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Geometry Click HERE for full text of Alaska Mathematics Standards Instructional Focus Course Clusters • Congruence, proof, and construction Similarity, proof and trigonometry Extending to three dimensions • • • • Understand similarity in terms of similarity transformations. (G-‐SRT.1 a-‐b, G-‐ SRT.2, G-‐ SRT.3) • • Prove theorems involving similarity. (G-‐SRT.4, G-‐SRT.5) • • Apply geometric concepts in modeling situations. (G-‐MG.1, G-‐MG.2, G-‐MG.3) Apply trigonometry to general triangles. (G-‐SRT.9, G-‐SRT.10, G-‐SRT.11) • Explain volume formulas and use them to solve problems. (G-‐GMD.1, G-‐ GMD.3) Visualize the relation between two-‐dimensional and three-‐dimensional objects. (G-‐ GMD.4) Apply geometric concepts in modeling situations. (G-‐MG.1) • • Connecting algebra and geometry through coordinates Circles with and without coordinates Application of probability Experiment with transformations in the plane. (G-‐CO.1, G-‐CO.2, G-‐CO.3, G-‐ CO.4, G-‐ CO.5) Understand congruence in terms of rigid motions. (G-‐CO.6, G-‐CO.7, G-‐CO.8) Prove geometric theorems. (G-‐CO.9, G-‐CO.10, G-‐CO.11) Make geometric constructions. (G-‐CO.12, G-‐CO.13) • • Define trigonometric ratios and solve problems involving right triangles. (G-‐ SRT.6, G-‐ SRT.7, G-‐SRT.8) Use coordinates to prove simple geometric theorems algebraically. (G-‐GPE.4, G-‐GPE.5, G-‐GPE.6, G-‐GPE.7) Translate between the geometric description and the equation for a conic section. (G-‐ GPE.2) • Understand and apply theorems about circles. (G-‐C.1, G-‐C.2, G-‐C.3, G-‐C.4) • • Find arc lengths and areas of sectors of circles. (G-‐C.5) Translate between the geometric description and the equations for a conic section. (G-‐ GPE.1) • • Use coordinates to prove simple geometric theorem algebraically. (G-‐GPE.4) Apply geometric concepts in modeling situations. (G-‐MG.1) • Understand independence and conditional probability and use them to interpret data. (S-‐ CP.1, S-‐CP.2, S-‐CP.3, S-‐CP.4, S-‐CP.5) • Use the rules of probability to compute probabilities of compound events in a uniform probability model. (S-‐CP.6, S-‐CP.7, S-‐CP.8, S-‐CP.9) • Use probability to evaluate outcomes of decisions. (S-‐MD.6, S-‐MD.7) Pacing Guide Standards for Mathematical Content Emphasis Average Chapter 1 Foundations for Geometry Pace: 8 days + 6 assess Advanced 1-‐1 Understanding Points, Lines, and Planes G-‐CO.1 1-‐2 Measuring and Constructing Segments G-‐CO.1, G-‐CO.12 high 1-‐3 Measuring and Constructing Angles G-‐CO.1, G-‐CO.12 high 1-‐4 Pairs of Angles G-‐CO.1 1-‐5 Using Formulas in Geometry G-‐GMD.1, G-‐GMD.3 1-‐6 Midpoint and Distance in the Coordinate Plane 1-‐7 Transformations in the Coordinate Plane high G-‐GPE.4, G-‐GPE.7, G-‐ GMD.3 G-‐CO.2, G-‐CO.3, G-‐ CO.4, G-‐CO.5 high Chapter 2 Geometric Reasoning Pace: 7 days + 5 asses 2-‐1 Using Inductive Reasoning to Make Conjectures 2-‐2 Conditional Statements 2-‐3 Using Deductive Reasoning to Verify Conjectures 2-‐4 Bi-‐conditional Statements and Definitions 2-‐5 Algebraic Proof 2-‐6 Geometric Proof G-‐CO.9, G-‐CO.11, G-‐CO.10 high 2-‐7 Flowchart and Paragraph Proofs G-‐CO.9, G-‐CO.11, G-‐CO.10 