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Course Description
This course covers essential introductory components of geometry. It implements appropriate grade level strands, unifying ideas, and standards from the Content Standards for California Public Schools, (California Department of Education, 2001), and Principals and Standards for School Mathematics Grades 9-12, (National Council of Teachers of Mathematics, 2000). A scientific calculator is required and the use of computers and graphing calculators are encouraged. Problems are designed to engage higher order thinking processes in a collaborative environment. These learning opportunities are provided equitably for all students through the use of mathematical ideas, tools, and techniques.
Credits: 5
Released: 2005
Course Content
Distance Formula Unit 6
- Derive Distance Formula
- Use Distance Formula to solve problems
- Write geometric proof relating to Distance Formula
- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
- Write geometric proofs, including proofs by contradiction
- Use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles
- Prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles
- Write Midpoint Formula
- Use Midpoint Formula to solve problems
- Write geometric proof relating to Midpoint Formula
- Find slope of a line
- Use slope to determine whether lines are parallel or perpendicular
- Use inductive and deductive reasoning to write proofs
- Apply understanding of distance, midpoint, and slope to write equations of lines
- Solve problems involving parallel and perpendicular lines
- Know and Use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles
Trigonometry Unit 7
- Review six basic terms of trigonometry with respect to the coordinate plan
- Define and use inverse trigonometric functions
- Solve special right triangles using trigonometric ratios
- Solve problems using trigonometric identities
- Solve problems using circular functions
- Use special angles to compute trigonometric functions
- Write geometric proofs
- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
- Write geometric proofs, including proofs by contradiction
- Construct and judge the validity of a logical argument
- Know the definitions of the basic trigonometric functions defined by the angles of a right triangle. Know and use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x))2 + (cos(x))2 = 1
- Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side
- Know and use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles
- Define and apply Law of Sines
- Derive the Law of Sines
- Define and apply Law of Cosines
- Derive Law of Cosines
- Apply understanding of trigonometry to real world applications
- Use inductive reasoning to develop vocabulary and geometric properties of trigonometry
- Know definitions of trigonometric ratios and use them to solve problems
- Know definitions of trigonometric identities and functions, and use them to solve problems
- Students explore formulas for the Laws of Sines and Cosines and use them in problem-solving exercises
- Students use inductive and deductive reasoning to write proofs
Areas of Plain Figures Unit 8
- Define polygons of n number of sides
- Define and use Postulates of Area and Addition
- Define and use formulas to calculate areas of various polygons
- Use formula to find the number of diagonals in any polygon
- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
- Write geometric proofs, including proofs by contradiction
- Know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures
- Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids
- Prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles
- Know and use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles
- Use trigonometry and formulas to calculate areas of triangles
- Define and use formulas to calculate areas of regular polygons
- Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side
- Define terms relating to circle
- Compute circumference and area of circle using formulas
- Compute arc lengths and areas of sectors of a circle
- Compute areas of inscribed and circumscribed circles
- Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles
- Determine scale factor of similar figures
- Compute perimeter and area of similar figures
- Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids
- Apply understanding of area to real world applications
Surface Areas and Volumes of Solids Unit 9
- Recognize parts of prisms and pyramids.
- Determine lateral areas and surface areas of prisms and pyramids.
- Use formulas to find lateral areas and surface areas of prisms and pyramids
- Write geometric proof for surface area of a prism
- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
- Write geometric proofs, including proofs by contradiction
- Know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures
- Compare the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders
- Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids
- Know and are use angles and side relationships in problems with special right triangles, such as 30o, 60o, and 90o triangles and 45o, 45o, and 90o triangles
- Recognize parts of circular solids
- Determine surface areas of circular solids
- Use formulas to find surface areas of circular solids
- Compare lateral areas and surface areas of circular solids
- Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids
- Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles
- Use formulas to find volumes of solids
- Compare volumes of solids
- Derive formula for finding volume of sphere
- Apply understanding of areas and volumes of solids to real-world applications
Transformational Geometry Unit 10
- Define terms relating to transformational geometry
- Solve mapping, transformation, and reflection problems
- Identify isometries and preimages
- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning
- Know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections
- Solve glide reflection and rotation problems
- Solve translation problems using special right triangles
- Identify transformations
- Know and use angles and side relationships in problems with special right triangles, such as 30o, 60o, and 90o triangles and 45o, 45o, and 90o triangles
- Solve transformation problems of dilation
- Know definitions and terms related to transformational geometry
- Solve problems related to transformational geometry
- Use special right triangles to solve problems related to transformational geometry
- Apply properties of scale to solve problems relating to transformational geometry
- Apply understanding of transformational geometry to mosaics
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