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# NCTM 2012 Presentation 2

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This is the second of the presentations we gave at the TI booth at NCTM 2012 in Philly.

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### NCTM 2012 Presentation 2

1. 1. Geometry Resources for the TN-Nspire Media4Math includes a variety of free and premium resources, including short video tutorials on the Nspire, Math in the News, and other tutorials.
2. 2. Geometry Resources for the TN-Nspire In this presentation we will focus on our DVD Library, Geometry Applications.
3. 3. Geometry Resources for the TN-NspireVolume 1 Volume 21. Points and Lines 6. Circles2. Angles and Planes 7. 3D Geometry3. Triangles 8. Area and Volume4. Quadrilaterals 9. Coordinate Geometry5. Polygons 10. Transformations
4. 4. Geometry Resources for the TN-Nspire Geometry Applications: Circles visits ancient Rome.
5. 5. Geometry Resources for the TN-Nspire We look at the architecture of the Roman Coliseum.
6. 6. Geometry Resources for the TN-Nspire The Coliseum was not circular, but elliptical.
7. 7. Geometry Resources for the TN-Nspire Of the surveying tools at the time, the main one was the Groma.
8. 8. Geometry Resources for the TN-Nspire The Groma was ideal for constructing circular structures, since it was simple to define the locus of points from the center.
9. 9. Geometry Resources for the TN-Nspire The Romans built lots of circular structures. So why did they choose the more challenging elliptical shape for the Coliseum?
10. 10. Geometry Resources for the TN-Nspire Rather than a center, the ellipse has the two foci and variable lengths for L1 and L2, even though their sum is constant.
11. 11. Geometry Resources for the TN-Nspire You can easily construct an ellipse using push pins, string, and a pencil.
12. 12. Geometry Resources for the TN-Nspire But translating this into the language of surveying and construction is not straightforward and introduces a great deal of error.
13. 13. Geometry Resources for the TN-Nspire Because of the condition of the Coliseum, it’s not known exactly how it was built.
14. 14. Geometry Resources for the TN-Nspire One theory involves using circles and piecewise arcs to construct the elliptical shape. We’ll use the Nspire to show how this is done.
15. 15. Geometry Resources for the TN-Nspire Create a Graph Window. We’ll be using the Geometry Tools within this environment.
16. 16. Geometry Resources for the TN-Nspire Turn on the background grid, which will make it easier to construct the circles.
17. 17. Geometry Resources for the TN-Nspire Construct a circle of radius 6. Use the background grid as your guide.
18. 18. Geometry Resources for the TN-Nspire Construct a line from (0, – 6) to (2, 0). Extend the line past the circle.
19. 19. Geometry Resources for the TN-Nspire Find the midpoint of the segment between (0, –6) and (2, 0).
20. 20. Geometry Resources for the TN-Nspire Construct a line from the midpoint to (4, 0). Extend this line past the circle.
21. 21. Geometry Resources for the TN-Nspire Construct a second circle centered at (4, 0) of radius 2.
22. 22. Geometry Resources for the TN-Nspire Construct an intersection point where the small circle and the second line intersect.
23. 23. Geometry Resources for the TN-Nspire Construct another circle centered at the midpoint and intersecting the small circle. Notice that these two circles have the same tangent at that point.
24. 24. Geometry Resources for the TN-Nspire Construct an intersection point where the third circle and the first line intersect.
25. 25. Geometry Resources for the TN-Nspire Construct a circle centered at (0, 6) and intersecting the third circle. These two circles also share a tangent at that point.
26. 26. Geometry Resources for the TN-Nspire Notice that the three arcs from the three circles form a seamless elliptical arc.
27. 27. Geometry Resources for the TN-Nspire Here are the three arcs pieced together, forming a seamless elliptical arc. But this was all constructed with circles.
28. 28. Geometry Resources for the TN-Nspire You can now see how this elliptical arc—constructed solely from circles— can be used to construct the Coliseum.
29. 29. Geometry Resources for the TN-Nspire You can also turn this into a hands-on activity.
30. 30. Geometry Resources for the TN-Nspire Using a compass, ruler, paper, and pencil, you can sketch out the pattern for constructing the elliptical outline.