Using TI-nspire CAS Technology in Teaching Engineering Mathematics:

Calculus

TIME 2012 Technology and its Integration in Mathematics Education

10th Conference for CAS in Education & ResearchJuly 10-14, Tartu, Estonia

Geneviève Savard Michel BeaudinGilles Picard Chantal Trottier

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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.

Abstract

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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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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

About ETS : École de technologie supérieure

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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?

• 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 featuresin calculus courses

Compared to Voyage 200

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First Example: Optimization Problem

Question Which point on the parabola y = x2 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 questionLet 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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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.

Example 2: An Exam Question

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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.

  

Example 3: Intersection of Two Surfaces

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Let us consider a sphere and a plane:

Students should be able to produce this:

2 2 2( 1) ( 3) 16x y z 2 2.x y z

Example 3: Intersection of Two Surfaces

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Example 3: Intersection of Two Surfaces

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They still need to find the parametric equations themselves. Here are some: 

(using the “completeSquare” function!)

And they can plot this curve:

2 690 cos( ) 115 sin( ) 1015

115 sin( ) 8 0 23

690 cos( ) 2 115 sin( ) 5

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t tx

ty t

t tz

Example 3: Intersection of Two Surfaces

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Example 3: Intersection of Two Surfaces

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Example 4: Position and Velocity Vectors

• The position vector goes from the origin to a point on the trajectory.

• The starting point of the velocity vector is at the end of the vector (convention).

• vector is tangent to the trajectory.

With this animation of a particle movement we can observe that

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• What will happen to the velocity vector (in red) at a corner point?

• Let us find the exact x-value of this corner point.

Example 4: Position and Velocity Vectors

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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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Question 1 a) Plot the trajectory 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.

Students’ Works (homework)

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Differentiate a vector function.

Use a CAS calculator: store variables and functions.

Plot parametric 2D curves, points and vectors.

Students’ Works (homework)

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Non-respect of the convention

Students’ Works: errors

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This velocity vector is not tangent.

Students’ Works: errors

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Students showed by computation that

is constant.

But this norm doesn’t seem constant on the students’ graph:

why?

Students’ Works: errors

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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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AitähThank you

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Chokrane

Michel Beaudin Geneviève Savard Gilles Picard Chantal Trottier

ÉTS, Montréal, Québec, Canada