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Geometry Course Title: Grade Range: Geometry Grade 8 [Accelerated students], 9 [Level 5 students], and 10 [Levels 2, 3, and 4 students] Length of Course: One Year (5 credits) Prerequisites: Algebra Description: "Mathematics is not about answers, it's about processes." (Robert H. Lewis, mathematician) This course is designed to enable students to develop the logical reasoning that is the foundation of mathematical proof. A primary goal of this Geometry curriculum is to frame experiences that enable students to develop Geometric Habits of Mind. According to Driscoll, these habits include: reasoning with relationships, generalizing geometric ideas, investigating invariants, and balancing exploration and reflection. The mathematical content is delineated in learning objectives and content outline. Objectives are coded to demonstrate alignment with the NJ Core Curriculum Content Standards, and include all topics that students should learn in preparation for HSPA and the SATs. Evaluation: Student performance will be measured using a variety of assessment instruments, including computer-based investigations, classroom labs, projects and challenges, and paper and pencil assessments. These will include instructor-generated quizzes and text-generated and related software-based tests as well as a common departmental Midterms and Final Exams. Assessments will reflect the balance between process and content, concepts, and skills, with the overarching development of reasoning threaded throughout. Scope and Sequence: Unit sequencing is designed to ensure that students at every level have the opportunity to learn the essential concepts and skills. A pacing guide for Levels 2 through 5 identifies the common priorities and specific distinctions in content development. The teams will adequately pace the course so that all the material necessary to achieve the goal is taught. Text: Discovering Geometry (Key Curriculum Press, 2008) 4 May 2017 1 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 1. Develop definitions of geometric terms using visual representations and written descriptions. NJCCC 4.2 A 3, 4 Content Outline Concepts/Reasoning: 1. Connect visual diagrams with written descriptions 2. Distinguish between examples and counterexamples (Counterexamples are important for inductive reasoning) 3. Identify the characteristics of a good definition (Lays a foundation for the properties aspect of proof) 4. Use graphic organizers to relate and distinguish geometric terms and models (Shifting the emphasis from definitions of whole figures and directing students’ attention to components of figures) Skills: 1. Classify lines, angles, and shapes 2. Measure segments and angles with geometric tools 3. Find the measure of complementary and supplementary angles 4. Find the measure of angles formed by intersecting lines Instructional Materials Text: DG: Ch. 1 Printed Materials: Investigations: Math Models Virtual Pool Triangles Special Quads Def. Circles Conjectures / Notes Develop these approaches, and then guide the development of subsequent terms using: Beginning Steps to Create a Good Definition (p48 ) and Things you may assume:/ Things you may not assume (p59) Sketchpad DG labs Exploring Geometry p143-4; Appendix A Labs: 1, 2, 3, 4, 5 Supplies: rulers, protractors, patty paper, compass, straightedge 2 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 2. Use inductive reasoning to identify patterns and solve problems. Use deductive reasoning to justify conclusions. Content Outline Concepts/Reasoning: 1. Make conjectures (Recognize the importance of the inductive process in conjecture formulation) 2. Determine if a conjecture is true (Give deductive arguments for the truth of conjectures) 3. Generalize number or picture patterns 4. Write a converse of a statement and determine if it is true. 5. Write a deductive argument NJCCC 4.2 A 3, 4 Skills: 1. Identify vertical angles and linear pairs 2. Determine the relationship of angles formed by a transversal cutting parallel lines: a. Identify relationships between lines. b. Identify angles formed by a transversal c. Find congruent angles formed when a transversal cuts parallel lines 3. Determine traversable networks (p.120) Concept Check: When two parallel lines are cut by a transversal, which angles are supplementary and which angles are congruent? Instructional Materials Text: DG: Ch. 2 Printed Materials: Conjectures / Notes C-1 Linear Pair p 122 C-2 Vertical Angles p. 123 Investigations: C-3 Parallel Lines p.129: Party Handshakes a. Corresponding Angles Overlap. Segments b. Alt. Interior Angles Angle Relationships c. Alt. Exterior Angles Is the converse true? Exploration: C-4 Converse of Parallel Lines 7 Bridges