Mathematics Scope & Sequence Geometry

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1 Mathematics Scope & Sequence Geometry Readiness Standard(s) First Six Weeks (29 ) Coordinate Geometry G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons G.7.C [derive and] use formulas involving length, slope, and midpoint G.4.A select an appropriate representation ([concrete,] pictorial, graphical, verbal, or symbolic) in order to solve problems Use slope to determine the relationship between perpendicular and parallel lines. (G.7.B) Use the slope formula to determine if lines are parallel, perpendicular or intersecting. (G.7.C) Given the equations of two lines, determine if the lines are perpendicular, parallel, intersecting, or coinciding. (G.7.B) Write the equations of parallel and perpendicular lines in slope intercept form and point-slope form given a point and a line, a slope and a point, etc. (G.7.B) Make connections between multiple representations of a function including concrete, pictorial, graphical, verbal, and symbolic. (G.4.A) 6-8 Textbook Sections: 3.5, 3.6 Readiness Standard(s) G.7.C [derive and] use formulas involving length, slope, and midpoint G.1.B recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes Define Euclidian Geometry (study of flat space) and non- Euclidean Geometries (such as spherical or hyperbolic geometries) as distinct axiomatic systems (G.1.C) Recognize the development of geometric systems (G.1.B) Identify, name, and draw points, lines, segments, rays and planes (G.7.A) Foundations in Geometry G.1.C compare and contrast the structures and implications of Euclidean and non-euclidean geometries G.2.A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.7.A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures Solve application problems using the Segment Addition Postulate, including algebraic expressions representing dimensions (G.3.B) Use the coordinate plane to find properties such as midpoint and distance between two points (G.3.B) Use appropriate mathematical markings to show congruence in diagrams (G.3.B) Construct midpoints, congruent segments, and segment bisectors using a variety of tools such as a compass and straight edge, paper folding, and dynamic geometric software (G.2.C) Relate the distance formula to the Pythagorean Theorem (G.7C) 8-10 Solve application problems involving distance and midpoint formulas, including algebraic expressions representing dimensions (G.7.C) Simplify radicals Textbook Sections: 1.1, 1.2, 1.6 First six weeks curriculum continued on next page.

2 More Foundations in Geometry G.2.B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.2.A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships Name and identify congruent, adjacent, complementary, supplementary, linear pairs, and vertical angles in complex figures. (G.2.B) Solve application problems involving vertical angles, bisected angles, linear pairs, complementary angles, and supplementary angles, including algebraic expressions representing dimensions. (G.2.B) Solve application problems using the Angle Addition Postulate, including algebraic expressions representing dimensions. (G.3.B) Construct congruent angles and angle bisectors using a variety of tools such as a compass and straight edge, paper folding, and dynamic geometric software (G.2.B) Explore attributes to make connections and verbalize how and why constructions work. (G.2.B) 6-8 Textbook Sections: 1.3, 1.4, 2.6 (linear pair and rt angle thm), 2.7 (vertical angles thm)

