The Common Core in Maryland

1 Congruence, Proof, and Constructions

• Experiment with transformations in the plane.

  • 1.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

    • 1.G.CO.1.a Ability to fluently use mathematical vocabulary accurately

      • Angle vocabulary (G-C.1)

  • 1.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; 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 (e.g., translation versus horizontal stretch).

    • 1.G.CO.2.a Ability to make connections between function transformations (F.BF.3) and geometric transformations

      • Classify congruence transformations (G-H.1)
      • Translations: graph the image (G-H.2)
      • Translations: find the coordinates (G-H.3)
      • Translations: write the rule (G-H.4)
      • Reflections: graph the image (G-H.5)
      • Reflections: find the coordinates (G-H.6)
      • Rotate polygons about a point (G-H.7)
      • Rotations: graph the image (G-H.8)
      • Rotations: find the coordinates (G-H.9)
      • Sequences of congruence transformations: graph the image (G-H.12)
      • Congruence transformations: mixed review (G-H.16)
      • Dilations: graph the image (G-I.1)
      • Dilations: find the coordinates (G-I.2)
      • Dilations: find the scale factor (G-I.4)
      • Identify transformation matrices (A2-TT.15)
      • Transformation matrices: write the vertex matrix (A2-TT.16)
      • Transformation matrices: graph the image (A2-TT.17)

    • 1.G.CO.2.b Knowledge that rigid transformations preserve the shape of a figure

      • Translations: graph the image (G-H.2)
      • Translations: find the coordinates (G-H.3)
      • Translations: write the rule (G-H.4)
      • Reflections: graph the image (G-H.5)
      • Reflections: find the coordinates (G-H.6)
      • Rotate polygons about a point (G-H.7)
      • Rotations: graph the image (G-H.8)
      • Rotations: find the coordinates (G-H.9)
      • Sequences of congruence transformations: graph the image (G-H.12)
      • Congruence transformations: mixed review (G-H.16)

  • 1.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

    • 1.G.CO.3.a Ability to use appropriate vocabulary to describe rotations and reflections

      • Transformations that carry a polygon onto itself (G-H.15)

    • 1.G.CO.3.b Ability to use the characteristics of a figure to determine and then describe what happens to the figure as it is rotated (such as axis of symmetry, congruent angles or sides….)

  • 1.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

    • 1.G.CO.4.a Ability to construct a definition for each term based upon a synthesis of experiences

      • Rotate polygons about a point (G-H.7)

  • 1.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

    • 1.G.CO.5.a Ability to interpret and perform a given sequence of transformations and draw the result

      • Translations: graph the image (G-H.2)
      • Reflections: graph the image (G-H.5)
      • Rotations: graph the image (G-H.8)
      • Sequences of congruence transformations: graph the image (G-H.12)

    • 1.G.CO.5.b Ability to accurately use geometric vocabulary to describe the sequence of transformations that will carry a given figure onto another

      • Classify congruence transformations (G-H.1)

• Understand congruence in terms of rigid motions.

  • 1.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

    • 1.G.CO.6.a Ability to recognize the effects of rigid motion on orientation and location of a figure

      • Translations: find the coordinates (G-H.3)
      • Translations: write the rule (G-H.4)
      • Reflections: find the coordinates (G-H.6)
      • Rotations: find the coordinates (G-H.9)

    • 1.G.CO.6.b Ability to use rigid motions to map one figure onto another

      • Transformations that carry a polygon onto itself (G-H.15)

    • 1.G.CO.6.c Ability to use the definition of congruence as a test to see if two figures are congruent

  • 1.G.CO.7 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.

    • 1.G.CO.7.a Knowledge of vocabulary corresponding parts and the connection to the given triangles

    • 1.G.CO.7.b Ability to identify the corresponding parts of two triangles

  • 1.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

    • 1.G.CO.8.a Ability to recognize why particular combinations of corresponding parts establish congruence and why others do not

• Prove geometric theorems.

  • 1.G.CO.9 Prove theorems about lines and angles.

    • 1.G.CO.9.a Ability to construct a proof using one of a variety of methods

      • Proofs involving angles (G-C.9)
      • Proofs involving parallel lines I (G-D.7)
      • Proofs involving parallel lines II (G-D.8)

  • 1.G.CO.10 Prove theorems about triangles.

