1. NEKSDC CCSSM HS GEOMETRY February 12, 2013

2. PRESENTATION WILL INCLUDE… Overview of K – 8 Geometry Overarching Structure of HS Geometry Content Standards Closer Look at Several Key Content Standards Discussion and Activities around Instructional Shifts and Tasks to engage students in Geometry Content Standards and reinforce Practice Standards

3. K – 6 GEOMETRY STUDENTS BECOME FAMILIAR WITH GEOMETRIC SHAPES THEIR COMPONENTS (Sides, Angles, Faces) THEIR PROPERTIES (e.g. number of sides, shapes of faces) THEIR CATEGORIZATION BASED ON PROPERTIES (e.g. A square has four equal sides and four right angles.)

4. K – 6 GEOMETRY COMPOSING AND DECOMPOSING GEOMETRIC SHAPES The ability to describe, use, and visualize the effects of composing and decomposing geometric regions is significant in that the concepts and actions of creating and then iterating units and higher-order units in the context of construction patterns, measuring, and computing are established bases for mathematical understanding and analysis. K-6 GEOMETRY PROGRESSIONS

5. SPATIAL STRUCTURING AND SPATIAL RELATIONS IN GRADE 3 Students are using abstraction when they conceptually structure an array understand two dimensional objects and sets of objects in two dimensional space as truly two dimensional. For two-dimensional arrays, students must see a composition of squares (iterated units) and also as a composition of rows or columns (units of units)

6. SPATIAL STRUCTURING AND SPATIAL RELATIONS IN GRADE 5 Students must visualize three-dimensional solids as being composed of cubic units (iterated units) and also as a composition of layers of the cubic units (units of units).

7. CLASSIFY TRIANGLES IN GRADE 4 By Side Length Equilateral Isosceles Scalene

8. CLASSIFY TRIANGLES IN GRADE 4 By Angle Size Acute Obtuse Right

9. ANGLES, IN GRADE 4, STUDENTS Understand that angles are composed of two rays with a common endpoint Understand that an angle is a rotation from a reference line and that the rotation is measured in degrees

10. PERPENDICULARITY, PARALLELISM IN GRADE 4, STUDENTS Distinguish between lines and line segments Recognize and draw Parallel and perpendicular lines

11. COORDINATE PLANE Plotting points in Quadrant I is introduced in Grade 5 By Grade 6, students understand the continuous nature of the 2-dimensional coordinate plane and are able to plot points in all four quadrants, given an ordered pair composed of rational numbers.

12. ALTITUDES OF TRIANGLES In Grade 6, students recognize that there are three altitudes in every triangle and that choice of the base determines the altitude. Also, they understand that an altitude can lie… Outside the triangle On the triangle Inside the triangle

13. POLYHEDRAL SOLIDS In Grade 6, students analyze, compose, and decompose polyhedral solids They describe the shapes of the faces and the number of faces, edges, and vertices

14. VISUALIZING CROSS SECTIONS In Grade 7, students describe cross sections parallel to the base of a polyhedron.

15. SCALE DRAWINGS In Grade 7, students use their understanding of proportionality to solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Scale: ¼ inch = 3 feet

16. UNIQUE TRIANGLES In Grade 7 students recognize when given conditions will result in a UNIQUE TRIANGLE They partake in discovery activities, and form conjectures, but do not formally prove until HS.

17. IMPOSSIBLE TRIANGLES In Grade 7 students recognize when given side lengths will or will not result in a triangle The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. If the sum of the lengths of A and B is less than the length of C, then the 3 lengths will not form a triangle. If the sum of the lengths of A and B are equal to the length of C, then the 3 lengths will not form a triangle, since segments A and B will lie flat on side C when they are connected.

18. GRADE 7 FORMULAS FOR CIRCLES Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. C = 2πr A = πr 2

19. GRADE 7 ANGLE RELATIONSHIPS Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

20. GRADE 7 PROBLEMS INVOLVING 2-D AND 3-D SHAPES Solve real-world and mathematical problems involving area, volume and surface area of two- and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Find the volume and surface area

21. GRADE 8 TRANSFORMATIONS Understand congruence and similarity using physical models, transparencies, or geometry software. Verify experimentally the properties of rotations, reflections, and translations:  Lines are taken to lines, and line segments to line segments of the same length.  Angles are taken to angles of the same measure.  Parallel lines are taken to parallel lines.

