TIME 2012
Technology and its Integration in Mathematics Education
10th Conference for CAS in Education & Research
July 10-14, Tartu, Estonia
Using TI-nspire CAS Technology in
Teaching Engineering Mathematics:
Calculus
Abstract
Using TI-nspire CAS Technology in Teaching Engineering Mathematics: Calculus
In September 2011, our engineering school has adopted the TI-nspire CAS CX calculator (and software) for all
new students entering bachelor level. With both platforms and integrated facilities, nspire CAS is becoming a
more complete solution than Voyage 200. Although we started to use this new device in the classroom, we will
face a transition period where both the TI Voyage 200 and the new CX handheld calculators can be found on
the students’ desks. This talk will demonstrate examples of how this technology is used in classrooms for our
courses in calculus.
The nspire CAS CX handheld with its faster CPU and extended graphing capabilities offers us even more options
for doing math in our classes. Here are some topics where we can benefit from the use of this new CAS
calculator in the single variable course: defining the derivative (at some point f (a)) and using a slider bar to
create the function f (x); solving equations, plotting a function defined by an integral and computing Taylor
series in a much faster way; the new possibility of animating some optimization problems. For the multiple
variables course, we can now work graphically with 2D vectors and plot multiple 3D surfaces; again, the faster
CPU allows us to face heavy optimization problems and easily compute multiple integrals. For both calculus
courses, the fact that we can now plot different 2D graphs in the same window is a great advantage for
teaching and understanding mathematical concepts.
As teachers, we are using this technology in two different and complimentary ways: we use it to introduce and
explore mathematical concepts and we also want our students to learn some skills in regards to technology.
This is now part of the curriculum and will be presented in the talk; also, we will show some examples (taken
from exams or take-home work) where the use of nspire CAS (or Voyage 200) technology is required. Pencil and
paper techniques are still important and students should also learn to validate manual results using these tools
and understand unexpected results obtained by CAS systems. We will show examples where the benefit of
using this approach is obvious.
Keywords TI-nspire CAS technology, calculator, calculus, optimization, animation.
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Overview
• About ETS : our students, our tools
• What’s new with nspire CAS?
(Since Voyage200)
• Examples in our calculus courses
– distance from a point to a parabola
– short exam question
– finding and plotting a space curve from 2
intersecting surfaces
– animation of vectors and homework question
• Conclusion
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About ETS : École de technologie supérieure





Engineering school in Montréal, Québec, Canada
Our students come from college technical programs
“Engineering for Industry”
More than 6300 students, 1500 new students each year
All maths teachers and students have the same calculator
and textbook
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About ETS : Our Tools
•
•
•
•
1999: TI-92 Plus CAS handheld
2002 : TI Voyage 200
2011 : TI-nspire CAS CX
Different softwares (Derive, Maple,
Matlab, DPGraph, Geogebra)
 Only CAS calculators are allowed
during exams.
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What’s New with TI-nspire?
Compared to Voyage 200
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What’s New with TI-nspire?
Compared to Voyage 200
•
•
•
•
2 platforms
managing documents
list and spreadsheet
faster processor
(better for solve, Taylor, special functions, …)
• some CAS improvements
• new graphical capabilities
• animations : powerful tool for teaching
• interactive geometry : « experimental mathematics »
• multiple 2D plot window
(functions, parametric, scatter plot, etc.)
• 3D parametric surfaces and curves (OS 3.2)
We will show how we can use some of these new features
in calculus courses
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First Example: Optimization Problem
Question
Which point on the parabola
2
y = x is closest to the point
P=(½,2)?
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A Classic Optimization Problem
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Ex. 1 Which point on the
parabola y = x2 is closest
to the point P=(½,2)?
1. Explore the problem
graphically
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Ex. 1 Which point on the
parabola y = x2 is closest
to the point P=(½,2)?
1. Explore the problem
graphically
2. Conceptualise the function
that should be minimised *
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Ex. 1 Which point on the
parabola y = x2 is closest
to the point P=(½, 2)?
1. Explore the problem
graphically
2. Conceptualise the function
that should be minimised *
3. Give the algebraic
definition of this function
4. Find the minimum value
using calculus
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Ex. 1 Which point on the
parabola y = x2 is closest
to the point P=(½, 2)?
1. Explore the problem
graphically
2. Conceptualise the function
that should be minimised *
3. Give the algebraic
definition of this function
4. Find the minimum value
using calculus
5. Compare algebraic and
graphical approaches
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Example 2: An exam question
Let C(t) be the concentration of salt in grams per liter
(g/L) of a liquid at time t, where t is measured in
hours:
C(t) = 1+(4t − t2) e−t
a) Plot the graph of C(t) for 0 ≤ t ≤ 5.
b) Find the moment, during the 5 first hours, where
the concentration is decreasing the most rapidly.
Use calculus to find the exact value.
Don’t forget to show your steps.
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Example 2: An Exam Question
Let C(t) be the concentration of salt in
grams per liter (g/L) of a liquid at time t,
where t is measured in hours :
C(t) = 1+(4t − t2) e−t
(a) Plot the graph of C(t)
for 0 ≤ t ≤ 5.
(b) Find the moment, during the 5 first
hours, where the concentration is
decreasing the most rapidly.
Use calculus to find the exact value.
Don’t forget to show your steps.
Technological Goals of Syllabus
• Define a function, plot its graph and
analyse it.
• Use the handheld for computing
derivatives and integrals.
• Solve an equation, symbolically and
numerically.
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Example 3: Intersection of Two Surfaces
With OS 3.2, we can now plot
parametric 3D curves and surfaces.

