Download Common Curriculum Map Discipline: Math Course: AP Prep Geometry

Common Curriculum Map
Discipline: Math
Course: AP Prep Geometry
Chapter 1:
Standards:
9.A.1b Draw two-dimensional shapes.
9.A.2c Describe and draw representations of geometric relationships, patterns, symmetries, and designs in
two- and three-dimensions with and without technology.
9.A.3a Draw or construct two- and three- dimensional geometric figures including prisms, pyramids,
cylinders and cones.
9.B.3 Identify, describe, classify and compare two- and three- dimensional geometric figures and models
according to their properties.
9.B.4 Recognize and apply relationships within and among geometric figures.
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
What are the main components of a coordinate plane?
What are the applications and relationships of points, lines, and planes?
How do you establish a plan for solving problems, including the use of formulas?
How are parts of segments related to one another?
What are the key angle relationships?
Content:
Points, Lines, Segments, and Planes
Skills:
1. Identify the Coordinate Plane and its parts
2. Define basic terms of Geometry (point, line, plane, space)
3. Explore relationships between the points, lines, and planes
4. Apply formulas to solve problems.
5. Measure Segments
6. Examine relationships among segments
7. Apply the Distance Formula
8. Apply the Midpoint Formula
9. Explore Angles
10. Identify the parts of angles
11. Classify angles by measurement and location (in relation to one another)
12. Explore Angle Relationships
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 2:
Standards:
9.C.1 Draw logical conclusions and communicate reasoning about simple geometric figures and patterns
using concrete materials, diagrams and contemporary technology.
9.C.2 Formulate logical arguments about geometric figures and patterns and communicate reasoning.
9.C.3a Construct, develop and communicate logical arguments (informal proofs) about geometric figures
and patterns.
9.C.3b Develop and solve problems using geometric relationships and models, with and without the use
of technology.
9.C.4b Construct and communicate convincing arguments for geometric situations.
9.C.5a Perform and describe an original investigation of a geometric problem and verify the analysis and
conclusions to an audience.
Essential Questions:
What is a conjecture, and how do I go about making one?
What is a counterexample?
How do I apply the Law of Detachment and the Law of Transitivity of Implication?
What is a proof?
How do I make a proof?
Content:
Reasoning and Proof
Skills:
1. Employ the use of Truth Tables
2. Define and Apply: Conjunction, Disjunction, Negation, Implication, Biconditional, Tautology,
Converse, Inverse, Contrapositive, Counterexample
3. Use Syllogisms
4. Define Proof
5. Define and Make Valid Connections
6. Develop a logically valid inference scheme in order to prove geometric theorems in a syllogistic proof.
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 3:
Standards:
9.A.2c Describe and draw representations of geometric relationships, patterns, symmetries, and designs in
two- and three-dimensions with and without technology.
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar,
congruent and line symmetry.
9.B.3 Identify, describe, classify and compare two- and three- dimensional geometric figures and models
according to their properties.
9.B.4 Recognize and apply relationships within and among geometric figures.
9.C.3a Construct, develop and communicate logical arguments (informal proofs) about geometric figures
and patterns.
9.C.3b Develop and solve problems using geometric relationships and models, with and without the use
of technology.
9.C.4b Construct and communicate convincing arguments for geometric situations
9.C.4c Develop and communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements.
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
What does it mean for two lines to be parallel? Skew? Perpendicular?
How can I show two lines are parallel given coordinates of points on the lines?
What are the special angle relationships formed when two lines are cut by a transversal?
What do I know about those angles when the lines happen to be parallel?
Content:
Parallel and Perpendicular Lines
Skills:
1. Identify examples of different sets of lines
2. Identify special angle names for pairs of angles formed by transversal cutting lines (interior, exterior,
alternate interior, alternate exterior, consecutive interior)
3. Learn and apply theorems focusing on the special relationship between angles when the transversal
cuts parallel lines
4. Determine slope of a line
5. Find the distance from a point to a line
6. Classify triangles by angles
7. Classify triangles by sides
8. Apply the Angle Sum Theorem
9. Apply the Exterior Angle Theorem
10. Identify Congruent Triangles
11. Prove triangles congruent
12. Identify specific pieces of an isosceles triangle
13. Compare pieces of an isosceles triangle
14. Identify congruent parts of an isosceles triangle
15. Solve problems dealing with isosceles triangles
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 4:
Standards:
9.A.1b Draw two-dimensional shapes
9.A.2c Describe and draw representations of geometric relationships, patterns, symmetries, and designs in
two- and three-dimensions with and without technology.
