Exploring Patterns and Functions

Transcription

1 Technology and other electronic media have become popular with students of all ages. Game programs, virtual-reality helmets, and online environments are just some of the electronic tools that fascinate school-age students. Yet numerous engaging applications of technology for learning school mathematics exist. This article describes how to use virtual pattern blocks, a virtual graphing tool, and a spreadsheet to explore the concept of function in algebra in grades 6 through 8. By using these electronic tools, students have the opportunity to generate and compare various representations of a function. The technology enables students to focus on the mathematics of the tasks by eliminating the distraction of tedious computations. Students in the middle grades study more abstract concepts and move away from the concrete tasks of elementary school. Sometimes they need extra help and scaffolding to make this transition. The use of contextualized problems with multiple representations can help students connect these concrete and abstract mathematical ideas. Middle school students sometimes see using concrete materials as less sophisticated than using a technological tool. Therefore, using a virtual mathematics applet may be more appealing to students, yet still will provide the necessary scaffolding and support that moves students into abstract algebraic concepts. Important goals of the NCTM Standards for middle school students are to investigate, make conjectures, verify those conjectures, and generalize concepts (NCTM 2000). These investigations should focus on linear and non-linear relationships that develop students understandings of characteristics of each relationship and that allow students to distinguish between the two types of relationships. Students can explore these relationships in a variety of ways, using tables, graphs, and symbols so that they then can compare the different representations. The ability to move flexibly among various representations increases students mathematical power. Exploring Patterns and Functions The following investigation includes five different tasks that focus on helping students in grades 6 8 develop a comprehensive view of functions. These tasks have the following objectives: Incorporate the use of technology (virtual pattern blocks, spreadsheets, and a graphing tool) into the familiar frames problem Explore the algebraic, graphic, and numeric representations of functions Introduce students to concepts of functions and function notation Task 1 allows students to create visual models of square frames that are used to generate data representing both linear and quadratic relationships. In Task 2 students examine the data they have recorded and look for patterns in the number of tiles that form the frame, the number of interior tiles, the total number of tiles, and the perimeter. In Task 3, students have the opportunity to verify the rules they created in Task 2 by using the spreadsheet. Task 4 is designed to allow students to explore a graphical representation of their functions. And in Task 5 students analyze the graph and solidify and extend their understanding through articulation of their thinking.

2 Task 1: Generating, Gathering and Recording Data The first task allows students to create visual models of square frames that are used to generate data representing both linear and quadratic relationships. Students begin the task by creating a 2 x 2 frame, using squares from the virtual pattern blocks. Virtual pattern blocks can be found at several locations on the Web, including the NCTM Web site ( repository/6165/applet/shapetool/index.html ), the National Library of Virtual Manipulatives (www. matti.usu.edu/nlvm), and the Arcytech site (www. arcytech.org). Using the square blocks on one of these sites, students create a 3 x 3, 4 x 4, and 5 x 5 (and so on) frame using the virtual squares (see fig. 1). The created shapes are called frames because the blocks used to create the frames are on only the outer edge of the shape; that is, there is empty space in the interior of the frame. For example, a 3 x 3 frame would have three blocks on each side of the frame with an open space the size of one block in the middle; therefore, the frame would contain eight blocks. A 4 x 4 frame would have four blocks on each side of the frame with an open space the size of four blocks in the middle; therefore, the frame would contain twelve frame tiles. Interior tiles are created by using a square tile that is the same size as those in the frame, but are a different color. After creating the frames, students should record the following data about each frame using a standard spreadsheet program: Frame dimensions: (for example, 2 x 2, 3 x 3, 4 x 4) Side length: the length of one side of the frame (for example, two would be the side length for a 2 x 2 frame) Frame tiles: the number of tiles that form the frame of the square and whose edges lie on the perimeter of the frame (for example, a 3 x 3 frame would have eight frame tiles) Interior tiles: the tiles that would fit on the interior of the square that forms the frame (for example a 3 x 3 frame would have one interior tile) Total tiles: the sum of the frame tiles and the interior tiles (for example a 3 x 3 frame would have nine total tiles) Perimeter the distance around the outside of the frame (for example a 3 x 3 frame would have a perimeter of twelve units). Task 2: Look for Patterns and Create a Formula In this task, students examine the data they have recorded and look for patterns in each of the categories: Frame Tiles, Interior Tiles, Total Tiles, and Perimeter. Students can begin by describing what happens from one stage to the next in each sequence. For example, in the Frame Tiles sequence, students should notice that numbers increase by four in each stage. The Perimeter sequence also grows by a common difference of four. In the other two sequences, no common difference exists. Students should discuss these patterns and look for other characteristics or ways to describe the number sequences. To help students with this task, ask questions such as What is happening to the numbers (increasing, decreasing)? By how much are they increasing? Is there anything the numbers have in common? How do these numbers relate to the length of each frame? After students have the opportunity to explore the iterative pattern, they can consider what would happen if they needed to generate data for a 50 x 50 square or a 100 x 100 square. Clearly, continuing the pattern is one way to find the information, but it is not the most efficient way. Through discussion, encourage students to define an explicit rule for each sequence, using the variable n to represent the length of the side of the frame. By creating this rule, students can generate data for frames of any length.

