Computer Algebra in Education
ACA 2011, Houston, TX, USA, 27-30 June
TI-Nspire CAS for Teaching at University Level
Michel BEAUDIN
École de technologie supérieure
1100 Notre-Dame Street West, Montréal, Québec, Canada, H3C 1K3
[email protected]
ABSTRACT
Users of the Derive program were shocked when they learned that Derive would be
discontinued and replaced by a new product ─ it was at the TIME-2006 conference in
Dresden, Germany. This new product is TI-Nspire CAS, with 2 platforms: handheld and
software. In order to satisfy the needs of Derive users, two meetings were organized in
Stockerau, Austria (April 2007 and March 2008). It was a good opportunity to tell TIdevelopers which features of Derive should be included in future versions of Nspire CAS. At
the TIME-2008 conference in South Africa, many features (namely 3D plotting) were still
missing. In fact, the progress of Nspire CAS was a long achievement: the format of the
calculator could not be the one of Voyage 200 and it was difficult to find an alternative that
would satisfy the “old” Derive users. Also, at the TIME-2010 conference in Malaga, Spain,
many colleagues were not yet convinced by Nspire CAS: it was not clear at all if TI really
wanted to attract university level teachers or leave the space to the big CAS like Maple or
Mathematica.
It seems that the new handheld color screen offers many advantages and the new operating
system (OS 3) has the potential to attract those who are teaching engineering students, at
university level. The talk will show some of these advantages, using examples in multiple
variable calculus, differential equations and complex analysis.
1. Introduction
At ÉTS, a CAS calculator (TI-92 Plus, now Voyage 200 and, beginning September 2011, TINspire CX CAS) is used by every undergraduate student in the classroom since September 1999:
students are allowed to use it in (most) parts of all exams and many questions are based on their use.
So, an answer to the question “why do we need a CAS calculator?” will be given at the beginning of
this paper. Unfortunately, a CAS calculator like Voyage 200 can’t compete with computers, not only
for some heavy computations, but for 3D plotting for example. At ÉTS, we are still using CAS like
Maple and Derive. A dream would be to switch from the CAS calculator to the CAS on computer
with no effort. As a Derive user, I was able to do this easily with Voyage 200 and Derive: but if my
Voyage 200 was able to find the critical points of a scalar field, it took a long time to plot the surface
and analyze these points; or if my Voyage 200 was able to plot a slope field, there was no way to drag
the initial condition point and observe the effect on the solution. Now, with the CX CAS handheld,
this becomes possible! In other words, teaching mathematics at university level to engineering
students is now a reality with Nspire CAS (both software and handheld).
2. CAS at ÉTS: a brief recap of the last 20 years
The particular situation of ÉTS has been described in many talks (see, for example, [1] and
[3]). ÉTS stands for École de technologie supérieure, an engineering school in Montréal, Québec,
Canada. At the beginning of 1991, few mathematics teachers were starting to use CAS (namely
Derive by the author of this paper, Maple for some others). They were rapidly convinced by the
pedagogical capabilities of these software and they started to bring a computer in the classroom (or
moving with the students to the computer lab) in order to illustrate particular concepts in some courses
(namely calculus courses). When the TI-92 came out in 1996, many students at ÉTS decided to buy
their own unit: this machine was a CAS machine, so easy to use despite its power, and the famous
David Stoutemyer (one author of Derive) was the man behind the CAS engine. But the boost we really
needed to move to mandatory CAS calculator was the flash technology and the TI-92 Plus in summer
of 1999: no need to go to the computer lab (and use a computer) for computing Fourier coefficients or
plotting a slope field! Students can remain in the classroom at their own desk, using their own
machine and doing these computations so easily! Having only one machine helps a lot: resources can
be put into a website, exams questions can be updated with more complex calculations and students
are not able to communicate during exams using the CAS calculator. The following examples are
given in order to show the audience what kind of problems/questions have been asked to our students
during exams in the past 11 years while the CAS handheld was allowed.
