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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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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.
Geometry Resources for the TN-Nspire
In this presentation we will focus on our DVD Library, Geometry Applications.
Geometry Resources for the TN-Nspire
Volume 1
1. Points and Lines
2. Angles and Planes
3. Triangles
4. Quadrilaterals
5. Polygons
Volume 2
6. Circles
7. 3D Geometry
8. Area and Volume
9. Coordinate Geometry
10. Transformations
Geometry Resources for the TN-Nspire
Geometry Applications: Circles visits ancient Rome.
Geometry Resources for the TN-Nspire
We look at the architecture of the Roman Coliseum.
Geometry Resources for the TN-Nspire
The Coliseum was not circular, but elliptical.
Geometry Resources for the TN-Nspire
Of the surveying tools at the time, the main one was the Groma.
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.
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?
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.
Geometry Resources for the TN-Nspire
You can easily construct an ellipse using push pins, string, and a pencil.
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.
Geometry Resources for the TN-Nspire
Because of the condition of the Coliseum, it’s not known exactly how it was built.
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.
Geometry Resources for the TN-Nspire
Create a Graph Window. We’ll be using the Geometry Tools within this environment.
Geometry Resources for the TN-Nspire
Turn on the background grid, which will make it easier to construct the circles.
Geometry Resources for the TN-Nspire
Construct a circle of radius 6. Use the background grid as your guide.
Geometry Resources for the TN-Nspire
Construct a line from (0, – 6) to (2, 0). Extend the line past the circle.
Geometry Resources for the TN-Nspire
Find the midpoint of the segment between (0, –6) and (2, 0).
Geometry Resources for the TN-Nspire
Construct a line from the midpoint to (4, 0). Extend this line past the circle.
Geometry Resources for the TN-Nspire
Construct a second circle centered at (4, 0) of radius 2.
Geometry Resources for the TN-Nspire
Construct an intersection point where the small circle and the second line intersect.
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.
Geometry Resources for the TN-Nspire
Construct an intersection point where the third circle and the first line intersect.
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.
Geometry Resources for the TN-Nspire
Notice that the three arcs from the three circles form a seamless elliptical arc.
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.
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.
Geometry Resources for the TN-Nspire
You can also turn this into a hands-on activity.
Geometry Resources for the TN-Nspire
Using a compass, ruler, paper, and pencil, you can sketch out the pattern for constructing the elliptical outline.
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