With V200, 3D plotting was
restricted to a SINGLE explicit
function of type z = z(x, y).
So plotting a sphere (spherical
coordinates) and a plane in the SAME
window can be done.

And if someone can find
parametric equations for the
curve of intersection of these
two surfaces, it can be plotted.
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Example 3: Intersection of Two Surfaces
Let us consider a sphere and a plane:
Students should be able to produce this:
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Example 3: Intersection of Two Surfaces
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Example 3: Intersection of Two Surfaces
They still need to find the parametric
equations themselves. Here are some:
(using the “completeSquare” function!)
And they can plot this curve:
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Example 3: Intersection of Two Surfaces
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Example 4: Position and Velocity Vectors
With this animation of a particle movement
we can observe that
• The position vector 𝑟 goes
from the origin to a point
on the trajectory.
• The starting point of the
𝑑𝑟
𝑑𝑡
velocity vector 𝑣 =
is at the end of the vector 𝑟
(convention).
• vector 𝑣 is tangent to the
trajectory.
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Example 4: Position and Velocity Vectors
• What will happen to the velocity vector (in red) at a
corner point?
• Let us find the exact x-value of this corner point.
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How to Create this Animation?
•
•
•
•
Parametric curve
Slider u
Scatter plot (3 points)
Vectors (by connecting
the points)
• Labels and colors
Scatter Plot
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Students’ Works (homework)
Question 1 a) Plot the trajectory 𝑟 = 85 𝑡 − sin 𝑡 , 1 − cos 𝑡 . Moreover, compute and
plot the position, velocity and acceleration vectors when x = 100, x = 500 and x = 550.
Define a vector function.
Solve a single equation.
Work with vectors and matrices.
r(t)[1, 1] : element of r(t) in row #1 and column #1.
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Students’ Works (homework)
Differentiate a vector function.
Use a CAS calculator: store variables and
functions.
Plot parametric 2D curves, points and
vectors.
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Students’ Works: errors
Non-respect of
the convention
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Students’ Works: errors
This velocity vector
is not tangent.
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Students’ Works: errors
Students showed by
computation that
𝑎
is constant.
But this norm
doesn’t seem constant
on the students’ graph:
why?
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Conclusion
• We appreciate this new technology.
– 2 platforms: very useful for us (handheld for exams)
– lessens the necessity to use different software
• New graphical possibilities:
– easier exploration
– easier validation
of results obtained by calculus … or without calculus!
However, we must be more specific when designing tasks.
• To promote teaching/learning with this technology:
– it is required in some common exam questions
– specific technological goals are added in the curriculum
• But we will never forget the beloved QWERTY keyboard
of the old Voyage200!
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Aitäh
Thank you
Merci
Kiitos
Дзякую
Gracias
Hvala
Aitäh
ευχαριστώ
Grazie
Dziękuję
Cпасибо
Danke
Děkuji
Chokrane
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