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CADGME 2010 Hluboká nad Vltavou, near České Budějovice Unexpected answers offered by computer algebra systems to school equations Eno Tõnisson University of Tartu Estonia 1 Plan • • • • Background Unexpected answers CASs Equations – Quadratic – Trigonometric • Could the unexpected answer be useful? How? 2 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. 1999 – 542 problems – 68 as usually taught at schools – another 34 advanced math classes. 3 4 Unexpected answer • Differently than – student/teacher/textbook – expects/waits for/presents • Expectations could vary – curriculum – teacher – textbook • Not incorrect but according to different standards • Classification and mapping of the unexpected answers • What are the equations/answers that have more didactical potential? 5 CASs • (Relatively) easily available – – – – – OpenAxiom Maxima Sage (Maxima??) WIRIS WolframAlpha • Quite different – – – – Computer Algebra System Open Scientific Computation Platform Computational Knowledge Engine … • If necessary it is possible to use some of them • We do not compose the rating. We do not focus on shortcomings. 6 Commands • Command solve – first choice for solving equations • equation solution process answer – solution process (answer) is impressionable by change of command, additional arguments, form of argument • solve radicalSolve – small difference in the expression could change the situation • 1 1.0 7 Equations • • • • • • • • • linear quadratic fractional equations that contain an absolute value irrational exponential logarithmic trigonometric literal equations 8 Plan for particular equation type • Initial set of examples – textbook etc. classification • sometimes simple, sometimes more complicated – simpler non-trivial examples • sometimes a bit more complicated – expressive examples from literature • Solving the example equations by all CASs • Tentative mapping – "zoom" if needed – detail the boundaries if needed • Special focus on the phenomena that are (could be) more meaningful to students and teachers 9 Classification of Quadratic Equations • Natural (textbook) classification is suitable as a base. • Classification by – (manual) solution process ax bx0 2 ax2 c 0 2 ax bx c0 – number of (real) solutions b2 4ac0 b2 4ac0 b2 4ac0 10 Quadratic Equations Type Example ax2 bx0 10 x217 x0 ax2 c 0 2x2 80 Phenomena form (fraction) ±, form (radical) 2 ax bx c0 x2px q0 b2 4ac0 2x24x50 b2 4ac0 x2 2x10 b2 4ac0 x2 x20 form (radical) multiplicity imaginary 11 Phenomena from Quadratic Equations • Form of the solution, equivalence of solutions – decimal fraction – common fraction 2 – form of solution b 4ac0 • Imaginary numbers, domain • Equal solutions, multiplicity of solutions • x 1 or x1 1, x2 1 • Choice of command – solve 12 Form • Radical 13 Imaginary • • • • no solution includes i includes 7 includes decimal numbers and i 14 Trigonometric Equations • Different range – – – – only sin, cos, tan or also cot or even sec, cosec how complicated? • Different order (in textbooks) – all basic equations at first, then more complicated – basic equations with sine at first, then more complicated with sine, then basic with cosine etc etc • General solution, one solution or solutions in the interval – Find all solutions in the interval [0;2π) • Radians or degrees 15 Classification of Trigonometric Equations • Different classifications are possible • Basic equations 3 cotx 1 cos x – "nice" answer sin x 0 1 2 sinx 0.1 – "not-so-nice" answer sinx 10 – impossible (in school) sin x 2 cosx 2 • Advanced equations ("one-function") – – – – 2 cos( 2 x ) 6 2 more complicated argument factorization sin x( 1 sin x ) 0 2 quadratic equations sin x 2 sin x 3 0 4 2 biquadratic equations 2 tan 3 x 3 tan 3 x 1 0 • More advanced ("function-change") – change function tan x 3 cot x4 – homogeneous 2 sin x 3 cos x 0 – … 2 2 2 2 cos ( 2 x ) ( 2 sin x ) 16 Basic trigonometric equations Type "Nice" answer Example sin x 0 cosx "Not-so-nice" answer Impossible sinx 3 2 1 10 cosx 2 Phenomena choice of solution number of solutions approximate-exact when inverse function choice of solution number of solutions approximate-exact when inverse function when inverse function17 Phenomena from Basic Trigonometric Equations • choice of solution • number of solutions 4 3 4 – 1 / 2 / infinitely • approximate-exact • when inverse function is in the answer 18 Number of solutions • • • • one solution one solution and warning two solutions general solution 19 Added by advanced trigonometric equations • Solutions are more complicated, checking correctness is more difficult – Textbook x n CAS x( 1 )n n 2 • Biquadratic (trigon.) equation could be too complicated for the CAS 4 2 2 tan 3 x 3 tan 3 x 1 0 – could be possible to solve by parts 2t4t 3210 20 21 22 From other equations • Mainly same phenomena – equivalence – number domain – approximate-exact – branches • Symbolic expressions in case of literal equation • Sometimes a CAS could not solve the equation 23 So? • What phenomena could appear? • When the phenomenon appears? • So what? – – – – ignore avoid explain use • even evoke • unexpected didactic, instructive 24 Why equations at all? • The 12th ICMI Study The Future of the Teaching and Learning of Algebra – The activities of school algebra can be said to be of three types: • generational – forming of expressions and equations • transformational – rule-based activities: collecting like terms, solving equations, simplifying expressions etc, etc • global/meta-level – problem solving, modelling, noticing structure, justifying, proving etc 25 Pilot Study / Pilot Course??? • Course for – students – pre-service teachers – in-service teachers • Topics – – – – equivalence number domain approximate-exact branches • The topics are very important but could be somewhat behind the scenes • Detailed mapping gives good examples • Something for everyday maths teaching? 26 Unexpected answer in instrumentation • Instrument = Artifact + Schemes and Techniques • Unexpected answer? – instrumental genesis – orchestration • As a base for discussion? – "Real life" example • Computer tells that … 27 • There could be more than one (correct?) answer! • In mathematics???!!!! 28