Overview  About ETS: our students, our tools.  What’s New with TI-Npsire CAS?  Different Objects in the Same 2D Plot Window.  Why is Parametric 3D.

Slides:



Advertisements
Similar presentations
Vector Functions and Space Curves
Advertisements

Parametric Equations Local Coordinate Systems Curvature Splines
1 TIME 2012 Technology and its Integration in Mathematics Education 12th ACDCA Summer Academy July 10-14, Tartu, Estonia Adopting TI-Nspire CAS technology.
Exponential Functions Logarithmic Functions
Technology and its Integration into Mathematics Education July 6 th -10 th, 2010 E. T. S. I. Telecomunicaciones, Málaga, Spain.
Chapter 7: Vectors and the Geometry of Space
TIME 2012 Technology and its Integration in Mathematics Education 10 th Conference for CAS in Education & Research July 10-14, Tartu, Estonia.
17 VECTOR CALCULUS.
Lecture # 32 (Last) Dr. SOHAIL IQBAL
Chapter 16 – Vector Calculus
Intermediate Math Parametric Equations Local Coordinate Systems Curvature Splines.
Cylindrical and Spherical Coordinates
Vectors and the Geometry of Space 9. 2 Announcement Wednesday September 24, Test Chapter 9 (mostly )
Concepts and value of TI-Nspire™ Technology
TIME 2014 Technology in Mathematics Education July 1 st - 5 th 2014, Krems, Austria.
Geometry Equations of a Circle.
Concepts and value of TI-Nspire™ Technology Module A.
James Matte Nicole Calbi SUNY Fredonia AMTNYS October 28 th, 2011.
ACA 2014 Applications of Computer Algebra Session: Computer Algebra in Education Fordham University New York, NY, USA, July 9-12.
Concepts and value of TI-Nspire™ Technology Module A.
Math Menu: Using Nspire CAS in the Classroom Day 2.
Computer Graphics: Programming, Problem Solving, and Visual Communication Steve Cunningham California State University Stanislaus and Grinnell College.
Welcome to AP Calculus BC with Mathematica. You may be wondering: What will I learn? How will I be graded? How much work will I have to do? What are the.
To teach mathematics in the school of tomorrow Thomas Lingefjärd University of Gothenburg.
1.4 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for.
1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH.
TIME 2012 Technology and its Integration in Mathematics Education 10 th Conference for CAS in Education & Research July 10-14, Tartu, Estonia.
CS 376 Introduction to Computer Graphics 04 / 23 / 2007 Instructor: Michael Eckmann.
Section 17.5 Parameterized Surfaces
7.4 Lengths of Curves. 2+x csc x 1 0 If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the.
1.4 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for.
Module A Concepts and value of TI-Nspire™ Technology.
Plane Curves and Parametric Equations New Ways to Describe Curves (10.4)
Slide 9- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Introducing a fully integrated mathematics learning platform a fully integrated mathematics learning platform.
TI-NSPIRE™ TECHNOLOGY V. 3.0 RELEASE MATH IMPROVEMENTS February 2011.
1998 AB Exam. 1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH.
Syllabus for Analytic Geometry of Space It is an introductory course, which includes the subjects usually treated in rectangular coordinates. They presuppose.
1 Integrating CAS casmusings.wordpress.com Chris Harrow Atlanta, GA Twitter:
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
The Pluses and Minuses of Technology in a Math Classroom Madeline Dillner.
Geogebra Introduction. Geogebra Download and Installation
CS 376 Introduction to Computer Graphics 04 / 25 / 2007 Instructor: Michael Eckmann.
11/6/ :55 Graphics II Introduction to Parametric Curves and Surfaces Session 2.
Equations of Circles. Vocab Review: Circle The set of all points a fixed distance r from a point (h, k), where r is the radius of the circle and the point.
Cylinders and Quadric Surfaces
8.1 The Rectangular Coordinate System and Circles Part 2: Circles.
Parametric Surfaces and their Area Part I
Clicker Question 1 What is the area enclosed by f(x) = 3x – x 2 and g(x) = x ? – A. 2/3 – B. 2 – C. 9/2 – D. 4/3 – E. 3.
15 Copyright © Cengage Learning. All rights reserved. Vector Analysis.
CURVES Curves in cartesian coordinates Curves in 2D and 3D: explicit, implicit and parametric forms Arc length of a curve Tangent vector of a curve Curves.
TIME 2012 Technology and its Integration in Mathematics Education 10 th Conference for CAS in Education & Research July 10-14, Tartu, Estonia.
Then/Now You wrote equations of lines using information about their graphs. Write the equation of a circle. Graph a circle on the coordinate plane.
Calculus 3 The 3-D Coordinate System. The 3D coordinate plane.
Bell Ringer Solve even #’s Pg. 34.
Cylindrical and Spherical Coordinates
Copyright © Cengage Learning. All rights reserved.
Using the Real Power of Computer Algebra
Vector Functions and Space Curves
Static and dynamic surfaces representation
Section 17.1 Parameterized Curves
1.4 Parametric Equations Mt. Washington Cog Railway, NH.
Communication and Coding Theory Lab(CS491)
Geogebra: Dynamic Math Worksheets for the K-12 Classroom
1.4 Parametric Equations Mt. Washington Cog Railway, NH
The quadratic formula.
Cylindrical and Spherical Coordinates
Find the following limit. {image}
Presentation transcript:

Overview  About ETS: our students, our tools.  What’s New with TI-Npsire CAS?  Different Objects in the Same 2D Plot Window.  Why is Parametric 3D Plotting so Important?  Some Examples of Parametric 3D Plotting.  Conclusion. 2

 Engineering school in Montréal, Québec, Canada.  Our students come from college technical programs.  “Engineering for Industry”.  More than 7000 students, 1700 new students each year.  All math teachers and students have the same calculator and textbook. 3 About ETS: our Students, our Tools

1999: TI-92 Plus CAS handheld : TI Voyage : TI-nspire CX CAS. Different softwares (Derive, Maple, Matlab, DPGraph, Geogebra).  Only CAS calculators are allowed during exams. 4 About ETS: Our Tools

5 Compared to Voyage 200 What’s New with TI-Nspire CAS?

6 2 platforms managing documents list and spreadsheet faster processor (better for solving, Taylor polynomials, special functions, …) some CAS improvements new graphical capabilities animations : powerful tool for teaching interactive geometry : « experimental mathematics » multiple 2D plots window (functions, parametric, scatter plot, etc.) 3D parametric surfaces and curves (OS 3.2) Today, we will mainly focus on parametric 3D plotting. Compared to Voyage 200 What’s New with TI-Nspire CAS?

Different Objects in the Same 2D Plot Window  (In OS 3.2) The 2D plot window graph Entry/Edit accepts up to 7 different types but 2D implicit plots are still missing.  Slider bars, animations, dynamic geometry, styles and colors make each of these 2D plot windows very attractive and useful for teaching mathematics and sciences. 7

Different Objects in the Same 2D Plot Window  Slider bars/animations, dynamic geometry and calculus are now used by teachers for showing many concepts: this was never done before. Here is a simple example.  Two objects are moving in the plane, with respective given position at time t. When will the distance between them be the smallest?  The next slide shows the situation. 8

Different Objects in the Same 2D Plot Window 9

 A word on implicit plotting: it is a must for multiple variable calculus (but probably not so important for college level).  Here is a trick to “fake” implicit 2D plotting in Nspire CAS.  To plot the graph of f(x, y) = 0, use f1(x) = zeros(f(x, y), y). Of course, the equation should be solvable for y. 10

Different Objects in the Same 2D Plot Window  Moreover, the “complex” format must be selected.  Here are some examples.  Let us take a look at this circle:  And at this cubic polynomial equation: 11

Different Objects in the Same 2D Plot Window  Let us show, using Nspire CAS, how to plot the circle using different 2D plot windows.  And let’s also show how to plot the implicit curve 12

Why is Parametric 3D Plotting so Important?  First of all, where do I find the 3D plot window? And what can I plot?  How can I plot a surface defined by an equation of the form x = f(y, z) or by an equation of the form y = f(x, z)?  How can I plot a circular cylinder? A sphere? 13

Why is Parametric 3D Plotting so Important?  For the sphere, the “trick” using “zeros” can be used but the graph won’t be attractive…  Since OS 3.2, a new 3D plot type is available: PARAMETRIC.  One or 2 parameters? A space curve or a surface? 14

