Arkansas

RT Right Triangles

• Special Right Triangles & Pythagorean Theorem: Students explore right triangles and apply the Pythagorean Theorem.

  • G.RT.1 Apply the properties of special right triangles (30° -60° -90° and 45° -45° -90°) to solve real-world and mathematical problems.

    • Special right triangles (G-Q.5)
    • Solve a special right triangle (G-Q.7)

  • G.RT.2 Prove and apply the Pythagorean Theorem and its converse.

    • Pythagorean theorem (G-P.1)
    • Prove the Pythagorean theorem (G-P.2)
    • Converse of the Pythagorean theorem (G-P.3)
    • Pythagorean Inequality Theorems (G-P.4)

  • Checkpoint opportunity

    • Checkpoint: The Pythagorean theorem (G)

• Trigonometry Ratios: Students apply trigonometric ratios to solve problems.

  • G.RT.3 Explain how the definitions for trigonometric ratios are developed by similarity and how the side ratios in right triangles are properties of the angles in the triangle.

    • 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)

  • G.RT.4 Explain the relationship between the sine and cosine of complementary angles and use them to solve problems.

    • Sine and cosine of complementary angles (G-Q.9)

  • G.RT.5 Determine the sine, cosine, and tangent ratios of acute angles given the side lengths of right triangles.

    • Trigonometric ratios: sin, cos, and tan (G-Q.1)
    • Trigonometric ratios with radicals: sin, cos, and tan (G-Q.2)
    • Trigonometric ratios in similar right triangles (G-Q.4)

  • G.RT.6 Use trigonometric ratios (sine, cosine, and tangent) to calculate missing side lengths and angle measures in a right triangle, including applications of angles of elevation and depression; include real-world and mathematical problems.

    • 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)

  • Checkpoint opportunity

    • Checkpoint: Right triangle trigonometry (G-Q.14)

CIR Circles

• Circle Relationships: Students explore and use circle relationships to solve problems.

  • G.CIR.1 Apply the precise definition and standard geometric notation for a circle to understand geometric relationships.

    • Parts of a circle (G-W.1)

  • G.CIR.2 Recognize and apply relationships between angles, radii, and chords, tangents, and secants including: 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; and the relationship of angles and segments formed by chords, secants and/or tangents to a circle.

    • Central angles and arc measures (G-W.2)
    • Arcs and chords (G-W.9)
    • Tangent lines (G-W.10)
    • Inscribed angles (G-W.12)
    • Angles in inscribed right triangles (G-W.13)
    • Angles in inscribed quadrilaterals I (G-W.14)
    • Angles in inscribed quadrilaterals II (G-W.15)
    • Angles formed by chords, secants, and tangents (G-W.16)
    • Segments formed by chords, secants, and tangents (G-W.17)

  • G.CIR.3 Use the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.

    • Understand arc length and sector area of a circle (G-W.3)
    • Arc length (G-W.4)
    • Convert between radians and degrees (G-W.5)
    • Radians and arc length (G-W.6)

  • G.CIR.4 Use the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

    • Understand arc length and sector area of a circle (G-W.3)
    • Area of sectors (G-W.7)

  • G.CIR.5 Explain why the formulas for the area and circumference of a circle work using dissection and informal limit arguments.

  • Checkpoint opportunity

    • Checkpoint: Angles and lines in circles (G-W.18)
    • Checkpoint: Arc length and area of sectors (G-W.20)

• Equation of a Circle: Students solve problems involving the equation of a circle.

  • G.CIR.6 Write the equation of a circle, given the radius and center, where the center is at the origin or another point.

    • Write equations of circles in standard form from graphs (G-Y.5)
    • Write equations of circles in standard form using properties (G-Y.6)
    • Graph circles from equations in standard form (G-Y.9)

  • G.CIR.7 Identify the center and radius of a circle, given the equation of a circle, where the center is at the origin or another point.

    • Find the center of a circle (G-Y.1)
    • Find the radius or diameter of a circle (G-Y.2)
    • Convert equations of circles from general to standard form (G-Y.7)
    • Find properties of circles from equations in general form (G-Y.8)

  • G.CIR.8 Apply the equation of a circle to solve real-world problems.

GF Geometric Figures

• Three-Dimensional: Students explore and solve problems involving three-dimensional figures.

  • G.GF.1 Find the volume and surface area of complex three-dimensional figures composed of prisms, pyramids, cones, cylinders, and spheres.

    • Surface area and volume of rectangular prisms (G-U.3)
    • Surface area of prisms and cylinders (G-U.4)
    • Surface area of pyramids and cones (G-U.5)
    • Surface area of spheres (G-U.6)
    • Volume of prisms and cylinders (G-U.7)
    • Volume of pyramids and cones (G-U.8)
    • Volume of spheres (G-U.9)
    • Volume of compound figures (G-U.10)
    • Surface area and volume review (G-U.15)

  • G.GF.2 Use three-dimensional geometric figures and their measures to model real-world objects and solve problems.

    • Nets and drawings of three-dimensional figures (G-T.3)
    • Cross sections of three-dimensional figures (G-T.4)

  • G.GF.3 Explain why the formulas for the volume and surface area of a cylinder, pyramid, and cone work.