3 Task 3: Verify Your Formulas In this task, students have the opportunity to verify the rules they created in Task 2 by using the spreadsheet. Students should create four new columns on their spreadsheet with the following headings: Formula (Frame), in which they will test their rule describing the pattern for the frame tiles sequence Formula (Interior), in which they will test their rule describing the pattern for the interior tiles sequence Formula (Total), in which they will test their rule describing the pattern for the total tiles sequence Formula (Perimeter), in which they will test their rule describing the pattern for the perimeter sequence (see fig. 3) Using the spreadsheet s formula capabilities, students can enter the formula once under each column heading and cut and paste it to the remaining cells in the column to quickly generate data. For example, the formula for Frame Tiles, using n as the length of the side of the frame, is 4n 4. In the first cell under the new heading titled Formula (Frame), students click on the formula button and enter 4*B2 4 (where B2 is the cell in which the length of the 2 x 2 square, 2, is entered). The number 4 should appear, which shows the number of tiles in the frame. Selecting the cell in which the formula was entered, copying it, and then pasting it into the remaining cells in the column will generate results using each of the side lengths 3, 4, 5, 6, 7, and 8 for n. These numbers should match those that the student entered under Frame Tiles, verifying the student s rule. If the numbers do not match, the rule is incorrect and the student can analyze the incorrect results to determine adjustments to make the rule correct. Task 4: Explore the Function This task is designed to allow students to explore a graphical representation of their functions. Students can enter data as a set of ordered pairs and as a function, and view both representations on the same graph. For example, to view the Frame Tiles function graphically, students might first enter their data as a set of ordered pairs in which x represents the length of the side of the frame and y represents the number of frame tiles in that frame. The entry for the third frame, 4 x 4, would be (4, 12). Clicking on PLOT creates a line graph of this information. Students can toggle between a connected graph and scatterplot view. Next, students can enter their rule in the function box (4*x 4). Clicking on PLOT shows both graphs concurrently. Students can compare the two graphs and begin to get a sense of how to illustrate the data (see fig. 4). Task 5: Analyze the Graph Once students have the opportunity to gather data, look for patterns, create and test formulas, and explore graphs, they need opportunities to solidify and extend their understanding through articulation of their thinking. This can occur through written and oral communication in small groups as well as wholeclass settings. Teachers may focus on a variety of topics and questions during these discussions. Depending on the level of the class, these discussions can lead to exploring such concepts as linear versus quadratic functions, function notation, and domain and range.

4 Conclusion This investigation allows students to look for patterns in data they have generated in the context of a problem, a fundamental skill in the development of algebraic thinking. Through these tasks, students can see how developing an algebraic expression or rule that describes the pattern allows them to generate data quickly. By creating their own numbers, students are able to observe the behavior of the pattern as it emerges how the numbers change and how the relationship develops between the input and output variables. As a result, students are able to use an arithmetic or procedural approach, in which operations are carried out on numbers, rather than a structural approach, in which operations are carried out on algebraic expressions. Research on the development of beginning algebraic thinking indicates that students find the former type of approach more meaningful than the latter (Kieran 1992). Exploring the functions generated by this problem through several different views numeric, algebraic, and graphical helps students begin to develop their understanding of variables. Research indicates that exposure to a curriculum that focuses on the use of multiple representations helps students develop algebraic reasoning skills (Moseley and Brenner 1997). Students using multiple representations display more flexible usage and a deeper understanding of variables when solving problems than do students exposed to a more traditional curriculum. In addition, this investigation allows students the opportunity to explore the difference between linear and quadratic functions graphically while they relate their explorations to the situation and data that the graph describes. For example, students can compare the difference in the consecutive entries for the frame tiles to the differences in the consecutive entries for the interior and total tiles. They will discover that the frame tiles display a constant difference, whereas the differences for the other two are not constant. Through discussion, students can compare this finding to the graphical representations and explore the relationship between the rate of change whether it is constant or not and the behavior of the graph. Finally, students can begin to develop an understanding of domain and range by comparing the graph of the data set and the graph of the function. For example, in looking at the graph of both the data set and function for Frame Tiles, students can explore the possible values for x and y represented in each, discussing the differences and what each represents. In the data set, values for x, the domain of the function, are limited to the set {2, 3, 4, 5, 6, 7, 8} because they represent the side lengths of the frames actually built. A similar result occurs for the range. In the graph of the function, the domain and range are both continuous, representing the result for frames with a side length of any real number. Introducing students to the terms domain and range in this context establishes a more meaningful learning experience by allowing them to connect these concepts to a familiar situation.

