| Measurement and Geometry - Grade 9 | |
| 1 Students solve problems involving geometric measures. | |
| 1.1 Students know, use, derive formulas for, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures | |
| 1.2 Students describe how changes in the dimensions of an object affect the perimeter, area, and volume | tripling the radius of a sphere multiplies its volume by 27 |
| 2 Students identify, find missing measures, and solve problems involving angles, right triangles, other polygons, circles, planes and solid geometric objects. (Draft 1.0 Grade 9/10 Measurement and Geometry 3 ; TIMSS 1.3.3) | |
| 2.1 Students find and use measures of sides, interior and exterior angles of triangles and polygons to classify figures and solve problems (Draft 1.0 Grade 9/10 Measurement and Geometry 3.1) |
classify, for example, as isosceles, obtuse, convex, regular etermine the number of degrees in a central angle of a regular polygon |
| 2.2 Students describe the relationships between vertical angles, angles that are supplementary and complementary, and angles formed when parallel lines are cut by a transversal and use these to find missing angle measures in such systems | |
| 2.3 Students use the Pythagorean Theorem, its converse, properties of special right triangles, and right triangle trigonometry to find missing information about triangles | Special right triangles include, for example, those with sides in the ratios 3 to 4 to 5 or angles of 30-60-90 degrees |
| 2.4 Students develop and implement a plan for obtaining indirect measures using similarity, proportions, and trigonometric ratios | |
| 2.5 Students compare, contrast, classify, and solve problems involving quadrilaterals (square, rhombus, rectangle, parallelogram, trapezoid, kite) on the basis of their definitions and properties | |
| 2.6 Students apply the properties of angles, arcs, chords, radii, tangents, and secants to solve problems involving circles | |
| 2.7 Students apply the triangle inequality properties (given information concerning the lengths of sides and/or measures of angles) to determine whether a triangle exists and to order sides and angles | |
| 2.8 Students compare, contrast, and classify geometric figures based on their characteristics, and solve problems based on these features. | |
| 3 Students demonstrates mastery of basic constructions and uses combinations of these to generate more complicated figures | |
| 3.1 Students construct, using a compass and straightedge, a line segment congruent to a given line segment, the bisector of a line segment, a perpendicular to a given line from a point not on the line, a perpendicular to a given line at a point on the line, the bisector of a given angle, an angle congruent to a given angle and use combinations of these basic constructions to create more complex (center of a circle, tangents to circles, etc.) constructions. | |
| 4 Students visualize and describe objects, paths, and regions in space. | |
| 4.1 Students use geometric language to describe the intersection of two planes or of a plane and a solid object | |
| 4.2 Students use geometric language to describe how the conic sections are derived as cross sections of a cone | |
| Mathematical Reasoning - Grade 9 | |
| 1 Students demonstrate understanding of an axiomatic system, and the nature of proof. | |
| 1.1 Students identify and give examples of undefined terms, axioms, theorems, inductive, and deductive reasoning | |
| 1.2 Students prove the Pythagorean Theorem using algebraic and geometric arguments. | |
| 1.3 Students prove basic theorems involving congruence and similarity | Assumed postulates, like SSS, SAS and ASA will be given. |
| 1.4 Students prove theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles | |
| 2 Students constructs and judges the validity of a logical argument consisting of a premise and a set of conclusions | |
| 2.1 Students use valid forms of deductive reasoning, including the law of syllogism, and translates short verbal arguments into symbolic form | |
| 2.2 Students identify the converse, inverse, and contra-positive of a conditional statement | |
| 2.3 Students diagram arguments involving quantifiers (all, no, none, some) | |
| 2.4 Students use counterexamples to disprove a statement |