#### GEOMETRY STANDARDS

#### Problem Solving Strand

Students willbuild new mathematical knowledge through problem solving

Use a variety of problem solving strategies to understand new mathematical content

Students will solve problems that arise in mathematics and in other contexts

Observe and explain patterns to formulate generalizations and conjectures

Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations)

Students will apply and adapt a variety of appropriate strategies to solve problems

Construct various types of reasoning, arguments, justifications and methods of proof for problems

Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)

Use a variety of strategies to extend solution methods to other problems

Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving

Students will monitor and reflect on the process of mathematical problem solving

Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions

Interpret solutions within the given constraints of a problem

Evaluate the relative efficiency of different representations and solution methods of a problem

#### Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Recognize that mathematical ideas can be supported by a variety of strategies

Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies

Students will make and investigate mathematical conjectures

Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion

Students will develop and evaluate mathematical arguments and proofs

Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments

Present correct mathematical arguments in a variety of forms

Evaluate written arguments for validity

Students will select and use various types of reasoning and methods of proof

Construct a proof using a variety of methods (e.g., deductive, analytic, transformational)

Devise ways to verify results or use counterexamples to refute incorrect statements

Apply inductive reasoning in making and supporting mathematical conjectures

#### Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form

Explain relationships among different representations of a problem

Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

Support or reject arguments or questions raised by others about the correctness of mathematical work

Students will analyze and evaluate the mathematical thinking and strategies of others

Read and listen for logical understanding of mathematical thinking shared by other students

Reflect on strategies of others in relation to one's own strategy

Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others

Students will use the language of mathematics to express mathematical ideas precisely

Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures

Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams

Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing

#### Connections Strand

Students will recognize and use connections among mathematical ideas

Understand and make connections among multiple representations of the same mathematical idea

Understand the corresponding procedures for similar problems or mathematical concepts

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Model situations mathematically, using representations to draw conclusions and formulate new situations

Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

Understand how quantitative models connect to various physical models and representations

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and apply mathematics to situations in the outside world

Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

Develop an appreciation for the historical development of mathematics

#### Representation Strand

Students will create and use representations to organize, record, and communicate mathematical ideas

Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

Recognize, compare, and use an array of representational forms

Use representation as a tool for exploring and understanding mathematical ideas

Students will select, apply, and translate among mathematical representations to solve problems

Select appropriate representations to solve problem situations

Investigate relationships between different representations and their impact on a given problem

Students will use representations to model and interpret physical, social, and mathematical phenomena

Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank)

Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation)

Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent)

#### Geometry Strand

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes

Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them

Know and apply that through a given point there passes one and only one plane perpendicular to a given line

Know and apply that through a given point there passes one and only one line perpendicular to a given plane

Know and apply that two lines perpendicular to the same plane are coplanar

Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane

Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane

Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines

Know and apply that if two planes are perpendicular to the same line, they are parallel

Know and apply that the lateral edges of a prism are congruent and parallel

Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal

Know and apply that the volume of a prism is the product of the area of the base and the altitude

Apply the properties of a regular pyramid, including: lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude

Apply the properties of a cylinder, including: bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base

Apply the properties of a right circular cone, including: lateral area equals one-half the product of the slant height and the circumference of its base; volume is one-third the product of the area of its base and its altitude

Apply the properties of a sphere, including: the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so i

Construct a bisector of a given angle, using a straightedge and compass, and justify the construction

Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction

Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction

Construct an equilateral triangle, using a straightedge and compass, and justify the construction

Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles

Solve problems using compound loci

Graph and solve compound loci in the coordinate plane

Students will identify and justify geometric relationships formally and informally

Determine the negation of a statement and establish its truth value

Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true

Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences

Write a proof arguing from a given hypothesis to a given conclusion

Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles

Identify corresponding parts of congruent triangles

Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle

Investigate, justify, and apply the isosceles triangle theorem and its converse

Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem

Investigate, justify, and apply the triangle inequality theorem

Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle

Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines

Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons

Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons

Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals

Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals

Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals

Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids

Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle

Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1

Establish similarity of triangles, using the following theorems: AA, SAS, and SSS

Investigate, justify, and apply theorems about similar triangles

Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle

Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the

Investigate, justify, and apply the Pythagorean theorem and its converse

Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords; the relative lengths of chords as compared to their distance from the center of the circle

Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles

Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is: inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two

Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines

Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two inte

Students will apply transformations and symmetry to analyze problem solving situations

Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation

Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections

Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism

Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)

Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)

Investigate, justify, and apply the properties that remain invariant under similarities

Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism

Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90° and 180°, reflections over the lines x = 0, y =0, and y = x , and dilations centered at the origin

Students will apply coordinate geometry to analyze problem solving situations

Find the slope of a perpendicular line, given the equation of a line

Determine whether two lines are parallel, perpendicular, or neither, given their equations

Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line

Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line

Find the midpoint of a line segment, given its endpoints

Find the length of a line segment, given its endpoints

Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment

Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas

Solve systems of equations involving one linear equation and one quadratic equation graphically

Write the equation of a circle, given its center and radius or given the endpoints of a diameter

Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer

Find the center and radius of a circle, given the equation of the circle in center-radius form

Graph circles of the form ( x - h )

^{2}+ ( y - k )^{2}= r^{2}**Information Source:**NYSED.GOV CS&IT (standards pdf)