“ a → e ” “ a pproaching…but not e qual to…”

04/19/23

“A→E”

“Approaching…but not Equal to…”

A Dynamic Visualization Tool for introducing the Concept of Limits to

Pre-calculus Students

Yunn Chyi Chao

04/19/23

Given y = 4(1 – e-x),

When x gets larger, what is y?

x

y4

“Infinity-ness” !!

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

x

y4

“Infinity-ness” !!

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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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Press run to start…

Input PanelInput Panel

Numerical Data Numerical Data PanelPanel

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Perspective 3D Visual Perspective 3D Visual DisplayDisplay

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Focus (close inspection)

Design Principles:

Perspective 3D Visual DisplayPerspective 3D Visual Display

Extend to infinity

Make explicit the experts’ experience!

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AnimationAnimation

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Processing…

00.10.20.30.4

02.3333.444.5..6610.44

AnimationAnimation

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Processing…

00.10.20.30.42.4344

02.3333.444.3.6663.7773.88883.8899

AnimationAnimation

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Processing…

00.10.20.30.42.4344344455E3e234

02.3333.444.3.6663.7773.88883.88993.9003.9913.6777234455

AnimationAnimation

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x1

y1

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.X1

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.y1

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Tight - couplingTight - coupling

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x1

y1

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.X1

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.y1

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Tracking and Tracking and DetailsDetails

Given the functionY = 4(1- e-x)When x = x1,Y = 4(1- e-x1)

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Direct Direct ManipulationManipulation

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Grid – Grid – interval analysisinterval analysis

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Conclusion:

A tool will not be useful if used without

• Meaningful learning activities• Passionate teachers• Right learning environment

“ a → e ” “ a pproaching…but not e qual to…”

Documents