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Mathematics Standards Comparison GEOMETRY Next Generation Sunshine State Standards Body of Knowledge: Geometry Standard 1: Points, Lines, Angles, and Planes BENCHMARK CODE MA.912.G.1.1 Common Core State Standards Conceptual Categories: Geometry BENCHMARK DOMAIN CLUSTER STANDARD Find the lengths and midpoints of line segments in two-dimensional coordinate systems. G-GPE Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinates to prove simple geometric theorems algebraically MA.912.G.1.2 Not Assessed MA.912.G.1.3 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and justifying the process used. G-CO Congruence Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Identify and use the relationships between special pairs of angles formed by parallel lines and transversals. G-CO Congruence Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints *In 8th grade: 8.G.5 Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal Body of Knowledge: Geometry Standard 2: Polygons BENCHMARK CODE MA.912.G.2.1 Assessed with MA.912.G.2.3. BENCHMARK Identify and describe convex, concave, regular, and irregular polygons. Conceptual Categories: Functions DOMAIN CLUSTER STANDARD There is no Benchmark – Common Core Alignment MA.912.G.2.2 Determine the measures of interior and exterior angles of polygons, justifying the method used. G-CO Congruence Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point *In 8th grade: 8.G.5 5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles. MA.912.G.2.3 Also assesses MA.912.G.2.1, MA.912.G.4.1/4.2 MA.912.G.4.4/4.5 MA.912.G.2.4 Use properties of congruent and similar polygons to solve mathematical or realworld problems. Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. G-SRT Similarity, Right Triangles, & Trigonometry G-CO Congruence Understand similarity in terms of similarity transformations 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures Experiment with transformations in the plane. 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). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-SRT Similarity, Right Triangles, & Trigonometry Understand congruence in terms of rigid motions 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. Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. b. 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. The dilation of a line segment is longer or shorter in the ratio given by the scale factor 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MA.912.G.2.5 Also assesses MA.912.G.2.7 MA.912.G.2.6 Honors MA.912.G.2.7 Assessed with MA.912.G.2.5, MA.912.G.7.7 Explain the derivation and apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.). Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane. G-GPE Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G-GPE Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). *In 7th Grade: 7.G.1 Solve problems involving scale drawings of geometric figures, Determine how changes in dimensions affect the perimeter and area of common geometric figures. including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Body of Knowledge: Geometry Standard 3: Quadrilaterals BENCHMARK CODE MA.912.G.3.1 Assessed with MA.912.G.3.4. MA.912.G.3.2 Assessed with MA.912.G.3.4. MA.912.G.3.4 Also assesses MA.912.D.6.4, MA.912.G.8.5. Conceptual Categories: Algebra/Functions BENCHMARK DOMAIN CLUSTER STANDARD Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. G-CO Congruence Prove geometric theorems 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals Compare and contrast special quadrilaterals on the basis of their properties. Prove theorems involving quadrilaterals. MA.912.G.3.3 Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals. G-GPE Expressing Geometric Properties with Equations Body of Knowledge: Geometry Standard 4: Triangles BENCHMARK CODE MA.912.G.4.1 Assessed with MA.912.G.2.3. BENCHMARK Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Conceptual Categories: Algebra/ Functions/Geometry/Statistics DOMAIN CLUSTER STANDARD *Classifying triangles is done in 4th grade: 4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. *Did not find mention of classification based on side lengths: scalene, isosceles, equilateral. *Constructing triangles done in 7th grade: 7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. MA.912.G.4.2 Assessed with MA.912.G.2.3. Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter. G-CO Congruence Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. *Did not find mention of orthocenter, centroid, incenter, and circumcenter in CCSS. MA.912.G.4.3 Not assessed. MA.912.G.4.4 Assessed with MA.912.G.2.3. There is no Benchmark – Common Core Alignment Construct triangles congruent to given triangles. Use properties of congruent and similar triangles to solve problems involving lengths and areas. G-SRT Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides MA.912.G.4.5 Assessed with MA.912.G.2.3. MA.912.G.4.6 Also assesses MA.912.D.6.4 and MA.912.G.8.5. Apply theorems involving segments divided proportionally. Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures G-SRT Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity G-GPE Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-CO Congruence Understand congruence in terms of rigid motions 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. