THE DERIVE - NEWSLETTER #76 USER GROUP
Contents
1. Taking the day as basis the subsequent units of time where e the uinal or month of 20 kins days e the tun or year of 18 uinals 18 20 360 days e the katun or period of 20 tuns 20 360 7200 days e the baktun or period of 20 katuns 20 20 360 144 000 days e the pictun or period of 20 baktuns 20 20 20 360 2 880 000 days e etc As it seems obvious Mayan astronomers knew the corrections that had to be made after considering a year of 360 days instead of the solar year of 365 25 days Grube 2000 E Roanes Mayan Numbering System in DERIVE With this system and counting long periods of time they estimated the phases of Venus which they rounded up to 584 days Mercury and Mars the duration of a lunar month in what today we would write as 29 53020 days and they even registered periods up to 300 000 000 years We can confirm all these through their commemorative steles and the pages of the few Mayan books codices which have being preserved Landa 1937 where we find an original numbering system with three basic features it had a base 20 used the place value system and introduced the zero In fact in association to its religious origin numerical quantities registered by the Mayas where strictly joined to the god corresponding to each order the god carrying kins the god carrying uinals the god carrying tuns etc Their conception of the zero was based upon this religious nature 1t was a symbol created2. Zx 56 check x wrong 4 El Lfex 119 co3 119 6 4 173 54 so 17 6 Do you find the mistake DERIVE has a wonderful feature its STEPWISE SIMPLIFICATION Unfortunately no other CAS which is on the market now has a similar option If the student has a wrong inte gral and he she does not know how to obtain the right answer then he she can ask DERIVE to stepwise integrate 2 50 Ha ax De 1 x dx aF dx a Fix dx Il illustrate this applying stepwise integration on the second problem of the list of integrals by substitution from above e I 2 Z x dee 270 ix l dx a Fixi a Fix e e Fix di a 2 52 2x 4 x There is a similar DERIVE program for exercising repeating or deepening the differentiation rules Most of my programs which were produced in the last years show the same structure The user hast to load the program as a utility file and then simplify the command start Then he is presented the instructions how to run the tool It might be a good moment now to stop demonstrating more examples but asking which fields in school mathematics might need some exercising Which manipulating and other skills techniques competences could could not should should not be trained deepened and or extended supported by CAS or other technolo gies My list is more or less CAS oriented but hope that can extend it to other technologies and
3. Center ing matrices are used for example in multivariate statistics Screenshots 4 1 and 4 2 show what cen tering matrices look like and also that the 2x2 and 3x3 centering matrices are indeed idempotent CNTR 2 1 CNTR 2 CNTR 2 CNTR 2 Screenshot 4 1 Screenshot 4 2 Obviously the CNTR function can only generate one particular idempotent matrix of a specific dimen sion Therefore the utility file contains another function for the computation of an idempotent matrix the IDEM function which uses a property that holds for the Moore Penrose inverse A of a matrix nxm A mxn A A and AA as well as Z A A and J AA are idempotent matrices Karsten Schmidt Introductory Linear Algebra Course p 45 The Moore Penrose inverse of any matrix A is the unique matrix A satisfying the four condi tions AA A A A AA A A A A A AA AAN As DERIVE does not include a function for the computation of the Moore Penrose inverse the utility file includes the two functions MPIV a computes the Moore Penrose inverse of any nxl vector a 8 MPICA computes the Moore Penrose inverse of any mxn matrix A 9 Both functions are described in detail in Schmidt 2003 If A is a square non singular matrix the Moore Penrose inverse and the inverse A coincide and we have A A A A I and I A A I A A 0 ic we are back to the two standard examples for idempotent matrices from section 2
4. 20 6 de element is 19 de element is 146 8 de som is 7 de element is 6 de element s 12 9 de som is f amp de som 15 48 COMMAND MANS Build Calculus Declare Expand Factor Help Jump soLve Manage Options Plot Quit Remove Simplify Transfer Unremove moVe Window appro Enter option 7 de element s 117 Simp lt 19 D DOKUS SCHULENSKI Free 189 Derive Algebra 3 de som is 47 REKRIJ with DERIVE 3 14 and with DERIVE 6 1 left the file in its original Flemish Dutch version All problems are connected with arith metic series Examples like these could and still can be found in nearly all textbooks The first example should be read as the 15 element of an arithmetic series is 270 and the sum of the first 15 elements is 2055 Find the series 1 element and difference Solutions were not provided will come back to a modern version of this tool later in the pa per There was another attempt providing training tools for elementary calculation skills presented by Heinz Rainer Geyer 38 36 66 55 19 TO 97 TO 96 63 MIXEDC10 1005 34 37 66 103 59 55 4 16 19 35 wrong right right right right right right right right right COMMAND aU ayer Build Calculus Declare Expand Factor Help Jump soLve Mana ptions Plot Quit Remove Simplify Transfer Unremove mole Window approx User fimp User MENTAL MTH Free 97 Derive
5. Therefore the more interesting cases are when A is either square but singular or non square The matrix A from section 2 is such a singular matrix The Moore Penrose inverse of A is computed in Screenshot 4 3 MPICA Screenshot 4 3 The utility file includes the function IDEM A generates an idempotent nxn matrix from any mxn matrix A 10 IDEM A Screenshot 4 4 pa Karsten Schmidt Introductory Linear Algebra Course The formula used in this function is A A Screenshot 4 4 shows the idempotent matrix which is gen erated when matrix A is passed as the parameter and also the one generated by passing the 3x2 ma tr x of ones Note that finding the rank of an idempotent matrix is a relatively easy task as rank and trace are iden tical in this case for the first example in Screenshot 4 4 we get 2 4 2 2 for the second 5 1 We can also quickly double check this on the PC Screenshot 4 5 RANKCIDEM A 2 TRACE IDEM A 2 RANKCIDEM J 3 2 1 TRACECIDEM I 3 2 1 Screenshot 4 5 5 Orthogonal Matrices Finally we want to consider orthogonal matrices any square matrix A with the property A A The utility file includes the function ORTH a generates an orthogonal nxn matrix from any nx1 vector 0 11 The formula used in this function I 2aa nxn nx11xn b b 1 ORTH b Screenshot 5 1 generates an orthogonal matrix if a a 1 Hence the vector a whi
6. 1 LOADCO DOKUSSCHULE sk711s conics mth Fe start Condes Trainer Trainer f r Kege lschnitte conl Center 1n origin unrotated Kegelschnitt 1n Ursprungs lage cong shifted origin unrotated werschobener Kege lschnitt con3 conc shifted and rotated Kegelschnitt in allgemeiner Lage con random choice from above Zufallsaufgabe aus den ob igen ans gives the analysis ant liefert die Analyse 2 2 4 con3 20 x 38 x y 14 x 14 y 15 y 1 Type Center Real axis 2a Imaginary axis 2b 5 ans Hyperbola 0 06942 0 4414 0 7052 0 6266 Real Vertices Asymptotes 0 07450 0 1195 1 3 2 t 0 06942 0 6192 t 0 4414 0 2133 0 7634 0 4535 t 0 06942 1 435 t 0 4414 The student performs the calculation possibly supported by CAS and checks the ob tained results As an add on he she can plot the conic and its center the vertices and the asymptotes in case of a hyperbola 2 1 0 6 0 6 0 4 0 Josef B hm CAS Tools for Exercising p 37 presented a lot of examples from various fields of school mathematics performed with vari ous pieces of soft and hardware So you might have made up your own opinion about Pros and Contras for using technology in this very special way What are my personal Pluses and Minuses I ll start with the MINUSES Very too close to traditional fundamental mathematics education There is a danger that training could become a main activity in math
7. Screenshot 3 1 has full rank is nonsingular the other two have rank 1 As any element of a matrix generated by the RNDU 2 2 1 function is either 0 or 1 and as there are only 16 different 2x2 matrices with two different elements the probability of getting a matrix having full rank is lower than it is of getting a singular matrix This allows an interesting discussion of the rank of a matrix In general the probability of getting a rank deficient matrix decreases if the number of rows the num ber of columns or the max value is increased For example all three matrices in Screenshot 3 2 are of full row rank RNDU 2 3 9 RNDU 2 3 9 RNDU 3 9 Screenshot 3 2 Note that there is also a built in DERIVE function similar to the RNDU function the RANDOM_MATRIX m n s function generates an mxn matrix of random integers from the set s 1 s 2 1 0 1 s 1 pa Karsten Schmidt Introductory Linear Algebra Course 4 Idempotent Matrices As we have now already come across two examples for idempotent matrices any square matrix A with the property AA A namely square zero matrices and identity matrices we should spend a little more time with these matrices The utility file includes the function CNTR n generates an idempotent nxn matrix 7 The formula used in this function I J nxn nxn generates a so called centering matrix using an identity matrix and a square matrix of ones
8. and the vector a already defined we want to check the following property see Screenshot 2 3 A I IT A A mxn nxn mxm mxn I 3 A I 2 a cal 10 1 2 1 2 O14 Screenshot 2 3 It is also shown that the product of the 2x2 identity matrix with itself is the 2x2 identity matrix As this is true for any identity matrix which follows immediately from the above mentioned property by choosing A identity matrices are another standard example for idempotent matrices Karsten Schmidt Introductory Linear Algebra Course p 43 3 Random Matrices Many students appreciate having the option to use additional example matrices to practise certain top ics in matrix algebra The utility file therefore includes the function RNDU m n max generates an mxn matrix of random nonnegative integers 6 The random numbers are from the set 0 A max Le the third parameter determines the highest possible value for example if max 1 the generated matrix contains only Os and Is cf Screenshot 3 1 The RNDU function should only be called with a positive integer as the third parameter If max lt Q an error message is printed if max gt Q but not an integer no random matrix is generated Note that the RNDU function uses the built in DERIVE function RANDOM n with n max 1 0 RNDU 2 1 1 RNDU 2 2 1 1 1 RNDU 2 2 1 1 Screenshot 3 1 Only the last of the three matrices generated in
9. as well We will demonstrate this in 2010 We have good news for you Bernhard Kutzler gave permission to put all Conference Proceedings which were published by bk teachware on the ACDCA website www acdca ac at We will start uploading as soon as possible Philip Yorke who pub lished the Proceedings of the famous Krems Conferences 1992 and 1993 also per mitted to upload the Proceedings This will be more work because we don t have them in electronic form I Il scan the books and produce pdf files This will need some time so please be patient Have a look at the recommended websites I checked them just now they all are valid at the moment We wish a Merry Christmas and a Happy New Year 2010 Noor and Josef Bohm Jose Luis Galan and Josef are planning TIME 2010 right Finally I d like to remind you that dead line for submitting a paper for the TIME2010 conference is 15 March The website provides some information about accomodation and registration fees The Conference website is www time2010 uma es Download all DNL DERIVE and TI files from http www austromath at dug EN EDITOR IA OL N L 76 The DERIVE NEWSLETTER is the Bulle tin of the DERIVE amp CAS TI User Group It is published at least four times a year with a contents of 40 pages minimum The goals of the DNL are to enable the ex change of experiences made with DERIVE TI CAS and other CAS as well to create a group to discuss the possibili
10. b 2 5 If Pita bof2 eFC 0 fa bi f b la bof2 Ca bose BISECC a b ni ITERATES Hiv w I vw le bl m HO i ee Taken from an exam in my University April 2009 Topic Environmental techniques The goal To find the minimum height of a chimney of a thermoelectric central n a rural area to be concordant with the quality environmental standards m The equation to be solved for finding the height H 1 6 575 29 1 3 10 588 H 2 3 5 1 3 2 3 1 6 575 29 10 588 x 7 F x _ X 5 1 3 2 3 1 6 575 29 10 588 H 5 F H im H 5 600 MIA a a cr cl ne Kinn 300 600 400 200 _ gt po 400 600 80 A 400 V 600 l 800 Sketch the graph to approach the initial values of bisection method 9 BISECC 200 500 10 200 500 350 500 350 425 387 5 425 387 5 406 25 10 396 875 406 25 401 5625 406 25 401 5625 403 90625 401 5625 402 734375 402 1484375 402 734375 402 1484375 402 4414062 Figure 7 The height of a chimney For engineering purposes the chimney height is around 400 meters Conclusion The use of a toolbox adapted to each student s individual needs containing instructions that the stu dents themselves find useful must we believe surely reinforce the learning process since students participate actively and the use of the CAS is not limited to what is sometimes blind use of the CAS instructions References 1 Council of Europe
11. gives a task for cubing a binomial tr gives a task for squaring a trinomial pro asks for a product of two binomials di sum times difference of two monomia ls quiza presents a random problem of the above tasks sgben cubin quizbin gives n tasks of the same type res gives the correct answer and chfanswerj analyses single results The students are offered the instructions This might by the start of a session exercising cubing binomials 3 4 cu 2 n 3 u 3 u 5 ch 8 n 36 n u Sd m u 8 check cubes 3 Z 2 3 5 res Aen S6 n u 54 n u eu 3 p 7 5 3 7 cuba 2 g v 3 Am t 3 2 2 3 p lep 5 14 ps 343 5 3 Z 3 RA fest Sg tle geyvy O amp gevt yv 3 Z 2 3 Beem l182 m t Zd m t t As you can see tried to include some error analysis for the most common types of errors This is a nice programming challenge specially collecting the common types of errors together with the students Josef B hm CAS Tools for Exercising p 31 1 LOADCO VODRUSSCHULE ski115 factor mth 2 start Program for training factorizing expressions TOS ax Taetor QUE eT se Daria lly Factor aur sd complete square cu complete cubic di difference of squares co sum or difference of cubics 1S 355 TP noni quiz random problem With Toben prbin soben quizbin n tasks of the same type will be offered ch shows the correct res
12. hm Programmieren in Derive bk teachware 2002 3 J B hm Programmieren mit dem TI NspireCAS bk teachware 2008 4 H Heugl E Lehmann W Herget B Kutzler Unverzichtbare handwerkliche Rechenkompentenzen im CAS Zeitalter Indispensable Competencies in the Era of CAS http www acdca ac at material vortrag kompetl htm 5 H Heugl Helmut 1999 The necessary fundamental algebraic competence in the age of Computeralge bra Systems Proceedings of the 5th ACDCA Summer Academy 1999 http www acdca ac at 6 W Herget Rettet die Ideen Rettet die Rezepte Hischer Horst ed Rechenfertigkeit und Begriffsbil dung Zu wesentlichen Aspekten des Mathematikunterrichts vor dem Hintergrund von Computeralgeb rasystemen Hildesheim Franzbecker pp156 169 7 W Herget Wie viel Termumformung braucht der Mensch Taschencomputer und Mathematikunter richt Amelung Udo ed Der TI 92 im Mathematikunterricht Pfingsttagung 1998 Zentrale Koordinati on Lehrerausbildung ZKL Texte Nr 7 Westfalische Wilhelms Universitat Miinster pp 3 19 8 Derive Manual Texas Instruments 9 Voyage 200 Manual Texas Instruments Several articles published in the Derive amp CAS TI Newsletters DNL 10 Jan Vermeylen and J Bohm Vieta at Random DNL 20 1996 11 J Bohm The Trigonometric Superbox DNL 23 1996 12 H R Geyer MENTAL MTH DNL 27 1997 13 J B hm DERIVE as Problem Generator DNL 40 2000 14 J Wiesenb
13. many rows as orders of units including a very special symbol reminding us of a shell for filling up an order that could not be left empty of dots or lines lI The pictures were taken at my 3 weeks travel through Mexico in November 2009 Josef E Roanes Mayan Numbering System in DERIVE the Mayan zero Through this notation the Mayas could write as follows what in our sys tem would be a number m m w 20 x 20 y 20 z 20 For example the number 1 368 280 8 20 11 20 0 20 14 20 0 20 was written using dots lines and shells as it is shown in Figure 3 lt 0 0600 4 Edzna Temple of the five Stores Figure 3 Example of Mayan numerals For those interested in further mathematical developments of the Mayas see the book by Romero 2004 3 What DERIVE does about number base changes Regarding number base changes the CAS DERIVE only provides the possibility to choose the input and output bases from 2 8 10 and 16 in Options gt Mode Settings Therefore we ll try to complete its possibilities implementing the corresponding proce dures so that 1t can convert any given integer number into Mayan notation The mathematical background of each procedure is previously taught or retraced and the students are guided through the development of the implementations these details are omitted here for the sake of brevity As said above this unit was successfully experimented dur
14. might help to visualize the following set operations Sketch the Yenn diagrams for the given set operations by paper and pencil or using the technique described above 1 AuBiaC 2 AnCjuB 3 AuB n Buc 4 AnBju AnC 5 An Buc 6 AuCin AuBin BuC 7 Au CAB 8 An BYu An O 9 ANIBuC 10 A BjufAnC 11 AVC u C B 12 AuBuUC 13 AuBuC An Bn 14 AUBIn CaB 15 AVC u Bic 16 AnBlulAnC iuiBACIWANBAC The correct answer will be given by plat of the task ea pl3 12 plots A o Bru C You can visualize your own diagrams but you have to take In account that DERIVE cannot plot set operations but their eqivalent as Boolean expressions tis easy to convert the set operations to Boolean expressions Au A or B o AvB AnB gt A and B or AaB AY not A or AA AAB gt A and not B or AA aB Let s assume that student Josef wants to treat problem 10 He marks the disjunct subsets q2 and q5 A B together with q8 AABAC by entering and plotting q2 q5 q8 because he thinks that this is the solution Then he plots pl3 10 to check if he is right or not a2 95 08 p13 10 EA 2D plot 1 1 2D plot 1 1 will skip the Solving Triangles Training will follow in another DNL and remind you that the utility for working with times was published in DNL 74 What s the TIME Grandie One of my latest products is a tool for analysing conics p 36 Josef Bohm CAS Tools for Exercising