of Konigsberg* Sketchpad:DG labs Exploring Geometry p 17-18; Appendix A Labs: 6, 7 Supplies: patty paper, compass, straightedge 3 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 3. Make conjectures based on investigations using geometric constructions. NJCCC 4.2 A 3, 4, 5 Content Outline Concepts/Reasoning: 1. Recognize, match, identify, and construct drawings for conjectures. (Give deductive arguments for the truth of conjectures) 2. Distinguish examples from non-examples of specified constructions. (Constructions raise the level of abstraction since the focus shifts from a specific example to all possible examples) 3. Make conjectures related to the effect of a change in an angle or side on the points of concurrency. Skills: 1. Develop skills using a straightedge, compass, patty paper, and geometric software. 2. Construct segments, angles, midpoints and points of concurrency 3. Bisect a segment 4. Find the coordinate of the midpoint of a segment 5. Bisect an angle 6. Identify the medians in a triangle. 7. Use triangle measurements to decide which side is the longest and which angle is the largest Concept Check: What is the relationship between the median of a triangle and the triangle’s centroid? Instructional Materials Text: DG: Ch. 3 Printed Materials: Investigations: Duplicating Lines Duplicating Angles Bisectors Concurrence Circumcenter Incenter Centroid Exploration: The Euler Line Sketchpad: DG labs Exploring Geometry Appendix A Labs 814 Conjectures C-5 Perpendicular Bisector p150 C-6 Converse of Perpendicular Bisector p 151 C-7 Shortest Distance C-8 Angle Bisector p159 Concurrence p 178-80 C-9 Angle Bisector Concurrency C-10 Perpendicular Bisector Concurrency C-11 Altitude Concurrency C-12 Circumcenter C-13 Incenter C-14 Median Concurrency p 185 C-15 Centroid p 186 Supplies: rulers, protractors, patty paper, compass, straightedge 4 COLUMBIA HIGH SCHOOL Geometry Curriculum Learning Objectives The student will … 4. Investigate the properties of triangles, analyze relationships between their sides and angles, and articulate the conditions that guarantee that two triangles are congruent. NJCCC 4.2 A 3, 4 Content Outline Key Definitions, Skills and Concepts Concepts/Reasoning: 7. Demonstrate a working knowledge that physical determination is tied to logical determination or implication: (Essential to all deductive reasoning) a. Distinguish parts that determine a unique triangle b. Identify the information/conditions that determines the congruence of two triangles 7. Sketch counterexamples for false statements about triangle relationships 7. Distinguish biconditional conjectures (if and only if) 7. Recognize and demonstrate justification, organization and communication as essential in proof 7. Distinguish sequential and non-sequential steps Skills: 1. Classify triangles by angle measures and side lengths 2. Complete statements and/or summarize findings of investigations related to triangles, relationships between their angles and sides, and conditions that guarantee congruence 3. Apply findings to determine: missing angle measures in triangles, correctness of an application, or constructing a triangle based on given information 4. Use angle and perpendicular bisectors to prove congruent, and to compute angle measures and segment lengths 5. Apply CPCT in a variety of problems and examples that 6. Complete, analyze or build a paragraph, flow chart proof Instructional Materials Text: DG: Ch. 4 L5: Take Another Look activities Printed Materials: Investigations: Triangle Sum Where are the largest and smallest angles? Exterior Angle Congruence Shortcuts Conjectures C-17 Triangle Sum p 201 C-18 Isosceles Triangle p207 C-19 Converse of the Isosceles Triangle p 208 C-20 Triangle Inequality p216 C-21 Side-Angle Inequality p217 C-22 Triangle Exterior Anglep218 Triangle Congruence pp 222- 228 23)SSS 24)SAS 25)ASA 26)SAA Special Triangles pp 244C-27 Vertex Angle Bisector C-28 Equilateral/Equiangular Tri. Sketchpad: DG labs Appendix A Labs: 15-17 Supplies: rulers, protractors, patty paper, compass, straightedge, uncooked spaghetti NJCCC requirements for proof: Use reasoning and some form of proof to verify or refute conjectures and theorems: a) Verification or refutation of proposed proofs b) Simple proofs involving congruent triangles c) Counterexamples to incorrect conjectures 5 COLUMBIA HIGH SCHOOL Geometry Curriculum Learning Objectives The student will … 5. Investigate, analyze and articulate the properties of quadrilaterals, and the relationships between their sides and angles. Use visual models and reasoning in some form of proof to verify or refute conjectures. (Developing simple proofs or providing counterexamples to incorrect conjectures can achieve this.) NJCCC 4.2 A 3, 4 Content Outline Key Definitions, Skills and Concepts Concepts: 1. Draw a concept map relating different kinds of quadrilaterals 2. Recognize that properties of one category are inherited by all subcategories 3. Use the symmetries of various quadrilaterals to identify properties 4. Identify and distinguish relationships between polygons using all, some, or no (or always, sometimes, never) Skills: 1. Identify and classify polygons. 