Math Geometry

Math: Geometry Active component (content knowledge)

Comprehend

Learner outcomes Demonstrate by solving:

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Properties of similarity congruence and parallel lines cut by a transversal Basic theorems about congruent and similar triangles Definitions and basic properties of a circle Concepts behind simple geometric proofs Basic formulas for volume and surface area for threedimensional objects Geometric properties of lines (e.g., slope and midpoint of a line segment) The formula for the distance between two points Through recognition of geometric translations and transformations algebraically That geometric objects and figures can also be described algebraically The algebra and geometry of circles The algebra and geometry of parabolas and ellipses as a prerequisite to the study of calculus That there are geometries other than Euclidean geometry Similarities of figures and use scale factor to solve problems That numerical values associated with measurements of physical quantities must be assigned units of measurement or dimension That the effect of a scale factor k on length are and volume is to multiply each by k, k2 and k3, respectively

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Problems involving proofs through the use of geometric constructions Mathematical and real world problems that involve the properties of special right triangles with the Pythagorean theorem and its converse

Develop ability to: • • • • • • •

Figure area and perimeter of basic figures Develop and write simple geometric proofs Use similar triangles to find unknown angle measurements and lengths of sides Visualize solids and surfaces in three-dimensional space State and prove key basic theorems in geometry Use trigonometry for examples of algebraic/geometric relationships, including Law of Sines/Cosines Identify and apply properties of theorems about: ○

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• • • College Readiness Standards

Parallel lines – use to prove theorems such as two lines parallel to a third are parallel to each other and perform constructions such as a line parallel to a given line through a point not on a line Perpendicular lines – use to prove theorems such as the perpendicular bisectors of line segments are the set of all points equidistant from the two end points and to perform constructions such as the perpendicular bisector of a line segment Angles – use to prove theorems such as two lines are parallel exactly when the alternate interior angles they make with a transversal are equal and to perform constructions such as the bisector of an angle

Describe a line by a linear equation Find the distance between two points using their coordinates and the Pythagorean theorem Find an equations of a circle given its center and radius and, 22082293 9/15/2009

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given an equations of a circle, find its center and radius Use rigid motions to determine whether tow geometric figures are congruent and to create and analyze geometric designs Determine the perimeter of ○ ○ ○ ○

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A polygon and the circumference of a circle The area of a rectangle, a circle, a triangle and polygon with more than four sides by decomposing it into triangles The surface area of a prism, a pyramid, a cone and a sphere The volume of a rectangle box, a prism, a pyramid, a cone and a sphere

Represent geometric objects and figures algebraically using coordinates Express the intuitive concept of “slant” of a line in terms of the precise concept of slope, us the coordinates of two points on a line to define its slope, and use slope to express the parallelism and perpendicularity of lines

(Conley, 2003; 2005; 2007) (The American Diploma Project, 2004)

College Readiness Standards

22082293 9/15/2009

Standards for Success (S4S) Math: Algebra Successful students: A. Understand and use both basic plane and solid geometry’ B. Know analytic (i.e., coordinate geometry) C. Understand basic relationship between geometry and algebra

Cognitive Strategies Emphasized •

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Habits of the mind such as: ○ Time management – budgeting time to complete reading tasks ○ Understanding expectations of readings ○ Academic persistence Critical thinking skills such as: ○ Ability to discuss materials in-depth by asking engaging questions ○ Problem solving Understanding the connection between reading comprehension skills and disciplines: writing, speaking and research Self-analysis – learning from constructive criticism and feedback Developing comfort with ambiguity of readings and assignments

Bibliography Conley, D. T. (2005). College Knowledge. San Francisco: Jossey-Bass. Conley, D. T. (2003). Understanding University Success: A Project of the Association of American Universities and The Pew Charitable Trusts. Eugene: Center for Educational Policy Research. Conley, D. (2007). Towards a More Comprehensive Comprehension of College Readiness. Eugene, OR: Educational Policy Improvement Center. The American Diploma Project. (2004). Ready or Not: Creating a High School Diploma that Counts. Achieve, Inc.

College Readiness Standards

22082293 9/15/2009