MaplePrimes - Questions and Posts tagged with teaching
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSat, 23 Aug 2014 03:34:36 GMTSat, 23 Aug 2014 03:34:36 GMTThe most recent questions and posts on MaplePrimes tagged with teachinghttp://www.mapleprimes.com/images/mapleprimeswhite.jpgMaplePrimes - Questions and Posts tagged with teaching
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Interesting application
http://www.mapleprimes.com/posts/200172-Interesting-Application?ref=Feed:MaplePrimes:Tagged With teaching
<p>I'd like to pay attention to <a href="http://www.maplesoft.com/applications/view.aspx?SID=144592">an application "Periodicity of Sunspots " by Samir Khan</a>, where a real data is analysed. That application can be used in teaching statistics.</p>
<p>PS. The code by Samir Khan works well for me.</p><p>I'd like to pay attention to <a href="http://www.maplesoft.com/applications/view.aspx?SID=144592">an application "Periodicity of Sunspots " by Samir Khan</a>, where a real data is analysed. That application can be used in teaching statistics.</p>
<p>PS. The code by Samir Khan works well for me.</p>200172Fri, 24 Jan 2014 06:05:09 ZMarkiyan HirnykMarkiyan Hirnyk14 Additional Clickable Calculus Solutions Posted
http://www.mapleprimes.com/maplesoftblog/152504-14-Additional-Clickable-Calculus-Solutions-Posted?ref=Feed:MaplePrimes:Tagged With teaching
<p>Fourteen Clickable Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> area of the Maplesoft web site. Four are sequence and series explorations taken from algebra/precalculus, four are applications of differentiation, four are applications of integration, and two are problems from the lines-and-planes section of multivariate calculus. By my count, this means some 111 Clickable Calculus examples have now been posted to the section.<p>Fourteen Clickable Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> area of the Maplesoft web site. Four are sequence and series explorations taken from algebra/precalculus, four are applications of differentiation, four are applications of integration, and two are problems from the lines-and-planes section of multivariate calculus. By my count, this means some 111 Clickable Calculus examples have now been posted to the section.</p>
<p>Two of the applications of differentiation are related-rate problems: a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=105">sliding-ladder</a> and a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=106">shadow problem</a>. The sliding-ladder problem is a classic, in which the calculus enters after the correct expression to differentiate has been found. (Readers might also recall the <a href="http://www.maplesoft.com/applications/view.aspx?SID=148714">The Sliding Ladder, a Classroom Tips and Techniques article</a> that appeared in the June, 2013, Maple Reporter. In that article, the trajectory of an arbitrary point on the sliding ladder was found and graphed.)</p>
<p>The shadow problem asks for the rate at which a shadow, cast by a person walking away from a light, grows. A variant, not considered, can ask for the speed at which the head of the shadow moves along the ground. But again, what starts out as an illustration of the utility of the derivative often ends up as an algebraic challenge where the bottleneck is formulating the underlying model.</p>
<p>In the example "<a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=104">Minimize Area of Triangle</a>", the area of the triangle whose vertices are the origin, and the intersection of a (movable) line with a parabola, is to be minimized. Finding the two intersections, and writing an expression for the area of a triangle in terms of its vertices, is algebra; the minimization step is the intended application of the calculus. The situation is a bit better for the example "<a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=103">Curvature of an Ellipse</a>" in which the calculation of the curvature requires differential calculus to obtain the underlying model, and then continues using the calculus to find the points of maximum and minimum curvature along the ellipse.</p>
<p>The four applications of integration are more about evaluating definite integrals than about model-building. <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=107">Calculating the average value of a function</a>, verifying the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=108">Mean Value theorem for integrals</a>, finding <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=109">arc length</a> and the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=110">surface area of a surface of revolution</a> - these applications test integration skills more than they test algebraic and geometric training.</p>
<p>The two examples from the lines-and-planes section of the multivariate calculus course require the construction of a plane satisfying given conditions. Here, these planes are found with the standard vector and algebraic tools used in the typical calculus text. However, <a href="http://www.maplesoft.com/applications/view.aspx?SID=144642">New Tools for Lines and Planes</a>, the Classroom Tips and Techniques article in the March, 2013, Maple Reporter, shows how these, and other exercises from this section of the course can be solved with a set of tools new to Maple 17. Constructors for a line or plane object, and manipulators for these objects are available through the Context Menu system so that the standard exercises in this section of the multivariate calculus course can be solve with syntax-free calculations.</p>152504Fri, 04 Oct 2013 01:07:46 ZRobert LopezRobert LopezAdditional Clickable Calculus Solutions Posted
http://www.mapleprimes.com/maplesoftblog/149841-Additional-Clickable-Calculus-Solutions-Posted?ref=Feed:MaplePrimes:Tagged With teaching
<p>Thirteen Clickable Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts/">Teaching Concepts with Maple</a> section of the Maplesoft web site. The additions include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, and linear algebra. By my count, this means some 97 Clickable Calculus examples are now available.</p>
