(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 268881, 11273]*) (*NotebookOutlinePosition[ 269994, 11309]*) (* CellTagsIndexPosition[ 269950, 11305]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Mathematica-aided Education of Science-major Students", "Subtitle", PageBreakAbove->False, TextAlignment->Center, FontFamily->"Palatino"], Cell["\<\ K. Nakagami, F. Takeutchi, F. Ushitaki and M. Yasugi Faculty of Seience, Kyoto Sangyo University, Kyoto, Japan http://www.kyoto-su.ac.jp/~yasugi\ \>", "Text", PageBreakAbove->False, TextAlignment->Center, FontFamily->"Palatino"], Cell["\<\ We will make a comprehensive report of our experience with \ Mathematica-aided education. Some students belonging either to Mathematics \ or Computer ScienceDepartment do graduation works using Mathematica as a \ language. We take up three subjects in those works.\ \>", "SmallText", PageBreakAbove->False, FontFamily->"Palatino"], Cell[CellGroupData[{ Cell["Helping students to understand mathematics. ", "Section", PageBreakAbove->False, FontFamily->"Palatino", FontWeight->"Bold"], Cell["\<\ We have seen there are three ways of helping students to \ understand mathematics by Mathematica: (1) graphics, (2) reading and writing \ programs, and (3) experiments. (1) The converging process of intervals to Cantor set and linear \ transforms of figures in the plane and the space by matrices and an initial \ value problem using the conservation law: With graphics, one can obtain a \ vivid picture of a limiting process, can get an idea of what a matrix is and \ dependence of a n initial value on conservation quantity. (2) Mathematical property of Cantor set and the condition of a Mersenne \ number to be a prime: By reading and writing programs, which gives procedual \ aspect of the mathematical notions, students understand easily those \ properties and proofs. (3) The fractional expression of left out points of Cantor set: We present \ a conjecture on the form of such fractions, and let the students experiment \ on it. They then wrote a program of \"quasi Cantor set\" (dividing the unit \ interval into four equal parts and eliminating the second and the fourth \ subintervals).\ \>", "Text", PageBreakAbove->False, FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell[" Making teaching materials. ", "Section", PageBreakAbove->False, FontFamily->"Palatino"], Cell["\<\ Students can also help highschool students and freshmen at the \ university by inventing teaching materials with Mathematica. Graphics in \ linear algebra as in I above can be used also for this purpose, and a \ systematic drill for matrix computations is being programmed. In order to \ help understanding of quadratic functions, for example, one can let children \ draw a picture of a face with three parabolas. Through such works, with trial \ and error, they learn some properties of quadratic functions such as the fact \ that the shape of a parabola is determined by the coefficient of its highest order. In those trials, non-trivial devices were required in coloring by \ parts and solving inequalities.\ \>", "Text", PageBreakAbove->False, FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Programming to solve puzzles and games.", FontFamily->"Palatino", FontWeight->"Bold"]], "Section", PageBreakAbove->False], Cell["\<\ \tWe let students solve logical puzzles or make up strategies for \ winning a game by Mathematica. The principal objective of this activity is \ to help them analyze the puzzle or game, and materialize the algorithm in a \ relatively transparent way using Mathematica. \tAfter one year of training of using Mathematica as a programming language, \ each student chooses a puzzle or a game of appropriate difficulty for him/her \ to study for another year, assisted by the teacher. \tThe advantages of Mathematica as a tool are, its well thought-out structure \ as a programming language and the pattern matching feature of the language \ which is extremely useful for solving this kind of problems. Students are \ often amazed by the simplicity of the coding using this feature. \tWe would like to present here, as an example, two subjects chosen in the \ past by students. The first one is a game and the second a puzzle.\ \>", "Text", PageBreakAbove->False, PageBreakBelow->True, FontFamily->"Palatino"], Cell[CellGroupData[{ Cell["Yashima-game", "Subsubsection", FontFamily->"Palatino"], Cell["\<\ \tTwo players play alternately. On a sheet of paper, the players \ draw arbitrary number of nodes as points (meaning islands), and then connect \ arbitrarily the nodes by straight lines (meaning bridges). An example is \ shown is the following figure.