15. n 2 2 5 12 chc ab Qay as G x L af Having some experience in programming it is not really a problem to transfer the idea s to TI Nspire The screen shot shows the FACTOR utility The students have to do factorization by hands and then check their results by using the provided functions expand dib 3 25 m v2z 400 m factor expand AAl yz 240 hm p 540 h m p 560k n 22 875en gt 2 gt 25 m v2 z 400m vz 240 2 m p2 540 h m p 560 k n 29 875 ken 2 gt 25 m v2 22 2 2 m z 2 m 22 4m2 60 h2 3 h2 2m 3 h 2 m mp 35 en2 23 5 2 4 5 24 4 k 900 8 h 900 g h 900 g h factorl900 89 h 900 g 4A 900 8 gt p3 expandlsk 8 2 factorl224 c kez 28 c 2 nz ER 2 2 2 28 c hez z 2 c fz 2 0c z 4 C expandlzub 4 E A SPEER ee A pp 305 Ap dE eS 240n9 9 9 540m 9 10 1 2008 m 3 800 g m 3 99 Bre P k 24 S Pk don t want to demonstrate the Long Division because this can be found in DERIVE News letter 71 72 from 2008 Josef B hm CAS Tools for Exercising p 33 Many many years ago wrote a BASIC program for teaching and training set theory It was an intelligent program because implemented a parser in order to perform any set opera tion with 2 3 or four sets of numbers or characters given or randomly chosen e mau JAW Lie iRU L UUJ UJ UJ MENGE 30 10 1990 02 01 l
16. not commonly agreed amount seem to remain necessary in mathematics education The teacher can provide programs and functions to encourage self responsible learn ing training and repeating these skills It is the teacher and sometimes the cur riculum and the given standards which level of difficulty is appropriate will show some examples reaching from calculating GCD and LCM via factoriz ing and expanding expressions up to basics of calculus using various computer algebra systems This could raise a discussion about possible fields in math cur riculum which might be suitable for CAS supported exercising And this could also raise a discussion about advantages and disadvantages of using these or similar tools Let me start with a view some years back It was in 1994 when received a diskette from my friend Jan Vermeylen from Belgium no email at these times containing three files QUADR_EO MTH POLY_FAC MTH and REKRIJ MTH a QUADR_EQ 1KB MTH Datel 16 07 1994 15 02 al REKRI KB MIH Datei 14 07 2008 10 35 Let s try if REKRIJ works with the recent version of DERIVE too Josef B hm CAS Tools for Exercising p 23 19 SERCS3 D DOKUS SCHUL E skills DER314 EXE K 15 de element is 270 La de element ie 15 de som is 2055 15 de som is i de element is 38 12 de element is 30 gt de som is 5 de som is 60 4 de element is 19 de element is 4 de element 15 26
17. teaching am quite sure that the reader will add some other objections which probably will share My PLUSES are The tools could should be developed tested and improved in cooperation with the students which asks for deeper understanding of the algorithms and techniques Encourages self responsible exercising parallel to math education if neces sary Hesitating teachers could be led to the use of CAS and hopefully will not use CAS for training manipulation skills only but will later change or at least consider to change their methods teaching maths Students like it Let me add one more aspect which is very important from my point of view The kids pupils students have to learn a lot of rules believe that nobody will teach calculus without talking about the chain rule The students then know the rule and have in their minds Outer derivative times inner derivative which is ok But it is my strong opinion that a special competence is needed to apply this rule in the ap propriate way and to know when just this rule must be applied And this is valid for many cases applying rules also in daily life Knowing the rule is not sufficient enough One must be able to recognise when and how to apply the rule and not in all cases does the com puter all the work Exercised enough p 38 Josef Bohm CAS Tools for Exercising References 1 J B hm Mathe Trainer I bk teachware 2000 2 J B
18. the basic ideas can be realized using almost all tool tried to pro gram my VIETA with WIRIS a CAS which recently has become popular in some Austrian schools It is not the tool which decides what can be done it is always the user teacher who forms the software according to his or her didactical intentions and needs see the WIRIS screen shot on the next page Id like to know in how many CA systems will program VIETA in the future Josef Bohm CAS Tools for Exercising pas 14 WIRIS Desktop 2 D DOKUS wiris vieta wiris File Edit Tools Help El WIRIS CAS i Edit Operations Symbols Analysis Matrix Units Combinatorics Geometry Greek Progran v_ a b c d e f g h i j k l m n o p q r s t u v w x y z vieta begin local prob 21 1 Fandom 1 2 random 1 15 22 4 Fandom 1 2 random 1 15 23 random 1 25 VP V z3 sol z1 z2 prob expand vp 2z1 vp z2 0 end vieta gt r24 16 r 63 0 sol gt 7 9 vieta gt n2 19 n 84 0 sol gt 12 7 vieta gt e2 22 e 105 0 sol gt 7 15 fi Let s have another jump Even in times of Fir Fer Foxy F r o Fer JAT Info Ende WFF linear Iinkggra I CAS the basic rules of differentiation and integration should be mastered by the stu dents Calc a program package for the Tl handheld devices CAN help It is in your responsibility as teacher up to which level
19. time whenever they are needed during the course This paper demonstrates how beneficial it is for students to sit in front of a PC in an introduc tory linear algebra course from the very basic to the more advanced topics 1 Some preliminaries A matrix 1s a rectangular array of elements in m rows and n columns We denote a matrix by a bold face capital letter e g A In DERIVE a matrix can be defined simply by clicking on E setting the number of rows and columns and entering the elements An element of a matrix 1s denoted by the corresponding lowercase letter with a double index the first index being the row and the second the column index For example the element in the second row and first column of A is denoted by a A matrix with only one column n 1 is a column vector Hence throughout the course a vector has to be defined in DERIVE as a matrix with one column the vector data type e g defined by click ing on is not used We denote a vector by a boldface lowercase letter e g x Row vectors are not defined transposed column vectors are used instead A matrix is transposed by taking its rows col umns and writing these as columns rows of the transposed matrix The transpose of a matrix or vec tor is denoted by a prime e g A Clearly the transpose of an mxn matrix is an nxm matrix In DERIVE the transposition sign is a grave accent which can be entered for example by clicking on MI in the symbol list
20. A NGABE 19 17 12 20 15 2 3 29 7 10 16 11 6 22 120 16 17 11 6 7 15 12 25 22 2 9 3 10 128 2 6 15 3 3 11 10 112 23 11 160 Aufgabe AN BIN AN C AN BD SN ANC 9 17 153 Das stimmt leider nicht AN B S ANC 16 12 29 CENTER gt The set operations could be visualized Problem AUB n AUCI n CEUC Fur die Hengen nur den Buchstaben fur die Teilmengen nur die Ziffer eingeben Teilmengen werden grin gefullt Beleg te Teilmengen konnen durch nochmaliges n sprechen geleert wer den Die Eingabe kann ab geschlossen werden wenn nur mehr dunkle Teile vorkommen r ich t i yg xX ENTER 7 lost das Problem The nice thing is that students in my former school are still using this antique program which is running in the DOS environment It was a challenge for me to produce a similar tool with DERIVE without facing the problem to implement again a parser CAS does the job for me This is the recent version of set theory exercising and learning The students have the choice between basics and more complex operations with 2 3 or 4 sets The program provides a random generated universal set with the respective subsets and a list of problems Here is the section for tasks involving three subsets p 34 Josef Bohm CAS Tools for Exercising Given 15 a universal set together with 4 subsets A B C and D Find the following sets Simplify sols for the solutions u E n Bu
21. AGED 6 A DERIVE implementation of a Mayan number system converter Once we have the procedures of Sections 4 and 5 1t is very easy to implement a Mayan number system converter The idea is to convert first the given number to base 20 to trans form the numerals to Mayan notation and finally to stack these numerals maya n Prog maya_aux base_bl n 20 RETURN maya_aux R Prog If R mayal9 Ry1 If R DISPLAYC If R maya_aux REST R RETURN p20 E Roanes Mayan Numbering System in DERIVE Example 2721 6 x 20 x 20 16x 20 1 mayal27 21 7 Acknowledgments This work was partially supported by the research projects TIN2006 06190 Ministerio de Educaci n y Ciencia Spain and UCM2008 910563 UCM BSCH Gr 58 08 research group ACEIA Spain 8 Conclusions We do not know of any other similar implementations in any CAS apart from URL 2009 that uses special facilities for inserting graphics and the very similar Roanes Lozano amp Gonzalez Redondo 2009 written in Maple We believe this is an interesting example of synergy among different branches of knowl edge Mathematics History of Mathematics and Computer Science that can increase the interest of students for different topics The only negative issue we find is that parts of the implementation are a bit tricky Section 5 something that can distract students from the mathematical goals of the unit Perhaps this part cou