9.A.3c Use concepts of symmetry, congruency, similarity, scale, perspective, and angles to describe and
analyze two- and three-dimensional shapes found in practical applications (e.g., geodesic domes, Aframe houses, basketball courts, inclined planes, art forms, blueprints).
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar,
congruent and line symmetry.
9.B.4 Recognize and apply relationships within and among geometric figures
9.C.3a Construct, develop and communicate logical arguments (informal proofs) about geometric figures
and patterns.
9.C.3b Develop and solve problems using geometric relationships and models, with and without the use
of technology.
9.C.4b Construct and communicate convincing arguments for geometric situations
9.C.4c Develop and communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements.
Essential Questions:
How do I classify a triangle by its sides and by its angles?
What makes triangles congruent?
What is the minimum amount of information I need in order to show two triangles are congruent?
What are the pieces of an isosceles triangle?
Content:
Triangle Congruence
Skills:
1. Classify triangles by angles
2. Classify triangles by sides
3. Apply the Angle Sum Theorem
4. Apply the Exterior Angle Theorem
5. Identify Congruent Triangles
6. Prove triangles congruent
7. Identify specific pieces of an isosceles triangle
8. Compare pieces of an isosceles triangle
9. Identify congruent parts of an isosceles triangle
10. Solve problems dealing with isosceles triangles
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 5:
Standards:
8.A.3b Solve problems using linear expressions, equations and inequalities.
9.A.1b Draw two-dimensional shapes
9.A.2c Describe and draw representations of geometric relationships, patterns, symmetries, and designs
in two- and three-dimensions with and without technology.
9.A.3c Use concepts of symmetry, congruency, similarity, scale, perspective, and angles to describe and
analyze two- and three-dimensional shapes found in practical applications (e.g., geodesic domes, Aframe houses, basketball courts, inclined planes, art forms, blueprints).
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular,
similar, congruent and line symmetry.
9.B.4 Recognize and apply relationships within and among geometric figures.
9.C.3a Construct, develop and communicate logical arguments (informal proofs) about geometric
figures and patterns.
9.C.3b Develop and solve problems using geometric relationships and models, with and without the use
of technology.
9.C.4b Construct and communicate convincing arguments for geometric situations
9.C.4c Develop and communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements.
Essential Questions:
What are the special segments in triangles?
What is the minimum amount of information that I need in order to show that two right triangles are
congruent?
What is the relationship among the sides of a triangle?
Content:
Triangle Congruence
Skills:
1. Identify and define special segments of a triangle (median, angle bisector, perpendicular bisector,
altitude)
2. Identify theorems of congruence for right triangles
3. Solve problems involving right triangles and their congruence
4. Use indirect reasoning
5. Perform proof by contradiction
6. Define inequality
7. Use relationships of inequality for triangles
8. Identify relationships between sides and angles of a triangle (inequality)
9. Identify inequality relationship for the sides of a triangle (Triangle Inequality Theorem)
10. Calculate the possible values of a side or a variable representing the side of a triangle
11. Indentify inequalities involving two triangles
12. Solve problems dealing with inequalities between two triangles
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 6:
Standards:
9.A.2c Describe and draw representations of geometric relationships, patterns, symmetries, and designs in
two- and three-dimensions with and without technology.
9.B.1a Identify and describe characteristics, similarities and differences of geometric shapes
9.B.1b Sort, classify and compare familiar shapes.
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar,
congruent and line symmetry.
9.B.4 Recognize and apply relationships within and among geometric figures.
9.C.3a Construct, develop and communicate logical arguments (informal proofs) about geometric figures
and patterns.
9.C.3b Develop and solve problems using geometric relationships and models, with and without the use
of technology.
9.C.4b Construct and communicate convincing arguments for geometric situations
9.C.4c Develop and communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements.
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
What are the names of the special quadrilaterals?
What are the properties of each special quadrilateral?
How do you determine if a quadrilateral is one of the special quadrilaterals?
How do the properties of each special quadrilateral compare to one another?
Content:
Quadrilaterals
Skills:
1. Identify and define a parallelogram
2. Indentify and describe relationships among pieces of a parallelogram
3. Determine tests for parallelograms
4. Identify a rectangle and its properties.
5. Define a rhombus and its properties.
6. Define a square and its properties.
7. Define a trapezoid and its properties
8. Review all quadrilateral material
9. Define a ratio and a proportion
10. Solve proportions by cross multiplication
11. Define similar polygons
12. Solve problems involving similar polygons
13. Identify similar triangles
14. Define postulates and theorems for similar triangles (AA, SSS, SAS)
15. Parallel lines and proportional parts of those lines
16. Define parts of similar triangles
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 7:
Standards:
8.D.3b Propose and solve problems using proportions, formulas and linear functions.