22. GRADE 8 TRANSFORMATIONS Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

23. GRADE 8 CONGRUENCE VIA RIGID TRANSFORMATIONS Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

24. GR. 8 SIMILARITY VIA NON-RIGID AND RIGID TRANSFORMATIONS Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Enlarge PQR by a factor of 2.

25. GRADE 8 ANGLES Use informal arguments* to establish facts about: the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal the angle-angle criterion for similarity of triangles. *For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. b c a

26. GRADE 8 PYTHAGOREAN THEOREM  Understand and apply the Pythagorean Theorem.  Explain a proof of the Pythagorean Theorem and its converse. Here is one of many proofs of the Pythagorean Theorem. How does this prove the Pythagorean Theorem?

27. GRADE 8 PYTHAGOREAN THEOREM  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. From Kahn Academy

28. GRADE 8 VOLUME Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. http://www.math.com

29. TURN AND TALK TO YOUR NEIGHBOR What concepts and skills that HS Geometry have traditionally spent a lot of time on are now being introduced in middle school? How does that change your ideas for focus in HS Geometry? What concepts and skills do you predict will be areas of major focus in HS Geometry?

30. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Congruence (G-CO) Similarity, Right Triangles, and Trigonometry (G-SRT) Circles (G-C) Expressing Geometric Properties with Equations (G-GPE) Geometric Measurement and Dimension (G-GMD) Modeling with Geometry (G-MG)

31. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Congruence (G-CO) Experiment with transformations in the plane Understand congruence in terms of rigid motions Prove geometric theorems (required theorems listed) Theorems about Lines and Angles Theorems about Triangles Theorems about Parallelograms  Make geometric constructions (variety of tools and methods…by hand and using technology) (required constructions listed)

32. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Similarity, Right Triangles, and Trigonometry (G-SRT) Understand Similarity in terms of similarity transformations Prove theorems involving similarity Define trigonometric ratios and solve problems involving right triangles (+) Apply trigonometry to general triangles Law of Sines Law of Cosines

33. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Circles (G-C) Understand and apply theorems about circles All circle are similar Identify and describe relationships among inscribed angles, radii, and chords. Relationship between central, inscribed, and circumscribed angles Inscribe angles on a diameter are right angles The radius of a circle is perpendicular to the tangent where the radius intersects the circle Find arc lengths and sectors of circles

34. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Expressing Geometric Properties with Equations (G-GPE) Translate between the geometric description and the equation for a conic section Use coordinates to prove simple geometric theorems algebraically

35. STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS Geometric Measurement and Dimension (G-GMD) Explain volume formulas and use them to solve problems Visualize relationships between two-dimensional and three-dimensional objects Modeling with Geometry (G-MG) Apply geometric concepts in modeling situations

36. HS GEOMETRY CONTENT STANDARDS Primarily Focused on Plane Euclidean Geometry Shapes are studied Synthetically & Analytically Synthetic Geometry is the branch of geometry which makes use of axioms, theorems, and logical arguments to draw conclusions about shapes and solve problems Analytical Geometry places shapes on the coordinate plane, allowing shapes to defined by algebraic equations, which can be manipulated to draw conclusions about shapes and solve problems.

37. FINDING ANGLES Work through this “synthetic” geometry problem. What definitions, axioms, and theorems do students need to know? What algebraic skills?

38. FINDING ANGLES The next three shapes and the previous one were taken from a site filled with rich Geometry problems. http://donsteward.blogspot.com/ In addition to being used to find angles, students can be asked to create a copy of each shape using GeoGebra, which reinforces many of the Practice Standards as well as knowledge of transformations.

39. FINDING ANGLES

41. FORMAL DEFINITIONS AND PROOF HS Students begin to formalize the experiences with geometric shapes introduced in K – 8 by Using more precise definitions Developing careful proofs When you hear the word “proof”, what do you envision?

42. FORMAL DEFINITIONS AND PROOF In a triangle, the segment connecting the midpoints of two sides is parallel to the third side and has a length that is half the length of the third side. Given the verbal statement of a theorem, what are the steps that students need to take in order to prove the theorem?

