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Geometry Year-at-a-Glance Leander ISD 2nd six weeks 1st six weeks 2 weeks Content Topics Essential Units of Study 2 weeks 2 weeks 01 02 03 Parallel and Essentials Reasoning Perpendicular of Geometry and Proof Lines Points, lines, planes, coplanar, collinear, segment addition postulate, midpoint and distance formulas. Angle pair relationship s. Angle addition postulate. Focus G.1A, G.7AC Conjectures from patterns, inductive reasoning, conditional statements, converse, inverse, contrapositi ve, deductive reasoning. Proofs and angle relationship s. Focus G.1A, G.3ACDE Pairs of lines and angles, parallel lines and transversal, def of corresponding , alternate interior angles, etc.., perpendicular lines, write and graph equations of lines. 5 weeks 04 Triangles 3rd six weeks 3 weeks 06 05 Similarity Transformat ions Ratios, Classifying Translation, proportions, reflections triangles, scale. SSS, on the Similar SAS,HL, coordinate ASA, &AAS, polygons, plane. proportionali Rotations isosceles, equilateral, ty about origin perpendicula theorems. or vertex of r bisectors, Dilations a figure. and scale angle Line and factor. bisectors, rotational Similarity medians, symmetry. transformati altitudes, inequalities in ons. a triangle. Focus Focus G.7AB, Focus G.9A G.2A, G.3E, G.5B, G.7B, G.10B G.11AB TEKS Support Support Support G.1B, G.2B G.1B, G.2B, G.2B,G.4A, G.3B G.5A Metal Tag Geometry Assessment Resources McDougal Text Resources 1.1. to 1.5 Connjecture What's My Line as Discovery Assessment and Proof as Explanantion 2.1, 2.2, 2.4, 2.3, 2.5 to 2.7 3.1 to 3.3, 3.6, 3.4, 3.5 2 weeks Focus G.5C, G.10A, G.11A 2007-08 4th six weeks 4 weeks 3 weeks 3 weeks 07 Right Triangles 08 Quadrilaterals 09 Measuring Length and Area Angle Radicals measures in review, Pythagorean polygons. Classifying Thm, Pythagoren polygons, convex, triples, converse of concave, Pyth Thm, 30- regular. 60-90, 45-45- Properties of parallelogram 90, right triangle trig. s, rhombuses, rectangles, squares, trapezoids. Focus G.2A, G.5D, G.8C, G.11BC Support Support Support Support G.2B, G.4A, G.2B, G.4A, G.4A, G.7A G.2B,G.4A, G.5A G.5AB G.5A Congruent Triangles Assessment 4.1 to 4.8 and 5.1 to 5.6 Focus G.2A, G.5B, G.9B 6th six weeks 5th six weeks 3 weeks 3 weeks 10 Surface 11 Area and Properties of Volume of Circles Solids Area of triangles, parallelograms, trapezoids, rhombuses, regular polygons. Circumference, arc length, areas of circles and sectors. Effect of dimension changes. Geometric probability. Surface area, volume of prisms, cylinders, pyramids, cones, spheres. Include nets, lateral and total area, composite solids, spherical geometry. Tangents , chords, secants, arc measures, minor and major arcs, central angles, inscribed angles, equations and graphs of circles. Focus G.5B, G.8A, G.9B, G.11D Focus G.1C, G.5B,G.6AB C, G.8BD, G.9D,G.11D Support G.2B Focus G.5B, G.7B, G.8AB, G.9C Support G.2B, Support G.2B, G.7A G.4A Support G.2B, G.4A, G.5A Ancient Ruins Living Room Sightseeing Assessment Transformatio Walk Assessment ns Assessment 6.1 to 6.3, 6.6, 6.7, 9.1 and 9.3 to 9.6 7.1 to 7.7 8.1 to 8.6 11.1, 11.2, 11.4, 12.1 to 12.6 11.6, 11.5, 11.7 10.1 to 10.7 7 Aug 2007 Geometry Essential Units of Study 2007-08 01 EUS - Essentials of Geometry (2 Weeks) Focus TEKS Content Description (G.1) Geometric structure: The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: (A) develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Identify points, lines , and planes and become familiar with the notation and sketches to describe them. Know coplanar and collinnear. 1.1 Use segment addition postulates to identify congruent segments. 1.2 Find lengths of segments in the coordinate plane. Also use segment bisector/midpoints to solve for lengths. 1.3 Name, measure, and classify angles. Understand the Angle addition Postulate and angle bisectors. 1.4 Use special angle relationships like complementary, supplementary, and vertical to find angle measures. 1.5 Support TEKS (G.7) Dimensionality and the geometry of location: The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to: (A) use one- and two-dimensional coordinate systems to represent points, lines, ray, line segments, and figures. (TAKS 7) (C) derive and use formulas involving length, slope, and midpoint. (TAKS 7) Textbook Key Vocabulary - indefined term (points, (G.1) Geometric structure: The student understands the structure of, and relationships within, an axiomatic system. The student lines, plane) is expected to: - defined terms (B) recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes. - line segment - endpoints (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: - rays (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the - opposite rays conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. - postulate, axiom - congruent segments - midpoint - segment bisector - acute, right, obtuse - straight angles - congruent angles - angle bisector - linear pair - vertical angles Resources Common Assessment Metal Tag Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-08 02 EUS - Reasoning and Proof (2 Weeks) Focus TEKS (G.1) Geometric structure: The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: (A) develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Support TEKS (G.3) Geometric structure: applies logical reasoning to justify and prove mathematical statements. The student is expected to: (A) determine the validity of a conditional statement, its converse, inverse, and contrapositive. (C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false. (D) use inductive reasoning to formulate a conjecture. (E) use deductive reasoning to prove a statement. Content Description Textbook Describe patterns and use inductive reasoning to make conjectures 2.1 Write definitions as conditional statements. Also include converse, inverse, and contrapositive. 