2. Find the measures of interior and exterior angles of polygons. 3. Find the angle measures of quadrilaterals 4. Use the properties of a parallelogram to find the lengths of the sides and the measures of the angles 5. Show that a quadrilateral is a parallelogram 6. Use the properties of special types of parallelograms to find angle measures and side lengths 7. Use the properties of a trapezoid to find angle measures and side lengths 8. Identify special quadrilaterals based on limited information 9. Find the length of the midsegment of a trapezoid Formulas: 1. Sum of the int. angles of a polygon = (n – 2)180° 2. Sum of the ext. angles of a polygon = 360° Instructional Materials Text: DG: Ch. 5 Printed Materials: Investigations: Polygon Sum Ext. Angle Sum Property of Kites Trapezoids Midsegment Properties Parallelogram Properties Sketchpad: DG labs Appendix A Labs: 2025 Supplies: rulers, protractors, patty paper, compass, straightedge, graph paper, scissors Conjectures Polygon Sum p 258-261 C-29 Quadrilateral Sum C-30 Pentagon Sum C-31 Polygon Sum Exterior Angles of a Polygon p 263 C-32 Exterior Angle Sum C-33 Equiangular Polygon Kite and Trapezoid Properties p269 C-34 Kite Angles C-35 Kite Diagonals C-36 Kite Diagonal Bisector C-37 Kite Angle Bisector C-38 Trapezoid Consecutive Angles C-39 Isosceles Trapezoid C-40 Isosceles Trapezoid Diagonals Properties of Midsegments p 275 C-41 Three Midsegments C-42 Triangle Midsegment C-43 Trapezoid Midsegment Properties of Parallelograms p 281 C-44 Parallelogram Opposite Angles C-45 Parallelogram Consecutive Angles C-46 Parallelogram Opposite Sides C-47 Parallelogram Diagonals C-48 Double-Edge Straightedge C-49 Rhombus Diagonals C-50 Rhombus Angles C-51 Rectangle Diagonals C-52 Square Diagonals 6 COLUMBIA HIGH SCHOOL Geometry Curriculum Learning Objectives The student will … 6. Use geometry tools to explore, recognize and articulate relationships among angles and line segments, in and around circles. NJCCC 4.2 A 3, 4, D1 Content Outline Key Definitions, Skills and Concepts Concepts: 1. Connect basic properties of circles with visual representations. (list p 310) 2. Use points of tangency to recognize a) the relationship between radius and tangent b) the congruency of segments drawn outside a circle from a common point. 3. Chain two if-then statements into one if-and-only-if sentence. 3. Discover and recognize articulate properties of central angles, inscribed angles, chords, and arcs of circles. 4. Understand pi as the relationship between the circumference of a circle and its diameter Skills: 1. Identify segments and lines related to circles 2. Use properties of a tangent to a circle to find the lengths of segments 3. Use the properties of arcs of circles to find the measures of angles and arcs 4. Use the properties of chords of circles to find the measures of arcs and angles, and to determine other relationships 5. Use the properties of inscribed angles to find the measures of arcs and angles. 6. Apply the formula of circumference to calculate diameter, radius, or circumference Formula: 1. Arc length of AB = m AB 2 r 360 2. If <ADB is an inscribed angle, then m<ADB = 1 m AB 2 Instructional Materials Text: DG: Ch. 6 Investigations: Going Off on a Tangent Tangent Segments Define Angles in a Circle Chords & Their Central Angles Chords & the Center of the Circle Inscribed Angle Property Inscribed Angles Intercepting the Same Arc Angles Inscribed in a Semicircle Cyclic Quadrilaterals Arcs by Parallel Lines A Taste of Pi Finding the Arcs Exploration Intersecting Lines Through a Circle (All activities) Sketchpad: DG labs Exploring Geometry Appendix A Labs: 26-31 Supplies: tape measures, circular objects, protractors, patty paper, compass, straightedge Conjectures C-53 Tangent C-54 Tangent Segments C-55 Chord Central Angle C-56 Chord Arcs C-57 Perpendicular to a Chord C-58 Chord Distance to Center C-59 Perpendicular Bisector of a Chord C-60 Inscribed Angle C-61 Inscribed Angle Intercepting Arcs C-62 Angles Inscribed in a Semicircle C-63 Cyclic Quadrilateral C-64 Parallel Lines Intercepted Arcs C-65 Circumference C-66 Arc Length 3. .Ccircle = πd or 2πr 7 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 7. Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations. Determine whether two or more given shapes can be used to generate a tessellation. NJCCC 4.2 B 1, 3, 4, C2 Content Outline Concepts: 1. Use geometric transformations to define symmetry and isometry. 