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Topic: Geometry 

Common Core Mathematics 9-12

 Cluster  Experiment with transformations in the plane
  Grade 9-12
   G.CO.1.Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
   G.CO.2.Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
   G.CO.3.Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
   G.CO.4.Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
   G.CO.5.Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
 Cluster  Understand congruence in terms of rigid motions
  Grade 9-12
   G.CO.6.Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
   G.CO.7.Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
 Cluster  Understand similarity in terms of similarity transformations
  Grade 9-12
   G.SRT.1.Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
   G.SRT.2.Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
   G.SRT.3.Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
 Cluster  Prove theorems involving similarity
  Grade 9-12
   G.SRT.4.Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
   G.SRT.5.Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
 Cluster  Define trigonometric ratios and solve problems involving right triangles
  Grade 9-12
   G.SRT.6.Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
   G.SRT.7.Explain and use the relationship between the sine and cosine of complementary angles.
   G.SRT.8.Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
 Cluster  Apply trigonometry to general triangles
  Grade 9-12
   G.SRT.9.(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
   G.SRT.10.(+) Prove the Laws of Sines and Cosines and use them to solve problems.
   G.SRT.11.(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
 Cluster  Understand and apply theorems about circles
  Grade 9-12
   G.C.1.Prove that all circles are similar.
   G.C.2.Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
   G.C.3.Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
   G.C.4.(+) Construct a tangent line from a point outside a given circle to the circle.
 Cluster  Find arc lengths and areas of sectors of circles
  Grade 9-12
   G.C.5.Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
 Cluster  Translate between the geometric description and the equation for a conic section
  Grade 9-12
   G.GPE.1.Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
   G.GPE.2.Derive the equation of a parabola given a focus and directrix.
 Cluster  Use coordinates to prove simple geometric theorems algebraically
  Grade 9-12
   G.GPE.4.Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0, 2).
   G.GPE.5.Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
   G.GPE.6.Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
   G.GPE.7.Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
 Cluster  Explain volume formulas and use them to solve problems
  Grade 9-12
   G.GMD.1.Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
   G.GMD.3.Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
 Cluster  Apply geometric concepts in modeling situations
  Grade 9-12
   G,MG.1.Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
   G.MG.2.Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
   G.MG.3.Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
 Cluster  Understand independence and conditional probability and use them to interpret data
  Grade 9-12
   S.CP.1.Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
   S.CP.2.Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
   S.CP.3.Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
   S.CP.4.Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
   S.CP.5.Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.