<p>In the Algebra/Precalculus section, examples of an <p>Thirteen Clickable Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts/">Teaching Concepts with Maple</a> section of the Maplesoft web site. The additions include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, and linear algebra. By my count, this means some 97 Clickable Calculus examples are now available.</p>
<p>In the Algebra/Precalculus section, examples of an <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=87">induction proof</a>, the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=88">binomial theorem</a>, and an <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=86">inverse function</a> have been added. For differential calculus, we've added an example of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=91">finding an antiderivative</a>, and two related-rates examples, one of which is a version of the classic <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=90">"Conical Sand-Pile" problem</a>. For integral calculus, three examples of separable differential equations appear, two of which arise from the standard models: <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=93">Newton's Law of Cooling</a>, and <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=94">Logistic Growth</a>.</p>
<p>The three new examples in multivariate calculus (<a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=95">Plane through Three Points</a>, <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=96">Plane from Point and Normal</a>, and <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=97">Plane Containing a Direction and Two Points</a>) are taken from the "lines and planes" section that generally opens the typical course in this subject. The solutions given use the expected vector methods, algebraic methods, or a task template. However, in Maple 17 there are new commands for defining and manipulating lines and planes, commands that have been fully integrated into the Context Menu system. For an illustration of how standard exercises from this section of the multivariate calculus course can be solved with these new tools, see the Tips and Techniques article avaialble <a href="http://www.maplesoft.com/applications/view.aspx?SID=144642">here</a> in the Maple Application Center.</p>
<p>The final addition, in linear algebra, shows how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=98">simultaneously diagonalize two symmetric real matrices</a> A and B, one of which is positive definite, and then details the connection to the generalized eigenvalue problem <img style="vertical-align: middle;" src="/view.aspx?sf=149841/465943/Capture.JPG" alt="">. The algorithm used starts with finding a Cholesky decomposition of the positive-definite matrix, and is taken from the text Numerical Analysis, 3rd edition, by Melvin J. Maron and Robert J. Lopez, Wadsworth, 1991. </p>149841Thu, 25 Jul 2013 18:04:15 ZRobert LopezRobert Lopezannouncement of textbook for mathematics with Maple, new edition
http://www.mapleprimes.com/posts/143488-Announcement-Of-Textbook-For-Mathematics?ref=Feed:MaplePrimes:Tagged With teaching
<p> A powerful approach to the teaching and learning of mathematics for students of science and engineering has been made practical through the development of powerful general mathematical software, of which Maple provides the least steep learning curve. Accordingly, it is timely to produce an interactive electronic textbook that, for students of chemistry -- also biochemistry and chemical engineering, has as its objective in part I,<br>Mathematics for...<p> A powerful approach to the teaching and learning of mathematics for students of science and engineering has been made practical through the development of powerful general mathematical software, of which Maple provides the least steep learning curve. Accordingly, it is timely to produce an interactive electronic textbook that, for students of chemistry -- also biochemistry and chemical engineering, has as its objective in part I,<br>Mathematics for Chemistry, to teach all the mathematics that an instructor of chemistry might wish his undergraduate students to learn and to understand on the basis of courses typically delivered in departments of<br>mathematics and statistics. Of nine chapters in part I, the titles are<br> 0 Exemplary illustrations of use of Maple<br> 1 Numbers, symbols and elementary functions<br> 2 Plotting, geometry, trigonometry, complex analysis and functions<br> 3 Differentiation<br> 4 Integration<br> 5 Calculus with multiple independent variables<br> 6 Linear algebra<br> 7 Differential and integral equations<br> 8 Probability, statistics, regression and optimization<br>The content of these chapters is almost entirely mathematical, with some examples and exercises having a chemical or physical basis.<br> In part II, intended to serve as Mathematics of Chemistry, in the sense of the standard volumes by Margenau and Murphy, the intention is to present the mathematical basis of several topics typically taught within<br>chemistry courses at undergraduate or post-graduate level. The titles of chapters available in Part II within edition 4.0 are<br> 9 Chemical equilibrium<br> 10 Group theory<br> 11 Graph theory<br> 12 Introduction to quantum mechanics<br> 13 Introduction to molecular optical spectrometry<br>Further content to be added to Part II is under active development.<br> This book is made available, gratis, through Centre for Experimental and Constructive Mathematics at Simon Fraser University, Burnaby British Columbia Canada at <a href="http://www.cecm.sfu.ca/research/chemistry.html">http://www.cecm.sfu.ca/research/chemistry.html</a> and requires a recent release of Maple for its use.<br><br><br></p>143488Thu, 14 Feb 2013 22:36:50 ZJ F OgilvieJ F OgilvieCan anyone suggest some non-routine class projects for a Calculus II course using Maple?
