Digital Fabrication and Aunt Sarah: Enabling Quadratic Explorations via Technology. Michael L. Connell University of Houston - Downtown

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1 Digital Fabrication and Aunt Sarah: Enabling Quadratic Explorations via Technology Michael L. Connell University of Houston - Downtown Sergei Abramovich State University of New York at Potsdam Introduction When viewed as a content area, mathematics has a split personality. To use an example from language, there are parts of mathematics that function very much like a noun (the concepts of mathematics), while others function more like a verb (procedures, which many think of as actually doing math ). Such ideas form the basis for later, more formalized procedures. The role technology can play in visualizing these ideas for learners should not be overlooked. The following graph, for example, was created using a computer-based spreadsheet but could just as easily have been done using a smartphone app, an online graphics program or a handheld calculator. Figure 1. Graph of Y = 2X + 3. Things get a little complicated when the mathematics described has both noun and verb-like features (i.e., requiring understanding of both content and process components). For example, the number 2 can be a noun describing a position in a sequence or how many of something one might have. In this case, we are clearly using the noun-like features. In a different context, however, 2 can describe: how many times something appears as in the case of filling a bowl of cereal 2 times; a base used by computers to represent other numbers (this is also called Binary); or the power to which a quantity is raised as shown in X For students to develop meaningful mathematical understandings, they should have many rich experiences in mathematics from these two markedly different perspectives. So, as we select appropriate technology we need to allow them to experience mathematical structures containing both concepts to think about the noun-like content features, and processes to think with the verb-like procedural features. Once a teacher can see this dualism about mathematics, it has major impacts on potential roles of technology in the mathematic classroom. This can be shown very clearly when considering multiplication strategies. Multiplication is used to compute area, and area can be used to illustrate multiplication so both the concept and procedure can be illustrated at once. The Algebra Tiles application from the National Library of Virtual Manipulatives (found at ml ) was used in Figure 2 to model (X+1)(Y+2).

2 Figure 2. (X+1)(Y+2) In Figure 2 we see a rectangle being formed from placing representative tiles along two dimensions: X+1 in the vertical direction, and Y+2 in the horizontal direction. The resulting algebraic product is shown by the area itself. To fill this rectangle the student needs to use an XY piece, two X pieces, one Y piece, and two single squares. When this is written out in standard form it shows that (X+1)(Y+2) = XY + 2X + Y +2. In order to get to this point, however, students need to be able to utilize both the conceptual and procedural aspects of the representation created through interaction with this application. Digital Fabrication. One of the major tasks in the high school environment lies in developing the ability to work with abstract concepts. With its ability to provide an interface between functions and graphs technology can be an invaluable tool in this effort. The figures shown in this section were created with the Graphing Calculator (version 4.0) produced by Pacific Tech (Avitzur, 2011). There are a number of alternatives which could be used to do traditional graphing (Figure 3); yet, there exist very few tools (e.g., Wolfram Alpha) capable of graphing segments or their borders (Figures 4 and 5). It is a tribute to the power of technology that once the math is understood there are a variety of tools which might be used appropriately. The opposite is also true, however. If the math is not understood it does not matter what kind of tools you have access to. Consider the question of constructing the graphs of the functions y = x and y = x 2 in a single drawing. This construction is shown in Figure 3 and leads to the question of constructing just the parabolic segment and its reflection on the line X=1 as shown in Figure 4. Figure 3. Y=X and Y=X 2 Figure 4. A parabolic segment and its reflection in the line X=1. In order to construct the parabolic segment, one has to describe the points inside it in the form of inequalities. First, an x-coordinate of any point (x, y) that belongs to the parabolic segment satisfies the inequalities 0 < x <1, where x = 0 and x =1 are the points of intersection of the graphs y = x and y = x 2. Second, its y-coordinate satisfies the inequalities f (x) < y < g(x) where f (x) = x 2 and g(x) = x. These properties of the points that belong to the parabolic segment can be expressed in the form of simultaneous inequalities x - y > 0, y - x 2 > 0,x > 0,x <1.