3 Second Six Weeks (29 ) Readiness Standard(s) G.2.B make conjectures about angles, lines, polygons, circles, and three dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.3.C use logical reasoning to prove statements are true and find counter examples to disprove statement that are false Use inductive reasoning to make conjectures from number or picture patterns, specific examples, or events. (G.3.D) Use inductive reasoning to develop algebraic expressions representing geometric properties (G.5.A) Use examples and counter-examples that are mathematical and non-mathematical. (G.2.B) Use counter-examples to prove statements false. (G.3.C) Geometric Reasoning G.5.A use numeric and geometric patterns to develop algebraic expressions representing geometric properties G.1.A [develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems] G.3.A determine the validity of a conditional statement, its converse, inverse, and contrapositive G.3.D use inductive reasoning to formulate a conjecture G.3.E use deductive reasoning to prove a statement Use facts, definitions, postulates, theorems, and properties to prove statements true or false. (G.3.C) Identify and write the converse, inverse, and contrapositive of conditional statements and determine the truth values, using examples and counter-examples that are mathematical and non-mathematical. (G.3.A) Make, interpret, and/or understand statements such as if p, then q as applied to attributes of geometric drawings, figures, etc. (G.3.E) Draw conclusions by using deductive reasoning, including the Law of Detachment and the Law of Syllogism. (G.3.E) Introduce geometric proof by use of algebraic proof in twocolumn, flow chart, and paragraph forms. (G.3.E) Use two-column, flow chart, and paragraph proof to solve geometric proofs involving segment and angle relationships. (G.3.E) Textbook Sections: 2.1, 2.2, 2.3, 2.5, 2.6, 2.7 Readiness Standard(s) Parallel and Perpendicular Lines G.2.A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.3.E use deductive reasoning to prove a statement G.9.A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and [concrete] models Construct parallel lines, and a line parallel or perpendicular to a given line through a point not on the line using a variety of tools such as a compass and straight edge, paper folding, and dynamic geometric software. (G.2.A) Classify then construct and formulate conjectures about special angles formed by a pair of parallel lines and a transversal. (G.9.A) Solve application problems using the relationships between angles formed when transversals cross parallel lines. (G.9.A) Use two-column, flow chart, and paragraph proof to solve geometric proofs involving the angles formed by a parallel lines and a transversal. (G.3.E) Use two-column, flow chart, and paragraph proof to prove lines are parallel or perpendicular given information about angle pairs. (G.3.E) 7-9 Textbook Sections: 3.1, 3.2, 3.3, 3.4 Second six weeks curriculum continued on next page.

4 Triangle Congruence G.2.B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.10.B justify and apply triangle congruence relationships G.2.A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.3.E use deductive reasoning to prove a statement G.7.A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures Given three points on the coordinate plane, use the distance formula and slope formula to classify the triangle by its side lengths and angle measures. (G.1.A) Given the equations of three lines on the coordinate plane, use the intersections of the lines to classify the triangle by its side lengths and angle measures.(g.3.b) Use the Triangle Sum Theorem and Exterior Angle Theorem to find angle measures of interior and exterior angles of triangles. (G.2.B) Use two-column, flow chart, and paragraph proof to solve geometric proofs involving the Triangle Sum Theorem and Exterior Angle Theorem. (G.3.E) Use constructions and properties of congruence to justify the following triangle congruence postulates: SAS, SSS, ASA, AAS, and HL. (G.2.A) Use two-column, flow chart, and paragraph proof to solve geometric proofs involving congruent triangles. (G.10.B) Use CPCTC to prove parts of triangles are congruent. (G.10.B) 6-8 Textbook Sections: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6

5 Third Six Weeks (29 ) Triangle Properties and Attributes G.5.A use numeric and geometric patterns to develop algebraic expressions representing geometric properties G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons G.2.A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.5.B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles Use congruent triangles to prove the Isosceles Triangle Theorem. (G.3.B) Generate and solve equations to find a missing angle or side length of an isosceles or equilateral triangle. (G.3.B) Construct perpendicular bisectors using a variety of tools such as a compass and straight edge, paper folding, and dynamic geometric software. (G.2.A) Develop algebraic expressions representing midsegment based on a geometric pattern. (G.5A) Write the equation of the line representing the midsegment, perpendicular bisector, median and altitude of triangles. (G.7.B) Solve application problems using midsegments, angle bisectors, perpendicular bisectors, medians, and altitudes of triangles, including algebraic expressions representing dimensions. (G.3.B) Use constructions to explore and define centroid, circumcenter, incenter, and orthocenter of triangles. (G.2.A) Use constructions and counter-examples to justify the Triangle Inequality Theorem. (G.2.A) 8-10 Solve application problems using the Triangle Inequality Theorem. (G.3.B) Textbook Sections: 4.8, 5.1, 5.2, 5.3, 5.4, 5.5 Third Six weeks continued on next page

