“Approaching…but not Equal to…” “A→E” “Approaching…but not Equal to…” A Dynamic Visualization Tool for introducing the Concept of Limits to Pre-calculus Students Yunn Chyi Chao 11/14/2018

When x gets larger, what is y? Given y = 4(1 – e-x), When x gets larger, what is y? x y 4 “Infinity-ness” !! 11/14/2018

Learning Problem: How to help students to understand the asymptotic property of “approaching…but not equal to….” x y 4 “Infinity-ness” !! 11/14/2018

Approach An interactive software tools “A→E” Using perspective 3D visualization Learning activities Using this tool as the media for exploration 11/14/2018

Input Panel Press run to start… Numerical Data Panel 11/14/2018

Perspective 3D Visual Display Press run to start… 11/14/2018

Design Principles: Perspective 3D Visual Display Extend to infinity Focus (close inspection) Make explicit the experts’ experience! 11/14/2018

Animation Press run to start… 11/14/2018

Animation Processing… 11/14/2018 0.1 2.333 0.2 3.444. 0.3 5..66 0.4 0.1 0.2 0.3 0.4 2.333 3.444. 5..66 10.44 11/14/2018

Animation Processing… 11/14/2018 0.1 2.333 0.2 3.444. 0.3 3.666 0.4 0.1 0.2 0.3 0.4 2.4 344 2.333 3.444. 3.666 3.777 3.8888 3.8899 11/14/2018

Animation Processing… 11/14/2018 0.1 2.333 0.2 3.444. 0.3 3.666 0.4 0.1 0.2 0.3 0.4 2.4 344 34 4455 E3e 234 2.333 3.444. 3.666 3.777 3.8888 3.8899 3.900 3.991 3.6777 234455 11/14/2018

Tight - coupling y1 . X1 . y1 x1 11/14/2018

Tracking and Details y1 x1 Given the function Y = 4(1- e-x) When x = x1, Y = 4(1- e-x1) . X1 . y1 x1 11/14/2018

Press run to start… Direct Manipulation 11/14/2018

Grid – interval analysis Press run to start… Grid – interval analysis 11/14/2018

Conclusion: A tool will not be useful if used without Meaningful learning activities Passionate teachers Right learning environment 11/14/2018