Abstract
Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. The theoretical framework we adopt is the Anthropological Theory of Didactics (Chevallard 19:221–266, 1999), in combination with semiotic aspects from the instrumental approach to tool use. The analysis we present is based on interviews carried out with a 10th grade student who participated in our research. Our findings highlight the mathematical knowledge (technological discourse) constructed in the process of confronting, differentiating, and articulating the several mathematical techniques and theoretical ideas (pertaining to the numeric perspective and the syntactic perspective on algebraic equivalence) related to the designed equivalence tasks.
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Notes
WIKIPEDIA defines a Computer Algebra System (CAS) as a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form. The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include real, complex, interval, rational, and algebraic.
The term denotation is drawn from the work of the renowned mathematician and logician, Frege, who distinguished sense and denotation. For example, the expressions 4x + 2 and 2(2x + 1) would have different senses, but denote the same functional object. According to Arzarello et al. (1994), “the ‘denotation’ of a symbolic expression in algebra refers to the number set that is represented by the expression; it is determined by the symbolic expression and by the universe in which the expression is considered (for example the equation x 2 = − 1 denotes the empty set when it is considered in R and the set {+i, −i} when considered in C)” (p. 42).
The set of restrictions of a rational function \( f(x)=\frac{p(x) }{q(x) } \) (p and q polynomials in x) of real variables with real values is \( A = \left\{ {x \in R:q(x) = 0} \right\} \). The function f cannot be evaluated on the elements of the set A; its domain is R − A.
The definitions that will follow are given for rational fractions, considering that the ring of polynomials R[X] is embedded in the field of rational fractions R(X). The same is done for rational functions and polynomial functions.
The field of rational fractions in one indeterminate R(X) is the field of fractions of the polynomial ring in one indeterminate R[X]. See, for example, Grillet (2007).
The definition of syntactical equivalence can also be given in the following way: Two expressions are equivalent if their cross products are equal. For example, \( G(X)=\frac{1}{X} \) and \( H(X)=\frac{X-2 }{{{X^2}-2X}} \) are syntactically equivalent because their cross products are equal: \( (1)\left( {{X^2}-2X} \right)=(X)\left( {X-2} \right) \).
This is a more formal definition from the mathematical point of view, which follows the formal construction of the field R(X), but is not sensitive to the multiple variants to be considered in a student’s productions.
For rewriting H(X), one could also perform the division indicated by: \( \frac{X-2 }{X-2 }=\left( {X-2} \right)\div \left( {X-2} \right)=1 \), when H(X) is considered to represent a quotient of polynomials.
By substituting a number into the indeterminate symbol of a rational fraction, we obtain arithmetic expressions that can be simplified into a number or into impossibility (1/0). This allows considering the corresponding function of the rational fraction f(X), that we will call the rational function; that is to say a set of ordered pairs (x, f(x)) where x is in a set of values whose substitution gives a number and f(x) the number obtained by substitution.
This definition allows us to have a well-defined transitive property of the equivalence relationship: If f and g are equivalent on R − F 1 and g and h are equivalent on R − F 2, then f and h are equivalent on \( \mathbf{R}-\left( \mathrm{{F_1}\cup {F_2}} \right) \) (for F 1 and F 2 finite sets).
As do polynomials R[X] and polynomial functions R[x]. When the coefficients are taken in the field of real numbers R, for every polynomial function there exists one unique polynomial, and vice versa. But this is not always true. If the coefficients of the polynomials are in a field of characteristic different from zero, different polynomials may give rise to the same polynomial function. For example, the polynomial X 2 − X∈Z 2[X] corresponds to the zero function.
See the project web site for the entire set of activities: http://www.math.uqam.ca/APTE/TachesA.html
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Acknowledgments
We express appreciation to our co-researchers André Boileau and Denis Tanguay, to the student who participated in the research, to the teachers and administrators of his school, and to those who, along with C. Kieran, A. Boileau, and D. Tanguay, collaborated in designing the task sequences: F. Hitt, J. Guzmán, and L. Saldanha. We also acknowledge the support of the Social Sciences and Humanities Research Council of Canada (Grant #410-2007-1485). The majority of this article was conceptualized and written while Armando Solares was a post-doctoral fellow at the Université du Québec à Montréal, working with Carolyn Kieran and her research group. We also wish to thank the reviewers of an earlier draft of this article for their helpful suggestions.