\ \>", "Text", FontFamily->"Palatino"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .95106 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 0.5 0.333798 0.427716 0.333798 [ [ 0 0 0 0 ] [ 1 .95106 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .95106 L 0 .95106 L closepath clip newpath 0 g .03 w .5 .76151 Mdot .18254 .53087 Mdot .3038 .15767 Mdot .6962 .15767 Mdot .81746 .53087 Mdot .5 .92841 Mdot .02381 .58244 Mdot .2057 .02264 Mdot .7943 .02264 Mdot .97619 .58244 Mdot .5 Mabswid [ ] 0 setdash .5 .76151 m .3038 .15767 L .81746 .53087 L .18254 .53087 L .6962 .15767 L .5 .76151 L s .5 .92841 m .02381 .58244 L .2057 .02264 L .7943 .02264 L .97619 .58244 L .5 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3P02og40001b0003o`0000d000?o00003@000ol0001a0000L`02o`d000?o00002`02og@0001e0003 o`0000X000?o00002P000ol0001d0000MP000ol000090003o`0000T000?o0000M@0007L00_l90003 o`0000L00_mh0000N@000ol000060003o`0000H000?o0000N00007X00_l60003o`0000D000?o0000 N@0007`000?o00000`000ol0000300;oO00007d000?o00000P000ol000020003o`0007`0001n00;o 00D0oooo0002ogl0002000GoP@0007l01on00000O`07oh00001o00OoP00008001On10000P@03oh80 003o00L0003o00L00000\ \>"], ImageRangeCache->{{{0, 261.625}, {248.688, 0}} -> {-1.49873, -1.28138, 0.00916568, 0.00916568}}], Cell["\<\ \tAfter that, each player chooses his/her own starting node, and \ put a bead on those nodes. First player now moves from his/her node to \ another along one of the straight lines, and then deletes the strait line \ used (burns the bridge he/she just crossed). Second player does the same. \ The objective of the game is to make the opponent frozen in an island so that \ no straight line is existing from the node where the opponent is. \tThis game, although simple, is fairly amusing when played. To analyze this \ game, the student who chose this subject has given a unique name to each \ node. Then she made a matrix of N by N whose elements are either True or \ False. N is the number of nodes, and the element True means two nodes are \ still connected with a straight line. \tThen she made a program using the backtrack algorithm. She has started \ from the Backtrack function in the Skiena's book. She changed the function \ such that the length of the total solution is not a fixed number. She could \ then tell that given the initial board, she can win even the opponent plays \ flawlessly, or she can loose if the opponent plays flawlessly. \tIn solving this problem with Mathematica, she was impressed by the \ flexibility of Mathematica. The number of nodes as well as the number of \ lines can be anything. Also she could follow the movement of backtracking \ function at work in a very transparent way. Unless the board is very \ complicated, the speed of Mathematica is enough so that it is fun to play \ against the computer.\ \>", "Text", FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["Nonogram", "Subsubsection", FontFamily->"Palatino"], Cell["\<\ \tThe second example we show here is a puzzle called Nonogram. A \ very simple example is shown in the next figure. This example is entitled as \ Puppy.\ \>", "Text", FontFamily->"Palatino"], Cell[BoxData[ \(problem\)], "Input", PageBreakAbove->True], Cell[BoxData[GridBox[{ { ButtonBox["3", ButtonFunction:>process$r[ 1], ButtonEvaluator->Automatic, Active->True], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]]}, { ButtonBox["2", ButtonFunction:>process$r[ 2], ButtonEvaluator->Automatic, Active->True], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", Background->GrayLevel[0.500008]]}, { ButtonBox["4", ButtonFunction:>process$r[ 3], ButtonEvaluator->Automatic, Active->True], ButtonBox[" ", Background->GrayLevel[0.500008]], ButtonBox[" ", 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The player should either draw in a dot in the square or paint \ it black. The hint like 1 . 1 in the bottom row means in that row of two \ non-adjacent single squares somewhere should be painted black. The dot in \ the hints means that those painted squares are non adjacent. The hint of the \ left-most column is just 2. This means in this column, somewhere two \ consecutive squares should be painted black, and the rest (3 squares) should \ carry each a dot. \tThe rule is simple. Problems are very carefully prepared so that the \ player can reach the unique final solutions without cut-and-try method. \tThe real problems are a little more complicated with larger number of \ squares. 