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CREATING A
MATHEMATICAL LABORATORY USING A SPREADSHEET TO INVESTIGATE THE CONNECTION
BETWEEN MATRICES AND GEOMETRIC TRANSFORMATIONS
Overview / Rationale
This activity uses spreadsheets and an investigative teaching strategy (Eggen and Kauchak, 1996), to introduce students to matrices, matrix multiplication, matrix addition and geometric transformations. The lesson is the result of several years of refinement in my classrooms of an idea from the Systemic Initiative for Montana Mathematics and Science project (MCTM SIMMS, 1996). It is a valuable investigation for students from middle school through college. Students, using a spreadsheet, will investigate connections between matrix multiplication and addition, and geometric transformations, thereby creating bonds between algebra and geometry. This activity is applicable to any school with a computer and modern spreadsheet. The spreadsheet used in this activity is Excel (Microsoft, 1995), although it should be adaptable to most spreadsheet programs. Students are first introduced to matrices, appropriate terminology, and how matrices can be used to represent points in a coordinate system by using the older technology of blackboard and chalk. A matrix can be represented on a spreadsheet and graphed by using a scatterplot with connected points (see Figure 1). Matrix Flag is represented by the scatterplot of the preimage. Several more examples like the "flag" matrix are used to reinforce students' understanding. Students are next introduced to matrix multiplication. You may choose to do this before the spreadsheet activities, or after exploration and investigating with the spreadsheet. Another matrix, "Matrix Transform" is entered into the spreadsheet to the right of the preimage matrix "Matrix Flag" (See Figure 2). These two matrices are then multiplied using the built-in matrix multiplication function of Microsoft Excel (See Figures 2 and 3). If your spreadsheet does not have a built-in matrix multiplication function, it can be performed by following the algorithm for matrix multiplication and using regular spreadsheet addition and multiplication. Figure 4 shows how to setup the spreadsheet in Clarisworks (1995), which does not have a built-in matrix multiplication function. The resulting product matrix (the image), is graphed on the same set of axes in Microsoft Excel, using the capabilities of the spreadsheet (See Figure 5 for directions). Spreadsheets other than Excel may require two separate charts-one of the preimage and another for the image (See Figure 4). Teachers may opt to setup the spreadsheet for students, or let the students do it themselves, depending on the technological sophistication of the class. The spreadsheet as set up in Figure 2, is excellent for use with a single computer and overhead LCD panel, for classroom presentation. MAKING CONNECTIONS We have created a mathematical laboratory where students can explore and discover. When the teacher or students change numbers in "Matrix Transform," the spreadsheet will automatically "move" the image and students can immediately see the change in the image matrix (see Figure 6). To develop understanding, students are required to work through a set of questions designed to allow them to establish patterns and make generalizations (see Sheets 1 and 2). The questions require the students to formulate hypotheses and test them. Students now can investigate how changes in Matrix Transform result in transformations in the product matrix and corresponding image. They can make conjectures and immediately test and see the results of their hypotheses. Throughout the lesson emphasis is placed upon proper use of mathematical vocabulary and connections between algebra and geometry. Geometric mapping notation such as (x, y)-> (x, -y) is connected to the transformations and the matrices. After completing guided activities and assignments, the students are eventually asked to generalize how multiplication by certain matrices results in specific transformations, by summarizing what they have learned in writing. Students are required to state what happens when multiplying an "n x 2" matrix by matrices of the form Students complete guided group activities, assignments and quizzes (see Sheets 3, 4, 5, and 6). They are evaluated on class participation, answers on worksheets, homework assignments, quizzes, tests and a writing assignment summarizing what they have learned. Through this guided investigation, students will begin to understand,
how to produce a reflection, rotation and dilation, by using matrix multiplication.
