Geometry
Course Title: Geometry
Grade Range: 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)
28 January 2010
16
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline / Instructional Materials / Conjectures / Notes
1. Develop definitions of geometric terms using visual representations and written descriptions.
NJCCC 4.2 A 3, 4 / 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 / Text:
DG: Ch. 1
Printed Materials:
Investigations:
Math Models
Virtual Pool
Triangles
Special Quads
Def. Circles
Sketchpad DG labs
Exploring Geometry
p143-4;
Appendix A Labs: 1, 2, 3, 4, 5
Supplies: rulers, protractors, patty paper, compass, straightedge / 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)
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline / Instructional Materials / Conjectures / Notes
2. Use inductive reasoning to identify patterns and solve problems.
Use deductive reasoning to justify conclusions.
NJCCC 4.2 A 3, 4 / 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
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? / Text:
DG: Ch. 2
Printed Materials:
Investigations:
Party Handshakes
Overlap. Segments
Angle Relationships
Is the converse true?
Exploration:
7 Bridges of Konigsberg*
Sketchpad:DG labs
Exploring Geometry
p 17-18; Appendix A Labs: 6, 7
Supplies: patty paper, compass, straightedge / C-1 Linear Pair p 122
C-2 Vertical Angles p. 123
C-3 Parallel Lines p.129:
a. Corresponding Angles
b. Alt. Interior Angles
c. Alt. Exterior Angles
C-4 Converse of Parallel Lines
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline / Instructional Materials / Conjectures
3. Make conjectures based on investigations using geometric constructions.
NJCCC 4.2 A 3, 4, 5 / 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? / 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 8-14
Supplies: rulers, protractors, patty paper, compass, straightedge / 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
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials / Conjectures
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 / 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 / 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
Sketchpad: DG labs
Appendix A Labs:
15-17
Supplies: rulers, protractors, patty paper, compass, straightedge, uncooked spaghetti / 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 244-
C-27 Vertex Angle Bisector
C-28 Equilateral/Equiangular Tri.
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
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials / Conjectures
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 / 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° / 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: 20-25
Supplies: rulers, protractors, patty paper, compass, straightedge, graph paper, scissors / 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
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials / Conjectures
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
/ 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 =
2. If <ADB is an inscribed angle, then mADB =
3. .Ccircle = πd or 2πr / 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 / 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
Columbia High School
Geometry Curriculum
Learning ObjectivesThe student will … / Content Outline / Instructional Materials / Conjectures
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 / 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. / 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?
Sketchpad: DG labs
Appendix A Labs: 32-38
Supplies: rulers, protractors, patty paper, compass, straightedge, miras / 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
Columbia High School
Geometry Curriculum