CURRICULUM MAP
Subject: Geometry SY: 2009 - 2010 Course Description: This comprehensive course integrates algebra with the skills and concepts of geometry and features coverage of geometric terms and processes. It introduces points and lines, angles, perpendicular and parallel lines, polygons, triangle congruence and similarity, measurement, three dimensional figures, areas and volumes, coordinate geometry, logic and indirect reasoning. Various forms of proof are studied. Emphasis is placed upon deductive and inductive reasoning, investigative strategies in drawing conclusions, and problem solving. Use of these geometric concepts and skills will be applied towards relevant life situations as well as those of professional significance. Course Goal: The different lessons and activities throughout this course aim to develop in the students the following skills and complex thinking processes: 1. Communication – The ability to read, write, listen, ask questions, think, and communicate about math will develop and deepen students’ understanding of mathematical concepts. Students should also be able to explain answers, justify mathematical reasoning, and describe problem-solving strategies using correct mathematical vocabulary. 2. Representation – The language of mathematics is expressed in words, symbols, equations, figures and graphs. The students should be able to correctly interpret them. Activities in the course would require them to correctly interpret these representations, as well as make their own by drawing figures of graphs of their answers or conclusions. 3. Supported Logical Reasoning – The students should be able to give valid arguments using inductive or deductive reasoning supported by previous knowledge or even new information gained from investigation using manipulatives. 4. Problem-Solving – Students will meet new and unfamiliar situations that challenge them to apply what they have previously learned and come up with a solution. Their creativity and critical; examination of all the possibilities would allow them to “think outside the box”, a skill that is valued even in other aspects of life. General Objectives: At the end of the school year, the students should be able to: 1. verify and construct different types of proofs of theorems, congruency and similarity using the axiomatic system, 2. perform basic constructions of lines, segments, angles, polygons and circles using a straight-edge and compass, 3. graph representations of lines, circles, functions and conics in the coordinate plane, 4. create different artworks, game boards, models, and other projects by applying geometric concepts, and 5. solve real-life application problems that has great implication on the importance of geometry in different possible professions or career. Level: Third Year High School Teacher: Fidelfo C. Moral Jr.
Time Frame
Topics The Language and Logic of Geometry Using Patterns and Inductive Reasoning Overview of Geometry History Organization: the Axiomatic System Importance of Geometry Points, Lines and Planes If-Then Statements Properties from Algebra Proving Theorems Segments and Rays Measuring Length of Segments Construction of Congruent Segments Betweeness and Midpoint Segment Bisectors Opposite Rays Angles Parts Measuring Angles Types Adjacent Angles Angle Bisectors Angle Pairs Complementary and Supplementary Angles Linear Pairs Vertical Angles
Learning Objectives At the end of this unit, the students will be able to: produce a valid conjecture from a pattern/situation using inductive reasoning, trace the history and development of geometry by citing significant developments in the discipline, evaluate the practical applications of geometry in nature, art and other professions such as engineering and architecture, illustrate representations of real-life applications of the concepts points, lines, and planes, and their relationships, determine coordinates of the endpoints, midpoint and/or other points in a line using the distance formula, construct an angle/segment congruent to a given angle/segment and an angle bisector using a compass and a straight-edge, measure angles using the protractor and/or mathematical computations to correctly classify them, apply the different characteristics of angle pairs in finding unknown measures in given situations, identify convex and disjoint sets, and halfplanes in given figures, recognize and make use of algebraic properties as justification for steps in geometric problems, and create formal proofs of theorems and other statements.
