Pacing Guide Geometry Year Long Robert E. Lee High School Staunton City Schools Staunton, Virginia June 2006
Geometry Semester I (1st 9-weeks) Week(s) Organizing Topics Related Standards of Learning Inverse, Converse, Contrapositive Statements 1-2 1.) Logic and Law of Syllogism and Valid Arguments Reasoning Venn Diagrams Inductive and Deductive Reasoning 3 4-4.5 2.) Basics of Geometry 3) Bisection and Angles Points, Lines and Planes Intersections Acute, Right, Obtuse, Straight Angles Segment and Angle Addition Postulates Congruency and Bisection Complementary and Supplementary Angles Vertical Angles 4.5 1 st Benchmark Test Common Assessment for all Geometry Year Long Classes (18q s: Wk 1-4.5) 5-7 8-9 4) Parallel and Perpendicular Lines 5) Coordinate Geometry Parallel, Perpendicular and Skew Properties of Perpendicular Angle Relationships of parallel cut by transversal Showing lines are parallel Parallel and Perpendicular Slopes Midpoint of two ordered pairs Distance Formula 9 2 nd Benchmark Test Common Assessment for all Geometry Year Long Classes (39q s: Wk 1-9)
Geometry Semester I (2 nd 9-weeks) 10-11 12-13.5 6.) Triangle Relationships 7.) Congruent Triangles Classifying Triangles Angle Measures Properties of Isosceles and Equilateral Pythagorean Theorem and Converse Triangle Congruence SSS and SAS ASA, AAS, HL Overlapping Triangles Proof and Properties 13.5 3 rd Benchmark Test Common Assessment for all Geometry Year Long Classes (21q s: Wk 1-13.5) 14-16 8) Similar Triangles 17-18 9) Right Triangles and Trigometry Ratios and Proportions AA Similarity SSS and SAS Similarity Similar Proportions Application of Similarity Simplifying Square Roots Special Right Triangles Tangent Ratios Sine and Cosine Ratios Solving Right Triangles 18 Mid-Term Exam Common Assessment for all Geometry Year Long Classes (60q s: Wk 1-18)
Geometry Semester II (3rd 9-weeks) 1-1.5 1.) Polygons 1.5-4.5 2)Quadrilaterals Polygon Classification Similar Polygons Angles in Polygons Angles of Regular Polygons Paralellogram Properties Rhombuses, Rectangles and Squares Trapezoids and Kites 4.5 4 th Benchmark Test Common Assessment for all Geometry Year Long Classes (24 q s: Wk 1-22.5) 5-7 3) Cirlcles 8-9 4.) Area and Volume Parts of a Circle Properties of Tangents Central, Inscribed and Other Angles of a Circle Properties of Arcs and Chords Properties of Secants Area of Squares, Rectangles and Parallelograms Area of Triangles, Regular Polygons Area of Trapezoids and Circles Volume of Prisms and Pyramids Volume of Cylinders, Cones and Spheres 9 5 th Benchmark Test Common Assessment for all Geometry Year Long Classes (45 q s: Wk 1-27)
Geometry Semester II (4 th 9-weeks) 1-2.5 2.5-4.5 5.) Solid Figures and Surface Area 6.) Transformations and Constructions Solid Figures Nets and 3-D Visualization Surface Area of Prisms and Pyramids Surface Area of Cylinders, Cones and Spheres Translations, Reflections and Rotations Construction of Bisectors Construction of Perpendicular and Parallel Copying an Angle Tesselations 4.5 6 th Benchmark Test Common Assessment for all Geometry Year Long Classes (45 q s: Wk 1-31.5) 4.5-??-9 Full Year Review and SOL preparation Project or other end of year activities Projects vary, but must be weighted the Same, according to consensus of Geometry Teachers 9 FINAL EXAM Common Assessment for all Geometry Year Long Classes (60 q s: Wk 1-36)
Blueprint Summary Table Reporting Categories No. of SOL Items Lines and Angles 11 G.3 G.4 G.11 Triangles and Logic 12 G.1a,b,c,d G.5a,b G.6 G.7 Polygons and Circles 10 G.8a,b,c G.9 G.10 Three-Dimensional Figures 6 G.12 G.13 G.14a,b Coordinate Relations and Transformations 6 G.2a,b,c Total Number of Operational Items 45 Field Test Items* 10 *These field test items will not be used to compute students scores on the test Total Number of Items 55