http://www.mapleprimes.com/questions/141151-Can-Anyone-Suggest-Some-Nonroutine?ref=Feed:MaplePrimes:Tagged With teaching
<p>I'm looking for several challenging projects for a Calculus II course using Maple which I'll be teaching in Spring 2013. By challenging I mean that the project will have several steps including both conceptual and computational aspects. I'd also like them to be suitable for group work if possible. I'd be willing to devote from 3-5 class periods for each of three or four different projects. Our Calc I and Calc II are five hour...<p>I'm looking for several challenging projects for a Calculus II course using Maple which I'll be teaching in Spring 2013. By challenging I mean that the project will have several steps including both conceptual and computational aspects. I'd also like them to be suitable for group work if possible. I'd be willing to devote from 3-5 class periods for each of three or four different projects. Our Calc I and Calc II are five hour courses which meet daily for 50 minutes. Topics for these projects could be applications of integrals, techniques of integration, first order separable d.e.'s, series, and polar coordinates and conics, i.e. all of the standard Calc II projects.</p>141151Thu, 06 Dec 2012 00:30:35 Zjheidel@unomaha.edujheidel@unomaha.eduYet more Clickable Calculus solutions!
http://www.mapleprimes.com/maplesoftblog/141070-Yet-More-Clickable-Calculus-Solutions?ref=Feed:MaplePrimes:Tagged With teaching
<p>Ten more Clickable Calculus solutions have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft web site. Solutions to problems include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, linear algebra, and vector calculus.<br><br>The algebra additions include an example illustrating how a <p>Ten more Clickable Calculus solutions have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft web site. Solutions to problems include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, linear algebra, and vector calculus.<br><br>The algebra additions include an example illustrating how a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=76">piecewise function</a> can be defined and manipulated in Maple, and an example showing how to graph a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=77">curve defined parametrically</a>. In differential calculus, we explore <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=78">Newton's method</a> for root-finding; look at the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=79">approximations by means of the differential</a>; and examine the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=80">Mean Value theorem</a>. In integral calculus, we look at two of the "big three" methods of integration, namely, <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=82">partial fractions</a> and <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=81">trig substitution</a>. In multivariate calculus, we've added the problem of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=83">finding the distance from a point to a line</a>, and in linear algebra, we've added an example on the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=84">generalized eigenvalue problem</a>. Finally, in vector calculus, we've added an example showing how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=85">find a vector potential</a>.<br><br>For the most part, these examples continue to unfold the pedagogical approach of "resequencing of skills and concepts." Where possible, the most direct Maple solution is presented first, followed by a Maple implementation of any underlying algorithms. Direct solutions include calculations in task templates, Assistants, Tutors, or Context Menu. The implementation of algorithms is by point-and-click techniques, so the additional insights generated are syntax-free. In all these solutions, the emphasis remains on making the concept clear, without having to first learn a software tool.</p>141070Tue, 04 Dec 2012 01:02:23 ZRobert LopezRobert LopezTeaching Calculus with Maple: A Complete Kit
http://www.mapleprimes.com/posts/139346-Teaching-Calculus-With-Maple-A-Complete-Kit?ref=Feed:MaplePrimes:Tagged With teaching
<p>We have just released Teaching Calculus with Maple: A Complete Kit. Leveraging both Maple and Maple T.A., <em>Teaching Calculus with Maple</em> includes lecture notes, student worksheets, Maple demonstrations, Maple T.A. homework, and more – everything you need to teach Calculus 1 and Calculus 2. <em>Teaching Calculus with Maple</em> was developed at the University of Guelph under the leadership of an award-winning teacher and field-tested in classes with hundreds of students.<p>We have just released Teaching Calculus with Maple: A Complete Kit. Leveraging both Maple and Maple T.A., <em>Teaching Calculus with Maple</em> includes lecture notes, student worksheets, Maple demonstrations, Maple T.A. homework, and more – everything you need to teach Calculus 1 and Calculus 2. <em>Teaching Calculus with Maple</em> was developed at the University of Guelph under the leadership of an award-winning teacher and field-tested in classes with hundreds of students.</p>
<p>This is a free product. For more information and to download a copy, visit <a href="http://www.maplesoft.com/contact/webforms/CalculusKit.aspx?ref=mapleprimes">Teaching Calculus with Maple.</a></p>139346Thu, 08 Nov 2012 23:58:57 ZbryonbryonA Simple Thank You
http://www.mapleprimes.com/maplesoftblog/138227-A-Simple-Thank-You?ref=Feed:MaplePrimes:Tagged With teaching
<p>Recently, a Maplesoft customer service representative received an e-mail from one of our users with the subject line: A Simple Thank You. We wanted to share this message with you, as it demonstrates how the power and flexibility of Maple helped one student get ahead in his studies.<br> <br>The following is an actual email we received from Eli E., which describes his experience using Maple as a university student.</p>
