Mathematics Competency Assurance Documents

Geometry

Number/Computation Strand

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 1: Student will apply number sense and order relations in problem solving situations to perform estimations and/or calculations with equations, matrices, and sequences involving complex numbers (counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, etc.) with and without calculators and will communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Geometry: Number/Computation Standards

NC/G.1 Apply inductive reasoning to find patterns in geometric figures, polygons (growing shapes), and fractals to generalize first and second degree sequences by giving the rule for the nth term and defending the generalization.

NC/G.2 Apply appropriate strategies to solve equations and use formulas to find measures (missing lengths; formulas for circumference, area, volume, etc.) of two-dimensional and three-dimensional geometric figures or diagrams.

Skills, Concepts & Relationships

• Apply inductive reasoning to recognize, create, continue, and generalize patterns in arithmetic (linear) and geometric (quadratic) sequences by giving the rule for the nth term and defending the generalization (ch1)
• Introduce patterns in fractals and recursion in number patterns (ch1.2&1.3)
• Use geometric models to model physical situations and generalize patterns in problem solving (determine figurate numbers that correspond to geometric figures, determine the number connecting n random points, the number of intersections of n random lines, and the number of diagonals in an n-gon) (ch1.4-1.6)
• Introduce finding the coordinates of points of concurrency by estimating the coordinates from its graph and solving the equations of two appropriate lines simultaneously. (ch4.6)
• Review finding and approximating square roots using a calculator (recognize perfect squares to 169) (ch10.4)
• Review defining and using ratio, proportions and percents to solve problems (ch12.1)
• Review recognizing that measures of similar figures have equal ratios (ch12.1)

Geometry

Geometry/Measurement Strand

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 2: Student will apply properties of measurement (ratio measures including slope, rate, indirect measurement, similarity; surface area and volume of prisms, pyramids, cylinders, cones, and spheres, etc.) and will use geometric concepts, properties and relationships (prove, use and apply theorems/conjectures involving lines, angles, triangles, quadrilaterals, regular, and non-regular polygons, circles, and transformations, etc.) in problem solving situations and communicate the inductive and deductive reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.8 Procedures, 2.9 Space and Dimensionality, 2.10 Measurement, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Geometry: Geometry/Measurement Standards

GM/G.1 Classify, analyze, and draw visual representations of two-dimensional and three-dimensional figures with accurate standard measures (convert within a measurement system) using construction tools and instruments (compass, MIRA, patty paper, paper folding, protractor, angle ruler, isometric dot paper, Geometer’s Sketchpad, Peanut Geometry, etc.) to solve problems and support basic theorems/conjectures with inductive reasoning.

GM/G.2 Understand, prove, and apply theorems/conjectures involving space (points, lines, planes, betweeness, etc.), lines (slopes, parallel, perpendicular, transversal, equations of lines, etc.) & angles (interior, exterior, vertical, complementary, supplementary, etc.) to solve problems.

GM/G.3 Classify triangles (acute, right, obtuse, equilateral, scalene, isosceles), recognize and prove relationships in triangles: triangle sum theorem, triangle inequalities, triangles are congruent or similar, altitude, median, and use concept of corresponding parts of congruent triangles when appropriate.

GM/G.4 Classify quadrilaterals (square, rectangle, parallelogram, kite, special trapezoids) and polygons (regular and non-regular, convex and concave) based on number of angles, sides, and properties; determine and apply theorems/conjectures involving these properties and formulas.

GM/G.5 Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, concentric circles, and inscribed & circumscribed polygons of circles.

GM/G.6 Describe, draw and determine changes in properties of figures and their images involving symmetry and transformations (rotations, translations, reflections, and dilations) in tessellations and the coordinate plane.

GM/G.7 Determine and apply the formulas for volume and surface area of prisms, pyramids, cylinders, cones and spheres in real world problems.

GM/G.8 Determine and apply relationships of angles and sides in right triangles including: proving the Pythagorean Theorem (and its Converse) in more than one way, basic trigonometric ratios (sine, cosine, tangent), special right triangles (30-60-90, 45-45-90), proportional relationships to make scale drawings, angles of elevation & depression, and using similar triangle methods for finding indirect measurement (shadow method, mirror method, clinometer).

