1. 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 Day 10

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

3. 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. 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. 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. 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 The Iron Chef week 2 ∙ Tow tables obtained 14% each. (8/54)

7. 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 It really leaves opportunity for students to approach the problem in numerous ways, all leading to a specific solution, however allowing all of them to be considered appropriate ways of solving it. It lends itself to many greater concepts and opens up for much discussion. I chose this groups poster because it was a very well thought out simple question that had a very large range of different ways to come up with an answer. I could see students of all ages working on this problem and coming up with solutions that were very unique and interesting. A very simply stated question with not so simple solutions! This group gave a typical problem--changed to be open-ended. They were able to show numerous ways to exhibit the answer. I wish they had suggested some extensions; however, the extensions are easy to write.

8. 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 A winner of the Iron Chef week 2 is the Table 1

9. 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 This problem had many different entry points and allowed the students to use several methods. The students responses included Guess and Check. For many students this is the only way they approach a problem. This gives the teacher an opportunity to point out that 'guess and check' does not guarantee that you have found all of the cases. This provides an opportunity to explore other ways of doing the problem. A combination of teacher and students could do the explanations. There is a fun way of looking at the problem. This group chose a typical textbook problem, whereas some other groups chose ones that were less typical. In addition, this group chose a routine problem. This group changed the problem in an impressive way. They didn't make it harder - or more abstract. In fact they made it more practical and applied. This group did an outstanding job of anticipating student responses. The multitude of solutions were, in my mind, realistic possible ones by students - at the appropriate level. Other groups had solutions which were more advanced (by far) than those that should have been anticipated. Changing the problem by using the phrase "find all of the ways of solving the problem" does not, in my mind, meet the goals of the assignment. Rather, the goal was to change the problem in such a way as to inspire other solutions. This group did exactly this.

10. 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 Another winner of the Iron Chef week 2 is the Table 6

11. 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 Three Major Types of the Open-Ended Approach 1)A lesson using problems with multiple solutions. Week1 2)A lesson using problems with multiple solution methods. Week2 a typical problem--changed to be open-ended 3)A lesson using an activity called ‘problem to problem’ Week3

12. 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 problem to problem Problem Solving Students create his/her own problems by changing the original problem Comparing and Discussing their problems Provide students with opportunities to become active constructors, rather than passive recipients

13. 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

14. 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

15. 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 The Toothpick Problem One square four toothpicks Two squares seven toothpicks Three squares ten toothpicks Four squares ………… How many toothpicks do you need to make ten squares? Create you own problems by changing the toothpick problem.