(b) Geometric structure: knowledge and skills and performance descriptions.

(1) The student understands the structure of, and relationships within, an axiomatic system. Following are performance descriptions.

(A) The student develops an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems.
(B) Through the historical development of geometric systems, the student recognizes that mathematics is developed for a variety of purposes.
(C) The student compares and contrasts the structures and implications of Euclidean and nonEuclidean geometries.

Interactive Student

Interactive Classroom
(C) Dan Pedoe's Observation

(2) The student analyzes geometric relationships in order to make and verify conjectures. Following are performance descriptions.

(A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships.
(B) The student makes and verifies conjectures about angles, lines, polygons, circles, and threedimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

Interactive Student
(B) Cutting Corners
(B) Circle Ratio

Interactive Classroom
(B) Regular Tiling
(B) Looking at Polygons

(3) The student understands the importance of logical reasoning, justification, and proof in mathematics. Following are performance descriptions.

(A) The student determines if the converse of a conditional statement is true or false.
(B) The student constructs and justifies statements about geometric figures and their properties.
(C) The student demonstrates what it means to prove mathematically that statements are true.
(D) The student uses inductive reasoning to formulate a conjecture.
(E) The student uses deductive reasoning to prove a statement.

Interactive Student
(B) Light Bounce

Interactive Classroom

(4) The student uses a variety of representations to describe geometric relationships and solve problems.

Following is a performance description. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

Interactive Student

Interactive Classroom

(c) Geometric patterns: knowledge and skills and performance descriptions.
The student identifies, analyzes, and describes patterns that emerge from two and threedimensional geometric figures. Following are performance descriptions.

(B) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.
() The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations or fractals.
(3) The student identifies and applies patterns from right triangles to solve problems, including special right triangles (454590 and 306090) and triangles whose sides are Pythagorean triples.

Interactive Student

Interactive Classroom

(d) Dimensionality and the geometry of location: knowledge and skills and performance descriptions.

(1) The student analyzes the relationship between threedimensional objects and related twodimensional representations and uses these representations to solve problems. Following are performance descriptions.

(A) The student describes, and draws cross sections and other slices of threedimensional objects.
(B) The student uses nets to represent and construct threedimensional objects.
(C) The student uses top, front, side, and corner views of threedimensional objects to create accurate and complete representations and solve problems.

Interactive Student

Teacher

(2) The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. Following are performance descriptions.

(A) The student uses one and twodimensional coordinate systems to represent points, lines, line segments, and figures.
(B) The student uses slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons.
(C) The student develops and uses formulas including distance and midpoint.

Interactive Student


(e) Congruence and the geometry of size: knowledge and skills and performance descriptions.

(8) The student extends measurement concepts to find area, perimeter, and volume in problem situations. Following are performance descriptions.

(A) The student finds areas of regular polygons and composite figures.
(B) The student finds areas of sectors and arc lengths of circles using proportional reasoning.
(C) The student develops, extends, and uses the Pythagorean Theorem.
(D) The student finds surface areas and volumes of prisms, pyramids, spheres, cones, and cylinders in problem situations.

Interactive Student
(C) Pythagorean Puzzles


(2) The student analyzes properties and describes relationships in geometric figures. Following are performance descriptions.

(A) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties of parallel and perpendicular lines.
(B) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of polygons and their component parts.
(C) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of circles and the lines that intersect them.
(D) The student analyzes the characteristics of threedimensional figures and their component parts.

Interactive Student


(3) The student applies the concept of congruence to justify properties of figures and solve problems. Following are performance descriptions.

(A) The student uses congruence transformations to make conjectures and justify properties of geometric figures.
(B) The student justifies and applies triangle congruence relationships.

Interactive Student
(B) Congruence Theorems


(f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems. Following are performance descriptions.

(1) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures.
(2) The student uses ratios to solve problems involving similar figures.
(3) In a variety of ways, the student develops, applies, and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples.
(4) The student describes the effect on perimeter, area, and volume when length, width, or height of a threedimensional solid is changed and applies this idea in solving problems.

(3) Square in a Right Triangle 

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Geometry  Adopted 2012
(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

(A) apply mathematics to problems arising in everyday life, society, and the workplace;
(B) use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution;
(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
(E) create and use representations to organize, record, and communicate mathematical ideas;
(F) analyze mathematical relationships to connect and communicate mathematical ideas; and
(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Interactive Student

Interactive Classroom

(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one and twodimensional coordinate systems to verify geometric conjectures. The student is expected to:

(A) determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one and twodimensional coordinate systems, including finding the midpoint;
(B) derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines; and
(C) determine an equation of a line parallel or perpendicular to a given line that passes through a given point.

Interactive Student

Interactive Classroom

(3) Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and nonrigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to:

(A) describe and perform transformations of figures in a plane using coordinate notation;
(B) determine the image or preimage of a given twodimensional figure under a composition of rigid transformations, a composition of nonrigid transformations, and a composition of both, including dilations where the center can be any point in the plane;
(C) identify the sequence of transformations that will carry a given preimage onto an image on and off the coordinate plane; and
(D) identify and distinguish between reflectional and rotational symmetry in a plane figure.

Interactive Student

Interactive Classroom

(4) Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:

(A) distinguish between undefined terms, definitions, postulates, conjectures, and theorems;
(B) identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse;
(C) verify that a conjecture is false using a counterexample; and
(D) compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

Interactive Student

Interactive Classroom (D) Dan Pedoe's Observation

(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:

(A) investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools;
(B) construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge;
(C) use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships; and
(D) verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems.

Interactive Student (A) Cutting Corners (A) Circle Ratio
(B) Light Bounce

Interactive Classroom (A) Regular Tiling (A) Looking at Polygons

(6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as twocolumn, paragraph, and flow chart. The student is expected to:

(A) verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems;
(B) prove two triangles are congruent by applying the SideAngleSide, AngleSideAngle, SideSideSide, AngleAngleSide, and HypotenuseLeg congruence conditions;
(C) apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles;
(D) verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems; and
(E) prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

Interactive Student (D) Square in a Right Triangle (D) Pythagorean Puzzles

Teacher

(7) Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:

(A) apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles; and
(B) apply the AngleAngle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

Interactive Student


(8) Similarity, proof, and trigonometry. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as twocolumn, paragraph, and flow chart. The student is expected to:

(A) prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems; and
(B) identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

Interactive Student


(9) Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:.

(A) determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems; and
(B) apply the relationships in special right triangles 30°60°90° and 45°45°90° and the Pythagorean theorem, including Pythagorean triples, to solve problems.

Interactive Student


(10) Twodimensional and threedimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two and threedimensional figures. The student is expected to:

(A) identify the shapes of twodimensional crosssections of prisms, pyramids, cylinders, cones, and spheres and identify threedimensional objects generated by rotations of twodimensional shapes; and
(B) determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and nonproportional dimensional change.

Interactive Student


(12) Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:

(A) apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve noncontextual problems;
(B) apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems;
(C) apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems;
(D) describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle; and
(E) show that the equation of a circle with center at the origin and radius r is x2 + y2 = r2 and determine the equation for the graph of a circle with radius r and center (h, k), (x  h)2 + (y  k)2 =r2.



(13) Probability. The student uses the process skills to understand probability in realworld situations and how to apply independence and dependence of events. The student is expected to:

(A) develop strategies to use permutations and combinations to solve contextual problems;
(B) determine probabilities based on area to solve contextual problems;
(C) identify whether two events are independent and compute the probability of the two events occurring together with or without replacement;
(D) apply conditional probability in contextual problems; and
(E) apply independence in contextual problems.



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