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar,
congruent and line symmetry.
9.B.4 Recognize and apply relationships within and among geometric figures.
9.B.5 Construct and use two- and three-dimensional models of objects that have practical applications
(e.g., blueprints, topographical maps, scale models).
9.C.4a Construct and test logical arguments for geometric situations using technology where appropriate.
9.C.4b Construct and communicate convincing arguments for geometric situations
9.C.4c Develop and communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements.
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
How do we define similarity?
How can we easily identify similar triangles?
How do parallel lines passing through a triangle relate to similarity?
Content:
Similarity
Skills:
1. Define a ratio and a proportion
2. Solve proportions by cross multiplication
3. Define similar polygons
4. Solve problems involving similar polygons
5. Identify similar triangles
6. Define postulates and theorems for similar triangles (AA, SSS, SAS)
7. Parallel lines and proportional parts of those lines
8. Define parts of similar triangles
9. Work with similar triangles.
10. Solve problems involving similar triangles.
11. Worked with proportions formed when a triangle is cut by a line parallel to one of its sides.
12. Defined and discussed the parts of similar triangles (medium, altitude, angle bisector, perpendicular
bisector).
13. Solved problems involving the parts of similar triangles discussed above.
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 8:
Standards:
9.A.1b Draw two-dimensional shapes
9.A.5 Use geometric figures and their properties to solve problems in the arts, the physical and life
sciences and the building trades, with and without the use of technology.
9.B.4 Recognize and apply relationships within and among geometric figures
9.C.4a Construct and test logical arguments for geometric situations using technology where appropriate.
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
9.D.4 Analyze and solve problems involving triangles (e.g., distances which cannot be measured directly)
using trigonometric ratios
Essential Questions:
What are the special ratios in right triangles that can save time when applied?
How can trigonometry be used to assist in solving problems?
What are the special laws of trigonometry that can be used to find missing pieces of triangles?
Content:
Right Triangle Trigonometry
Skills:
1. Defined and explored geometric mean.
2. Looked at how an altitude formed similar triangles within a right triangle, and how several geometric
means are created.
3. Solved problems involving geometric mean as it relates to right triangles.
4. Define special right triangles (45-45-90) (30-60-90)
5. Define special side ratios for the special right triangles
6. Trigonometric ratios for right triangles
7. Define angles of elevation and depression and their relationship
8. Law of sines
9. Law of Cosines
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 9:
Standards:
9.A.1b Draw two-dimensional shapes.
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar,
congruent and line symmetry.
9.B.4 Recognize and apply relationships within and among geometric figures
9.C.4b Construct and communicate convincing arguments for geometric situations
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
What are the parts of a circle, and how are those parts related?
What is the relationship between angles and segments formed in and around circles and the arcs of the
circle that are intercepted?
How do you write the standard equation of a circle?
Content:
Circles
Skills:
1. Define the parts of a circle
2. Calculate Circumference given radius or diameter
3. Discuss central angles and arcs
4. Calculate arc measure and arc length
5. Discuss arcs and chords and their relationship
6. Solve problems using relationships of arcs and chords
7. Define inscribed angles and how to find their measure
8. Solve problems involving inscribed angles
9. Define and examine tangents and their properties
10. Define and examine secants and their properties
11. Examine the relationship between tangents, secants, and the angles formed by them (interior and
exterior)
12. Solve problems using information about tangents, secants, and the angles formed by them
13. Determine the relationship between special segments in a circle (secants in the interior, secants
intersecting in the exterior, secant and tangent in the exterior)
14. Explore the standard equation of a circle
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 10:
Standards:
9.A.1b Draw two-dimensional shapes
9.B.1a Identify and describe characteristics, similarities and differences of geometric shapes.
9.B.3 Identify, describe, classify and compare two- and three- dimensional geometric figures and
models according to their properties.
9.B.4 Recognize and apply relationships within and among geometric figures
9.C.4b Construct and communicate convincing arguments for geometric situations
9.D.3 Compute distances, lengths and measures of angles using proportions, the Pythagorean Theorem
and its converse.
Essential Questions:
What is a polygon and how are polygons named?
How do you find the areas of polygons, both regular and irregular?
How is geometric probability found?
Content:
Polygons
Skills:
1. Define a polygon
2. Name common polygons up to 12 sides
3. Define convex, concave, and regular as they apply to polygons
4. Explore the interior and exterior angle measures of regular polygons, and solve problems involving
those
5. Determine the formula for area for parallelograms, triangles, rhombi, and trapezoids; solve problems
involving all of these
6. Define apothem, and determine formula for area of a regular polygon
7. Discuss formula for area of a circle
8. Apply knowledge of area to solve problems looking for area of shaded regions (partial areas)
9. Discuss geometric probability and its applications
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 11:
Standards:
9.A.1a Identify related two- and three-dimensional shapes including circle-sphere, square-cube,
triangle-pyramid, rectangle-rectangular prism and their basic properties
9.A.3a Draw or construct two- and three- dimensional geometric figures including prisms, pyramids,
cylinders and cones.
