GEOMETRY ACCELERATED
COURSE DESCRIPTION & OUTCOMES
This course follows generally the description of the traditional geometry course but provides extensive experience from
early stages with the devising, presentation, and defense of student proofs and the theoretical consideration of the nature
of proof (direct and indirect; in two-column, flow, and analytical paragraph form). The Honors course includes a more
extensive coverage of solid geometry, an introduction to analytic geometry and trigonometry, and opportunities for
curricular enrichment in problem-solving. Additional topics include vectors, trigonometric identities, conic sections,
common and natural logarithms, sequences and series, and the study of trigonometric and circular functions. A Texas
Instruments TI-83 or TI-84 series graphing calculator is required.
In addition to all the essential questions and outcomes described in the course entitled “Geometry,” the student enrolled in
“Geometry Honors” will be able to address additional essential questions and achieve additional outcomes, as listed
below. Units designated as optional are presented at the teacher’s discretion as time permits.
Geometry Accelerated Enduring Understandings
1. Geometry is omnipresent in the physical world; it can be used to solve problems in real life.
2. Geometry knowledge is used in many branches of mathematics.
3. Geometry uses standard vocabulary and symbols to communicate facts and relationships about geometric figures.
4. Geometric figures are ruled by known relationships of measures, often expressed as theorems and/or algebraic
formulas.
5. Proofs, constructions and visual observations demonstrate why geometric relationships are true
6. Logic, in combination with facts, theorems and formulas can be used to draw conclusions about geometric figures.
7. A proof is a formal argument supported by postulates, theorems and definitions; it uses logical reasoning to come to its
conclusion.
8. Technology can be useful to illustrate and examine geometric relationships.
Geometry Accelerated Essential Questions and
Performance Objectives by Unit
BASIC SKILLS:
At the end of this unit, students will be able to:
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Describe the origin of area formulas for the triangle, parallelogram, rectangle, and trapezoid
Use the Trapezoid Rule to find the area under a curve.
Use the Midpoint Rule to find the area under a curve.
REASONING:
At the end of this unit, students will be able to:
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Use an arrow diagram to illustrate a theorem or postulate.
PROPERTIES OF TRIANGLES:
At the end of this unit, students will be able to:
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Prove theorems requiring the use of auxiliary sets.
TRIGONOMETRY:
1. How can trigonometry be used outside of right triangles to solve a variety of problems?
At the end of this unit, students will be able to:
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Determine exact values of trigonometric expressions.
Use the Law of Cosines to solve triangles.
Use the Law of Sines to solve triangles.
Choose an appropriate method to solve an ambiguous triangle.
Use trigonometry to solve problems.
Prove trigonometric identities using reciprocal, quotient, and Pythagorean relationships.
Add vectors in component form.
Add vectors using the Law of Cosines.
CONIC SECTIONS (optional):
1. What is the connection between the equation and the graph of a conic section?
At the end of this unit, students will be able to:
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Graph conic sections: Circle, Parabola, Ellipse and Hyperbola.
State and use the relationship between the center and radius of a circle and the equation of a circle.
State and use the relationship between the focus, directrix, vertex, endpoints of the latus rectum, eccentricity, and
the axis of symmetry of a parabola and the equation of a parabola.
State and use the relationship between the center, foci, intercepts, endpoints of the latus rectum, and eccentricity
of an ellipse and the equation of an ellipse.
State and use the relationship between the center, foci, intercepts, endpoints of the latus rectum, eccentricity, and
asymptotes of an hyperbola and the equation of an hyperbola.
Graph conic sections on a graphing calculator.
Work with the Geometry of conic sections.
LOGARITHMS (optional):
1. How can logarithms and exponents be used to solve problems?
At the end of this unit, students will be able to:
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Solve an exponential equation with any positive constant base, using base 10 and natural logarithms.
Learn the definition of logarithm by finding the logarithm, base, or argument, if the other two are given.
Learn the properties of logarithms by transforming expressions and solving equations.
Describe the similarities and differences between common and natural logarithms.
SEQUENCES AND SERIES (optional):
1. How can arithmetic and geometric sequences be sued to solve problems?
2. How can aithmetic and geometric series be used to solve problems?
At the end of this unit, students will be able to:
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Identify a sequence or series as arithmetic, geometric, or neither.
