Presentation at BCAMT 2014, Guildford Park Secondary, Surrey, BC

Overview

How "fundamental" is the Fundamental Theorem of Calculus, and what does it really mean? In this session, I discuss a teaching approach that has helped students learn the FTC visually, conceptually and enthusiastically with the use of Dynamic Geometry. This session will challenge you to reflect upon the teaching and learning of functions, pre-calculus and calculus in engaging and "dynamic" ways.

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Flow of Presentation

0 Warm-up discussion and question

0 Introduction to DGE

0 Share my teaching approach:

0 Derivative and antiderivative functions

0 Area and distances

0 Area functions & FTC (Part 1 and 2)

0 Some reflections

0 Wrap-up & Questions

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Warm-up

On your white boards and in groups, discuss:

0 Talk about the concept of FTC:

0 What is it?

0 How did you learn it?

0 How would (did) you teach it?

0 How are these concepts connected?

0 Indefinite integrals, definite integrals, FTC etc.

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Warm Up (Cont’d)

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What is DGE?

0 DGE allows students to create and physically transform, through on-screen dragging, mathematical objects. Its dynamic nature enables students to attend to the mathematical relationships more readily.

0 Research shows that using DGE can be effective for teaching calculus due to its dynamic nature (Ferrara, Pratt, & Robutti, 2006). As one student puts it: “I never understood what it meant to say that the derivative of sinx is cosx until I saw it grow on the computer” (p.261).

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Why DGE?

0 The study of calculus is a study of change

0 Mathematics is “dynamic”; for example:

0 Linear & quadratic functions (Grades 10-12)

0 Max & min problems (Pre-Calc 11, 12 & Calculus)

0 Trigonometric functions in a unit circle (Pre-Calc 12)

0 Mathematics, i.e. the study of functions, can be represented in multiple ways.

0 However, traditionally, only algebraic representation is emphasised in our curriculum

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Teaching FTC: 1) Derivative & Antiderivative

0 Use DGE to show that a derivative function is not just an algebraic equation, but also a set of points and a graph that relate “x” and the tangent slope at “x”

0 Antiderivative is a function F(x) such that F’(x) = f(x)

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2) Area and Distances

0 An introductory topic: finding area under curve

0 Also: define “definite integral”, 𝑎𝑏𝑓 𝑥 𝑑𝑥 , as area under

curve

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3) Area-Functions and FTC

0 Making connections…

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3) Area-Functions and FTC

0 The area under f can be plotted as a function! This is just like the way we plotted tangent slopes at different values of x.

0 Let 𝐴 𝑥 = 𝑎

𝑥𝑓 𝑡 𝑑𝑡, “as x varies, the area under f varies.”

0 It turns out…

…that this area-function A(x) is the antiderivative of f!!!11

FTC (Part 1)

0 Hence, 𝑑

𝑑𝑥 𝑎𝑥𝑓 𝑡 𝑑𝑡 =

𝑑

𝑑𝑥𝐹 𝑥 = 𝐹′ 𝑥 = 𝑓(𝑥)

0 The relationship among the 3 functions: f ’(x), f(x) and F(x) can be represented geometrically and graphically:

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FTC (Part 2)

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0 Since: 𝑘𝑥𝑓 𝑡 𝑑𝑡 = F(x) + C

0 It follows that: 𝑎𝑏𝑓 𝑡 𝑑𝑡

= 𝑘𝑏𝑓 𝑡 𝑑𝑡 − 𝑘

𝑎𝑓 𝑡 𝑑𝑡

= F(b) + C – [F(a) + C]

= F(b) – F(a)

Reflecting on Lessons

0 “Calculus is the study of change. It is different from math because it is dynamic and involves the concept of time.”

0 “In calculus, we work with rates of change and not the static value.”

0 “Calculus uses limits, where we can use numerical ways to approximate the solution (graphing limits and tangents), and also get the answer algebraically.”

0 “I knew more about the nature of calculus and I knew more about why calculus is created and why they are important.”

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Reflecting on Lessons (Cont’d)

0 “As we went along, we found that unlike courses like math and physics, where you follow a set of prescribed rules, calculus requires the understanding of a few important concepts, from which virtually any problem can be solved.

0 “We realized that there is actually real life situations that only calculus can solve.”

0 “Understanding the concepts is very easy, as well as interesting. Even though we understand everything perfectly conceptually, we find that our biggest problem on tests and quizzes is making simple arithmetic errors.

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Try it out

0 iPads Passcode: 1-1-1-1

http://www.sfu.ca/people/oilamn.html

0 Resources for Teachers

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Wrap-up

I have found that the use of DGE for teaching Pre-calculus and Calculus to be effective for:

0 Fostering dynamic thinking

0 Building connections and multiple representations

Also, students learn math:

0 Visually and conceptually

0 Interactively and engaged

See: Video assignment on constructing derivative of sine

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Wrap-up (Problem revisited)

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Questions and Comments?

Anything else?

Thank You!

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Resources

0 Contact me:

0 http://ilikemath.co.nr

0 oilamn@sfu.ca

0 GeoGebra Tube: http://geogebratube.com/

0 Sketch Exchange: http://sketchexchange.keypress.com/

0 Desmos: http://desmos.com

0 Google Search with keyword: [math topic] + “applet, GSP, java, ggb, geogebra”

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