<p style="padding-left: 30px;"><em>Hello, my name is Eli...</em><p>Recently, a Maplesoft customer service representative received an e-mail from one of our users with the subject line: A Simple Thank You. We wanted to share this message with you, as it demonstrates how the power and flexibility of Maple helped one student get ahead in his studies.<br> <br>The following is an actual email we received from Eli E., which describes his experience using Maple as a university student.</p>
<p style="padding-left: 30px;"><em>Hello, my name is Eli E. and I am a junior Mathematics undergraduate at Transylvania University. I have been using your Maple products since entering college.</em><br> <br><em>I now consider Maple to be a better interpreter than its competitors. I have bought and read Maple 5: First leaves / Language reference manual, and learned how to write Maple procedures, and they are beautiful in their design. I had not taken any Pascal, which is supposed to be very close in design to Maple's language, but I have programmed in LisP and C, as well as QBasic, and I think the Maple procedure language is a fantastic unity of simplicity in structure, and relatability in design. </em><br><br><em>I have also seen the power of the Maple kernel! There were textbooks that could not do certain problems in [competitor product], they would refer to a more powerful kernel--namely Maple! This was an enlightening prospect for someone thinking that other interpreters could do better--a common theme in my university class.</em><br><br><em>Anyway, I would just like to thank the development team for the power of Maple, and the immense(!) help filesection of Maple. It is an irreplaceable tool for the precocious and perspicacious. This is a program I plan to use for years to come.</em></p>
<p>We hope this user story illustrates the potential Maple has for providing an innovative and enriching experience for students that allows them to become more engaged in learning.<br> <br>If you’ve had similar experiences, we’d love to hear from you! Please feel free to email <a href="mailto:customerservice@maplesoft.com">customerservice@maplesoft.com</a> or post your experiences here on MaplePrimes.</p>138227Fri, 12 Oct 2012 00:29:55 ZKathleen McNicholKathleen McNicholImplicitplot3d bounded portion of surface
http://www.mapleprimes.com/questions/137603-Implicitplot3d-Bounded-Portion-Of-Surface?ref=Feed:MaplePrimes:Tagged With teaching
<p>I want to graph the portion of the plane 2x + 3y + z = 6 that are located in the first octant of a xyz coordinate system. The following implicitplot3d should in principle do it:</p>
<p><p>I want to graph the portion of the plane 2x + 3y + z = 6 that are located in the first octant of a xyz coordinate system. The following implicitplot3d should in principle do it:</p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=616d2842c57ba793a4f6951459e92e91.gif" alt="implicitplot3d(piecewise(`and`(`and`(x >= 0, y >= 0), z >= 0), 2*x+3*y+z = 6, undefined), x = -3 .. 3, y = -3 .. 3, z = -1 .. 7, color = red, axes = normal, scaling = constrained, style = surface)"></p>
<p>When I execute the command I actually see the expected triangular part of the plane for a split second, but then it disappears. I can of course solve the equation with respect to z and use plot3d, which works fine. The problem is that I want to use implicitplot3d to illustrate the concept of implicit surfaces to the students that I am teaching. So for pedagogical I want to keep the Maple code as close as possible to the way that the problem is stated in the text book. What I mean is that I am not looking for technical workarounds that tend to move the focus away from the core mathematics.</p>
<p><a href="/view.aspx?sf=137603/443190/Graph_bounded_plan.mw">Graph_bounded_plan.mw</a></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -71;" src="/view.aspx?sf=137603/443190/8c669b5fe20ea88a29e9b39508fa9a6c.gif" alt="implicitplot3d(piecewise(`and`(`and`(x >= 0, y >= 0), z >= 0), 2*x+3*y+z = 6, undefined), x = -3 .. 3, y = -3 .. 3, z = -1 .. 7, color = red, axes = normal, scaling = constrained, style = surface)" width="576" height="104" align="middle"></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -71;" src="/view.aspx?sf=137603/443190/e1a2a9d95a9c392d52a52960c42fea43.gif" alt="plot3d(piecewise(`and`(`and`(x >= 0, y >= 0), 6-2*x-3*y >= 0), 6-2*x-3*y, undefined), x = -3 .. 3, y = -3 .. 3, view = -1 .. 7, color = red, axes = normal, scaling = constrained, style = surface)" width="576" height="104" align="middle"></p>
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<p><a href="/view.aspx?sf=137603/443190/Graph_bounded_plan.mw">Download Graph_bounded_plan.mw</a></p>137603Sat, 22 Sep 2012 13:10:51 Zhsogaardhsogaard11 New Clickable Calculus Examples added to Teaching Concepts with Maple
http://www.mapleprimes.com/maplesoftblog/137377-11-New-Clickable-Calculus-Examples-Added?ref=Feed:MaplePrimes:Tagged With teaching
<p>Eleven new Clickable-Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft website. That means some 74 of the 154 solved problems in my data-base of syntax-free calculations are now available. Once again, these examples and associated videos illustrate point-and-click computations in support of the pedagogic message of resequencing skills and concepts.<br><br>This message has been articulated in ...<p>Eleven new Clickable-Calculus examples have been added to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft website. That means some 74 of the 154 solved problems in my data-base of syntax-free calculations are now available. Once again, these examples and associated videos illustrate point-and-click computations in support of the pedagogic message of resequencing skills and concepts.<br><br>This message has been articulated in <a href="http://www.mapleprimes.com/maplesoftblog/134197-Teaching-Concepts-With-Maple">earlier blog posts</a>. Suffice it to say here that Maple's technologies allow concepts to be taught first, before emphasis is placed on manipulative skills. Moreover, Maple can be used to implement stepwise solutions so that the relevance of such skills becomes clear. In my 15 years' experience with Maple in the classroom, this proved to be an effective strategy, one that helped students master mathematics more efficiently than any other paradigm I had previously used.<br><br>The new additions to the site include two problems in algebra, and two in trig, three in differential calculus, two in multivariate calculus, and one each in integral calculus and linear algebra. In algebra, we obtain the coordinates of the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=66">intersection of two parabolas</a> as a way of learning how to solve pairs of quadratic equations. A second algebraic problem is the analysis of a complete <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=67">quadratic relation</a> whose graph is an ellipse.</p>
