Developing Connections Between Algebra and Geometry,

Aided by the Power of the TI-92:

A Study of Affine Geometry

Daniel R. Miller

Illinois State University

Mathematics Department

Campus Box 4520

Normal, IL 61790

September, 1996

As one reviews the current literature in mathematics education an overriding theme is apparent. That is the development of connections between the various components that separate the current curriculum into distinct discipline. As an example of this separation, one could examine most secondary curricula guides to find sections on algebra, geometry and advanced mathematics. Yet this disjoint approach not only impedes the process of developing the connection between areas of mathematical content, it denies the fact that many important advances in mathematics have come as a result of applying algebraic methods to geometric concepts.

Currently, there are at least two forces driving this call to integrate curriculum. They are the NCTM Standards along with other NCTM publications and that of current research in mathematics education. In particular the NCTM acknowledges "[t]he interplay between geometry and algebra strengthens students' ability to formulate and analyze problem situations." (NCTM p. 161) Additionally, within the area of geometry it suggests the integration of "synthetic, coordinate and transformation geometry, [allowing] as many opportunities as possible to compare, contrast and translate among these systems." (p. 161) Furthermore, current research suggests that integration of other components such as computer programming produces positive effects on students' mathematical achievement, their problem solving abilities, and their attitudes toward computers. In particular, Lee (1990) reports "[t]he use of computer programming in mathematics instruction can be expected to shift the position of the average treated student from the 50th to the 64th percentile of the control group [in achievement]." (p.61)

Certainly this call to integrate computers into the mathematics curriculum is not new. As early as 1967 NCTM states, "[t]here is evidence that computer-assisted problem solving can be used effectively as an instructional aid in teaching mathematics." (p.2) But by the early 1970s' the promise of enhanced instruction through the use of computers was being drowned out by the problems associated with its implementation. (Conference Board of the Mathematical Sciences Committee on Computer Education, 1972)

In the last three decades, many of the problems associated with the implementation of computer programming as an integral part of mathematics education have been overcome due to the rapid advances in electronic technology; size, cost, functionality, and availability. It is now practical , with the new technology, to begin to implement curriculum changes which integrate computer programming and mathematics instruction. However, this integration will only be successful if teachers are exposed to materials which are both mathematically significant and allow the student to gain additional insight and abilities that were not easily acquired through conventional instructional methods.

We will now examine integrating new technology with newly proposed curriculum concepts in algebra and geometry. In particular, we will look at the application of the programming features of the TI-92 calculator with respect to developing insights into affine geometry.

The study of affine geometry includes the examination of similarity transformations and isometries. In general an affine transformation in an invertible linear transformation which preserves collinearity, betweeness, parallelism and ratio of division of segments. It is represented analytically in the form x'=ax+by+h and y'=cx+dy+k where ad-bc0. Additionally, if (a=d and b=-c) or (a=-d and b=c) the transformation is a similarity. Adding the restriction that the absolute value of (ad-bc)=1 causes the similarity to become an isometry.

So why examine Affine transformations? The Standards (NCTM, 1989) calls for increased emphasis on both similarity transformations and isometries with particular attention on their relationship to congruence. (p. 162) Additionally, affine transformations allow a direct connection between traditional high school algebra, geometry, and matrix algebra. This connection can be further clarified by examining the images of figures produced by affine transformations. Finally, the equations describing affine transformations are linear, allowing for the efficient use of matrix algebra in the analysis of these equations and the subsequent use of technology. The remainder of the article will focus on affine transformations which produce similarities and isometries. In particular, the focus will be on the problem solving strategies needed to develop computer programs that analyze these forms of affine transformations.

A simple, not complete, flowchart (see figure 1) serves as a introduction to both the general sequence of analysis while providing a foreshadowing of more detailed decision making to come without being saturated by them. In analyzing this flowchart it is apparent that the first decision to make is whether or not the given transformation is an affine transformation.

What does it mean to be an affine transformation? From the prior description, an affine transformation is a function defined by x'=ax+by+h and y'= cx+dy+k where a,b,h,c,d,k are real numbers and ad-bc does not equal zero.

