(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 3774942, 93458]*) (*NotebookOutlinePosition[ 3775665, 93484]*) (* CellTagsIndexPosition[ 3775621, 93480]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ ATCMA 2001, Melbourne, Au\ \>", "Subsubtitle", CellMargins->{{14, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["Computer Screens and Toys.", "Subtitle", CellMargins->{{193, Inherited}, {Inherited, Inherited}}], Cell["\<\ Ivan Cnop, Vrije Universiteit Brussel, Brussels, Belgium icnop@vub.ac.be\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Abstract.", "Subsubsection"], Cell[TextData[{ "In this paper we show that experimenting with simple motion on a computer \ screen leads to a surprising variety of visual objects in the plane, and \ allows to describe natural objects in space as well. Using commercial toys \ for building moving objects makes these constructions accessible to pupils \ from lower secondary on. Co-ordinate systems in the plane as well as \ trigonometry can be introduced rather early in this way. \nUsing a CAS we can \ generate much faster the orbits of moving objects, together with their \ animation, and it gives the student a good visual impression of 3D objects. A \ CAS moreover provides a link to the underlying theories. Advanced theorems \ and applications can be motivated before they arise in the tertiary education \ curriculum. \nPart of this work was written in Mathematica under the \n", StyleBox["\"Exploot\" (Experimental Learning Environment using CAS)", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "\nproject of the Flemish Ministry of Education at the Vrije Universiteit \ Brussels, Belgium and part of this material is visible on the projects' web \ site " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction.", "Subsection"], Cell["\<\ Is \"Computer Screens and Toys\" the right title for addressing an \ international conference? Let us start by some questions concerning\ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`1 + \[Sum]\+\(j = 1\)\%\(n - 1\)cos(\(\(2\)\(\ \)\(\ \[Pi]\)\(\ \)\(j\)\(\ \ \)\)\/n)\)], "Input", Evaluatable->False], Cell["\<\ Q1: what is this sum? Q2: can you prove this result? Q3: can you explain the meaning of this result in secondary school? If we do not know the answer to the first question,our CAS will \ immediately provide the answer:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`1 + \[Sum]\+\(j = 1\)\%\(n - 1\)cos(\(\(2\)\(\ \)\(\ \[Pi]\)\(\ \)\(j\)\(\ \ \)\)\/n)\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Closed]], Cell["\<\ For the second question, we can use trigonometric identities \ or reformulate the problem in terms of de Moivre\[CloseCurlyQuote]s formula \ and points on the unit circle. \tDo we know properties of a regular n-gon? \tDo we know what the sum of the n-th roots of unity is? \tDo we know facts about geometric series?\ \>", "Text"], Cell[TextData[{ " Answering positively to the third question seems ", StyleBox["impossible", FontColor->RGBColor[1, 0, 0]], ". Nobody will ever think of teaching trigonometry to pupils age twelve. \ The reason is that the formulation in terms of trigonometry hides the facts \ from elementary geometry that are behind most trigonometric identities.\n\ These elementary facts remain hidden because the mathematics curriculum \ imposes complicated formulations and many mathematicians seem to enjoy this \ because it gives them an aura of importance." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Multiplication of points in the plane.", "Subsection"], Cell[TextData[{ "\tLet us look at a computer screen.\nIt is a plane and it is therefore \ natural to start ", StyleBox["computer", FontColor->RGBColor[1, 0, 0]], "-assisted learning in the plane instead of accumulating numbers on a line. \ Successful packages such as Cabri II and Geometer\[CloseCurlyQuote]s \ Sketchpad do exactly this. \n\tEverything in trigonometry has to do with a \ point moving along the ", StyleBox["unit circle", FontColor->RGBColor[1, 0, 0]], " and the way the angle, over which it moves, is measured. Writing cosines \ is like looking at the picture through glasses completely covered with the \ exception of an horizontal slit, writing sines is looking at the picture \ through glasses with only a vertical slit. One can expect that the vision is \ impaired by imposing such restrictions on ourselves and on the pupils. \ Looking at the plane gives a lot more information.\n\tSelect a point on the \ computer screen as the origin,and a horizontal axis on which we choose a unit \ and put real numbers. Each other point is completely determined by a ", StyleBox["direction", FontColor->RGBColor[1, 0, 0]], " and its distance to the origin. Directions are measured counter-clockwise \ from the positive horizontal axis. A simple way to measure the (polar) angle \ with arbitrary precision is to measure the length it cuts off along the unit \ circle. Navigation chooses orientation clockwise from North (or the magnetic \ north pole), and measures angles in degrees, minutes and seconds (with \ precision limited to .1\" ). Both systems have evolved through history and \ there is no reason to replace one of these standards by the other. If we need \ a shorthand notation we can write\n\tcis(t) , ", StyleBox["unit(t)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontWeight->"Bold"], "\nor the Euler notation" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[ImaginaryI]\ t\)\)], "Input", Evaluatable->False, FontSize->14, FontColor->RGBColor[1, 0, 0], FontVariations->{"CompatibilityType"->0}], Cell["for the direction making an angle t from the positive axis.", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.4 0.4 0.4 0.4 [ [.8 .3875 -3 -9 ] [.8 .3875 3 0 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .8 .4 m 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This product gives the \ corresponding images under ", StyleBox["similarities and rotations", FontColor->RGBColor[0, 0, 1]], ". The theorem of ", StyleBox["Thales", FontColor->RGBColor[0, 0, 1]], " can be reformulated using products and many results from plane geometry \ can be obtained in this way. Let students experiment with multiplication of a \ (all points in a) geometric figure by a point, and see what properties are \ preserved.\n\tWhat we have defined under product of points is in fact the \ product of complex numbers in polar form. I want to avoid the word \ \"complex\" since I am only talking about points in the plane, and these \ points are very real. It is unfortunate that multiplying points or vectors in \ the plane is ", StyleBox["forbidden", FontColor->RGBColor[1, 0, 0]], " in classical curriculum. It is indeed a very natural operation. It is a \ special case of the matrix multiplication in the plane.\n\tDoubling angles, \ or reversing angles is even simpler. A double angle means that we square \ directions, a third power is given by three times the angle, and so on. A \ reversed angle is obtained by the division operation, and we can continue in \ this way. Each of the directions ", StyleBox["unit(2\[Pi] j/n)", FontColor->RGBColor[1, 0, 0]], " in the original question becomes 1 when you take its nth power. 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