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### 9-12.G Geometry

- 9-12.G-CO Congruence
- Experiment with transformations in the plane.
- 9-12.G-CO.1 Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc.
- 9-12.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).
- 9-12.G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
- 9-12.G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
- 9-12.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.

- Checkpoint opportunity
- Understand congruence in terms of rigid motions.
- 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.
- 9-12.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.
- 9-12.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions.

- Checkpoint opportunity
- Prove geometric theorems.
- 9-12.G-CO.9 Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles.
- 9-12.G-CO.10 Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles.
- 9-12.G-CO.11 Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms.

- Checkpoint opportunity
- Make geometric constructions.
- 9-12.G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
- 9-12.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

- Checkpoint opportunity

- Experiment with transformations in the plane.
- 9-12.G-SRT Similarity, Right Triangles, and Trigonometry
- Understand similarity in terms of similarity transformations.
- 9-12.G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
- 9-12.G-SRT.1.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.
- 9-12.G-SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

- 9-12.G-SRT.2 Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar.
- 9-12.G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

- 9-12.G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
- Checkpoint opportunity
- Prove theorems involving similarity.
- 9-12.G-SRT.4 Prove theorems about triangles.
- 9-12.G-SRT.5 Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures.

- Checkpoint opportunity
- Define trigonometric ratios and solve problems involving right triangles.
- 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.
- 9-12.G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
- 9-12.G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

- Checkpoint opportunity
- Apply trigonometry to general triangles.
- 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.
- 9-12.G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.
- 9-12.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).

- Checkpoint opportunity

- Understand similarity in terms of similarity transformations.
- 9-12.G-C Circles
- Understand and apply theorems about circles.
- 9-12.G-C.1 Prove that all circles are similar.
- 9-12.G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.
- 9-12.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- 9-12.G-C.4 Construct a tangent line from a point outside a given circle to the circle.

- Checkpoint opportunity
- Find arc lengths and areas of sectors of circles.
- 9-12.G-C.5 Use and apply the concepts of arc length and areas of sectors of circles. Determine or 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.

- Checkpoint opportunity

- Understand and apply theorems about circles.
- 9-12.G-GPE Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section.
- 9-12.G-GPE.1 Determine or 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.
- 9-12.G-GPE.2 Determine or derive the equation of a parabola given a focus and directrix.

- Checkpoint opportunity
- Use coordinates to prove simple geometric theorems algebraically.
- 9-12.G-GPE.4 Perform simple coordinate proofs. 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).
- 9-12.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).
- 9-12.G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- 9-12.G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

- Checkpoint opportunity

- Translate between the geometric description and the equation for a conic section.
- 9-12.G-GMD Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems.
- 9-12.G-GMD.1 Explain how to find the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
- 9-12.G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

- Checkpoint opportunity
- Visualize relationships between two-dimensional and three-dimensional objects.
- 9-12.G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

- Checkpoint opportunity

- Explain volume formulas and use them to solve problems.
- 9-12.G-MG Modeling with Geometry
- Apply geometric concepts in modeling situations.
- 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).
- 9-12.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).
- 9-12.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).

- Checkpoint opportunity

- Apply geometric concepts in modeling situations.

### 9-12.S Statistics and Probability

- 9-12.S-CP Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data.
- 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”).
- 9-12.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.
- 9-12.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.
- 9-12.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.
- 9-12.S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

- Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- 9-12.S-CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
- 9-12.S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
- 9-12.S-CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
- 9-12.S-CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

- Checkpoint opportunity

- Understand independence and conditional probability and use them to interpret data.
- 9-12.S-MD Using Probability to Make Decisions
- Use probability to evaluate outcomes of decisions.
- 9-12.S-MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- 9-12.S-MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

- Use probability to evaluate outcomes of decisions.