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 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 Looking for a map

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 Instruction as Interaction Adding it up, (National Research Council, 2001)

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 Geoboard

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 1 ∙ One table obtained 23% of the votes (13/56) ∙ I enjoyed reading the comments – Unfortunately some of the comments were truncated ∙ Three votes with no reasons ∙ Another vote with ‘Colorful!’ as the reason

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 1 clear, attractive poster, lots of levels from which to enter, somewhat limited in extensions Had multiple solutions and created interesting patterns. could be used to create greater understanding about area. I like how accessible this is to all students. I especially like that this group anticipated all types of student responses, not just the best responses.

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 1 I think this problem is very rich in mathematical content. I also believe it is appropriate for the stated grade level. There seem to be many possible extensions with this problem. I really liked their careful inclusion of a wide variety of student responses. It was an intriguing problem and I wanted to do the problem. It was clearly stated and well organized.

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 1 Table X‘s activity is appropriate to grade level indicated. The task is clearly stated. What's great about this activity is that it requires and reinforces Math skills that are tested at this grade level in Texas. It also offers possibilities for more explorations. This group's poster was the best because it was rich in challenge - an "elegant" question. It lends itself nicely to many extensions. It is a question that students at many different levels can learn from, yet also motivating to all.

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 1 This seemed to be basic and challenging al at the same time. The extensions are endless and even young students can learn about distance from a fixed point. Most of all I believe the students would be interested in this problem because it was turned into a competition, that is always good! It was impressive that this group took into account a large variety of possible student responses, including ways that students with different prior knowledge about the topic might approach and solve the problem. It was also interesting to see how they intended to support students who might get "stuck." Finally, unlike many of the groups, the question was clear and easy to jump into. It wasn't scaffolded and didn't need to be.

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 1 (1) It is a rich problem mathematically. (2) It is extensible. (3) it is thought provoking (4) It uses many elementary concepts and formulas of geometry. (5) appeals to students at different levels of mathematical knowledge. Those who know more may find more solutions. The competitive nature of the problem would draw students into wanting to work it. (nice challenge). The idea of seeking clear evidence for each response forces students to consider each response in depth. Very appropriate for grade level.

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 1 While many of the posters would be very good activities for students, the one from Table X seemed to offer the best combination of arithmetic, algebraic and geometric thinking of all. I was very intrigued by the wide variety of visualizations that they anticipated. I chose Going the Distance because it is very accessible to students. I think I will use it in my classes (without the competition). I envision a lot of mathematical discussion related to the mathematics involved.

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 winner of the Iron Chef week 1 is the Table 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 A lesson using problems with multiple solution methods

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 lesson using problems with multiple solution methods l m A B P x 40° 30° l // m

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 Developing an open-ended problem for your students 1)Determine if the problem is appropriate a)Is the problem rich in mathematical content and valuable mathematically? b)Is the mathematical level of the problem appropriate or the students? c)Does the problem include some mathematical features that lead to further mathematical development? 2)Anticipating students’ responses to design a lesson. 3)Making the purpose of using the problem clear. 4)Make the problem as attractive as possible.