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G-SRT Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MA.912.G.4.7 Apply the inequality theorems: triangle inequality, inequality in one triangle, and the Hinge Theorem. MA.912.G.4.8 Honors Use coordinate geometry to prove properties of congruent, regular, and similar triangles. There is no Benchmark – Common Core Alignment G-GPE Expressing Geometric Properties with Equations Body of Knowledge: Geometry Standard 5: Right Triangles BENCHMARK CODE MA.912.G.5.1 Assessed with Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Conceptual Categories: Algebra/Functions BENCHMARK DOMAIN CLUSTER STANDARD Prove and apply the Pythagorean Theorem and its converse. G-SRT Similarity, Right Triangles, & Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Trigonometry MA.912.G.5.4. Define trigonometric ratios and solve problems involving right triangles 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems *Students also work with the Pythagorean Theorem in 8 th Grade. MA.912.G.5.2 Assessed with MA.912.G.5.4. MA.912.G.5.3 Assessed with MA.912.G.5.4. There is no Benchmark – Common Core Alignment State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Use special right triangles (30° - 60° - 90° and 45° - 45° - 90°) to solve problems. G-SRT Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MA.912.G.5.4 Solve real-world problems involving right triangles. *Found in 4th year HS math course: F-TF Extend the domain of trigonometric functions using the unit circle. Also assesses MA.912.G.5.1, MA.912.G.5.2, MA.912.G.5.3. 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – × in terms of their values for x, where x is any real number. (+) = Honors Body of Knowledge: Geometry Standard 6: Circles BENCHMARK CODE MA.912.G.6.1 Conceptual Categories: Number and Quantity/Algebra BENCHMARK DOMAIN CLUSTER STANDARD Determine the center of a given circle. Given three points not on a line, construct the circle that passes through them. Construct tangents to circles. Circumscribe and inscribe circles about and within triangles and regular polygons. G-C Circles Understand and apply theorems about circles 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. (+) Construct a tangent line from a point outside a given circle to the circle. (+) = Honors G-CO Congruence Make geometric constructions 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle MA.912.G.6.2 Assessed with MA.912.G.6.5. MA.912.G.6.4 Assessed with MA.912.G.6.5. MA.912.G.6.5 Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles. G-C Circles Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). Understand and apply theorems about circles 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Find arc lengths and areas of sectors of circles 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Solve real-world problems using measures of circumference, arc length, and areas of circles and sectors. MA.912.G.6.3 Honors Prove theorems related to circles, including related angles, chords, tangents, and secants. G-C Circles Understand and apply theorems about circles 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MA.912.G.6.6 Also assesses MA.912.G.6.7. Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. G-GPE Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MA.912.G.6.7 Given the equation of a circle in center-radius form or given the center and the radius of a circle, sketch the graph of the circle. Found in Algebra I: A-REI Solve systems of equations. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3 Body of Knowledge: Geometry Standard 7: Polyhedra and Other Solids BENCHMARK CODE BENCHMARK Conceptual Categories: Algebra DOMAIN CLUSTER STANDARD *In 6th grade: Solve real-world and mathematical problems involving area, surface area, and volume. 4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. MA.912.G.7.1 Also assesses MA.912.G.7.2. Describe and make regular, non-regular, and oblique polyhedra, and sketch the net for a given polyhedron and vice versa. MA.912.G.7.2 Describe the relationships between the faces, edges, and vertices of polyhedra. MA.912.G.7.3 Honors Identify, sketch, find areas and/or perimeters of cross sections of solid objects. G-GMD Geometric Measurement & Dimension Visualize relationships between twodimensional and threedimensional objects 4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects MA.912.G.7.4 Assessed with MA.912.G.7.5. Identify chords, tangents, radii, and great circles of spheres. G-GMD Geometric Measurement & Dimension Explain volume formulas and use them to solve problems 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. MA.912.G.7.5 Explain and use formulas for lateral area, surface area, and volume of solids. MA.912.G.7.6 Assessed with MA.912.G.7.5. Identify and use properties of congruent and similar solids. MA.912.G.7.7 Determine how changes in dimensions affect the surface area and volume of common geometric solids. Also assesses MA.912.G.2.7. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. There is no Benchmark – Common Core Alignment Body of Knowledge: Algebra Standard 8: Mathematical Reasoning and Problem Solving BENCHMARK CODE MA.912.G.8.1 Embedded throughout. BENCHMARK Analyze the structure of Euclidean geometry as an axiomatic system. Distinguish between undefined terms, definitions, postulates, and theorems. Conceptual Categories: Number and Quantity DOMAIN CLUSTER STANDARD MA.912.G.8.2 Embedded throughout. MA.912.G.8.3 Embedded throughout. MA.912.G.8.4 MA.912.G.8.5 Assessed with MA.912.G.3.4 MA.912.G.4.6. MA.912.G.8.6 Not assessed. 1 Make sense of problems and persevere in solving them. Use a variety of problem solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Determine whether a solution is reasonable in the context of the original situation. Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. G-CO Congruence Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two column, and indirect proofs. Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals G-CO Congruence Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Body of Knowledge: Trigonometry Standard 2: Trigonometry in Triangles BENCHMARK CODE MA.912.T.2.1 Conceptual Categories: Functions/ Algebra BENCHMARK DOMAIN CLUSTER STANDARD Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. G-SRT Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Mathematical Practice Standard 1 Body of Knowledge: Discrete Standard 6: Logic BENCHMARK CODE MA.912.D.6.2 Also assesses MA.912.D.6.3. BENCHMARK MA.912.D.6.3 Determine whether two propositions are logically equivalent. MA.912.D.6.4 Use methods of direct and indirect proof and determine whether a short proof is logically valid. Assessed with MA.912.G.3.4 MA.912.G.4.6. DOMAIN CLUSTER STANDARD There is no Benchmark – Common Core Alignment Find the converse, inverse, and contrapositive of a statement. G-CO Congruence Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals