1. “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

2. 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” !!
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3. 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

4. Approach An interactive software tools “A→E”
Using perspective 3D visualization
Learning activities
Using this tool as the media for exploration
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5. Input Panel
Press run to start…
Numerical Data Panel
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6. Perspective 3D Visual Display
Press run to start…
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7. Design Principles: Perspective 3D Visual Display
Extend to infinity
Focus (close inspection)
Make explicit the experts’ experience!
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8. Animation
Press run to start…
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9. 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
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10. 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
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11. 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
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12. Tight - coupling y1 . X1 . y1 x1
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13. Tracking and Details y1 x1 Given the function Y = 4(1- e-x)
When x = x1,
Y = 4(1- e-x1) . X1 . y1 x1
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14. Press run to start…
Direct Manipulation
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15. Grid – interval analysis
Press run to start…
Grid – interval analysis
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16. Conclusion: A tool will not be useful if used without
Meaningful learning activities
Passionate teachers
Right learning environment
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