<p>For trigonometry, we show how to invert graphically the inverse of a trig function, resulting in a graph of the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=68">principle branch of that function</a>. Then, the Standard Functions tutor is used to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=69">illustrate the effects</a> of varying <em>a, b, h, k</em> in the transformation <em>a f(b (x - h)) + k </em>for<em> f</em>, any one of the elementary trig functions.</p>
<p>In differential calculus, the Rational Function tutor illustrates <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=70">graphing a rational function</a> and finding its asymptotes. <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=71">Taylor polynomials</a> are accessed from the Context Menu and in the Taylor Approximation tutor. The <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=72">optimization problem</a> of finding a point on <em>f(x) = sinh(x) - x e<sup>-3x</sup></em> closest to the point (1, 7) is solved. But linked to this blog is another post that <a href="http://www.mapleprimes.com/maplesoftblog/137375-Further-Analysis-Of-A-Minimization-Problem">explores that problem further</a>, past just the "solution."</p>
<p>For integral calculus, we again tackle the question of "What does Maple do for methods of integration?" when we show how the Integration Methods tutor helps implement the method of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=73">trigonometric substitution</a>. For the lines-and-planes section of multivariate calculus, one problem shows how to implement the equation of a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=50">line determined by a point and a given direction</a>. Another problem comes to grips with <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=74">skew lines</a>.<br><br>Finally, in linear algebra, a curve describing a linear model is fit to data via a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=37">least-squares technique</a>. (Shortly, the Maple Reporter will carry a Tips & Techniques article addressing the many ways that both linear and nonlinear least-squares fitting can be implemented in Maple.)<br><br>Once again, we hope that these examples illustrate the potential for changing the classroom dynamic, shifting math courses more towards the conceptual, by using Maple's syntax-free paradigm to support a resequencing of skills and concepts.</p>137377Fri, 14 Sep 2012 18:58:10 ZRobert LopezRobert LopezTen new 'Teaching Concepts with Maple' examples added
http://www.mapleprimes.com/maplesoftblog/136207-Ten-New-Teaching-Concepts-With-Maple?ref=Feed:MaplePrimes:Tagged With teaching
<p>With the addition of ten new Clickable-Calculus examples to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft website, we've now posted 63 of the 154 solved problems in my data-base of syntax-free calculations. Once again, these examples and associated videos illustrate point-and-click computations, but more important, they embody the <p>With the addition of ten new Clickable-Calculus examples to the <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a> section of the Maplesoft website, we've now posted 63 of the 154 solved problems in my data-base of syntax-free calculations. Once again, these examples and associated videos illustrate point-and-click computations, but more important, they embody the <a href="http://www.mapleprimes.com/maplesoftblog/134197-Teaching-Concepts-With-Maple">pedagogic message of resequencing skills and concepts</a>.<br><br>Instead of having students learn and master computational skills that are then used to manipulate and explore concepts in the hope that the concepts will be absorbed, the idea of resequencing is to implant the concept first, using technology to do any "heavy lifting" and to introduce the necessary manipulative skills afterwards, when their role is more readily apparent. The obstacle to this approach to using technology in the classroom would be the need to master the tool first, if the tool were not itself transparent. And that's where "syntax-free" computing comes to the aid of the pedagogical approach of resequencing skills and concepts.<br><br>Look for this reorganization in the newest ten examples posted to our website. There are two new Algebra/Precalculus examples, one for <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=56">solving a quadratic equation</a>, and one for exploring the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=64">parameter-dependence of the zeros of a polynomial</a>. Two new examples appear in the Trig section, one being a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=57">linear trig equation that has to be converted to a quadratic</a> in order to solve, and the other is an <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=65">equation that turns out to be an identity</a>.<br><br>In differential calculus, there's now an example showing how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=58">apply the limit-definition of the derivative to obtain the derivative of the square-root function</a>, and an example showing how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=59">obtain graphs of a function and its first two derivatives</a>.