Download Unexpected answers offered by computer algebra systems to school

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 bx0
2
ax2 c 0
2
ax
bx
c0
– number of (real) solutions
b2 4ac0 b2 4ac0
b2 4ac0
10
Quadratic Equations
Type
Example
ax2 bx0
10
x217
x0
ax2 c 0
2x2 80
Phenomena
form (fraction)
±, form (radical)
2
ax
bx
c0
x2px
q0
b2 4ac0
2x24x50
b2 4ac0
x2 2x10
b2 4ac0
x2 x20
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 4ac0
• 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
x4
– 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
2t4t
3210
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21
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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
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???!!!!
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