Geometry Skills Topic Outline Course Description and Philosophy Geometry Skills is the second course in the 3-year skills sequence, following Algebra Skills, and preceding Algebra II Skills. This course will cover the basics of geometry: defining, classifying, and measuring points, lines, planes, angles, triangles, quadrilaterals, polygons, circles and solid figures. Properties and theorems of congruence, equality, and similarity will be explored. Proofs and logical reasoning are interwoven in the lesson structure, introducing conjectures, counterexamples, theorems, corollaries, hypotheses, and postulates. Some time is devoted to transformations, touching on translations, rotations, dilations, and reflections. Trigonometry will be introduced, covering sine, cosine, tangent, the inverse ratios, and special right triangles. Other topics include circles, arcs, chords, and their theorems. The course is primarily hands-on and focuses on using geometry and algebra in real contexts to establish a value for the use of algebraic and geometric logic, reasoning and processes. This course is intended for college bound students who have completed Algebra Skills or Algebra. Evaluation of students performance will include various types of assessment strategies such as teacher-generated quizzes, homework, class participation, and occasional lab work. Text Reference: Larson, Boswell and Stiff, Geometry Concepts and Skillls, copyright 2003 by MacDougal Littell Inc, Evanston, Illinois Written 2007 1
Unit I: Geometry Basics (Points, lines, planes, angles, segments) Essential Questions: Why are rules necessary? Objectives: Students will be able to: Sketch points, lines, planes, and their intersections Measure segments and add their lengths Measure angles and classify them based on their measures Analyze segment bisectors and angle bisectors Identify complementary angles, supplementary angles, vertical angles, and linear pairs Use the properties of equality and congruence to justify mathematical statements Identify relationships between the angles formed by two lines and a transversal, especially when the two lines are parallel Use properties of parallel and perpendicular lines to find angle measures Identify the angle relationships necessary for showing that two lines are parallel 2
Unit I: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit I Topic 1: Patterns and Reasoning Test/Quizzes Text Lectures Geometer s Sketchpad Homework Manipulatives Hands-On Activities Class Participation Lab Work Projects Finding and Describing Patterns Undefined terms Conjectures Counterexamples Postulate Theorem If-then Hypothesis, conclusion Converse Corollary Topic 2: Points, Lines, and Planes Collinear, coplanar Segment, ray,endpoint Intersections Parallel, Perpendicular Skew Construction Topic 3: Segments Coordinates Distance Length Congruent Segments Bisection Topic 4: Angles Sides, vertex Measure, Degree Congruent angles Acute, right, obtuse, and straight angle Bisection Supplementary, Complementary Adjacent, Vertical, Linear Pair 4.2A1,3,4 4.2C1 4.2D1,2 4.3A3 4.5A1,2,3,4 4.5B1,2,4 4.5C 1,2,3,4 4.5D1,2,3,5 4.5F5 Topic 5: Properties of Equality and Congruence Reflexive 3
Symmetric Transitive Addition, subtraction, multiplication, division Substitution Topic 6: Transversals Corresponding angles Alternate interior Alternate exterior Same-side interior Differentiated Learning Activities Strategic learner: Review basic components of coordinate plane and how to plot points in a plane. Break problems into discrete steps. Verbalize reasoning process. Provide concrete examples furniture design, construction, jewelry design. Model transversal angle pairs with students and ropes or tape. Advanced learner: Have student create patterns of their own, with more than one characteristic changing. Make connections with other math topics. Use challenge problems Pascal s triangle, arrange 10 points in a plane in five lines with four points per line, etc. Ethical Decision Making/Character Education Paper clip game motivates the existence and importance of rules in society, school, and technology use (achieve order, protect property, and ensure privacy). Game acts as springboard for discussing ethics, and relevant case studies exemplifying ethical issues. Fits in with essential question of why we need rules in mathematics. 4
Unit II: Triangles Essential Question: What does it mean to say two triangles are congruent? What about similar? Objectives: Students will be able to: Classify triangles according to angle measures and side lengths Use the Triangle Sum Theorem, the Base Angles Theorem, the Pythagorean Theorem, and the Triangle Inequality Theorem Discover the relationship between the medians of a triangle and its centroid Identify corresponding parts of congruent triangles Show triangles are congruent using the SSS, SAS, and ASA Congruence Postulates, and the AAS and HL Congruence Theorems Use Angle bisectors and perpendicular bisectors to compute angle measures and segment lengths in situations involving triangles Reflect figures over lines and use reflections to discover lines of symmetry in a figure Use ratios and solve proportions, especially as they relate to similar polygons Identify similar polygons, and use postulates and theorems to show that two triangles are similar Identify and draw dilations of polygons Identify rotational symmetry and rotations in a plane 5
