the geometry of piles of salt thinking deeply about simple things. university of utah teacher’s...
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![Page 1: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/1.jpg)
The Geometryof Piles of SaltThinking Deeply About
Simple Things.University of Utah
Teacher’s Math Circle
February 4, 2008
Troy Jones
Waterford School
![Page 2: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/2.jpg)
• “The real voyage of discovery lies not in finding new lands, but in seeing with new eyes.”
Marcel Proust, French novelist/philosopher 1871-1922
![Page 3: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/3.jpg)
![Page 4: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/4.jpg)
Work on handout
• Read and answer questions
• You might want to work individually, but compare solutions and methods.
• Take time to think about why the constructions yield the desired results
• Stop at the end of page 6 and don’t look ahead (so you won’t ruin the surprise)
![Page 5: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/5.jpg)
Angle of Repose• The angle of repose is the
slope angle of granular material in a state of rest, measured from horizontal
• Table salt has an angle of repose of about 32°
![Page 6: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/6.jpg)
Circle
• Predict what will happen when salt is poured on a circular shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 7: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/7.jpg)
Salt Cone Modeled with Cabri 3D
![Page 8: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/8.jpg)
Triangle
• Predict what will happen when salt is poured onto a triangular shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 9: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/9.jpg)
Salt Tetrahedron Modeled with Cabri 3D
![Page 10: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/10.jpg)
Triangle Salt Ridges Modeled withGeometer’s Sketchpad
A
B
C
![Page 11: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/11.jpg)
Quadrilateral
• Predict what will happen when salt is poured onto a quadrilateral shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 12: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/12.jpg)
Quadrilateral Salt Ridges Modeled with Cabri 3D
![Page 13: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/13.jpg)
Quadrilateral Salt Ridges Modeled with Geometer’s Sketchpad
A
B
C
D
![Page 14: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/14.jpg)
Circle near edge
• Predict what will happen when salt is poured near the edge of a surface with a hole cut near the edge.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 15: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/15.jpg)
Parabolic Salt Ridges Modeled with Geometer’s Sketchpad
A B
C
D
![Page 16: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/16.jpg)
Circle inside circle
• Predict what will happen when salt is poured onto a circle with a hole cut in it.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 17: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/17.jpg)
Elliptical Salt Ridge Modeled with Cabri 3D
![Page 18: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/18.jpg)
Elliptical Salt Ridge Modeled with Geometer’s Sketchpad
A
B
C
D
![Page 19: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/19.jpg)
Hyperbola
• How would you model with salt an hyperbola ridge?.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
![Page 20: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/20.jpg)
Hperbolic Salt Ridge Modeled with Geometer’s Sketchpad
A
B
C
D
![Page 21: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/21.jpg)
Hoberman Sphere Model of Elliptical Salt Ridge with Cabri 3D
![Page 22: The Geometry of Piles of Salt Thinking Deeply About Simple Things. University of Utah Teacher’s Math Circle February 4, 2008 Troy Jones Waterford School](https://reader031.vdocument.in/reader031/viewer/2022032523/56649d815503460f94a65bd0/html5/thumbnails/22.jpg)
Elliptical Salt Ridge Modeled with Cones in Cabri 3D