Learning Activities PowerPoint presentation Game: Developing logical and inductive reasoning Lecture with discussion: Tracing the History, Organization and Importance of Geometry Student presentation on the existence of geometry in the arts and nature Project: Origami-making using recycled materials Class discussion facilitated using an open, used crate Small group discussion Rearranging a sequence of steps in a twocolumn direct proof Hands-on: Measuring segments and angles using ruler and protractor Project: Making clock replicas using recycled materials Hands-on: Teacher and student demonstrations on geometric constructions Interpreting figures and identifying geometric concepts Constructing formal proofs of some postulates, theorems and statements using algebraic properties and geometric concepts Exercises and seatwork
Resources Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Values Integrated Team work Visualization Patience Precision Listening Skills Respect Excellence Logical Thinking Deductive Reasoning Inductive Reasoning Good Work Ethics
First Semester
1st – 3rd Week
Perpendicular, Skew and Parallel Lines Intersecting Lines Perpendicular Lines Perpendicular Bisectors Skew Lines Parallel Lines Angle Relationships with Transversals Parallel Lines and Triangles
At the end of this unit, the students will be able to: differentiate perpendicular, skew and parallel lines by giving illustrative examples, construct perpendicular bisectors, perpendicular lines and parallel lines using a compass and a straight-edge, investigate the relationship between the angles formed when a pair of parallel lines is intersected by a third line, and apply these relationships in proving two lines parallel, and in solving real problems. At the end of this unit, the students will be able to: correctly classify given triangles based on the measure of their sides and angles, devise a creative method to prove the Triangle Angle Sum and the Exterior Angle Theory using cut-outs of triangles, utilize these theorems in solving relevant problems, construct a triangle given some of its parts using a compass and a straight-edge, and draw a triangles’ median, altitude and angle bisectors to construct circles inscribed in it and circles circumscribing it.
First Semester
3rd – 4th Week
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Hands-on: Teacher and student demonstrations on geometric construction Investigation: finding the relationship of angles when a pair of parallel lines is cut by a transversal Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork PowerPoint presentation Lecture with discussion Class discussion Small group discussion Investigation: Using manipulatives to investigate the relationship of a triangle’s interior and exterior angles Flash Presentation: Angle Sum Theorem Hands-on: Teacher and student demonstrations on geometric constructions Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.trenton.k12.nj.us www.wikipedia.org
Team work Positivity Sense of Direction Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
First Semester
5th Week
Triangles Types of Triangles Angle Measurements Sum of Interior Angles Exterior Angle of a Triangle Theorem Medians, Altitudes and Bisectors
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.hotmath.com www.wikipedia.org
Self-study Team work Positivity Sense of Direction Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Congruent Triangles Correspondence and Congruence Proving Triangles Congruent Using Congruent Triangles Proving Right Triangles Congruent Congruence in a Single Triangle: Isosceles Triangle Theorem
At the end of this unit, the students will be able to: illustrate the corresponding parts of two congruent triangles, prove two triangles congruent using SSS, SAS, ASA and SAA Theorem using the two-column proof method, relate these theorems in proving right triangles congruent, and for congruency in an isosceles triangle, and apply the concepts of triangle congruency in solving real problems.
First Semester
6th – 8th Week
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Investigation: Implication of the congruence theorems using manipulatives Constructing formal proofs of some postulates, theorems and statements using algebraic properties and geometric concepts Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork PowerPoint presentation Lecture with discussion Investigation: Using licorice lace to discover the theorem on triangle length inequalities of a triangle Class discussion Small group discussion Rearranging a sequence of steps in a twocolumn indirect proof Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Self-study Team work Positivity Sense of Direction Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
First Semester
9th – 10th Week
Inequalities in a Triangle Properties of Inequality Indirect Proof and Inequalities Mid-Segment Properties in Triangles Side and Angle Inequalities Hinge Theorem
At the end of this unit, the students will be able to: formulate the side and angle inequalities in a triangle through inquiry, apply properties of inequality to measures of segments and angles, write indirect proofs involving inequalities, and apply the inequality relations for one and two triangles.
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
11th – 12th Week
First Semester
Polygons Naming Polygons Regular Polygons Convex and Non-convex Diagonals Angle Measurements Central Angles Interior Angles Exterior Angles
At the end of this unit, the students will be able to: differentiate convex and non-convex, and regular and non-regular polygons, and deduce the mathematical relationship between a (regular) polygon’s number of sides and the measure of its central, interior and exterior angles; the sum of all its interior and exterior angles; and it’s number of diagonals.
