Geometry: Goals and Course Outline

 

 

 

DEPARTMENT: Mathematics                                         DATE: 2007-2008

COURSE TITLE: Geometry                                             COURSE NUMBER:  5210

o YEAR                                                                       QPA: 4.5

o SEMESTER                                                            CREDITS: 5

 

 

 

TEXTBOOK: (Title, Author, Publisher, Edition)

McDougal Littell Geometry (Both print and electronic editions), Larson, Boswell, Kanold, & Stiff, McDougal Littell Inc., 2007

 

MATERIALS USED:  TI-84 Plus graphing calculators, IBM ThinkPad, supplementary

      materials as supplied by McDougal Littell, protractor and compass

 

GOALS:  

1.      To provide a basic comprehensive course in geometry that is very concrete and visual.

2.      To discern patterns and recognize properties of two-dimensional geometric shapes.

3.      To reinforce the fundamentals of algebra in a geometric setting.

4.      To prepare students for the Math portions of the SAT.

5.      To use visualization, spatial reasoning, and geometric modeling to solve pro-blems.

6.      To use graphing calculators and measurement tools to analyze, solve, visualize, and clarify geometric concepts.

 

 

CONTENT OF COURSE:

 

FIRST SEMESTER

 

Unit 1: Fundamentals of Geometry

1.      Naming points, lines, rays, line segments and planes

2.      Segment addition and congruence

3.      Finding the midpoint and length of a line segment on a coordinate plane

4.      Measuring and classifying angles

5.      Angle addition

6.      Special angle pair relationships

7.      Classifying polygons

8.      Perimeter, circumference, and area measurements

Unit 2: Reasoning and Proof

1.      Using inductive reasoning to arrive at conjectures about observed events

2.      Testing the validity of conjectures

3.      Converting factual statements into conditional statements

4.      Writing the converse of conditional statements

5.      Point, line, and plane postulates

6.      Review algebraic properties of equality

7.      Right Angle Congruence Theorem

8.      Linear Pair Postulate

9.      Vertical Angles Congruence Theorem

 

Unit 3:  Parallel and Perpendicular Lines

  1. Review the concepts of parallel and perpendicular lines and planes
  2. Skew lines
  3. Special angle pairs formed when a transversal intersects a pair of lines
  4. Measurements of special angle pairs formed when the lines intersected by a transversal are parallel
  5. Review of slopes of lines
  6. Review of writing and graphing equations of lines
  7. Perpendicular transversal theorem

 

SECOND SEMESTER

 

Unit 4:  Congruent Triangles

1.      Classification of triangles by side lengths

2.      Classification of triangles by angle measurements

3.      Triangle Sum Theorem

4.      Exterior Angle Theorem

5.      Properties of congruent figures

6.      Prove triangles are congruent by SSS, SAS, ASA, and AAS

7.      Real life applications of triangle congruence

8.      Isosceles and equilateral triangles

9.      Congruence transformations

 

Unit 5:  Relationships within Triangles

1.      Midsegment Theorem

2.      Perpendicular bisectors

3.      Angle bisectors

4.      Medians

5.      Altitudes

6.      Relationship between side length and angle measurement in a triangle

7.      Triangle Inequality Theorem

 

Unit 6:  Similarity

1.      Ratios

2.      Proportions

3.      Geometric mean

4.      Using proportions to solve problems

5.      Similar polygons

 

Unit 7:  Right Triangles

1.      Pythagorean Theorem

2.      Pythagorean triples

3.      Converse of Pythagorean Theorem

4.      Special right triangles

 


 

Geometry Unit Proficiencies

 

Students will be able to…

 

Unit 1:   Fundamentals of Geometry

  1. Identify points, lines, line segments, rays and planes.
  2. Symbolically name a point, a line, a line segment, a ray and a plane.
  3. Recognize problematic situations that call for segment addition.
  4. Calculate the midpoint and length of a line segment.
  5. Identify segment congruence.
  6. Name, measure, and classify angles.
  7. Recognize problematic situations that call for angle addition.
  8. Identify special angle pairs: complementary, supplementary, linear pair and vertical angles.
  9. Classify polygons.
  10. Find the area and perimeter of rectangles and triangles.
  11. Find the circumference and area of circles.