3 In addition, the reflection of the parabolic segment in the line x = 1 can be expressed through another set of inequalities by substituting 2 x for x (2 - x) - y > 0, y - (2 - x) 2 > 0,(2 - x) > 0,(2 - x) <1. Figure 5. Digital fabrication of e -thick borders of the parabolic segment and its reflection. Likewise, the set of points that belong to the border of the parabolic segment can be described through inequalities. First, the graph of the upper border (a part of the line y = x) can be described as a set of points (x, y) for which the values of the coordinates x and y are e - close to each other; that is, y - x < e. Second, the graph of the lower border (a part of the parabola y = x 2 ) can be described as a set of points (x, y) for which the values of y are e - close to the values of x 2. Finally, once again, the inequalities 0 < x <1 characterize the points that belong to the border. In the context of the Graphing Calculator these properties of the points that belong to the border of the parabolic segment can be expressed in the form of the union of simultaneous inequalities é y - x < e, x > 0, x <1; ê ë y - x 2 < e, x > 0, x < 1. Adding another union of simultaneous inequalities é y - (2 - x) < e, 2 - x > 0, 2 - x < 1; ê ë y - (2 - x) 2 < e, 2 - x > 0, 2 - x <1 yields the right-hand side of the digital fabrication shown in Figure 5. Note that in Figures 4 and 5 e = Using technology to enable students to construct graphs of areas in the plane and their borders by using two-variable inequalities illustrates the way in which software can embody a mathematical definition (Conference Board of the Mathematical Sciences, 2001, p. 132). Aunt Sarah and the Farm. Aunt Sarah wants to help her nephew Jack. However, she does not want to simply give him money. Instead she will provide him with a 10 dkm x 10 dkm plot of land provided he keeps it fenced. At the end of the year she will reduce the width by 1 dkm and increase the length by 1 dkm so that in the second year he will have an 11dkm x 9 dkm plot. This will be done each year until there is nothing left but a fence (i.e., 20 dkm x 0 dkm). This way it will be up to Jack to work hard and make the most of this opportunity. Help Jack explore what to expect over the next 10 years. As a start, for each year find: 1. How much land will Jack lose from the preceding year?

4 2. How much land will Jack lose from the first year? 3. How will the shape of his farm change over time? 4. How many feet of fencing will it take to fence it in? The mathematics which underpins the Aunt Sarah problem allows for multiple competencies, both on the part of the teacher and that of the student, to be addressed. Without the use of supporting technology it typically takes several days of tedious calculations for sufficient data to be generated to get to the richer underlying mathematics. Thanks to the spreadsheet, the explorations of Aunt Sarah s farm allow more time to be spent on building connections between deeper levels of mathematical content than was previously possible including a powerful link forward from pre-algebra into limits and pre-calculus. It is recommended that initial problem setting and procedure choices be done prior to the introduction of the spreadsheet. A good teacher question to ask at this point is: Is there a way to predict what happens in the 5th year? The 6th? With the addition of spreadsheet an excellent bridging question is, How can we organize our work to make prediction easier? This last question quite commonly leads to a row and column layout which can be directly translated into a spreadsheet later. In exploring Aunt Sarah and the Farm problem, prior to the introduction of the spreadsheet, some fascinating mathematics can be shown. If you draw out what the farm would look like each year on a single figure, one will get the following: Figure 6. The changing shape of the land From here, the possibilities for exploration open up. For example, to show the land lost for any given year relative to the beginning year (the fourth year is shown), take the rectangle of land gained for that year (A), rotate it (B and C), and place it inside the original figure to show the total amount lost (D). Figure 7. Where the land is going from Year One It can quickly be shown that the land lost for each year that this is done will be a perfect square which certainly hints as some interesting patterns to come! To show the land lost for any given year relative to the preceding year (the difference between the fourth and fifth year is shown) take the rectangle of land gained for that year (A), rotate it (B and C), and place it inside the preceding figure to show the total amount lost (D).

5 Figure 8. Where the land is going from the preceding year These sketches both represent specific processes and the solution to problems. This preliminary exploration provides a context for the following spreadsheet explorations as well as providing important clues for exploration. The following screenshot from shows one possible way of representing the problem situation. Figure 9. Data and calculations confirmed using a spreadsheet When relationships between cells are observed and can be generalized the formula bar can be used to create many of the cells, for example cell B12 was defined as being: Figure 10. The Formula (Function) bar. This makes it easy to copy cell B12, together with its attributes, and easily copy these filling in the respective columns. This ability to copy relationships between cells, including functional relationships, helps in the students understanding and exploration of the mathematical situation. In a like fashion each of the following cells can be defined using the formula bar as being: Figure 11. Using the Formula (Function) bar. The increment in year was then defined in cell A13 as being: Figure 12. Last step in creating the spreadsheet