Part II-2: Quadrilaterals (Ch 6) Week(s) 4.1-4.2 G.8 a investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals; G.8.b prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; G.8.c use properties of quadrilaterals to solve practical problems. Unit Projects/ Activities/ Resources Geo Sketchpad Resource: M:\Instructor\Nussbaum\Quadrilaterals.gsp Proofs: Z:\Math\Geometry\Geometry Part II\Proofs Lessons Strategies/Activities to Engage the Learner Make sure you 6.1 Polygons 6.2 Properties of Parallelograms Define parallelogram. Use inductive reasoning to make conjectures about the properties of parallelograms. Discovering Activity: Z:\Math\Geometry\Other Geo Stuff\Geometry notes from Kevin\Notes\Notes 9.1a.doc Notes: Include additional problems with algebraic expressions for angles and segments (see Prentice Hall p458) 6.3 Showing Quadrilaterals are Parallelograms Prove the properties of parallelograms using deductive arguments as well as algebraic or coordinate methods. Geometry Part II\notes\Ch 9\Notes 9.2.doc Proofs: (see above) Verify the converses of the properties of parallelograms and use the converses to prove that a quadrilateral is a parallelogram. Coordinate methods may involve using slope to show that lines are parallel or perpendicular.
Lessons 6.4 Rhombuses, Rectangles, and Squares 6.5 Trapezoids 6.6 Reasoning About Special Quadrilaterals Identify rhombi, squares, and rectangles as special parallelograms and prove their properties using deductive arguments as well as algebraic and coordinate methods. Week(s) 4.1-4.2 Strategies/Activities to Engage the Learner Make sure you (see above)
Part I-9: Trigonometry (Ch 10) Week(s) 4.1-4.2 G.7 The student will solve practical problems involving properties of special right triangles, and right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations Unit Projects/ Activities/ Resources Fun handout: Z:\Math\Geometry\Honors Geometry Fall 2005\Notes_Handouts\Ch 8\sohcahtoa.doc Interactive website: http://www.explorelearning.com Trig notes in grid: Z:\Math\Geometry\Honors Geometry Fall 2005\Notes_Handouts\Ch 8\Trig Notes.doc Trig grid: Z:\Math\Geometry\Geometry Part II\notes\Ch 11\Trigonometry Grid.doc Lessons 10.1 Simplifying Square Roots 10.2 45 o -45 o -90 o Triangles 10.3 30 o -60 o -90 o Triangles Investigate the side lengths of isosceles right triangles and 30-60-90 triangles. Use inductive reasoning to conjecture about the relationships among the side lengths. Use the properties of special right triangles to solve practical problems. Use a calculator to find decimal approximations of solutions. 10.4 Tangent Ratio Define sine, cosine, tangent as trigonometric ratios in a right triangle. 10.5 Sine and cosine Ratios Discuss exact values for trigonometric ratios and decimal approximations. Strategies/Activities to Engage the Learner Simplifying Radicals: Z:\Math\Geometry\Honors Geometry Fall 2005\Notes_Handouts\simplifying radicals.doc Note: Geometry Part I\Accel Geo Pt. I\notes for notebooks\ch 5\Notes 5.4.doc Notes: \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part II\notes\Ch 11\Itentifying Trig sides.doc \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part Make sure you don t skip this section Go slow it will take awhile for students to catch on Cover this well, it can tie in with Trig and Pyth. Thm. later.
Lessons 10.6 Solving Right Triangles Use right triangle trigonometry to solve right triangles. Strategies/Activities to Engage the Learner II\notes\Ch 11\Notes 11.1.doc \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part II\notes\Ch 11\Notes 11.2.doc \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part II\notes\Ch 11\Notes 11.3.doc \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part II\notes\Ch 11\Notes 11.6.doc \\Leemen\teachers\Teacher Resources\Math\Geometry\Geometry Part II\notes\Ch 11\PRACTICE WITH TRIGONOMETRY.doc Week(s) 4.1-4.2 Make sure you
Week(s) 4.1-4.2 Part I-3,4,5: Tools of Geometry (Coordinate Relations and Transformations/ Lines and Angles) G.2 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope G.3 The student will solve practical problems involving complementary, supplementary, and congruent angles that include vertical angles, angles formed when parallel lines are cut by a transversal, and angles in polygons. Related SOL Mastery*(See Notes) G.3 Identify types of angle pairs: complementary angles supplementary angles vertical angles linear pairs of angles alternate interior angles consecutive interior angles corresponding angles. Use inductive reasoning to determine the relationship between complementary angles, supplementary angles, vertical angles, and linear pairs of angles. G.2a Define and identify parallel lines. Find the slope of a line given the graph of the line, the equation of the line, or the coordinates of two points on the line. Investigate the relationship between the slopes of parallel lines.
