1 Nadezhda Velikanova Eno Tonisson University of Tartu Estonia ANSWERS TO EQUATIONS FROM SCHOOL TEXTBOOKS OFFERED BY COMPUTER ALGEBRA SYSTEMS

2 Plan Background Classification of equations Classification of obtained answers Summary, Future work

3 Background CASs –In the beginning were designed mainly to help professional users of mathematics –Nowadays more suitable for schools There are still some differences. How do different CASs solve problems? Michael Wester. Computer Algebra Systems. A Practical Guide –542 problems –68 as usually taught at schools –another 34 advanced math classes.

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5 Scope Equations from school mathematics –mainly from textbooks Immediate solving (student enters the equation and the program gives the answer) –command solve (symbolic and numerical) 8 CASs Attempt to develop the strategy and methods that allow to investigate and to compare the answers offered by CASs with the schools’ answers

6 CASs Derive 6 Maple 8 Mathcad 2001 Mathematica 5.2 MuPAD 3.1 TI-92 Plus TI-nspire CAS (prototype) WIRIS

7 Classification of equations In school (textbooks) the (explicit) classification is not detailed enough A bit more detailed, but simple enough In order to check representatives of each smaller subtype 35 subtypes, 60 equations –a couple of the representatives of each subtype (one simple and one more complicated) –one representative, if very easy and very similar

8 Equations 8 most important types of equations: –linear, –quadratic, –fractional, –equations that contain absolute value, –irrational, –exponential, –logarithmic, –trigonometric equations.

9 Can be improved Given classification is not complete and can be improved. Considered as a brief sketch for approximate delimitation and listing the problems. Defining concrete boundaries requires much more thorough experiments and analysis.

10 Quadratic equations

11 Irrational equations

12 Tables of answers, 8 CASs, > 45 pages

13 Dimensions Is the result equivalent to the answer required at school? Is the result easily transformable to the form required at school? –Command simplify –Transformation that retains expressions equivalence with the original expression A bit subjective terms –Equivalent –Easily transformable –Required at school

14 Types of obtained answers TypeEquivalent?Easily transformable? 1. Answer is not equivalent to the answer required at school noyes/no Anyway keeps non-equivalence 2. Answer is equivalent but can not be easily transformed to the required form yesno 3. Correct answer that is easily transformed to the required form yes okyesNot needed Already in suitable form

15 Exists answer in particular type “ok” is not indicated

16 Why there are many “2”-s and “3”-s? 2. Difference in the form of the answer – answer is equivalent but can not be easily transformed to the required form –Notation questions: e. g. v (OR) 3. Correct answer that is easily transformed to the required form –Solve + Simplify 4^x  ln(64)/ln(4) –Sometimes necessity of transformation is questionable

17 Some examples Derive x = ±∞, x = 6 Mathcad, MuPAD, Mathematica x = 80 School x = -109, x = 80 Derive, MuPAD x = 4, x = 1, x = 0 School x = 4, x = 1

18 Mathematica, TI ‑ 92 Plus, TI-nspire give warning that some roots may be lost. or {{x = },{x =2.618}}

19 Possible actions of the teacher

20 Summary Strategy for comparing and investigating CASs –Choose area of mathematics –Classification of problems, choose examples –List of types of examples –Choose CASs –Experiments, classification of output examples –Outline special interesting cases Brief overview how well 8 computer algebra systems handle equations from school textbooks –Mostly systems are reliable and give reasonable answers –Work similarly for tested equations and give practically same results –The bigger differences are in the way of writing down the answers or indicating some known situations –Most problematic are irrational, logarithmic and trigonometric

21 Related Future Works Real and Compex Domains –(Saturday, CADGME) Branches –(July 1, ICTMT8, Hradec Kralove) Infinity Equivalence … Lexicon or dictionary that concisely compares the many CASs that are available.

22 Questions? Suggestions? Related works?