22. Algebra Sometimes like to reanimate my old DOS versions of DERIVE pas Josef Bohm CAS Tools for Exercising Allow a time step to the handheld devices My school was the first College for Business Administration in Austria where CAS was used in all classes When the TI 92 appeared on the market we the math teachers asked the parents to buy a device for their children for enabling a modern and attractive math educa tion It was in these times when remembered another old BASIC program which was later replaced by a DERIVE tool for exercising solving quadratic equations applying the rule of Vieta 9 2 v etal z 14 2 13 antl 13 richtig Fi letal z 5 2 0 ant O0 5 richtig 2 vieta 1 19 7 48 ant 3 16 falsch sondern 3 16 2 v etal lv w 72 antl 9 richtig Aufgaben gestellt 4 results davon richtig das sind in And here is the Tl Version Problems 1 Problems 3 g 18 4 80 0 c 10 c 11 0 1 Solution E Solution rd Solution right sorry false an nea el ee ee End ESC next any End ESC next any The idea remained the same the tools changed The pupils used the program during the breaks in school often in the bus or railway on their way to or from school and sometimes only just for fun One female pupil told me that she used her VIETA to concentrate before learning for other subjects Just to demonstrate that
23. C A n B u A n D Bu cn D An Cc uD A uv CO n Au D n Cu D Cu D A An B aD B D CB A AY Bu C 41 B u A D B Cc u BnD A uc uD Cn CA n B D u A 0 A u BY n CC n D Cu B C D Universal Set 2 8 11 13 16 21 24 29 30 Subset A 2 4 13 16 21 30 15 three subset B 8 11 16 21 301 Subset C 2 8 14 13 16 21 24 25 30 subset D 18 11 15 21 29 30 The students shall solve the tasks by paper and pencil and then compare their results with the given solutions Own operations can be entered It is also possible to generate sets of random generated sets for self made exercises call the solutions for the tasks given above 12 8 11 13 16 21 30 8 16 21 30 8 11 16 21 29 30 2 8 13 16 21 30 2 8 11 13 16 21 29 30 111 24 29 17 sals it 2 13 8 11 16 21 30 12 11 13 24 29 12 11 13 24 25 8 211 165 21 29 30 24 u Providing exercises with Venn diagrams was not so easy but finally found a way to realize this in a satisfying way show the part with three subsets embedded in a universal set First see the instructions together with a list of provided problems Josef B hm CAS Tools for Exercising p 35 Exercising set diagrams Venn diagrams with three sets Double click on the provided graph to open the 2D plot window Editing 4 or E or and plotting shades the respective sets editing and plotting g1 through g shades the distinct disjoint subsets This
24. COE 1997 Convention on the Recognition of Qualifications Concerning Higher Education in the European Region Lisbon 2003 http www bologna berlin2003 de pdf Lisbon_convention pdf 2 European Ministers of Education The European higher education area Joint declaration of the European Ministers of Education convened in Bologna 19 June 1999 http www crue org decbolognaingles htm 3 GARC A A GARC A F L PEZ A RODRIGUEZ G DE LA VILLA A C lculo I Teor a y problemas de An lisis Matem tico en una variable CLAG 2008 4 KUTZLER B Improving Mathematics teaching with DERIVE Chartwell Bratt 1996 5 STEWART J Calculus Thomson Brooks Cole 2009 6 Tunning Educational Structures in Europe project Approaches to teaching learning and assess ment in competence based degree programs 7 VALC RCEL M Ed La preparaci n del profesorado universitario espa ol para la convergencia europea en educaci n superior Universidad de C rdoba 2004 8 VEZ J M y MONTERO L La formaci n del profesorado en Europa El camino de la convergencia Revista espa ola de pedagog a 230 101 122 2005 9 WEIMER M Learned Centered Teaching Five Key Changes to Practice San Francisco Jossey Bass 2002 Contents of the Toolbox complex numbers dfw staringles dfw realproblem dfw parabolalength dfw numericalmethods dfw integralcalculus dfw differentialcalculus dfw E Roanes Mayan Numbering System in
25. Canada Some portions of this paper are also published in the Proceedings of the 7 Delta Conference in Gordon s Bay South Africa References Schmidt K 2003 An Introduction to the Moore Penrose Inverse of a Matrix The DERIVE Newsletter 50 12 18 Schmidt K 2009 Teaching Matrix Algebra in the PC Lab Proceedings of the 7 Delta Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics 216 224 ACAO9 DERIVE session presenters part 2 Josef Bohm Michel Beaudin Many thanks to Jose Luis Galan and to Gilles Picard for providing the pictures
26. DERIVE An Implementation of the Mayan Numbering System in DERIVE Eugenio Roanes Lozano Francisco A Gonzalez Redondo Algebra Dept School of Education Universidad Complutense de Madrid Spain eroanes mat ucm es faglezr edu ucm es Abstract The Mayan number system is a base 20 positional to be read from top to bottom not from left to right system that makes use of a symbol representing zero It has slightly dif ferent variations when used for counting days in religious and astronomical contexts Therefore 20 symbols are needed to represent 0 2 9 Of these the zero was denoted by a shell and the positive ones were represented using dots and horizontal segments If a number is greater than 20 the symbols corresponding to units twentieths 400 s 8000 s are stacked from bottom units to top in pure base 20 We have implemented a procedure that allows to convert numbers between any bases and that returns the output in row vector style another procedure that builds the 20 Mayan symbols for 0 1 2 19 and yet another procedure that uses the previously mentioned procedures converts any number from base 0 to base 20 and represents it in the Mayan numbering system We believe this is an interesting example of synergy among different branches of knowledge Mathematics History of Mathematics and Computer Science that can increase the in terest of the students for different topics 1 Introduction Let us tr
27. THE DERIVE NEWSLETTER 76 ISSN 1990 7079 THE BULLETIN OF THE DIT rv LZ vst coue ner Contents Letter of the Editor Editorial Preview User Forum Windows 7 and DERIVE ACA 09 The DERIVE Session 2 Agustin de la Villa a o A Toolbox with DERIVE Eugenio Roanes Lozano 6 Francisco A Gonz lez Redondo An Implementation of the Mayan Numbering System Josef B hm CAS Tools for Exercising Karsten Schmidt Making Life in an Introductory Linear Algebra Course Easier with DERIVE December 2009 D N L 76 Information D N L 76 Interesting and recommended websites Lehrstuhl f r Didaktik der Mathematik Uni Erlangen Prof Thomas Weth www didmath ewf uni erlangen de Homepage hp_weth htm Die Zeitschrift f r MathematiklehrerInnen Universit t Salzburg und PH Salzburg www mathematikimunterricht at Newsletter index_news html The Journal of Symbolic Geometry journal geometryexpressions com Documents of Maths amp Stats amp OR www ltsn gla ac uk headocs Among others a resource for background pictures staff spd dcu ie oreillym geometry htm Teach Engineering Resources for K12 www teachengineering com index php A publication of the Institute of Electrical and Electronics Engineers www ieee org web education preuniversity tispt lessons html Online proceedings of the CADGME 2009 Conferrence is available at www risc uni linz ac at about conferences cadgme2009 The first Journal of Mathematical Model
28. a Francisco Garcia Universidad Polit cnica de Madrid Departamento de Matematica Aplicada E U Informatica alfonsa garcia eui upm es gmazario eul upm es Gerardo Rodriguez Universidad de Salamanca Departamento de Matematica Aplicada E P S de Zamora gerardo usal es Agustin de la Villa Universidad Pontificia Comillas Departamento de Matem tica Aplicada y Computaci n ETSI CAT Universidad Polit cnica de Madrid Departamento de Matem tica Aplicada E U I T Industrial avilla upcomillas es Abstract An analysis of some of the characteristics of the European Higher Educa tion Area EHEA its difference in relation to the current University teaching sys tem and the role that new technologies might play in this new scenario has been performed This paper suggests a new possibility in use of technologies The de sign of a toolbox with DERIVE instructions about topics in a usual Calculus course Introduction The implementation of the EHEA see 1 2 implies new teaching methods taking into account that the students now are the centre of the learning process The role of teachers changes and they must be able to guide their students work see 6 7 8 9 Teachers are currently being required to change the traditional teaching model in order to adapt to learning based on competences It is necessary to define the competences to be acquired by the stu dents after attending a course on a certain subject and to de
29. a REP y f a DR a x a f has no derivative FIX Ifx lt 0 x x2 ANGENTE2 O f has no derivative With the DERIVE instruction TANGENT the result 1s not correct 8 TANGENT TEx x 0 9 Figure 2 TANGENT and TANGENT2 functions Taking advantage of DERIVE graphical capacities and of the structure of the IF instruction the monotonicity and concavity or convexity of a sufficiently differentiable function can be analyzed For example with the instruction CRECE x IFCF x gt 0 F x it is possible to represent the curve y F x in the intervals where F is increasing The graph of figure 3 has been obtained by applying the instruction CRECE x with F x x x Figure 3 Intervals where F x x x is increasing Other functions about Differential Calculus in the toolbox may be ROLLE_POINT for finding a point according Rolle s Theorem hypotheses LAGRANGE_POINT for finding a point according Lagrange s Theorem hypotheses p 10 Agustin de la Villa A Toolbox with DERIVE These functions can work in exact or approximate way because it is necessary solving equations 2 3 Integral calculus In order to introduce the Riemann integral we can use see 4 the DERIVE instructions AreaUnder Curve and LEFT_RIEMANN Furthermore we have included in the toolbox the DER_RIEMANN for calcu lating the sum for right rectangles associated to Riemann sums We also have implemented the rect_izq and rect_der func
