Facilitating the genesis of functional working spaces in guided explorations

Abstract

Approximating given real-valued functions by affine functions is among the most basic activities with functions. In this study we examine two contexts in which two such approximations are performed. The first involves a microscopic representation of functions for the study of tangents; the second a macroscopic representation of functions for the study of asymptotes. In the proposed research, we conducted three sessions to observe how small groups of college freshmen worked in a setting of multiple dynamical representations including algebraic, graphic and CAS (computer algebra system) views. This enabled the observation of individual Mathematical Working Spaces (iMWS). The analysis of students’ answers leads us to propose an enrichment of the MWS model. Specifically, this analysis suggests that educational resources could foster the geneses described in the MWS model: observation for visualizing, drawing for constructing and justification for proving.

Appendices

Appendix 1: A compact presentation of worksheets

Notes: (1) On the original worksheet (in Spanish), blank spaces were left after the questions for allowing the students to give their written answers. (2) A second part of the questionnaire (not given here) presented questions about the impact of the proposed activities: learning effects, difficulty, appreciation (or rejection) of the way in which the sessions were organized; work in groups with interactive examples, before introducing formal definitions.

BELIEVE IT OR NOT!

Work in groups of two or three students. Each participant has to complete his/her own worksheet.

1.1 Activity A: Instructions

Open the Grafica1.ggb file and perform the following activities (Note for the reader . In its initial state, this GeoGebra file displays: –in the Graphics View the graph of the function f(x) = $\frac{{1 - x^{4} }}{{1 - x^{3} }}$ and a point A on this graph;

• in the Algebra View the symbolic writing of these objects).

• (A1) Move the point A and observe that it disappears when its abscissa becomes very close to 1. Explain in detail this phenomenon.

• (A2) In the Menu bar go to the View menu and open the CAS View. In the opened CAS cell insert the letter f and then press the Enter key . What objects appear on the screen?

• (A3) Right click on the function f(x) for opening the Properties dialog and hide this object (click on Show object). Do the same with the point A and hide it. In a second row of CAS insert again the letter f, but this time by using the Factor icon . Copy (in the clipboard) the resulting expression. In the Input bar first insert “y=” and then paste this expression. You’ll see in the Algebra View a new function g(x). Open the Properties dialog menu by a right click on this function and, in the Color menu, select Blue.

  • (a) Compare both graphics of f(x) and g(x). Is there any difference between these curves?

  • (b) Create a point B on the graph of g(x). When moving B along this graph, do you observe that the point disappears for any value of its abscissa? Explain the causes of your observations.

• (A4) Traveling: Close the CAS View. In the Graphics View hide the graph of g(x). Then select the icon Zoom out and click many times on the Graphics View, until you see on the axes the graduation marks on thousands. On the graph of f(x), create the points C and D with respective abscissas −4000 and 4000 and draw the line passing through these two points. Choose the red color for this line. Then, by using Zoom in, return to the initial size of graphics with axes graduated on units.

  • (a) What is the equation of the line CD in Algebra View? Detail your observations about the relationship between the algebraic writing and the geometric figure of this line.

  • (b) In the Input bar insert the function y = x. Open the menu under the command and select the Relation icon . What result do you obtain when applying this command to the two lines CD and y = x? Detail your observations and comment on the obtained results.

  • (c) With respect to the graph of f(x), what do you state after comparing the lines y = x and CD?

1.2 Activity B: Instructions

Open the Grafica2.ggb file and perform the following activities (Note for the reader : In its initial state, this GeoGebra file displays only the equation and the graph of the function f(x) = $\frac{{1 + x^{3} + x^{4} }}{{1 - x^{3} }}$).

• (B1) Observe the graph of the function f(x) and complete the following variation table with numerical values, symbols and arrows

• (B2) Referring to the graph of f(x) do the following.

  • (a) Create a point A on the graph and move it till its abscissa is −0.5. Now use the Zoom in command repeatedly, until you can create on the graph a new point B whose abscissa is −0.4995. Draw the line AB and paint it in red. What is the equation of the line AB? Detail your observations about the relationship between the algebraic writing and the geometric figure of this line.

  • (b) Traveling for studying a tangent. Hide the point B and the line AB, and return to the initial Graphics View with the Zoom out command, until the axes be graduated in units. Go to the Toolbar at the Perpendicular line menu and use the Tangents command for producing the tangent line through A to the graph of f(x). What is its equation? Describe the relation between the algebraic and graphic representation of this line.

  • (c) Show the line AB. What result do you obtain when applying the Relation command to the line AB and the preceding tangent line? Detail your observations about all obtained representations.

• (B3) Traveling for studying an asymptote. Select the Zoom out icon and click many times on the Graphics View, until the axes be graduated in thousands. On the graph of f(x) locate the two points C and D of respective abscissas –5000 and 5000. Color in red the line passing through C and D. Return to the initial graphics View with axes graduated in units, using the Zoom in command. Then insert in the Input bar the lineal function y = – x – 1. Is this line coincident with the line CD visually and mathematically? Describe the results of your observations.

