Download B. 4.2.12 E. Measuring Geometric Objects Grade 12

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