Students will be able to predict what will happen when points in the plane
are transformed by matrix multiplication of the forms Matrix addition and geometric translations are covered in another spreadsheet, but are handled in a similar fashion. Translations and matrix addition can be done before or after the other transformations depending on your preference. The spreadsheet is setup for addition with Matrix Flag (the preimage matrix) and Matrix Transform added using typical array adding procedures for spreadsheets (See Figures 7 and 8). Matrix Transform must of course be the same size as the preimage matrix, and is best constructed by referencing an " x" and "y" cell. These two cells are at the top of Figure 7 and 8. Students can change the two cells for "x" and "y" and have the spreadsheet automatically make the changes in Matrix Transform (See Figure 7). REAL WORLD APPLICATIONS One application of this lesson that students today are familiar with is computer environments and graphics. They are comfortable with seeing an "object" move on the screen. Many are familiar with "copying and pasting" or "selecting and dragging" computer objects. This method of teaching matrices and transformations provides a foundation for how computers store and manipulate information, and thus how computers can perform graphic operations. The lesson provides a relevant application of mathematics which makes sense to them in their everyday life. COMMUNICATION The spreadsheet environment provides more opportunity for students to explore concepts than with traditional methods of teaching matrices, stimulating reflection and abstraction, which will further enhance student's construction of the desired concepts (Glaserfeld, 1995). The spreadsheet activities provide an opportunity for social interaction when students work in groups of two rather than alone, since the activity naturally generates discussion, which will assist in students' construction of concepts. If two students disagree about the outcome of a hypothesis, it opens the door for verbalization and increased student understanding (Hoyles, 1992; Hoyles, 1985; Piaget, 1969; Vygotsky, 1986). The spreadsheet environment can provide a medium and language through which students can express and communicate their ideas, which assist in developing a formal mathematical understanding (]Hoyles, 1992; Hoyies and Noss, 1992). It assists the teacher by giving students a way to express their ideas, and provides a medium through which students can investigate and discover mathematical concepts and systems, before having a formal mathematical background (Hoyles, 1992; Hoyles and Noss, 1992; Hoyles, Sutherland and Healy, 199 1; Sutherland, 1993; Suther land and Rojano, 1993). The use of the spreadsheet creates an environment in which students have much greater access to the mathematics than before the use of technology in the classroom. The spreadsheet provides a medium through which students can see multiple representations of a concept, in this case the numerical form of the matrix and its graphical representation. Students can form conceptual connections between the representations, their attention is constantly moving between the matrix, the graph, and the changes in the matrix they have made. They can see and participate in a dynamic environment, and watch the immediate consequences of their actions. Students can see many specific instances of a larger abstraction very quickly and they can see more of a variety of examples, thus creating a means by which students can make the leap from the specific to the general, which is central to the understanding of mathematics (Esty and Teppo,1996; Mason and Pimm, 1984). Clarisworks, Version 2.1, (1995). [Computer Software], Santa Clara, CA: ClarisCorporation. Eggen, P. D. & Kauchak, D. P. (1996). Strategies for teachers: Teaching content and thinking skills. Needham Heights, MA: Allyn & Bacon. Esty, W. W. & Teppo, A. R. (1996). Algebraic thinking, language and word problems. In P. Elliot (Eds.), Communication in mathematics, K-12 and beyond (pp. 45-53). Reston, VA: National Council of Teachers of Mathematics. Hoyles, C. (1985). What is the point of group discussion in mathematics. Educational Studies in Mathematics, 16 , 205-214. Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study. For the Learning of Mathematics, 12 (3), 32-44. Hoyles, C. & Noss, R. (1992). A pedagogy for mathematical microworlds. Educational Studies in Mathematics, 23, 31-57. Hoyles, C., Sutherland, R., Healy, L. (1991). Children talkining in computer environments: New insights into the role of discussion in mathematical learning. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Researchand practice (pp. 162-175). Bristol, PA: Open University Press. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the par- ticular. Educational Studies in Mathematics, 15, 277-289. MCTM SIMMS, (1996). Integrated mathematics: A modeling approach using technology. Levels 1-6. Needham Heights, MA: Simon and Schuster. Microsoft Excel, Version 5.0, (1995). [Computer software], Redmond. WA: Mi- crosoft Corporation. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (I 992). Professional standards forteaching mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (I 995). Assessment standards forschool mathematics. Reston, VA: Author. Sutherland, R. (1993). Consciousness of the unknown. For the Learning of Mathematics , 13 (1), 43 -46. Sutherland, R. & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior , 12, 353-382. Piaget, J. (1955/1969). The language and thought of the child. Cleveland, OH:The World Publishing Company. Von Glaserfeld, E. von (1995). Radical Constructivism: A way of knowing and learning. Washington, DC: The Falmer Press. Vygotsky, L. S. (1986). Thought and language. Cambridge,
Mass: The MIT Press.
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along with other relevant possibilities. Students can make inferences concerning
the effect of the magnitude of the matrix elements, describing how a transformation
using a number between "0" and "1", is different than a transformation
using a number larger than "1."
for all values of "a." Students will begin to ask questions about combining
and undoing transformations, and how a composition of two reflections can
be equivalent to a rotation. Teachers can then assist in developing the
formal mathematical meanings of function composition and inverse functions.
By practicing with the spreadsheet and going through this investigative
lesson, students will be able to set up matrix equations describing a given
geometric transformation. Conversely, they will be able to describe geometric
transformations given the matrix equations.