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Example 1: question from the final exam in Mat 472 (“Linear Algebra and geometry of space” for
students in software engineering). A triangle has its vertices located at the following points:
( 2, 1), (1, 2) and (
, 2). A 120° rotation with respect to the point (2, 3) is applied. Find
the coordinates of the new vertices. What students still have to be able to do by hand is to
find the matrices needed to perform this operation. So, they still have to know that a
translation by the vector [ 2, 3] will be used first ─ of course, they will use homogeneous
coordinates because translation is not a linear operator ─, then a rotation matrix around the zaxis and, finally a translation by the vector [2, 3]. How will they organize their work? They
will use their CAS calculator and type the following matrices (some of them will have stored
these important matrices into their calculator):
M1
1 0
2
cos(120 )
0 1
3 , M2
sin(120 )
cos(120 )
0
0
0 0
1
12
3 2 0
1 0 2
0
3 2
12
0 , M3
0 1 3 .
1
0
0
1
0 0 1
sin(120 ) 0
The matrix product M=M3 M 2 M1 will do the job and the coordinates of the image triangle
will be M T where T is the matrix of the vertices (in homogeneous coordinates):
T=
2
1
1
1
2
2 .
1
1
1
I was using Derive to show this (figure 1a). Students will now be able to use Nspire CAS and
directly obtain the new vertices of the rotated triangle (figure 1b).
Figure 1a (Derive)
Figure 1b (Nspire CAS software)
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Example 2: in ODE, when an explicit solution to a particular DE is found, we often ask our students
to plot it and to compare with the numerical solution curve obtained by Euler’s method.
Unfortunately, Voyage 200 did not allow 2 kind of different curves to be plotted into the same
window.
This example will continue at example 4 where OS3 of Nspire CAS will be used
successfully!
Example 3: As we said earlier, Voyage 200 3D capabilities were quite limited but the CAS machine
has been used many times to check some answers found by using paper and pencil techniques. With
Voyage 200, you just can’t plot 2 surfaces at the same time and you are limited to z = f (x, y) explicit
function ─ for the moment OS3 is also limited to explicit functions but more than one can be plotted in
the same window and the speed of the processor makes analysis quite easy: also, David Parker would
add soon parametric 3D plotting and space curves. So how did we use Voyage 200 in multiple variable
calculus? Critical points of a 2 variables function were analyzed using the solve command but
because 3D graphics were so slow with Voyage 200, we were forced to use the CAS part of the
machine without graphical analysis (switching to Derive in my case). This is now possible with OS3
of Nspire CAS and example 5 will follow in this direction. Here, let us give a different example.
Students are sometimes asked to find parametric equations for the curve of intersection of 2 surfaces.
A concrete example is to find the parametric equation for the curve of intersection of a parabolic
cylinder and a paraboloid, namely
z
4
x2
2
5x
, z
4
x2
5 y2
4
6,
finding, first the projection in the xy-plane of this curve, using their trig identities and then, being able
to parametrize the space curve. All of these are done by hand!
The following figures outline the situation (done with Derive: figure 2a can now be illustrated using
Nspire CAS OS3; still waiting for space curves defined by parametric equations on figure 2b).
Figure 2a
Figure 2b
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The Voyage 200 device is helping them only after some of them have found these equations:
x
y
z
985
5
cos t
12
12
1182
sin t
12
985 2 5 985
sin t
cos t
288
72
0 t
2
73
72
Our students have laptops, cell phones and many other gadgets: but for maths and sciences, in
the classroom, they have been using the same device, since 1999: TI-92 Plus/Voyage 200. Probably
the biggest disadvantage was the fact that no communication between the device and a CAS running
on computer was possible (in fact, when Derive 5 came out in 2000, a communication with the TI-92
Plus was possible, quite limited: it never became popular and, with the discontinuation of Derive, in
2007, this dream ended).
Nspire CAS was introduced at the TIME-2006 conference in Dresden,
Germany. Proclaimed as the “successor of Derive”, Nspire CAS took a long time to convince “old
derivers”. But 5 years later, OS3 of Nspire CAS can be considered as a serious university level
computer algebra system: with 2 platforms, users can do the same manipulation/calculation ─ but it is
quite easy on the computer! ─ and they can transfer their files from the handheld to the software and
vice-versa.