6 Polygons and Quadrilaterals G.5.A use numeric and geometric patterns to develop algebraic expressions representing geometric properties G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons G.3.E use deductive reasoning to prove a statement G.5.B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles G.7.A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures G.9.B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and [concrete] models Classify polygons based on their sides and angles, including regular/irregular and concave/convex. (G.3.B) Given four points on the coordinate plane, use the distance formula and slope formula to classify the quadrilateral by its side lengths and angle measures. (G.7.B) Given the equations of four lines on the coordinate plane, use the intersections of the lines to classify the quadrilateral by its side lengths and angle measures. (G.7.B) Use exploration of polygons to develop the Polygon Angle Sum Theorem and Polygon Exterior Angle Sum Theorem. (G.9.B) Use numeric and geometric patterns to develop algebraic expressions representing the Polygon Angle Sum Theorem and Polygon Exterior Angle Sum Theorem. (G.5.A) Find angle sums or missing angle measures using Polygon Angle Sum and Polygon Exterior Angle Sum theorems. (G.3.B) Solve application problems using Polygon Angle Sum Theorem and Polygon Exterior Angle Sum Theorem, including algebraic expressions representing dimensions. (G.3.B) Use constructions to make and justify conjectures about relationships between side lengths, angle measures, and diagonals of parallelograms, rectangles, squares, rhombi, and kites. (G.9.B) 9-12 Develop algebraic expressions and generalizations based on numeric and geometric patterns representing the relationships between side lengths, angle measures, and diagonals of parallelograms, rectangles, squares, rhombi, and kites. (G.5.A), (G.5.B) Solve application problems using the properties of quadrilaterals, including algebraic expressions representing dimensions. (G.3.B) Use two-column, flow chart, and paragraph proof to solve geometric proofs involving the properties of quadrilaterals. (G.3.E) Textbook Sections: 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 Third Six weeks continued on next page

7 Identify and draw translations given an algebraic expression, such as (x+2, y-3). (G.10.A) Transformations G.5.C use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations G.7.A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures G.10.A use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane Given the preimage and image of a figure, write or identify the algebraic expression representing the transformation. (G.5.C) Identify and draw reflections over any horizontal or vertical line, y = x, and y = -x. (G.10.A) Identify and draw rotations about the origin. (G.10.A) Identify and draw compositions of transformations that include two or more translations, reflections, or rotations. (G.10.A) Use properties of congruence and transformations to determine if a two-dimensional figure will tessellate on the plane. (G.5.C) 4-6 Solve real world problems using properties of transformations, including tessellations. (G.5.C) Textbook Sections: 1.7, 12.1, 12.2, 12.3, 12.4, 12.6

8 Fourth Six Weeks (29 ) Similarity G.11.C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods G.5.B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles G.11.A use and extend similarity properties and transformations to explore and justify conjectures about geometric figures G.11.B use ratios to solve problems involving similar figures Determine whether two figures are similar; include twoand three-dimensional figures. (G.11.A) Use proportions to find missing dimensions of similar figures, include overlapping figures. (G.11.B) Use constructions and properties of similarity to justify the following triangle similarity postulates/theorems: AA, SSS, and SAS. (G.11.C) Use two-column, flow chart, and paragraph proof to prove triangles are similar using AA, SSS, and SAS. (G.11.C) Solve application problems using similar triangles and polygons. (G.11.B) Use constructions and properties of similarity to prove the Triangle Proportionality Theorem and the Triangle Angle Bisector Theorem. (G.11.C) 6-8 Solve application problems using the Triangle Proportionality Theorem and the Triangle Angle Bisector Theorem, including algebraic expressions representing dimensions. (G.11.C) Use proportional relationships to solve problems involving indirect measurements and scale drawings. (G.11.B) Use dilations to apply similarity properties in the coordinate plane. (G.11.A) Textbook Sections: 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 12.7 Readiness Standard(s) Right Triangle Properties and Attributes G.5.D identify and apply patterns from right triangles to solve meaningful problems, including special right triangles and triangles whose sides are Pythagorean triples G.8.C [derive,] extend, and use the Pythagorean Theorem G.11.C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios and Pythagorean triples using a variety of methods Use the Pythagorean Theorem to solve real-world problems.(g.8.c) Use Pythagorean inequalities to classify triangles. (G.8.C) Extend the Pythagorean Theorem by identifying and using Pythagorean triples in problem solving situations. (G.5.D) Extend the Pythagorean Theorem by solving problems using compound figures in which one of the figures is a right triangle. (G.8.C) Use an equilateral triangle and the Pythagorean Theorem to develop the right triangle relationships. (G.11.C) Use a square and the Pythagorean Theorem to develop the right triangle relationships. (G.11.C) Solve application problems using special right triangle relationships in real-world situations. (G.5.D) Make connections to concrete models, pictures, and figures.(g.5.d) Use a square and the Pythagorean Theorem to develop the right triangle relationships. (G.11.C) 6-8 Solve application problems using special right triangle relationships in real-world situations. (G.5.D) Textbook Sections: 5.7, 5.8 Forth Six weeks continued on the next page

