Digital Fabrication and Aunt Sarah: Enabling Quadratic Explorations via TechnologyMI C HA EL L . C ON NEL LUN I V ERS I TY OF HOU ST ON - DOW NT OW N SE RG E I A BRA M OV IC HSTATE U N I V E RS I TY OF NE W YORK AT P OTS DA M

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. This graph of f(x)=2X+3, for example, could be viewed as either depending upon context.

Graph of Y = 2X + 3.

Content and Process 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 X2 + 3.

A Multiplication Example 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*. Here 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

This construction shown in 1, leads to the question of constructing of just the parabolic segment and its reflection on the line x=1 as shown in 2.

1 2

Consider the question of constructing the graphs of the functions y=x and y=x2 in a single drawing.

Needed Skills What skills are necessary to create these two diagrams? 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=x2.

Secondly, the Y coordinate must satisfy the inequalities f(x)<y<g(x) where f(x)=x2 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-x2>0 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

Digital Fabrication of -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 -close to each other; that is |y-x| <

Secondly, the graph of the lower border (a part of the parabola y=x2) can be described as a set of points (x, y) for which the values of y are -close to the values of x2. Finally, once again, the inequalities 0<x<1 and 0<2-x<1 characterize the points that belong to the border.

The Calculator View 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

Adding another union of simultaneous inequalities

yields the right-hand side of the digital fabrication shown on the next slide.

Software Embodying Mathematical Ideas

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).

Note: in these figures =0.02

Aunt Sarah and the Farm 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 calculus.

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? 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 Changing Shape of the Land

If you draw out what the farm would look like each year on a single figure, one will get the following:

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).

The Total Land Lost

The Land Lost from the Preceding Year

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).

Spreadsheet Confirmation

From Data to Function to Graph to Story

Using these columns the following graphs were generated:

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

A Technology Enabled Extension

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.

Now we can reconstruct the earlier graphs using this more finely tuned set of measurements. The results of this is shown in the figure to the right.

A Link to Limits 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 here, 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 "think-with" 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 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 245146, 9 pages, doi: 10.1155/2014/345146. 

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. 143–171). Binghamton, NY: Haworth Press. 

 *National Library of Virtual Manipulatives. http://nlvm.usu.edu