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not-so-well-skilled player can solve this problem in less than \ one hour. \tTo solve this problem, the pattern matching feature of the Mathematica \ language is very effectively used. Man can pick up the hints on columns and \ rows to be used in a right sequence. If one makes a program which picks up \ randomly the hints to be used, it needs usually an enormous amount of time to \ find out one square to be painted in a say row of 25 squares. So a \ simple-minded program needs many and many hours to solve such a problem as \ the one shown in Fig. 2. A very careful programming is needed to speed up \ the solving. In this case, a program containing some 400 lines of codes \ solved the second problem with a speed comparative to a man. \tIn doing this, we find that still a major disadvantage of Mathematica is \ its slow speed, due to the fact that it is an interpreter language. One \ should be able to use the compiled function by sacrificing some flexibility, \ still this part seems a problem with this language. Just for fun, the author \ coded the same program in Lisp. It was about 30 times faster than the \ program in Mathematica. Should a smart automatic background compilation \ mechanism be incorporated in Mathematica, this activity would certainly be \ more fun and productive.\ \>", "Text", FontFamily->"Palatino"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Acquiring mathematical concpets and skills", "Section", FontFamily->"Palatino"], Cell["\<\ In this section, we report some effects of graduation works by \ students of the third year and the fourth year. They do the work in \ continuation for two years. Our students, being computer students, do not \ have much chance of acquiring mathematical background, but with some help of \ Mathematica, they were able to make progress without pain.\ \>", "Text", FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["Mersenne numbers:4 students", "Subsubtitle", FontFamily->"Palatino", FontWeight->"Bold"], Cell[TextData[{ "By reading a program of finding Mersenne numbers, the students learned \ elements of number theory.\nThis work was motivated by experiments on the \ distribution of prime numbers in the set of natural numbers. By using \ \"ListPlot\" one can plot the distribution of prime numbers up to the \n \ 100th, 1000th, 1000000th, ...\nprimes, and see that the frequency \ decreases. The students, on the other hand, know that there are infinitely \ many prime numbers by the method of Euclid. \nThey then became interested \ in possible algorithms of finding large prime numbers, and found a \ well-known program named \"MersenneExponentQ\" in a textbook. It is no \ easy job to justify the \"Do\" part of the program, which stands\n \ Do[r=Mod[", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " -2,N],{p-2}]\nwhere the program tries to judge whether ", Cell[BoxData[ \(TraditionalForm\`2\^p\)]], " -1 is a prime number for a prime number p. The output is \"True\" if \ r=0 when the \"Do\" part is repeated (p-2) times, and is \"False\" \ otherwise.\n Upon suggestion in the text, they began to challenge the proof \ of an old theorem (1935) by D.H.Lehmer, in Journal of the London Mathematical \ \n Society. It is a short article which assumes deep knowledge in number \ theory, and so, in order to give a detailed proof, the students first \ studied elements of number theory and then completed the proof. \n Lehmer's \ theorem states as follows." }], "Text", FontFamily->"Palatino"], Cell[TextData[{ "For any prime number n, put\n N=", Cell[BoxData[ \(TraditionalForm\`2\^n\)]], " -1 ", Cell[BoxData[ \(TraditionalForm\`S\_1\)]], " =4, ", Cell[BoxData[ \(TraditionalForm\`S\_k\)]], " =", Cell[BoxData[ \(TraditionalForm\`S\_\(k - 1\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\^2\)]], " -2\nThen N divides", Cell[BoxData[ \(TraditionalForm\`S\_\(n - 1\)\)]], " if and only if N is a prime number." }], "Text", FontFamily->"Palatino"], Cell[TextData[{ "In the program, the condition that N divides ", Cell[BoxData[ \(TraditionalForm\`S\_\(n - 1\)\)]], " is reduced to a computation of Mod[", Cell[BoxData[ \(TraditionalForm\`S\_n\)]], "] modulo ", Cell[BoxData[ \(TraditionalForm\`2\^n\)]], " -1. The students were able tomathematically justify this reduction of \ computation.\nAll in all, starting with experiments with plotting, the \ students found pleasure in number theory and drew out their intelligence for \ it. Thay also learned that an old mathematical theorem can contribute to \ programming a mathematical algorithm, and that a mathematical theorem has to \ be adjusted to efficient computations in writing a program." }], "Text", FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["Cantor set:6 students", "Subsubtitle", FontFamily->"Palatino", FontWeight->"Bold"], Cell["\<\ The purpose of assigning this theme to the students was to let them \ understand ternary expressions of real numbers as well as the limiting \ process of sets. By plotting the construction process of the Cantor set using graphics, and \ doing experiments, the students were able to prove some mathematical \ properties of the points removed and the points unremoved as well as the \ length and cardinality of the Cantor set. \ \>", "Text", FontFamily->"Palatino"], Cell[TextData[{ "Let J_{2,2} denote the second removed subinterval at the second step.