Calculate term values of arithmetic and geometric sequences.
Given a term value of an arithmetic or geometric sequence, identify the term number.
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Evaluate a partial sum.
Represent a series using sigma notation.
Determine whether a geometric series is divergent or convergent.
Calculate the value to which a convergent series converges.
PROBABILITY(optional):
1. What is the connection between area and probability?
At the end of this unit, students will be able to:
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Communicate basic terminology involving geometric probability
Use segments in geometric probability.
Use area in geometric probability.
Communicate basic terminology and words associated with probability.
Use counting principles to determine the number of outcomes in a sample space.
Calculate probabilities of various permutations.
Calculate probabilities of various combinations.
Use the properties of probability in calculations.
Calculate the mathematical expectation of a random experiment.
GRAPHING PERIODIC FUNCTIONS(optional):
1. What is the connection between the equation and the graph of a periodic function?
At the end of this unit, students will be able to:
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Draw a graph of a periodic function accurately, by pointwise plotting.
Draw a graph of a periodic function quickly, by finding certain critical points.
Identify the amplitude, period, vertical shift, and phase displacement of a sinusoidal given its equation.
Given the graph of a sinusoidal function or information about the graph, write the particular equation.
Given a situation from the real world in which something varies sinusoidally, derive an equation and use it as a
mathematical model to make predictions and reach conclusions.
Assignments, Assessment and Instructional Strategies
Key Assignments:
• In-class individual and group guided exploration of new topics
• In-class practice of problem solving
• At-home practice of problem solving
• Chapter review problem sets and practice tests
Instructional Methods and/or Strategies:
• Guided inquiry and exploration, fostering persuasive arguments and clear communication about
geometric/mathematical thinking
• Instructor-as-role-model for mathematical thinking
• Warm-up problems to review or to pique interest
• Daily review of homework & re-teaching as needed
• Regular student articulation to peers, both in small groups and before the whole class
• Lecture, class discussion and note-taking
• Guided notes
• Formative assessments such as homework, group problem-solving & board work
• Problem topics designed to engage teenagers
• Graphing calculator and/or computer activities
• Instructional videos
• Hands-on activities/inquiries involving geometric shapes and/or measurement
• Compass & Straightedge constructions
• Individual tutoring by the teacher outside of class as needed
Assessments, including Methods and/or Tools:
Summative Assessments will include:
• Chapter Tests
• Quizzes
• Homework
• Two semester Final Exams
• Individual projects such as physical models, geometric art or chapter portfolios
Formative assessments are listed above as instructional methods.
Suggested Geometry Pacing (Accelerated)
(Based on Larson Geometry text, 2004)
Chapter
# of class days,
Counting assessments
& Review days
Emphasis
1
Notation; segments; angles (mostly independent)
2
Bisectors; two-column proof; reasoning
3
Parallel Lines; two-column proof; paragraph proof
First quarter midterm prep and midterm (2 days)
4
Congruent triangles with proof; coordinate proof
5
Special segments in triangles; some construction; auxiliary sets
6
Interior angles (all polygons); parallelograms; area; area under a curve;
coordinate proof
Fall semester Final Exam prep and final exam (2 days)
7
Rotations; reflections; translations; most can be done via Sketchpad
8
Similar triangles; proportion; proofs
9
Pythagorean Theorem, Special Rt. Triangles, Reciprocal Functions
10
Equation of a circle with center NOT at the origin
Third quarter midterm prep and midterm exam (2 days)
11
Exact values for regular hexagon’s area
12
Volume of composite figures
Trig Supp I
6
Radians, Rotations, Unit Circle, Exact trig values; emphasis on trig
expressions and memorizing
Calc. Use
2
Trig Inverse
Trig ID
5
Pythagorean; reciprocal; quotient; emphasis on algebraic manipulation
Vectors
8
Vectors, Law of Sines; Law of Cosines; Ambiguous Case; Models
Spring semester Final Exam prep and final exam (2 days). This exam will include as many topics as were
covered in the course. Trig ID can be shortened or moved to later.
Logs/Exp
6
Solving equations
Conics
3
Parabola and ellipse only
Seq/Ser
4
Using formulas
Probability
2
Composite figures
Periodics
3
Transformations