1. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Building Math in the classroom - Teaching Through Problem-Solving - Day 6

2. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi

3. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi

4. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi

5. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi

6. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi What we can learn from all your posters. 1)Three triangles are similar triangles (table 1,3,4,5,6,7,10,12) 2)Area changes (table 2, 8, 9,eM) 3)Fold, symmetry, slope (table 11)

7. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Three triangles are similar Conditions for similarity congruent angles or proportional sides (table 6) Hands-on approach Cut out three triangles and line up to see the similarity (table 7) Comparing three triangles Describe similarities and differences among the triangles (table 10) What commonalities do all three triangles share (table 12). Conjecture and proof (table 3) Specific to general What changes and what stays the same (table 5). How does moving your fold change or affect your findings (table 1) Can you fold differently so the angles are no longer the same (table 4)

8. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Designing a lesson for teaching through problem solving Teaching through problem solving Curriculum & Textbook Students Problem

9. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi From Curriculum NCTM Curriculum Focal Points Grade 4: Explore congruence and similarity – Examine the congruence, similarity, and line or rotational symmetry of objects using transformations

10. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi PPSM & Curriculum Focal Points PSSM: 6-8 1)Transformations can also be used to help students understand similarity and symmetry. Work with magnifications and contractions, called dilations, can support students' developing understanding of similarity. For example, dilation of a shape affects the length of each side by a constant scale factor, but it does not affect the orientation or the magnitude of the angles.

11. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi PPSM & Curriculum Focal Points NCTM Curriculum Focal Points Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.

12. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi PPSM & Curriculum Focal Points PSSM: 9-12 explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;

13. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Similarity Define (4th, 7th) Develop the concept through transformation (magnifications, enlargement, reduction) -informal definition Properties (7th) dilation of a shape affects the length of each side by a constant scale factor, but it does not affect the orientation or the magnitude of the angles Application (7th) Solve problems by using scale factors that relate corresponding lengths of the objects Logical Reasoning (9th-12th) make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others

14. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi From Students 1)Questioning is a powerful tool for teachers to lead students to the goal – Asking students to find “relationship” might be a bit too broad. Right triangles Can you make 3-4-5 triangles?

15. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi NCTM’s view of problem solving 1) Problem solving means engaging in a task for which the solution method is not known in advance. 2) Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. 3) Choosing worthwhile problems and mathematical tasks – There are many, many problems that are interesting and fun but that may not lead to the development of the mathematical ideas that are important for a class at a particular time.

16. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Problem solving is an integral part of all mathematics learning It might be easy to say but difficult to practice….. We do not have time to spend a whole period on problem solving. We do not have time to spend a whole period on problem solving. Our students do not want to solve a problem by themselves. Our students do not want to solve a problem by themselves. We need to be ready for the test. We need to be ready for the test. I am not comfortable teaching that way. I am not comfortable teaching that way.

17. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Examples of teaching through problem solving AC=a, CB=b

18. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Tokyo Shoseki 5A

19. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Designing a lesson for teaching through problem solving Teaching through problem solving Curriculum & Textbook Students Problem

20. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi The next step (a) what the students need to learn in the lesson [according to the standards and/or curriculum]; (b) what the students have learned previously; (c) what is the major focus of this lesson [by comparing (a) and (b), the objective/goal of this lesson should be clearly stated]; and (d) the way to help students accomplish the above objective [Posing a problem/ questioning]

21. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Begin with a mathematical situation that your table chose to design a lesson for teaching through problem solving Another paper folding (rectangular sheet of paper)

22. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Begin with a mathematical situation that your table chose to design a lesson for teaching through problem solving Triangular pool table Start from the midpoint of a side. The ball moves parallel to another side.

23. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Begin with a mathematical situation that your table chose to design a lesson for teaching through problem solving Paper Pool NCTM Illumination web site http://illuminations.nctm.org/ActivityDetail.aspx?ID=28

24. Presentation is prepared for The Park City Mathematics Institute, Secondary School Teachers Program, July 9-July 20, 2007 by Akihiko Takahashi Task for each table -Tuesday and Wednesday- Short sketch of teaching through problem solving a)Purpose of the problem solving (goal of the lesson) What mathematics, beside developing problem solving skills, would you teach by using this situation? b)Questioning How would you pose the problem? What question(s) would you ask to your students to learn mathematics? c)Beyond show and tell Anticipate students’ responses to your questions that including misunderstandings to facilitate discussion. Briefly describe how you would facilitate discussion.