\n x \ \[Epsilon] ", StyleBox[Cell[BoxData[ \(TraditionalForm\`J\_\(2, 2\)\)]]], " \[DoubleLongLeftRightArrow] x= 2/3 + 1/", StyleBox[Cell[BoxData[ \(TraditionalForm\`3\^2\)]]], "+ ", StyleBox[Cell[BoxData[ FormBox[ StyleBox[\(\[Sum]\+\(n = 3\)\%\[Infinity]\ \), ScriptLevel->0], TraditionalForm]]]], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_n\)]]], "/3 < 2/3 + 2/", StyleBox[Cell[BoxData[ \(TraditionalForm\`3\^2\)]]], "\nwhere ", StyleBox[Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_i\)]]], " = 0,1,2, is but one example. (Before the experiments, my lecture had \ passed over their heads!) \nThe students next attempted to write a program \ of the following. Instead of dividing into three parts, divide the closed \ interval [0,1] into four equal parts and remove the second and the fourth \ subintervals; repeat the same process to all the remaining subintervals. \ They were able to set up some propositions on the points in the intervals \ and characterize a Cantor-like set thus created. \nWith programming of the \ construction process of this set, they found one snag, that is, in this \ case, one has to refer to the interval [3/4,1]. Mathematica regards 1 as \ \"Integer\", but not as \"Rational\", and a uniform expression of the \ removed intervals as in the case of the Cantor set breaks down. One has to \ make this initial step as an extra case. \nInspite of lack of training in \ the limit preocesses, especially of that of sets, the students thus \ understood the essence of the Cantor set stimulated by a programming of the \ Cantor-like set." }], "Text", FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["Educational materials by Mathematica:8 students", "Subsubtitle", FontFamily->"Palatino", FontWeight->"Bold"], Cell["\<\ Those students wanted to take the advantage of Mathematica in \ helping highschool and freshman level students in linear algebra by \ presenting experimental tools and self-training tools by Mathematica. They \ have written the following programs.\ \>", "Text", FontFamily->"Palatino"], Cell["\<\ Showing the effect of a matrix or a successive application of \ matrices by animation. The motion of a point or a set of points in the plane \ when applied a matrix or several matrices in succession is animated, so that \ a learner will understand what kind of transformation a matrix does to the \ plane.\ \>", "Text", FontFamily->"Palatino"], Cell["\<\ Self learning kits for matrix translations and solving simultaneous \ linear equations. This project aimed at making self-learning kits on translations of two by two \ or three by three matrices and solving simultaneous linear equations. The \ program first creates randomly matrices either in integers or in reals and \ then let a learner choose a problem. If a student does the problem, then the \ program will output \"True\" or \"False\" and, if \"False\" then a correct answer is \ output.\ \>", "Text", FontFamily->"Palatino"], Cell["\<\ The students not only came up with many ideas on programming, but \ also learned linear algebra a lot better than before. They also became \ deeply concerned with what be good educational materials.We would like to do \ experiments on the effect of learning by using these kits with highschool \ students and freshmen.\ \>", "Text", FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "On initial value problem of the KdV-equation and Mathematica", FontSize->14, FontWeight->"Bold"]], "Subsubtitle", TextAlignment->Left, FontFamily->"Palatino", FontSize->10], Cell[BoxData[ \(TextForm\`\(\t\ We\ consider\ the\ KdV - equation\)\)], "Text", TextAlignment->Left, FontFamily->"Palatino", FontSize->10, FontWeight->"Plain"], Cell[BoxData[ \(TextForm \`\[PartialD]\_t u = 6 u\ \ \ \[PartialD]\_x u - \[PartialD]\_xxx u\ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((1)\)\)], "Text", TextAlignment->Center, FontFamily->"Palatino", FontSize->10, FontWeight->"Plain"], Cell[BoxData[{ FormBox[ RowBox[{ \(which\ is\ known\ as\ a\ nonlinear\ differential\ equation\ \ with\ solitary\ - \ wave\ \ \ solutions . \ \ \ \ We\ \ rewrite\)}], TextForm], FormBox[ RowBox[{ FormBox[\(\ equation\ \ \((1)\)\ \ in\ \ the\ \ form\), "TextForm"], " "}], TextForm]}], "Text", TextAlignment->Left, FontFamily->"Palatino", FontSize->10, FontWeight->"Plain"], Cell[BoxData[ \(TextForm\`\(\ \[PartialD]\_t u = \[PartialD]\_x\ \((3\ u\^2 - 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2 At - x\_0)\)\ \ \ , \ \ \ \ \((3)\)\)], "Text", TextAlignment->Center, FontFamily->"Palatino", FontSize->10], Cell[BoxData[ FormBox[ RowBox[{ FormBox[ \(where\ \ \ A\ \ is\ \ an\ \ amplitude . \ \ Thus\ \ the\ \ solitary\[Dash]\ wave\ \ solution\ \ is\ \ determined\ by\ it' s\ amplitude\ \ \n\tA\ \texcept\ for\ the\ phase\), "TextForm"], " ", RowBox[{\(x\_0\), ".", FormBox[\(\n\t\t Suppose\ \ that\ \ the\ \ initial\ \ wave\ \ function\ \ is\ given \ by\), "TextForm"]}]}], TextForm]], "Text", FontFamily->"Palatino", FontSize->10], Cell[BoxData[ \(TextForm\`\(\ u \((x, t = 0)\) = \(-\ A\_0\)\ \(sech\^2\) \((x/\[Alpha]\ )\)\ . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((4)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Palatino", FontSize->10], Cell[BoxData[{ FormBox[ RowBox[{\(In\ \ the\ \ following, we\ \ restrict\ \ ourselves\ \ to\ \ \ the\ case\ where\ \ \ \ 0 < \[Alpha]\^2\ A < 2\ . \ \ Then\ the\ initial\ wave\), " "}], TextForm], FormBox[ RowBox[{ RowBox[{ \(function\ tends\ to\ \ a\ \ solitary - 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As a result of such \ drawings, students learn various properties of the fucntions, and it \ enables them to comprehend \nsome important concepts in mathematics. \n \ On the other hand, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " graphics is also useful in teaching elementary mathematics. In this \ section, we propose a device for it. Of course, it is meaningless to plot \ the graphs of functions given as exercises in a text-book. Our idea is to \ let highschool students draw some simple pictures on the coordinate plane by \ arranging some parts of the graphs of elementary functions. \n They \ become engrossed in this kind of work because it is a creative work. They \ thus become familiar with some mathematical notitons without even realizing \ it.\n As an example, we will explain how to use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " for teaching the theory of quadratic functions and parabolas. With our \ device, students can help highschool children to understand essence of \ these functions. \n Quadratic function is one of the most basic functions \ for learners of mathematics. Its graph parabola is also important. However, \ when learning it in school, they must plot a lot of graphs from given \ functions or conditions, or solve various exercises in order to acquire basic \ properties of such a function. Instead of such a traditional method, let \ us use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ".\n Only by drawing pictures of human faces with three parabolas as \ follows, they can learn most of the properties of quadratic functions and \ parabolas.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Palatino"], Cell[CellGroupData[{ Cell[BoxData[ \(righteye = Plot[\(-\((x - 2)\)^2\)\ + \ 2, \ {x, \ 1, 3}]; \n lefteye = Plot[\(-\((x + 2)\)^2\)\ + \ 2, \ {x, \ \ \(-3\), \(-1\)}]; \n mouth = Plot[1/2\ x^2\ - \ 2, \ {x, \ \(-2\), \ 2}]; \n Show[{righteye, \ lefteye, \ \ mouth}, \ AspectRatio -> Automatic]\)], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .66667 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 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After explanation, let them change the coefficients in the functions one \ by one as in the following example. They observe how the graph changes \ accordingly. Through such works, with trial and error, they discover some \ characteristics of quadratic functions such as the fact that the shape of a \ parabola is determined by the coefficient of its highest order. 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However, he/she does it because he/she wants to complete the work. For \ example, in the process of drawing the following picture, he/she learns the \ relation between two parabolas, how to compute the coordinate of the \ intersection point of two parabolas, and the meaning of the region which is \ surrounded by several curves. Moreover, if he/she thinks that the outline of \ the face does not look like the human one, he/she is invited to study more \ advanced functions.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Palatino", FontSize->10] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell["Solve[-5 (x-3/2)^2 + 3 == (x-2)^2 + 7/4, x]", "Input", PageBreakWithin->Automatic, GroupPageBreakWithin->Automatic, AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> 7/6}, {x -> 2}} \ \>", "\<\ 7 {{x -> -}, {x -> 2}} 6 \ \>"], "Output", Evaluatable->False, PageBreakWithin->Automatic, GroupPageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Solve[1/2 x^2 - 2 == -1/2 x^2 - 3/2]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> -2^(-1/2)}, {x -> 2^(-1/2)}} \ \>", "\<\ 1 1 {{x -> -(-------)}, {x -> -------}} Sqrt[2] Sqrt[2] \ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], 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