1. Reasoning Activities Using 0.999... = 1 in Developmental Mathematics Chris L. Yuen, CLYUEN@buffalo.edu SUNY University at Buffalo November 13, 2014

2. Cross Examination O Inherent the Wind (1960) https://www.youtube.com/watch?v=l5Kdc0 LLSW8 O Witness for the Prosecution (1957) https://www.youtube.com/watch?v=Fq3UK 04pNrY https://www.youtube.com/watch?v=Fq3UK 04pNrY O Legally Blonde (2001) Start the clip at 3:14 https://www.youtube.com/watch?v=ytWGiO uzpe4

3. .999… as an Object of Interest O Why use.999…? Yopp et al. (2011) studied in-service teachers’ perception toward.999…

4. Another Finding…

5. More findings from Yopp

6. And More… O Some teachers modify “truth” to fit their subjective beliefs. O Some teachers dismiss conflict as unimportant or uninteresting as it relates to their teaching. O Some teachers teach that approximations are good enough and “small” does not matter.

7. Developmental Math O Much instruction directed to development math is pedagogically delivered differently from higher math. O Emphasis of procedures O Emphasis of formulas O Emphasis of performing well on traditional tests and exams

8. Proving.999… = 1

9. A Multiplication Proof

10. An algebraic proof

11. Proof by Limit

12. How to Engage Students? O The proofs alone are not going to motivate students. O Developmental students often do not know much about the concepts of “limits.” O One can show an application of proof in legal arguments – Reductio ad absurdum O So, involving a trial for the students to craft arguments may be a productive alternative.

13. What is Reductio ad absurdum? O Big Bang Theory https://www.youtube.com/watch?v=ytWGiOuz pe4 https://www.youtube.com/watch?v=ytWGiOuz pe4

14. Reductio ad absurdum O Reduction to absurdity O To demonstrate a statement is true by showing that a false, untenable, or absurd result follows from its denial O This sounds very much like “Proof by Contradiction

15. The Trial

16. Analysis of the Activity O The trial encompasses all the proofs that a developmental mathematics student could comprehend. O The trial co-create mathematical discourse with students in a social manner, as opposed to proving a mathematical fact in a solitary setting. O Mathematics can be system/postulate specific, and the court room setting is analogous to a specific system.

17. Application

18. Questions/Comments Contact Information: Chris L. Yuen, Ed.D., CLYUEN@buffalo.eduCLYUEN@buffalo.edu EOC Assistant Professor of Mathematics University at Buffalo Educational Opportunity Center 555 Ellicott Street Buffalo, NY 14203