Undefined notions of point, line, distance along a line, and distance around a circular arc, Use known and developed definitions and logical connections to develop new definitions. Properties of a mathematical definition, i.e. the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept). Students know: Students are able to: Characteristi cs of transformations (translations, rotations, reflections, and dilations), Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane Methods for with and without technology, representing transformations, Communicate Characteristi the results of cs of functions, performing transformations on objects and their Conventions Understanding Resources Students understand that: Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition, A precise definition should allow for the inclusion of all examples of the concept, and require the exclusion of all non-examples. Click below to access all ALEX resources aligned to this standard. ALEX Resources Students understand that: Mapping one point to another Click below to through a series of access all ALEX transformations can resources be recorded as a aligned to this function, standard. Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment 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)], Teacher Vocabulary Knowledge of functions with corresponding mapping coordinates in the notation. coordinate plane, including when the transformation preserves distance and angle, Use the language and notation of functions as mappings to describe transformations. Compare transformations that preserve distance and angle to those that do not. 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Congruence Experiment with transformations in the plane. Geometry G-CO.3 Students: Given a collection of figures that include rectangles, parallelograms, trapezoids, or regular polygons, 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 Franklin County Schools Skills Students know: Students are able to: Characteristi cs of transformations (translations, rotations, reflections, and dilations), Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane Characteristi with and without cs of rectangles, technology, parallelograms, trapezoids, and Communicate regular the results of polygons. performing transformations on objects and their corresponding coordinates in the coordinate plane. Understanding Resources then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image, Distortions, such as only a horizontal stretch, preserve neither. Students understand that: Mapping one point to another through a series of transformations can Click below to be recorded as a access all ALEX function, resources aligned to this Since rotations standard. and reflections preserve distance ALEX and angle measure, Resources the image is then congruent. Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources carry the object back to itself, Describe the relationship of these findings to symmetry. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Congruence Experiment with transformations in the plane. Geometry G-CO.4 Students: Students know: Students are able to: 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, Characteristi cs of transformations (translations, rotations, reflections, and dilations), Use these descriptions to communicate precise definitions of rotation, reflection, and translation. 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 Congruence Experiment with transformations in the plane. Geometry G-CO.5 Franklin County Schools Students: Given a geometric figure, Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or Students understand that: Accurately Geometric perform rotations, definitions are reflections, and developed from a translations on few undefined objects with and notions by a logical Click below to without technology, sequence of access all ALEX connections that resources aligned to this Properties of Communicate lead to a precise definition, standard. a mathematical the results of definition, i.e., performing the smallest transformations on A precise ALEX amount of objects, definition should Resources information and allow for the properties that Use known and inclusion of all are enough to examples of the developed determine the concept and require definitions and concept. (Note: logical connections the exclusion of all may not include to develop new non-examples. all information definitions. related to concept). Students know: Students are able to: Students understand that: Click below to access all ALEX Characteristi resources Accurately The same cs of aligned to this transformations perform rotations, transformation may standard. (translations, reflections, and be produced using rotations, translations on a variety of tools, ALEX reflections, and objects using graph but the geometric Resources dilations), paper, tracing sequence of steps paper, or geometry that describe the transformation is Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID will carry a given figure onto another. Evidence of Student Attainment Teacher Vocabulary geometry software, Techniques for producing images under transformations using graph paper, tracing paper, or geometry software. Describe and justify the sequence of transformations that will carry a given figure onto another. 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. Congruence Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for development of concept of geometric proof.) Geometry G-CO.6 Students: Given geometric descriptions of rigid motions, 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, 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. Franklin County Schools Knowledge Rigid motion Skills Understanding software, consistent, Communicate the results of performing transformations on objects. Any distance preserving transformation is a combination of rotations, reflections, and translations. Students know: Students are able to: Students understand that: Characteristi cs of translations, rotations, and reflections including the definition of congruence, Use geometric descriptions of rigid motions to accurately perform these transformations on objects, Any distance preserving transformation is a combination of rotations, reflections, and translations, Communicate the results of performing transformations on objects. If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent. Techniques for producing images under transformations using graph paper, tracing paper, compass, or geometry software, Geometric terminology (e.g., angles, circles, perpendicular lines, parallel lines, and line segments) which describes the series of steps necessary to Resources Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources produce a rotation, reflection, or translation. 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. If and only if Students: Given a triangle and its image under a sequence of rigid motions (translations, on rigid motions reflections, and as a familiar translations), Congruence Understand congruence in terms of rigid motions. (Build starting point for development of concept of geometric proof.) Geometry G-CO.7 8. Explain how the criteria for triangle congruence, angleside-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. Congruence Understand congruence in terms of rigid motions. (Build Verify that corresponding sides and corresponding angles are congruent. Use geometric descriptions of rigid motions to accurately perform these transformations on objects, Communicate the results of performing transformations on objects. Students understand that: If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent. Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation. Triangle congruence Use rigid motions ASA and the basic properties of rigid on rigid motions motions (that they SAS as a familiar preserve distance and starting point for angle), which are development of assumed without proof SSS concept of to establish that the geometric usual triangle proof.) congruence criteria Franklin County Schools Characteristi cs of translations, rotations, and reflections including the definition of congruence, Techniques for producing images under transformations, Given two triangles that have the same side lengths and angle measures, - Find a sequence of rigid motions that will map one onto the other. Students: Students know: Students are able to: Students know: Students are able to: Basic properties of rigid motions (that they preserve distance and angle), Use logical reasoning to connect geometric ideas to justify other results, Perform rigid Methods for motions of Students understand that: It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Geometry G-CO.8 Evidence of Student Attainment make sense and can then be used to prove other theorems. Teacher Vocabulary Knowledge presenting logical reasoning using assumed understandings to justify subsequent results. Skills geometric figures, Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation), Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion), Apply the SSS, Franklin County Schools Understanding congruence). Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources SAS, and ASA criteria to verify whether or not two triangles are congruent. 9. Prove theorems about lines and angles. Theorems Congruence Students: Prove geometric theorems. Make, explain, and include vertical angles (Focus on justify (or refute) are congruent; when validity of conjectures about a transversal crosses underlying geometric relationships parallel lines, reasoning while with and without alternate interior using variety of technology, 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. ways of writing proofs.) Geometry G-CO.9 Prove Students know: Students are able to: Students understand that: Transversal Requirement Communicate Proof is s for a logical reasoning in necessary to Alternate interior mathematical proof, a systematic way to establish that a angles present a conjecture about a mathematical proof relationship in Techniques Corresponding mathematics is for presenting a of geometric angles theorems, always true, and Explain the proof of may also provide requirements of a geometric insight into the Generate a mathematical proof, theorems. mathematics being conjecture about Click below to addressed. geometric access all ALEX Present a complete relationships that resources mathematical proof of calls for proof. aligned to this geometry theorems standard. including the following: vertical angles are congruent; when a ALEX transversal crosses Resources 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. 10. Prove theorems about triangles. Congruence Students: Prove geometric Franklin County Schools Students know: Students are able to: Students understand that: Click below to access all ALEX Geometry CCRS Standard Theorems include measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, the medians of a triangle meet at a point. Mathematics CCRS Standards and Alabama COS Standard ID theorems. (Focus on validity of underlying reasoning while using variety of ways of writing proofs.) Geometry G-CO.10 Evidence of Student Attainment 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 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point, Teacher Vocabulary Knowledge Skills Understanding Resources resources aligned to this Communicate Proof is standard. logical reasoning in necessary to a systematic way to establish that a present a conjecture about a ALEX mathematical proof relationship in Resources of geometric mathematics is Techniques always true and for presenting a theorems, may also provide proof of insight into the Generate a geometric mathematics being theorems. conjecture about addressed. geometric Requirement s for a mathematical proof, relationships that calls for proof. Critique proposed proofs made by others. 11. Prove theorems Congruence Students: about parallelograms. Prove geometric Theorems include theorems. Make, explain, and opposite sides are (Focus on justify (or refute) congruent, opposite validity of conjectures about angles are congruent, underlying geometric relationships the diagonals of a reasoning while with and without parallelogram bisect using variety of Franklin County Schools Students know: Students are able to: Students understand that: Requirement Communicate Proof is s for a mathematical logical reasoning in necessary to proof, a systematic way to establish that a present a conjecture about a mathematical proof relationship in Techniques Click below to access all ALEX resources aligned to this standard. ALEX Geometry CCRS Standard each other, and conversely, rectangles are parallelograms with congruent diagonals. Mathematics CCRS Standards and Alabama COS Standard ID ways of writing proofs.) Geometry G-CO.11 Evidence of Student Attainment technology, Teacher Vocabulary Knowledge Skills Understanding for presenting a of geometric proof of theorems, geometric theorems. Generate a conjecture about geometric relationships that calls for proof. mathematics is always true and may also provide insight into the mathematics being addressed. Students: Students know: Students are able to: Students understand that: 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 Methods for accurately using tools to perform geometric constructions, including compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software, 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, Resources Resources Critique proposed proofs made by others. 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, etc. Congruence Make geometric constructions. (Formalize and explain processes.) Geometry G-CO.12 Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, Franklin County Schools Choose and use Limiting oneself appropriate to a specific tool or construction tools set of tools to strategically to perform a perform geometric geometric constructions, construction illuminates important Use logical mathematical reasoning and properties of and features of the object being relationships between geometric constructed, figures to justify geometric Methods for constructions. Different tools for geometric Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Evidence of Student Attainment Teacher Vocabulary 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, Knowledge Skills justifying a geometric construction using geometric properties. Compare and contrast different methods for doing the same construction, and identify geometric properties that justify steps in the constructions. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Congruence Make geometric constructions. (Formalize and explain processes.) Geometry G-CO.13 Students: Similarity, Right Construct Use tools (e.g., Inscribed compass, straight edge, geometry software) and geometric relationships to construct regular polygons inscribed in circles, Franklin County Schools Students: Given a center of Resources constructions may offer different levels of precision in the construction and the purpose of the construction should help determine the tool of choice, Properties of geometric figures can and should be used to verify the correctness of geometric constructions regardless of the construction tool or method used. Explain and justify the sequence of steps taken to complete the construction. 