G-CO: Congruence

1.1: Experiment with transformations in the plane.

G-CO.1: State and apply precise definitions of angle, circle, perpendicular, parallel, ray, line segment, and distance based on the undefined notions of point, line, and plane.

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Inscribed Angles
Parallel, Intersecting, and Skew Lines

G-CO.2: Represent transformations in the plane. (e.g., using transparencies and/or geometry software);

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

G-CO.2.a: Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

G-CO.2.b: Compare transformations that preserve distance and angle to those that do not (e.g., translation versus dilation).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that map the figure onto itself.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures

G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Circles
Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

G-CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure, (e.g., using graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will map a given figure onto another.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

1.2: Understand congruence in terms of rigid motions.

G-CO.6: Use geometric descriptions of rigid motions to transform figures.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

G-CO.6.a: Predict the effect of a given rigid motion on a given figure.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

G-CO.6.b: Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

G-CO.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

1.3: Prove geometric theorems.

G-CO.9: Prove theorems about lines and angles. Theorems must include but not limited to: vertical angles are congruent; when a transversal intersects parallel lines, alternate interior angles are congruent and same side interior angles are supplementary (using corresponding angles postulate); points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints.

Investigating Angle Theorems

G-CO.11: Prove theorems about parallelograms. Theorems must include but not limited to: 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

1.4: Make geometric constructions.

G-CO.12: Perform geometric constructions with a compass and straightedge. including copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines/segments, constructing a line parallel to a given line through a point not on the line.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles
Triangle Inequalities

G-SRT: Similarity, Right Triangles and Trigonometry

2.1: Understand similarity in terms of similarity transformations.

G-SRT.1: Verify experimentally and apply the properties of dilations as determined by a center and a scale factor.

Dilations
Similar Figures

G-SRT.2: Determine whether figures are similar, using the definition of similarity and using similarity transformations.

Circles
Dilations
Similar Figures
Similarity in Right Triangles

G-SRT.3: Use the properties of similarity transformations to establish similarity theorems. Theorems must include AA, SAS, and SSS.

Similar Figures

2.2: Prove theorems involving similarity.

G-SRT.4: Prove theorems about triangles involving similarity. Theorems must include but not limited to: a line parallel to one side of a triangle divides the other two proportionally, and its converse; the Pythagorean Theorem proved using triangle similarity.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures

G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs
Congruence in Right Triangles
Constructing Congruent Segments and Angles
Dilations
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

2.3: Define trigonometric ratios and solve problems involving right triangles.

G-SRT.6: Define, using similarity, that side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios (sine, cosine, and tangent) for acute angles.

Sine, Cosine, and Tangent Ratios

G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

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

G-C: Circles

3.1: Understand and apply theorems about circles.

G-C.2: Identify and describe relationships among central angles, inscribed angles, circumscribed angles, radii, and chords.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

G-C.3: Construct, using a compass and straight edge, the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

3.2: Find arc lengths and areas of sectors of circles.

G-C.5: Derive using similarity the length of the arc intercepted by an angle is proportional to the radius.

Chords and Arcs

G-C.5.a: Define the radian measure of the angle as the constant of proportionality;

Chords and Arcs

G-C.5.b: Derive and apply the formula for the area of a sector.

Chords and Arcs

G-GPE: Expressing Geometric Properties with Equations

4.1: Translate between the geometric description and the equation for a conic section.

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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

4.2: Use coordinates to prove simple geometric theorems algebraically.

G-GPE.5: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Distance Formula

G-GMD: Geometric Measurement and Dimension

5.1: Explain volume and surface area formulas and use them to solve problems.

G-GMD.1: Give an informal argument for the formulas for the volume of a cylinder, pyramid, sphere, and cone. Use dissection arguments, and informal limit arguments.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

G-GMD.2: Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Prisms and Cylinders
Pyramids and Cones

G-GMD.3: Know and apply volume and surface area formulas for cylinders, pyramids, cones, and spheres for composite figures to solve problems.

Prisms and Cylinders
Pyramids and Cones
Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones

S-CP: Statistics and Probability-Conditions Probability and Rules of Probability

7.1: Understand independence and conditional probability and use them to interpret data.

S-CP.1: Describe events as subsets of a sample space or as unions, intersections, or complements of other events.

Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability

S-CP.2: Determine whether two events A and B are independent.

Independent and Dependent Events

S-CP.3: Determine conditional probabilities and interpret independence by analyzing conditional probability.

Independent and Dependent Events

S-CP.4: Construct and interpret two-way frequency tables of data. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Histograms

S-CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and situations.

Independent and Dependent Events

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 result.

Independent and Dependent Events

S-CP.8: Apply the general Multiplication Rule, P(A and B), and interpret the result.

Independent and Dependent Events