2. Recognize that properties (including parallelism, angle measurement, distance and area) are preserved by all isometries. 3. Recognize that similarity transformations (dilations) preserve angle measurement (including perpendicularity) but do not necessarily preserve distance or area. 4. Describe or demonstrate how to compose transformations to make other transformations. 5. Classify and identify monohedral, regular, and semiregular tessellations. Skills: 1. Identify and perform rotations; recognize, draw and apply rotational symmetry, and articulate its properties 2. Identify and perform reflections, and recognize, draw and apply reflections and articulate its properties. 3. Identify, distinguish, and draw translations. 4. Identify and draw dilations of polygons. 5. Distinguish rigid and nonrigid transformations. Concept Check: How do translations relate to parallel lines? How do reflections relate to congruence? What kinds of symmetry do reg. polygons have? Give examples of parallelism and perpendicularity in transformations. Use polygons as examples to describe the symmetries of a figure under each transformation. Instructional Materials Text: DG: Ch. 7 Printed Materials: Investigations: The Basic Property of a Reflection Transformations on a Coordinate Plane Reflections Across Two Parallel Lines Reflections Across Two Intersecting Lines The Semiregular Tessellations Do All Triangles Tessellate? Do All Quadrilaterals Tessellate? Conjectures C-67 Reflection Line C-68 Coordinate Transformation C-70 Reflection Across Parallel Lines C-71 Reflection Across Intersecting Lines C-72 Tessellating Triangles C-73 Tessellating Quadrilaterals Definitions p 370 TE: Symmetry p 371 TE : Transformation, line of reflection, image, rotational symmetry, symmetries of a figure Sketchpad: DG labs Appendix A Labs: 3238 Supplies: rulers, protractors, patty paper, compass, straightedge, miras 8 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … Content Outline 8. Use a variety of strategies to estimate and determine perimeter and area of plane figures and surface area of 3D figures. Concepts: 1. Derive formulas and methods for finding and relating areas. 2. Recognize, apply and describe using the sub concepts of area: unit iteration, additivity and invariance. 3. Determine a generalizable way to find the area of any polygon. 4. Determine a generalizable way to approximate the area of an irregular figure. NJCCC 4.2 D2, E 2 Skills: 1. Estimation of area, perimeter, volume, and surface area 2. Find the area of any given: rectangle, triangle, parallelogram, rhombus, composite polygons, similar polygons, circle, and sector 3. Find the surface area of prisms and cylinders. 4. Find the lateral area of prisms and cylinders. 5. Find the slant height of pyramids and cones. 6. Find the surface area of pyramids and cones. 7. Determine which shape has minimal (or maximal) area and perimeter, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets Concept Checks: 1. How is finding the area of a rhombus different from finding the area of other parallelograms? 2. Describe the similarities and differences in finding the area of a circle and finding the area of a polygon. 3. Do all rectangles with the same perimeter have the same area? 4. If the area of a square foot looks like this[drawing], draw the area of a square yard. Then state how many you need of each in an equivalent area. Conjectures / Notes Instructional Materials Text: DG: Ch. 8 Printed Materials: Investigations: Area Formula for Parallelograms Area Formula for Triangles Area Formula for Trapezoids Area Formula for Kites Solving Problems with Area Formulas Area Formula for Regular Polygons Area Formula for Circles Surface Area of a Regular Pyramid Surface Area of a Cone Sketchpad: DG labs Appendix A Labs: 3941 Supplies: rulers, protractors, patty paper, compass, straightedge, scissors, graph paper, prisms, pyramids and other solids C-74 Rectangle Area C-75 Parallelogram Area C-76 Triangle Area C-77 Trapezoid Area C-78 Kite Area C-79 Regular Polygon Area C-80 Circle Area Formulas: A square = side2 A rectangle = base height (also the area of a parallelogram) A triangle = ½ base height A rhombus = ½ (product of the diagonals) A trapezoid = ½ height (sum of the bases) A circle = πr2 SAprism = 2(area of the base) + (perimeter of the base)(height) SAcylinder = 2(area of the base) + (circumference of the base)(height) Lateral Area = surface area – area of the base(s) SApyramid = (area of the base) + ½ (perimeter of the base)(height) SAcone = (area of the base) + ½ (circumference of the base)(height) 9 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 9. Dissect and apply the Pythagorean Theorem, recognizing its conditions, and focusing on the squares of sides, and then generalizing the relationship to other applications for similar figures. NJCCC 4.2 A 1,E 1 Content Outline Concepts: 1. Given a right triangle, identify a and b as the lengths of legs of a right triangle and c as the hypotenuse. 