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.

The Geometryof Piles of SaltThinking Deeply About

Simple Things.University of Utah

Teacher’s Math Circle

February 4, 2008

Troy Jones

Waterford School

• “The real voyage of discovery lies not in finding new lands, but in seeing with new eyes.”

Marcel Proust, French novelist/philosopher 1871-1922

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)

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°

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?

Salt Cone Modeled with Cabri 3D

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?

Salt Tetrahedron Modeled with Cabri 3D

Triangle Salt Ridges Modeled withGeometer’s Sketchpad

A

B

C

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?

Quadrilateral Salt Ridges Modeled with Cabri 3D

Quadrilateral Salt Ridges Modeled with Geometer’s Sketchpad

A

B

C

D

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?

Parabolic Salt Ridges Modeled with Geometer’s Sketchpad

A B

C

D

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?

Elliptical Salt Ridge Modeled with Cabri 3D

Elliptical Salt Ridge Modeled with Geometer’s Sketchpad

A

B

C

D

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?

Hperbolic Salt Ridge Modeled with Geometer’s Sketchpad

A

B

C

D

Hoberman Sphere Model of Elliptical Salt Ridge with Cabri 3D

Elliptical Salt Ridge Modeled with Cones in Cabri 3D