912.HSG Geometry

912.HSGCO Congruence

912.HSGCO.A Experiment with transformations in the plane

912.HSGCO.A.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.

912.HSGCO.A.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).

912.HSGCO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

912.HSGCO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

912.HSGCO.A.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

912.HSGCO.B Understand congruence in terms of rigid motions

912.HSGCO.B.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.

912.HSGCO.B.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.

912.HSGCO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Checkpoint opportunity

912.HSGCO.C Prove geometric theorems

912.HSGCO.C.9 Prove theorems about lines and angles.

912.HSGCO.C.10 Prove theorems about triangles.

912.HSGCO.C.11 Prove theorems about parallelograms.

Checkpoint opportunity

912.HSGCO.D Make geometric constructions

912.HSGCO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

912.HSGCO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Checkpoint opportunity

912.HSGSRT Similarity, Right Triangles, and Trigonometry

912.HSGSRT.A Understand similarity in terms of similarity transformations

912.HSGSRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:

912.HSGSRT.A.1a 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.

912.HSGSRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

912.HSGSRT.A.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.

912.HSGSRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Checkpoint opportunity

912.HSGSRT.B Prove theorems involving similarity

912.HSGSRT.B.4 Prove theorems about triangles.

912.HSGSRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Checkpoint opportunity

912.HSGSRT.C Define trigonometric ratios and solve problems involving right triangles

912.HSGSRT.C.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.

912.HSGSRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

912.HSGSRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Checkpoint opportunity

912.HSGSRT.D Apply trigonometry to general triangles

912.HSGSRT.D.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.

912.HSGSRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.

912.HSGSRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).

Checkpoint opportunity

912.HSGC Circles

912.HSGC.A Understand and apply theorems about circles

912.HSGC.A.1 Prove that all circles are similar.

912.HSGC.A.2 Identify and describe relationships among inscribed angles, radii, and chords.

912.HSGC.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

912.HSGC.A.4 Construct a tangent line from a point outside a given circle to the circle.

Checkpoint opportunity

912.HSGC.B Find arc lengths and areas of sectors of circles

912.HSGC.B.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.

Checkpoint opportunity

912.HSGGPE Expressing Geometric Properties with Equations

912.HSGGPE.A Translate between the geometric description and the equation for a conic section

912.HSGGPE.A.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.

912.HSGGPE.A.2 Derive the equation of a parabola given a focus and directrix.

Checkpoint opportunity

912.HSGGPE.B Use coordinates to prove simple geometric theorems algebraically

912.HSGGPE.B.4 Use coordinates to prove simple geometric theorems algebraically.

912.HSGGPE.B.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).

912.HSGGPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

912.HSGGPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Checkpoint opportunity

912.HSGGMD Geometric Measurement and Dimension

912.HSGGMD.A Explain volume formulas and use them to solve problems

912.HSGGMD.A.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.

912.HSGGMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Checkpoint opportunity

912.HSGGMD.B Visualize relationships between twodimensional and threedimensional objects

912.HSGGMD.B.4 Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.

Checkpoint opportunity

912.HSGMG Modeling with Geometry

912.HSGMG.A Apply geometric concepts in modeling situations

912.HSGMG.A.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).

912.HSGMG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

912.HSGMG.A.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
912.HSS Statistics and Probability

912.HSSCP Conditional Probability and the Rules of Probability

912.HSSCP.A Understand independence and conditional probability and use them to interpret data

912.HSSCP.A.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).

912.HSSCP.A.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.

912.HSSCP.A.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.

912.HSSCP.A.4 Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities.

912.HSSCP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Checkpoint opportunity

912.HSSCP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model

912.HSSCP.B.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.

912.HSSCP.B.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.

912.HSSCP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model.

912.HSSCP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

Checkpoint opportunity

912.HSSMD Using Probability to Make Decisions

912.HSSMD.B Use probability to evaluate outcomes of decisions

912.HSSMD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

912.HSSMD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
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