Grade 8 Pythagorean Theorem

1. 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?

2. 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

3. 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

4. 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?

5. 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)

6. 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)

7. 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

8. 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

9. 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

10. 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

11. 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.

12. 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?

13. 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

14. 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

15. 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

16. 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.

17. 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?

18. Find Scale Factors Given a Transformation www.ck12.org Similarity Transformations Created by: JacelynO'Roark

19. 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 x2 + y2 – 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?

20. 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 = x2 + 5x + 7

21. Visualize relationships between 2-D and 3-D objects • Identify the shapes of 2-dimensional cross sections of 3-dimensional objects

22. 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

23. North country Inservice outline • Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences • Review of CCSSM Practice Standards – Share Experiences • Presentation of How Geometry Unfolds over K – 12 in CCSSM • Focus on Volume Standard in HS Geometry • Develop one unit focusing on HS Volume Standard and Practice Standards

24. HS.GMD.A.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. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

25. Informal Argument for Area Of Circle • http://www.youtube.com/watch?v=7zoqL2iOpvo • Area of Circle GeoGebra Applet • http://www.geogebratube.org/student/m279?mobile=true

26. From Don Steward

27. HS.GMD.A.3 • Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ • Dan Meyer 3 Act: Popcorn Picker • Dan Meyer 3 Act: The Coffee Carrier • Dan Meyer 3 Act: You Pour, I Choose • Andrew Stadel 3 Act: Trashketball

29. Mathematics Assessment Project • H.G-GMD: Geometric measurement and dimension • Explain volume formulas and use them to solve problems • Equations of Circles 2 • Evaluating Statements About Enlargements (2D and 3D) • Calculating Volumes of Compound Objects

30. Illustrative Math G-GMD • G-GMD.3 Centerpiece • G-GMD.3 Doctor’s Appointment

31. Mathematics Assessment Project • Visualize relationships between two-dimensional and three- dimensional objects • 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: Rolling Cups • 2D Representations of 3D Objects

32. Volume Animations • Charles A. Dana Center • Mathematics Unfolding: Volume • http://ccsstoolbox.agilemind.com/animations/standards_content_mathematics_volume.html

33. HS.GMD.B.4 • Identify the shapes of two-dimensional cross-sections of three-dimensional objects generated by rotations of two-dimensional objects. • G-GMD.4 Tennis Balls in a Can

34. North country Inservice outline • Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences • Review of CCSSM Practice Standards – Share Experiences • Presentation of How Geometry Unfolds over K – 12 in CCSSM • Focus on Volume Standard in HS Geometry • Develop one unit focusing on HS Volume Standard and Practice Standards