high Chapter 3 Parallel and Perpendicular Lines Pace: 8 days + 5 assess 3-‐1 Lines and Angles G-‐CO.9, G-‐CO.12, G-‐ CO.9, G-‐CO.1 high 3-‐2 Angles formed by Parallel Lines and Transversals G-‐CO.9, G-‐MG.3, G-‐CO.12, G-‐CO.1 high 3-‐3 Proving Lines Parallel G-‐CO.9, G-‐CO.12 high 3-‐4 Perpendicular Lines G-‐CO.12, G-‐CO.9, G-‐MG.3, G-‐CO.12, G-‐CO.1 high 3-‐5 Slopes of Lines G-‐GPE.5, G-‐MG.3 high 3-‐6 Lines in the Coordinate Plane G-‐GPE.5 high Chapter 4 Triangle Congruence Pace: 11 days + 5 assess 4-‐1 Classifying Triangles G-‐CO.1, G-‐CO.13 high 4-‐2 Angle Relationships in Triangles G-‐CO.10 high 4-‐3 Congruent Triangles G-‐CO.8, G-‐CO.7, G-‐CO.6 high 4-‐4 Triangle Congruence: SSS and SAS G-‐CO.8, G-‐MG.3, G-‐ CO.10 high 4-‐5 Triangle Congruence: ASA, AAS, and HL G-‐CO.8, G-‐MG.3, G-‐CO.10 high 4-‐6 Triangle Congruence: CPCTC G-‐CO.8, G-‐CO.10 high 4-‐7 Introduction to Coordinate Proof G-‐GPE.4, G-‐CO.10 high 4-‐8 Isosceles and Equilateral Triangles G-‐CO.1, G-‐CO.10 high Chapter 5 Properties and Attributes of Triangles Pace: 11 + 6 assess 5-‐1 Perpendicular and Angle Bisectors G-‐CO.10, G-‐CO.1 high 5-‐2 Bisectors of Triangles G-‐CO.12, G-‐C.3, G-‐ MG.3, G-‐CO.10, G-‐ CO.1 high 5-‐3 Medians and Altitudes of Triangles G-‐CO.12, G-‐MG.3, G-‐ CO.10, G-‐CO.1 high 5-‐4 The Triangle Mid-‐segment Theorem G-‐CO.10, G-‐CO.12, G-‐GPE.4, G-‐MG.3, G-‐CO.1 high 5-‐5 Indirect Proof and Inequalities in One Triangle G-‐CO.10 5-‐6 Inequalities in Two Triangles 5-‐7 The Pythagorean Theorem G-‐SRT.8, G-‐SRT.6, G-‐MG.3 high 5-‐8 Applying Special Right Triangles G-‐SRT.6, G-‐MG.3, G-‐CO.1 high Chapter 6 Polygons and Quadrilaterals Pace: 6 days + 4 assess 6-‐1 Properties and Attributes of Polygons G-‐CO.1, G-‐CO.13 high 6-‐2 Properties of Parallelograms G-‐CO.11, G-‐MG.2 high 6-‐3 Conditions of Parallelograms G-‐CO.11, G-‐GPE.5, G-‐GPE.4 high 6-‐4 Properties of Special Parallelograms G-‐GPE.4, G-‐MG.2, G-‐ GPE.5 high 6-‐5 Conditions for Special Parallelograms G-‐GPE.4, G-‐GPE.5 high 6-‐6 Properties of Kites and Trapezoids G-‐MG.2 high B l o c k 7 Chapter 9 Extending Perimeter, Circumference, Area Pace: 10 days + 4 assess 9-‐1 Developing Formulas for Triangles and Quadrilaterals G-‐MG.3, G-‐MG.2, G-‐CO.1 high 9-‐2 Developing Formulas for Circles and Regular Polygons G-‐MG.3, G-‐MG.2, G-‐GMD.1, G-‐CO.1 high 9-‐3 Composite Figures G-‐MG.3, G-‐MG.2 high 9-‐4 Perimeter and Area in the Coordinate Plane G-‐GPE.7, G-‐MG.2, G-‐ GPE.7 high 9-‐5 Effects of Changing Dimensions Proportionally G-‐GPE.7, G-‐CO.2 high 9-‐6 Geometric Probability S-‐CP.1, S-‐CP.2, S-‐ CP.3, S-‐CP.4, S-‐CP.5 high Chapter 10 Spatial Reasoning Pace: 11 days + 5 assess 10-‐1 Solid Geometry G-‐CO.1, G-‐GMD.4 high 10-‐4 Surface Area of Prisms and Cylinders G-‐MG.2, G-‐MG.3, G-‐ MG.1 high 10-‐5 Surface Area of Pyramids and Cones G-‐MG.2, G-‐MG.3, G-‐MG.1 high 10-‐6 Volume of Prisms and Cylinders G-‐GMD.3, G-‐MG.2, G-‐MG.3, G-‐MG.1, G-‐ GMD.1, G-‐GMD.2 high 10-‐7 Volume of Pyramids and Cones G-‐GMD.1, G-‐GMD.3, G-‐MG.2, G-‐MG.3, G-‐ MG.1, high 10-‐8 Spheres G-‐GMD.3, G-‐MG.2, G-‐MG.1, G-‐GMD.2 high Chapter 7 Similarity Pace 6 days + 4 assess 7-‐1 Ratio and