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# Strand Trace Geometry

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. PK.G.1 Shapes Match shapes, first with same size and orientation, then with different sizes and orientation. PK.G.2 Shapes Informally play with solids (e.g., building blocks). K.G.1 Shapes Describe characteristics and relationships of geometric objects. 1.G.1 Shapes Match shapes and parts of shapes to justify congruency. 1.G.2 Shapes Recognize, name, describe, create, sort, and compare two -dimensional and three-dimensional shapes. 2.G.1 Shapes Experiment with slides, flips, and turns to compare two-dimensional shapes. 2.G.2 Shapes Identify and appropriately name two-dimensional shapes: circle, square, rectangle, and triangle (both regular and irregular). 2.G.3 Shapes Compose (put together) and decompose (break apart) two-dimensional shapes. 3.G.1 Shapes Define and use correct terminology when referring to shapes (circle, triangle, square, rectangle, rhombus, trapezoid, and hexagon). 3.G.2 Shapes Identify congruent and similar figures. 3.G.3 Shapes Name, describe, compare, and sort three-dimensional shapes: cube, cylinder, sphere, prism, and cone. 3.G.4 Shapes Identify the faces on a three-dimensional shape as two-dimensional shapes. 4.G.1 Shapes Identify and name polygons, recognizing that their names are related to the number of sides and angles (triangle, quadrilateral, pentagon, hexagon, and octagon). 4.G.2 Shapes Identify points and line segments when drawing a plane figure. 4.G.3 Shapes Find perimeter of polygons by adding sides . 4.G.4 Shapes Find the area of a rectangle by counting the number of squares needed to cover the rectangle. 4.G.5 Shapes Define and identify vertices, faces, and edges of three-dimensional shapes. 5.G.1 Shapes Calculate the perimeter of regular and irregular polygons. 6.G.1 Shapes Calculate the length of corresponding sides of similar triangles, using proportional reasoning. 6.G.2 Shapes Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas. 6.G.3 Shapes Use a variety of strategies to find the area of regular and irregular polygons. 6.G.4 Shapes Determine the volume of rectangular prisms by counting cubes and develop the formula. 6.G.5 Shapes Identify radius, diameter, chords and central angles of a circle. 6.G.6 Shapes Understand the relationship between the diameter and radius of a circle.
 6.G.7 Shapes Determine the area and circumference of a circle, using the appropriate formula. 6.G.8 Shapes Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle. 6.G.9 Shapes Understand the relationship between the circumference and the diameter of a circle. 7.G.1 Shapes Calculate the radius or diameter, given the circumference or area of a circle. 7.G.2 Shapes Calculate the volume of prisms and cylinders, using a given formula and a calculator. 7.G.3 Shapes Identify the two-dimensional shapes that make up the faces and bases of three-dimensional shapes (prisms, cylinders, cones, and pyramids). 7.G.4 Shapes Determine the surface area of prisms and cylinders, using a calculator and a variety of methods. 8.G.0 Constructions Construct the following, using a straight edge and compass: Segment congruent to a segment, Angle congruent to an angle, Perpendicular bisector, Angle bisector.

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. A.G.1 Shapes Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only). A.G.2 Shapes Use formulas to calculate volume and surface area of rectangular solids and cylinders. G.G.1 Shape 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. G.G.2 Shape Know and apply that through a given point there passes one and only one plane perpendicular to a given line. G.G.3 Shape Know and apply that through a given point there passes one and only one line perpendicular to a given plane. G.G.4 Shape Know and apply that two lines perpendicular to the same plane are coplanar. G.G.5 Shape 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. G.G.6 Shape 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. G.G.7 Shape Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane. G.G.8 Shape Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines. G.G.9 Shape Know and apply that if two planes are perpendicular to the same line, they are parallel. G.G.10 Shape Know and apply that the lateral edges of a prism are congruent and parallel. G.G.11 Shape Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal. G.G.12 Shape Know and apply that the volume of a prism is the product of the area of the base and the altitude. G.G.13 Shape Apply the properties of a regular pyramid, including: 1) lateral edges are congruent; 2) lateral faces are congruent isosceles triangles; and 3) volume of a pyramid equals one-third the product of the area of the base and the altitude. G.G.14 Shape Apply the properties of a cylinder, including: 1) bases are congruent; 2) volume equals the product of the area of the base and the altitude; and 3) lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base. G.G.15 Shape Apply the properties of a right circular cone, including: 1) lateral area equals one-half the product of the slant height and the circumference of its base; and 2) volume is one-third the product of the area of its base and its altitude. G.G.16 Shape Apply the properties of a sphere, including: 1) the intersection of a plane and a sphere is a circle; 2) a great circle is the largest circle that can be drawn on a sphere; 3) two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; 4) surface area is ; and 5) volume is . G.G.17 Constructions Construct a bisector of a given angle, using a straightedge and compass, and justify the construction. G.G.18 Constructions Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction. G.G.19 Constructions Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction. G.G.20 Constructions Construct an equilateral triangle, using a straightedge and compass, and justify the construction. G.G.21 Locus Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles. G.G.22 Locus G.G.23 Locus Solve problems using compound loci . Graph and solve compound loci in the coordinate plane .

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will identify and justify geometric relationships, formally and informally. K.G.2 Relationships Sort groups of objects by size and size order (increasing and decreasing). 2.G.4 Relationships Group objects by like properties . 4.G.6 Relationships Draw and identify intersecting, perpendicular, and parallel lines. 4.G.7 Relationships Identify points and rays when drawing angles. 4.G.8 Relationships Classify angles as acute, obtuse, right, and straight. 5.G.2 Relationships Identify pairs of similar triangles. 5.G.3 Relationships Identify the ratio of corresponding sides of similar triangles. 5.G.4 Relationships Classify quadrilaterals by properties of their angles and sides. 5.G.5 Relationships Know that the sum of the interior angles of a quadrilateral is 360 degrees. 5.G.6 Relationships Classify triangles by properties of their angles and sides. 5.G.7 Relationships Know that the sum of the interior angles of a triangle is 180 degrees. 5.G.8 Relationships Find a missing angle when given two angles of a triangle. 5.G.9 Relationships Identify pairs of congruent triangles. 5.G.10 Relationships Identify corresponding parts of congruent triangles. 7.G.5 Relationships Identify the right angle, hypotenuse, and legs of a right triangle. 7.G.6 Relationships Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem. 7.G.7 Relationships Find a missing angle when given angles of a quadrilateral. 7.G.8 Relationships Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle. 7.G.9 Relationships Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator. 8.G.1 Relationships Identify pairs of vertical angles as congruent. 8.G.2 Relationships Identify pairs of supplementary and complementary angles. 8.G.3 Relationships Calculate the missing angle in a supplementary or complementary pair. 8.G.4 Relationships Determine angle pair relationships when given two parallel lines cut by a transversal. 8.G.5 Relationships Calculate the missing angle measurements when given two parallel lines cut by a transversal.
 8.G.6 Relationships Calculate the missing angle measurements when given two intersecting lines and an angle. G.G.24 Proofs Determine the negation of a statement and establish its truth value. G.G.25 Proofs Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true. G.G.26 Proofs Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences. G.G.27 Proofs Write a proof arguing from a given hypothesis to a given conclusion. G.G.28 Proofs 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. G.G.29 Proofs Identify corresponding parts of congruent triangles. G.G.30 Proofs Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle. G.G.31 Proofs Investigate, justify, and apply the isosceles triangle theorem and its converse. G.G.32 Proofs Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem. G.G.33 Proofs Investigate, justify, and apply the triangle inequality theorem. G.G.34 Proofs 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. G.G.35 Proofs 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.

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will identify and justify geometric relationships, formally and informally. G.G.36 Proofs Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons. G.G.37 Proofs Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons. G.G.38 Proofs Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals. G.G.39 Proofs Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals. G.G.40 Proofs Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals. G.G.41 Proofs Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids. G.G.42 Proofs 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. G.G.43 Proofs Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1. G.G.44 Proofs Establish similarity of triangles, using the following theorems: AA, SAS, and SSS. G.G.45 Proofs Investigate, justify, and apply theorems about similar triangles. G.G.46 Proofs 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. G.G.47 Proofs Investigate, justify, and apply theorems about mean proportionality: 1) the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; and 2) the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg. G.G.48 Proofs Investigate, justify, and apply the Pythagorean theorem and its converse. G.G.49 Proofs Investigate, justify, and apply theorems regarding chords of a circle: 1) perpendicular bisectors of chords; and 2) the relative lengths of chords as compared to their distance from the center of the circle. G.G.50 Proofs Investigate, justify, and apply theorems about tangent lines to a circle: 1) a perpendicular to the tangent at the point of tangency; 2) two tangents to a circle from the same external point; and 3) common tangents of two non-intersecting or tangent circles. G.G.51 Proofs 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: 1) inside the circle (two chords); 2) on the circle (tangent and chord); and 3) outside the circle (two tangents, two secants, or tangent and secant). G.G.52 Proofs Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines. G.G.53 Proofs Investigate, justify, and apply theorems regarding segments intersected by a circle: 1) along two tangents from the same external point; 2) along two secants from the same external point; 3) along a tangent and a secant from the same external point; and 4) along two intersecting chords of a given circle.

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will apply transformations and symmetry to analyze problem solving situations 1.G.3 Transform Experiment with slides, flips, and turns of two-dimensional shapes. 1.G.4 Transform Identify symmetry in two-dimensional shapes. 2.G.5 Transform Explore and predict the outcome of slides, flips, and turns of two-dimensional shapes. 2.G.6 Transform Explore line symmetry. 3.G.5 Transform Identify and construct lines of symmetry. 5.G.11 Transform Identify and draw lines of symmetry of basic geometric shapes. 8.G.7 Transform Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations). 8.G.8 Transform Draw the image of a figure under rotations of 90 and 180 degrees. 8.G.9 Transform Draw the image of a figure under a reflection over a given line. 8.G.10 Transform Draw the image of a figure under a translation. 8.G.11 Transform Draw the image of a figure under a dilation. 8.G.12 Transform Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation. G.G.54 Transform Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation. G.G.55 Transform Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections. G.G.56 Transform Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism. G.G.57 Transform Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections). G.G.58 Transform Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries). G.G.59 Transform Investigate, justify, and apply the properties that remain invariant under similarities. G.G.60 Transform Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism. G.G.61 Transform Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines x = 0 , y = 0 , and y = x , and dilations centered at the origin.

Performance Indicators Organized by Grade Level and Band under Major Understandings

 Students will apply coordinate geometry to analyze problem solving situations. K.G.5 Coordinate Understand and use ideas such as over, under, above, below, on, beside, next to, and between. 1.G.5 Coordinate Recognize geometric shapes and structures in the environment. 5.G.12 Coordinate Identify and plot points in the first quadrant. 5.G.13 Coordinate Plot points to form basic geometric shapes (identify and classify). 5.G.14 Coordinate Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes. 6.G.10 Coordinate Identify and plot points in all four quadrants. 6.G.11 Coordinate Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths). 7.G.10 Coordinate Graph the solution set of an inequality ( positive coefficients only ) on a number line (See 7.A.5). 8.G.13 Coordinate Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change. 8.G.14 Coordinate Determine the y- intercept of a line from a graph and be able to explain the y-intercept. 8.G.15 Coordinate Graph a line using a table of values. 8.G.16 Coordinate Determine the equation of a line given the slope and the y-intercept. 8.G.17 Coordinate Graph a line from an equation in slope-intercept form ( y = mx + b ). 8.G.18 Coordinate Solve systems of equations graphically (only linear, integral solutions, y = mx + b format, no vertical/horizontal lines). 8.G.19 Coordinate Graph the solution set of an inequality on a number line. 8.G.20 Coordinate Distinguish between linear and nonlinear equations ax2 + bx + c; a=1 (only graphically). 8.G.21 Coordinate Recognize the characteristics of quadratics in tables, graphs, equations, and situations. A.G.3 Coordinate Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations. A.G.4 Coordinate Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions. A.G.5 Coordinate Investigate and generalize how changing the coefficients of a function affects its graph. A.G.6 Coordinate Graph linear inequalities. A.G.7 Coordinate Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10). A.G.8 Coordinate Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions. A.G.9 Coordinate Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
 A.G.10 Coordinate Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value. G.G.62 Coordinate Find the slope of a perpendicular line, given the equation of a line. G.G.63 Coordinate Determine whether two lines are parallel, perpendicular, or neither, given their equations. G.G.64 Coordinate Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line. G.G.65 Coordinate Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line. G.G.66 Coordinate Find the midpoint of a line segment, given its endpoints. G.G.67 Coordinate Find the length of a line segment, given its endpoints. G.G.68 Coordinate Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment. G.G.69 Coordinate Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas. G.G.70 Coordinate Solve systems of equations involving one linear equation and one quadratic equation graphically. G.G.71 Coordinate Write the equation of a circle, given its center and radius or given the endpoints of a diameter. G.G.72 Coordinate Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer. G.G.73 Coordinate Find the center and radius of a circle, given the equation of the circle in center-radius form. G.G.74 Coordinate Graph circles of the form(x − h)2 + ( y − k)2 = r2 .
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