Algebra I/Geometry Honors: Goals and Course Outline

 

 

 

DEPARTMENT: Mathematics                                        DATE: 2007-2008

COURSE TITLE: Algebra I/Geometry Honors               COURSE NUMBER:  5150

o YEAR                                                                       QPA: 5.0

o SEMESTER                                                            CREDITS: 5

 

 

 

TEXTBOOKS: (Title, Author, Publisher, Edition)

McDougal Littell Algebra I (Both print and electronic editions), Larson, Boswell, Kanold, & Stiff, McDougal Littell Inc., 2001

 

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 and other resources, protractor

 

GOALS:

1.      To provide a challenging honors level approach preparatory for future honors level courses.

2.      To complete the mastery of the symbolic language of algebra begun in middle school mathematics courses.

3.      At the completion of Algebra I to immediately move into Geometry.

4.      To discern and appreciate the characteristics of two-dimensional geometric shapes.

5.      To competently use formulas defining relationships between two or more real-life quantities in algebraic and geometric scenarios.

6.      To construct formal logical arguments and proofs in  a geometric setting.

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

8.      To demonstrate the relevance of algebraic and geometric problem solving to the real world.

9.      To use geometric manipulatives, graphing calculators and modern computer graphics to analyze, solve, visualize, and clarify mathematical concepts.

 

CONTENT OF COURSE:

 

FIRST SEMESTER - Algebra

 

Unit 1: Review of Algebraic Fundamentals

1.      Definition of a function                                                         

2.      Creating models that represent real-life situations with concomitant unit analysis

3.      Solving linear equations at all levels of difficulty using inverse operations

4.      Formulas

5.      Probability of an Event

 

Unit 2: Graphing Linear Equations and Functions

1.      Plotting points in a coordinate plane

2.      Graphing by means of an input-output table

3.      Interpreting data presented graphically

4.      Intercept points as a quick method of graphing

5.      Investigating the slope of a line

6.      Graphing linear equations by means of the slope-intercept form of a linear equa-tion

7.      Direct variation

8.      Functions and relations

9.      Function notation

 

Unit 3: Writing Linear Equations and Linear Inequalities

1.      Writing linear equations given pertinent information about the graph of the function

2.      Point-slope form of a linear equation

3.      Standard form of a linear equation

4.      Predicting with linear models

5.      Solving linear inequalities and compound inequalities

6.      Graphing linear inequalities in two variables

7.      Solving linear systems by graphing, substitution, or linear combination

 

Unit 4: Exponents and Exponential Functions

1.      Multiplication and division properties of exponents

2.      Zero and negative exponents

3.      Graphing exponential functions

 

Unit 5: Polynomials

1.      Definition and standard form of a polynomial

2.      Classifying polynomials by degree and number of terms

3.      Addition, subtraction and multiplication of polynomials

4.      Using the distributive property to factor polynomials

5.      Factoring trinomials

6.      Factoring difference of two squares pattern

 

Unit 6: Applications of Factoring

1.      Using factoring to solve quadratic equations

2.      Using factoring to simplify rational expressions

3.      Using factoring to multiply and divide rational expressions

 

SECOND SEMESTER – Geometry

Unit 7: 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 8: Reasoning and Proof

1.      Inductive reasoning

2.      Testing the validity of conjectures

3.      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 9:  Parallel and Perpendicular Lines

1.  Types of angles formed when a pair of lines is intersected by a transversal

2.  Angle measurements when parallel lines are intersected by a transversal   

3.  Proving that lines are parallel

4.  Perpendicular transversal theorem

5.  Distance from a point to a line

 

Unit 10:  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.      Prove right triangles are congruent by HL

8.      Real life applications of triangle congruence

9.      Isosceles and equilateral triangles

10.  Congruence transformations

 

Unit 11:  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 12:  Similarity

1.      Ratios, proportions, and the geometric mean

2.      Using proportions to solve problems

3.      Similar polygons

4.      Proving triangles similar

5.      Proportionality theorems

 

Unit 13:  Right Triangles

1.      Pythagorean Theorem

2.      Pythagorean triples

3.      Converse of Pythagorean Theorem

 

 


 

Algebra I/Geometry Honors Proficiencies

 

Core course proficiencies (present in all units)

 

Students will be able to…

  1. Communicate mathematical ideas correctly in oral and written form.
  2. Read for comprehension, demonstrating conceptual understanding.
  3. Make mathematical connections to other subjects and real-life situations.
  4. Confidently problem solve by thinking critically, logically, analytically and ethically.
  5. Use technology and accurate geometric measurements to confirm and enhance analytical techniques presented.
  6. Acquire the mathematical skills and understanding needed to be successful in

their daily lives and future math courses.

 

Algebra I/Geometry Unit Proficiencies:

 

Students will be able to…

 

Unit 1:  Review of Algebraic Fundamentals

1.      Correctly translate verbal phrases into algebraic expressions/equations/inequa-lities.

  1. Create an algebraic model to problem solve in real-life situations.
  2. Select the correct inverse operations in logical sequence for solving linear equa-tions.
  3. Use the “LCD” move to solve rational equations.
  4. Recognize when an equation has one solution, infinite solutions or no solution.
  5. Check the accuracy of a solution by substituting it in the original equation.
  6. Rewrite a formula to solve for any one of its variable components.
  7. Use formulas to solve real-life problems.
  8. Rewrite a two-variable equation in function form.
  9. Identify when a relation is a function.
  10. Define a function’s domain and range and organize this data in table form and as a mapping.
  11. Determine the probability of an event as a tool for predicting the future occur-rence of this event.

 

Unit 2:  Graphing Linear Equations and Functions

  1. Plot a point given its coordinates and identify the quadrant it lies within.
  2. Interpret graphed information by carefully reading axes, understanding the dependent and independent quantities involved, and how the behavior of the graph reveals the relationship between these quantities.
  3. Determine if a point is a solution of a function both graphically and algebraically.
  4. Algebraically determine the x- and y- intercepts when given a linear function and use these points to produce a quick graph.
  5. Algebraically calculate the slope of the graph of a linear function when given two points that lie on its graph.
  6. Appreciate that slope reveals the rate at which one quantity changes with respect to another in real-life application problems.
  7. Recognize the impact of slope signage as an indicator of how a line runs or how the value of the function changes.
  8. Identify the slopes of parallel and perpendicular lines.
  9. Use the slope formula to calculate rates of change in real-life application pro-blems.
  10. Discover that, when a linear equation is written in function form, the slope and y-intercept are revealed; function form can now be viewed as the slope-intercept form of a linear equation.
  11. Use the slope-intercept form to quickly and confidently graph a linear function.
  12. Recognize direct variation and be able to calculate the constant of variation and

write the resulting direct variation equation.

  1. Use the vertical line test to determine if a graphed relation is a function.
  2. Competently use and react to function notation.

 

Unit 3: Writing Linear Equations and Linear Inequalities

  1. Write a linear equation when given:  the slope and y-intercept, the actual graph of the function, the slope and a point on the graph of the function, two points on the graph of the function.
  2. Use the point-slope form to write a quick linear equation when given the slope and a point on the graph of the function or when given two points on the graph of the function.
  3. Convert linear equations from one form (slope-intercept, point-slope, standard) to another.
  4. Model a real-life situation by designing linear equations derived from given data.
  5. Make predictions using these models.
  6. Choose the most appropriate form of a linear equation given the problematic situation.
  7. Write, solve, and graph linear inequalities and apply these skills to solving real-life problems.
  8. Write, solve, and graph compound inequalities and apply these skills to solving real-life problems.
  9. Graph and interpret a linear inequality in two variables.
  10. Solve systems of linear equations both graphically and algebraically; choose the best method given the system.
  11. Recognize problematic situations that lend themselves to a systems approach.

 

Unit 4:  Exponents and Exponential Functions

  1. Use properties of exponents to multiply and divide exponential expressions.
  2. Use technology to expedite exponentiation.
  3. Recognize the graph of an exponential function.

Unit 5:  Polynomials

  1. Recognize when polynomials are to be added, subtracted, or multiplied.
  2. Correctly add, subtract and multiply polynomials.
  3. Execute the complete factorization of quadratic expressions.
  4. Use technology to confirm the accuracy of factorization.

