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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. 2 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 3 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 4 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. 5 What’s New with TI-nspire? Compared to Voyage 200 6 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 7 First Example: Optimization Problem Question Which point on the parabola 2 y = x is closest to the point P=(½,2)? 8 A Classic Optimization Problem 9 Ex. 1 Which point on the parabola y = x2 is closest to the point P=(½,2)? 1. Explore the problem graphically 10 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 * 11 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 12 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 13 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. 14 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. 15 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. 16 Example 3: Intersection of Two Surfaces Let us consider a sphere and a plane: Students should be able to produce this: 17 Example 3: Intersection of Two Surfaces 18 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: 19 Example 3: Intersection of Two Surfaces 20 Example 4: Position and Velocity Vectors With this animation of a particle movement we can observe that • 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. 21 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. 22 How to Create this Animation? • • • • Parametric curve Slider u Scatter plot (3 points) Vectors (by connecting the points) • Labels and colors Scatter Plot 23 Students’ Works (homework) Question 1 a) Plot the trajectory = 85 − sin , 1 − cos . 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. 24 Students’ Works (homework) Differentiate a vector function. Use a CAS calculator: store variables and functions. Plot parametric 2D curves, points and vectors. 25 Students’ Works: errors Non-respect of the convention 26 Students’ Works: errors This velocity vector is not tangent. 27 Students’ Works: errors Students showed by computation that is constant. But this norm doesn’t seem constant on the students’ graph: why? 28 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! 29 Aitäh Thank you Merci Kiitos Дзякую Gracias Hvala Aitäh ευχαριστώ Grazie Dziękuję Cпасибо Danke Děkuji Chokrane 30