Geometry
Approved January 2012
1
Points, Lines, Planes and Angles
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
Essential Question: How do algebraic concepts relate to geometric concepts? How do patterns and functions help us represent data and
solve real-word problems?
Learning Goals: Students will:
Know the undefined terms point, line, and plane.
Know the terms collinear, coplanar, intersection, and equidistant.
Know appropriate symbols for lines, segments, rays, and distances.
Know and apply the Segment Addition Postulate.
Be able to name angles and find angle measures.
Know and apply the Angle Addition Postulate.
Understand and apply postulates and theorems relating to points, lines and planes.
Understand and use patterns and inductive reasoning.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Using graphing calculators
Using CFA data
DI according to student readiness
Small group and whole class instruction
Quizzes
CFA’s
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Acute, angle, angle bisector, collinear, congruent, conjecture, coplanar, counterexample, inductive reasoning,
line, linear pair, midpoint, obtuse, opposite rays, parallel lines, perpendicular lines, plane, point, postulate, ray,
right angle, segment, space, straight angle, theorem
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Core Ethical Values
3
Deductive Reasoning
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
Essential Question: How is deductive reasoning used to make, investigate, and prove conjectures in the real world? How do reasoning
and proofs lead to a better understanding of geometric concepts?
Learning Goals: Students will:
Recognize the hypothesis and conclusion of a conditional.
Be able to state the converse, inverse, and contrapositive of a conditional.
Be able to use the properties of real numbers in proofs. (A level Only)
Be able to use the midpoint and angle bisector theorem.
Apply the definitions of complementary, supplementary, and adjacent angles.
Know and use the vertical angle theorem.
Know and apply the definitions and theorems of perpendicular lines.
Be able to write a bi-conditional statement.
Be able to recognize a good definition.
Apply the Law of Detachment and the Law of Syllogism. (A level only)
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher-facilitated class discussions
Cooperative learning
Venn diagrams
Replacing words with symbols
Project–logic poster
Performance tasks
Writing assignment
Union/intersection worksheet(s) (A level)
CFA(s)
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
Alice in Wonderland video clip: http://fliiby.com/file/412746/7tr3jbb4k7.html
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
2-column proof, adjacent, biconditional, complementary angles, conclusion, conditional, converse, deductive
reasoning, hypothesis, inverse, Law of Detachment, Law of Syllogism, negation, proof, supplementary angles,
symmetric property, theorem, truth value, vertical angles
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
5
Parallel Lines and Planes
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
Essential Question: How do geometric relationships help solve real-world problems?
Learning Goals: Students will:
Be able to distinguish between intersecting lines, parallel lines, and skew lines.
Identify the angles formed when lines are cut by a transversal.
Know and apply postulates and theorems about parallel lines and planes.
Classify triangles according to sides and to angles.
Know and apply theorems about the sum of measures of interior and exterior angles of a triangle.
Be able to find the measure of interior and exterior angles of a polygon.
Be able to find the number of diagonals of a polygon.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher-facilitated class discussions
Student inquiry through guided investigations
Cooperative learning
Constructions
Writing assignment
CFA(s)
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
Compass
Ruler
Protractor
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Alternate interior angles, corresponding angles, concave polygon, convex polygon, equiangular polygon,
equilateral polygon, exterior angle, parallel lines, parallel planes, polygon, regular polygon, remote interior
angles, same-side interior angles, skew lines, transversal
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Core Ethical Values
7
Triangles
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Understand and use algebraic skills and concepts
Essential Question: How does writing 2-column proofs relate to drawing conclusions in real life? How do geometric relationships help
solve real-world problems?
Learning Goals: Students will:
Identify the corresponding parts of congruent figures.
Be able to prove two triangles congruent by using postulates and theorems.
Deduce information about segments and angles after proving that two triangles are congruent.
Apply theorems about isosceles and equilateral triangles.
Be able to prove multiple triangles congruent.
Apply definitions of median and altitude of a triangle and the perpendicular bisector of a segment.
Know and apply theorems about a point on a perpendicular bisector of a segment and a point on the bisector of an angle.
Apply the triangle midsegment theorem.
Identify points of concurrency in triangles.
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Suggested Strategies
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Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
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Teacher facilitated class discussions
Student inquiry through guided investigations:
o Congruent triangles investigation
o Paper tearing and paper folding to discover and/or reinforce concepts
o Coffee stirrer and gumdrop (or marshmallow) bridge design
Games for reinforcement and review
o Jeopardy
o Hotseat
o Board races
o Chutes and Ladders
Teacher observations
Performance tasks
Homework/classwork
Cooperative learning
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
Susan B. Anthony dollar (for novelty)
Protractor
Ruler
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Acute triangle, altitude, base angles of an isosceles triangle, base of an isosceles triangle, centroid, circumcenter,
circumscribed about concurrent, congruent triangles, corollary, CPCTC, distance from a point to a line,
equiangular triangle, equilateral triangle, equivalent statements, incenter, inscribed in, isosceles triangle, median,
midsegment, obtuse triangle, orthocenter, point of concurrency, right triangle, scalene triangle
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Lifelong Learning/21st Century
Skills
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Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
10
Quadrilaterals
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Understand and use algebraic skills and concepts
Essential Question: How do the properties of quadrilaterals relate to real-world problems?