Where can we make connections within the context of this description of an affine transformation. First, explore specific examples of the mappings created by these kind of functions. How can these mappings be represented? By a chart, by a graph? Can we build a connection between the algebraic representation, ordered pairs, and the geometric representation of the plotted points( see figure 2)? Are the original figure and its image similar or congruent? Does there appear to be a reflection, rotation, translation, or dilation involved?

From this simple repeated mapping, we certainly provide the opportunity to explore the concepts of both similarity and congruence. Additionally, the opportunity to discover many of the properties of affine transformations are possible if enough examples are explored and technology allows this exploration to occur a great deal faster.

The remainder of the article will deal with the development of a TI-92 program that will generate information about Affine transformations.

The first step is to decide whether or not the given transformation is affine. What information do we have to base the decision on and how can we program the TI 92 to make this kind of decision. First we have the values of a,b,h,c,d,and k. Second the program for the TI 92 is as follows:

Input "Enter the value of a",a

Input "Enter the value of b",b

Input "Enter the value of h",h

Input "Enter the value of c",c

Input "Enter the value of d",d

Input "Enter the value of k",k

If a*d-b*c=0

Then

Disp "The transformation is not Affine"

Else

Disp "The transformation is Affine"

EndIf

The real power of using the TI-92 algebraic features becomes apparent when it performs the calculations necessary to find the fixed points in the transformation. In theory it is easy set x'=x and y'=y and solve the system of linear equations in two unknowns (see figure 3). The coding is given below:

continue from above:

[[a,b,h][c,d,k]]-[[1,0,2*h][0,1,2*k]]->f

rref(f)->redtrans

Disp redtrans

Here rref() is the TI-92 command that returns the reduced echelon form of the matrix. For a further explanation of the coding see figure 4.

The next requirement is to determine whether the affine transformation is a similarity. This requires that one evaluates a series of conditions. If (a=+-d and b=c) or (a=d and b=+-c) then the affine transformation will be a similarity. The coding for these conditions may seem long and cumbersome, but remember the goal here is to design a program that will be repeatedly used. Besides, the discussion of the logic necessary to produce the code is a worthwhile classroom activity. One solution is as follows. You may want to convince yourself of its validity.

continue from above:

If a^2<>d^2 then

Goto notsimil

EndIf

If b^2<>c^2 Then

Goto notsimil

EndIf

det([[a,b][c,d]])->trandet

If trandet=-(a^2+b^2) Then

Goto simila

EndIf

If trandet=(a^2+b^2) Then

Goto simila

EndIf

Lbl notsimil

Disp "Not a similarity"

Goto done

To continue, our next step is to decide if the similarity transformation is an isometry. Here again we discovery the efficiency of programming for we have already calculated all the information needed to make this decision.

Lbl simila

If abs(trandet)=1 then

Goto isometry

EndIf

At this point, if the program passed the If statement we know that the transformation is a similarity that is not an isometry.

Finally, we can determine which isometry is represented by the transformation. Again we have already calculated all the necessary information, i.e., the determinant value and the number of fixed points. Rather than continuing to describe the logic and coding aspects (see figure 5 for the completed program) of this activity let us return to reflect on the original goals of this process.

In the outset our goal was to develop an activity which would integrate mathematical concepts from algebra, geometry, and matrix algebra. Clearly, this activity does that, but it also incorporates additional concepts of computer programming and logic. In this activity, through the addition of technology which allows for repeated trials, we incorporate an opportunity for our students to discover many of the properties of affine transformations while developing problem solving strategies such as computer programming.

In conclusion, this activity has a variety of classroom application. First, at the middle school level, the completed program could be used with specific test data to discover transformation that yields similarities and isometries. Additionally for high school algebra students, the discussion of alternatives methods for solving systems of equations is quite appropriate. Finally, the activity can be used with more advance students to develop the connections between algebra and geometry while concurrently developing skills in logic and computer programming.