2. Mara Alagic IMECT 3 Cambridge July 2002 Technology-Based Representations of Some Mathematical Ideas Mara Alagic Wichita State University Wichita, Kansas, USA

3. Mara Alagic IMECT 3 Cambridge July 2002 Teaching and Learning Mathematics  Standards-based teaching (NCTM, 2000)  Current findings about the nature of human learning (NRC, 2000)  Development of the IC technologies  Technology Principle (NCTM, 2000)

4. Mara Alagic IMECT 3 Cambridge July 2002 Mathematics Teaching and Technology  Technology Principle:  "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (NCTM, 2000)  Technology in context

5. Mara Alagic IMECT 3 Cambridge July 2002 Representations  Representing knowledge (Langer, 1989)  Problem-based representations  Representation standard (NCTM, 2000)  Things before words, concrete before abstract. (Pestalozzi,1803)

6. Mara Alagic IMECT 3 Cambridge July 2002

7. Conceptual understanding via multiple representations  Perception-based representations  Multiple representations  External and internal representations  Problem-based representations  Translations among representations

8. Mara Alagic IMECT 3 Cambridge July 2002 MENTAL MODELS/ INTERNAL REPRESENTATIONS EXTERNAL REPRESENTATIONS & TRANSLATIONS

9. Mara Alagic IMECT 3 Cambridge July 2002 "... the defining characteristic of knowledge workers is that they are themselves changed by the information they process" Kidd (1994, p. 186)

10. Mara Alagic IMECT 3 Cambridge July 2002 Questions & Tasks: (Jaelyn & Barbie) 1. DEFINE TESSELLATION(S). 2. ARE THERE DIFFERENT TYPES OF TESSELLATIONS? EXPLAIN YOUR ANSWER. http://www.mathacademy.com/pr/minitext/escher/index.asp http://www.coolmath.com/tesspag1.htm

11. Mara Alagic IMECT 3 Cambridge July 2002 Have Fun Creating Tessellations (Jaelyn & Barbie) Escher Web Sketch Link http://www-sphys.unil.ch/escher/

12. Mara Alagic IMECT 3 Cambridge July 2002 Cultural Influences (Jaelyn & Barbie) IN THE WEB SITES PROVIDED FIND 3 EXAMPLES OF TESSELLATIONS USED IN OTHER CULTURES. Cultural references http://mathforum.org/sum95/suzanne/historytess.html Cultural tilling examples http://www2.spsu.edu/math/tile/grammar/index.htm

13. Mara Alagic IMECT 3 Cambridge July 2002 Graphs and Slope Robin Racette  y = -4x b = 0; m = -4 -4/1 = down 4, right 1 4/-1 = up 4, left 1 2.y = -1/4x b = 0; m = -1/4 -1/4 = down 1, right 4 1/-4 = up 1, left 4

14. Mara Alagic IMECT 3 Cambridge July 2002 More Slope Robin Racette Try these on your own.  y = 1/4x - 4 b = -4, m = 1/4 2.y = -3x b = 0, m = 3 Graphing y = mx + b on the web

15. Mara Alagic IMECT 3 Cambridge July 2002 Adam the early bird Starts at age 22 Invests $3000 for 6 years At 11% interest Millionaire Meredith, LuAnne & Dan

16. Mara Alagic IMECT 3 Cambridge July 2002 Brenda the procrastinator Starts investing at age 28 Invests $3000 every year thereafter Until age 63 At 11% interest Millionaire Meredith, LuAnne & Dan

17. Mara Alagic IMECT 3 Cambridge July 2002 Who retires with the most? Meredith, LuAnne & Dan Adam? Brenda? And how much difference between the final results? Plan A and B.xls

18. Mara Alagic IMECT 3 Cambridge July 2002 Meredith, LuAnne, & Dan

19. Mara Alagic IMECT 3 Cambridge July 2002 Invests $18,000 Retires at age 63 with $1,128,249 Invests $108,000 Retires at age 63 with $1,265,947 Meredith, LuAnne, & Dan

20. Mara Alagic IMECT 3 Cambridge July 2002 Graph of Sin  and Csc  Sarah  Things to Consider Period? Amplitude? Asymptotes?  Differences? Between function and its inverse  Green Globs Green Globs

21. Mara Alagic IMECT 3 Cambridge July 2002 Sarah

22. Mara Alagic IMECT 3 Cambridge July 2002 Cos and Sin Wave Geometer Sketchpad Sarah

23. Mara Alagic IMECT 3 Cambridge July 2002 Tangent on a Circle B A time C D Tan D. Bennett 3/91 Tangent Tracer Geometer Sketchpad Sarah

24. Mara Alagic IMECT 3 Cambridge July 2002 What if I still need help?  Ask for help  Websites http://www.drmath.com/ http://www.gomath.com/  Play Green Globs  AND….. Sarah

25. Mara Alagic IMECT 3 Cambridge July 2002 TO BECOME AN EXPERT OF TRIG FUNCTIONS Sarah