6 Once relationships are recognized and their underlying functions identified, it becomes easy to create meaningful tables of values. In this example, we can see this by copying cell A13 into cells A14 through A22. In a like fashion it is possible to copy cells B12, C12 and D12 into cells B13 through B22, C13 through C22 and D13 through D22. This is a bit different than the typical use of data tables serving as the basis for function identification. In this case, the function is created first and used to create a table of data for exploration. An examination of the formula bar for Column B (fx=10+a12), Column C (fx=10-a12), and Column D (fx=b12*c12) provides a possible avenue to explore the concept of difference of squares (i.e., the length (10+A12) and width (10-A12)) being used in the area calculation. In this case, Column D s function is equivalent to B12*C12 which in turn is equivalent to (10+A2)*(10-A12). Depending upon the classroom, this may not be followed up, but it does provide an important clue which could be utilized in further exploration into the mathematics underlying the Aunt Sarah and the Farm problem. By making explicit the relationships between cells the formula bar can often be used in this fashion to gain hints as to potential mathematical underpinnings. The mathematics which may be found beneath the rules can then be made available for student explorations. The remaining columns look at some of the other interesting interactions immediately springing from the problem situation. Each cell in Column E, E13 for example, was computed using the following convention: Figure 13. Differences from the preceding year When this is done the sequence of odd numbers is generated, leading to questions concerning where this shows up in the graphical and functional representations generated in the group activity. Column F was generated using the following: Figure 14. Differences from the first year The $ sign preceding the D and the 12 indicates that this location will be locked in and used as the reference for each of the cells generated by copying it. This ensures that each subsequent years difference will be computed taking the first year as the comparison. Now a sequence of squares is generated, once again leading to questions concerning where this shows up in the graphical and functional representations generated. Using these columns the following graphs were generated: Figure 15. Data to function to graph to story! It is now up to the students to describe which series gives rise to each graph and why. They should also be able to link their graphic representation created prior to the use of the spreadsheet (typically, done using graph paper) to these graphs.

7 An important conclusion that one can draw from this investigation is that given perimeter of rectangle, no smallest area exists whereas square (that is, rectangle with congruent adjacent sides) has the largest area. However, as noted by Kline (1985), A farmer who seeks the rectangle of maximum area with given perimeter might, after finding the answer to his question, turn to gardening, but a mathematician who obtains such a neat result would not stop there (p. 133). This note motivates extending Aunt Sarah and the Farm problem using the computational power of a spreadsheet. Technology enabled extensions. Of course, technically, in order for a line graph to be properly used a case must be made that there will not be any changes in the line as the difference between sampling times becomes infinitely small. This provides an easy link to the calculus which may be made via the spreadsheet. This can be shown by first changing the spreadsheet so that the change point occurs every month instead of every year. This action effectively changes the difference between points on the line graphs by 1/12. This is easily done by changing cell A13 to be: Figure 16. Links to advanced math Now we can reconstruct the earlier graphs using this more finely tuned set of measurements. When this is done the graphs created look like the following: Figure 17. Identical curves. This is the identical shapes as shown in the earlier set of graphs. The underlying equivalency can be better shown by changing the chart type to not plot the locations of the individual data points. In a like fashion we can narrow the limit to the day, the hour, the minute to any degree we might choose in each case since the underlying functions are the same the graphs will maintain the same shape! Technology has enabled us to develop in a very intuitive fashion the notions of limit which underpin calculus. Without technology this amazing development is not possible. Summary If we take the student-centered and meaning driven approach to mathematics education advocated in this paper, the question becomes what tools and abilities are necessary for success and how can educational technology be used as a tool in acquiring these? This is a crucial question as the nature of the "tools" which are provided to students to "thinkwith" come to significantly shape their performance and cognitive styles (Connell, 2001). For example, two-digit division may constitute a legitimate problem when paper and pencil are the only tools available

8 for the student to use but are no longer a problem when calculators are available. When a computer is available for the students use, the situation shifts again. A legitimate problem with a computer might involve the identification and selection of what data to include in the problem, identification of the problem goals, and selection of appropriate procedures and control statements to obtain and verify the desired results. Let us be careful not to transfer a misplaced belief that mathematics education is solely about developing speed of process over to our thinking about technology uses. Modern technology is capable of blinding speeds of process so this cannot be viewed as our end goal. If a student is to internalize and construct meanings from experiences, there must be time to reflect upon the nature of the experiences and how they connect with the students' existing mathematical knowledge (Abramovich & Connell, 2014). Great care must be taken to allow students to construct their own knowledge and representations and then establish the linkages with other (also student constructed) tools, representations, and concepts many of which are technology dependent. References Abramovich, S., & Connell, M. L. (2014). Using technology in elementary teacher education: A sociocultural perspective. ISRN (International Scholarly Research Network) Education, Article ID , 9 pages, doi: /2014/ Avitzur, R. (2011). Graphing calculator (Version 4.0). Berkeley, CA: Pacific Tech. Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: The Mathematical Association of America. Connell, M. L. (2001). Actions upon objects: A metaphor for technology enhanced mathematics instruction. In D. Tooke & N. Henderson (Eds.), Using information technology in mathematics education (pp ). Binghamton, NY: Haworth Press. Kline, M. (1985). Mathematics for the non-mathematician. New York: Dover.