    • 1.G.CO.10.a Ability to construct a proof using one of a variety of methods

      • SSS Theorem in the coordinate plane (G-L.7)
      • Proofs involving isosceles triangles (G-L.11)
      • Proofs involving triangles I (G-N.9)
      • Proofs involving triangles II (G-N.10)

  • 1.G.CO.11 Prove theorems about parallelograms.

    • 1.G.CO.11.a Ability to construct a proof using one of a variety of methods

      • Proofs involving triangles and quadrilaterals (G-O.14)
      • Proofs involving quadrilaterals (G-O.15)

• Make geometric constructions.

  • 1.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

    • 1.G.CO.12.a Ability to use understanding of geometric concepts to establish a rationale for the steps/procedures used in completing a construction

      • Construct the midpoint or perpendicular bisector of a segment (G-B.15)
      • Construct an angle bisector (G-C.7)
      • Construct a congruent angle (G-C.8)
      • Construct a perpendicular line (G-D.2)
      • Construct parallel lines (G-D.6)
      • Construct an equilateral triangle or regular hexagon (G-G.5)
      • Construct a square (G-G.6)
      • Construct the circumcenter or incenter of a triangle (G-N.6)
      • Construct the centroid or orthocenter of a triangle (G-N.7)
      • Construct a tangent line to a circle (G-X.1)
      • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

  • 1.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    • 1.G.CO.13.a Ability to use understanding of geometric concepts to establish a rationale for the steps/procedures used in completing a construction

      • Construct an equilateral triangle inscribed in a circle (G-X.2)
      • Construct a square inscribed in a circle (G-X.3)
      • Construct a regular hexagon inscribed in a circle (G-X.4)

2 Similarity, Proof and Trigonometry

• Understand similarity in terms of similarity transformations.

  • 2.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

    • 2.G.SRT.1.a 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.

      • 2.G.SRT.1.a.1 Ability to connect experiences with dilations and orientation to experiences with lines

        • Dilations and parallel lines (G-I.6)

    • 2.G.SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

      • 2.G.SRT.1.b.1 Ability to develop a hypothesis based on observations

  • 2.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

    • 2.G.SRT.2.a Ability to make connections between the definition of similarity and the attributes of two given figures

      • Similar figures: side lengths and angle measures (A1)
      • Similarity statements (G-M.2)
      • Side lengths and angle measures in similar figures (G-M.4)
      • Similar triangles and similarity transformations (G-M.9)
      • Similarity of circles (G-M.11)

    • 2.G.SRT.2.b Ability to set up and use appropriate ratios and proportions

      • Similar figures: side lengths and angle measures (A1)
      • Ratios in similar figures (G-M.1)
      • Side lengths and angle measures in similar figures (G-M.4)

  • 2.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

    • 2.G.SRT.3.a Ability to recognize why particular combinations of corresponding parts establish similarity and why others do not

      • Similar triangles and indirect measurement (A1)

• Prove theorems involving similarity.

  • 2.G.SRT.4 Prove theorems about triangles.

    • 2.G.SRT.4.a Ability to construct a proof using one of a variety of methods

      • Proofs involving similarity in right triangles (G-M.17)
      • Proofs involving triangles II (G-N.10)
      • Prove the Pythagorean theorem (G-P.2)

  • 2.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

    • 2.G.SRT.5.a Ability to use information given in verbal or pictorial form about geometric figures to set up a proportion that accurately models the situation

      • Congruency in isosceles and equilateral triangles (G-L.10)
      • Hypotenuse-Leg Theorem (G-L.12)
      • Ratios in similar figures (G-M.1)
      • Identify similar figures (G-M.3)
      • Side lengths and angle measures in similar figures (G-M.4)
      • Similar triangles and indirect measurement (G-M.5)
      • Perimeters of similar figures (G-M.6)
      • Triangle Proportionality Theorem (G-M.12)
      • Areas of similar figures (G-M.14)

• Define trigonometric ratios and solve problems involving right triangles.

  • 2.G.SRT.6 Understand 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.