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Appendix
Appendix
Equivalence of expressions
Part I (with CAS): Comparing expressions by numerical evaluation
I (A) The table below displays three algebraic expressions and two possible values for x.
Using the two given values of x (i.e., \( \frac{1}{3} \) and –5) and two others of your own choosing, calculate the resulting values for each expression by means of the evaluation tool of your calculator [i.e., the “with operator,” (|)].
Important: Proceed one complete row at a time when filling in the table.
Record your choice of additional x values in the table’s top row, and record the results in the appropriate cells below.
For x = | \( \frac{1}{3} \) | −5 | ||
Expression | Result | Result | Result | Result |
2. \( \left( {{x^2}+x\text{-}\ 20} \right)\left( {3{x^2} + 2x\text{-}\ 1} \right) \) | ||||
3. \( \left( {3x\text{-}\ 1} \right)\left( {{x^2}\text{-} x\text{-}\ 2} \right)\left( {x + 5} \right) \) | ||||
5. \( \frac{{({x^2}+3x-10)(3x-1)({x^2}+3x+2)}}{(x+2) } \) |
I (B) Compare the results obtained for the various expressions in the table above. Record what you observe in the box below.
I (C) Reflection question:
Based on your observations with regard to the results in the table above (in I(A)), what do you conjecture would happen if you extended the table to include other values of x?
Part II (with paper and pencil): Comparing expressions by algebraic manipulation
II (A) Based on your observations in Part I A, make a conjecture as to which of the above set of given expressions might be re-expressed in a common form?
II (B) To test your above conjecture by means of paper and pencil algebra, re-express the given expressions below in another form (not the expanded form). Show all your work in the table’s right-hand column.
Given expression | Re-expressed form of given expression |
2. \( \left( {{x^2}+x\text{-}\ 20} \right)\left( {3{x^2} + 2x\text{-}\ 1} \right) \) | |
3. \( \left( {3x\text{-}\ 1} \right)\left( {{x^2}\text{-} x\text{-}\ 2} \right)\left( {x + 5} \right) \) | |
5. \( \frac{{({x^2}+3x-10)(3x-1)({x^2}+3x+2)}}{(x+2) } \) |
II (C) In Part I C, you made some conjectures based on numerical evaluations of expressions. Explain in what way the algebraic manipulations in Part II B supported (or not) each of those conjectures.
For any conjectures of Part I C not supported by your algebraic manipulations in Part IIB, how do you account for the discrepancy?
Part III (with CAS): Testing for equivalence by re-expressing the form of an expression—using the EXPAND command
The left-hand column of the table below contains the expressions from the previous lesson. Using your calculator, fill in the right-hand column with the expression produced by the EXPAND command (see F2 menu in the calculator).
Syntax: EXPAND (expression)
Given expression | Result produced by EXPAND |
2. \( \left( {{x^2}+x\text{-}\ 20} \right)\left( {3{x^2} + 2x\text{-}\ 1} \right) \) | |
3. \( \left( {3x\text{-}\ 1} \right)\left( {{x^2}\text{-} x\text{-}\ 2} \right)\left( {x + 5} \right) \) | |
5. \( \frac{{({x^2}+3x-10)(3x-1)({x^2}+3x+2)}}{(x+2) } \) |
Part IV (with CAS): Testing for equivalence without re-expressing the form of an expression—using a test of equality
It is possible to explore whether two expressions are equivalent without having to re-express their forms. An alternative approach is to use a CAS test of equality:
IV (A) Enter directly into your calculator’s entry line the equation formed by expressions 3 and 5:
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1.
What does the calculator display as a result?
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2.
How do you interpret this result?
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3.
Use your calculator’s “with operator” (|) to replace x by –2 in the above equation. Interpret the result displayed by the calculator.
IV (B) Enter directly into your calculator’s entry line the equation formed from the two given expressions 2 and 3:
-
1.
What does the calculator display as a result?
-
2.
How do you interpret this result?
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Solares, A., Kieran, C. Articulating syntactic and numeric perspectives on equivalence: the case of rational expressions. Educ Stud Math 84, 115–148 (2013). https://doi.org/10.1007/s10649-013-9473-7
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DOI: https://doi.org/10.1007/s10649-013-9473-7