<br><br>In integral calculus, we've added the <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=60">Riemann-sum calculation of the definite integral of x sin(x), integrated over the interval [a, b]</a>. By forming and evaluating the limit of a Riemann sum, the connection between area under a curve and an antiderivative is illustrated.<br><br>We've also added a second problem in the "lines-and-planes" section of the typical multivariate calculus course. This example is that of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=61">finding the vector equation for a line between two points</a>. For linear algebra, we've added an example illustrating the meaning and calculation of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=62">eigenvalues and eigenvectors</a>.<br><br>Finally, in vector calculus, there's now a problem illustrating how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=63">find a scalar potential for a conservative vector field</a>. There's a Context Menu option for this, but the underlying technique of evaluating a line integral is implemented both with Maple's LineInt command and from first principles.</p>136207Fri, 03 Aug 2012 20:04:25 ZRobert LopezRobert LopezNew Teaching Examples Added
http://www.mapleprimes.com/maplesoftblog/135292-New-Teaching-Examples-Added?ref=Feed:MaplePrimes:Tagged With teaching
<p>My list of problems solved with Clickable-Calculus syntax-free techniques now numbers 154, spread over eight subject areas. Recently, Maplesoft posted to its website 44 of these problems, along with videos of their point-and-click solutions. Not only do these solutions demonstrate Maple functionalities, but they also have a pedagogic message, that is resequencing skills and concepts. They show how Maple can be used to obtain a solution, then show how Maple can be used to implement...<p>My list of problems solved with Clickable-Calculus syntax-free techniques now numbers 154, spread over eight subject areas. Recently, Maplesoft posted to its website 44 of these problems, along with videos of their point-and-click solutions. Not only do these solutions demonstrate Maple functionalities, but they also have a pedagogic message, that is resequencing skills and concepts. They show how Maple can be used to obtain a solution, then show how Maple can be used to implement the calculations stepwise, a process that injects insight and stresses concept development before it worries about building manipulative skills.<br><br>All the 154 Clickable-Calculus examples have been recorded. An additional nine are being added to the <a href="http://www.maplesoft.com/teachingconcepts/">Teaching Concepts with Maple</a> section of the Maplesoft website. The initial 44 were distributed across the six subject areas of Differential Calculus, Integral Calculus, Multivariate Calculus, Linear Algebra, Differential Equations, and Vector Calculus. This next ten introduce two new categories, that of Algebra and Precalculus, and Trigonometry. There are two each in these new categories.<br><br>The first Algebra/Precalculus problem is a solution of a simple <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=46">one-variable linear equation</a>; it's solution illustrates both the new Smart Pop-Ups and the older Equation Manipulator. Both of these tools implement a stepwise solution, thereby carrying out the pedagogic paradigm of "resequencing." This paradigm is continued in the second problem in this category, namely, the solution of a system of <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=54">two linear equations in two unknowns</a>. Here, a graphical solution is investigated, then three different stepwise solutions are given.<br><br>The two trig problems are trig equations that are quadratic in a trig function. In the first problem, just <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=47">one trig function</a> appears; but in the second, <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=55">two different ones</a> appear. Of course, graphical and numeric solutions come first, but then analytic solutions, and a general solution are also given. Finally, a stepwise solution aided again by Maple's new Smart Pop-Up technology demonstrates the traditional "by-hand" approach found in textbooks.<br><br>In differential calculus, we've added a <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=48">limit problem</a> that illustrates the use of the Limit Methods tutor for obtaining a stepwise solution. In integral calculus, we've added a problem on <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=49">Riemann sums and the connection between area and antiderivatives</a>. This problem illustrates the Riemann Sum tutor and the RiemannSum command embedded in a task template. In differential equations, we <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=51">classify an ODE</a> with a task template that implements the DEtools odeadvisor command. In linear algebra, <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=52">extracting a maximal linearly independent set (i.e., a basis) from a set of vectors</a> is done with a task template that implements the Basis command from the LinearAlgebra package. Finally, in vector calculus, we show how to <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=53">find a directional derivative stepwise</a>, and with the DirectionalDiff command from the VectorCalculus package.</p>135292Wed, 20 Jun 2012 18:47:04 ZRobert LopezRobert LopezTeaching Concepts with Maple
http://www.mapleprimes.com/maplesoftblog/134197-Teaching-Concepts-With-Maple?ref=Feed:MaplePrimes:Tagged With teaching
<p>Being easy to use is nice, but being easy to learn with is better. Maple’s ease-of-use paradigm, captured in the phrases “Clickable Calculus” and “<a href="http://www.maplesoft.com/products/maple/new_features/clickablemath.aspx">Clickable Math</a>” provides a syntax-free way to use Maple. The learning curve is flattened. But making Maple easy to use to use badly in the classroom helps neither student nor instructor.</p>
