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“ A → E ” “ A pproaching…but not E qual to…”

“ A → E ” “ A pproaching…but not E qual to…” A Dynamic Visualization Tool for introducing the Concept of Limits to Pre-calculus Students Yunn Chyi Chao. y. 4. x. Given y = 4(1 – e -x ), When x gets larger, what is y?. “Infinity-ness” !!. y. 4. “Infinity-ness” !!. x.

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“ A → E ” “ A pproaching…but not E qual to…”

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  1. “A→E” “Approaching…but not Equal to…” A Dynamic Visualization Tool for introducing the Concept of Limits to Pre-calculus Students Yunn Chyi Chao

  2. y 4 x Given y = 4(1 – e-x), When x gets larger, what is y? “Infinity-ness” !!

  3. y 4 “Infinity-ness” !! x Learning Problem: How to help students to understand the asymptotic property of “approaching…but not equal to….”

  4. Approach • An interactive software tools “A→E” • Using perspective 3D visualization • Learning activities • Using this tool as the media for exploration

  5. Input Panel Press run to start… Numerical Data Panel

  6. Perspective 3D Visual Display Press run to start…

  7. Focus (close inspection) Extend to infinity Design Principles: Perspective 3D Visual Display Make explicit the experts’ experience!

  8. Animation Press run to start…

  9. 0 2.333 3.444. 5..66 10.44 0 0.1 0.2 0.3 0.4 Animation Processing…

  10. 0 2.333 3.444. 3.666 3.777 3.8888 3.8899 0 0.1 0.2 0.3 0.4 2.4 344 Animation Processing…

  11. Animation Processing… 0 2.333 3.444. 3.666 3.777 3.8888 3.8899 3.900 3.991 3.6777 234455 0 0.1 0.2 0.3 0.4 2.4 344 34 4455 E3e 234

  12. Tight - coupling y1 . . . y1 . . . . . . X1 . . . x1

  13. Given the function Y = 4(1- e-x) When x = x1, Y = 4(1- e-x1) Tracking and Details y1 . . . y1 . . . . . . X1 . . . x1

  14. Press run to start… Direct Manipulation

  15. Press run to start… Grid – interval analysis

  16. Conclusion: • A tool will not be useful if used without • Meaningful learning activities • Passionate teachers • Right learning environment

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