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

2. 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.

3. Geometry Resources for the TN-Nspire
In this presentation we will focus on
our DVD Library, Geometry
Applications.

4. Geometry Resources for the TN-Nspire
Volume 1 Volume 2
1. Points and Lines 6. Circles
2. Angles and Planes 7. 3D Geometry
3. Triangles 8. Area and Volume
4. Quadrilaterals 9. Coordinate Geometry
5. Polygons 10. Transformations

5. Geometry Resources for the TN-Nspire
Geometry Applications: Circles visits
ancient Rome.

6. Geometry Resources for the TN-Nspire
We look at the architecture of the
Roman Coliseum.

7. Geometry Resources for the TN-Nspire
The Coliseum was not circular, but
elliptical.

8. Geometry Resources for the TN-Nspire
Of the surveying tools at the time, the
main one was the Groma.

9. 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.

10. 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?

11. 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.

12. Geometry Resources for the TN-Nspire
You can easily construct an ellipse
using push pins, string, and a pencil.

13. 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.

14. Geometry Resources for the TN-Nspire
Because of the condition of the
Coliseum, it’s not known exactly how
it was built.

15. 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.

16. Geometry Resources for the TN-Nspire
Create a Graph Window. We’ll be
using the Geometry Tools within this
environment.

17. Geometry Resources for the TN-Nspire
Turn on the background grid, which
will make it easier to construct the
circles.

18. Geometry Resources for the TN-Nspire
Construct a circle of radius 6. Use the
background grid as your guide.

19. Geometry Resources for the TN-Nspire
Construct a line from (0, – 6) to (2, 0).
Extend the line past the circle.

20. Geometry Resources for the TN-Nspire
Find the midpoint of the segment
between (0, –6) and (2, 0).

21. Geometry Resources for the TN-Nspire
Construct a line from the midpoint to
(4, 0). Extend this line past the circle.

22. Geometry Resources for the TN-Nspire
Construct a second circle centered at
(4, 0) of radius 2.

23. Geometry Resources for the TN-Nspire
Construct an intersection point where
the small circle and the second line
intersect.

24. 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.

25. Geometry Resources for the TN-Nspire
Construct an intersection point where
the third circle and the first line
intersect.

26. 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.

27. Geometry Resources for the TN-Nspire
Notice that the three arcs from the
three circles form a seamless
elliptical arc.

28. 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.

29. 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.

30. Geometry Resources for the TN-Nspire
You can also turn this into a hands-on
activity.

31. Geometry Resources for the TN-Nspire
Using a compass, ruler, paper, and
pencil, you can sketch out the
pattern for constructing the elliptical
outline.