    • 2.G.SRT.6.a Ability to generalize that side ratios from similar triangles are equal and that these relationships lead to the definition of the six trigonometric ratios

      • Trigonometric ratios: sin, cos, and tan (G-Q.1)
      • Trigonometric ratios with radicals: sin, cos, and tan (G-Q.2)
      • Trigonometric ratios: csc, sec, and cot (G-Q.3)
      • Trigonometric ratios in similar right triangles (G-Q.4)
      • Find trigonometric functions using a calculator (G-Q.8)
      • Trigonometric ratios: sin, cos, and tan (A2-II.3)
      • Trigonometric ratios: csc, sec, and cot (A2-II.4)
      • Trigonometric ratios in similar right triangles (A2-II.5)
      • Find trigonometric functions using a calculator (A2-II.10)
      • Find trigonometric ratios using right triangles (PC-R.5)
      • Trigonometric ratios: sin, cos, and tan (A1)
      • Find trigonometric ratios using a calculator (A1)

  • 2.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

    • 2.G.SRT.7.a See the skills and knowledge that are stated in the Standard.

      • Sine and cosine of complementary angles (G-Q.9)
      • Complementary angle identities (A2-LL.1)
      • Complementary angle identities (PC-T.1)

  • 2.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

    • 2.G.SRT.8.a See the skills and knowledge that are stated in the Standard.

      • Pythagorean theorem (G-P.1)
      • Converse of the Pythagorean theorem (G-P.3)
      • Pythagorean Inequality Theorems (G-P.4)
      • Trigonometric ratios: find a side length (G-Q.11)
      • Trigonometric ratios: find an angle measure (G-Q.12)
      • Solve a right triangle (G-Q.13)
      • Pythagorean Theorem and its converse (A2-II.1)
      • Trigonometric ratios: find a side length (A2-II.16)
      • Trigonometric ratios: find an angle measure (A2-II.17)
      • Solve a right triangle (A2-II.18)
      • Trigonometric ratios: find a side length (PC-R.12)
      • Trigonometric ratios: find an angle measure (PC-R.13)
      • Solve a right triangle (PC-R.14)
      • Pythagorean theorem: word problems (A1)
      • Trigonometric ratios: find a side length (A1)
      • Trigonometric ratios: find an angle measure (A1)
      • Solve a right triangle (A1)

• Apply geometric concepts in modeling situations.

  • 2.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).

    • 2.G.MG.1.a See the skills and knowledge that are stated in the Standard.

      • Area and perimeter: word problems (A1-D.1)

  • 2.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

    • 2.G.MG.2.a See the skills and knowledge that are stated in the Standard.

  • 2.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).

    • 2.G.MG.3.a See the skills and knowledge that are stated in the Standard.

• Apply trigonometry to general triangles.

  • 2.G.SRT.9 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.

    • 2.G.SRT.9.a Ability to make connections between the formula A = 1/2 (base)(height) and right triangle trigonometry

      • Area of a triangle: sine formula (A2-JJ.4)
      • Area of a triangle: Law of Sines (A2-JJ.5)
      • Area of a triangle: sine formula (PC-R.18)

  • 2.G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.

    • 2.G.SRT.10.a Ability to recognize when it is appropriate to use the Law of Sines and the Law of Cosines

      • Law of Sines (G-R.1)
      • Law of Cosines (G-R.2)
      • Law of Sines (A2-JJ.1)
      • Law of Cosines (A2-JJ.2)
      • Area of a triangle: Law of Sines (A2-JJ.5)
      • Law of Sines (PC-R.15)
      • Law of Cosines (PC-R.16)

  • 2.G.SRT.11 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).

    • 2.G.SRT.11.a See the skills and knowledge that are stated in the Standard.

      • Solve a triangle (G-R.3)
      • Solve a triangle (A2-JJ.3)
      • Solve a triangle (PC-R.17)

3 Extending to Three Dimensions

• Explain volume formulas and use them to solve problems.

  • 3.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.

    • 3.G.GMD.1.a See the skills and knowledge that are stated in the Standard.

  • 3.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

    • 3.G.GMD.3.a See the skills and knowledge that are stated in the Standard.

      • Volume (A1-D.2)
      • Volume of prisms and cylinders (G-U.7)
      • Volume of pyramids and cones (G-U.8)
      • Volume of spheres (G-U.9)

• Visualize relationships between two-dimensional and three-dimensional objects.

  • 3.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

    • 3.G.GMD.4.a Ability to make connections between two-dimensional figures such as rectangles, squares, circles, and triangles and three?dimensional figures such as cylinders, spheres, pyramids and cones

      • Cross sections of three-dimensional figures (G-T.4)
      • Solids of revolution (G-T.5)

• Apply geometric concepts in modeling situations.

  • 3.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).