  • G.GF.4 Apply the Pythagorean Theorem to determine missing measurements in a three-dimensional figure.

  • G.GF.5 Identify the three-dimensional figure generated by rotating a two-dimensional figure.

    • Solids of revolution (G-T.5)

  • Checkpoint opportunity

    • Checkpoint: Cross sections and solids of revolution (G-T.6)
    • Checkpoint: Volume (G-U.17)
    • Checkpoint: Geometric modeling and design (G)

• Two-Dimensional: Students explore and solve problems involving two-dimensional figures.

  • G.GF.6 Apply theorems about quadrilaterals, including those involving angles, diagonals, and sides to solve problems.

    • Properties of parallelograms (G-O.6)
    • Properties of rhombuses (G-O.8)
    • Properties of squares and rectangles (G-O.9)
    • Properties of kites (G-O.11)
    • Classify quadrilaterals I (G)
    • Classify quadrilaterals II (G)
    • Properties of trapezoids (G)
    • Review: properties of quadrilaterals (G)

  • G.GF.7 Prove that a given quadrilateral is a parallelogram, rhombus, rectangle, square, kite, or trapezoid, and apply these relationships to solve problems.

    • Proving a quadrilateral is a parallelogram (G-O.7)
    • Proofs involving quadrilaterals (G-O.15)

  • G.GF.8 Prove and apply theorems about triangles including: Angle-Sum Theorem, Exterior Angle Theorem, Isosceles Triangle Theorem and its converse, Midsegment Theorem, Proportionality Theorem, Inequality Theorem and its converse, and Geometric Mean Theorem.

    • Triangle Angle-Sum Theorem (G-F.2)
    • Exterior Angle Theorem (G-F.3)
    • Proofs involving isosceles triangles (G-L.11)
    • Triangle Proportionality Theorem (G-M.12)
    • Midsegments of triangles (G-N.1)
    • Triangle Inequality Theorem (G-N.5)
    • Proofs involving triangles I (G-N.9)
    • Proofs involving triangles II (G-N.10)

  • G.GF.9 Calculate the perimeter of polygons when given the vertices, including using the distance formula.

    • 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)

  • G.GF.10 Calculate the area of triangles and rectangles when given the vertices, including using the distance formula and decomposing figures.

    • 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)

  • G.GF.11 Describe reflectional and rotational symmetry as they apply to a rectangle, parallelogram, trapezoid, or regular polygon.

    • Line symmetry (G-J.1)
    • Rotational symmetry (G-J.2)

  • Checkpoint opportunity

    • Checkpoint: Parallelogram theorems (G-O.16)
    • Checkpoint: Area and perimeter in the coordinate plane (G-S.16)
    • Checkpoint: Triangle theorems (G)

• Geometric Probability: Students determine probability in geometric contexts.

  • G.GF.12 Calculate probabilities as a proportion of area in a geometric context.

    • Geometric probability (G-AA.14)

LA Lines & Angles

• Define & Construct: Students use precise definitions and various construction tools to create geometric figures.

  • G.LA.1 Use precise definitions and standard geometric notation for angles, perpendicular lines, parallel lines, and line segments based on the undefined notions of point, line, and distance along a line.

    • Angle vocabulary (G-C.1)

  • G.LA.2 Make formal geometric constructions with a variety of tools and methods including: congruent segments and angles; segment and angle bisectors; perpendicular lines and perpendicular bisectors of a line segment; parallel lines; and an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    • Construct a congruent segment (G-B.8)
    • 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 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)
    • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

  • Checkpoint opportunity

    • Checkpoint: Definitions of geometric objects (G-D.9)
    • Checkpoint: Inscribed and circumscribed circles (G-W.19)
    • Checkpoint: Geometric constructions (G-X.6)

• Coordinate Geometry: Students reason about geometric figures using the coordinate plane.

  • G.LA.3 Determine the point that cuts a line segment into a specified ratio on a number line and a coordinate plane, including finding the midpoint.

    • Midpoints (G-B.6)
    • Midpoint formula: find the midpoint (G-B.10)
    • Partition a line segment in a given ratio (G-B.11)
    • Midpoint formula: find the endpoint (G-B.12)

  • G.LA.4 Derive the distance and midpoint formulas and use the formulas, including the slope formula, to verify geometric relationships on a coordinate plane.

    • Classify shapes on the coordinate plane: justify your answer (G-O.13)
    • Determine if a point lies on a circle (G-Y.3)

  • Checkpoint opportunity

    • Checkpoint: Partition a line segment (G-B.16)
    • Checkpoint: Coordinate proofs (G-E.8)

• Parallel & Perpendicular Lines: Students solve problems involving parallel and perpendicular lines.

  • G.LA.5 Prove and apply slope criteria of parallel and perpendicular lines to solve problems.

    • Slopes of parallel and perpendicular lines (G-E.4)
    • Find the distance between a point and a line (G-E.6)
    • Find the distance between two parallel lines (G-E.7)

  • G.LA.6 Write an equation of a line that is parallel or perpendicular to a given line and passing through a given point.