• (B4) Referring to the graph of f(x) do the following.

  • (a) Open the CAS view, insert the letter f in the first cell and then press the Enter key . What objects appear on the screen?

  • (b) In a second row of CAS insert again the letter f, but this time by using the Factor icon . What objects appear on the screen?

  • (c) In the Input bar insert y = f(x) + x +1. What objects appear in the Algebra view? Explain what the system did in the Algebra view and in the CAS view.

  • (d) In a third row of the CAS view, copy and paste the preceding object. What expression do you obtain? Explain what the system did in the Algebra view and in the CAS view.

  • (e) We say that the line y = – x – 1 is an asymptote of the graph of f(x). Some dictionaries give the following definition of an asymptote: “A line whose distance to a given curve tends to zero. An asymptote may not intersect its associated curve.” Does the line y = – x – 1 intersect the graph of f(x)? Are you satisfied by the definition of an asymptote given above? If not, propose your own definition.

1.3 Activity C: Instructions

Open the Grafica3.ggb file and perform the following activities (Note for the reader : In its initial state, this GeoGebra file displays only the equation and the graph of the functionf(x) = $x - \frac{{\sqrt {\left| x \right|} - 2}}{x - 1}$).

• (C1) Asymptotes

  • (a) If you consider that the Zoom commands are useful, you can use them after observing and analyzing Grafica3. Describe the features of the asymptote(s) of the curve that you see and write its (their) equation(s).

  • (b) Write all justifications that strengthen what you asserted in the item (a).

• (C2) Tangents

  • (a) Mark the point M (4, 4), which lies on the graph of f(x). Obtain the tangent line at M to this curve using the Tangents command in the Perpendicular Line menu . Describe the features of this tangent and give its equation, and connect these two representations of the object.

  • (b) What happens when you try to obtain the tangent to the curve at P (0, –2) with the same steps as in item a? If the tangent exists, describe the features of this line, write its equation, and connect these two representations of the object.

  • (c) If you gave a positive answer for item b, compare geometric representation and equation of the tangents in the items a and b, and specify similarities and differences between those mathematical objects and their representations.

  • (d) Now create in GeoGebra, with the command, a slider which you will name r, designed for covering the numbers from 0.001 to 0.5 with increment 0.001. The width of the slider may be 500 px (pixels). Then, for the particular value r = 0.5, draw the circle of center P and radio r, and name Q and R the two points where the circle and the graph of f(x) intersect. Draw the lines PQ and PR. Then describe all you can observe when you reduce the radio of the circle to 0.001 using the slider.

  • (e) From the point M (4, 4) follows the same steps as in item d, with a circle of center M and radio r. Name A and B the points where the circle and the graph of f(x) intersect. What result is obtained when using the Relation command for comparing the lines MA and MB? Detail your comments about this situation and compare the results of items d and e.

• (C3) Cubic roots With the File menu, open a New Window and perform the following activities.

  • (a) In the Input Bar insert the cubic root function y = cbrt(x). Does the graph of this function have a tangent at O (0, 0)? Detail your answer writing your observations.

  • (b) Hide the graph of the preceding function (deactivating the Show object command) and insert in the Input Bar the function y = cbrt(x ²). Does the graph of this function have a tangent at O (0, 0)? Detail your answer writing your observations.

Appendix 2: Coding instructions

The following table shows the general coding principle that we used.

Answer	Successful	Incomplete without error	Wrong	Missing
Code	1	2	3	0

We did not code the items whose answer consists only in executing a given instruction and writing the obtained result. We were interested in specific consideration of visual elements (V), elements linked with instruments of construction or formulation (I) and elements related to proof or reasoning (R). In what follows, we specify only special instructions for coding; otherwise the general principle was applied.

2.1 Activity A: Main purpose of study: function defined by $f\left( x \right) = \frac{{1 - x^{4} }}{{1 - x^{3} }}$

• A1: Gives rise to three variables.

• A1V. Case x = 1, graphic and algebraic vision

  • 1 Graphical and algebraic correct observation (Disappearance and “undefined”)

  • 2 Only one of the two aspects observed correctly

  • 3 Presence of one or more errors (e.g. confusion between abscissa and ordinate)

• A1I. Case x = 1, problem formulation

  • 1 “Indetermination” or 0/0

  • 3 “Infinity”

• A1R. Case x = 1, factorization

  • 1 Reference to the factor (1 – x)

• A2: Not coded

• A3

• A3V. Comparison between f(x) and g(x) = (1 + x)(1 + x ²)/(1 + x + x ²)

  • 1 Overlapping except for x = 1

  • 2 Only “overlapping”

  • 3 One of the functions f or g wrong

• Other item not coded

• A4

• Item a not coded

• A4bI. Relationship between the lines CD, where C(–4000, f(–4000)) and D(4000, f(4000)), and y = x.

