Kansas Geometry Math Standards

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N Number and Quantity

• N.Q Quantities
  • Reason quantitatively and use units to solve problems.
    • N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
    • N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
    • N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A Algebra

• A.SSE Seeing Structure in Expressions
  • Interpret the structure of expressions.
    • A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
      • A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.
      • A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)ⁿ as the product of P and (1 + r)ⁿ.
    • A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
• A.CED Creating Equations
  • Create equations that describe numbers or relationships.
    • A.CED.1 Apply and extend previous understanding to create equations and inequalities in one variable and use them to solve problems.
    • A.CED.2 Apply and extend previous understanding to create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
    • A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
    • A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
• A.REI Reasoning with Equations and Inequalities
  • Understand solving equations as a process of reasoning and explain the reasoning.
    • A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • Solve equations and inequalities in one variable.
    • A.REI.2 Apply and extend previous understanding to solve equations, inequalities, and compound inequalities in one variable, including literal equations and inequalities.
  • Represent and solve equations and inequalities graphically.
    • A.REI.8 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

F Functions

• F.IF Interpreting Functions
  • Understand the concept of a function and use function notation.
    • F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
    • F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • Interpret functions that arise in applications in terms of the context.
    • F.IF.4 For a function that models a relationship between two quantities, interpret key features of expressions, graphs and tables in terms of the quantities, and sketch graphs showing key features given a description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
    • F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  • Analyze functions using different representations.
    • F.IF.9 Compare properties of two functions using a variety of representations (algebraically, graphically, numerically in tables, or by verbal descriptions).
• F.TF Trigonometric Functions
  • Extend the domain of trigonometric functions using the unit circle.
    • F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
    • F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
    • F.TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.
    • F.TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

G Geometry

• G.CO Congruence
  • Experiment with transformations in the plane.
    • G.CO.1 Verify experimentally (for example, using patty paper or geometry software) the properties of rotations, reflections, translations, and symmetry:
      • G.CO.1a Lines are taken to lines, and line segments to line segments of the same length.
      • G.CO.1b Angles are taken to angles of the same measure.
      • G.CO.1c Parallel lines are taken to parallel lines.
      • G.CO.1d Identify any line and/or rotational symmetry within a figure.
    • G.CO.2 Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of translations, rotations, and reflections on two-dimensional figures.
    • Checkpoint opportunity
  • Understand congruence in terms of rigid motions.
    • G.CO.3 Given two congruent figures, describe a sequence of rigid motions that exhibits the congruence (isometry) between them using coordinates and the non-coordinate plane.
    • G.CO.4 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
    • G.CO.5 Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
    • G.CO.6 Demonstrate triangle congruence using rigid motion (ASA, SAS, and SSS).
    • Checkpoint opportunity
  • Construct arguments about geometric theorems using rigid transformations and/or logic.
    • G.CO.7 Construct arguments about lines and angles using theorems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. (Building upon standard in 8th grade Geometry.)
    • G.CO.8 Construct arguments about the relationships within one triangle using theorems. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point; angle sum and exterior angle of triangles.
    • G.CO.9 Construct arguments about the relationships between two triangles using theorems. Theorems include: SSS, SAS, ASA, AAS, and HL.
    • G.CO.10 Construct arguments about parallelograms using theorems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (Building upon prior knowledge in elementary and middle school.)
    • Checkpoint opportunity
  • Make geometric constructions.
    • G.CO.11 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
    • G.CO.12 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
    • Checkpoint opportunity
• G.SRT Similarity, Right Triangles, and Trigonometry
  • Understand similarity in terms of similarity transformations.
    • G.SRT.1 Use geometric constructions to verify the properties of dilations given by a center and a scale factor:
      • G.SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
      • G.SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
    • G.SRT.2 Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of dilations on two-dimensional figures.
    • G.SRT.3 Given two similar figures, describe a sequence of transformations that exhibits the similarity between them using coordinates and the non-coordinate plane.
    • G.SRT.4 Understand the meaning of similarity for two-dimensional figures as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
    • Checkpoint opportunity
  • Construct arguments about theorems involving similarity.
    • G.SRT.5 Construct arguments about triangles using theorems. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity, and AA.
    • G.SRT.6 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
    • Checkpoint opportunity
  • Define trigonometric ratios and solve problems involving right triangles.
    • G.SRT.7 Show that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
    • G.SRT.8 Explain and use the relationship between the sine and cosine of complementary angles.
    • G.SRT.9 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
    • Checkpoint opportunity
  • Apply trigonometry to general triangles.
    • G.SRT.10 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
    • G.SRT.11 Prove the Laws of Sines and Cosines and use them to solve problems.
    • G.SRT.12 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
    • Checkpoint opportunity
• G.C Circles
  • Understand and apply theorems about circles.
    • G.C.1 Construct arguments that all circles are similar.
    • G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
    • G.C.3 Construct arguments using properties of polygons inscribed and circumscribed about circles.
    • G.C.4 Construct inscribed and circumscribed circles for triangles.
    • G.C.5 Construct inscribed and circumscribed circles for polygons and tangent lines from a point outside a given circle to the circle.
    • Checkpoint opportunity
  • Find arc lengths and areas of sectors of circles.
    • G.C.6 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
    • Checkpoint opportunity
• G.GPE Expressing Geometric Properties with Equations
  • Translate between the geometric description and the equation for a conic section.
    • G.GPE.1 Write the equation of a circle given the center and radius or a graph of the circle; use the center and radius to graph the circle in the coordinate plane.
    • Checkpoint opportunity
  • Use coordinates to prove simple geometric theorems algebraically.
    • G.GPE.6 Use coordinates to prove simple geometric theorems algebraically, including the use of slope, distance, and midpoint formulas.
    • G.GPE.7 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
    • G.GPE.8 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, including the use of the distance and midpoint formulas.
    • Checkpoint opportunity
• G.GMD Geometric Measurement and Dimension
  • Explain volume formulas and use them to solve problems.
    • G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments and informal limit arguments.
    • G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a solid figure.
    • Checkpoint opportunity
• G.MG Modeling with Geometry
  • Apply geometric concepts in modeling situations.
    • G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
    • G.MG.2 Apply concepts of density and displacement based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
    • G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
    • Checkpoint opportunity

S Statistics and Probability

• S.CP Conditional Probability and the Rules of Probability
  • Understand independent and conditional probability and use them to interpret data.
    • S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
    • S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
    • S.CP.3 Understand the conditional probability of A given B as P(AandB)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
    • S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
    • S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
  • Use the rules of probability to compute probabilities of compound events in a uniform probability model.
    • S.CP.6 Find the conditional probability of A given B as the fraction of B‘s outcomes that also belong to A, and interpret the answer in terms of the model.
    • S.CP.7 Apply the Addition Rule, P(AorB) = P(A) + P(B) – P(AandB), and interpret the answer in terms of the model.
    • S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P(AandB) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
    • S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.
    • Checkpoint opportunity
• S.MD Using Probability to Make Decisions
  • Use probability to evaluate outcomes of decisions.
    • S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
    • S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).