Skills, Concepts & Relationships

• Review defining and using basic geometric terms: point, line, plane, space, segment, ray, angle, collinear, and coplanar points (ch2.1)
• Review using a protractor or angle ruler to measure and draw angles (ch2.2)
• Use and express appropriate symbols for marking figures to show measurements and congruence relationships (ch2.2)
• Recognize and apply incoming and outgoing angles in practical situations (billiards, miniature golf, light striking and reflecting from a mirror, etc.) (ch2.2)
• Identify, write and interpret conditional statements including: converse, if-then, biconditional, and counterexample (ch2.3)
• Review recognizing and applying special line (parallel, skew, perpendicular, midpoint of a segment, etc.) and angle relationships (right, acute, obtuse, complementary, supplementary, vertical, linear pair, angle bisector, etc.) in problem solving situations (ch2.4)
• Review defining, classifying, and applying polygons (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, undecagon, dodecagon, n-gon) and related terms (angle, side, vertex, convex, concave, congruent, perimeter, diagonal, equilateral, equiangular, regular) in problem solving situations (ch2.5)
• Review defining, classifying, and applying triangles (acute, obtuse, scalene, isosceles, etc.) and their related parts (median, altitude, height) in problem solving situations (ch2.6)
• Review defining, classifying, and applying quadrilaterals (trapezoid, kite, parallelogram, rhombus, rectangle, square) in problem solving situations (ch2.7)
• Describe the classifications of triangles and quadrilaterals (ch2.7)
• Review visualizing, drawing and recognizing relationships in two and three dimensions (cross sections and solids: prism, pyramid, cylinder, cone, sphere, hemisphere) (ch2.8)
• Review translating descriptions and word problems into drawings and diagrams (ch2.9)
• Use construction tools and methods (straightedge & compass, patty papers, and computer software-Geometer’s Sketchpad, Peanut Geometry, etc.) to duplicate segments, angles and polygons; to construct perpendicular bisectors & midpoints, and make conjectures; to construct a perpendicular from a point not on a line, using the shortest path from a point to a line; to construct an angle bisector and determine that a point on the bisector of an angle is equally distant from the sides of the angles (ch3)
• Determine the measure of each angle of an equilateral triangle is 60 degrees (ch3.4)
• Use construction tools and methods to construct parallel lines (equidistant method, rhombus method) (ch3.5)
• Determine through construction whether on not a triangle can be determined given certain parts (ch3.6)
• Determine, identify and apply relationships between points of concurrency and: angle bisectors, perpendicular bisectors, altitudes of triangles, medians, and inscribed & circumscribed circles (ch3.4&3.8)
• Determine, recognize, and apply relationships between special pairs of angles (complementary, supplementary, linear pair, vertical) in problem solving situations (ch4.1)
• Determine, recognize and apply relationships of the angles of parallel lines cut by a transversal (corresponding, alternate interior, alternate exterior, consecutive interior, etc.) in problem solving situations (ch4.2)
• Review that the sum of the angles of a triangle is 180 degrees (ch5.1)
• Use geometric tools and inductive reasoning to recognize, determine, and apply relationships among the sides & angles of a triangle (ch5)
• Use geometric tools and problem solving to recognize, determine, and apply valid shortcut methods for deciding whether triangles are congruent (SSS, SAS, ASA, & SAA are good shortcuts; SSA & AAA not necessarily congruent) (ch5.4 & 5.5)
• Use the definition of congruent triangles to show that corresponding parts of congruent triangles are congruent (CPCTC) (ch5.6)
• Use logical and visual thinking skills to create flow chart proofs to prove geometric relationships (introduce deductive reasoning) (ch5.6)
• Review that the sum of the measures of the four angles of any quadrilateral is 360 degrees (ch6.1)
• Use geometric tools and inductive reasoning to recognize, determine, and apply the properties of polygons (ch6)
• Introduce using parallelograms in vector diagrams and to find resultant vector, or vector sums (ch6.5)
• Define, recognize, and apply properties of circles and their parts (radius, diameter, chord, secant, tangent, arc, minor arc, major arc, inscribed angle, central angle, etc.), and figures related to circles (congruent circles, concentric circles, etc.) (ch7)
• Review recognizing, determining, and applying the relationship between the circumference of a circle and the length of its diameter as p (C=pd because D=2r and C=2pr) (ch7.5&7.6)
• Use geometric tools and inductive reasoning to recognize, determine and apply a formula for finding the length of an arc of a circle (arc length=degree of the measure of the are divided by 360 degrees and multiplied by the circumference of the circle) (ch7.7)
• Describe, draw, and determine changes in properties of figures and their images involving transformations: translations (slide), rotations (turn), and reflections (flip) in the coordinate plane (ch8.1)
• Recognize, determine and apply concepts of reflectional, rotational, translational, and glide-reflectional symmetry to polygons, nature, etc. (ch8.3)
• Recognize, create, classify and apply monohedral, regular, semiregular, and demiregular tessellations (ch8.4)
• Recognize, create and apply Escher-type tessellations using translations, rotations, and glide reflections (ch8.6-8.8)
• Review determining and applying formulas for areas of rectangles, parallelograms, triangles, trapezoids, and circles (ch9.1-9.3&9.5&9.8)
• Determine and apply formulas for areas of kites and regular polygons (ch9.2&9.4)
• Determine and apply formulas and methods for calculating areas of annuli, sectors, and segments of circles (ch9.6)
• Determine and apply formulas for surface area of solids (prism, pyramid, cylinder, cone, etc.) (ch9.7-9.8)
• Determine and apply the Pythagorean Theorem and its converse (ch10)
• Determine, recognize and apply the relationship among the sides of special right triangles (45-45-90 & 30-60-90) (ch10.4)
• Determine, recognize and apply a relationship in the multiples of Pythagorean triples (multiply the lengths of all three sides of any right triangle by the same number and the resulting triangle will also be a right triangle) (ch10.5)
• Determine and apply the Pythagorean relationship to problems involving circles (ch10.8)
• Identify and use appropriate vocabulary for polyhedra (regular polyhedron, prism, right prism, oblique prism, pyramid, base, lateral faces, lateral edges, vertex, altitude, height, tetrahedron, etc.)
• Circular solids (sphere, radius, center, hemisphere, great circle, cylinder, right cylinder, oblique cylinder, cone, right cone, oblique cone, bases, axis) (11.1&11.2)
• Determine and apply formulas for finding the volumes of prisms, cylinders, pyramids and cones (V=BH where B is the area of the base and H is the height of the solid; V=(1/3)BH where B is the are of the base and H is the height of the solid) (ch11.3-11.5)
• Recognize the 5 Platonic Solids (ch11)
• Determine how changes in dimensions affect perimeter, area, and volume of common geometric figures and solids (maximum volume of a box -- graphing calculator, can problem, etc.) (ch11)
• Determine and apply the formulas for finding the volumes and surface areas of spheres and hemispheres (Volume of a sphere where V=(4/3)r³ and a hemisphere is half the volume of a sphere) (ch11.7-11.9)
• Determine, recognize and apply the concept of similar polygons and dilations (expand & contract) to solve problems (if one polygon is the image of another polygon under a dilation, then the polygons are similar) (ch12.2)
• Use geometric & measurement tools, proportions, and problem solving to recognize, determine, and apply valid shortcut methods for deciding whether triangles are similar (SSS, AA, & SAS are valid shortcuts) (ch12.3)
• Calculate indirect measurement (height of flagpole, mirror method, shadow knows method project, use the clinometer from project in ch4, etc.) with similar triangles (ch12.4)
• Determine, recognize, and apply the relationship between corresponding parts of similar triangles (ch12.5)
• Use geometric tools and problem solving to determine, recognize, and apply that angle bisector in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the two sides forming the angle (ch12.5)
• Derive and recognize the relationship between areas of similar figures (for similar polygons m and n the ratio of the areas is m²/n²), and volumes of similar solids (for similar solids m and n the ratio of the volumes is m³/n³) (ch12.6)
• Recognize and apply similarity conjectures to problems involving area and volume (ch12.6)
• Use similarity to develop, evaluate, and use the trigonometric ratios (sine, cosine, and tangent) (ch13.1)
• Apply right triangle trigonometry to pratical situations including finding indirect measurement (determine angles of evaluation & depression, use clinometer from project in ch4) (ch13.2)

