Killingly Public Schools. Grades Draft Sept. 2002

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1 Killingly Public Schools Grades Draft Sept. 2002

2 ESSENTIALS OF GEOMETRY Grades Language of Plane Geometry CONTENT STANDARD EG 1: The student will use the properties of points, lines, and planes to define and construct other basic geometric figures. The student is able to understand the basic geometric 10-12EG 1.1 figures and their properties names, knows the properties of, draws and writes the symbols for: points, lines, segments, rays, and planes identifies collinear, non-collinear, coplanar, and non-coplanar points. The student is able to understand angles and their various 10-12EG 1.2 forms identifies the two rays that form the sides of the angle and the vertex of the angle names an angle is a variety of ways identifies right angles, perpendicularity, opposite rays, and angles. The student is able to understand parallel lines, parallel 10-12EG 1.3 planes, intersecting lines, and intersecting planes identifies the intersection of two lines as a point identifies the intersection of two planes as a line defines parallels and identifies the symbols for parallel names lines and planes that are parallel identifies and names skew lines.

3 The student is able to use the metric system to measure EG 1.4 length names a length using symbols identifies the common units of length in the metric system converts between the various metric units identifies the best unit of metric measurement applied to real-world objects. The student is able to construct congruent segments and 10-12EG 1.5 segment bisectors identifies the symbols for congruence identifies and constructs congruent segments defines midpoint identifies and constructs bisectors and perpendicular bisectors. The student is able to use different types of reasoning 10-12EG 1.6 techniques to solve problems identifies deductive and inductive reasoning techniques applies inductive and deductive reasoning to solving problems solves problems by drawing a diagram or making a model.

4 Angles CONTENT STANDARD EG 2: The student will measure and classify angles. The student is able to use amount of turn to measure angles 10-12EG 2.1 and use the relationships among parts of circles defines and identifies a circle, its center and all radii, chords, and diameters calculates radius or diameter using d = 2r identifies the amount of turn of a central angle EG 2.2 The student is able to measure angles in degrees calculates the change between amount of turn and degree measure uses a protractor to measure and draw angles identifies adjacent angles adds and subtracts angle measures. The student is able to inderstand the relationships between 10-12EG 2.3 pairs of angles identifies, labels, and constructs congruent angles, angle bisectors, vertical angles, complementary angles, and supplementary angles solves word problems by working backwards.

5 Polygons and Polyhedrons CONTENT STANDARD EG 3: The student will recognize, categorize, and/or name polygons and polyhedrons. The student is able to recognize concave and convex 10-12EG 3.1 polygons identifies figures that are polygons, the vertices, and sides of a polygon recognizes convex and concave polygons identifies polygons by their number of sides and names a polygon by the vertices identifies consecutive angles, adjacent (consecutive) sides, adjacent (consecutive) vertices, and the diagonals of a polygon EG 3.2 The student is able to classify triangles identifies equilateral, equiangular, isosceles, scalene, acute, right, and obtuse triangles by the number of congruent sides or by its angles identifies the base, legs, vertex angle, and base angles of a triangle. The student is able to recognize types of quadrilaterals and 10-12EG 3.3 other polygons names, defines, and draws a trapezoid, parallelogram, rhombus, rectangle and a square identifies regular polygons.

6 10-12EG 3.4 The student is able to name polyhedrons and prisms names the faces, edges, and vertices of a polyhedron names the bases and lateral faces of a prism identifies a prism by the bases accurately draws a prism and identifies points of view draws figures using one-point and two-point perspective.

7 Introduction to Transformations CONTENT STANDARD EG 4: The student will examine congruence and its relationship to transformations and symmetry. The student is able to understand congruence and 10-12EG 4.1 corresponding parts names corresponding sides and angles of congruent polygons writes congruence statements for congruent sides, angles, and polygons. The student is able to understand various congruence EG 4.2 preserving transformations names and draws reflection, translation, and rotation images identifies the line of reflection and clockwise, counter-clockwise, and reverse orientation solves problems involving reflections EG 4.3 The student is able to identify different types of symmetry identifies and draws lines of symmetry identifies line, rotational, and point symmetry.