9 Trigonometry G.11.C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods Use exploration to justify trigonometric ratios in right triangles. (G.11.C) Select and use appropriate trigonometric ratios to find angle measures and side lengths of right triangles. (G.11.C) Solve application problems using trigonometric ratios in real-world situations, including algebraic expressions representing dimensions. (G.11.C) 6-8 Solve problems involving angles of elevation and angles of depression. (G.11.C) Textbook Sections: 8.2, 8.3, 8.4

10 Fifth Six Weeks (28 ) G.7.C [derive and] use formulas involving length, slope, and midpoint and apply this idea in solving problems G.8.A find areas of regular polygons, circles, and composite figures G.11.D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems Find the perimeter of polygons and composite figures with side lengths expressed as algebraic expressions. (G.5.A) Write and solve equations involving perimeter of polygons and composite figures. (G.8.A) Find perimeter of composite figures where students measure dimensions. (G.8.F) Find perimeter of polygons and circumference of circles in a coordinate plane. (G.7.C) Perimeter and Area G.8.E use area models to connect geometry to probability and statistics G.8.F use conversions between measurement systems to solve problems in real-world situations Describe the effect on perimeter when one or more dimensions are changed. (G.11.D) Compare the effects on perimeter of a figure to a different figure when dimensions are changed. (G.11.D) Use the circumference formula to solve problems (express answers in terms of and by approximate values). (G.7.C) Solve application problems using perimeter and circumference. (G.7.C) Find the area of polygons and composite figures with side lengths expressed as algebraic expressions. (G.5.A) Find area of composite figures where students measure dimensions. (G.8.A) Find area of regular figures. (G.8.A) 9-12 Find the area of a circle (express answers in terms of and by approximate values). (G.8.A) Estimate areas of irregular shapes on a coordinate grid. (G.8.A) Connect area problems to perimeter and vice versa. (G.11.D) Describe the effect on area when one or more dimensions are changed. (G.11.D) Compare the effects on area of a figure to a different figure when dimensions are changed. (G.11.D) Solve application problems using area of polygons, circles, and sectors in real-world situations. (G.8.A) Find geometric probabilities involving area. (G.8.E) Textbook Sections:1.5, 7.5, 9.1, 9.2 and lab, 9.3, 9.4, 9.5, 9.6 Fifth six weeks curriculum continued on next page.