2.2 Use postulates involving points, lines, and planes. 2.4 Use deductive reasoning to form a logical argument. ( This is the beginning of the proof section.) 2.3 Use algebraic properties in logical arguments. "Algebra Proofs" 2.5 Write proofs using geometric theorems such as the segment addition property". Begin with fill in the blank. 2.6 Use properties of special pairs of angles (supplementary, complementary, linear pair, vertical angles.) 2.7 (G.1) Geometric structure: The student understands the structure of, and relationships within, an axiomatic system. The student Key Vocabulary is expected to: - conjecture (B) recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes. - inductive reasoning - counterexample (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: - conditional statement (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the - converse conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. - inverse - contrapositive (G.3) Geometric structure: applies logical reasoning to justify and prove mathematical statements. The student is expected to: - if-then form (B) Construct and justify statements about geometric figures and their properties - hypothesis - conclusion - negation - equivalent statements - if and only if (biconditional statement) - deductive reasoning - proof - two-column proof - theorem Resources Common Assessment Conjecture as Discovery and Proof as Explanantion Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-008 03 EUS - Parallel and Perpendicular Lines (2 Weeks) Focus TEKS Content Description (G.7) Dimensionality and the geometry of location: The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to: (A) use one- and two-dimensional coordinate systems to represent points, lines, ray, line segments, and figures. (TAKS 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. (TAKS 7) Identify angle pairs formed by three intersecting lines. Include definitions of corresponding angles, alternate interior and exterior angles, same side interior and exterior. 3.1 Use angles formed by parallel lines and transversals. 3.2 Use angle relationships to prove that lines are parallel 3.3 Prove Theorems about perpendicular lines. (optional?) 3.6 Find and compare slopes of lines including the relationship between parallel and perpendicular lines. 3.4 Support TEKS (G.9) Congruence and the geometry of size: analyzes properties and describes relationships in geometric figures. Write and graph the equations of lines The student is expected to: (A) formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models. Textbook Resources 3.5 (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: Vocabulary (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the - parallel lines conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. - skew lines - parallel planes (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student - transversal is expected to : - corresponding angles (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). - alternate interior angles - alternate exterior angles (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is - consecutive interior angles expected to: - paragraph proof (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. - slope-intercept form - standard form Common Assessment What's My Line Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-08 04 EUS - Triangles (5 Weeks) Focus TEKS (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Content Description 4.1 Identify congruent figures. Use CPCTC. 4.2 4.3 4.4 4.5 4.6 4.7 Use side lengths to prove triangles congruent. SSS Use side lengths and angles to prove congruence. SAS & HL Use angles and side lengths to prove congruence. ASA & AAS (G.3) Geometric structure: applies logical reasoning to justify and prove mathematical statements. The student is expected to: (E) use deductive reasoning to prove a statement Use congruent triangles to prove corresponding parts congruent. CPCTC (G.7) Dimensionality and the geometry of location: understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to: (B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. (TAKS 7) Use properties of midsegments and write coordinate proofs Support TEKS (G.10) Congruence and the geometry of size: applies the concept of congruence to justify properties of figures and solve problems. The student is expected to: (B) justify and apply triangle congruence relationships. Textbook Classify triangles by side length and angles. Use theorems about isosceles and equilateral triangles. Create an image congruent to a given triangle. More congruence transformations in Chap 6 Use perpendicular bisectors to solve problems. Use angle bisectors to find distance relationships. Use medians and altitudes of triangles Find possible side lengths in a