Proportion 7-‐2 Ratios in Similar Polygons G-‐SRT.5, G-‐SRT.2 high 7-‐3 Triangle Similarity: AA, SSS, SAS G-‐SRT.3, G-‐SRT.5, G-‐SRT.2, G-‐MG.3 high G-‐SRT.5, G-‐MG.3, GSRT.4, G-‐SRT.6 high 7-‐5 Using Proportional Relationships G-‐MG.2, G-‐MG.3 high 7-‐6 Dilations and Similarity in the Coordinate Plane G-‐SRT.1 high 7-‐4 Applying Properties of Similar Triangles Chapter 8 Right Triangle Pace: 7 days + 4 assess 8-‐1 Similarity in Right Triangles G-‐SRT.6 8-‐2 Trigonometric Ratios 8-‐3 Solving Right Triangles G-‐SRT.6, G-‐SRT.7, G-‐MG.3, GSRT.8, G-‐SRT.7 G-‐MG.3, G-‐MG.2, G-‐SRT.11, G-‐SRT.8, G-‐SRT.7 high high B l o c k 7 8-‐4 Angles of Elevation and Depression G-‐MG.3, G-‐MG.2, G-‐ SRT.11, G-‐SRT.8 high G-‐SRT.9, G-‐SRT.10, G-‐MG.3, G-‐SRT.11, G-‐SRT.9 high 8-‐5 Law of Sines and Law of Cosines Chapter 11 Circles Pace: 9 days + 5 assess high 11-‐1 Lines that Intersect Circles G-‐C.4, G-‐CO.1 11-‐2 Arcs and Chords G-‐CO.1 11-‐3 Sector Area and Arc Length G-‐MG.3, G-‐MG.2, G-‐C.5, high 11-‐4 Inscribed Angles G-‐C.2, G-‐C.13, G-‐CO.1 high 11-‐5 Angle Relationships in Circles G-‐C.2 11-‐6 Segment Relationships in Circles G-‐C.2 11-‐7 Circles in the Coordinate Plane G-‐GPE.1, G-‐GPE.4 Chapter 12 Extending Transformational Geometry high Pace: 7 days + 4 assess 12-‐1 Reflections G-‐CO.5, G-‐CO.4, G-‐CO.3, G-‐ CO.2, G-‐CO.1 high 12-‐2 Translations G-‐CO.5, G-‐CO.4, G-‐CO.3, G-‐ CO.2, G-‐CO.1 high 12-‐3 Rotations G-‐CO.5, G-‐CO.4, G-‐CO.3, G-‐ CO.2, G-‐CO.1 high 12-‐4 Compositions of Transformations G-‐CO.5, G-‐CO.4, G-‐CO.3, G-‐CO.2, G-‐CO.1 high 12-‐5 Symmetry 12-‐6 Tessellations 12-‐7 Dilations G-‐CO.5, G-‐CO.4, G-‐ CO.3, G-‐CO.1 Mathematical Practice Standards 1. 2. 3. 4. 5. 6. 7. 8. in solving them. Make sense of problems and persevere Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. high Geometry Alaska Mathematics Standards Instructional Materials (Core): Holt McDougal Geometry, 2011 Standards for Mathematical Content Where to find in Holt McDougal Geometry ©2011 Geometry Congruence G-‐C0 Experiment with transformations in the plane. G-‐CO.1 Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc. Ch 1, Ch 3, Ch 5, Ch 9, Ch 11, Ch 12 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). Ch 1, Ch 9, Ch 12 G-‐CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Ch 1, Ch 12 G-‐CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. Ch 12 G-‐CO.5 Ch 1, Ch 12 Understand congruence in terms of rigid motions. 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. Ch 4, Ch 12 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. Ch 4 G-‐CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions. Ch 4 Prove geometric theorems. G-‐CO.9 Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles. 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. Ch 2, Ch 3 G-‐CO.10 Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles. 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. Ch 4, Ch 5, Ch 2 G-‐CO.11 Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms. 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. Ch 6, Ch 2 Make geometric constructions. 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.). 