Also, OS3 contains features absolutely needed in engineering mathematics (like
differential equations plotting and 3D plotting). With animations/geometry and programming
environment, Nspire CAS is not only devoted to college level teaching. This is why at ÉTS, we have
decided to move to this technology (the fact that Voyage 200 OS has been stopped at OS 3.10 since
July 2005 is also a reason). Students will be able to use their calculator during exams ─ no way to
communicate between them, so we don’t need to wait to have the internet access blocked ─, they will
be able to continue their work from the handheld to the computer at home using the software. In the
classroom, the teacher will be using the software with the handheld emulator, showing the students
how to use their own handheld, but using the big computer screen. The cost of a Voyage 200 handheld
was almost 300CAD and the cost of a Nspire CAS combo (software and handeld) will be less than
200CAD!
3. Using this technology: what we have, what is still missing, what is
coming.
In [1], we wrote the following (it was in fall 2008). “We need implicit 2D plotting, 3D
plotting (especially parametric 3D plotting); we also need numerical differential equations plotting”.
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For 2D implicit plotting, we still have to wait but OS 3 has 3D (explicit) plotting and, in a demo,
Gosia Brothers showed us that 3D parametric and implicit plotting should come soon ─ with David
Parker behind the 3D, everything is possible! Also, OS 3 now has differential equation plotting, using
either Euler or RK[3,2] method, exactly what was included in the Voyage 200. Slider bar (available in
Nspire since version 1.4) is now better with animation easy to perform. We wrote also: “Also, we were
so well served by the ability of Derive to integrate piecewise continuous functions that we just can’t
understand why this is not carried over Nspire.” It seems that this ability is lost and we think that it
will never be exported to Nspire CAS. We wrote: “Voyage 200 has many 2D plot windows, not only
one. This can also be an advantage”. Here, we can have also many 2D plot windows and, using
different pages, it is easy to switch from one to another. But also we can plot in the same window 2
different kind of curves ─ for example, Euler’ method produced a numerical curve and if the solution
of the DE is an explicit function, it can be plotted in the same window in order to compare with
Euler’s method: that was impossible with Voyage 200.
“Finally, the 14 digits limitation for
computation is not acceptable at university level. This limitation represents another illustration of
leaving higher mathematics subjects to the competitors.” Unfortunately, we are still limited to 14
digits: as long as the software version will be the same as the handheld, how could it be different?
Let us now give some concrete examples of how OS3 will be used, at university level, at ETS, when
teaching to engineering students.
Example 4: (continuation of example 2)
It is well known that, except for linear equations, when
the solutions of a first order ODE is available by usual techniques, it is given by an implicit equation,
almost all the time. This is why we are still asking for a 2D plot window that will allow implicit
plotting. So, for the moment, we will restrict to a separable ODE whose implicit solution can be
converted into 2 explicit functions ─ one of the 2 branches will be the “unique solution” according to
the existence and uniqueness theorem. Consider the following separable first order ODE and let’s
compare the implicit solution and the generated curve obtained by Euler’s method, in a neighbourhood
of the initial condition point (0, 3):
dy
dx
x2 e 7 x
,
6 y
y (0)
3.
The implicit solution can be shown here because we have plotted the 2 parts of the quadratic implicit
solution! But the new thing is that we can also use the same window, switching to DE graphic mode
and plot the numerical solution generated by Euler’s method (or even RK). Figure 3a shows the 2
parts of the implicit solution, where trace mode was used to recall the initial condition. And figure 3b
on the other screen shows the slope field along with Euler’s numerical solution.
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Figure 3a
Figure 3b
Here we can say that figure 3b is close to what Derive can do: when 2D explicit plotting will be
available, that will be nice!
Example 5: (continuation of example 3) Finding and classifying the critical points of the function of
2 independent variables is something calculus students learn in their multiple variable calculus course.
With Voyage 200, if a polynomial expression f of variables x and y was given, it was easy to find the
critical points because of the implemented Gröbner basis: so the device will be able to solve the
polynomial system ─ at least numerically ─
f
x
0,
f
y
0.
It would have been nice to visualize these critical points. With OS3 of Nspire CAS, this is possible
(but we still have to learn how to plot these points in the space: with Derive, simply highlight the point
and use the plot command!). The very simple example will make use of the new command “Complete
the Square” where a saddle point can be located directly by algebra: the next figure shows a
screenshot of Nspire CAS handheld and the expression we want to analyze:
Figure 4
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We can now assert that the scalar field 2 x 2
saddle point located at
5 7 217
, ,
4 6
24
5x 3 y2
7 y 10 has only one critical point, namely a
( 1.25,1.17, 9.04). This helps for finding an appropriate
window. The trace mode shows also the plane z = 9 intersecting the hyperboloid almost at the
critical point. Windows parameters used were: 3 < x < 2, 2 < y < 4, 13 < z < 5. For a better view,
figure 5 comes from the software instead of handheld.
