CPC: Congruence, Proof and Constructions

(Framing Text): Experiment with transformations in the plane.

CPC.M.GHS.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.

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines

CPC.M.GHS.2: Represent transformations in the plane using, for example, 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).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

CPC.M.GHS.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

CPC.M.GHS.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

(Framing Text): Understand congruence in terms of rigid motions.

CPC.M.GHS.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.

Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Translations

CPC.M.GHS.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Proving Triangles Congruent

(Framing Text): Prove geometric theorems.

CPC.M.GHS.9: Prove theorems about lines and angles. Theorems include: 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.

Investigating Angle Theorems

CPC.M.GHS.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.

Isosceles and Equilateral Triangles
Proving Triangles Congruent
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Triangle Angle Sum
Triangle Inequalities

CPC.M.GHS.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Parallelogram Conditions
Special Parallelograms

(Framing Text): Make geometric constructions.

CPC.M.GHS.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.

Concurrent Lines, Medians, and Altitudes
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Parallel, Intersecting, and Skew Lines
Segment and Angle Bisectors

SPT: Similarity, Proof, and Trigonometry

(Framing Text): Understand similarity in terms of similarity transformations.

SPT.M.GHS.14: Verify experimentally the properties of dilations given by a center and a scale factor.

SPT.M.GHS.14.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations
Similar Figures

SPT.M.GHS.15: 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.

Similar Figures

(Framing Text): Prove theorems involving similarity.

SPT.M.GHS.17: 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.

Isosceles and Equilateral Triangles
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures
Triangle Angle Sum
Triangle Inequalities

SPT.M.GHS.18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Dilations
Perimeters and Areas of Similar Figures
Similarity in Right Triangles

(Framing Text): Define trigonometric ratios and solve problems involving right triangles.

SPT.M.GHS.19: 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.

Sine, Cosine, and Tangent Ratios

SPT.M.GHS.21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

ETD: Extending to Three Dimensions

(Framing Text): Explain volume formulas and use them to solve problems.

ETD.M.GHS.25: 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.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

ETD.M.GHS.26: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones

CAG: Connecting Algebra and Geometry Through Coordinates

(Framing Text): Use coordinates to prove simple geometric theorems algebraically.

CAG.M.GHS.32: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem.

Circles
Distance Formula

(Framing Text): Translate between the geometric description and the equation for a conic section.

CAG.M.GHS.33: Derive the equation of a parabola given a focus and directrix.

Parabolas

CWC: Circles With and Without Coordinates

(Framing Text): Understand and apply theorems about circles.

CWC.M.GHS.35: 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.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

(Framing Text): Find arc lengths and areas of sectors of circles.

CWC.M.GHS.38: 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.

Chords and Arcs

(Framing Text): Translate between the geometric description and the equation for a conic section.

CWC.M.GHS.39: 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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

AP: Applications of Probability

(Framing Text): Understand independence and conditional probability and use them to interpret data.

AP.M.GHS.42: 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”).

Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability

AP.M.GHS.43: 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.

Independent and Dependent Events

AP.M.GHS.44: Recognize 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.

Independent and Dependent Events

(Framing Text): Use the rules of probability to compute probabilities of compound events in a uniform probability model.

AP.M.GHS.47: 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.

Independent and Dependent Events

AP.M.GHS.49: 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.

Independent and Dependent Events

AP.M.GHS.50: Use permutations and combinations to compute probabilities of compound events and solve problems.

Binomial Probabilities
Permutations and Combinations

(Framing Text): Use probability to evaluate outcomes of decisions.

AP.M.GHS.51: Use probabilities to make fair decisions (e.g., drawing by lots and/or using a random number generator).

Lucky Duck (Expected Value)
Probability Simulations
Theoretical and Experimental Probability

AP.M.GHS.52: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).

Estimating Population Size
Probability Simulations
Theoretical and Experimental Probability