Geometry

 1.0 REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS L1.1 Number Systems and Number Sense L1.1.6 Explain the importance of the irrational numbers and in basic right triangle trigonometry, and the importance of because of its role in circle relationships. L1.2 Representations and Relationships L1.2.3 Use vectors to represent quantities that have magnitude and direction, interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors. 2.0 CALCULATION, ALGORITHMS, AND ESTIMATION L2.3 Measurement Units, Calculations, and Scales L2.3.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly. 3.0 MATHEMATICAL REASONING, LOGIC, AND PROOF L3.1 Mathematical Reasoning L3.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each. L3.1.2 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic. L3.1.3 Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each. L3.2 Language and Laws of Logic L3.2.1 Know and use the terms of basic logic. L3.2.2 Use the connectives "not," "and," "or," and "if..., then," in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives. L3.2.3 Use the quantifiers "there exists" and "all" in mathematical and everyday settings and know how to logically negate statements involving them. L3.2.4 Write the converse, inverse, and contrapositive of an "If..., then..." statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not. L3.3 Proof L3.3.1 Know the basic structure for the proof of an "If..., then..." statement (assuming the hypothesis and ending with the conclusion) and that proving the contrapositive is equivalent. L3.3.2 Construct proofs by contradiction. Use counter-examples, when appropriate, to disprove a statement. L3.3.3 Explain the difference between a necessary and a sufficient condition within the statement of a theorem. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied. 4.0 FIGURES AND THEIR PROPERTIES G1.1 Lines and Angles; Basic Euclidean and Coordinate Geometry G1.1.1 Solve multistep problems and construct proofs involving vertical angles, linear pairs of angles, supplementary angles, complementary angles, and right angles. G1.1.2 Solve multistep problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass. G1.1.4 Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass. Given a line and a point, construct a line through the point that is perpendicular to the original line. Justify the steps of the constructions. G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint. G1.1.6 Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions, and theorems. G1.2 Triangles and Their Properties G1.2.1 Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles. G1.2.2 Construct and justify arguments and solve multistep problems involving angle measure, side length, perimeter, and area of all types of triangles. G1.2.3 Know a proof of the Pythagorean Theorem, and use the Pythagorean Theorem and its converse to solve multi-step problems. G1.2.4 Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles. G1.2.5 Solve multistep problems and construct proofs about the properties of medians, altitudes and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle. Using a straightedge and compass, construct these lines. G1.3 Triangles and Trigonometry G1.3.1 Define the sine, cosine, and tangent of acute angles in a right triangle as ratios of sides. Solve problems about angles, side lengths, or areas using trigonometric ratios in right triangles. G1.3.2 Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle q using the formula Area = (1/2) absin q. G1.3.3 Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various contexts. G1.4 Quadrilaterals and Their Properties G1.4.1 Solve multistep problems and construct proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids. G1.4.2 Solve multistep problems and construct proofs involving quadrilaterals using Euclidean methods or coordinate geometry. G1.4.3 Describe and justify hierarchical relationships among quadrilaterals. G1.4.4 Prove theorems about the interior and exterior angle sums of a quadrilateral. G1.5 Other Polygons and Their Properties G1.5.1 Know and use subdivision or circumscription methods to find areas of polygons. G1.5.2 Know, justify, and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums. G1.6 Circles and Their Properties G1.6.1 Solve multistep problems involving circumference and area of circles. G1.6.2 Solve problems and justify arguments about chords and lines tangent to circles. G1.6.3 Solve problems and justify arguments about central angles, inscribed angles, and triangles in circles. G1.6.4 Know and use properties of arcs and sectors, and find lengths of arcs and areas of sectors. G1.8 Three-dimensional Figures G1.8.1 Solve multistep problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres, and spheres. G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres. 5.0 RELATIONSHIPS BETWEEN FIGURES G2.1 Relationships Between Area and Volume Formulas G2.1.1 Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram, and the area formula of a trapezoid. G2.1.2 Know and demonstrate the relationships between the area formulas of various quadrilaterals. G2.1.3 Know and use the relationship between the volumes of pyramids and prisms. G2.2 Relationships Between Two-dimensional and Three-dimensional Representations G2.2.1 Identify or sketch a possible three-dimensional figure, given two-dimensional views. Create a two-dimensional representation of a three-dimensional figure. G2.2.2 Identify or sketch cross sections of three-dimensional figures. Identify or sketch solids formed by revolving two-dimensional figures around lines. G2.3 Congruence and Similarity G2.3.1 Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria and that right triangles are congruent using the hypotenuse-leg criterion. G2.3.2 Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates. G2.3.3 Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity. G2.3.4 Use theorems about similar triangles to solve problems with and without use of coordinates. G2.3.5 Know and apply the theorem stating that the effect of a scale factor of k relating one two-dimensional figure to another or one three-dimensional figure to another, on the length, area, and volume of the figures, is to multiply each by k, k2, and k3, respectively. 6.0 TRANSFORMATIONS OF FIGURES IN THE PLANE G3.1 Distance-preserving Transformations Isometries G3.1.1 Define reflection, rotation, translation, and glide reflection and find the image of a figure under a given isometry. G3.1.2 Given two figures that are images of each other under an isometry, find the isometry and describe it completely. G3.1.3 Find the image of a figure under the composition of two or more isometries and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure. G3.2 Shape-preserving Transformations: Dilations and Isometries G3.2.1 Know the definition of dilation and find the image of a figure under a given dilation. G3.2.2 Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation.