Try our Free Online Math Solver!

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 threedimensional shapes. 
2.G.1 Shapes  Experiment with slides, flips, and turns to compare twodimensional shapes. 
2.G.2 Shapes  Identify and appropriately name twodimensional shapes: circle, square, rectangle, and triangle (both regular and irregular). 
2.G.3 Shapes  Compose (put together) and decompose (break apart) twodimensional 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 threedimensional shapes: cube, cylinder, sphere, prism, and cone. 
3.G.4 Shapes  Identify the faces on a threedimensional shape as twodimensional 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 threedimensional 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 twodimensional shapes that make up the faces and bases
of threedimensional 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, semicircles, quartercircles, 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 onethird 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 onehalf the product of the slant height and the circumference of its base; and 2) volume is onethird 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 nonintersecting 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 twodimensional shapes. 
1.G.4 Transform  Identify symmetry in twodimensional shapes. 
2.G.5 Transform  Explore and predict the outcome of slides, flips, and turns of twodimensional 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 yintercept. 
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 yintercept. 
8.G.17 Coordinate  Graph a line from an equation in slopeintercept 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 centerradius form. 
G.G.74 Coordinate  Graph circles of the form(x − h)^{2} + ( y − k)^{2} = r^{2} . 
Prev  Next 