1. Day 2
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

2. The Water-Flask Problem
A transparent flask in the shape of a right rectangular prism is partially filled with water. When the flask is placed on a table and tilted, with one edge (or one point) of its base being fixed, several geometric shapes of various sizes are formed by the cuboid’s face and surface of the water. The shapes and sizes may vary according to the degree of tilt or inclination. Try to discover as many invariant relations (rules) concerning these shapes and sizes as possible. Write down all your findings.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

3. Advantages of the open-ended approach
Students participate more actively in the lesson and express their ideas more frequently.
Students have more opportunities to make comprehensive used of their mathematical knowledge and skills.
Even low-achieving students can respond to the problem in some significant ways of their own.
Students are intrinsically motivated to give proofs.
Students have rich experiences in the pleasure of discovery and receive the approval of fellow students.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

4. Developing an open-ended problem for your students
Determine if the problem is appropriate
Is the problem rich in mathematical content and valuable mathematically?
Is the mathematical level of the problem appropriate or the students?
Does the problem include some mathematical features that lead to further mathematical development?
Anticipating students’ responses to design a lesson.
Making the purpose of using the problem clear.
Make the problem as attractive as possible.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

5. Each week, after the surprise ingredient is revealed, the challenger and the Iron Chef face off in a frenetic culinary battle. The guest panel judges the menus to determine who is victorious and who is vanquished.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

6. Geoboard (Virtual Maniplatives)
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

7. By using a line segment AB as one of the sides, make an isosceles triangle ABC on your geoboard. How many isosceles triangles can you make on your Geoboard? Find as many isosceles triangles as possible. A B
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

8. ＡＢ＝ＡＣ ＡＣ＝ＢＣ ＡＢ＝ＢＣ ＧＳＰ B B B A A A
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

9. Developing an open-ended problem for your students
Determine if the problem is appropriate
Is the problem rich in mathematical content and valuable mathematically?
Is the mathematical level of the problem appropriate or the students?
Does the problem include some mathematical features that lead to further mathematical development?
Anticipating students’ responses to design a lesson.
Making the purpose of using the problem clear.
Make the problem as attractive as possible.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL

10. Developing an open-ended problem for your students
Determine if the problem is appropriate
Is the problem rich in mathematical content and valuable mathematically?
Is the mathematical level of the problem appropriate or the students?
Does the problem include some mathematical features that lead to further mathematical development?
Anticipating students’ responses to design a lesson.
Making the purpose of using the problem clear.
Make the problem as attractive as possible.
Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, June 27-July 15, 2005
by Akihiko Takahashi, DePaul University, Chicago IL