Figure 5
But students, with their device will also have a good view:
Figure 6
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Example 6: Derive is famous when time comes to integrate symbolically piecewise continuous
functions (using the indicator CHI function to define it). This ability has been lost in Nspire. Let us
be more concise: Nspire CAS can integrate a piecewise continuous function as long as you don’t
multiply it by another one. So, if someone wants to define a “Fourier series” function like the one we
have in Derive, it won’t work symbolically if the signal is piecewise but it will work numerically.
And graphs are pretty and fast, even on the handheld. For example, let us use the square wave
f ( x)
1 0
x 1
1 1 x
2
f ( x 2)
f ( x).
Because the floor function is implemented into Nspire CAS (also the case in Voyage 200), we can
plot f defined as ( 1)floor( x ) . But we can also define f using the piecewise template ─ much more
easier than using the “when” command of Voyage 200 ─ and, in order to extend it periodically, we
can use the mod function, as shown below in figure 7a (note also that we have used each of 3 different
ways to define an expression or a function: finally the “:=” of Derive is allowed!). Figure 7b shows
both graphs f1 and f3: these are the same:
Figure 7a
Figure 7b
If we decide to define our own Fourier series, the device will compute the Fourier coefficients by
using approximate arithmetic. But the fast processor yields a satisfactory graph!
Figure 8
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4. Conclusion
Will Nspire CAS finally become the “successor of Derive”? Instead of asking this question,
let us say that, with OS3, Nspire CAS has reached an important step enabling it to be used at
university level. In this sense, we are looking forward to using it with great pleasure and we won’t
forget the Derive features that we would like to be added in future versions. Moreover, the “CAS
part” of the Nspire system is our favourite subject: we are working in collaboration with friends at TI
in order to make it better and better. The foundations are already very good. Future looks good!
It seems that TI has not decided to leave “higher mathematics to the competitors”. In fact, the
following example comes from complex analysis and shows that Nspire CAS has the potential to deal
with university level mathematics.
Example 7: Computation of a line integral in the complex plane, using residue integration method and
also the definition of line integral. Let C be the circle z
I
f ( z ) dz , where
f ( z)
2 , let us compute
cos( z ) sinh z
z2 1
C
3
.
Residue integration method and the Nspire CAS will give this:
I
2 i B1 B2 where B1 B2
e 7e 1
. So, I
16
e 7e
16
1
2 i
0.056i.
This last numerical value will be confirmed by the powerful numerical integrator of the system.
Figure 9a
Figure 9b
Figures 9 shows the use of the generalized series function with compute Laurent expansion, so we can
find directly the residues at the triple poles +1 and 1. Adding these 2 values we obtain the value of
the integral I (see figure 10a). And we confirm it numerically by choosing z
2eit , 0 t
2
for a
parametrization of the path (figure 10b):
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Figure 10a
Figure 10b
5. References
[1] Beaudin, M.. Teaching Mathematics with CAS to Future Engineers: Some Examples of What We
(Absolutely) Need. Lecture, TIME 2008 conference, Buffelspoort conference centre, South Africa.
22-26 September 2008.
[2] Beaudin, M. and Picard, G.. Teaching Mathematics wit to Engineering Student: To Use or Not To
Use TI-Nspire CAS. Lecture, TIME 2008 conference, Buffelspoort conference centre, South Africa.
22-26 September 2008.
[3] Beaudin, M. and Picard, G. Ten Years of Using Symbolic TI Calculators at ETS. Lecture,
Education session, ACA 2009 conference, Montreal, Qc, Canada, 25-28 June 2009.
[4] Brothers, G. and Stulens, K. Interactive Explorations of Mathematics with TI-Nspire Technology.
Lecture, TIME 2010 conference. E.T.S.I. Telecomunicaciones e Informatica, Malaga, Spain, 6-10
July 2010.
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