9.A.4b Make perspective drawings, tessellations and scale drawings, with and without the use of
technology.
9.B.1a Identify and describe characteristics, similarities and differences of geometric shapes
9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular,
similar, congruent and line symmetry.
9.B.3 Identify, describe, classify and compare two- and three- dimensional geometric figures and
models according to their properties.
9.B.4 Recognize and apply relationships within and among geometric figures
9.D.4 Analyze and solve problems involving triangles (e.g., distances which cannot be measured
directly) using trigonometric ratios
Essential Questions:
How do you find the lateral area and surface area of a solid?
How do you find the volume of a solid?
How are surface area and volume of similar solids related?
Content:
Surface Area and Volume
Skills:
1. Define polyhedra and discuss the properties of polyhedra.
2. Define Platonic Solids and their properties.
3. Work on spatial recognition and construction of 3-D figures and solids.
4. Determine and apply the formula for surface area of a prism
5. Determine and apply the formula for surface area of a cylinder
6. Determine and apply the formula for surface area of a pyramid
7. Determine and apply the formula for surface area of a cone
8. Determine and apply the formula for volume of a prism
9. Determine and apply the formula for volume of a cylinder
10. Determine and apply the formula for volume of a pyramid
11. Determine and apply the formula for volume of a cone
12. Determine and apply the formula for surface area of a sphere
13. Determine and apply the formula for volume of a sphere
14. Discuss and define similar and congruent solids
15. Define scale factor of similar solids
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 12:
Standards:
8.D.3a Solve problems using numeric, graphic or symbolic representations of variables, expressions,
equations and inequalities.
8.D.3c Apply properties of powers, perfect squares and square roots
Essential Questions:
What are the standard equations of a circle, parabola, ellipse, and hyperbola?
How do you shift the location of the conic section around a Cartesian Coordinate plane?
Content:
Conic Sections
Skills:
1. Derive standard equation for a circle.
2. Work with algebraic manipulation to reach standard equation of a circle, including shifting the center
of the circle.
3. Derive standard equation of a parabola.
4. Complete the square to arrive at standard equation of a parabola, including shifting the vertex of the
parabola.
5. Derive the standard equation of an ellipse.
6. Complete the square multiple times to arrive at the standard equation of an ellipse, including shifting
the location of the center.
7. Derive the standard equation for a hyperbola.
8. Complete the square multiple times to arrive at the standard equation of a hyperbola, including
shifting the location of the center.
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.
Chapter 13:
Standards:
8.B.5 Use functions including exponential, polynomial, rational, parametric, logarithmic, and
trigonometric to describe numerical relationships.
8.C.5 Use polynomial, exponential, logarithmic and trigonometric functions to model situations.
8.D.3c Apply properties of powers, perfect squares and square roots
Essential Questions:
How do you define a logarithm?
What is an exponential function?
How can logarithms and exponential functions be applies to real world situations?
Content:
Logarithmic and Exponential Functions
Skills:
1. Work with rational exponents
2. Work with all real number exponents
3. Set up and solve exponential equations
4. Defined the operation of composition as it relates to two functions.
5. Worked with composition of functions.
6. Defined the inverse of a function.
7. Solved problems involving solving for and applying the inverse of a function.
8. Defined a logarithmic function.
9. Defined a logarithm.
10. Discussed and worked with the properties of logs.
11. Solved problems involving the simplification of logarithms.
12. Solved problems involving solving exponential equations using logs.
13. Determined and discussed the laws of logs.
14. Algebraically proved the laws of logs.
15. Applied the laws of logs to problem solving situations.
16. Defined common logs.
17. Used calculators to solve problems involving logs, including logs of different bases.
18. Discussed and proved the change of base formula.
19. Applied the change of base formula to problems to simplify.
20. Discussed and worked with logarithmic formulas, including compound interest, half life decay, and
doubling time growth.
21. Applied knowledge of logarithmic formulas and the properties and laws of logs to problem solving
situations and word problems.
22. Defined the natural logarithmic function.
23. Discussed similarities and differences between natural log and other logs with different bases.
24. Solved problems using natural logs.
Assessment:
Assignments from textbook.
Homework Quizzes.
Student presentation of work on board.
Observation during group work/individual work in class.
Class discussions, formal and informal.
Tests and Quizzes.