Why is Parametric 3D Plotting so Important?  Parametric 3D plotting is important because: It yields nice graphs! We need to use mathematics in order to get something. This shows students the importance of some trig identities, some formulae, some techniques. 15

Why is Parametric 3D Plotting so Important?  Parametric 3D plotting is important but students will never master it if they never use it by themselves!  This is why, at ETS, we want our students to use computer algebra in the classroom (with their handheld).  Of course, 3D plotting is easier on the software version. But students can learn the basics with the handheld. 16

Some Examples of Parametric 3D Plotting Planes: the general equation is of the form Ax+ By + Cz = D. If we can solve for z, we can use the “classic” editor “z1(x, y) = ”. If C = 0, then parametric plotting is used: x can be t, z can be u and y depends on x (t). 17

Some Examples of Parametric 3D Plotting We will show the graph of the 2 intersecting planes: The first plane can be defined as: xp1(t, u) = t yp1(t, u) = 2  t zp1(t, u) = u 18

Some Examples of Parametric 3D Plotting The second can also be described in parametric form (replacing x by t and y by u) or we can use the standard editor z1(x, y) = 2x  3y + 1. Here is the result. 19

Some Examples of Parametric 3D Plotting  To plot a circular cylinder like  The first trig identity can be used: xp1(t, u) = u yp1(t, u) =  1+ 2cos(t) zp1(t, u) =  2+ 2sin(t)

Some Examples of Parametric 3D Plotting  We want a nice plot of the sphere  We need to use spherical coordinates: 21

Some Examples of parametric 3D plotting  Students can produce animations: a “bumpy sphere” for example: 22

Some Examples of Parametric 3D Plotting  Here are some plots of space curves.  Suppose an object moves along the helix  Here are the helix and the position vector (a line segment). 23

Some Examples of Parametric 3D Plotting  Let us generate a torus: we will rotate a circle around the z-axis (a “rotation matrix” will be used).  The circle is centered at (3, 0, 0), in the plane y = 0, and has a radius of 1.  We should obtain this: 24

Some Examples of Parametric 3D Plotting 25

Some Examples of Parametric 3D Plotting  Let us obtain a space curve by intersecting 2 surfaces, a sphere and a plane.  The sphere will be and the plane will be 26

Some Examples of Parametric 3D Plotting  We will use the (very useful) function “Complete the Square”.  Moreover, students need to understand what an equation like “f(x, y) = 0” means in 2D and in 3D.  We should obtain this: 27

Some Examples of Parametric 3D Plotting 28

Some Examples of Parametric 3D Plotting  Let us obtain a space curve by intersecting 2 surfaces (no need of parametric plotting for the surfaces).  The surfaces will be a parabolic cylinder and a paraboloïd: 29

Some Examples of Parametric 3D Plotting  Before Nspire CAS, all of these last examples were done (by me) using Derive (and students could not do it themselves before Nspire CAS CX). 30

Conclusion  When “implicit 3D plotting” will be available, it will be easier to plot any surface defined by an equation of the form f(x, y, z) = 0.  But this is not so urgent because many important surfaces can be parametrized. 31

Conclusion  Also, when time comes to solve optimization problems (2 variables), the functions we are dealing with are of the form z = f(x, y).  Since OS 3.2, my students are much more familiar with cylindrical and spherical coordinates than before. 32

Conclusion  They plot more 3D graphs than before and are proud to do it themselves.  Mathematics ─ especially for future engineers ─ is a tool they will always need; and if it can have a little taste of “experimental sciences”, they don’t disagree! 33

Conclusion  The 3D plot window of Nspire CAS represents a big progress compared to what we used to have with V200.  But, it would be nice if TI could add the possibility to plot space curves with different styles (line weight: thin, medium, thick as we have in 2D). 34

Conclusion  Also, if someone plots a point in space (using parametric plotting), the point is almost invisible!  Unless you use the trick of plotting a “small” sphere!  But, in any case, this does not affect the pleasure we have using Nspire CAS!!! 35

Conclusion  Website for Nspire CAS at ETS (in French):  Many documents on using Nspire CAS in teaching can be found.  My personal homepage:  A library of functions is useful for engineering mathematics. 36

37