 

Unit 6: Applications of Factoring

  1. Solve quadratic equations by means of factoring and the zero product rule.
  2. Confirm the accuracy of solution(s) by means of the graphing calculator.
  3. Recognize the graph of a quadratic function.
  4. Simplify rational expressions by means of factoring.
  5. Multiply and divide rational expressions and present final result in simplest form.

 

Unit 7:  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. Appreciate the derivation of the midpoint and distance formulas.
  5. Calculate the midpoint and length of a line segment.
  6. Simplify radicals that result from applying the distance formula.
  7. Identify segment congruence.
  8. Name, measure, and classify angles.
  9. Recognize problematic situations that call for angle addition.
  10. Identify special angle pairs: complementary, supplementary, linear pair and vertical angles.
  11. Classify polygons.
  12. Find the area and perimeter of rectangles and triangles.
  13. Find the circumference and area of circles.

 

Unit 8:  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 and be able to draft biconditional statements.
  6. Appreciate the pertinence of conditional statements to geometric investigations.
  7. Interpret two-dimensional and three-dimensional geometric diagrams by means of point, line, and plane postulates.
  8. Substantiate steps used to solve an algebraic equation by means of algebraic pro-perties of equality.
  9. Make clear distinctions between angle relationships in right angles, linear pair angles, and vertical angles.


 

Unit 9:  Parallel and Perpendicular Lines

  1. Distinguish between corresponding, alternate interior, alternate exterior, and con-

      secutive interior angles.

  1. Determine the measurements of all angles formed when a pair of parallel lines is

      intersected by a transversal given the measurement of only one of the angles.

  1. Use angle relationships to prove that lines are parallel.
  2. Apply the perpendicular transversal theorem.
  3. Accurately determine the distance from a point to a line.

 

Unit 10:  Congruent Triangles

  1. Distinguish between scalene, isosceles, and equilateral triangles.
  2. Apply the Distance Formula to determine the classification of a triangle drawn on a coordinate plane.
  3. Distinguish between acute, right, obtuse, and equiangular triangles.
  4. Determine the measurement of a missing angle of a triangle by means of the Tri-angle Sum Theorem.
  5. Understand how the measurement of an exterior angle of a triangle is related to

      measurements of its interior angles.

  1. Identify all pairs of congruent corresponding parts of congruent figures.
  2. Demonstrate that figures are congruent.
  3. Determine which triangle congruence theorem/postulate would appropriately demonstrate triangle congruence.
  4. Appreciate that demonstrating the congruence of right triangles is a bit more streamlined in comparison to other types of triangles.
  5. Understand how congruent triangles can  be used to find distances that are diffi-cult to measure directly.
  6. Appreciate how the sides and angles of a triangle are related if there are two or three congruent sides.
  7. Create an image congruent to a given figure in the coordinate plane.
  8. Distinguish between the three main types of transformations.

 

Unit 11:  Relationships within Triangles

  1. Identify and draw the various types of segments within a triangle’s interior.
  2. Appreciate the relationship between a midsegment and the sides of a given trian-gle.
  3. Use the Midsegment Theorem to find lengths within a triangle.
  4. Understand the properties of points located on perpendicular or angle bisectors.
  5. Explain how triangle side lengths relate to angle measurements.
  6. List the sides of a triangle in order when given its angle measurements and vice versa.
  7. Determine possible lengths of the third side of a triangle when the lengths of its two other sides are known.

 

Unit 12:  Similarity

  1. Simplify a ratio.
  2. Use ratios to find dimensions.
  3. Understand the nature of a proportion and solve it using the cross products pro-

      perty.

  1. Determine when a proportion approach is appropriate for solving real world pro-

      blems.

  1. Calculate the geometric mean of two numbers and present it in simplest radical form.
  2. Identify similar polygons by investigating angle measurements and proportional-ity of corresponding sides.
  3. Determine the scale factor of similar polygons and use it to determine lengths of missing dimensions.
  4. Appreciate that similar polygons also exhibit proportionality of perimeters and in-ternal segment lengths.

 

Unit 13:  Right Triangles

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

      right triangle.

  1. Recognize common Pythagorean triples and their multiples.
  2. Demonstrate how Pythagorean triples can be put to use to quickly arrive at the length of a missing side of a right triangle.
  3. Use the Pythagorean Theorem to classify a triangle as right, acute, or obtuse.