  • 1 Lines are parallel

  • 2 Lines are equal

  Questions about opinions (A5a, A5b, A5c, A5d, A5e): Each answer is coded as an integer between –2 (very negative opinion) and 2 (very positive opinion).

2.2 Activity B: Main purpose of study: function defined by $f\left( x \right) = \frac{{1 + x^{3} + x^{4} }}{{1 - x^{3} }}.$

• B1: Variation table giving rises to four variables

• B1aV. Values of a and f(a).

  • 1 Precise value: a∈[−1.13, −1.10] and f(a).

  • 2 Imprecise value: a∈[−1.3, −1.1]\[−1.13, −1.10] and f(a).

  • 3 Error; for instance of sign.

• B1bV. Values of b and f(b).

  • 1 Precise value: b∈[1.90, 1.93] and f(b).

  • 2 Imprecise value: b ∈ [1.79, 2]\[1.90, 1.93] and f(b).

  • 3 Error

• B1fI. Arrows

  • 1 Correct

  • 3 Wrong

• B1iI. Symbol ∞

  • 1 – ∞ when x → ∞.

  • 3 Wrong use of symbol

• B2: Only item b is coded

• B2bI. Tangent to the curve at point of abscissa –0.5

  • 1 y = 0.7778x + 1.2222.

  • 2 Approximate equation

  • 3 Error

• B3: Two variables are considered: direct observation (V) and deepening (R).

• B3V. Secant line through two distant points and asymptote y = – x – 1.

  • 1 Overlapping and equations [almost] confounded

  • 2 Overlapping, without consideration of Eqs. 

  • 3 Error.

• B3R. Distinction between secant line through two distant points and asymptote y = – x – 1.

  • 1 Lines declared secant with more accuracy (Rounding to more decimals)

  • 2 Lines declared secant without justifying

• B4: Only are items d y e coded, this last one giving rise to one observation variable (V) and one reasoning variable (R).

• B4dI. Formation of f(x) + x + 1

  • 1 (–x − 2)/(x ³ − 1).

  • 2 Discourse, e.g. factorize without formula

  • 3 Error

• B4 eV. Intersection of the curve y = f(x) and the line y = −x − 1

  • 1 Line and curve intersect

  • 3 Line and curve do not intersect

• B4eR. Is correct RAE’ s definition, which excludes intersection between curve and asymptote?

  • 1 No, because curve and asymptote can intersect

  • 3 Yes

• Questions about opinions (B5a, B5b, B5c, B5d, B5e):: Each answer is coded by an integer between –2 (very negative opinion) and 2 (very positive opinion).

2.3 Activity C: Main purpose of study: function defined by $f(x) = x - \frac{{\sqrt {\left| x \right|} - 2}}{x - 1}.$

• C1: Only the item a is coded, with one variable for each asymptote

• C1a1V. Asymptote x = 1.

  • 1 x = 1.

  • 2 Vertical asymptote without its equation

  • 3 Error (e.g. error of expression, only writing 1 and not x).

• C1a2V. Asymptote y = x.

  • 1 y = x with observation about intersection with the gaph.

  • 2 Only one of equation or intersection.

• C2.

• C2aI. Tangent at M(4, 4).

  • 1 y = 0.92x + 0.33 or with more decimal digits

• C2bI. Tangent at P(0, −2)? What is seen on the screen

  • 1 There is no tangent at point P.

• C2bR. Tangent at P(0, −2)? Mathematical situation

  • 1 There is a tangent: Oy.

  • 3 There is no tangent at (0, –2).

• C2dI. Secant lines through P and neighbor points.

  • 1 Lines almost overlapping with y-axis, may be with the equations −0.001x = 0 and −0.00101x = 0 (or other similar)

  • 2 Reference to lines without precision about their position with regard to y-axis

• C3.

• C3aR. Tangent at (0, 0) to the graph of $y = \sqrt[3]{x}$ ?

  • 1. Tangent: Oy

  • 3. No tangent at (0, 0)

• C3bR. Tangent at (0, 0) to the graph of $y = \sqrt[3]{{x^{2} }}$ ?

  • 1 Tangent: Oy

  • 3 No tangent at (0, 0)

• Questions about opinions (C4a, C4b, C4c, C4d, C4e): Each answer is coded by an integer between –2 (very negative opinion) and 2 (very positive opinion).

Appendix 3: Macroscopic view of a function obtained with GeoGebra

The image shows the obtained view from the Grafica2.ggb file after Zooming out and performing the choice of points C and D on the graph and drawing the line through C and D.

Note 1: The represented function f(x) is the same as in Fig. 5.

Note 2: The vertical asymptote is not displayed!