8 Triangles and Inequalities CONTENT STANDARD EG 5: The student will explore in-depth triangles and their related segments. The student understands the Isosceles Triangle Theorem 10-12EG 5.1 and related theorems identifies the line of symmetry of an isosceles triangle and recognizes that it bisects the vertex angle and is the perpendicular bisector of the base recognizes the base angles of an isosceles triangle are congruent identifies the two sides opposite two congruent angles in a triangle are congruent. The student is able to construct or name several types of 10-12EG 5.2 concurrent lines that occur with triangles identifies or draws the altitudes of a triangle and the orthocenter identifies or draws the medians of a triangle and labels the centroid constructs the circumcenter of a triangle using the perpendicular bisectors of the sides constructs the incenter of a triangle using angle bisectors demonstrates that a point on the perpendicular bisector of segment is equidistant from the ends of the segment. The student is able to understand inequalities involving 10-12EG 5.3 side lengths and angle measure in triangles orders the sides/angles of a triangle by size given the measure of the angles/sides identifies if three given lengths of segments can be the sides of a triangle identifies included angles and sides applies the Hinge Theorem solves problems using the guess-and-check technique.

9 Congruent Triangles CONTENT STANDARD EG 6: The student will explore congruent triangles and their properties. The student understands various postulates and theorems 10-12EG 6.1 that prove that two triangles are congruent identifies the postulates (SSS, SAS, ASA) and theorems (AAS, HL) applies the appropriate postulate of theorem to show that two triangles are congruent identifies congruent triangles and sides of triangles based on the postulate or theorem being applied. The student understands the relationship between congruent 10-12EG 6.2 triangles and their corresponding sides and angles identifies Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.) applies C.P.C.T.C. to determine which angles and sides are congruent given two congruent triangles. The student is able to use the understanding of congruent 10-12EG 6.2 triangles to solve real-world applications estimates distances using congruent triangles.

10 Parallel Lines CONTENT STANDARD EG 7: The student will explore the various pairs of angles that are formed when two lines are cut by a transversal. The student is able to identify the angles formed when two 10-12EG 7.1 lines are cut by a transversal identifies interior, exterior, alternate interior/exterior, corresponding, vertical, adjacent supplementary and same side interior/exterior angles. The student is able to understand angle relationships that 10-12EG 7.2 occur when parallel lines are cut by a transversal recognizes that the alternate interior, vertical, and corresponding angles are congruent recognizes that same side interior, same side exterior, and adjacent angles are supplementary identifies two lines are parallel by the angle measure relationships. The student is able to find interior and exterior angle 10-12EG 7.3 measures of polygons calculates various interior and exterior angle measurements using the Triangle Angle-Sum Theorem, vertical, adjacent supplementary, exterior and remote interior angles calculates the sum of the measures of the interior of a polygon with n sides as (n-2) calculates exterior angle measure using the Exterior Angle-Sum Theorem interprets and solves problems using contour maps.

11 Properties of Quadrilaterals CONTENT STANDARD EG 8: The student will explore and apply properties of quadrilaterals. The student is able to use the properties of parallelograms, 10-12EG 8.1 including special parallelograms (rhombuses, rectangles, and squares) identifies the properties of each type of parallelogram applies the properties to determine the measures of the sides and angles of a given parallelogram. The student is able to identify the parts of trapezoids and 10-12EG 8.2 isosceles trapezoids and apply the properties of each identifies the bases, legs, base angles, and median of trapezoids and isosceles trapezoids identifies the properties of trapezoids and isosceles trapezoids applies the properties to determine the measures of the sides and angles of a given trapezoid or isosceles trapezoid. The student is able to understand the relationship between 10-12EG 8.3 uadrilaterals, parallelograms, and special parallelograms (rhombuses, rectangles, and squares) identifies theorems that determine if a quadrilateral is a parallelogram, and if a parallelogram is a special parallelogram applies the theorems to classify quadrilaterals as parallelograms or as special parallelograms (rectangles, rhombuses, or squares).

12 Perimeter and Area CONTENT STANDARD EG 9: The student will focus on the concepts of perimeter and area of geometric figures. The student is able to determine the perimeter and area of 10-12EG 9.1 various polygons measures or calculates the perimeter of various convex and concave polygons determines the area of a polygon by counting squares calculates the area of rectangles, parallelograms, triangles, trapezoids, rhombuses, and regular polygons using appropriate formulas solves formulas for unknown variables calculates the area of a complex figure by breaking the problem down into smaller pieces and using appropriate formulas. The student is able to determine the circumference and area 10-12EG 9.2 of a circle identifies pi (π) and the commonly used approximations calculates circumference using C= πd=2πr calculates the area of a circle using A= πr writes solutions to circumference and area of circle problems as exact and/or approximate solutions.

