Modern Geometry

Mr. Al Bierschbach

Al.Bierschbach@k12.sd.us

Madison High School

 

Textbook: Geometry. Authors: Larson, Boswell, Kanold, Stiff.  McDougal Littell (2007).    ISBN# 978-0-618-59557-0

 

Description of course

 

Modern geometry integrates plane geometry, solid geometry and algebra onto a unified mathematics course that is primarily designed to make the individual think.  The course begins by explaining and demonstrating falsities that occur in everyday thinking.  Students discover that intuitive and inductive thinking are nothing but calculated guessing and that deductive reasoning, which the course dwells on, is the only absolute method with which one should work to draw valid conclusions.  Modern geometry is a fine foundation for Algebra and is required as a pre-requisite for taking Advanced Algebra.  Trigonometry and coordinate geometry are both introduced is this course, and more for purpose explaining their operations than to master the content.

 

STATE  OF SOUTH DAKOTA MATHEMATICS STANDARDS COVERED

 

9-12.G.1.1. Apply the properties of triangles and quadrilaterals to find unknown parts.

 

9-12.G.1.2.  Identify and apply relationships among triangles.

 

9-12.G.2.1.  Recognize the relationship between a three-dimensional figure and its two-dimensional representation.

 

9-12.G.2.2. Reflect across vertical or horizontal lines, and translate two-dimensional figures.

 

9-12.G.2.3. Use proportions to solve problems.

 

9-12.G.1.1.A. Justify properties of geometric figures.

 

9-12.G.1.2A.  Determine the values of the sine, cosine, and tangent ratios of right triangles

 

9-12.G.1.3A. Apply properties associated with circles.

 

9-12.G.1.4A. Use formulas for surace area and volume to solve problems involving three-dimensional figures.

 

9-12.G.2.1A. Use Cartesian coordinates to verify geometric properties.

 

9-12.M.1.1.  Choose appropriate unit label, scale and precision.

 

9-12.M.1.2.  Use suitable units when describing rate of change.

 

9-12.M.1.3.  Use formulas to find perimeter, circumference, and area to solve problems involving common geometric figures.

 

9-12.N.1.1.  Identify multiple representations of a real number.

 

9-12.A.1.1.  Write equivalent forms of algebraic expressions using properties of the set of real numbers.

 

9-12.A.1.2.  Use algebraic properties to transform multi-step, single variable, and first degree equations.

 

9-12A.4.1.  Use graphs, tables, and equations to represent linear functions.

 

Topics covered:

 

  1. Essentials of Geometry
    1. Identify Points, Lines, and Planes
    2. Use Segments and Congruence         
    3. Use Midpoint and Distance Formulas
    4. Measure and Classify Angles
    5. Describe Angle Pair Relationships
    6. Classify Polygons
    7. Find Perimeter, Circumference and Area      

 

  1. Reasoning and Proof
    1. Use Inductive Reasoning
    2. Analyze Conditional Statements
    3. Apply Deductive Reasoning
    4. Use Postulates and Diagrams
    5. Reason Using Properties and Diagrams
    6. Prove Statements about Segments and Angles
    7. Prove Angle Pair Relationships

 

 III.            Parallel and Perpendicular Lines

    1. Identify Pairs of Lines and Angles
    2. Use Parallel Lines and Transversals
    3. Prove Lines Parallel
    4. Find and Use Slopes of Lines
    5. Write and Graph Equations of Lines
    6. Prove Theorems about Perpendicular Lines

 

  1. Congruent Triangles
    1. Applying Triangle Sum Properties
    2. Apply Congruence and Triangles
    3. Prove Triangles Congruent by SSS
    4. Prove Triangles Congruent by SAS and HL
    5. Prove Triangles Congruent by ASA and AAS
    6. Use Congruent Triangles
    7. Use Isosceles and Equilateral Triangles
    8. Perform Congruence Transformations

 

    V.             Relationships within Triangles

a.       Midsegment Theorem and Coordinate Proof

b.      Use Perpendicular Bisectors

c.       Using Angle Bisectors of Triangles

d.      Use Medians and Altitudes

e.       Use Inequalities in a Triangle

f.       Inequalities in Two Triangles and Indirect Proof

 

 VI.             Similarity

a.       Ratios, Proportions, and Geometric Mean

b.      Use Proportions to Solve Geometry Problems

c.       Use Similar Polygons

d.      Prove Triangles Similar by AA

e.       Prove Triangles Similar by SSS and SAS

f.       Use Proportional Theorems

g.      Perform Similarity Transformations

 

VII.             Right Triangles and Trigonometry

a.       Apply the Pythagorean Theorem

b.      Use the Converse of the Pythagorean Theorem

c.       Use Similar Right Triangles

d.      Special Right Triangles

e.       Apply the Tangent Ratio

f.       Apply the Sine and Cosine Ratios

g.      Solve Right Triangles

 

VIII.             Quadrilaterals

a.       Find Angle Measures in Polygons

b.      Use Properties of Parallelograms

c.       Show that a Quadrilateral is a Parallelogram

d.      Properties of Rhombuses, Rectangles, and Squares

e.       Use Properties of Trapezoids and Kites

f.       Identify Special Quadrilaterals

 

 IX.             Properties of Transformations

a.       Translate Figures and Use Vectors

b.      Use Properties of Matricies

c.       Perform Reflections

d.      Apply Compositions of Transformations

e.       Identify Symmetry

f.       Identify and Perform Dilations

 

    X.             Properties of Circles

a.       Use Properties of Tangents

b.      Find Arc Measures

c.       Apply Properties of Chords

d.      Use Inscribed Angles and Polygons

e.       Apply Other Angle Relationships in Circles

f.       Find Segment Lengths in Circles

g.      Write and Graph Equations and Circles

 

 XI.             Measuring Length and Area

a.       Areas of Triangles and Parallelograms

b.      Areas of Trapezoids, Rhombuses and Kites

c.       Perimeter and Area of Similar Figures

d.      Circumference and Arc Length

e.       Areas of Circles and Sectors

f.       Areas of Regular Polygons

g.      Use Geometric Probability

 

XII.            Surface Area and Volume of Solids

a.       Explore Solids

b.      Surface Area of Prisms and Cylinders

c.       Surface Area of Pyramids and Cones

d.      Volumes of Prisms and Cylinders

e.       Volume of Pyramids and Cones

f.       Surface Area and Volume of Spheres

g.      Explore Similar Solids

 

 

Grading of Course

10% - Notetaking Guide – Students will be expected to take notes and turn them in using a notetaking guide.

15% - Homework Grade – Homework will be graded for every assignment.  It is expected to be done within certain time parameters.

25% - Quiz Grade – Quizzes will be taken after every two or three topics (sections) in each chapter.

40% - Test Grade – A chapter test will be taken at the end of each chapter.

10% - Computer Grade – Students will be given 10 point per day for following computer usage rules. 

 

Rules/Guidelines of the class:

Students will have respect for the classroom, its property, their fellow students and the teacher.  Pop or food is not allowed in the classroom.