<p>In the mid to late ‘80s,...<p>Being easy to use is nice, but being easy to learn with is better. Maple’s ease-of-use paradigm, captured in the phrases “Clickable Calculus” and “<a href="http://www.maplesoft.com/products/maple/new_features/clickablemath.aspx">Clickable Math</a>” provides a syntax-free way to use Maple. The learning curve is flattened. But making Maple easy to use to use badly in the classroom helps neither student nor instructor.</p>
<p>In the mid to late ‘80s, as the movement to put computers in the classroom began, at least three researchers zeroed in on the idea to “resequence concepts and skills.” The traditional approach to math instruction, an approach written into countless texts, stresses skill development in the service of concept development. By-hand manipulative skills are necessary for exploring concepts when the only tools for that exploration are pencil and paper. They are a prerequisite for the acquisition of conceptual understanding.</p>
<p>Tools like Maple allow concepts to be explored before the manipulative skills are acquired. Maple draws the graphs, solves the equations, simplifies the expressions. Concepts can be presented and studied, using Maple in place of the not-yet-developed skills. More than that, Maple can be used to implement the steps of the relevant algorithms. Thus, Maple allows a student to see the big picture first, followed by a look at how the details, the steps of relevant calculations, fit into the big picture. All the while, it’s Maple doing the heavy-lifting; the student is learning where the bits fit, and why certain manipulations are needed.</p>
<p>Classroom experience shows that students learn the necessary skills more efficiently and effectively when they have a clearer idea of why they are necessary. They’ve seen what the “right answers” are supposed to look like, they know where the parts belong, and they understand what the goal is supposed to be. </p>
<p>Maple makes this resequencing of concepts and skills easier to implement because virtually no time is spent learning the tool. Maple’s point-and-click approach to computing in its new interface means that conceptual development can take place right from the start, without a pause to teach a computing language. The simplicity of the tool is one thing, but its use in service of a better pedagogy is far more important. And that better pedagogy is well served by the ease-of-use of the tool.</p>
<p>The collection of examples made available on the Maple web site, in a new section called <a href="http://www.maplesoft.com/teachingconcepts">Teaching Concepts with Maple</a>, illustrates this resequencing of concepts and skills. Each example shows how, after a statement of a problem, it can expeditiously be solved in Maple. Then, like peeling away the layers of an onion, various facets of the solution process can be explored, using Maple’s point-and-click technologies. From a big-picture conceptual approach, to a mastery of details, Maple helps the student learn more quickly, and with greater insight and understanding.</p>134197Mon, 14 May 2012 22:31:36 ZRobert LopezRobert LopezFactoring a Quadratic Polynomial
http://www.mapleprimes.com/maplesoftblog/120910-Factoring-A-Quadratic-Polynomial?ref=Feed:MaplePrimes:Tagged With teaching
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left">This is the <a href="http://www.maplesoft.com/applications/TipsAndTechniques/classroom.aspx">Classroom Tips & Techniques</a> article for the May, 2011 Maplesoft Reporter, which, after publication, finds...</td></tr></tbody></table><p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"> </p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left">This is the <a href="http://www.maplesoft.com/applications/TipsAndTechniques/classroom.aspx">Classroom Tips & Techniques</a> article for the May, 2011 Maplesoft Reporter, which, after publication, finds its way into the Maple Application Center. The article takes the liberty to rail against the stress placed on a particular manipulative skill in the precalculus curriculum, and likewise, I take the liberty to post it as a blog. The windmill at which I tilt is the "skill" of factoring a quadratic polynomial by inspection, a technique in which I find little intrinsic value.</p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> </span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">My guess is that for historic reasons, factoring a quadratic was the way to obtain its zeros. The essence of the concept one would want a student to absorb is the factor-remainder theorem, so finding zeros becomes important. But demanding that students learn about the factor-remainder theorem via the travail of factoring a quadratic by inspection seems to me rather senseless, given that shortly, the student learns to complete the square and thereby obtain the roots of a quadratic equation. In fact, the quadratic formula is derived by completing the square, and not by "factoring."</span></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">I remember in my high school math curriculum (mid 1950s) that I learned to multiply and divide large numbers via the addition and subtraction of their logarithms. This material disappeared from the curriculum as soon as calculators became a commodity in the 1970s. If the curriculum can change in one way because of technology, why, using the same technology, can't it change in another? The higher cognitive merit is in understanding the relationship between zeros and factors. It isn't really necessary to torment students with factoring-by-inspection as a way of finding zeros. There are other ways - either use an electronic technology or use the quadratic formula.</span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> </span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">Indeed, consider how a cubic is factored in the precalculus curriculum. First, zeros are found; then from the zeros, the factors are written. Just the opposite of the process imposed in the quadratic case!