triangles. Use inequalities to make comparisons in two triangles. Resources 4.8 5.1 5.2 5.3 5.4 5.5 5.6 Key Vocabulary - circumcenter - scalene - incenter - isosceles - median - equilateral - centroid - equiangular - altitude - interior angles (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: - exterior angles - orthocenter (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the - congruent figures conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. - corresponding parts - right triangle (legs, hypotenuse) (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student - isosceles triangle (legs, vertex angle, Common Assessment is expected to : base, base angles) (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). - congruence transformation - midsegment Congruent Triangles (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is - coordinate proof Assessment expected to: - perpendicular bisector (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. - equidistant (B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, - point of concurrency ratios in similar figures and solids, and angle relationships in polygons and circles. (TAKS 6) Aug 7, 2007 Geometry Essential Units of Study 2007-008 05 EUS - Similarity (3 Weeks) Focus TEKS Content Description (G.5) Geometric patterns: uses a variety of representations Solve problems by writing and solving proportions. (No geometric mean.) to describe geometric relationships and solve problems. The student is expected to: (B) use numeric and geometric patterns to make Use proportions to solve geometry problems including scale. generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, Use proportions to identify similar polygons. and angle relationships in polygons and circles. (TAKS 6) Support TEKS (G.11) Similarity and the geometry of shape: applies the Use proportions with a triangle or parallel lines. concepts of similarity to justify properties of figures and solve problems.. The student is expected to: (A) use and extend similarity properties and transformations Perform dilations. Know how to use and determine scale factor in these to explore and justify conjectures about geometric figures. similarity transformations. Supplement with TAKS Objective 8. (TAKS 8) (B) use ratios to solve problems involving similar figures. (TAKS 8) Textbook Resources 6.1 6.2 6.3 6.6 6.7 (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: Vocabulary (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. - ratio - proportion (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student - scale factor is expected to : - scale drawing (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). - similar polygons - dilation (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is - center of dilation expected to: - reduction (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. - enlargement Common Assessment Ancient Ruins Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-008 06 EUS - Transformations (2 Weeks) Focus TEKS Content Description Represent translations using coordinate geometry. (No vectors.) (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (C) use properties of transformations and their compositions Reflect a figure in any given line on and off the coordinate plane (no to make connections between mathematics and the real matrices). Know line of reflection and symmetry. world, such as tessellations. (TAKS 6) Rotate a figure about a point concentrating on 90 , 180, 270, and 360 (G.10) Congruence and the geometry of size: applies the degrees, rotating about the origin or vertex of the figure. concept of congruence to justify properties of figures and solve problems. The student is expected to: (A) use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. (TAKS 6) Textbook Resources Vocabulary Common Assessment 9.1 9.3 9.4 Apply a combination of two or more transformations to form a composition of transformations. 9.5 Identify line and rotational symmetry of a figure. 9.6 (G.11) Similarity and the geometry of shape: applies the concepts of similarity to justify properties of figures and solve problems.. The student is expected to: (A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. (TAKS 8) Support TEKS (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to : (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). - image - preimage - isometry (G.7) Dimensionality and the geometry of location: understands that coordinate systems provide convenient and efficient ways - line of reflection of representing geometric figures and uses them accordingly . The student is expected to: - center of rotation (A) use one- and two-dimensional coordinate systems to represent points, lines, ray, line segments, and figures. (TAKS 7) - angle of rotation - line of symmetry - rotational symmetry - composition of transformations - glide reflection - tesselation Living Room Transformations Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-08 07 EUS - Right Triangles (4 Weeks) Focus TEKS (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Content Description 7.1 Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle. 7.2 Use similar right triangles to solve problems. (No geometric mean theorems, use similarity of triangles to solve). 