MAT10.GM.26

10^{th} Grade (MAT) Targeted Standard
(GM) Geometry and Measurement
Learners will use visualization, spatial reasoning, geometric modeling, and measurement to investigate the characteristics of figures, perform transformations, and construct logical arguments.

MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
Proficiency Scale
Progressions
Unit Size and Scale
 MAT06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.
 MAT09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
 MAT09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
 MAT09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
 MAT09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
 MAT10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
 MAT12.NO.04 Use units to understand problems and to guide the solution of multistep problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
 MAT12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
 MAT12.AR.08 Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
Ratio and Proportional Relationships
 MAT06.AR.RP.01 Describe the concept of a ratio relationship between two quantities using ratio language and visual models.
 MAT06.AR.RP.03 Make and use tables of equivalent ratios, tape diagrams, double number line diagrams, and equations to reason about ratios, rates, and unit rates.
 MAT07.AR.RP.02 Analyze the relationships between the dependent and independent variables of a proportional relationship using graphs and tables. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, k) where k is the unit rate.
 MAT07.AR.RP.03 Identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by an equation of the form y = kx, where k is the constant of proportionality, and describe the meaning of each variable (y, k, x) in the context of the situation.
 MAT07.AR.RP.04 Use proportional relationships to solve multistep problems involving ratios, percents, and scale drawings of geometric figures, including authentic problems.
 MAT08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
 MAT08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
 MAT10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
 MAT10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
 MAT10.GM.16 Prove similarity theorems about triangles.
 MAT10.GM.18 Recognize how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.
 MAT10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°60°90° and 45°45°90°).
 MAT10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
 MAT10.GM.29 Determine the midpoint or endpoint of a line segment using coordinates. (+) Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
 MAT10.GM.36 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).
 MAT12.GM.03 Determine and apply appropriate formulas to solve right and nonright triangle problems in context.
Functional Relationships
 MAT08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
 MAT08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
 MAT08.AR.F.03 Compare and contrast linear and nonlinear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
 MAT08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
 MAT08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
 MAT09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
 MAT09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
 MAT09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
 MAT09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
 MAT09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
 MAT09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
 MAT09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
 MAT09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
 MAT09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
 MAT09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
 MAT10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus nonrigid motion).
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
 MAT12.AR.F.01 Write a function that describes a relationship between two quantities.
 MAT12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
 MAT12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
 MAT12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x  h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
 MAT12.AR.F.05 Find inverse functions.
 MAT12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
 MAT12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
 MAT12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
 MAT12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
 MAT12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
 MAT12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
 MAT12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
 MAT12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
 MAT12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
 MAT12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
 MAT12.AR.F.18 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.
 MAT12.AR.F.19 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.
 MAT12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
 MAT12.AR.F.21 Create a trigonometric function to model periodic phenomena.
 MAT12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
 MAT12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
 MAT12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.
Measure Length
 MAT01.GM.M.01 Measure the length of an object as a whole number of samesize, nonstandard units from end to end.
 MAT02.GM.M.01 Measure the length of an object using two different standard units of measurement. Describe how the two measurements relate to the size of the units chosen.
 MAT03.GM.M.01 Measure lengths using rulers marked with halves and fourths of an inch.
 MAT10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
Angles/Triangles
 MAT03.GM.G.01 In twodimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.
 MAT04.GM.G.01 Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).
 MAT04.GM.M.07 Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.
 MAT07.GM.GF.01 Draw triangles from given conditions using appropriate tools. Defend whether a unique triangle, multiple triangles, or no triangle can be constructed when given three measures of angles or sides.
 MAT07.GM.GF.02 Describe the anglepair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.
 MAT08.GM.GF.04 Describe the following anglepair relationships: interior and exterior angles of triangles and angles formed when a transversal cuts parallel lines or intersecting lines. Solve for an unknown angle in a figure by applying
facts about these angles.
 MAT08.GM.GF.05 Describe the relationship between the leg length and the hypotenuse length of a right triangle. Determine whether a triangle is a right triangle using this relationship.
 MAT08.GM.GF.06 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in two and three dimensions on and off a coordinate plane, including authentic problems.
 MAT10.GM.01 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and plane.
 MAT10.GM.09 Prove and apply theorems about lines and angles.
 MAT10.GM.10 Prove and apply theorems about triangles.
 MAT10.GM.18 Recognize how the properties of similar right triangles allow for trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.
 MAT10.GM.19 Explain and use the relationship between the sine and cosine of complementary angles.
 MAT10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, 60º, 90º, and 45º45º90º).
 MAT10.GM.21 Solve unknown sides and angles of nonright triangles using the Laws of Sines and Cosines.
 MAT10.GM.23 Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.
 MAT10.GM.24 Construct a tangent line from a point outside a given circle to the circle.
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
 MAT12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π/3, π/4, and π/6.
 MAT12.AR.F.17 Use the Pythagorean Identity sin² (θ) + cos² (θ) = 1 to find sin (θ), cos (θ), or tan (θ) given sin (θ), cos (θ), or tan (θ) and the quadrant of the angle.
 MAT12.GM.03 Determine and apply appropriate formulas to solve right and nonright triangle problems in context
Circle Measurements
 MAT07.GM.AV.01 Describe the relationship between the circumference and diameter of a circle (pi). Apply the given formula to calculate the area and circumference of a circle, including in authentic problems.
 MAT10.GM.22 Apply theorems about relationships between line segments and circles or angles and circles formed by radii, diameter, secants, tangents, and chords to find unknown lengths or angles.
 MAT10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.
 MAT10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
 MAT10.GM.31 Explain derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone.
 MAT12.GM.01 Write the equation of a conic section given its special features. Convert between the standard form and general form equations of conic sections.
 MAT12.GM.02 Identify key features of a conic section given its equation. Apply properties of conic sections in context.
 MAT12.AR.F.18 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.