2. Recognize that the Pythagorean Theorem is about areas of squares. 3. Demonstrate that the variables a, b, and c can be replaced by any other representations of those lengths. 4. Represent geometrically the side lengths of special triangles and determine the ratio of side lengths for each. 5. Generalize that if 2 figures are similar with a scale factor of s, then their areas have a scale factor of s2. 6. Compare common structure between and across applications. Skills: 1. Use the Pythagorean Theorem and the Distance Formula to find the lengths of sides of a triangle. 2. Use the converse of the Pythagorean Theorem to classify triangles according to angle measure. 3. Simplify radical expressions. Conjectures Instructional Materials Text: DG: Ch. 9 Printed Materials: Investigations: The 3 Sides of a Right Triangle Is the Converse True? Isosceles Right Triangles 300-600-900 Triangles The Distance Formula C-81 The Pythagorean Theorem C-82 Converse of the Pythagorean Theorem C-83 Isosceles Right Triangle C-84 300-600-900 Triangles C-85 Distance Formula Sketchpad: DG labs Appendix A Labs: 42 45 Supplies: rulers, protractors, patty paper, compass, straightedge 10 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 10. Develop and derive formulas for the volumes of solids and apply them to solve problems. Recognize 3-dimensional figures obtained through transformations of twodimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization. NJCCC 4.2 A 2, B 2 Content Outline Concepts: 1. Connect visual diagrams with written descriptions 2. Distinguish between examples and counter examples Skills: 1. Classify and distinguish solid shapes 2. Identify the characteristics of a good definition 3. Use graphic organizers to relate and distinguish geometric terms and models 4. Apply formulas and solve for any variable. Conjectures Instructional Materials Text: DG: Ch. 10 Printed Materials: Investigations: The Volume Formula for Prisms and Cylinders The Volume Formula for Pyramids and Cones Modeling the Platonic Solids The Formula for Volume of a Sphere The Formula for Surface Area of a Sphere C-86a Regular Prism Volume C-86b Right Prism-Cylinder Vol. C-86c Oblique Prism-Cylinder Vol C-86d Prism-Cylinder Volume C-87 Pyramid-Cone Volume C-88 Sphere Volume C-89 Sphere Surface Area Formulas: 1. Vprism = (A base)(height) 2. Vcylinder = (A base)(height) 3. Vpyramid = ⅓ (Abase)(height) 4. Vcone = ⅓ (Abase)(height) 5. Vsphere = Sketchpad: DG labs Appendix A Lab: 46 4 3 πr 3 Supplies: 3-D models, interlocking cubes, graph paper, nets of 3D solids 11 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 11. Identify the properties of similar figures. Recognize and apply similarity to scale drawings or dilations, and indirect measurement. NJCCC 4.2 E 1 Content Outline Concepts: 1. Based on a given model, identify the conditions that guarantee two polygons are similar. 2. Determine the connection between dilation, scale factor, proportionality and similar figures. 3. Distinguish between examples and nonexamples of similarity 4. Recognize the relationship between corresponding parts of similar triangles 5. Identify and generalize the relationship between the areas of similar figures, and between surface areas and volumes of solids. 6. Determine the relationship between the ratios of the parts into which parallel lines cut the sides of a triangle. Skills: 1. Use ratio and proportion to solve problems 2. Identify and create similar polygons 3. Figure out if 2 triangles are similar using the AA Similarity Shortcut. 4. Show that 2 triangles are similar using the SAS or SSS Similarity Shortcuts 5. Use the 3 Similarity Shortcuts to find the missing sides to triangles. 6. Use techniques of indirect measurement to represent and solve problems. Conjectures Instructional Materials Text: DG: Ch. 11 Investigations: What Makes Polygons Similar? Dilations on the Coordinate Plane Is AA a Similarity Shortcut? Is SSS a Similarity Shortcut? Is SAS a Similarity Shortcut? Mirror, Mirror Corresponding Parts Opposite Sides Ratios Area Ratios Surface Area Ratios Volume Ratios Parallels and Proportionality C-90 Dilation Similarity C-91 AA Similarity C-92 SSS Similarity C-93 SAS Similarity C-94 Proportional Parts C-95 Angle Bisector/Opposite Side C-96 Proportional Areas C-97 Proportional Volumes C-98 Parallel/Proportionality C-99 Extended Parallel/Proportionality Sketchpad: DG labs Appendix A Labs: 47 54 Supplies: rulers, protractors, patty paper, compass, straightedge 12 COLUMBIA HIGH SCHOOL GEOMETRY CURRICULUM Learning Objectives The student will … 12. Define the sine, cosine, and tangent ratios of the acute angles in a right triangle. Using trigonometric ratios, find the unknown lengths or angle measurements in a right triangle. NJCCC 4.2 E 1 Content Outline Conjectures Instructional Materials Concepts: 1. Derive and apply the Law of Sines 2. Recognize and apply the Law of Cosines 3. Use trig to solve problems involving right triangles. 