7.3 (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The Use the relationships among the sides of a 30-60-90 and 45-45-90 special student is expected to: right triangles. (D) identify and applies patterns from right triangles to solve meaningful problems, including special right triangles (45-45- Apply the tangent ratio for indirect measurement of a right triangle. 90 and 30-60-90) and triangles whose sides are Pythagorean triples. (TAKS 6) Use the sine and cosine ratios to solve right triangle problems. (G.8) Congruence and the geometry of size: uses tools to determine measurements of geometric figures and extends Use inverse tangent, sine, and cosine ratios to solve right triangle problems. measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (C) G.8C Derive, extend, and use the Pythagorean Theorem. (TAKS 8) (G.11) Similarity and the geometry of shape: applies the concepts of similarity to justify properties of figures and solve problems.. The student is expected to: (B) use ratios to solve problems involving similar figures. (TAKS 8) (C) develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods. (TAKS 8) (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (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. Support TEKS Textbook Apply the Pythagorean Theorem to find side lengths in right triangles. (Operations with radicals review). (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to : (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). Resources 7.4 7.5 7.6 7.7 Key Vocabulary -Pythagorean triple - trigonometric ratio - opposite side - adjacent side - sine - cosine - tangent - angle of elevation - angle of depression - inverse sine - inverse cosine - inverse tangent Common Assessment Sightseeing Walk Assessment (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Aug 7, 2007 Geometry Essential Units of Study 2007-08 08 EUS - Quadrilaterals (3 Weeks) Focus TEKS Content Description (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Finding angle measures in polygons. (Review section 1.6 material - convex, concave, classifying polygons, regular, interior, exterior, etc…) 8.1 Use properties of parallelograms to find angle and side measures in parallelograms. 8.2 Use properties to identify/show that a quadrilateral is a parallelogram. (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The Use properties of rhombuses, rectangles, and squares to solve problems. student is expected to: (B) use numeric and geometric patterns to make generalizations about geometric properties, including Use properties/characteristics (isosceles, midsegment) of trapezoids to properties of polygons, ratios in similar figures and solids, solve problems. (No kites). and angle relationships in polygons and circles. (TAKS 6) Identify special quadrilaterals. (G.9) Congruence and the geometry of size: analyzes properties and describes relationships in geometric figures. The student is expected to: (B) formulates and tests conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. Support TEKS (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (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.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to : (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). Textbook Resources Vocabulary Common Assessment 8.3 8.4 8.5 8.6 - diagonal - parallelogram - rhombus - rectangle - square - trapezoid - bases - base angles - isosceles trapezoid - midsegment of a trapezoid Aug 7, 2007 Geometry Essential Units of Study 2007-08 09 EUS - Measuring Length and Area (3 Weeks) Focus TEKS Content Description (G.5) Geometric patterns: uses a variety of representations Find areas of triangles and Parallelograms. to describe geometric relationships and solve problems. The student is expected to: (B) use numeric and geometric patterns to make Find areas of trapezoids and rhombuses. (No Area of kite). generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, Find circumference and arc length of circles. and angle relationships in polygons and circles. (TAKS 6) (G.8) Congruence and the geometry of size: uses tools to determine measurements of geometric figures and extends Find areas of regular polygons inscribed in circles. measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: Find the areas of circles and sectors of circles. (A) find areas of regular polygons, circles, and composite figures. (TAKS 8). Use lengths and areas to find geometric probability. Use this section as (G.9) Congruence and the geometry of size: analyzes springboard for review of probability TEKS of TAKS Objective 9. properties and describes relationships in geometric figures. The student is expected to: (B) formulates and tests conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. Textbook Resources 11.1 11.2 11.4 11.6 11.5 11.7 Key Vocabulary Support TEKS (G.11) Similarity and the geometry of shape: applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: (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. (TAKS 8) (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (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.7) Dimensionality and the geometry of location: The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to: (A) use one- and two-dimensional coordinate systems to represent points, lines, ray, line segments, and figures. (TAKS 7) - bases of a parallelogram - height of a parallelogram - height of a trapezoid - circumference - arc length - sector of a circle - regular polygons - center of a polygon - radius of a polygon - apothem - central angle of a regular polygon - probability - geometric probability Common Assessment Aug 7, 2007 Geometry Essential Units of Study 2007-08 10 EUS - Surface Area and Volume of Solids (3 Weeks + 1 week TAKS review) Focus TEKS (G.1) Geometric structure: understands the structure of, and relationships within, an axiomatic system. The student is expected to: (C) compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (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. (TAKS 6) Content Description Textbook Explore solids (definitions such as polyhedra, face, edge, base, vertex, convex, concave, regular, etc…) 12.1 Find surface area of prisms and cylinders. Discuss nets and lateral versus total surface area. 12.2 Find the surface area of pyramids and cones. 12.3 Find the volume of prisms and cylinders. Include composite solids. "B" represents the area of the base. 12.4 Find the volume of pyramids and cones. 12.5 (G.6) Dimensionality and the geometry of location: analyzes Find the surface areas and volumes of spheres. Include definitions of great the relationship between three-dimensional geometric circle and hemisphere. Study of spherical geometry after TAKS. figures and related two-dimensional representations and uses these representations to solve problems. The student Use properties of similar solids. (optional) is expected to: Resources 12.6 12.7 Key Vocabulary (A) describe and draw the intersection of a given plane with various three-dimensional geometric figures (B) use nets to represent and construct three-dimensional objects. (TAKS 7) (C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems. (TAKS 7) Support TEKS - polyhedron - face - edge - vertex of a solid - cross section - prism (G.8) Congruence and the geometry of size: uses tools to determine measurements of geometric figures and extends measurement - surface area concepts to find perimeter, area, and volume in problem situations. The student is expected to: - lateral area (B) find areas of sectors and arc lengths of circles using proportional reasoning. (TAKS 8) - net (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem - right prism situations. (TAKS 8) - oblique prism - cylinder (G.9) Congruence and the geometry of size: analyzes properties and describes relationships in geometric figures. The student is - pyramid expected to: (D) analyzes the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and - regular pyramid - cone concrete models. (TAKS 7) - volume (G.11) Similarity and the geometry of shape: applies the concepts of similarity to justify properties of figures and solve problems. The stud - sphere (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- great circle - hemisphere Common Assessment (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: (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. Aug 7, 2007 Geometry Essential Units of Study 2007-08 11 EUS - Properties of Circles (3 Weeks) Focus TEKS (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (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. (TAKS 6) Content Description Textbook Use properties of a tangent to a circle. Include basic definitions, chords, secants, etc.. 10.1 Use angle measures to find arc measures. Include central angle, minor vs. major arc. 10.2 Use relationships of arcs and chords in a circle. 10.3 (G.7) Dimensionality and the geometry of location: The Use inscribed angles of circles and polygons. student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected Find the measure of angles with vertices inside, outside, and on the circle. to: (B) Use slopes and equations of lines to investigate geometric relationships, including parallel lines, Find segment lengths in circles. perpendicular lines, and special segments of triangles and other polygons. (TAKS 7) Write and graph equations of circles in a coordinate plane. Resources 10.4 10.5 10.6 10.7 (G.8) Congruence and the geometry of size: uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures. (TAKS 8). (B) find areas of sectors and arc lengths of circles using proportional reasoning. (TAKS 8) Support TEKS (G.9) Congruence and the geometry of size: analyzes properties and describes relationships in geometric figures. The student is expected to: (C) Formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models. Key Vocabulary - circle - center - radius - diameter (G.2) Geometric structure: analyzes geometric relationships in order to make and verify conjectures.. The student is expected to: - chord (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the - secant conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. - tangent - central angle (G.4) Geometric structure: uses a variety of representations to describe geometric relationships and solve problems. The student - minor arc is expected to : - major arc (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. (TAKS 6). - semicircles - congruent circles (G.5) Geometric patterns: uses a variety of representations to describe geometric relationships and solve problems. The student is - congruent arcs expected to: - inscribed angle (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties. - intercepted arc - standard equation of a circle Common Assessment Aug 7, 2007