</span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> </span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">Finally, I would conjecture that in some appropriate space, the set of quadratics that can be factored by "inspection" has measure zero. Any quadratic worth factoring probably doesn't yield to "inspection" anyway.</span></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">OK, now that I've ranted, I'd like to finish with a tool I just wrote, a tool for helping a student with the task of factoring a quadratic polynomial. I never thought I'd find myself creating such an applet, but I get asked about this as part of many of the webinars I present for Maplesoft. Just recently, after again getting that question, I found I just couldn't let it go. I kept thinking about what it takes for a student to master the appropriate skill. So, the tool I built reflects the way I think about the task.</span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=120910/384460/0078cba1df637b8b8feae17858274ea6.gif" alt="" width="11" height="23"></p>
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<p style="margin: 0 0 0 0; padding-top: 8px; padding-bottom: 4px;" align="left"><span style="color: #000000; font-size: 150%; font-weight: bold; font-style: normal;">Factoring a Quadratic Polynomial by Inspection</span></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">To factor a polynomial in the form </span><img style="vertical-align: -6;" src="/view.aspx?sf=120910/384460/1bfbfdeb4d144b85d10fe1d5580a75f1.gif" alt="" width="99" height="27"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">, the "factor-pairs" of both </span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: italic;">a</span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> and </span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: italic;">c</span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> must be determined. It's not enough just to get the divisors of these coefficients; one must determine the </span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: italic;">pairs</span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> of divisors whose product equals that coefficient. Then, these factor-pairs must be arranged in linear factors in such a way as to reproduce the given quadratic. What I believe students struggle with is holding the various combinations in their heads while they test to see if the product of their linear factors equals the given quadratic.</span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> </span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">Indeed, if the linear factors are of the form </span><img style="vertical-align: -7;" src="/view.aspx?sf=120910/384460/96db09e313f832313bf53ed22ffadf22.gif" alt="" width="136" height="26"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">, then the mental arithmetic that the student must master is deciding if a sum or difference of the products </span><img style="vertical-align: -7;" src="/view.aspx?sf=120910/384460/7ba0231bc64e0b3e98bfbb85e86cd9f3.gif" alt="" width="26" height="26"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> and </span><img style="vertical-align: -7;" src="/view.aspx?sf=120910/384460/a285d827cbe99da05a571f57019a820a.gif" alt="" width="24" height="26"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> can equal the coefficient </span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: italic;">b</span><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">. (What this has to do with understanding the factor-remainder theorem is beyond me, but this is the stumbling block for many a weary student. Oops, there I am back on my high-horse again.)</span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;"> </span></p>
<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">The tool I constructed (shown below, and available in the Maple document) is not modeled on a "drill-and-practice" philosophy. (At one stage of my teaching career, I was a strong advocate of such a heavy-handed approach to learning, but as I "got older" I mellowed, realizing that bludgeoning a student into mastery of a skill was counter-productive.)</span></p>
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<p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><span style="color: #000000; font-size: 100%; font-weight: normal; font-style: normal;">Instead, the tool found below simply presents the factor-pairs, the linear factors so determined, and the expansion of the products of these factors. It's my belief that a student struggling with the task of factoring a quadratic will more likely see the "pattern" in these calculations than in a tool that asks for the right answer and simply tells the student that the answer is right or wrong.</span></p>
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<p style="margin: 0 0 0 0; padding-top: 8px; padding-bottom: 4px;" align="left"><span style="color: #000000; font-size: 150%; font-weight: bold; font-style: normal;">The Tool</span></p>
<p style="margin: 0 0 0 0; padding-top: 8px; padding-bottom: 4px;" align="left"><span style="color: #000000; font-size: 150%; font-weight: bold; font-style: normal;"><a href="/view.aspx?sf=120910/384460/blogtool.JPG"><img src="/view.aspx?sf=120910/384460/blogtool.JPG" alt=""></a><br></span></p>
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<p><a href="/view.aspx?sf=120910/384460/Factoring_a_Quadrati.mw">Download Factoring_a_Quadrati.mw</a></p>120910Wed, 01 Jun 2011 18:10:44 ZRobert LopezRobert LopezI want hlep me to slove non-linear function
http://www.mapleprimes.com/questions/119907-I-Want-Hlep-Me-To-Slove-Nonlinear-Function?ref=Feed:MaplePrimes:Tagged With teaching
<p><strong>Hi , I'm Alan GHafur , I'm student in master dagree in Statistic in Irbil / Iraq .I want learn how can slove non linear function in Maple.</strong></p>
<p><strong>thank you for hlep me.</strong></p><p><strong>Hi , I'm Alan GHafur , I'm student in master dagree in Statistic in Irbil / Iraq .I want learn how can slove non linear function in Maple.</strong></p>
<p><strong>thank you for hlep me.</strong></p>119907Mon, 16 May 2011 22:42:43 Zalan82statalan82stat