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Investigation: Deduce the characteristics of a polygon’s central, interior and exterior angles Flash Presentation: Exterior Angles of a Polygon Project: Making Geo-fan using recycled materials Project: Making Polygon Quilt using recycled materials Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork PowerPoint presentation Lecture with discussion Class discussion Small group discussion Project: Constructing geo-boards using recycled materials Investigation: Using geo-boards in examining properties of quadrilaterals Game: What’s my Quadrilateral? Investigation: Discovering the MidSegment Property of Triangles Online Interactive: Quadrilateral Quest Hands-on: Teacher and student demonstrations on geometric constructions Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://teams.lacoe.edu www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
12th – 14th Week
Quadrilaterals The Parallelogram – A Special Quadrilateral Special Parallelograms Mid-segment Theorem Proving Quadrilaterals as Parallelograms Trapezoid and Isosceles Trapezoid Trapezium and Kite Congruent Quadrilaterals
At the end of this unit, the students will be able to: demonstrate all possible shapes of quadrilaterals using geo-boards, correctly classify quadrilaterals using the unique properties of each type, describe the special characteristics of a trapezoids median and a kite’s diagonals, discover the mid-segment theorem by inquiry, and prove quadrilaterals congruent using appropriate theorems or properties.
First Semester
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://teams.lacoe.edu www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Transformational Geometry Size Transformation Scale factor Dilation Lines of Symmetry Symmetry Reflection Rotation Translation Tessellations
At the end of this unit, the students will be able to: perform size transformations given the figure and scale factors, identify the reflection and rotation symmetries of two- and three-dimensional figures, perform reflections and rotations using compass and straight-edge constructions, and show and describe the results of combinations of translations, reflections and rotations to create an artistic design.
PowerPoint presentation Balloon inflation Hands-on: Performing size transformations using scaling grids and isometric dot papers Lecture with discussion Game: Seating arrangement reflections and translations Class discussion Small group discussion Project: Making geometric artworks such as tessellations, Celtic and Arabic designs using recycled materials Using the Mira™ to understand reflection Line of symmetry with photographs Exercises and seatwork PowerPoint presentation Flash Presentation: Scale factors and Similar Triangles Lecture with discussion Class discussion Small group discussion Field Activity: Measuring heights of trees and building using mirrors (and similar triangles) Hands-on: Teacher and student demonstrations on dilation and scaling Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork
First Semester
14th Week
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Second Semester
1st – 3rd Week
Similarity Ratio and Proportion Geometric Mean Properties of Proportions Similar Polygons Proving Triangles Similar More on Similar Triangles Proportional Segments
At the end of this unit, the students will be able to: discover the key characteristics of similar triangles, congruent angles and proportional sides, use scale factors for similar figures to solve problems using proportional reasoning, prove triangles similar using AAA, AA, SAS and SSS Similarity Theorem, and describe similarity within one triangle.
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://hotmath.com www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Second Semester
4th – 6th Week
Right Triangles Pythagorean Theorem Converse of the Pythagorean Theorem Pythagorean Triples Special Right Triangles Right Triangle Similarity
At the end of this unit, the students will be able to: prove the Pythagorean Theorem using tangrams and other methods, and apply the properties of right triangles in reallife situations.
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Project: Making Tangram using recycled materials Hands-on: Proving the Pythagorean Theorem using tangrams and other manipulatives Flash Presentation: Proving Pythagorean Theorem Hands-on: Going outdoors and applying concepts of the right triangle Exercises and seatwork PowerPoint presentation Lecture with discussion Class discussion Small group discussion Hands-on: Teacher and student demonstrations on geometric constructions Rearranging a sequence of steps in a twocolumn direct proof Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://hotmath.com www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Circles Parts of a Circle Concentric and Congruent Circles Inscribed and Circumscribed Circles Arcs, Chords and Central Angles Inscribed Angles Tangents, Secants and Angles Circles and Segment Lengths
At the end of this unit, the students will be able to: describe the relationships between two chords that intersect in the interior of a circle, solve problems and prove statements involving inscribed and intercepted angles formed by chords, secants and tangents, and apply theorems relating lengths of chords, secant segments, and tangent segments.