5 student activities Task 1 Generating, Gathering, and Recording Data Open the virtual pattern blocks at matti.usu.edu/nlvm/nav/frames_asid_ 170_g_2_t_2.html?open=activities. Using the virtual squares, create a 2 x 2, 3 x 3, 4 x 4, and 5 x 5 (and so on) squares with a frame of one color and an interior of another color (see fig. 1). Open a spreadsheet and record the following information in the first cell of each column: In column A, Frame Dimensions ; in column B, Side Length ; C, Frame Tiles ; D, Interior Tiles ; E, Total Tiles ; and F, Perimeter. Under the Frame Dimensions column, enter 2 x 2, 3 x 3, 4 x4, 5 x 5, 6 x 6, 7 x 7, and 8 x 8 (see fig. 2). Using your virtual squares, record the side length, the number of frame tiles, the number of interior tiles, the total number of tiles, and the perimeter in the spreadsheet. Task 2: Look for Patterns and Create a Formula Study the data to find a pattern for the sequence of numbers in each category. Describe your pattern for each category. Frame Tiles _ Interior Tiles _ Total Tiles _ Perimeter Using your pattern, fill in the spreadsheet for a 10 x 10 square. Verify your pattern by using the virtual manipulatives to generate a 10 x 10 square. Were you right? Once you have determined each pattern, express it by using n to represent the number length of a side of the square frame; for example, for the 4 x 4 square, n would be 4. In the row following the 8 x 8 square frame on the spreadsheet, enter n x n under the Frame Dimensions column and type in a formula under each of the categories.

6 student activities Task 3: Verify Your Formulas In columns G, H, I, and J on your spreadsheet, type in the following: Formula (Frame), Formula (Interior), Formula (Total), and Formula (Perimeter). Use the spreadsheet s formula tool to enter the formula for the frame tiles in cell G2, under the Formula (Frame) column. Copy and paste this formula into cells G3 G8. Repeat the procedure for the remaining columns: Formula (Interior), Formula (Total), and Formula (Perimeter). Check your results with your original data. What do you notice? What adjustments do you need to make to the formulas, if any? Task 4: Explore the Function Open the Graphit! graphing tool ( org/interactivate/activities/graphit/index.html). Enter the data for the number of frame tiles in each frame as ordered pairs such that x represents the length of the side of the square and y represents the number of frame tiles in that frame. For example, the entry for the third frame, 4 x 4, would be (4, 12). In the function box, enter the formula for the number of frame tiles in each frame. Click on PLOT. The graph will show the data points in red and the function graph in blue. What do you notice? Toggle between the Scatter and Connected options under Plot Type to get a different view of the relationship between the graphs of the data points and the graph of the function. Repeat these procedures for the Interior Tiles, Total, and Perimeter.

7 student activities Task 5: Analyze the Graph What do you notice about the graphs? How are the graphs of the three functions similar? How are they different? Do you see anything in the formulas that might explain this? Do you see anything in the data that might explain this? Why does the function graph appear to be continuous but the data points do not? Figure 1: 2 _ 2, 3 _ 3, and 4 _ 4 Frames. Figure 3: Formula Columns Figure 2: Spreadsheet Figure 4: Function and Data Graph

8 references Kieran, Carolyn. The Learning and Teaching of School Algebra. Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp Reston, Va.: National Council of Teachers, Moseley, Bryan, and Mary E. Brenner. Using Multiple Representations for Conceptual Change in Pre-Algebra: A Comparison Of Variable Usage with Graphic and Text- Based Problems. Washington, D.C. Office of Educational Research and Improvement,1997. National (NCTM).. Principles and Standards for School Mathematics. Reston, Va. NCTM, 2000.