NEKSDC CCSSM HS Geometry February 12, 2013

1. NEKSDC CCSSM HS GeometryFebruary 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. They partake in discovery activities, and form conjectures, but do not formally prove until HS. Unique Triangles • In Grade 7 students recognize when given conditions will result in a UNIQUE TRIANGLE

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 = πr2

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 b a c • 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.  

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 Geometryis 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

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

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)