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. Ch 1, Ch 3 G-‐CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Ch 4, Ch 6, Ch 11 Similarity, Right Triangles, and Trigonometry G-‐SRT Understand similarity in terms of similarity transformations G-‐SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: 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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Ch 7 G-‐SRT.2 Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar. Ch 7 G-‐SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Ch 7 Prove theorems involving similarity. G-‐SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely. Ch 7 G-‐SRT.5 Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures. Ch 7 Define trigonometric ratios and solve problems involving right triangles 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. Ch 5, Ch 8 G-‐SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Ch 8 G-‐SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Ch 5, Ch 8 Apply trigonometry to general triangles G-‐SRT.9 Ch 8 (+) 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.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. Ch 8 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). Ch 8 Circles G-‐C Understand and apply theorems about circles. G-‐C.1 Prove that all circles are similar. Ch 11 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. Ch 11 G-‐C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Ch 5, Ch 11 Ch 11 (+) Construct a tangent line from a point outside a given circle to the circle. G-‐C.4 Find arc lengths and areas of circles. Ch 11 Use and apply the concepts of arc length and areas of sectors of circles. Determine or 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. G-‐C.5 Expressing Geometric Properties with Equations G-‐GPE Translate between geometric description and the equation for a conic section. G-‐GPE.1 Determine or 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. Ch 11 Use coordinates to prove simple geometric theorems algebraically. Perform simple coordinate proofs. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Ch 4, Ch 6, Ch 11 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). Ch 3, Ch 6 G-‐GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-‐GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Ch 9 G-‐GPE.4 G-‐GPE.5 Geometric Measurement and Dimension G-‐GMD Explain volume formulas and use them to solve problems. G-‐GMD.1 Explain how to find the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Ch 9, Ch 10 G-‐GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Ch 10 G-‐GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. For example: Solve problems requiring determination of a dimension not given.* Ch 10 Visualize relationships between two-‐dimensional and three dimensional objects 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. Modeling with Geometry Ch 10 G-‐MG 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). Ch 10 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). Ch 6, Ch7, Ch 8, Ch 9, Ch 10, Ch 11 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). Ch 1, Ch 3, Ch 4, Ch5, Ch 6, Ch 7, Ch 8, Ch 9, Ch 10, Ch 11
