Geometry

MAGE: Geometry

MAGE.A: Geometry

MAGE.B: Statistics and Probability

MAGE.A.1: make formal geometric constructions with a variety of tools and methods (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software); 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

MAGE.A.2: construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle

MAGE.A.3: know and apply the 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

MAGE.A.4: describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates

MAGE.A.5: 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)

MAGE.A.6: given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another

MAGE.A.7: describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself

MAGE.A.8: develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments

MAGE.A.9: 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

MAGE.A.10: 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

MAGE.A.11: explain how the criteria for triangle congruence (i.e., ASA, SAS, SSS, HL, AAS) follow from the definition of congruence in terms of rigid motions

MAGE.A.12: use informal arguments to establish facts about the angle sum and exterior angles of triangles and about the angles created when parallel lines are cut by a transversal

MAGE.A.13: prove theorems about lines and angles (i.e., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints)

MAGE.A.14: construct the inscribed and circumscribed circle of a triangle

MAGE.A.15: informally prove the Pythagorean Theorem and its converse geometrically (i.e., using area model)

MAGE.A.16: prove theorems about triangles (i.e., measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point)

MAGE.A.17: prove theorems about parallelograms (i.e., opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals)

MAGE.A.18: use congruence criteria for triangles to solve problems and to prove relationships in geometric figures

MAGE.A.19: verify experimentally the properties of dilations given by a center and a scale factor (i.e., a dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged; the dilation of a line segment is longer or shorter according to the ratio given by the scale factor)

MAGE.A.20: 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

MAGE.A.21: use the properties of similarity transformations to establish the AA criterion for two triangles to be similar

MAGE.A.22: prove theorems about triangles (i.e., a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity)

MAGE.A.23: use similarity criteria for triangles to solve problems and to prove relationships in geometric figures

MAGE.A.24: 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

MAGE.A.25: explain and use the relationship between the sine and cosine of complementary angles

MAGE.A.26: use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

MAGE.A.27: prove that all circles are similar (i.e., using transformations; ratio of circumference to the diameter is a constant)

MAGE.A.28: identify and describe relationships among inscribed angles, radii, chords, tangents, and secants (i.e., 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)

MAGE.A.29: prove properties of angles for a quadrilateral inscribed in a circle

MAGE.A.30: construct a tangent line from a point outside a given circle to the circle

MAGE.A.31: 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

MAGE.A.32: 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

MAGE.A.33: give informal arguments for geometric formulas (i.e., informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments; informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri's principle)

MAGE.A.34: give an informal argument using Cavalieri's principle for the formulas of the volume of a sphere and other solid figures

MAGE.A.35: use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

MAGE.A.36: explain and apply the distance formula as an application of the Pythagorean Theorem

MAGE.A.37: use coordinates to prove simple geometric theorems algebraically (i.e., prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that a point lies on a circle centered at the origin and containing a given point), including quadrilaterals, circles, right triangles, and parabolas

MAGE.A.38: 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)

MAGE.A.39: find the point on a directed line segment between two given points that partitions the segment in a given ratio

MAGE.A.40: use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula)

MAGE.A.41: identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects

MAGE.A.42: apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in three dimensions

MAGE.A.43: use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)

MAGE.A.44: apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)

MAGE.A.45: 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)

MAGE.B.46: describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (i.e., "or," "and," "not")

MAGE.B.47: understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent

MAGE.B.48: understand the conditional probability of A given B as P(A and B)/P(B); interpret independence of A and B in terms of conditional probability (i.e., 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)

MAGE.B.49: 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 (e.g., 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)

MAGE.B.50: recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (e.g., compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer)

MAGE.B.51: 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 context

MAGE.B.52: apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in context