14. Verify experimentally the Understanding Dilations Students know: Students are able to: Students understand that: Properties of Choose and use Properties of regular polygons, appropriate geometric figures construction tools can and should be used to verify the Characteristi strategically to perform geometric correctness of cs of inscribed constructions, geometric figures, constructions regardless of the Communicate Methods for construction tool or accurately using with logical method used. reasoning the tools to perform series of steps geometric necessary for constructions. constructing an inscribed figure and the justification for each step. Students know: Students are able to: Students understand that: Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID properties of dilations Triangles, & given by a center and Trigonometry a scale factor. Understand similarity in a. A dilation takes a terms of line not passing similarity transformations. through the Geometry center of the G-SRT.1 dilation to a b. parallel line and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Evidence of Student Attainment Teacher Vocabulary dilation, a scale factor, Center and a polygonal image, Scale factor 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 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, Knowledge Skills Understanding Resources resources A dilation uses aligned to this standard. a center and line segments through vertex points to ALEX create an image Resources which is similar to the original image Accurately find but in a ratio the length of line specified by the scale factor, Dilations segments and may be ratios of line The ratio of the performed on segments, polygons by line segment drawing lines Communicate formed from the through the center of dilation to with logical center of dilation reasoning a a vertex in the new and each vertex conjecture of image and the of the polygon generalization from corresponding then marking off experimental vertex in the a line segment results. original image is changed from equal to the scale the original by factor. the scale factor. Methods for finding the length of line segments (both in a coordinate plane and through measurement), Accurately create a new image from a center of dilation, a scale factor, and an image, Conjecture a generalization of these results for all dilations. 15. Given two figures, use the definition of similarity in terms of similarity Similarity, Right Triangles, & Trigonometry Franklin County Schools Students: Given two figures, Determine if they Similarity transformation Students know: Students are able to: Students understand that: Properties of Apply rigid rigid motions A figure that Click below to access all ALEX resources aligned to this Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 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. Understand similarity in terms of similarity transformations. Geometry G-SRT.2 Evidence of Student Attainment Teacher Vocabulary are similar by demonstrating whether one figure can be obtained from the other through a dilation and a combination of translations, reflections, and rotations. Knowledge Skills Understanding Resources and dilations, motion and dilation may be obtained standard. to a figure, from another through a dilation Definition of ALEX and a combination Explain and similarity in Resources terms of justify whether or of translations, similarity not one figure can reflections, and transformations, be obtained from rotations is similar another through a to the original, Techniques combination of rigid motion and dilation. When a figure for producing is similar to another images under a the measures of all dilation and rigid corresponding motions. angles are equal, and all of the corresponding sides are in the same proportion. 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. 16. Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. Geometry Franklin County Schools Students: Given two triangles, Explain why if the measures of two angles from one triangle are equal to the measures of two angles from another triangle, then measures of the third AA criterion Students know: Students are able to: The sum of the measures of the angles of a triangle is 180 degrees, Explain and justify why the third pair of corresponding angles of two Properties of triangles must be equal if each of the rigid motions Students understand that: It is beneficial to have minimal sets of requirements to justify geometric results (i.e., use AA instead of all sides Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID G-SRT.3 Evidence of Student Attainment Teacher Vocabulary angles must be equal to each other, Knowledge and dilations. Use this established fact and the properties of a similarity transformation to demonstrate that the Angle-Angle (AA) criterion for similar triangles is sufficient. 17. Prove theorems about triangles. Similarity, Right Theorems include a Triangles, & line parallel to one Trigonometry side of a triangle Prove theorems divides the other two involving proportionally, and similarity. conversely; the Geometry Pythagorean G-SRT.4 Theorem proved using triangle similarity. Students: Given a triangle and a line parallel to one of the sides, 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. Franklin County Schools Skills Students know: Students are able to: Properties of similar triangles and methods of showing that triangles are similar, Resources other two proportional and all corresponding pairs angles congruent are equal, for similarity), Justify through the use of rigid motion and dilation why corresponding sides of triangles are in the same proportion if the measures of two pairs of corresponding angles are equal. Theorem Understanding If the measures of two angles of one triangle are equal to the measures of two angles of another triangle, then the triangles are similar and the similarity of the triangles can be justified through similarity transformations. Students understand that: Apply Triangle properties of similar similarity may be triangles to justify used to justify relationships of the theorems involving sides of a triangle, the connection Click below to between the access all ALEX proportion of sides Properties of Explain and resources and whether or not equality (Table justify that a line aligned to this 4), passing through the the line dividing the standard. triangle divides the sides is parallel to the other side of sides Previously ALEX the triangle, proven theorems proportionally, if Resources and only if, the line including those is parallel to a side Through the concerning of the triangle, use of similar parallel lines. triangles, a right triangle may be Justify the divided into two Pythagorean Theorem through right triangles the use of similar which are similar to the original right triangles. Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills 18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. Geometry G-SRT.5 Use similar triangles and properties of equality (Table 4) to prove the Pythagorean Theorem. Students: Given a contextual situation involving triangles, 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, Establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles. Franklin County Schools Resources triangle; therefore the corresponding sides must be proportional and may be used to prove the Pythagorean Theorem, Given a right triangle, Understanding The same theorem may be proven in many different ways (i.e., the Pythagorean Theorem). Congruence and similarity criteria for triangles Students know: Students are able to: Criteria for congruent (SAS, ASA, AAS, SSS) and similar (AA) triangles and transformation criteria, Students understand that: Accurately Congruence solve a contextual and similarity problem by criteria for triangles applying the criteria may be used to find of congruent and solutions of similar triangles, contextual Click below to problems, access all ALEX Techniques Provide resources to apply criteria justification for the Relationships in aligned to this of congruent solution process, geometric figures standard. and similar may be proven triangles for through the use of Analyze the ALEX solving a solutions of others congruent and Resources contextual similar triangles. and explain why problem. their solutions are valid or invalid, Justify relationships in geometric figures through the use of congruent and similar triangles. Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 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. Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. Geometry G-SRT.6 Evidence of Student Attainment Students: Given a collection of right triangles, Construct similar right triangles of various sizes for each right triangle given, Teacher Vocabulary Knowledge Skills Understanding Side ratios Students know: Students are able to: Trigonometric ratios Techniques Accurately find The ratios of to construct similar triangles, the side ratios of the sides of right triangles, triangles are dependent on the Properties of size of the angles similar triangles. Explain and justify relationships of the triangle. Compare the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles, Resources Students understand that: between the side ratios of a right triangle and the angles of a right triangle. Click below to access all ALEX resources aligned to this standard. 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 similar triangles, remains the same, ALEX Resources Summarize these observations by defining the six trigonometric ratios. 20. Explain and use the relationship between the sine and cosine of complementary angles. Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve Franklin County Schools Students: Given a right triangle, Sine Cosine Explain why the two smallest angles Complementary must be complements, angles Students know: Students are able to: Students understand that: Click below to access all ALEX resources Methods for The sine of an aligned to this finding sine and Accurately standard. cosine ratios in a solve a contextual angle is equal to right triangle problem by using the cosine of the (e.g., use of the sine and cosine complement of the ALEX Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID problems involving right triangles. Geometry G-SRT.7 Evidence of Student Attainment 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, Teacher Vocabulary Skills Knowledge triangle properties: similarity; Pythagorean Theorem; isosceles and equilateral characteristics for 45-45-90 and 30-60-90 triangles and technology for others). ratios, Understanding Resources angle, Resources Justify solutions Switching and discuss other between using a possible solutions given angle or its through the use of complement and complementary between sine or angles and the sine cosine ratios may or cosine ratios. be used when solving contextual problems. Compare solutions to the situation using the sine of the given angle and the cosine of its complement. 21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. Geometry G-SRT.8 Students: Given a contextual situation involving right triangles, Create a drawing to model the situation, Find the missing sides and angles using trigonometric ratios and the Pythagorean Theorem, Use the above information to interpret results in the context of the situation. Franklin County Schools Students know: Students are able to: Methods of using the trigonometric ratios to solve for sides or angles in a right triangle, The Pythagorean Theorem and its use in solving for unknown parts of a right triangle. Create an accurate diagram to model a contextual situation involving right triangles and use it to solve the right triangles, Students understand that: Unknown parts of right triangles may be found through the use of trigonometric ratios, Pythagorean Theorem, or a combination of both, Identify the trigonometric ratio Right triangles useful to solve for a may be used to particular unknown model and solve part of a right contextual triangle and use situations. that ratio to accurately solve for the unknown part. Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources use the Pythagoren Theorem to find unknown sides of a right triangle explain the solution in terms of the given contextual situation 22. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. Geometry G-SRT.9 Click below to access all ALEX resources aligned to this standard. 23. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. Geometry G-SRT.10 Click below to access all ALEX resources aligned to this standard. 24. (+) 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). Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. Geometry G-SRT.11 Click below to access all ALEX resources aligned to this standard. 25. Prove that all Circles Franklin County Schools Students: Students know: Students are able Students ALEX Resources ALEX Resources ALEX Resources Click below to Geometry CCRS Standard circles are similar. Mathematics CCRS Standards and Alabama COS Standard ID Understand and apply theorems about circles. Geometry G-C.1 Evidence of Student Attainment Teacher Vocabulary Given a collection of circles, Skills Knowledge to: Techniques to create dilations, 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, Similar figures have the same shape, A figure transformed by a dilation and any combination of rigid motions will be similar to the image. 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, Understanding Resources understand that: access all ALEX resources Accurately Any circle can aligned to this create circles by be created through standard. making dilations of a dilation of a given circle, another circle and a ALEX combination of rigid Resources Communicate motions, logical reasoning in a systematic way to Any geometric present a figure that is fully mathematical proof defined by a single of geometric parameter will be theorems. similar to all other figures in that class (squares, equilateral triangles, circles, parabola, etc.) Explain with logical reasoning how a circle is fully defined by a single parameter "r" so the only nontranslational changes that can be made is alteration of "r", which changes the size and not the shape and therefore the circles are similar. 26. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, Circles Understand and apply theorems about circles. Geometry G-C.2 Franklin County Schools Students: Given circles with two points on the circle, Central angles Students know: Students are able to: Students understand that: Inscribed angles Definitions and characteristics of central, inscribed, and Relationships that exist among inscribed angles, radii, and chords Compare the Circumscribed measures of the angles angles (with and without Explain and justify possible relationships among central, Click below to access all ALEX resources aligned to this standard. ALEX Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID 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. Evidence of Student Attainment 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 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, Franklin County Schools Teacher Vocabulary Knowledge circumscribed angles in a circle, Techniques to find measures of angles including using technology (dynamic geometry software). Skills inscribed, and circumscribed angles sharing intersection points on the circle, Accurately find measures of angles (including using technology (dynamic geometry software)) formed from inscribed angles, radii, chords, central angles, circumscribed angles, and tangents Understanding may be used to find the measures of other angles when appropriate conditions are given, Identifying and justifying relationships exist in geometric figures. Resources Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources 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. 27. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Circles Understand and apply theorems about circles. Geometry G-C.3 Franklin County Schools Students: Given a triangle, Use tools (e.g., compass, straight edge, geometry software) to construct inscribed and circumscribed circles, Construct Students know: Students are able to: Inscribed and Techniques circumscribed circles to find of a triangle circumcenter and incenter of a triangle, and Quadrilateral to use this in inscribed in a circle inscribing or circumscribing Students understand that: Click below to access all ALEX resources Use appropriate Every triangle aligned to this tools to accurately has a point which is standard. construct inscribed equidistant from and circumscribed each vertex of the ALEX circles of a triangle, triangle and a point Resources which is equidistant from each side of Explain and Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Explain and justify the sequence of steps taken to complete the construction. Knowledge Skills Understanding Resources the triangle, justify the steps the triangle, that are used when Properties of creating the Opposite angles inscribed angles. construction, of a quadrilateral inscribed in a circle Apply the are supplementary. properties of inscribed angles to reach a conclusion. Given a quadrilateral inscribed in a circle, 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). 28. (+) Construct a tangent line from a point outside a given circle to the circle. Circles Understand and apply theorems about circles. Geometry G-C.4 Click below to access all ALEX resources aligned to this standard. 29. 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 Circles Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.) Geometry G-C.5 Franklin County Schools Students: Given an arc intercepted by an angle, Similarity Students know: Students are able to: Constant of proportionality Techniques to use dilations (including using dynamic geometry software) to create circles with arcs Use dilations to Sector create arcs intercepted by the same central angle with radii of various sizes (including Reason from progressive examples using dynamic geometry software to form conjectures about relationships Students understand that: Radians measure the ratio of the arc length to the radius for an intercepted arc, The ratio of the ALEX Resources Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID the area of a sector. Evidence of Student Attainment 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. Franklin County Schools Teacher Vocabulary Knowledge intercepted by same central angles, Skills Understanding Techniques to find arc length, among arc length, area of a sector to central angles, and the area of a circle the radius, is proportional to the ratio of the central angle to a Use logical reasoning to justify complete revolution. (or deny) these Formulas for area and circumference of a circle. conjectures and critique the reasoning presented by others, Interpret a sector as a portion of a circle, and use the ratio of the portion to the whole circle to create a formula for the area of a sector. Resources Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 30. 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. Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. Geometry G-GPE.1 Evidence of Student Attainment Students: Given the center (h,k) and radius (r) of a circle, 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 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. Teacher Vocabulary Knowledge Skills Students know: Students are able to: Key features Create a right of a circle, triangle in a circle using the horizontal The and vertical shifts Pythagorean from the center as Theorem, the legs and the radius of the circle The as the hypotenuse, technique of completing the Convert an square. equation of a circle from general form to standard form using the method of completing the square. Given an equation of a circle in general form, Understanding Resources Students understand that: Circles represent a fixed distance in all directions in a plane from a given point, and a right triangle may be created to show the relationship of the horizontal and vertical shift to the Click below to distance, access all ALEX resources Rewriting aligned to this algebraic standard. expressions or equations in ALEX equivalent forms Resources often reveals significant features of the expression, (i.e., circles written in standard form are useful for recognizing the center and radius of a circle). Complete the square to rewrite the equation in the form r2 = (x - h)2 + (y - k)2 and determine the center and radius. 31. Use coordinates to prove simple geometric theorems algebraically. Example: Prove or Expressing Geometric Properties with Equations Franklin County Schools Simple geometric Students know: Students are able Students: Given coordinates and theorems to: geometric theorems Relationship and statements defined s (e.g. distance, Accurately on a coordinate slope of line) determine what Students understand that: Modeling geometric figures Click below to access all ALEX resources aligned to this standard. Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 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). Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Evidence of Student Attainment system, Use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others. Geometry G-GPE.4 Teacher Vocabulary Knowledge Skills Understanding between sets of information is or relationships on points, needed to prove or a coordinate graph disprove a assists in statement or determining truth Properties of theorem, of a statement or geometric theorem, shapes, Resources ALEX Resources Accurately find Coordinate the needed information and graphing rules and techniques, explain and justify conclusions, Geometric theorems may be proven or disproven by examining the properties of the Techniques geometric shapes Communicate for presenting a logical reasoning in in the theorem proof of a systematic way to through the use of geometric appropriate present a theorems. mathematical proof algebraic techniques. of geometric theorems. 32. 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). Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Geometry G-GPE.5 Slope criteria for Students know: parallel and perpendicular lines Techniques Create lines parallel to find the slope to the given line and of a line, compare the slopes of parallel lines by Key features examining the rise/run needed to solve ratio of each line, geometric Students: Given a line, Create lines perpendicular to 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 Franklin County Schools Students are able to: Explain and justify conclusions reached regarding the slopes of parallel and perpendicular lines, Students understand that: Relationships exist between the slope of a line and any line parallel or Click below to perpendicular to access all ALEX that line, resources problems, aligned to this Apply slope Slope criteria standard. criteria for parallel for parallel and Techniques for presenting a and perpendicular perpendicular lines ALEX lines to accurately may be useful in proof of Resources find the solutions of solving geometric geometric geometric problems problems. theorems. and justify the solutions, Communicate logical reasoning in a systematic way to Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge have equal slopes and the slopes of perpendicular lines are negative reciprocals. Skills Understanding Resources present a mathematical proof of geometric theorems. Given a geometric problem involving parallel or perpendicular lines, 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. 33. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Geometry G-GPE.6 Franklin County Schools Directed line Students: Given two points and a segment ratio that partitions the segment between the Partitions points, 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 Students know: Students are able to: Techniques for finding the distance between two points and the equation of a line passing through two points, Accurately find the distance between two points and the equation of a line passing through two points, Students understand that: A radius of a circle may be used to show the distance between Click below to access all ALEX two points, resources aligned to this A dilation of a standard. Accurately find circle may be used the equation of a to partition a line dilation of a circle, segment by making c. ALEX Resources it the radius, in a given ratio. Find the Forms for writing the equation of a circle dependent intersection on the point(s) of a line information and a circle. given to find the equation of the dilation of a Geometry CCRS Standard 34. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ Mathematics CCRS Standards and Alabama COS Standard ID Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) Geometry G-GPE.7 Evidence of Student Attainment Teacher Vocabulary Knowledge Skills given points, circle, Justify and explain the reasons for each step in the process of finding a point that partitions a segment in a given ratio. Techniques to find the intersection between a line and a circle. Students: Given a contextual situation that requires the perimeter and/or area of a polygon as part of its solution, Students know: Students are able to: 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. Understanding Resources Students understand that: The distance formula and its Create applications, geometric figures on a coordinate Techniques system from a contextual for coordinate situation, graphing. Contextual situations may be modeled in a Cartesian coordinate system, Click below to access all ALEX resources Coordinate aligned to this Accurately find modeling is standard. the perimeter of frequently useful to polygons and the visualize a situation ALEX area of triangles and to aid in Resources and rectangles solving contextual from the problems. coordinates of the shapes, Explain and justify solutions in the original context of the situation. 35. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama) Geometric Measurement & Dimension Use coordinates to prove simple geometric theorems algebraically. (Alabama) Geometry Franklin County Schools Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources G-GMD.5 36. 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 Geometric Measurement & Dimension Explain volume formulas and use them to solve problems. Geometry dissection arguments, G-GMD.1 Cavalieri’s principle, and informal limit arguments. Students: Given a circle, Dissection arguments 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, Cavalieri’s Principle 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 Franklin County Schools Cylinder Pyramid Cone Students know: Students are able to: Students understand that: Techniques Geometric to find the area Accurately and perimeter of decompose circles, shapes may be parallelograms, cylinders, pyramids, decomposed into and cones into other shapes which may be useful in Techniques other geometric creating formulas, to find the area shapes, of circles or polygons. Explain and justify how the formulas for circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone may be created from the use of other geometric shapes. Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape. Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources height is the volume of the cylinder, and use Cavalieri’s Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume. Given a pyramid or cone, Explain that the shapes could be divided into crosssections, and the area of the cross-sections 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. 37. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ Geometric Measurement & Dimension Explain volume formulas and use them to solve problems. Geometry G-GMD.3 Students: Given a contextual situation that requires finding the volume of a cylinder, pyramid, cone, or sphere as part of its solution, Use an appropriate shape or 2-D drawing to model the situation, Solve using the Franklin County Schools Students know: Students are able to: Volume formulas for cylinders, pyramids, cones, and spheres, Students understand that: Click below to A contextual access all ALEX situation involving resources cylinders, pyramids, aligned to this cones, and spheres standard. may be modeled by shapes or 2-D Techniques ALEX drawings, and the for modeling 3-D Resources Use the model model may provide objects with shapes or 2-D or drawing to find insight into the drawings, and values needed for solution of the for using these use in the volume Accurately model a contextual situation with a cylinder, pyramid, cone, sphere, or a 2-D drawing, Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment appropriate formula, Justify and explain the solution and solution path in the context of the given situation. 38. Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama) ★ 39. Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects. Teacher Vocabulary Knowledge models to identify specific values for use in volume formulas. Skills Understanding formula, problem, Accurately find a solution to the given situation, and explain the solution in the context of the situation. Formulas are useful for efficiency when many problems of the same type need to be solved. Geometric Measurement & Dimension Explain volume formulas and use them to solve problems. Geometry G-GMD.6 Geometric Measurement & Dimension Visualize relationships between twodimensional and threedimensional objects. Geometry G-GMD.4 Click below to access all ALEX resources aligned to this standard. Students: Given 3-D objects, Conjecture about the characteristics of geometric shapes formed if a crosssection of a 3-D shape is taken, Take 2-D crosssections at different angles of cut, Explain the shape formed by taking 2-D cross-sections, Compare and contrast the figures formed when the angle Franklin County Schools Resources ALEX Resources Two-dimensional Students know: Students are able cross-sections to: Students understand that: Characteristi Two-dimensional cs of 2-D and 3- Conjecture objects D geometric about the objects, characteristics of geometric shapes Threedimensional objects Techniques formed from taking a cross-section of a for finding a 3-D shape, or cross-section of Rotations rotating a 2-D a 3-D object, shape about a line, 3-D objects can be created from 2D plane figures Click below to through access all ALEX transformations such as rotations, resources aligned to this Cross-sections standard. Techniques for rotating a 2- Accurately D object about a determine the geometric shapes line. formed from taking a cross-section of a 3-D shape, or rotating a 2-D shape about a line. of 3-D objects can be formed in a variety of ways, depending on the angle of the cut with the base of the object. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources of the cut changes. Given 2-D objects, 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 2D object is rotated about a line. 40. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★ Modeling with Geometry Apply geometric concepts in modeling situations. Geometry G-MG.1 Students: Given a real-world object, Students know: Students are able to: Techniques to find measures Model a realof geometric world object shapes, through the use of a geometric shape, Select an appropriate geometric shape to model the object, Properties of Justify the geometric shapes. model by connecting its measures and properties to the object. Provide a description of the object through the measures and properties of the geometric shape which is modeling the object, Franklin County Schools Geometric shapes may be used to model realClick below to world objects, access all ALEX resources Attributes of aligned to this geometric figures standard. help us identify the figures and find ALEX their measures, Resources therefore matching these figures to real world objects allows the application of geometric techniques to real world problems. Explain and justify the model which was selected. 41. Apply concepts of Modeling with Students: Students understand that: Density Students know: Students are able Students Click below to Geometry Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).★ Geometry Apply geometric concepts in modeling situations. Geometry G-MG.2 Evidence of Student Attainment Teacher Vocabulary Given a contextual situation involving density, Skills Knowledge to: Understanding understand that: Geometric concepts of area Accurately and volume, model a situation involving density, Situations involving density may be modeled through a Properties of Justify how the representation of a rates, concentration per model is an unit of area or unit accurate Modeling representation of of volume. techniques. the given situation. 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?), Resources access all ALEX resources aligned to this standard. ALEX Resources Explain and justify the model in terms of the original context. 42. 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).★ Modeling with Geometry Apply geometric concepts in modeling situations. Geometry G-MG.3 Students: Given a contextual situation involving design problems, Geometric methods Students know: Students are able to: Students understand that: Properties of Design problems geometric Accurately Design shapes, model and solve a problems may be Click below to access all ALEX Create a geometric design problem, modeled with resources method to model the geometric methods, Characteristi aligned to this situation and solve the Justify how cs of a standard. problem, Geometric mathematical their model is an model. accurate models may have ALEX Explain and justify representation of physical Resources the model which was the given situation. constraints, created to solve the problem. Models represent the mathematical core of a situation Franklin County Schools Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources without extraneous information, for the benefit in a problem solving situation. 43. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. (Link to data Conditional Students: Given scenarios probability involving two events A and B both when A and Independence B are independent and when A and B are dependent, Determine the probability of each individual event, then from simulations limit the sample space or experiments.) to those outcomes Statistics & where B has occurred Probability and calculate the S-CP.3 probability of A, compare the P(A) and the P(A given B), and explain the equality or difference in the original context of the problem, Students know: Students are able to: Methods to find probability of simple and compound events, Techniques to find conditional probability. Justify that P(A given B) = P(A∩B)/P(B). 44. Construct and interpret two-way frequency tables of data when two categories are associated with each Conditional Probability & the Rules of Probability Understand independence Franklin County Schools Two way Students: Given a situation in frequency tables which it is meaningful to collect categorical Sample space data for two categories Students understand that: Accurately determine the probability of simple and compound events, The independence of two events is determined by the effect that one event has on the outcome of another Accurately event, Click below to determine the access all ALEX conditional The occurrence resources probability P(A given B)from a of one event may aligned to this standard. sample space or or may not from the knowledge influence the of P(A∩B) and the likelihood that ALEX P(B). another event Resources occurs (e.g., successive flips of a coin - the first toss exerts no influence on whether a head occurs on the second, drawing an ace from a deck changes the probability that the next card drawn is an ace). Students know: Students are able to: Students understand that: Techniques to construct two-way frequency Two-way frequency tables show conditional Accurately construct a twoway frequency Click below to access all ALEX resources aligned to this standard. Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: Collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. and conditional probability and use them to interpret data. (Link to data Statistics & Probability S-CP.4 probabilities of simple events and conditional events from the table, explain whether the events are independent based on the context and the probability calculations. 45. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Example: Compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. (Link to data Conditional Students: Given a contextual probability situation and scenarios involving two events, Independence Collect data and create two-way frequency tables, Independent tables, Conditional probabilities Techniques to find simple and conditional probability in two-way frequency tables. from simulations or experiments.) Determine Explain the meaning of independence from a formula perspective P(A∩B) = P(A) x P(B) and from the intuitive from simulations notion that A occurring or experiments.) has no impact on Statistics & whether B occurs or Probability not, S-CP.5 Compare these two Franklin County Schools Skills Knowledge probability and can be used to test for Accurately find independence. simple and conditional probability from a two-way frequency table. Resources table, Students know: Students are able to: Possible relationships and differences between the simple probability of an event and the probability of an event under a condition. Understanding ALEX Resources Students understand that: Communicate The occurrence the concepts of of one event may conditional or may not probability and influence the independence using likelihood that everyday language another event by discussing the occurs (e.g., impact of the successive flips of a occurrence of one coin - first toss event on the exerts no influence likelihood of the on whether a head other occurring. occurs on the second, drawing an ace from a deck Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills interpretations within the context of the scenario. Understanding Resources changes the probability that the next card drawn is an ace), Events are independent if the occurrence of one does not affect the probability of the other occurring. 46. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. Statistics & Probability S-CP.6 Students: Given a contextual situation consisting of two events, Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A, compare the P(A) and the P(A given B), and explain the equality or difference in the original context of the problem, Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the P(A and B), compare the ratio of P(A and B) and P(B) to P(A given B), Franklin County Schools Conditional probability Students know: Students are able to: Possible relationships and differences between the simple probability of an event and the probability of an event under a condition. Accurately determine the probability of simple and compound events, Accurately determine the conditional probability P(A given B) from a sample space or from the knowledge of P(A∩B) and the P(B). Students understand that: Conditional probability is the probability of an event occurring given that another event has occurred. Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Skills Knowledge Understanding Resources and explain the equality or difference in the original context of the problem. 47. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. Statistics & Probability S-CP.7 Students: Given a contextual situation consisting of two events, Addition Rule Determine the simple probability of each event, Students know: Students are able to: Students understand that: Techniques for finding probabilities of simple and compound events. Formulas are useful to generalize regularities, but must be justified, Accurately determine the probability of simple and compound events. The Addition Rule may be used for finding compound probability. Determine the P(A or B) and P(A and B), Interpret the Addition Rule by counting outcomes in the four events A, B, A and B, A or B and showing the relationship to P(A or B) = P(A) + P(B) - P(A and B), Click below to access all ALEX resources aligned to this standard. ALEX Resources Interpret the Addition Rule in the case that the P(A and B) = 0. 48. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform Franklin County Schools Students: Given a contextual situation consisting of two events A and B, Use the definition of conditional probability P(B|A) = P(A and B)/P(A) to determine the probability of the Uniform probability model General Multiplication Rule Probability Students are able Students know: to: Techniques for finding probabilities of simple and conditional events. Determine the probability of a single event. Determine the Students understand that: The general Multiplication Rule for probability is a manipulation of the formula for conditional Click below to access all ALEX resources aligned to this standard. ALEX Resources Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID probability model. Statistics & Probability S-CP.8 49. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. Statistics & Probability S-CP.9 Evidence of Student Attainment Teacher Vocabulary compound event (A and B) when the P(A|B) and the P(A) are known or may be determined. Interpret the probability as it relates to the context. Simple events Skills probability of a conditional event. Conditional events Students are able Students know: to: Students: Given a contextual situation, Choose the appropriate counting technique (permutation or combination), Find the number of ways an event(s) can occur, Use these counts to determine probabilities of the event, including compound events. Franklin County Schools Knowledge Permutations Combinations Compound events Probability Possible outcomes Order is the determining factor in whether a event requires a permutation or a combination to count the number of possible outcomes of the event. Techniques for finding probabilities of simple and compound events. Techniques for finding the number of permutations or combinations of an event. Evaluate factorial expressions. Apply the multiplication and addition rules to determine probabilities. Interpret and apply the different notations for combinations and permutations. Perform procedures to evaluate expressions involving the number of combinations and Understanding Resources probability. The formula P(A and B) = P(A)P(B|A) will always apply regardless of whether the events are independent or dependent. Students understand that: There are contextual situations that can be interpreted through the use of combinations Click below to and access all ALEX permutations. resources aligned to this The contextual standard. situation determines ALEX whether Resources combinations or permutations must be utilized. Mathematics is a coherent whole. Structure within mathematics Geometry CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills permutations of n things taken r at a time. 50. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Using Probability to Make Decisions Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.) Statistics & Probability S-MD.6 Resources allows for procedures or models from one concept to be applied elsewhere (e.g., Pascal's triangle as it applies to the number of combinations). Students understand that: Students: Given a contextual situation in which a decision needs to be made, Use a random probability selection model to produce unbiased decisions. Franklin County Schools Understanding Fair decisions Probability Fair decisions Random Students are able to: Students know: The characteristics of a random sample. Randomly select a sample from a population (using technology when appropriate). Multiple factors may ultimately determine the decision one makes other than the probability of events, such as Click below to access all ALEX ethical resources constraints, social policy, or aligned to this standard. feelings of others. ALEX Probabiliti Resources es can be used to explain why a decision was considered to be fair or objective. Geometry CCRS Standard 51. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Using Probability to Make Decisions Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.) Statistics & Probability S-MD.7 Understanding Resources Students understand that: Students know: Students: Given a contextual situation in which a decision needs to be made, Franklin County Schools Skills Knowledge Use probability concepts to analyze, justify, and make objective decisions. Probability Techniq ues for finding probabi lities of simple, compou nd, and conditio nal events and from probabi lity distribu tions. Students are able to: Choose the appropriat e probability concept for the given situation. Use and apply the selected probability rule. Communic ate the reasoning behind decisions. 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