30. at the bottom right of the DERIVE window A matrix which has the same number of rows and columns m n is called a square matrix The diagonal from the upper left to the lower right of a square matrix 1s called the main diagonal pao Karsten Schmidt Introductory Linear Algebra Course 2 Matrices and Vectors Containing Os and Is It is important for students to develop a good understanding of e matrices and vectors containing only Os zero matrices and vectors e matrices and vectors containing only 1s matrices and vectors of ones and e square matrices containing Is on the main diagonal and otherwise Os identity matrices While zero matrices and vectors behave similarly to the number 0 ell matrices and vectors of ones do not behave similarly to the number le but rather identity matrices are the matrix algebra ana logues to le_ However it is rather tedious to verify this in exercises The utility file therefore in cludes the following five functions O m n generates an mxn Zero matrix 1 o n generates an n x1 zero vector 2 J m n generates an mxn matrix of ones 3 1 n generates an nx1 vector of ones 4 I n generates an nxn identity matrix 5 The names of the functions were chosen such that they are close to commonly used symbols for such matrices O for zero matrices J for matrices of ones J for identity matrices o for zero vectors and I lowercase L for vecto
31. auer A Macro for Solving Equations DNL 42 2001 15 J B hm Long Division Step by Step DNL 71 72 2008 16 J B hm What s the Time Grandie DNL 74 2009 There is some space left So cannot resist to add my Taylor Series Exercising Tool function xO degree 0 05 1 77 He tay 3 7 a 0 04 16 x 0 03 3 E 160 280 x 168 35 To cheek wrong 0 02 Fi 5 5 4 iT iT AT AT 7 01 3 E E 3 180 2 280 7ee 16 7 ox 35 77 al gt Hd ras zoa In 7 0 07 IT 5 0 04 The student is given a random generated function together with location xO where to develop a Taylor series of requested degree plot shows the respective graphic representation Karsten Schmidt Introductory Linear Algebra Course p 39 Making Life in an Introductory Linear Algebra Course Easier with DERIVE KARSTEN SCHMIDT Schmalkalden University of Applied Sciences Germany email kschmidt fh sm de The Schmalkalden University Faculty of Business and Economics moved its intro ductory linear algebra course from the classroom to the PC lab and purchased a DERIVE license that also allows its use on the students own PCs A DERIVE util ity file was then developed to facilitate exercises with special matrices throughout the course This utility file allows the computation of zero matrices and vectors matrices and vectors of ones random matrices as well as idempotent and orthogo nal matrices just in
32. ch is passed as the parameter will be transformed within the ORTH function such that it is of length 1 Let us nevertheless start with a vector b that has length 1 anyway as in Screenshot 5 1 Karsten Schmidt Introductory Linear Algebra Course p 47 In order to check if this matrix is indeed orthogonal cf Screenshot 5 2 we have to compute both its inverse and transpose and see if these two matrices are identical Since for any orthogonal matrix we have A A A A I a second method to prove that a matrix A is orthogonal is to show that A A T 1 ooo ORTH b ORTH b 0 0 J 10 ORTH b ORTH b Y 1 Screenshot 5 2 For a second example Screenshot 5 3 we choose the 2x1 vector of ones as the parameter of the ORTH function 0 1 ORTHC1 2 A 0 1 0 0 ORTH 1 2 ORTHC1 2 0 10 ORTHO1C299 ORTH 1 2 oO 1 Screenshot 5 3 To complete this section we take advantage of the graphical capabilities of DERIVE Any column of an orthogonal matrix has length 1 and any two are pairwise orthogonal Since both examples are two dimensional its column vectors lie on the unit circle and form a right angle Screenshot 5 4 The graph also shows the vector b from the first example Screenshot 5 4 Karsten Schmidt Introductory Linear Algebra Course Remarks This paper is linked to a presentation given on June 26 2009 at the ACA 2009 Conference in Mont real
33. eas to any other CAS which is programmable like MATHEMATICA MAPLE MAXIMA WIRIS TI NspireCAS In my abstract mentioned GCD and LCM So I ll start with exercising calculating GCD and LCM of numbers Josef Bohm CAS Tools for Exercising p29 11 start Trainer fur ggT kgV und kgV Trainer for ged and lcm ggtkgv liefert eine Aufgabe gcdlcm offers one task res gibt die richtige Antwort aus res delivers the correct answer Find gcd and Jem of 12 gcdicm 36828 468 1800 18972 GCD 36 13 res LOW 406949400 See next the the modern version of the arithmetic series program remember Jan Vermey len s file from the DOS times This version provides the solutions too Trainer for arithmetic series ar_s offers a problem the last row shows the questions ans gives the correct answer a 1 HY difference difference a a In 2000 bk teachware published my Mathe Trainer containing packages for the TI 92 There was one package algebra which was intended for my students to exercise among others expanding powers of binomials Some years later when could use DERIVE version 5 was on the market and with it the possibility for programming Two utility files were produced expand mth and factor mth p 30 Josef Bohm CAS Tools for Exercising 1 LOADED DOKUS SCHULE sk71715 expand mth 2 start Expanding Expressions sd gives a task for squaring a binomial cu
34. es with zero matrices 1 LOAD CC Program Files TI Education Derive 6 Math SpeM mth TET A 0 3 2 0 0 oo o 3 A 0 0 0 00 0 o 2 0 3 0 O DJ 0 2 2 0 2 2 Screenshot 2 1 We continue with matrices and vectors of ones For the previously defined matrix A we want to check some of the following properties n 2 2 j l a 1 la yYa Al i l lxn nxl mxn nxl n 2 a j l m m Az S 4 Ya lxm mxn i VAL Na J 11 J J nJ lxm mxn nxl A mxn mxlixn mxn nxp mx p Clearly the first four properties show that vectors of ones are useful for the computation of sums namely of the elements of a vector or the row or column sums of a matrix or the sum of all elements of a matrix Screenshot 2 2 shows a few exercises with matrices of ones paz Karsten Schmidt Introductory Linear Algebra Course 1 2 a al a2 u A 103 15 24 13 A 12 15 18 1 3 A 1G3 45 33313 JQ 3 1G 4 3333 Screenshot 2 2 For a change the summation of the elements of a vector a is done for a 2x1 vector of arbitrary ele ments which are denoted by al and a2 The result of the multiplication of two matrices of ones is unexpected by many students as it is not another matrix of ones After realizing that matrices of ones do not behave similarly to the number lel attention can im mediately be directed to identity matrices For the matrix A
35. ex number e Find the n vertices of a regular polygon The following example allows one to calculate the n roots of a complex number and taking advan tage of the graphic capacities of DERIVE plot a start Arche d r Pere mdu Dmplbhcar Peske aki Dsciones Veni Eds Cae RS aha ARH ser ras m J EN 1 4 Ehre 10 Ed Eip segle Complex numbers a bed 1m PHASE 2 Z kum Inf PHASE Z 2ekom starn m vorm z os Izl SIN t 0 fi j mi i Bei Regular Hexagon atard 1 ERFERBEIDHEEMEGEEBEREIDEIEIE 1 1 x 4 4 DENSR CBE BEN a IEEE LA Avelar 7 44d 4d dd doi Jn Figure 1 A regular hexagon with a six pointed start 2 2 Differential calculus The utilities performed allow the analysis of the continuity or discontinuity of the function at a point studying the one sided limits analyzing the equality of the obtained values and comparing with the value of the function at the point The differentiability of a function at a point is studied following the same strategy The calculation of the tangent line has been implemented with additional information to the TANGENT function of DERIVE since it tells us when the function is not differentiable at the point see figure 2 Agustin de la Villa A Toolbox with DERIVE po The tangent f x f a h f a DL a 11m h gt 0 h f a h fia DR a lim h gt 0 h TANGENTE2 a IF Ra Dita D
36. in order to fill any possible gap in the place where some numerical quantity should be supported by the god corresponding to the said order Ifrah 1999 In short this system conceived at the temples was perfect for representing dates elapsed days prepare ritual celebrations etc although with 1t no arithmetical operation could be performed For such operations which were well apart from religious rites and more related to administration and commerce the Mayas put aside the irregular tun year of 360 kins assum ing the natural order 20 20 400 proper of a base 20 numbering system While in our usual base 10 system we need nine numerals and the zero for writing their numerals the Mayas needed nineteen numerals But the symbols used for each of these nine teen units were very simple dots and horizontal lines see Figure 2 gt ee eee e eee 0 2 3 4 e a o eee 15O WI 5 6 7 8 9 E o eee ET 10 1 12 13 14 ee eee 6 0 0 0 m se ain wee s wee eee 15 16 17 18 19 Palenque Temple of Inscriptions Figure 2 Mayan numerals When we write today a positive four numerals integer n in our usual base 10 system n wxyz where w x y y z can vary from 1 to 9 or 0 we are really shortening the expression n w 10 x 10 y 10 z 10 For writing any number greater than nineteen the Mayas placed the symbols referred in a vertical column containing as
