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Eno Tõnisson

University of TartuEstonia

Answers Offered by Computer Algebra Systems to Expression

Transformation Exercises

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Plan

• Background– School– Computer algebra systems

• Scope• Is the answer as expected?

– No • How to ask?• Issues on Domain

– Yes• Branches

• Comparison with equations• Summary, Future work

Motivation

• Unexpected answers – according to school mathematics

• Could be differ in different countries and schools

• Answers to equations from school textbooks offered by Computers Algebra Systems – CADGME Pecs, Tonisson & Velikanova

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Transformation of expressions at school

• Mathematics lessons, textbooks and tests frequently require transformation of expressions

• Polynomials• Algebraic fractions• Exponents, Radicals• Trigonometry• …• …

• Dozens of exercises

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May or may not be mentioned explicitly in the text

• Explicitly (same words as in CAS)– Simplify– Factor– Expand

• Explicitly (even in more detail)– Multiply, using a distributive property– Take out factors common to all terms – Remove parentheses and simplify – Express in simplest radical form– Simplify, leaving answers with negative exponents

• As a part of more complicated exercise– Solve equation by factoring– Solve equation???

– Equivalence checking Simplify(expr1-expr2)0 ????

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CAS era

• CASs have a wide range of commands for expression transformation.

• Questions about the use of the CASs in transformation exercises in schools.

• A reappraisal of the time and effort – Dedicated in the curriculum to the training of manual

transformation skills.– Indispensable manual skills

• An overview of the answers offered by the CASs is necessary 6

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Classification of exercises with respect to the role a CAS plays in

the solution.

• J. Böhm, I. Forbes, G. Herweyers, R. Hugelshofer, G. Schomacker: The Case for CAS

• Lokar M. & M.: CAS and the Slovene External Examination. In "The International Journal of Computer Algebra in Mathematics Education", 2001. Vol. 8, No 1.

• C0 Exercises where the use of CAS is of little or no help. More importantly, the typing would take more time than solving the problem by-hand (this of course depends on the students' skill).

• C1 Traditional exercises (by-hand or using scientific calculators) that are solved faster or even trivialised by CAS (emphasis usually on manual skills).

• C2 Exercises that essentially test the ability of using

CAS competently.

• C3 Exercises starting from traditional ones that are extended to CAS-exercises (e.g. by including formal parameters or using realistic data).

• C4 Exercises that are difficult or time consuming to solve without CAS or those that can only be solved with the aid of CAS. 8

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Example of C4: Cebatorev Assumption• From The Case for CAS

• Factor xn −1 for n = 1, 2, ... , 10 with integer coefficients. All coefficients of the polynomial factors are 1. Is this true for all exponents n?

• Solution: The problem was set by the Russian mathematician N.G. Cebotarev in 1938 in the Russian journal “Progress in mathematical sciences (volume IV)” and solved by V. Ivanov in 1941. The first exponent which leads to integer coefficients > 1 is n = 105.

• The voyage 200 factors the expression in 8 seconds, but the problem cannot be solved by hand by high school students.

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CASs. 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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Wester, page 45

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Scope• Transformation (simplifying, expanding or factoring)

of expressions from school mathematics– mainly from textbooks

• Immediate solving (student enters the expression and the program gives the answer)– basic commands (simplify, expand, factor) and some

advanced ones.

• The aim is not to analyze the performance of particular CASs but to discover general trends.– In presentation only some issues

• Derive, Maple, Mathcad, Mathematica, Maxima, MuPAD, TI-92 Plus, TI-nspire, WIRIS

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Is the answer as expected? CAS = SCH?

• Yes!

–In very many cases– Is the expected school-like answer good enough?

• Yes?– Simplest form?– Order of terms?– …

• No– Ask differently, more precisely?– Explain– Beware

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How to ask?

• Command– Basic commands (simplify, expand, factor) and some advanced

ones• simplify (automatic???)

