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“ 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”

“Approaching…but not Equal 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

Learning Problem:

How to help students to understand the asymptotic property of “approaching…but not equal to….”


  • Approach

  • An interactive software tools “A→E”

  • Using perspective 3D visualization

  • Learning activities

  • Using this tool as the media for exploration


Input Panel

Press run to start…

Numerical Data Panel


Perspective 3D Visual Display

Press run to start…


Focus (close inspection)

Extend to infinity

Design Principles:

Perspective 3D Visual Display

Make explicit the experts’ experience!


Animation

Press run to start…


0

2.333

3.444.

5..66

10.44

0

0.1

0.2

0.3

0.4

Animation

Processing…


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…


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


Tight - coupling

y1

.

.

.

y1

.

.

.

.

.

.

X1

.

.

.

x1


Given the function

Y = 4(1- e-x)

When x = x1,

Y = 4(1- e-x1)

Tracking and Details

y1

.

.

.

y1

.

.

.

.

.

.

X1

.

.

.

x1


Press run to start…

Direct Manipulation


Press run to start…

Grid – interval analysis


  • Conclusion:

  • A tool will not be useful if used without

  • Meaningful learning activities

  • Passionate teachers

  • Right learning environment


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