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“ A → E ” “ A pproaching…but not E qual to…” PowerPoint PPT Presentation


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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 ” “ A pproaching…but not E qual to…”

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A e a pproaching but not e qual 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


A e a pproaching but not e qual to

y

4

x

Given y = 4(1 – e-x),

When x gets larger, what is y?

“Infinity-ness” !!


A e a pproaching but not e qual to

y

4

“Infinity-ness” !!

x

Learning Problem:

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


A e a pproaching but not e qual to

  • Approach

  • An interactive software tools “A→E”

  • Using perspective 3D visualization

  • Learning activities

  • Using this tool as the media for exploration


A e a pproaching but not e qual to

Input Panel

Press run to start…

Numerical Data Panel


A e a pproaching but not e qual to

Perspective 3D Visual Display

Press run to start…


A e a pproaching but not e qual to

Focus (close inspection)

Extend to infinity

Design Principles:

Perspective 3D Visual Display

Make explicit the experts’ experience!


A e a pproaching but not e qual to

Animation

Press run to start…


A e a pproaching but not e qual to

0

2.333

3.444.

5..66

10.44

0

0.1

0.2

0.3

0.4

Animation

Processing…


A e a pproaching but not e qual to

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…


A e a pproaching but not e qual to

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


A e a pproaching but not e qual to

Tight - coupling

y1

.

.

.

y1

.

.

.

.

.

.

X1

.

.

.

x1


A e a pproaching but not e qual to

Given the function

Y = 4(1- e-x)

When x = x1,

Y = 4(1- e-x1)

Tracking and Details

y1

.

.

.

y1

.

.

.

.

.

.

X1

.

.

.

x1


A e a pproaching but not e qual to

Press run to start…

Direct Manipulation


A e a pproaching but not e qual to

Press run to start…

Grid – interval analysis


A e a pproaching but not e qual to

  • Conclusion:

  • A tool will not be useful if used without

  • Meaningful learning activities

  • Passionate teachers

  • Right learning environment


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