• Discussion (brief) on the The Sphere  and the Torus.
  
• Coop work- 3-4 partners- record responses of group. 5 minutes, followed by discussion.
  


Question 1. How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface?

Question 2. How can one distinguish the sphere from a torus based solely on experiences on the surface?

• Measurement and the Pythagorean Theorem (PT)
  

a² + b² = c²

• Review Pythagorean Activity Sheet
• Virtual Manipulative for PT
• Discuss Pythagorean Theorem and proofs.
• Over 30 proofs of the Pythagorean theorem!
• Many Java Applets that visualize proofs of the Pythagorean Theorem

Show video on PT.
Outline:
Area of triangles = 1/2 bh
Area of parallelogram= bh
Scaling: a linear scale change of r gives area change of factor r².
3 questions: running, moat, wind power...
Proof of the PT: Similar right triangles: c= a² /c + b² /c .
applications and other proofs.
Prop. 47 of Euclid.
Dissection Proof.
Prop 31 Book  VI  Similar shapes.
Simple proof of PT using similar triangles of the triangle.
Use in 3 dimensional space.
 

Puzzles and Polygons

• Tangrams. More
  
• Tangrams – Use all seven Chinese puzzle pieces to make shapes and solve problems.
• Tangram Activity
  
  
  
  Use templates to cut out pieces from larger (blue) sheets. [#Partners=3.]
  

• Cutting and reassembling polygons. More next class. :)