    • Equations of parallel and perpendicular lines (G-E.5)

  • G.LA.7 Prove and apply theorems about lines and angles including: vertical angles, angles formed by parallel lines cut by a transversal, and points on a perpendicular bisector.

    • Perpendicular Bisector Theorem (G-B.9)
    • Angle bisectors (G-C.6)
    • Proofs involving angles (G-C.9)
    • Transversals of parallel lines: solve for x (G-D.5)
    • Proofs involving parallel lines I (G-D.7)
    • Proofs involving parallel lines II (G-D.8)

  • Checkpoint opportunity

    • Checkpoint: Line and angle theorems (G-D.10)
    • Checkpoint: Parallel and perpendicular lines (G-E.9)

TRF Transformations

• Coordinate Plane: Students transform figures on the coordinate plane.

  • G.TRF.1 Describe rotations, reflections, and translations as functions that take points in the coordinate plane as inputs and give other points as outputs; write in prime notation.

    • 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)
    • Rotations: graph the image (G-H.8)
    • Rotations: find the coordinates (G-H.9)
    • Reflections and rotations: write the rule (G-H.10)

  • G.TRF.2 Compare transformations that preserve distance and angle (rotations, reflections, and translations) to those that do not (dilations) to develop definitions for congruence and similarity.

    • Classify congruence transformations (G-H.1)
    • Dilations: graph the image (G-I.1)
    • Dilations: find the coordinates (G-I.2)
    • Dilations: find the scale factor (G-I.4)

  • Checkpoint opportunity

    • Checkpoint: Transformations in the plane (G-I.7)

• Plane: Students transform figures and make geometric constructions.

  • G.TRF.3 Apply understanding of angles, circles, perpendicular lines, parallel lines, and line segments to develop definitions for rotations, reflections, and translations.

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

  • G.TRF.4 Use geometric constructions to represent rotations, reflections, translations, and dilations in the plane with a variety of tools and methods.

    • Translations: graph the image (G-H.2)
    • Reflections: graph the image (G-H.5)
    • Rotations: graph the image (G-H.8)
    • Dilations: graph the image (G-I.1)

  • G.TRF.5 Given two congruent figures, identify the sequence of transformations that maps one figure to another.

    • Sequences of congruence transformations: graph the image (G-H.12)
    • Sequences of congruence transformations: find the rules (G-H.14)
    • Transformations that carry a polygon onto itself (G-H.15)
    • Congruence transformations: mixed review (G-H.16)

  • Checkpoint opportunity

    • Checkpoint: Congruence transformations (G-H.17)

SC Similarities & Congruence

• Similarity: Students use similarity criteria to solve problems.

  • G.SC.1 Given two figures, apply the definition of similarity in terms of a dilation to identify similar figures, proportional sides, and corresponding congruent angles.

    • Identify similar figures (G-M.3)
    • Side lengths and angle measures in similar figures (G-M.4)
    • Similar triangles and similarity transformations (G-M.9)
    • Determine if two figures are similar: justify your answer (G-M.10)

  • G.SC.2 Develop and apply the criteria of similarity for triangles (AA~, SAS~, and SSS~) to solve problems and prove geometric relationships.

    • Similar triangles and indirect measurement (G-M.5)
    • Angle-angle criterion for similar triangles (G-M.7)
    • Similarity rules for triangles (G-M.8)
    • Triangle Proportionality Theorem (G-M.12)
    • Similarity and altitudes in right triangles (G-M.13)
    • Prove similarity statements (G-M.15)
    • Prove proportions or angle congruences using similarity (G-M.16)
    • Proofs involving similarity in right triangles (G-M.17)

  • G.SC.3 Use transformations to prove all circles are similar.

    • Similarity of circles (G-M.11)

  • Checkpoint opportunity

    • Checkpoint: Similarity transformations (G-M.18)
    • Checkpoint: Prove circles are similar (G-M.20)

• Triangle Congruence: Students apply congruence criteria to solve problems.

  • G.SC.4 Explain, using rigid motion transformations, why two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

    • Prove triangles are congruent using rigid motions (G-L.1)

  • G.SC.5 Develop and apply the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) to solve problems and prove geometric relationships.

    • SSS and SAS Theorems (G-L.2)
    • Proving triangles congruent by SSS and SAS (G-L.3)
    • ASA and AAS Theorems (G-L.4)
    • Proving triangles congruent by ASA and AAS (G-L.5)
    • SSS, SAS, ASA, and AAS Theorems (G-L.6)
    • SSS Theorem in the coordinate plane (G-L.7)
    • Proving triangles congruent by SSS, SAS, ASA, and AAS (G-L.8)
    • Proofs involving corresponding parts of congruent triangles (G-L.9)
    • Congruency in isosceles and equilateral triangles (G-L.10)
    • Proofs involving isosceles triangles (G-L.11)
    • Hypotenuse-Leg Theorem (G-L.12)
    • Proofs involving triangles and quadrilaterals (G-O.14)

  • Checkpoint opportunity

    • Checkpoint: Rigid motion and congruence (G-L.13)
    • Checkpoint: Triangle similarity and congruence (G-M.19)