Unit II: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit II Topic 1: Classify Triangles Test/Quizzes Text Lectures Geometer s Sketchpad Homework Manipulatives Hands-On Activities Class Lab Work Participation Sides, angles, vertices Legs, base, base angles Equilateral, isosceles, scalene Equiangular, acute, right, obtuse Interior angles, exterior angles Medians, centroid Topic 2: Triangle Theorems Pythagorean, Distance Formula Converse of Pythagorean Intersection of Medians Triangle Inequality Triangle Proportionality Theorem and converse Midsegment Theorem Angle Bisector Theorem Perpendicular Bisector Theorem Topic 3: Congruence Corresponding parts SSS, SAS ASA, AAS Hypotenuse-Leg Proofs Topic 4: Transformations Translations Reflections Line of Symmetry Dilations Rotations Topic 5: Similarity Ratios and Proportions Scale AA Similarity Postulate SSS SAS 4.2A3,4 4.2B1 4.2E1 4.5A1,2,3,4 4.5B1,2,4 4.5C 1,2,3,4 4.5D1,2,3,5 4.5F5 Differentiated Learning Activities Strategic learner: Use visual models. Verbalize thinking process when solving problems. Have students tell you how to do the problems. Emphasize difference between segments and their measures. Clearly demonstrate different naming conventions. Review congruent versus similar. 6
Advanced learner:.make cross-curricular connections. Have students write own word problems. Use challenge problems -- Garfield s Pythagorean proof, tiling the plane, finding simpler figures imbedded in irregular figures, Ethical Decision Making/Character Education: Before the midterm, review standards for academic integrity. Have general discussion of cheating, have groups analyze various scenarios. 7
Unit III: Polygons, Circles, and Solid Figures Essential Question: How are polygons structured and represented? Objectives: Students will be able to: Find angle measurements of quadrilaterals Identify special quadrilaterals Use the properties of parallelograms, rhombuses, rectangles, squares, and trapezoids to find their side lengths and angle measures Investigate the midsegment of a trapezoid Find measures of interior and exterior angles of polygons Find the area of squares, rectangles, triangles, parallelograms, and trapezoids Find the circumference and area of circles Identify and name solid figures Find the volume of prisms, cylinders, pyramids, cones, and spheres 8
Unit III: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit III Topic 1: Classify Polygons Test/Quizzes Text Lectures Geometer s Sketchpad Homework Manipulatives Hands-On Activities Class Lab Work Participation Sides, vertices, diagonals Trangles, quadrilaterals, pentagons, hexagons, heptagons, and octagons Parallelograms Rhombus, Rectangles, Squares Concave, convex Equilateral, equiangular regular Topic 2: Quadrilateral Theorems Quadrilateral Interior Angles Parallelogram Opposite Sides Parallelogram Opposite Angles Parallelogram Consec Angles Converses Bisecting Diagonals Topic 3: Trapezoids bases, legs, base angles isosceles trapezoid midsegment of trapezoid Base Angle theorem Topic 4: Polygon Theorems Interior Angles Exterior Angles Topic 5: Area Squares Rectangles Complex Polygon Trangles Parallelograms Trapezoids Circles Topic 6: Solid Figures 4.2A1,3 4.2B2 4.2E2 4.5A1,2,3,4 4.5B1,2,4 4.5C 1,2,3,4 4.5D1,2,3,5 4.5F5 polyhedron rectangular prism triangular pyramid cylinder cone 9
sphere faces, edges platonic solids Topic 7: Volume prism cylinder pyramid cone sphere Differentiated Learning Activities Strategic learner: Play the why game, use graph paper to help understand measuring area in square units. provide cardboard models of plane figures. make scale drawing of house, estimate carpet square yardage, use unit cubes for volume, fill small cardboard boxes. Advanced learner: Investigate fractals, keep record of unanswered questions, discuss cash prizes for famous unsolved math problems, connect algebraic mean with geometric mean, challenge problems. 10
Unit IV: Trigonometry and Circles Essential Question: How can we use trigonometry to measure tall buildings and mountains? Objectives: Students will be able to: Simplify radicals Find the sine, cosine, and tangent rations of the acute angles in a right triangle Identify special right triangles Solve right triangles identify parts of a circle, such as arcs, and study their properties Identify inscribed angles and intersecting chords in circles, and use the properties related to them 11
Unit IV: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit IV Topic 1: Radicals Test/Quizzes Text Lectures Geometer s Sketchpad 4.1A1,2 Homework Manipulatives Hands-On Activities 4.2A3,4 Class Lab Work 4.2E1 Participation 4.5A1,2,3,4 simplify square roots multiply radicals find distances Topic 2: Special Triangles 45 45 90 triangles 30 60 90 triangles Topic 3: Trigonometric ratios opposite, adjacent tangent sine cosine inverse tangent inverse sine inverse cosine Topic 4: Circles chord diameter radius secant tangent arcs, minor and major semicircle inscribed angles intercepted arc circumscribed Topic 5: Circle Theorems Perpendicular Tangent Converse Congruent tangent segments Arc addition Arc length Perpendicular chord bisector Arc chord congruency Measure of inscribed angle Inscribed right triangle diameter Inscribed quadrilatieral Intersecting chord angles Intersecting chord segments 12 4.5B1,2,4 4.5C 1,2,3,4 4.5D1,2,3,5
Differentiated Learning Activities Strategic learner: Practice with graphing calculator. Try to memorize perfect squares. Advanced learner: Use challenge problems in groups, research Leaning Tower of Pisa. 13