4. Recognize the value and using techniques of indirect measurement to represent and solve problems. Skills: 1. Find the sine and cosine of an acute angle. 2. Find the tangent of an acute angle. 3. Determine lengths or angle measurements. Text: DG: Ch. 12 C-100 SAS Triangle Area C-101 Law of Sines C-102 Law of Cosines Printed Materials: Investigations: Trigonometric Tables Area of a Triangle The Law of Sines The Law of Cosines Sketchpad: DG labs Appendix A Labs: 55 56 Supplies: rulers, protractors, patty paper, compass, straightedge 13 REASONING STRATEGIES DRAW A LABELED DIAGRAM AND MARK WHAT YOU KNOW REPRESENT A SITUATION ALGEBRAICALLY APPLY PREVIOUS CONJECTURES AND DEFINITIONS ADD AN AUXILIARY LINE THINK BACKWARDS ASPECTS OF PROOF SKILLS PROPERTIES: from whole figures to components to relationships to hierarchies PURPOSE: from proofs for explaining to proofs for justifying and then to proofs for systematizing FORMAT: from oral arguments to deductive arguments to paragraph proofs to flowchart proofs to two-column proofs EVIDENCE: from being convinced by appearance to being convinced by measurement to requiring deductive proof THE SEQUENTIAL DEVELOPMENT OF EACH ASPECT OF PROOF DG Chapter van Hiele Level Properties Purpose Format Evidence 0 0 Whole figures 1-3 1-2 Components Explain Deductive argument Measurement 4-6 2 Relationships Explain, justify Paragraph, flowchart Measurement, Deduction 7-12 2-3 Relationships, Hierarchies Justify Paragraph, flowchart Deduction 13 3-4 Hierarchies Two-column Deduction Appearance Systematize 14 PROOF RUBRIC 5 POINTS: Given information is clearly stated. The diagram is labeled and marked correctly. The proof is clear and correct with all statements supported by reasons. 4 POINTS: Given information is stated. The diagram is labeled and marked. Some markings may be missing or incorrect. The proof is correct, perhaps with a few missing reasons or some redundancy. 3 POINTS: The given information is stated, but the diagram is not marked, or incorrectly marked. The proof contains some correct steps with reasons and the correct conclusion, but is missing one or more significant intermediate steps. Or, the proof is incomplete but has sound logic in the steps that are shown. 2 POINTS: The given information is not complete. A diagram is incomplete or incorrectly marked. Some true statements are given without justification. 1 POINT: The proof is largely incomplete. Statements are false or unrelated to the conclusion. 15 Appendix A: Geometer’s Sketchpad Labs Source: Exploring Geometry by Dan Bennett, Key Curriculum Press, 2002 *Please note: the labs marked with an asterisk are designed for students with little or no experience using Sketchpad. They offer more specific guidance on how to use Sketchpad as well as geometry content that is introductory. Once sketchpad is mastered, many of these labs can be combined, for example, two or more labs per class period. Chapter in Discovering Geometry (Text) Name of Lab (.gsp files must be loaded on to the Page Number in Exploring Geometry where the Lab is suggested for use computers) (Lab book) and in the Appendix (attached) *Introductory labs for the Geometer’s sketchpad application 1.1 1.2 1.5 1.6 1.7 2.5 2.6 3.2 3.3 3.7 3.7 3.7 3.8 3.9 4.2 4.3 4.4 4.5 *Introducing Points, Segments, Rays, and Lines *Introducing Angles *Defining Triangles (ClassifyTriangles.gsp) *Defining Special Quadrilaterals (Special Quads.gsp) *Introducing Circles Angles formed by intersecting lines Properties of parallel lines Constructing a perpendicular bisector Distance from a point to a line Perpendicular bisectors in a triangle (circumcenter) Altitudes in a triangle (orthocenter) Angle bisectors in a triangle Medians in a triangle (centroid) The Euler segment Properties of isosceles triangles Triangle inequalities Triangle congruence (Triangle Congruence.gsp) 3–6 7–9 63 – 64 89 – 90 119 – 120 15 – 16 17 – 18 19 25 73 – 74 75 – 76 77 71 – 72 78 - 79 69 67 68 16 Appendix A: Geometer’s Sketchpad Labs (Continued) 5.1 5.2 5 5 5 5 5 5 6.1 6.1 6.2 6.3 6.5 6.75 7 7 7 7 7 7 7 8.1 8.2 8.4 8.5 9.1 9.1 9.2 9.3 9.3 10 Polygon angle measure sums Exterior Angles in a polygon Properties of parallelograms Properties of rectangles