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Geometry

Approved January 2012

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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 process2. 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 Using graphing calculators Using CFA data DI according to student readiness Small group and whole class instruction

Suggested Assessments Quizzes CFA’s Unit test

Suggested Resources Text: Holt Geometry (B & C Level); Prentice Hall Geometry (A level) Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com

Suggested Tech Integration TI graphing calculators Geometers Sketchpad SmartBoard Elmo document camera

Content Vocabulary 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

Lifelong Learning/21st Century Skills Productive habits of mind Quality work Read critically Communicate effectively Collaborate and cooperate Core Ethical Values

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Deductive Reasoning



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 Teacher-facilitated class discussions Cooperative learning Venn diagrams Replacing words with symbols

Suggested Assessments Project–logic poster Performance tasks Writing assignment Union/intersection worksheet(s) (A level) CFA(s) Quizzes Unit test

Suggested Resources 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


Content Vocabulary 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

Lifelong Learning/21st Century Skills Productive habits of mind Quality work Read critically Communicate effectively Collaborate and cooperate Access and process information Core Ethical Values

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http://fliiby.com/file/412746/7tr3jbb4k7.html

Parallel Lines and Planes



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 Teacher-facilitated class discussions Student inquiry through guided investigations Cooperative learning Constructions

Suggested Assessments Writing assignment CFA(s) Quizzes Unit test

Suggested Resources 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


Content Vocabulary 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

Lifelong Learning/21st Century Skills Productive habits of mind Quality work Read critically Communicate effectively Collaborate and cooperate Core Ethical Values

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Triangles


Content Standards:1. Use a variety of strategies in the problem solving process2. Understand and use spatial relationships and basic concepts of geometry3. 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 Teacher facilitated class discussions Student inquiry through guided investigations:

o Congruent triangles investigationo Paper tearing and paper folding to discover and/or reinforce conceptso Coffee stirrer and gumdrop (or marshmallow) bridge design

Games for reinforcement and reviewo Jeopardyo Hotseato Board raceso Chutes and Ladders

Suggested Assessments Teacher observations Performance tasks Homework/classwork Cooperative learning Writing assignment Quizzes Unit test

Suggested Resources 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)

Suggested Tech Integration Protractor Ruler TI graphing calculators Geometers Sketchpad SmartBoard Elmo document camera

Content Vocabulary 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

Productive habits of mind Quality work Read critically Communicate effectively Collaborate and cooperate Access and process information Core Ethical Values

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Quadrilaterals



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 Teacher-facilitated class discussions Student inquiry through guided investigation

o Venn diagramo Card sorting activity

Graphic organizer Cooperative learning Games for reinforcement and review

o Jeopardyo Hotseat



Suggested Tech Integration SmartBoard Geometers Sketchpad Elmo document camera

Content Vocabulary Base, base angles of a trapezoid, consecutive angles ,isosceles trapezoid, kite, midsegment of a trapezoid, parallelogram, rectangle, rhombus, square, trapezoid

Lifelong Learning/21st Century Skills

Productive habits of mind Quality work Read critically Communicate effectively Collaborate and cooperate Access and process information Core Ethical Values

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Inequalities in Geometry (A level only)



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 Teacher-facilitated class discussions Student inquiry through guided investigation Cooperative learning Exposure to discrete mathematics proofs by contradiction



Suggested Tech Integration SmartBoard Geometers Sketchpad Elmo document camera

Content Vocabulary Indirect proof, indirect reasoningLifelong Learning/21st Century Skills Productive habits of mind

Quality work Read critically Communicate effectively Collaborate and cooperate Access and process information Core Ethical Values

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Similar Polygons


Content Standards:1. Use a variety of strategies in the problem solving process2. Understand and use spatial relationships and basic concepts of geometry3. Demonstrate number sense4. 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 Teacher facilitated class discussions Guided practice Cooperative learning

Suggested Assessments Teacher observations Performance tasks Homework/classwork Writing assignment Quizzes Unit test


Suggested Tech Integration Protractor Ruler TI graphing calculators Geometers Sketchpad SmartBoard Elmo document camera

Content Vocabulary Cross-product property, extended proportion, geometric mean, means-extremes product theorem, proportion, scale, similar, similarity ratio, ratio


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Right Triangles and Trigonometry




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 Teacher facilitated class discussions Review simplifying radicals Pneumonics Pythagorean dissection proofs with manipulatives Games for reinforcement and review

o Jeopardyo Matho (like Bingo)

Suggested Assessments Teacher observations Performance tasks

o Indirect measurement Homework/classwork Cooperative learning Writing assignment Quizzes Unit test



Content Vocabulary Adjacent side, cosine, hypotenuse ,inverse cosine, inverse sine, inverse tangent, legs of a right triangle, opposite side, Pythagorean theorem, Pythagorean triple, sine, tangent


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Circles



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 Assessments Teacher observations Performance tasks Homework/classwork Writing assignment Quizzes Unit test



Content Vocabulary 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


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Areas of Plane Figures


Content Standards:1. Use a variety of strategies in the problem solving process2. Understand and use spatial relationships and basic concepts of geometry3. Understand and use algebraic skills and concepts4. Understand and use basic concepts of probability and statistics


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 Teacher facilitated class discussions Review simplifying radicals Pneumonics Pythagorean dissection proofs with manipulatives


o Why do the bees choose the hexagon? Homework/classwork Cooperative learning CFA(s) CAPT released items Quizzes Unit test



Content Vocabulary 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


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Areas and Volumes of Solids




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 Assessments Teacher observations Performance tasks Homework/classwork CFA(s) CAPT released items Quizzes Unit test



Content Vocabulary 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


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Constructions



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 Teacher facilitated class discussions Cooperative learning Guided practice


o Euler line, Islamic art, 9-point circle construction project / writing assignment Classwork Cooperative learning Quizzes Unit test

Suggested Resources 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


Content Vocabulary Angle bisector, compass, construction, perpendicular bisector, reflection, straight edgeLifelong Learning/21st Century Skills Productive habits of mind

Quality work Read critically Communicate effectively Collaborate and cooperate Access and process information Core Ethical Values

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Coordinate Geometry(A level only)



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 Teacher facilitated class discussions Review simplifying radicals Pneumonics Pythagorean dissection proofs with manipulatives Cooperative learning Games for reinforcement and review

o Jeopardyo Matho

Suggested Assessments Teacher observations Performance tasks Homework/classwork Writing assignment Quizzes Tic-tac-toe analytic proofs take-home assessment Unit test

Suggested Resources Prentice Hall Geometry Internet sites: My.HRW.com (online companion website for Holt Geometry); purplemath.com


Content Vocabulary Analytic proof, coordinate, coordinate plane, coordinate proof, point-slope form of a liner equation, slope-intercept form of a linear equation, standard form of a linear equation


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