Geometry

Probability/Statistics Strand

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 3: Student will use data collection & analysis, graphing of single-variable and two-variable data (line, bar & circle graphs, histogram, stem and leaf plots, box and whisker plots, scatterplot, linear regression & curve fitting), statistics (mean, median, mode, range, outliers, quartiles), and designing probability experiments & simulations to test theories about real world problems and communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.11 Change, 2.12 Structure, 2.13 Probability & Statistics; Goal 5 Think & Solve Problems; & Goal 6 Integrate Knowledge

Geometry: Probability/Statistics Standards

PS/G.1 Use geometric principles and properties to construct geometric models (linear, area, volume) to analyze theoretical probabilities.

Skills, Concepts & Relationships

• Draw, visualize, and use geometric models to solve probability problems (ch4.7)

Geometry

Algebraic Ideas Strand

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 4: Student will model, analyze, compare and apply linear & nonlinear algebraic functions (quadratic, polynomial, exponential, etc.) using tables, graphs in the coordinate plane, variables, expressions, equations, formulas and inequalities in practical situations and communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.9 Space and Dimensionality, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Geometry: Algebraic Ideas Standards

AI/G.1 Understand and prove how geometric concepts are related to algebraic procedures by comparing, contrasting and translating among synthetic, coordinate, and transformational geometry (prove the Pythagorean Theorem; in the coordinate plane determine: distance, slope, midpoint, transformations, etc.).

Skills, Concepts & Relationships

• Determine and recognize the graph of a sequence with a constant difference (first degree) is a set of points that lie on a straight line (linear) (ch1.4)
• Determine and recognize that a sequence with two linear factors (no constant difference at the first level) is a quadratic function (second degree) (ch1.5)
• Review determining, recognizing, and applying the slope of a line in the coordinate plane (ch4.3)
• Determine, recognize and apply a formula for finding the midpoint of a segment in the coordinate plane (ch4.3)
• Review relationships between the slopes of parallel lines (equal slopes) and between the slopes of perpendicular lines (negative reciprocal slopes) (ch4.4)
• Review recognizing, determining, and applying slope-intercept form (y=mx+b) of the equation of a line (find y-intercept) and applying linear relationships to solve real world problems (ch4.5)
• Use the slope of a line that best fits a set of points to predict values in linear relationships (ch4)
• Review solving a system of linear equations graphically by finding the intersections of lines to model practical situations (ch4.6)
• Introduce determining and applying the Pythagorean relationship on the coordinate plane (distance formula) and deriving the equation of a circle from the distance formula (ch10.7)