37. in the previous section a supplementary toolbox has been created that can be used in a more specific way With the toolbox our aim is to extend the use of the CAS in our case DERIVE and we therefore try to use as far as possible its symbolic numerical and graphic capacities Some of the utilities are improvements to DERIVE instructions for example the TANGENT2 method where a distinction is made between differentiable and non differentiable functions Other tools are analogous to known instructions integrated in DERIVE whose syntax is complicated For this reason we prefer students to automate the algorithms according to their own criterion so that they get a more continuous use when they have to apply these concepts in other topics In addition this automation guarantees that they have understood the corresponding concepts Our toolbox could be designed with several compartments one for each section of Calculus of One Variable course e Complex Numbers e Limits and Continuity peo Agust n de la Villa A Toolbox with DERIVE e Differentiability e Integral Calculus including numerical integration 0 Numerical Methods for solving nonlinear equations 0 Below we briefly explain the tools 2 1 Complex Numbers The Complex Number compartment includes tools for e Convert a complex number to exponential form e Plot a complex number as a pair of real numbers e Compute a list with the n roots of a compl
38. ing and Application of the Reference Center for Mathematical Modeling in Teaching CREMM The first number is on line at furb br ojs index php modelling Teaching Math through Culture a very interesting site www rpi edu eglash csdt html Just recently I was informed by Philip Yorke former DERIVE dealer for UK about a very rich resource of A Level Math Exam questions and others Many thanks Philip for this valuable notice Have a look or more than only one www mathsnetalevel com Please inform me about interesting websites that we can share our resources Josef LETTER OF THE EDITOR Dear DUG Members This is DNL 76 containing the remaining lectures of the special DERIVE session at ACAO9 in Montreal With DNL 77 we will proceed publishing original contributions intended for publication in the DERIVE Newsletter The User Forum is not very extended in this issue because the articles needed a lot of pages I received a mail from Santa Claus about a lot of DERIVE bugs You will find them in the next DNL As now WINDOWS 7 is on the market I have been asked several times if DERIVE will work under WINDOWS 7 Yes it does The respective information provided by DUG Members is given in the User Forum Many thanks to Gunter and Peter Among many others Phil Todd joined the DUG this year Phil is author of Geometry Expressions which is an excellent piece of software GE enables exporting of re sults to DERIVE and to TI Nspire
39. ing the 2008 2009 course with the CAS Maple E Roanes Mayan Numbering System in DERIVE 4 A DERIVE implementation for number base changes It is straightforward to implement the auxiliary procedure or function integer quotient in DERIVE As DERIVE s command MOD returns the remainder of the integer division we can follow the integer division definition a MOD a b quo a b Now the previous procedure can be used as subprocedure by a short recursive procedure that converts numbers from base 10 to any base and returns the output in row vector style base_bl n b If n lt b En APPEND base_bl quo n b b MOD n b Examples base_b1 399 20 19 19 base_b1 400 20 1 0 0 5 A DERIVE implementation of the Mayan numerals Representing the Mayan numerals 0 to 19 is tricky and the less interesting procedure from the mathematical point of view mayal9 n Prog Ir ne 0 DISPLAYC 0 If n l DISPLAY N If n 2 DISPLAY If n 3 DISPLAY 5 If n 4 DISPLAY 0 If ng 5 DISPLAY _____ mayal9_aux n RETURN E Roanes Mayan Numbering System in DERIVE mayal9_aux n Prog If MOD n 5 gt O mayal9 MOD n 5 If quo n 5 1 mayal9 5 If quo n 5 2 maya_10 If quo n 5 3 maya_15 RETURN maya_10 PROG DISPLAY _____ DISPLAY ____ maya_15 PROG DISPLAY _____ DISPLAY ____ DISPLAY _____ Example 9 TEEN
40. ld be provided to the students implemented beforehand a The Observatory Chich n Itz Uxmal Temple of the Wizard E Roanes Mayan Numbering System in DERIVE References Grube N ed Maya Divine Kings of the Rainforest K nemann Kohln Germany 2000 Ifrah G The Universal History of Numbers John Wiley amp Sons New York 1999 Landa D de Relaci n de las cosas de Yucat n Yucatan before and after the Conquest The Maya Society Baltimore 1937 Roanes E Did ctica de los convenios en que se basan los sistemas de numeraci n Gaceta Matem tica XXIII 3 4 1971 3 13 Roanes Lozano E Gonzalez Redondo F A Implementaci n del sistema de numeraci n maya en Maple una experiencia interdisciplinar Bol Soc Puig Adam 83 2009 34 45 Romero Conde P Numerolog a matem tica maya Centro de Estudios del Mundo Maya M rida M xico 2004 URL 2009 http www mapleprimes com forum convertmayannumber These are the Spanish presenters of the ACAO9 DERIVE session F i 7 7 4 de a an Jose Luis Galan Agustin de la Villa Pedro Rodrigues Cielos The second part of the Gallery is on page 48 paz Josef Bohm CAS Tools for Exercising Josef Bohm Austrian Center for Didactics of Computer Algebra DERIVE User Group and Technical University of Vienna nojo boehm v at Abstract Even in times of a CAS a lot of basic manipulating skills up to a
41. luding the historic antecedents y Translating the problem into mathematical language mathematization y Solving the proble l Implementing the solution found There is a didactic antecedent to this unit at a lower educational level and without using technology Roanes 1971 We do not know of any other similar implementations in any Computer Algebra System CAS apart from URL 2009 that uses special facilities for in serting captured drawings and doesn t have a didactic orientation or the very similar Roanes Lozano amp Gonz lez Redondo 2009 by the same authors of this paper but written in the CAS Maple that was successfully experimented during the 2008 2009 course 2 The Mayan Numbering System The Mayan civilization developed a characteristic world of astronomical and mathemati cal knowledge in and from their temples the most appropriated place for a political system organized as priestly autocracy which based its power upon Astrology Without any observational instrument they did not know glass and consequently could not manufacture lenses without sand or water clocks for computing periods of time as hours minutes or seconds their smallest unit of time was the kin or day whose duration was meas ured through the repeated registration of the shadows projected by a wooden bar placed verti cally on the ground known as the gnomon also used by the Greeks for determining the azi muth and the height of the sun
42. merical Integration 2 4 Numerical calculus for solving nonlinear equations The methods usually taught to students for solving equations are Bisection Newton and fixed point DERIVE has the NEWTON and FIXED_POINT algorithms integrated so we can propose that students define a function to implement the bisection method An algorithm for this method could as follows Hia b If Fita b Fla lt 0 la a b 2 If Fita b F b lt 0 Ca b 2 b Ca b 2 a b f2 BISECC a b n ITERATESCH v v v la b n 1 2 Figure 6 Bisection Method The instruction BISSEC a b n see figure 6 computes the interval obtained after n iterations of Bi section method to solve the equation f x 0 in the interval a b 3 Using the toolbox to solve technical problems Students are encouraged to use the toolbox to solve technical problems Here we set out one example taken from an Environmental Sciences exam that has been used by our engineering students Specifically the equation to be solved which appears together with its solution in the figure 6 pro vides the height of the chimney of a thermoelectrical station in a rural area with the required quality standards p 12 Agustin de la Villa A Toolbox with DERIVE Numerical methods 1 Fix tc 2 PUNTO FIJO s0 in t TTERATESCHIS I amp 30 m 3 Fix tx Fix Ha Mwea mi tz rreeaTes ee ea Fie Hla by z If Pita biz bal 0 la La
43. olic or graphic capacities such as the TI92 or Casio ClassPad 300 The teacher should suggest a series of essential tools depending on the corresponding subject that the box should contain Students must to define the corresponding functions test them and add those they find appropriate In addition they must complete their work by writing a brief user s manual for their tools If they have created a good toolbox and they have also understood the algorithms well they will then have their own resource which they will be able to use in other subjects during the same or following years Before finishing with the general ideas on toolboxes it should be noted that nearly all CAS offer a very thorough toolbox and that the creation of new tools or the modification of those already available will only be necessary or convenient for reasons of ease of use or as a teaching strategy in order to adapt them to the user s specific needs 2 A toolbox of Calculus In this section we propose a toolbox that can be done by students of Calculus in the first course of Engineering whose reference text book may be 3 or 5 Before present our toolbox is necessary to make it clear that in DERIVE are implemented the most of the instructions for the study of Calculus of One Variable Thus the calculation of limits derivatives integrals or Taylor polynomials is basic using the DERIVE menu or instructions Following the strategy referred to