Week(s) 4.1-4.2 Part I-5,6: Tools of Geometry (Coordinate Relations and Transformations/ Triangles and Logic) G.2 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations. Related SOL Mastery*(See Notes) Triangl es Find the coordinates of the midpoint of a line segment. G.2 a G.7 Verify the Pythagorean Theorem and its converse using deductive arguments as well as algebraic and coordinate methods. Solve practical problems involving the Pythagorean Theorem and its converse. Use a calculator to find decimal approximations of solutions. G.2 a Use the Pythagorean Theorem to derive the distance formula. Use the distance formula to find the length of line segments when given the coordinates of the endpoints.
Part I-6, 9: Tools of Geometry (Triangles and Logic) Week(s) 4.1-4.2 G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations. Relat ed SOL Mastery*(See Notes) Tria ngles G.7 Investigate the side lengths of isosceles right triangles and 30-60-90 triangles. Use inductive reasoning to conjecture about the relationships among the side lengths. Use the properties of special right triangles to solve practical problems. Use a calculator to find decimal approximations of solutions. Define sine, cosine, and tangent as trigonometric ratios in a right triangle. Discuss exact values for trigonometric ratios and decimal approximations. Use right triangle trigonometry to solve right triangles. Use right triangle trigonometry to solve practical problems. Use a calculator to find decimal approximations of solutions.
Week(s) 4.1-4.2 Part I-7: Investigating Geometric Figures (Triangles and Logic) G.5 The student will a) investigate and identify congruence and similarity relationships between triangles; and b) prove two triangles are congruent or similar, given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs. Related SOL Mastery*(See Notes) Triang les G.5 a b Investigate and identify congruent figures. Define congruent figures. Map corresponding parts (angles and sides) of congruent figures onto each other. Discuss applications of congruence such as rubber stamps, manufacturing, and patterns. Understand the structure of Euclidean geometry: undefined terms defined terms postulates theorems. Verify that triangles are congruent using the following postulates: side-angle-side (SAS) angle-side-angle (ASA) side-side-side (SSS) angle-angle-side (AAS) hypotenuse-leg (HL).
Week(s) 4.1-4.2 Part I-6,7: Investigating Geometric Figures (Triangles and Logic) G.5 The student will a) investigate and identify congruence and similarity relationships between triangles; and b) prove two triangles are congruent or similar, given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs. Related SOL Mastery*(See Notes) Trian gles G.5 a b Plan proofs. Write deductive arguments as well as coordinate and algebraic demonstrations that triangles are congruent. Use the definition of congruent triangles (corresponding parts of congruent triangles are congruent) to plan and write proofs. Explore the constraints on the lengths of the sides of a triangle to develop the triangle inequality. Use the triangle inequality to determine if three given segment lengths will form a triangle.
Week(s) 4.1-4.2 Part II-1: Investigating Geometric Figures (Polygons and Circles) G.9 The student will use measures of interior and exterior angles of polygons to solve problems. Tessellations and tiling problems will be used to make connections to art, construction, and nature. G.2 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include c) determining whether a figure has been translated, reflected, or rotated. Relat ed SOL Mastery*(See Notes) G.9 Identify, name, and classify polygons. G.2 c Examine pre-image and image figures in the coordinate plane and determine whether a translation, reflection, or rotation has occurred. G.9 Use inductive reasoning to develop a formula for finding the sum of the measures of the interior angles of a convex polygon and the measure of each interior angle of a regular polygon. Investigate the sum of the measures of the exterior angles of any convex polygon and the measure of each exterior angle of a regular convex polygon. Use tessellations and tiling problems to make connections to art, architecture, construction, and the sciences.
