(GM) Geometry and Measurement Categories All categories Not categorized Prioritized Standards (AR) Algebraic Reasoning (DPS) Data, Probability, and Statistics (GM) Geometry and Measurement (NO) Number and Operations

#### MAT-08.GM

MAT-08.GM Domain

# (GM) Geometry and Measurement

## Sub-Categories

• (G) Geometry
Learners will compose and classify figures and shapes based on attributes and properties; represent and solve problems using a coordinate plane.
• (M) Measurement
Learners will represent and calculate measurement data, including time, money, and geometric measurement, and convert like measurement units within a given system.
• (AV) Area and Volume
Learners will use visualization and spatial reasoning to solve problems involving the area, surface area, and volume of geometric figures.
• (GF) Geometric Figures
Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-08.GM.AV.01

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (AV) Area and Volume Learners will use visualization and spatial reasoning to solve problems involving the area, surface area, and volume of geometric figures.

## Progressions

Volume

• MAT-05.GM.M.03 Recognize volume as an attribute of rectangular prisms and measure volume by counting unit cubes.
• MAT-06.GM.AV.02 Describe the concept of volume of a right rectangular prism. Apply given formulas to calculate the volume of right rectangular prisms, including fractional edge lengths, including authentic problems.
• MAT-07.GM.AV.03 Solve problems involving the volume of prisms and composite solids, including authentic problems.
• MAT-08.GM.AV.01 Apply given formulas to solve problems involving the volume of cones, cylinders, and spheres, including authentic problems.
• MAT-10.GM.32 Calculate the surface area for prisms, cylinders, pyramids, cones, and spheres to solve problems.
• MAT-10.GM.33 Know and apply volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.
• MAT-10.GM.35 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

#### MAT-08.GM.GF.01

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Congruence and Similarity

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-10.GM.07 Use the definition of congruence, based on rigid motions, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.
• MAT-10.GM.08 Prove two triangles are congruent using the congruence theorems.
• MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.

Transformations

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-08.GM.GF.03 Name the type of transformation(s) needed to map a pre-image to its image.
• MAT-09.AR.F.09 Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative). Find the values of a, h, and k given the graph of the function.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.03 Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.
• MAT-10.GM.04 Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.
• MAT-10.GM.05 Draw the image of a figure that has undergone a series of transformations [rotation(s),
• reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).
• MAT-10.GM.06 Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.
• MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• MAT-10.GM.16 Prove similarity theorems about triangles.
• MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.
• MAT-12.AR.F.4 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 values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.NO.18 Multiply a vector by a scalar.

#### MAT-08.GM.GF.02

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Congruence and Similarity

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-10.GM.07 Use the definition of congruence, based on rigid motions, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.
• MAT-10.GM.08 Prove two triangles are congruent using the congruence theorems.
• MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.

Transformations

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-08.GM.GF.03 Name the type of transformation(s) needed to map a pre-image to its image.
• MAT-09.AR.F.09 Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative). Find the values of a, h, and k given the graph of the function.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.03 Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.
• MAT-10.GM.04 Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.
• MAT-10.GM.05 Draw the image of a figure that has undergone a series of transformations [rotation(s),
• reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).
• MAT-10.GM.06 Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.
• MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• MAT-10.GM.16 Prove similarity theorems about triangles.
• MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.
• MAT-12.AR.F.4 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 values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.NO.18 Multiply a vector by a scalar.

#### MAT-08.GM.GF.03

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Transformations

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-08.GM.GF.03 Name the type of transformation(s) needed to map a pre-image to its image.
• MAT-09.AR.F.09 Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative). Find the values of a, h, and k given the graph of the function.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.03 Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.
• MAT-10.GM.04 Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.
• MAT-10.GM.05 Draw the image of a figure that has undergone a series of transformations [rotation(s),
• reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).
• MAT-10.GM.06 Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.
• MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• MAT-10.GM.16 Prove similarity theorems about triangles.
• MAT-10.GM.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.
• MAT-12.AR.F.4 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 values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.NO.18 Multiply a vector by a scalar.

#### MAT-08.GM.GF.04

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Angles/Triangles

• MAT-03.GM.G.01 In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.
• MAT-04.GM.G.01 Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).
• MAT-04.GM.M.07 Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.
• MAT-07.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.
• MAT-07.GM.GF.02 Describe the angle-pair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.
• MAT-08.GM.GF.04 Describe the following angle-pair 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.
• MAT-08.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.
• MAT-08.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.
• MAT-10.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.
• MAT-10.GM.09 Prove and apply theorems about lines and angles.
• MAT-10.GM.10 Prove and apply theorems about triangles.
• MAT-10.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.
• MAT-10.GM.19 Explain and use the relationship between the sine and cosine of complementary angles.
• MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, -60º, -90º, and 45º-45º-90º).
• MAT-10.GM.21 Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.
• MAT-10.GM.23 Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.
• MAT-10.GM.24 Construct a tangent line from a point outside a given circle to the circle.
• MAT-10.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.
• MAT-12.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.
• MAT-12.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.
• MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context

#### MAT-08.GM.GF.05

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Angles/Triangles

• MAT-03.GM.G.01 In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.
• MAT-04.GM.G.01 Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).
• MAT-04.GM.M.07 Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.
• MAT-07.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.
• MAT-07.GM.GF.02 Describe the angle-pair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.
• MAT-08.GM.GF.04 Describe the following angle-pair 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.
• MAT-08.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.
• MAT-08.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.
• MAT-10.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.
• MAT-10.GM.09 Prove and apply theorems about lines and angles.
• MAT-10.GM.10 Prove and apply theorems about triangles.
• MAT-10.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.
• MAT-10.GM.19 Explain and use the relationship between the sine and cosine of complementary angles.
• MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, -60º, -90º, and 45º-45º-90º).
• MAT-10.GM.21 Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.
• MAT-10.GM.23 Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.
• MAT-10.GM.24 Construct a tangent line from a point outside a given circle to the circle.
• MAT-10.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.
• MAT-12.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.
• MAT-12.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.
• MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context

#### MAT-08.GM.GF.06

 8th Grade (MAT) Targeted Standard     (GM) Geometry and Measurement (GF) Geometric Figures Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.

## Progressions

Angles/Triangles

• MAT-03.GM.G.01 In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.
• MAT-04.GM.G.01 Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).
• MAT-04.GM.M.07 Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.
• MAT-07.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.
• MAT-07.GM.GF.02 Describe the angle-pair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.
• MAT-08.GM.GF.04 Describe the following angle-pair 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.
• MAT-08.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.
• MAT-08.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.
• MAT-10.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.
• MAT-10.GM.09 Prove and apply theorems about lines and angles.
• MAT-10.GM.10 Prove and apply theorems about triangles.
• MAT-10.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.
• MAT-10.GM.19 Explain and use the relationship between the sine and cosine of complementary angles.
• MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, -60º, -90º, and 45º-45º-90º).
• MAT-10.GM.21 Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.
• MAT-10.GM.23 Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.
• MAT-10.GM.24 Construct a tangent line from a point outside a given circle to the circle.
• MAT-10.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.
• MAT-12.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.
• MAT-12.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.
• MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context