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Unexpected answers offered by computer algebra systems to

school equations

Eno Tõnisson

University of Tartu

Estonia

CADGME 2010 Hluboká nad Vltavou, near České Budějovice

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Plan

• Background

• Unexpected answers

• CASs

• Equations– Quadratic– Trigonometric

• Could the unexpected answer be useful? How?

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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.

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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?

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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.

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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

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Equations

• linear

• quadratic

• fractional

• equations that contain an absolute value

• irrational

• exponential

• logarithmic

• trigonometric

• literal equations

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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

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Classification of Quadratic Equations

• Natural (textbook) classification is suitable as a base.

• Classification by – (manual) solution process

– number of (real) solutions

02 bxax 02 cax 02 cbxax

042 acb 042 acb042 acb

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Quadratic Equations

Type Example Phenomena

form (fraction)

±, form (radical)

form (radical)

multiplicity

imaginary

02 qpxx

042 acb

02 cbxax

02 cax

02 bxax

042 acb

042 acb

01710 2 xx

082 2 x

022 xx

0542 2 xx

0122 xx

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Phenomena from Quadratic Equations

• Form of the solution, equivalence of solutions– decimal fraction – common fraction– form of solution

• Imaginary numbers, domain• Equal solutions, multiplicity of solutions

• or

• Choice of command – solve

042 acb

1x 1,1 21 xx

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Form

• Radical

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Imaginary

• no solution • includes i• includes• includes decimal numbers and i

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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

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Classification of Trigonometric Equations• Different classifications are possible

• Basic equations – "nice" answer – "not-so-nice" answer– impossible (in school)

• Advanced equations ("one-function")– more complicated argument – factorization– quadratic equations – biquadratic equations

• More advanced ("function-change")– change function – homogeneous– …

0sin x 2

3cos x 1cot x

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1sin x 1.0sin x

2sin x 2cos x

2

2)6

2cos(

x

0)sin1(sin xx03sin2sin 2 xx

222 )sin2()2(cos2 xx

0cos3sin2 xx

4cot3tan xx

013tan33tan2 24 xx

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Basic trigonometric equations

Type Example Phenomena

"Nice" answer choice of solutionnumber of solutionsapproximate-exactwhen inverse function

"Not-so-nice" answer

choice of solutionnumber of solutionsapproximate-exactwhen inverse function

Impossible when inverse function

0sin x

2

3cos x

10

1sin x

2cos x

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Phenomena from Basic Trigonometric Equations

• choice of solution

• number of solutions– 1 / 2 / infinitely

• approximate-exact

• when inverse function is in the answer

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4

3

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Number of solutions

• one solution

• one solution and warning

• two solutions

• general solution

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Added by advanced trigonometric equations

• Solutions are more complicated, checking correctness is more difficult– Textbook CAS

• Biquadratic (trigon.) equation could be too complicated for the CAS

– could be possible to solve by parts

013tan33tan2 24 xx

0132 24 tt

nx

nx n

2)1(

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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

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So?

• What phenomena could appear?• When the phenomenon appears?• So what?

– ignore– avoid– explain– use

• even evoke

• unexpected didactic, instructive

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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

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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?

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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 …

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• There could be more than one (correct?) answer!

• In mathematics???!!!!