Week(s) 4.1-4.2 Part II-2, 6: Investigating Geometric Figures (Polygons and Circles) G.2 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include b) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point G.8 The student will a) investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals; b) prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and c) use properties of quadrilaterals to solve practical problems. Related SOL Mastery*(See Notes) Other Polygons G.2 c Investigate symmetry. Determine if a geometric figure has point symmetry, line symmetry, or no symmetry, and justify the conclusion. G.8 Define parallelogram. Use inductive reasoning to make conjectures about the properties of parallelograms. Prove the properties of parallelograms using deductive arguments as well as algebraic or coordinate methods. Verify the converses of the properties of parallelograms and use the converses to prove that a quadrilateral is a parallelogram. Coordinate methods may involve using slope to show that lines are parallel or perpendicular. Identify rhombi, squares, and rectangles as special parallelograms and prove their properties using deductive arguments as well as algebraic and coordinate methods.
Part II-2: Investigating Geometric Figures (Polygons and Circles) Week(s) 4.1-4.2 G.8 The student will a) investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals; b) prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and c) use properties of quadrilaterals to solve practical problems. G.14 The student will a) use proportional reasoning to solve practical problems, given similar geometric objects; and b) determine how changes in one dimension of an object affect area and/or volume of the object. Related SOL Mastery*(See Notes) Other Polygons G.8 Identify trapezoids and isosceles trapezoids. Prove their properties using deductive arguments as well as algebraic and coordinate methods. Identify kites and prove their properties using deductive arguments as well as algebraic and coordinate methods. Use the properties of parallelograms and other quadrilaterals to solve practical problems. G.14 Given the areas of similar geometric figures, investigate the effect on the constant of proportionality of changing one dimension (multiplying by a constant). Generalize the change and use the generalization to solve practical problems.
Week(s) 4.1-4.2 Part II-3: Investigating Geometric Figures (Polygons and Circles) G.10 The student will investigate and solve practical problems involving circles, using properties of angles, arcs, chords, tangents, and secants. Problems will include finding arc length and the area of a sector, and may be drawn from applications of architecture, art, and construction. Relate d SOL Mastery*(See Notes) Circles G.10 Define the term circle. Differentiate between a circle and a circular region. Use the vocabulary associated with circles. Describe ways that circles intersect. Explore and state properties of tangents. Measure central angles, inscribed angles, and arcs of circles directly and indirectly Generalize the relationship between angle measure and arc measure.
Week(s) 4.1-4.2 Part II-3: Investigating Geometric Figures (Polygons and Circles) G.10 The student will investigate and solve practical problems involving circles, using properties of angles, arcs, chords, tangents, and secants. Problems will include finding arc length and the area of a sector, and may be drawn from applications of architecture, art, and construction. Relate d SOL Mastery*(See Notes) Circles G.10 Investigate properties of chords and arcs of circles. Measure angles formed by tangents, chords, and secants directly and indirectly. Generalize the relationship between angle measure and arc measure. Find the area of a sector and the area of a segment of a circle. Use the properties of angles, arcs, segments, and lines associated with circles to solve practical problems involving circles. Look at applications in art, construction, architecture, and the sciences.
Part II-6: Constructions (Lines and Angles) Week(s) 4.1-4.2 G.11 The student will construct a line segment congruent to a given line segment, the bisector of a line segment, a perpendicular to a given line from a point not on the line, a perpendicular to a given line at a point on the line, the bisector of a given angle, and an angle congruent to a given angle. Related SOL Mastery*(See Notes) G.11 Using a compass and straightedge only, construct the following: a line segment congruent to a given segment an angle congruent to a given angle the bisector of a given angle a perpendicular to a given line from a point not on the given line a perpendicular to a given line at a point on the given line.