43. Geometry, Proofs, and the Common Core Standards, Sue Olson, Ed.D, UCLA Curtis Center Mathematics Conference March 3, 2012 How has the proof of the theorem already been scaffolded at this step? SCAFFOLDING PROOFS

44. WAYS TO SCAFFOLD THIS SYNTHETIC* PROOF Easiest to Most Challenging: Provide a list of statements and a list of reasons to choose from and work together as a class The above, but no reasons provided The above, but done individually No list of statements or reasons and done individually *As opposed to Analytic (using coordinates)

45. CHANGE IT TO AN ANALYTIC APPROACH Easiest to Hardest Use the methods of coordinate geometry to prove that the segment connecting the midpoints of a triangle with vertices A (8, 10), B (14, 0), and C (0, 0) is parallel to the third side and has a length that is one-half the length of the third side. Start by drawing a diagram. Would this method result in a proof? Why or why not?

46. CHANGE IT TO AN ANALYTIC APPROACH Harder: Use the methods of coordinate geometry to prove that the segment connecting the midpoints of a triangle with vertices A (2b, 2c), B (2a, 0), and C (0, 0) is parallel to the third side and has a length that is one-half the length of the third side. Would this method result in a proof? Why or why not?

47. CHANGE IT TO AN ANALYTIC APPROACH Most Challenging Use the methods of coordinate geometry to prove that the segment connecting the midpoints of any triangle is parallel to the third side and has a length that is one-half the length of the third side. What could help make this less challenging?

48. INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE Congruence, Similarity, and Symmetry are understood from the perspective of Geometric Transformation extending the work that was started in Grade 8

49. INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE Rigid Transformations (translations, rotations, reflections) preserve distance and angle and therefore result in images that are congruent to the original shape. G-C0 Cluster Headings Revisited Experiment with transformations in the plane Understand congruence in terms of rigid motions Prove geometric theorems Make geometric constructions

50. TRANSFORMATIONS AS FUNCTIONS Using an Analytical Geometry lens, transformations can be described as functions that take points on the plane as inputs and give other points on the plane as outputs. What transformations do these functions imply? Will they result in congruent shapes? (x,y)  (x + 3, y) (x, y)  (y, x) (x,y)  (x,-y) (x, y)  (-y, x) (x,y)  (2x, 2y) (x, y)  (3x + 2, 3y + 2) (x, y)  (.5x, y) (x, y)  (x – 1, y – 1)

51. TRANSFORMATIONS AS FUNCTIONS (x,y)  (x + 3, y)* Turn and talk to your neighbor: *Compare and contrast the notation above that communicates a right shift of 3 and the function notation f(x – 3) used to indicate the function f(x) is shifted 3 to the right.

52. INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE Two shapes are defined to be congruent to each other if there is a sequence of rigid motions that carries one onto the other. Prove these triangles are congruent by writing the sequence of rigid transformations

53. CONGRUENCY BY TRANSFORMATION Prove these shapes are congruent by describing the sequence of rigid transformations

54. PROVING SIMILARITY VIA TRANSFORMATIONS Dilation is a Non-Rigid Transformation that preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape. G-SRT Cluster Headings dealing with Similarity: Understand Similarity in terms of similarity transformations Prove theorems involving similarity

55. PROVING SIMILARITY VIA TRANSFORMATIONS From a transformational perspective… Two shapes are defined to be similar to each other if there is a sequence of rigid motions followed by a non- rigid dilation that carries one onto the other. A dilation formalizes the idea of scale factor studied in Middle School.

56. ANIMATION SHOWING DILATIONS OF LINES AND CIRCLES Link to Charles A. Dana Center Mathematics Common Core Toolbox Click on the link Go to Standards for Mathematical Content Go to Key Visualizations Go to Geometry Discuss how this visualization could be used in the classroom. What would be a good follow-up activity?

57. PROVE SIMILARITY BY TRANSFORMATIONS What non-rigid transformation proves that these triangles are similar? What is the center of dilation? What is the scale factor of the Dilation?