Second Semester
7th – 10th Week
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.math.wichita.edu www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Measurements of Plane Figures Perimeter of Polygons Area of Squares and Rectangles Area of Triangles Area of Parallelograms and Rhombus Area of Trapezoids Area of Kites Area of Regular Polygons Circumference and Arc Lengths Area of Circles and Sectors Area of Similar Figures Measurement of Geometric Solids Polyhedron and Platonic Solids Euler’s Formula Prism Pyramids Cylinders Cones Spheres Areas and Volumes of Similar Solids
Third Semester
1st – 4th Week
At the end of this unit, the students will be able to: formulate equations for perimeter and areas of plane figures using the geo-board, compute the perimeter, circumference and area of plane figures, and apply these skill in solving real problems.
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Investigation: Using geo-boards to formulate equations for perimeter and areas of plane figures Investigation: Finding the value and universality of pi (π) using manipulatives Investigation: Slicing pizza and comparing sector area Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.math.wichita.edu www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Third Semester
5th – 8th Week
At the end of this unit, the students will be able to: create models for a variety of solid like prisms, pyramids, cylinders and cones, come up with formulas to find the surface area and volume of geometric solids from observing geometric models, and apply these skills in solving real problems.
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Project: Making models of geometric solids and platonic polyhedrons using recycled materials Investigation: Using marshmallows and toothpicks to validate Euler’s formula Investigation: Investigating models of geometric solids to determine how to look for the surface area Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.math.wichita.edu www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Pre-Drafting Orthographic Projection Isometric Projection Third Semester 8th Week
At the end of this unit, the students will be able to: represent the correct interpretations of orthographic projections using building blocks, demonstrates simple orthographic and isometric drawing of some geometric solids, and apply these skills in art and architectural design.
PowerPoint presentation Class discussion Small group discussion Orthogonal Projections with Building Blocks Examination of different works of art Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Team work Patience Precision Artistry Deductive Reasoning Inductive Reasoning Good Work Ethics
Third Semester
9th Week
Trigonometric Ratio Review on Right Triangle Properties Sine, Cosine and Tangent Ratios Application of Trigonometric Ratios
At the end of this unit, the students will be able to: express the trigonometric functions in ratio form, derive the relationship among sine, cosine and tangent ratios, and apply to obtain solutions to real world problems.
PowerPoint presentation Flash Presentation: Slope Triangles Lecture with discussion Class discussion Small group discussion Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://hotmath.com www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Coordinate Geometry The Cartesian Plane Distance Formula Midpoint Formula Slope of a Line Slope of Parallel and Perpendicular Lines Equations of a Line The Equation of a Circle Proving Statements Using Coordinate Geometry Translations in the Coordinate Plane
At the end of this unit, the students will be able to: describe the features of the Cartesian plane, determine the distance between two points and the midpoint of a line segment by applying coordinate geometry, write equations of lines and their parallels and perpendicular given sufficient information, find the general equation of a circle, and plot it in a graph, perform translations in the coordinate plane, and prove previous theorems using the graphical method. At the end of this unit, the students will be able to: describe the graphs of the three basic conics and its parts, and sketch the graphs of these conics given the curves’ equation.
10th – 11th Week
Third Semester
PowerPoint presentation Lecture with discussion Class discussion Small group discussion Game: The Distance Game Hands-on: examining pictures on the coordinate plane Game: Shuffling translation cards Hands-on: The Coordinate Challenge as aid in proving using the coordinate plane Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: www.wikipedia.org www.regentsprep.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
First Semester
Conics Parabola Ellipse Hyperbola 12th Week
PowerPoint presentation Flash Presentation: Generating Conic Shapes Lecture with discussion Class discussion Small group discussion Exercises and seatwork
Books: MSA Geometry Textbook Workbook XP Geometry Geometry III E-Math Geometry Prentice Hall Geometry TE Prentice Hall Geometry 1995 Internet Sources: http://hotmath.com www.wikipedia.org
Team work Patience Precision Deductive Reasoning Inductive Reasoning Good Work Ethics
Date submitted: June 26, 2009 Checked and Approved by:
MR. CARL JESTONI DAKAY Subject Head
MR. MART ANDREW MARAVILLAS Environment Integration
MS. LAURICE PACONLA Vocabulary Bank
MR. NEIL ARNOLD MONTESCLAROS Department Head