44. roblem solving Teachers often complain that outside the context of the subject for example in later years students are not able to use the mathematical skills acquired during the basic years of their training They seem to have a kind of mental laziness that prevents them from remembering and using what they have learnt and in many cases they do not have a fast and effective way to find the information or the ap propriate methods either The use of technology in the classroom is increasing and in certain cases students are even asked to define some tools that allow them to automate certain simple tasks such as for example the implemen tation of a function to calculate the tangent to a curve y f x at a point and subsequently use it in other problems In general however when teachers suggest to students that they should define a method aimed at automating a mathematical task they merely wish to help them understand the corre sponding algorithm and they do not usually expect students to design their own resources or use the implemented functions in following years Agustin de la Villa A Toolbox with DERIVE One educational activity is to encourage students to create their own well organized Toolbox for solving mathematical problems This toolbox is no more than a file or collection of files of utilities programmed in the characteristic programming language of a CAS Derive Maple Maxima or even on a calculator with symb
45. rs of ones since this resembles the symbol 1 which is typically used for vectors of ones It is important for the students to understand that if they use the utility file 1 e if the file is loaded into DERIVE they cannot use the above names for anything else e g for a matrix as this would overwrite the function in the utility file Note that the I n function was added simply to have a shortcut notation for DERIVE s built in function for the computation of identity matrices IDENTITY_MATRIX n The utility file also contains a function that generates an mxn random matrix of nonnegative integers cf next section The third parameter of this function determines the highest possible integer such that the following function is actually within the scope of this section RNDU m n 1 generates an mxn random matrix of zeros and ones 6a We define the matrix I 223 A 4 5 6 3x3 7 8 9 for which we want to check some of the following properties of zero matrices AO 0 OA 0 O 00 OO 0 mxnnxl mxl Ixmmxn Ixn mxn mxlixn mxnnxp mxp Karsten Schmidt Introductory Linear Algebra Course p 41 Note that the last of the above properties implies that the product of any square zero matrix with itself iS a Square zero matrix of the same dimension making square zero matrices a standard example for idempotent matrices any square matrix A with AA A is called an idempotent matrix cf section 4 Screenshot 2 1 shows a few exercis
46. rts amp ine substitution pfr partial fractions guiz random problem quiz random problem out of all levels n tasks of same type pol2bin pf2bin quiz2bin check Integralj or check Intl Int2 qlves the answer Ccorrect true ans returns the correct antiderivative Let s have 5 integrals of level 1 to exercise integration by substitution together with the cor rect answers The students should do the integration by hands and then double check their results 143 1 3 1 3 9942 G2 34x 7 2 3 16 16 33e 7 2 dix 20x fix Z x 5 e 20e 1 2 16 x DiX f 16 x Dax 7 amp SIN ee 48 subst1lb S q Efix 159 005 ees 3 2 5 3 2 5 49 ans 3 3 2 2 47x Sax rox Gox F SIN ll 7 94 x 27008 2i amp e 5 2 Soo 2 2 5 SINCI6 x 20 COS 16 x ST 4 The next figure will show a level 2 quiz 5 problems This is a random selection of all possi ble problems When was a teacher and needed problems for a test did not prepare the problems at home but ran my problem generator files and eg simplified twice quiz1b 10 for two groups of students in the class So they could not blame me for providing extra difficult tasks for the test the computer made the selection Josef B hm CAS Tools for Exercising p 27 15 x 155 2 1 de 3 dex 120 El Bux 5 5 3 13 x 4 2 3 Aux 39173 54 qu z2bi5
47. ry variable as follows Now 1t works as expected u E A AS a A le ED Se Prog A 1 7 dik i ee a If x lt a_ tr 2 1 2 8 If x evitr astro a tr If x lt vO tr A e 2 0 2 PAN AN ae Me UO oe er a ei ar es oe lt 10 7 IF x lt 2 2 1 OZ P Sere gt Judith Lindenberg Austria Dear Josef We define nv x m S the density function of the normal distribution and as pnv m s a b the area under the density between a and b TI Nspire has no problems solving pnv 400 10 400 d 400 d 0 95 d but it seems to be unable to solve for the mean nsolve pnv m 2 495 0 0 95 m Any advice DNL Dear Judith Numerical solving s sometimes tedious even for CAS without any special tricks DERIVE needs 12 5 sec to find m 498 2897 But you can assist your system by adding restrictions for the numeri cal search There are some possibilities for TI Nspire As 0 95 is greater 0 5 we should know that mean m is greater than 495 A rough estimation could be m 500 so enter your guess as follows nSolve pnv m 2 495 0 95 m 500 or nSolve pnv m 2 495 0 95 m 600 There is another even easier way remembering the wonderful operator from the TI 92 Voyage 200 you can enter nSolve pnv m 2 495 0 95 m m gt 495 m gt 400 does it too Merry Christmas Josef D N L 76 Agustin de la Villa A Toolbox with DERIVE A TOOLBOX WITH DERIVE Alfonsa Garci
48. s may cause troubles parts of the display are cut off and inserted at other places Looks very strange What to do Decrease the resolution then it should work properly Gunter Schodl Wr Neusdtadt Austria Hallo Josef Derive runs properly under Windows 7 All what one has to do is installing a patch for the Online Help This can be found at www microsoft at under KB917607 Windows6 1 KB917607 x64 msu for 64Bit Systems Windows6 1 KB917607 x86 msu for 32Bit Systems Best regards G nter Gerhard Hagen Friesach Austria Dear all I ask for help to solve a DERIVE mystery I found the following function in an interesting article of the Scientific News and wanted to reproduce it with DERIVE Vy Tp Tf a lt n T d d v T lt x we a tr vO If 0 lt x lt a 2 tr 2 J 2 a x Ifa ir 2 x We tr a 2 ur 2 1 a tr If vo tr a 2 tr lt x lt vO tr J 2 a vO tr 2 a x If vO tr lt x 0 pa DERIVE and CAS TI User Forum This looks quite good the mystery appears by substituting special values for the parameters eg a 7 tr 1 and v0 10 7 2 AVEX Fa tr W rlo AR a Fa a A Pee Tee ie ft oe 2 ib Ce eel aa F X 0 Within the first IF clause a 2 1s correctly replaced by 7 2 but starting with the second IF all remaining a 2 are replaced by 7 a Can you help me DNL I dont know why DERIVE is behaving so strange but there is help I rewrote the function using an auxilia
49. sign the activities according these compe tences Such change may fail if a considerable amount of effort imagination common sense and hope is not devoted to it The system inertia and the difficulty involved in designing effective activities must be borne in mind since teachers generally have extensive experience in preparing expository lessons with more or less of students and although we have worked hard to find the best way of introducing and presenting concepts and results we do not have any experience in guiding the search The new teaching model implies more autonomous work by students To see this it is merely neces sary to analyze the new structure of studies which are articulated in 60 yearly ECTS European Credit Transfer System credits where each ECTS credit must reflect between 25 and 30 hours of the stu 9 66 dents overall work only 10 to 15 of which must necessarily involve classroom attendance po Agust n de la Villa A Toolbox with DERIVE This is why teachers find it challenging to design a mathematical course for engineering students for example a course on Calculus of One Variable or a course on Linear Algebra taking these determin ing factors into account In our opinion we are doomed to design mathematical courses in which magisterial lessons theory and problems solved in detail by the teacher practical workshops problem solved and Mathematical laboratories based in a CAS and tutorial acti
50. that will be able to reach one or the other goals of higher competences than only manipu lating skills pas Josef Bohm CAS Tools for Exercising This is my list Working with fractions Expanding and Factorizing Expressions Power Rules Quadratic Equation Completing the Square Simultaneous Linear Equations Working with complex numbers Long Division of Polynomials GCD amp LCM Working with units time ength area Basic Problems with Linear Functions Working with Vectors and Matrices Set Theory Truth Tables Boolean Expressions Analysing Conics Basic Problems for Financial Mathematics Solving Triangles Limits Arithmetic amp Geometric Sequencies and Series Do you remember Jan Vermeylen 1990 Investigation of Sequences Differentiation and Integration Rules Discussion of Curves Taylor Series Fourier Series Typical Forms of Differential Equations Implicit Differentiation Conversion between number bases Recognising Function Types from their graphs Sketching derivative and or antiderivative to a given function graph Finding Polynomial Functions All the items in bold letters are more or less ready The underlined ones do not provide randomly generated problems but help solving given problems All the items in italics are on my TO DO list in the following will show a selection of screen shots All tools from above are done using my favourite CAS DERIVE But it is possible to convert all id