Properties of rhombuses Properties of isosceles trapezoids Midsegments of a trapezoid and a triangle Summarizing properties of quadrilaterals Tangents to a circle Tangent segments Chords in a circle Arcs and angles The circumference / diameter ratio Exploration on angles formed by intersecting lines in a circle *Introducing Transformations Properties of Reflection Translations in the Coordinate Plane Reflections over two Parallel lines Reflections over two Intersecting lines Symmetry in regular polygons Tessellating with regular polygons Areas of Parallelograms and triangles The area of a trapezoid Areas of regular polygons and circles 112 – 113 109 – 110 91 93 95 97 – 98 100 – 101 104 – 105 123 124 121 – 122 125 – 126 127 – 128 Last section in Chapter 6, Discovering Geometry Textbook 33 – 35 36 – 37 39 44 – 45 46 – 47 50 – 51 52 133 – 134 142 – 143 145 – 146 Visual Demonstration of the Pythagorean Theorem (shearing squares) Pythagorean Triples 155 The isosceles right triangle The 30-60 right triangle Constructing templates for the Platonic Solids 159 – 160 161 – 162 115 157 – 158 17 Appendix A: Geometer’s Sketchpad Labs (Continued) 11 11.1 11.2 11.2 The golden rectangle Similar polygons Similar triangles: AA Similar triangles SSS, SAS, SSA (Triangle Similarity.gsp) Finding the height of a tree Measuring height with a mirror Proportions with area Parallel lines in a triangle Trigonometric ratios Modeling a ladder problem 11.3 11.3 11.5 11.7 12.1 12.2 167 – 168 169 170 171 176 – 178 179 – 180 190 – 191 181 195 – 196 197 – 198 Topics on the SAT, as recorded in the SAT Preparation Book are: Geometric Notation and Perception Points, Lines, and Angles Quadrilaterals: Parallelograms Triangles: Rectangles Equilateral Squares Isosceles Right triangles Area and Perimeter: 30 – 60 – 90 Squares and Rectangles A. Triangles 45 – 45 – 90 Parallelograms 3–4–5 triangles Transformations Circles: Diameter and radius Circumference and area Arcs Tangent to the circle Solid Geometry: Volumes of solid figures Surface area of solid figures Coordinate Geometry Slopes of parallel and perpendicular lines The midpoint formula The distance formula Other Polygons: Angles in a polygon Congruent & similarity 18 Revised NJCCC: Geometry Standards (January 2008) 4.2.12 A. Geometric Properties Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will: 2. Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views). 1. Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway). 3. Apply the properties of geometric shapes. Parallel lines – transversal, alternate interior angles, corresponding angles Triangles a. Conditions for congruence b. Segment joining midpoints of two sides is parallel to and half the length of the third side c. Triangle Inequality Minimal conditions for a shape to be a special quadrilateral Circles – arcs, central and inscribed angles, chords, tangents Self-similarity 5. Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology). Perpendicular bisector of a line segment Bisector of an angle Perpendicular or parallel lines 3. Use reasoning and some form of proof to verify or refute conjectures and theorems. Verification or refutation of proposed proofs Simple proofs involving congruent triangles Counterexamples to incorrect conjectures 19 4.2.12 B. Transforming Shapes Grade 12 1. Determine whether two or more given shapes can be used to generate a tessellation. 1. Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic [object] representation, and, conversely, determine whether and how one [object]representation can be transformed to another by a transformation or a sequence of transformations. 2. Recognize three-dimensional figures obtained through trans-formations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization. 4. Generate and analyze iterative geometric patterns. Fractals (e.g., Sierpinski’s Triangle) Patterns in areas and perimeters of self-similar figures Outcome of extending iterative process indefinitely 4.2.12 C. Coordinate Geometry Grade 12 1. Use coordinate geometry to represent and verify properties of lines and line segments. Distance between two points Midpoint and slope of a line segment Finding the intersection of two lines Lines with the same slope are parallel Lines that are perpendicular have slopes whose product is –1 2. Show position and represent motion in the coordinate plane using vectors. Addition and subtraction of vectors 3. Find an equation of a circle given its center and radius and, given an equation of a circle in standard form, find its center and radius. 20 4.2.12 D. Units of Measurement 1. Understand and use the concept of significant digits. 2. Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation. Degree of accuracy of a given measurement tool Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements B. 4.2.12 E. Measuring Geometric Objects Grade 12 1. Use techniques of indirect measurement to represent and solve problems. Similar triangles Pythagorean theorem Right triangle trigonometry (sine, cosine, tangent) Special right triangles 2. Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. Approximation of area using grids of different sizes Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets Estimation of area, perimeter, volume, and surface area [Relate to indicator 4.2.12 B 2, recognizing three-dimensional figures obtained through trans-formations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude)]Finding surface area and volume of 3D figures is included in indicator 4.2.12 E 2 above.] 21 HSPA Test Specifications (New Jersey Department of Education) KNOWLEDGE: The student should have a conceptual understanding of: 1. 2. 3. 4. 5. 6. Geometric terms (e.g. point, ray, line, angle, plane, side, vertices, polygon, face, polyhedron, circle, sphere) Standard notations Properties of geometric figures Fundamental relationships between geometric figures (e.g., parallelism, perpendicularity, intersection, congruence, similarity) Inductive and deductive reasoning Spatial relationships (e.g., direction, orientation, and perspective of objects in space); The student should have a conceptual understanding of: 1. 2. 3. 4. Congruence Similarity Symmetry Transformations a. Rotations b. Reflections c. Translations d. Dilations 5. The rectangular coordinate system 6. Matrices 7. Tessellations 8. Vectors 9. Measurable attributes (e.g., perimeter, circumference, area, surface area, volume, angle measure) 10. Standard and non-standard units of measure 11. Dimensions, shapes, and properties of figures and objects 12. Right triangle relationships a. The Pythagorean Theorem b. Basic trigonometric ratios 22 HSPA Test Specifications (New Jersey Department of Education) The student should be able to: 7. Use properties, definitions, and relationships to identify, classify, and describe two-dimensional and three-dimensional geometric figures 8. Draw two-dimensional representations of three-dimensional objects by sketching shadows, projections, perspectives, and map views 9. Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings 10. Apply concepts of symmetry, similarity, and congruence to problem solving 11. Use coordinates, maps, tables, and grids 12. Use transformations 1. Given the pre-image & transformation, find image 2. Given the image and transformation, find the pre-image 3. Given the pre-image & image, determine the transformation 13. Draw a figure & tessellate it 14. Perform scalar multiplication on matrices 15. Use vectors to show the position of an object 16. Utilize appropriate formulas and label answers with appropriate units of measure 17. Measure geometric objects and determine the degree of accuracy needed when measuring them 18. Choose the appropriate techniques, tools, and units to measure quantities to achieve the desired level of accuracy 23 HSPA Test Specifications (New Jersey Department of Education) PROBLEM-SOLVING SKILLS: In problem settings, using abilities that comprise the power base, the student should be able to: 11. Analyze properties of three-dimensional geometric figures by using models and by drawing and interpreting two-dimensional representations of them 12. Use inductive and deductive reasoning to solve real-life problems and justify solutions 13. Solve real-world and mathematical problems using geometric models 14. Determine the sequence of transformations needed to map one figure onto another 15. Solve problems in geometry using transformations, coordinates, and vectors 16. Relate the concepts of symmetry, similarity, and congruence to transformations 17. Predict and represent resulting figures when combining, subdividing, and changing figure 8. Use basic trigonometric ratios to solve problems involving indirect measurement 9. Develop and apply a variety of strategies for determining perimeter, circumference, area, surface area, volume, and angle measure 10. Solve problems using the Pythagorean Theorem 11. Develop informal ways of approximating the measures of familiar objects Express mathematically and explain the impact of change in an object's dimensions on its surface area, volume, and/or perimeter 24 Strategies to use to teach proofs while teaching each objective: 1. 2. 3. 4. 5. 6. 7. Connect the given in a diagram with the statements. Categorize conjectures by what they prove. Do fill in the blank proofs. Offer an option for paragraph or flow chart proofs. Write conjectures and have students state what is needed to use them. Write logical arguments for answers to questions. Have peer review of arguments and proofs. 25