 

Unit 2:  Reasoning and Proof

  1. Use inductive reasoning to form conjectures about observed events.
  2. Determine the validity of a conjecture and present a counterexample to demon-strate the falsity of an invalid conjecture.
  3. Convert a factual statement into a conditional statement.
  4. Identify the hypothesis and conclusion of a conditional statement.
  5. Write the converse of a conditional statement.
  6. Interpret geometric diagrams by means of point, line, and plane postulates.
  7. Appreciate that algebraic properties of equality substantiate all of the steps used to solve an algebraic equation.
  8. Understand that all right angles are congruent.
  9. Understand that angles that form a linear pair are supplementary.
  10. Understand that vertical angles are congruent.

 

Unit 3:  Parallel and Perpendicular Lines

  1. Recognize parallel, perpendicular and skew lines and understand the distinctions between them.
  2. Distinguish between corresponding, alternate interior, alternate exterior and con-secutive interior angles formed when a transversal intersects a pair of lines.
  3. Determine the measurements of all eight angles formed when a pair of parallel lines are intersected by a transversal when only one of the angle measurements is known.
  4. Determine the slope of a line when viewing its graph.
  5. Calculate the slope of a line algebraically by means of the slope formula.
  6. Write the equation of a line when presented with its graph.
  7. Appreciate that the perpendicular transversal theorem is an extension of angle re-lationships present when a pair of parallel lines are intersected by a transversal.

 

Unit 4:  Congruent Triangles

1.      Distinguish between scalene, isosceles, and equilateral triangles.

2.      Distinguish between acute, right, obtuse, and equiangular triangles.

3.      Determine the measurement of a missing angle of a triangle by means of the Triangle Sum Theorem.

4.      Understand how the measurement of an exterior angle of a triangle relates to the measurements of its interior angles.

5.      Identify all pairs of congruent corresponding parts of congruent figures.

6.      Determine if figures are congruent.

7.      Appreciate that congruent triangles can be used to find distances that are difficult to measure directly.

8.      Create an image congruent to a given figure in the coordinate plane.

9.      Distinguish between the three main types of transformations.

 

Unit 5:  Relationships within Triangles

1.      Identify special triangle segments.

2.      Use the Midsegment Theorem to find lengths within a triangle.

3.      Understand the properties of points located on perpendicular or angle bisectors.

4.      Explain how triangle side lengths relate to angle measurements.

5.      List the sides of a triangle in order when given its angle measurements and vice versa.

6.      List possible lengths of the third side of a triangle when the lengths of its two other sides are given.

 

Unit 6:  Similarity

1.      Simplify a ratio.

2.      Use ratios to find missing dimensions.

3.      Recognize a proportion and use the cross products property to solve it.

4.      Recognize when a proportion approach is appropriate for solving real world pro-blems.

5.      Calculate the geometric mean of two numbers and present it in simplest radical form.

6.      Identify similar polygons by investigating angle measurements and proportional-ity of corresponding sides.

7.      Determine the scale factor of similar polygons and use it to determine lengths of missing dimensions.

8.      Appreciate that similar polygons also exhibit proportionality of perimeters and in-ternal segment lengths.

 

Unit 7:  Right Triangles

1.      Apply the Pythagorean Theorem to determine the length of a missing side of a right triangle.

2.      Recognize common Pythagorean triples and their multiples.

3.      Understand that these Pythagorean triples can be put to use to quickly arrive at the length of a missing side of a right triangle.

4.      Use the Pythagorean Theorem to classify a triangle as right, acute, or obtuse.

5.      Understand the relationship between side lengths in 45-45-90 and 30-60-90 trian-gles.

6.      Quickly present the length in simplest radical form of a missing side in 45-45-90 and 30-60-90 by means of these relationships.