– Commands, menus, buttons …

• Expression– notation questions

• division : /

– 3/1

??3 aa 2/1

??

aa

x2cos

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Commands

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Some examples

• Different CASs

• Factor

• Trigonometric expression

• Combine commands

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factor( )

babaa )25(3 2

232 1563 baabba

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Factor(x^105-1)

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C1 - are solved faster or even trivialised by CAS

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• In textbook 12(x-3)(x+3)2

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Simplest?

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What is the required answer?

1tan2 2cos

12][aSec

)(tan)2sin()2

3cos()(cos 22

Combine commands

• tcollect(texpand(expr))

• combine(simplify(expr))

• simplify(combine(expr))

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trigreduce, ratsimp, and radcan may be able to further simplify the result

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Some steps manually

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Validity of transformation rules could depend on the domain

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Issues of Domainvariable to be a real number

variable to be in the certain interval

Derive Author Variable Domain

Author Variable Domain

Maple AssumeRealDomain

Assume

Mathematica Assuming Assuming

Maxima Default Assume

MuPAD Assume Assume

TI-92+ Default | “with”

TI-nspire Default | “with”

WIRIS Default Not applicable

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Is the expected school-like answer good enough?

• Wester

• All CASs: success! (hurrah!)• x = -2 ???

2

2

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2

x

x

xx

x

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Forbidden branches• Operations and functions that are impossible (at least at

school) in the case of some values of arguments. – Division by zero is undefined.– There is no square root for a negative number. – The domain of a logarithmic function is the set of all positive real

numbers.– Tangent function is not defined if , .

• Demonstrate all “forbidden branches” separately in expression transformation exercises? Actually, the distinguishing of forbidden branches is discarded, and the practice is even “legalised”. For example, a textbook says: – Even though not always explicitly stated, we always assume that

variables are restricted so that division by 0 is excluded; Unless stated to the contrary all variables are restricted so that all quantities involved are real numbers.

2/)12( nx Zn

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CASs and Forbidden branches

• The CASs do not show “forbidden branches” either.

• Simplification of – all computer algebra systems give the answer

97, without recording the peculiarity of x = 0. • TI-nspire adds a warning message: Domain of the

result may be larger

• CAS=SCH<MATH

x

x97

How?

• Like in Gentzen-type calculi in mathematical logic???

• Only main branch with the condition(s)

• All branches separately

and

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9797

,0 x

xx

9797

,0 x

xx 0

097,0 x

xx

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Types of obtained answers Equations (CADGME, Pecs)

Type Equivalent? Easily transformable?

1. Answer is not equivalent to the answer required at school

no yes/no

Anyway keeps non-equivalence

2. Answer is equivalent but can not be easily transformed to the required form

yes no

3. Correct answer that is easily transformed to the required form

yes yes

ok yes Not needed

Already in suitable form

• In case of transformation exercises• What is the required form?• Combine commands – Type 3 • Order of terms – 2? 3? ok?• Trigonometry – 2? 3? Easily?

Some highlights

• Commands are simple – But sometimes we need special parameters or

special commands

• Notation• What is the required answer?

– If not equivalence

• Combination of commands• Domain• Forbidden branches

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Material for different types of exercises

– Indispensable skills?– Why?

– Is the answer equivalent to textbook answer?– Other domain?– …

• C0 use of CAS is of little or no help• C1 are solved faster or even trivialised by CAS • C2 test the ability of using CAS competently • C3 extended to CAS-exercises • C4 difficult or time consuming to solve without CAS or only be solved with the

aid of CAS37

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Papers are to be submitted by 15 September 2009

• More exercises– Absolute value– ..

• Automatic simplification

• Suggestions?

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Is the answer as expected? CAS = SCH?

• Yes!

–In very many cases– Is the expected school-like answer good enough?

• Yes?– Simplest form?– Order of terms?– …

• No– Ask differently, more precisely?– Explain– Beware