Learning Goals: Students will:
Apply the definition of a parallelogram and theorems about parallelograms.
Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square.
Apply the definitions and identify the properties of a trapezoid and an isosceles trapezoid.
Apply the definition and identify the properties of a kite.
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Suggested Strategies
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Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century
Skills
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Teacher-facilitated class discussions
Student inquiry through guided investigation
o Venn diagram
o Card sorting activity
Graphic organizer
Cooperative learning
Games for reinforcement and review
o Jeopardy
o Hotseat
Teacher observations
Performance tasks
Homework/classwork
Cooperative learning
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
SmartBoard
Geometers Sketchpad
Elmo document camera
Base, base angles of a trapezoid, consecutive angles ,isosceles trapezoid, kite, midsegment of a trapezoid,
parallelogram, rectangle, rhombus, square, trapezoid
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
12
Inequalities in Geometry
(A level only)
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Understand and use algebraic skills and concepts
Essential Question: How can proving one relationship help you prove another relationship? How can a variety of appropriate strategies
be applied to solving geometric problems?
Learning Goals: Students will:
Apply properties of inequalities to positive numbers, lengths of segments, and measures of angles.
Understand the relationship between logically equivalent statements.
Understand the concept of indirect proof.
Know and apply inequality theorems for one triangle.
Know and apply inequality theorems for two triangles.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher-facilitated class discussions
Student inquiry through guided investigation
Cooperative learning
Exposure to discrete mathematics proofs by contradiction
Teacher observations
Performance tasks
Homework/classwork
Cooperative learning
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
SmartBoard
Geometers Sketchpad
Elmo document camera
Indirect proof, indirect reasoning
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
14
Similar Polygons
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Demonstrate number sense
4. Understand and use algebraic skills and concepts
Essential Question: How do geometric relationships and measurements help solve real-world problems?
Learning Goals: Students will:
Be able to express a ratio in simplest form.
Be able to solve for an unknown term in a given proportion and write proportions in equivalent form.
Know and apply the properties of similar polygons.
Be able to use similarity postulates to prove triangles similar.
Be able to use similar triangles to deduce information about segments and angles.
Apply the triangle proportionality theorem.
Know and apply the triangle angle bisector theorem.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Guided practice
Cooperative learning
Teacher observations
Performance tasks
Homework/classwork
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
Protractor
Ruler
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Cross-product property, extended proportion, geometric mean, means-extremes product theorem, proportion,
scale, similar, similarity ratio, ratio
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
16
Right Triangles and Trigonometry
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Demonstrate number sense
4. Understand and use algebraic skills and concepts
Essential Question: How do geometric relationships and measurements help solve real-world problems?
Learning Goals: Students will:
Determine the geometric mean between two numbers.
Know and apply the Pythagorean Theorem.
Know and apply the converse of the Pythagorean Theorem and related theorems about triangles including Pythagorean triples.
Know and understand side relationships in 30–60–90 and 45-45-90 triangles.
Define tangent, sine, and cosine ratios.
Be able to solve right triangle problems using basic trigonometry, including application of angles of elevation/depression.
Be able to solve right triangle applications.
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Suggested Strategies
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Suggested Assessments
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Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Review simplifying radicals
Pneumonics
Pythagorean dissection proofs with manipulatives
Games for reinforcement and review
o Jeopardy
o Matho (like Bingo)
Teacher observations
Performance tasks
o Indirect measurement
Homework/classwork
Cooperative learning
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Adjacent side, cosine, hypotenuse ,inverse cosine, inverse sine, inverse tangent, legs of a right triangle,
opposite side, Pythagorean theorem, Pythagorean triple, sine, tangent
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
18
19
Circles
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Demonstrate number sense
4. Understand and use algebraic skills and concepts
Essential Question: How do circles and their parts relate to the physical world?
Learning Goals: Students will:
Define terms relating to circles and spheres.
Be able to recognize inscribed polygons and circumscribed circles.
Apply theorems related to tangents and radii.
Define and apply properties of arcs and central angles.
Apply theorems about the chords of a circle.
Be able to solve problems involving inscribed angles and angles formed by chords, secants, and tangents.
Be able to solve problems involving lengths of chords, secants, and tangent segments.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Guided practice
Cooperative learning
Teacher observations
Performance tasks
Homework/classwork
Writing assignment
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Arc, arc length, center of a circle, central angle, chord, circumscribed about, inscribed angle, inscribed in,
intercepted arc, point of tangency, secant, tangent to a circle
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
21
Areas of Plane Figures
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Understand and use algebraic skills and concepts
4. Understand and use basic concepts of probability and statistics
Essential Question: How do geometric relationships and measurements help solve real-world problems?