Week(s) 4.1-4.2 Part I-6,8: Triangle Relationships (Triangles and Logic) G.6 The student, given information concerning the lengths of sides and/or measures of angles, will apply the triangle inequality properties to determine whether a triangle exists and to order sides and angles. These concepts will be considered in the context of practical situations. G.5 The student will a) investigate and identify congruence and similarity relationships between triangles; and b) prove two triangles are congruent or similar, given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs. Relate d SOL G.6 Explore the relationship between the angle measures in triangles and the lengths of the sides opposite those angles Given side lengths in a triangle, identify the angles in order from largest to smallest or vice versa. Given angle measures in a triangle, identify the sides in order from largest to smallest or vice versa. Use indirect proof (proof by contradiction) to argue that all but one possible case in a given situation is impossible. Use properties of proportions to solve practical problems. G.5 Investigate and identify similar polygons. Define similar polygons. Use the following postulates to verify that triangles are similar. Deductive arguments as well as algebraic and coordinate methods may be used. angle-angle (AA) side-angle-side (SAS) side-side-side (SSS) Mastery*(See Notes)
Part II-4,5: Measuring in the Plane, Measuring in Space (Three-Dimensional Figures) Week(s) 4.1-4.2 G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to find decimal approximations for results. Relate d SOL Mastery*(See Notes) G.13 Use the appropriate formulas to find the surface area of cylinders, prisms, pyramids, cones, and spheres. Use the appropriate formulas to calculate the volume of cylinders, prisms, pyramids, cones, and spheres. Solve practical problems involving surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures.
Week(s) 4.1-4.2 Part I-4: Reasoning and Parallel Lines (Lines and Angles) G.3 The student will solve practical problems involving complementary, supplementary, and congruent angles that include vertical angles, angles formed when parallel lines are cut by a transversal, and angles in polygons. G.4 The student will use the relationships between angles formed by two lines cut by a transversal to determine if two lines are parallel and verify, using algebraic and coordinate methods as well as deductive proofs. Related SOL Mastery*(See Notes) G.3 Explore the relationship between alternate interior angles, consecutive interior angles, and corresponding angles when they occur as a result of parallel lines being cut by a transversal. State these angle relationships as conditional statements. Solve practical problems involving these angle relationships. G.4 Use the converses of the conditional statements about the angles associated with two parallel lines cut by a transversal to show necessary and sufficient conditions for parallel lines. Verify the converses using deductive arguments, coordinate, and algebraic methods.
Part II-5: Reasoning and Parallel Lines (Three-Dimensional Figures) G.12 The student will make a model of a three-dimensional figure from a two-dimensional drawing and make a two-dimensional representation of a three-dimensional object. Models and representations will include scale drawings, perspective drawings, blueprints, or computer simulations. Relate d SOL Mastery*(See Notes) G.12 Use properties of three-dimensional objects to make models. Make a model of a three-dimensional figure from a two-dimensional drawing. Make a two-dimensional representation of a three-dimensional object. Use scale drawings, perspective drawings, blueprints, or computer drawings as models of three-dimensional objects to solve problems. Identify a three-dimensional object from different positions such as the top view, side view, and front view.
Part II-4,5: Reasoning and Parallel Lines (Three-Dimensional Figures) G.14 The student will a) use proportional reasoning to solve practical problems, given similar geometric objects; and b) determine how changes in one dimension of an object affect area and/or volume of the object. Related SOL Mastery*(See Notes) G.14 Use proportions to compare surface area and volumes of three-dimensional geometric figures. Describe how a change in one measure affects other measures of an object. Solve practical problems involving similar objects. Additional Topics for Geometry Honors Prove with constructions and coordinate methods: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Investigate concurrency properties in triangles through paper folding, and form a conjecture. Perform and prove the concurrency constructions using only a straightedge and compass. angle bisectors and the incenter of the triangle perpendicular bisectors of the sides of the triangle and the circumcenter of the triangle medians from each vertex of the triangle and the centroid of the triangle altitudes to each side of the triangle from the opposing vertex and the orthocenter of the triangle.
Resources The Geometry Center http://www.umn.edu/ NASA http://spacelink.nasa.gov/.index.html The Math Forum http://forum.swarthmore.edu/ 4teachers http://www.4teachers.org Appalachia Educational Laboratory (AEL) http://www.ael.org/pnp/index.htm Eisenhower National Clearinghouse http://www.enc.org/ Geometry Instructional Modules The Geometry Center http://www.umn.edu/ NASA http://spacelink.nasa.gov/.index.html The Math Forum http://forum.swarthmore.edu/ 4teachers http://www.4teachers.org Appalachia Educational Laboratory (AEL) http://www.ael.org/pnp/index.htm Eisenhower National Clearinghouse http://www.enc.org/ Mathematics SOL Teacher Resource Guide http://www.pen.k12.va.us/vdoe/instruction/math_resource.html SOL Test Blueprints SOL Test Released Items