    • 3.G.MG.1.a Ability to connect experiences with this standard as it related to the two? dimensional shapes studied in Unit 2 to three?dimensional shapes

      • Area and perimeter: word problems (A1-D.1)

4 Connecting Algebra and Geometry Through Coordinates

• Use coordinates to prove simple geometric theorems algebraically.

  • 4.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.

    • 4.G.GPE.4.a Ability to use distance, slope and midpoint formulas

      • Midpoint formula: find the midpoint (A1-I.2)
      • Distance between two points (A1-I.4)
      • Find the slope of a graph (A1-K.1)
      • Find the slope from two points (A1-K.2)
      • Slopes of parallel and perpendicular lines (A1-L.18)
      • Midpoints (G-B.6)
      • Midpoint formula: find the midpoint (G-B.10)
      • Distance formula (G-B.13)
      • Slopes of lines (G-E.1)
      • Slopes of parallel and perpendicular lines (G-E.4)

  • 4.G.GPE.5 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).

    • 4.G.GPE.5.a See the skills and knowledge that are stated in the Standard.

      • Slopes of parallel and perpendicular lines (A1-L.18)
      • Write an equation for a parallel or perpendicular line (A1-L.19)
      • Slopes of parallel and perpendicular lines (G-E.4)
      • Equations of parallel and perpendicular lines (G-E.5)
      • Find the distance between a point and a line (G-E.6)
      • Find the distance between two parallel lines (G-E.7)

  • 4.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

    • 4.G.GPE.6.a Ability to use the slope formula

      • Find the slope of a graph (A1-K.1)
      • Find the slope from two points (A1-K.2)
      • Slopes of parallel and perpendicular lines (A1-L.18)
      • Slopes of lines (G-E.1)
      • Slopes of parallel and perpendicular lines (G-E.4)

  • 4.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

    • 4.G.GPE.7.a See the skills and knowledge that are stated in the Standard.

      • Distance between two points (A1-I.4)
      • Distance formula (G-B.13)
      • Area and perimeter in the coordinate plane I (G-S.7)
      • Area and perimeter in the coordinate plane II (G-S.8)

5 Circles With and Without Coordinates

• Understand and apply theorems about circles.

  • 5.G.C.1 Prove that all circles are similar.

    • 5.G.C.1.a See the skills and knowledge that are stated in the Standard.

      • Similarity of circles (G-M.11)

  • 5.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords.

    • 5.G.C.2.a See the skills and knowledge that are stated in the Standard.

      • Arcs and chords (G-W.9)
      • Inscribed angles (G-W.12)

  • 5.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

    • 5.G.C.3.a Ability to use concurrence of perpendicular bisectors and angle bisectors for the basis of the construction

      • Construct the circumcenter or incenter of a triangle (G-N.6)
      • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

  • 5.G.C.4 Construct a tangent line from a point outside a given circle to the circle.

    • 5.G.C.4.a See the skills and knowledge that are stated in the Standard.

      • Tangent lines (G-W.10)
      • Construct a tangent line to a circle (G-X.1)

• Find arc lengths and areas of sectors of circles.

  • 5.G.C.5 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.

    • 5.G.C.5.a See the skills and knowledge that are stated in the Standard.

      • Arc length (G-W.4)
      • Area of sectors (G-W.7)
      • Convert between radians and degrees (A2-HH.1)
      • Radians and arc length (A2-HH.2)
      • Convert between radians and degrees (PC-R.1)
      • Radians and arc length (PC-R.2)

• Translate between the geometric description and the equation for a conic section.

  • 5.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

    • 5.G.GPE.1.a See the skills and knowledge that are stated in the Standard.

      • Write equations of circles in standard form using properties (G-Y.6)
      • Graph circles from equations in standard form (G-Y.9)
      • Write equations of circles in standard form using properties (A2-EE.4)
      • Convert equations of circles from general to standard form (A2-EE.5)
      • Find properties of circles (PC-U.4)
      • Write equations of circles in standard form (PC-U.5)
      • Graph circles (PC-U.6)

• Use coordinates to prove simple geometric theorems algebraically.

  • 5.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.

    • 5.G.GPE.4.a Ability to connect experience with coordinate proofs from Unit 4 to circles

• Apply geometric concepts in modeling situations.

  • 5.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).

    • 5.G.MG.1.a Ability to connect experiences from Unit 2 and Unit 3 with two-dimensional and three-dimensional shapes to circles