Learning Goals: Students will:
Understand the meaning of area of a polygon.
Know and understand the area of parallelograms, triangles, rhombus, trapezoid, and kite.
Know and use the formula for the areas of regular polygons.
Know and use the formulas for the circumferences and areas of circles that are derived from formulas related to regular polygons.
Know and use the formulas for arc lengths and the areas of sectors of a circle.
Be able to find the ratio of areas of two triangles.
Understand and apply the notion of scale factor.
Be able to use areas to solve problems involving geometric probability.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Review simplifying radicals
Pneumonics
Pythagorean dissection proofs with manipulatives
Teacher observations
Performance tasks
o Why do the bees choose the hexagon?
Homework/classwork
Cooperative learning
CFA(s)
CAPT released items
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Altitude of a parallelogram, apothem, base of a parallelogram, base of a triangle, center of a regular polygon,
central angle, circumference, concentric, diameter, geometric probability, height of a parallelogram, height of
a trapezoid, height of a triangle, major arc, minor arc, pi, radius of a circle, radius of a polygon, sector,
segment, semi-circle
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
23
Areas and Volumes of Solids
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
3. Understand and use algebraic skills and concepts
Essential Question: How do geometric relationships and measurements help solve real-world problems?
Learning Goals: Students will:
Identify the parts of a prism.
Be able to find lateral areas, total areas, and volumes of right prisms.
Identify and solve lateral areas, total areas, and volumes of pyramids.
Be able to find the lateral areas, total areas, and volumes of cylinders and cones.
Be able to find the area and volume of a sphere.
State and apply the proportion of similar solids.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Guided practice
Cooperative learning
Teacher observations
Performance tasks
Homework/classwork
CFA(s)
CAPT released items
Quizzes
Unit test
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Altitude, base(s), center of a sphere, circumference of a sphere, cone, cross-section, cube, cylinder, diameter of
a sphere, edge, face, height, hemisphere, lateral area, lateral faces, oblique cylinder, oblique prism, prism,
pyramid, radius of a sphere, regular pyramid, right cone, right cylinder, right prism, slant height, sphere,
surface area, vertex, volume
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
25
Constructions
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
Essential Question: How do hands-on compass-and-straightedge constructions relate to postulates and theorems? How can a variety of
appropriate strategies be applies in solving geometric problems?
Learning Goals: Students will:
Be able to construct congruent segments and angles.
Be able to construct the perpendicular bisector of a segment.
Be able to construct the bisector of an angle.
Be able to construct a line parallel to a given line and through a given point that is not on the line.
Be able to construct the perpendicular to a given line at a given point on the line.
Be able to construct the perpendicular to a given line through a given point not on the line.
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Suggested Strategies
Suggested Assessments
Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Cooperative learning
Guided practice
Teacher observations
Performance tasks
o Euler line, Islamic art, 9-point circle construction project / writing assignment
Classwork
Cooperative learning
Quizzes
Unit test
Compass
MIRA
Straightedge
Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level)
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Angle bisector, compass, construction, perpendicular bisector, reflection, straight edge
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
27
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Coordinate Geometry
(A level only)
Essential Understandings:
1. The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Content Standards:
1. Use a variety of strategies in the problem solving process
2. Understand and use spatial relationships and basic concepts of geometry
Essential Question: When is an analytic proof more effective than a 2-column proof? How do alternative methods help lead to choosing
the most effective problem solving method? How does algebra relate to geometry graphically?
Learning Goals: Students will:
Apply the distance, midpoint, and slope formulas.
Be able to write a coordinate proof.
Be able to place figures on a coordinate plane using their properties.
Be able to write an analytic proof.
Be able to relate slope to parallel and perpendicular lines.
Compare and contrast 2-column and analytic proofs.
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Suggested Strategies
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Suggested Assessments
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Suggested Resources
Suggested Tech Integration
Content Vocabulary
Lifelong Learning/21st Century Skills
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Teacher facilitated class discussions
Review simplifying radicals
Pneumonics
Pythagorean dissection proofs with manipulatives
Cooperative learning
Games for reinforcement and review
o Jeopardy
o Matho
Teacher observations
Performance tasks
Homework/classwork
Writing assignment
Quizzes
Tic-tac-toe analytic proofs take-home assessment
Unit test
Prentice Hall Geometry
Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com
TI graphing calculators
Geometers Sketchpad
SmartBoard
Elmo document camera
Analytic proof, coordinate, coordinate plane, coordinate proof, point-slope form of a liner equation, slopeintercept form of a linear equation, standard form of a linear equation
Productive habits of mind
Quality work
Read critically
Communicate effectively
Collaborate and cooperate
Access and process information
Core Ethical Values
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