11 G.2.B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic Determine names and numbers of faces, edges, and bases of three-dimensional figures; include composite figures. (G.9.D) Apply Euler s formula to find the number of vertices, edges, and faces of a polyhedron. (G.2.B) G.8.D find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations Describe a three-dimensional figure given its net.(g.6.b) Describe cross sections of solids; cross section may be a point, line segment, or region of a plane. (G.6.A) Surface Area G.11.D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems G.6.A describe and draw the intersection of a given plane with various three-dimensional geometric figures G.6.B use nets to represent and construct three-dimensional geometric figures G.6.C use orthographic and isometric views of three-dimensional geometric figures and solve problems G.9.D analyze the characteristics of polyhedral and other three-dimensional figures and their component parts based on explorations and [concrete] models Find area of cross sections of prisms.(g.8.d) Draw orthographic and isometric views from a 3-D view of a model and draw a3-d view of a model given an orthographic or isometric view. (G.6.C) Determine a three-dimensional view from a drawing of the top view with number of cubes in each column. G.6.C) Find lateral surface area of prisms, pyramids, cones, and cylinders using the formulas on the Assessment Reference Materials. (G.8.D) Find total surface area of prisms, pyramids, cones, and cylinders using the formulas on the Assessment Reference Materials. (G.8.D) Find lateral or total surface area of 3-D figures given a net (with and without dimensions). (G.8.D) Solve application problems using surface area of prisms, pyramids, spheres, cones, cylinders and composite figures in real-world situations. (G.8.D) 7-10 Describe the effect on surface area when one or more dimensions are changed. (G.11.D) Textbook Sections: 10.1, 10.2, 10.3, 10.4, 10.5, 10.8 Readiness Standard(s) Volume G.8.D find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations G.11.D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems G.6.C Use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems Find volume of prisms, pyramids, cones, and cylinders using the formulas on the Assessment Reference Materials. (G.8.D) Find volume of 3-D figures given a net (with and without dimensions). (G.6.C) Solve application problems using volume of prisms, pyramids, spheres, cones, cylinders and composite figures in real-world situations. Describe the effect on volume when one or more dimensions are changed. (G.11.D) 4-6 Textbook Sections: 10.6, 10.7, 10.8

12 Sixth Six Weeks (33 ) G.2.B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic Identify tangents, secants, chords, diameters, radii, inscribed angles, central angles, and major/minor arcs. (G.9.C) Determine angle relationships in circles based on geometric patterns. (G.5.B) G.5.A use numeric and geometric patterns to develop algebraic expressions representing geometric properties Use constructions to explore the properties of intersecting chords, secants, and tangents and use these properties to solve problems. (G.2.A) Circles G.5.B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles G.8.B find areas of sectors and arc lengths of circles using proportional reasoning G.9.C formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and [concrete] models Use numeric and geometric patterns to develop algebraic expressions representing the properties of intersecting chords, secants, and tangents. (G.5.A) Explore the relationships among angles in circles and use these relationships to solve problems (include tangent line to radius, central angles, and inscribed angles). (G.9.C) Use numeric and geometric patterns to develop algebraic expressions representing the relationships among angles in circles. (G.5.A) Find the measure of a central angle if a circle were divided into a given number of equal sectors. (G.8.B) Apply central angles to circle graphs. (G.8.B) Find the length of segment formed by lines that intersect circles (G.2.B) Use proportions to find arc length. (G.8.B) Solve application problems using arc length in real-world situations. (G.8.B) Use proportions to find the area of a sector. (G.8.B) Solve application problems using area of sectors in realworld situations. (G.8.B) Textbook Sections: 11.1, 11.2,, 11.3, 11.4, 11.5, 11.6 Preparing For Functional Relationships with Area and Volume 2A.8.A analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems 2A.8.D solve quadratic equations and inequalities using graphs tables, and algebraic methods Solve application problems integrating geometry and algebra concepts such as: Solving for area or volume given figures whose sides are polynomials. (2A.8.D) Maximizing area of a rectangle with a fixed perimeter. (2A.8.A) Minimize the perimeter of a rectangle with a given area. (2A.8.A) Finding dimensions of an open box built from a net created by cutting off corners of a rectangle. (2A.8.D) Use an area model to represent a quadratic equation. (2A.8.A) 7-9 Textbook Sections: Semester Review Individualized review based on student needs. 4-6

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