58. FIND SCALE FACTORS GIVEN A TRANSFORMATION www.ck12.orgwww.ck12.org Similarity Transformations Created by: Jacelyn O'Roark

59. TOOLS FOR CREATING TRANSFORMATIONS Using Compass Ruler Protractor Transparencies Task: Leaping Lizards

60. TOOLS FOR CREATING TRANSFORMATIONS Using manipulatives such as a set of Tangrams What shapes do you see? How are they related? Can you compose the shapes to form other congruent or similar shapes? Rachel McAnallen's Tangram Activities

61. TANGRAM PARTNER ACTIVITY Switch partner roles between “creator” and “maker” Place a file folder between the partners so they can’t see each other’s shape. Each partner has a white sheet of paper marked N, S, E, W on the appropriate edges. 1 st couple of rounds: The creator creates a shape using all 7 pieces. Then stands up and gives directions while watching the “maker” create the shape. 2 nd couple of rounds: Creator doesn’t watch the maker. What Practice Standards are being used?

62. TANGRAM PARTNER ACTIVITY Using two sets of tangrams, show an illustration of the Pythagorean Theorem. What Practice Standards are being used?

63. M.C. ESCHER HTTP://WWW.MCESCHER.COM/HTTP://WWW.MCESCHER.COM/

64. GROUP ACTIVITY Go to the M.C Escher website and choose Picture Gallery and Symmetry. Choose a picture. Describe the transformations as clearly as you can. What transformations do you see. Are there more than one? What Practice Standards did you use?

65. TOOLS FOR CREATING TRANSFORMATIONS GeoGebra Geometer’s Sketchpad Other Dynamic Geometric Software Roman Mosaic Work with a partner or a group to create this mosaic using GeoGebra. Discuss the Practice Standards and Content Standards that were used.

66. C-C 5. ARC LENGTHS AND SIMILARITY 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. http://www.themathpage.com/atrig/arc-length.htm

67. C-C 5. ARC LENGTHS AND SIMILARITY http://www.themathpage.com/atrig/arc-length.htm

68. C-C 5. ARC LENGTHS AND SIMILARITY http://www.themathpage.com/atrig/arc-length.htm The arc length s is proportional to the radius r. The radian measure θ is the constant of proportionality

69. C-C 5. ARC LENGTHS AND SIMILARITY http://www.themathpage.com/atrig/arc-length.htm The arc length s is proportional to the radius r. The radian measure θ is the constant of proportionality

70. RIGHT TRIANGLE TRIGONOMETRY Understand that by similarity, side ratios in right triangles are properties of angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Explain and use the relationships between the sine and cosine of complementary angles. Relationship between sine and cosine in complementary angles

71. CIRCLES IN ANALYTIC GEOMETRY G-GPE (Expressing Geometric Properties with Equations)  Derive the equation of a circle given center (3,-2) and radius 6 using the Pythagorean Theorem  Complete the square to find the center and radius of a circle with equation x 2 + y 2 – 6x – 2y = 26 Think of the time spent in Algebra I on factoring Versus completing the square to solve quadratic Equations. What % of quadratics can be solved by factoring? What % of quadratics can be Solved by completing the square? Is completing the square using the area model more intuitive for students?

72. CONIC SECTIONS – CIRCLES AND PARABOLAS Translate between the geometric description and the equation for a conic section Derive the equation of a parabola given a focus and directrix Parabola – Note: completing the square to find the vertex of a parabola is in the Functions Standards (+) Ellipses and Hyperbolas in Honors or Year 4 Sketch and derive the equation for the parabola with Focus at (0,2) and directrix at y = -2 Find the vertex of the parabola with equation Y = x 2 + 5x + 7

73. VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS Identify the shapes of 2-dimensional cross sections of 3- dimensional objects

74. VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS Identify 3-dimensional shapes generated by rotations of 2-dimensional objects http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html

75. LINKS FOR TASKS ON CONSTRUCTION CCSSI Math Tools Part 1 CCSSI Math Tools Part 2 http://ccssimath.blogspot.fr/2012/12/mathematical-tools-part-3.html

76. RICH HS GEOMETRY TASK http://www.illustrativemathematics.org/illustrations/607 This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7). Teachers who wish to use this problem as a classroom task may wish to have students work on the task in cooperative learning groups due to the high technical demand of the task. If time is an issue, teachers may wish to use the Jigsaw cooperative learning strategy to divide the computational demands of the task among students while requiring all students to process the mathematics in each part of the problem.