13 Similarity CONTENT STANDARD EG 10: The student will explore the relationships between similarity, ratios, proportions, and scale factors EG 10.1 The student is able to express ratios in simplest form defines and identifies ratios rewrites ratios in various forms (a to b, a:b, and a/b) simplifies ratios EG 10.2 The student is able to solve proportions defines proportion, means, and extremes identifies the Means-Extremes Property and applies it to solve proportions. The student is able to recognize figures that appear to be 10-12EG 10.3 similar and identify corresponding parts of similar polygons defines similar figures and the notation (~) determines if figures appear to be similar uses proper notation when writing similarity statements. The student understands the relationship between 10-12EG 10.4 similarity, proportions, and scale factors sets up various proportions given similar polygons solves proportions to find missing lengths of similar polygons verifies that figures are similar using proportions defines scale factor and can identify the scale factor of similar polygons applies scale factors to calculate actual lengths from scale drawings.

14 Similar Triangles CONTENT STANDARD EG 11: The student will expand their knowledge of, and the relationships between, similar triangles. The student is able to show that pairs of triangles are 10-12EG 11.1 similar identifies similar triangles using the AA, SSS, and SAS Similarity Postulates writes similarity statements for triangles calculates angle and side measures using proportions and similarity statements. The student understands the Triangle Proportionality 10-12EG 11.2 Theorem and its consequences identifies the corresponding, proportional segments of two triangles calculates lengths of triangle segments given certain segment lengths solves problems using indirect measurement applies the fact that parallel lines divide transversals into proportional segments.

15 Squares, Square Roots, and Right Triangles CONTENT STANDARD EG 12: The student will explore various properties of squares, square roots, and right triangles EG 12.1 The student is able to find squares and square roots defines square, square root, perfect square, and radical notation ( ) evaluates squares and square roots estimates square roots of non-perfect squares uses a calculator to find exact or approximate square root values identifies the Product Property of Square Roots and applies it to simplify square roots. The student is able to use the Pythagorean Theorem and its 10-12EG 12.2 converse identifies the Pythagorean Theorem and its converse applies the Pythagorean Theorem to solve for the missing side of a right triangle applies the converse of the Pythagorean Theorem to determine if a triangle is right, acute, or obtuse defines Pythagorean Triples and is able to identify them applies the Pythagorean Theorem to find the shortest distance across a room. The student is able to manipulate products and quotients of 10-12EG 12.3 square root expressions applies the Product Property of Square Roots to find the product of a square root expression and write it in simplified form identifies the properties of simplest radical form rewrites square root expressions, including expressions with a square root in the denominator, in simplest radical form.

16 The student is able to understands the relationship between 10-12EG 12.4 triangles, sides, and altitudes of right triangles identifies and right triangles and their properties applies the properties of and right triangles to determine the lengths of the sides of the triangles defines geometric mean and identifies the properties involving right triangles and geometric mean applies the geometric mean properties to find various lengths within right triangles.

17 Area and Volume CONTENT STANDARD EG 13: The student will explore area and volume of various three-dimensional geometric figures EG 13.1 The student is able to find area and volume of right prisms identifies the bases and faces of a right prism calculates lateral and total area of a right prism calculates volume or a missing dimension (given the volume) of a right prism changes a measure of area or volume to another unit. The student is able to find area and volume of right circular 10-12EG 13.2 cylinders identifies the circular bases and curved lateral surface calculates the lateral area using L.A.=Ch=Βdh=2Βrh calculates total surface area calculates volume using V=Βr 2 h solves problems using area and volume of three-dimensional figures.

18 Trigonometry CONTENT STANDARD EG 14: The student will explore basic trigonometric ratios. The student is able to find the ratios of the basic 10-12EG 14.1 trigonometry ratios for any acute angle of a right triangle given the lengths of the sides of the triangle. A calculates sine, cosine and tangent ratios for acute πa in a right triangle as: lengthof leg opposite sin A = length of hypotenuse, lengthof leg adjacent to cos A = length of hypotenuse, and length of leg opposite A tan A = length of leg adjacent to A

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