51. ties of new methodical and didactical manners in teaching mathematics Contributions Please send all contributions to the Editor Non English speakers are encouraged to write their contributions in English to rein force the international touch of the DNL It must be said though that non English articles will be warmly welcomed nonethe less Your contributions will be edited but not assessed By submitting articles the author gives his consent for reprinting it in the DNL The more contributions you will send the more lively and richer in contents the DERIVE amp CAS TI Newsletter will be _ _ _ Editor Mag Josef B hm D Lust 1 A 3042 Wiirmla Austria Phone 43 0 6604070480 e mail nojo boehmOpgv at Neri March D016 Deadline 15 February 2010 Preview Contributions waiting to be published Some simulations of Random Experiments J Bohm AUT Lorenz Kopp GER Wonderful World of Pedal Curves J Bohm Tools for 3D Problems P Luke Rosendahl GER Financial Mathematics 4 M R Phillips Hill Encription J Bohm Simulating a Graphing Calculator in DERIVE J Bohm Henon Mira Gumowski amp Co J Bohm Do you know this Cabri amp CAS on PC and Handheld W Wegscheider AUT Steiner Point P Luke Rosendahl GER Overcoming Branch amp Bound by Simulation J Bohm AUT Diophantine Polynomials D E McDougall Canada Graphics World Currency Change P Charland CAN Cubics Quartics interesting fea
52. tions for plotting the left and right rectangles associated to Riemann sums We promote our students to define simple instructions for computing lengths areas and volumes with a syntax more recognizable that POLAR_ARC_LENGTH PARA_ARC_LENGTH POLAR_AREA VOLUME_OF_REVOLUTION AREA_OF_REVOLUTION etc For instance to calculate the length of an arc of the curve y f x it is possible to define the func tion LEXP and theoretically find the length of any arc of curve The figure 4 shows the DERIVE implementation and the calculation of the parabola s length y x between the abscissas 1 and 2 1 Fix b i 2 3 TIERES DE Jero O de a The parabola length E 3 Tix ic x 4 LEXP 1 2 LNG 85 217 4 5 8 5 5 SIE En 2 And the approximate value 6 Figure 4 The length of an arc of curve Finally students can define tools for numerical integration using the Composite Trapezoidal rule and Composite Simpson rule see figure 5 Agustin de la Villa A Toolbox with DERIVE p 11 1 b a f a fib n 1 1 b a 2 TRAP a b n 2 fla n 2 eL n b a nf2 2 1 2 b a nf2 1 2 1 b a 3 SIMPSON a b n i Fla f b 4 2 fla 2 E fia _ 3n 1 1 n 1 1 n 2 x xe 4 fx x 4 5 J ra dx fl 2 x xe 6 dx 2 x 4 1 7 TRAP 1 1 10 SIMPSON 1 1 10 8 0 4342879366 0 4215187544 Figure 5 Nu
53. tures T Koller amp J Bohm Logos of Companies as an Inspiration for Math Teaching Exciting Surfaces in the FAZ Pierre Charland s Graphics Gallery BooleanPlots mth P Schofield UK Old traditional examples for a CAS what s new J Bohm AUT Truth Tables on the TI M R Phillips Advanced Regression Routines for the Tls M R Phillips Where oh Where is IT GPS with CAS C amp P Leinbach USA Embroidery Patterns H Ludwig GER Mandelbrot and Newton with DERIVE Roman Hasek CZ Snail shells Piotr Trebisz GER A Conics Explorer J Bohm AUT Coding Theory for the Classroom J B hm AUT Tutorials for the NSpireCAS G Herweyers BEL Some Projects with Students R Schroder GER Runge Kutta Unvealed J Bohm AUT The Horror Octahedron W Alvermann GER RK6 Heinrich Ludwig GER and others Impressum Medieninhaber DERIVE User Group A 3042 Wiirmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm DERIVE and CAS TI User Forum DERIVE and Windows 7 Peter Hofbauer Horn Austria Hi Josef DERIVE is running under Windows 7 without any problems The Online Help cannot be shown like under Vista But there is a patch available Download the respective patch from http www microsoft com downloads details aspx familyid 258AA5EC E3D9 4228 8844 008E02B32A2C amp displaylang de for 64 bit or 32 bit operation systems Best regards Peter Note Some screen resolution
54. ult poly gives a factorizable polynomia po lybin z gives n polynomials ch Tactorizes for rational zeros chr for irrational and che for complex zeros 2 10 2 IS 4 A 630 p5 5 t 16 40 p 5 t 110 p t 6 2 4 G 2 5 4 2 6 350 h u w 180 h u w 540 h u ew 3 2 4 2 5 3 quizbi5 432 7 k 68 Tk 1 336 f k 1l 43 3 Bb 3 a 3 Br5 c em 1350 c 1 em Hr5 c n A d d 2 4 3 AD0 b u y 500 b u 4 2 3 a Dep Ss t C305 4 t J 4 2 4 15C h u w a h C2 hta w 2 2 3 3 4 ch 48 f k 1 9 f 15 k 2 1 J aos 3 Z 675 c n Le l J A 2 2 0 b u y C4 b 25 u J The instruction informs that this file can also be used for training factorizing polynomials on different levels of knowledge starting with only rational zeros one can proceed to irrational zeros and then finish with complex zeros depending on the age of the students p 32 Josef Bohm CAS Tools for Exercising 5 4 3 x Bx eX 6 5 4 3 x 15 x 48 x 64 x 6 5 4 3 Bu her Head 1LOK x 9 polybt5 j 3 z BX 104 x x olaaa 2 4 2 24 3 a A X 3 3 12 x an Ze 11 chr fi 2 8 6 x1 4 x 4 x 13 25 1 D x as G x L af 3 e x lx 5 8 3 Ze O X bx 4 6 ex3 Bete 10 ch 2 B Sex dex 402113 a x 9 Geet 27001 3 3 x laa 2 4 2 24 3 2 x l z X 3 3 l2 x x 6 3 x L 2 1 1 amp B S x poa
55. vities must be blended so that students can acquire the required competences Laboratory classes must be designed with clear goals Our proposal is that Computer Algebra System CAS could help in the automatic performance of certain tasks involved in the problem solving proc ess For this it would sometimes be necessary to use certain functions or commands which might al ready be integrated to the system or which might have been prepared by the teacher or even created by the students themselves Working in a more autonomous way allows students to access computing technologies outside con ventional training which is why it is essential that they acquire the skills to make optimum use of them The advantages of CAS must be boosted visualization computation facilities the possibility of experimenting avoiding possibly damaging effects such as the lack of a critical attitude when con sidering the computer response an inability to interpret the results etc In any case in order to be effective all the activities suggested must be designed without letting the intended goals out of sight mainly taking into account the students at whom they are aimed 1 A box of mathematical tools Mathematical subjects which are usually programmed within the first years of Engineering studies have as their main goal the initiation of students into the language of Science and Technology and their preparation in the correct use of certain algorithms in p
56. y to briefly describe the evolution of the Spanish educational system at Secondary Education as regarding Mathematics Until the mid 70 s the curriculum focused on classic plane geometry After the arrival of Modern Mathematics set theory relations correspon dences algebraic structures number base changes etc were studied instead Now the curricu lum of 10 16 year old students focuses on practice and experimentation oriented to achieve basic skills Nowadays possibly as a consequence of an incorrect implementation of the latter ideas many freshmen at university do not know or do not master basic concepts procedures and algorithms Some Schools like the School of Mathematics of our University have introduced 0 courses devoted to those students that can t follow the subjects of the first year at univer sity For example they can t perform number base changes because they have never studied this topic E Roanes Mayan Numbering System in DERIVE We propose here a unit at Teacher Training level organized in three sessions of 1 5 hours about number base changes Here the Mayan numbering system is introduced as a historic justification for the need of working in other bases Another justification is the use of base 2 and indirectly base 16 by computers Therefore this is a multidisciplinary unit involving Maths History and Computer Science The working scheme for the unit is the following Studying the problem inc
57. you think that the manipulating Skills in calculus are indispensable kr Fae intestat lintearan il TAN Fie F r FSr 1 Dal mami ne EN oir EA Intedral Integra ul 2 Fo lumomials pa 1 Po lu of i Les i OWeErtT UNEEI O 1 About AAT uE EE Product Rule 2 oWerfunct10 S Product Rule 4i Duotient Rule 9 Chain Rule iDiff Quiz End dire iie MN Intern u 1 Polunomials 1 Folumomials _ 2 2 Powerfunctions 2 Fowerfunctions gt i Substitution Substitution di int By parte f by parts wo 5 zitartia ractions Fi op PND z 1 Integration Quiz Partial Fractions All tasks are random generated II demonstrate the basic idea by using the respective DERIVE tool which was created when we had the opportunity to teach math in the PC lab working with DERIVE pas Josef Bohm CAS Tools for Exercising provided one program for training the differentiation rules and another one for exercising the integration rules Both of them are offering two levels of difficulty This is the start screen for integration 2 levels Of da treat dla cando Lewel 1 poll polynomials DELS 2 2 or power Tun ct ians substl Int by substitution partl Int by parts pfrl partial fractions quizl random problem n tasks of same type pollbin pHbin quizlbin Level 2 pol polynomials BES Gee Ar power Funckisns subste 4 mt By Substituting Par t2 00 Int iby parts parts Int pa
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