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Dragon Curve Fractal Project AMP Dr. Antonio Quesada – Director, Project AMP FRACTALS ROCK: INVESTIGATING THE DRAGON CURVE FRACTAL Keyword: Fractal (Dragon Curve) Rotation Fractals: A figure generated by repeating a special sequence of steps infinitely often. Fractals often exhibit self-similarity (the fractals clone themselves). Introduction: Exploring nature can be an intriguing and challenging topic for all students. Nature is all around us and provides students with many interesting designs. The designs that make up objects in nature are polygons. However, the polygonal designs may not be recognized as the traditional polygons many students have studied and are familiar with from their mathematics classes. The designs in nature are given the general name of fractals. A branch of geometry called fractal geometry explores the designs within nature and the patterns that exist in the designs. A few examples of such fractals are the Koch Snowflake, the Fern Fractal, the Tree Fractal, the Sierpinski Triangle and our topic of investigation, the Dragon Curve. The Dragon Curve will challenge students as they try to create the various levels of the fractal. Existing knowledge: Students should be familiar with the following concepts: length of a segment, rotating objects about a point, polygons and their characteristics, and fractals (Koch Snowflake, Tree Fractal and the Sierpinski Triangle). They should also have experience using Dynamic Geometry Software. NCTM Standards: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning, and geometric modeling to solve problems Learning Objectives: 1. Students will be able to define a polygon on a plane. 2. Students will be able to rotate a defined polygon about a point on the polygon. Page 1 Project AMP Dr. Antonio Quesada – Director, Project AMP Materials: 1. Piece of paper (for the pre-lab paper folding activity) 2. Worksheet from optional paper folding activity 3. Laboratory worksheet from Lab 1 4. Laboratory worksheet from Lab 2 5. Computer and/or graphing calculator equipped with Cabri Dynamic Geometry Software. Procedure: 1. Students should be grouped in pairs. Method of grouping is left to the individual instructor. 2. Pose the following challenge to students: “Suppose your school district is sponsoring a contest to design a new school logo. The logo will be placed on all school communication and apparel. The challenge is to use fractals, specifically the Dragon Curve. How could you take the following figure to create the Dragon Curve and a potentially creative design that will rock the fashion industry?” 3. Optional: Have students complete the pre-lab paper folding activity (see lab worksheet). 4. Have students complete Lab 1 (rotating objects about vertices) 5. Have students complete Lab 2 (Dragon Curve Fractal: Creating the Dragon Curve) 6. Choices for the types of assessment are left to the discretion of the instructor. Some examples might include: the completed lab questions, class participation (student explanations of the lab, conjectures, answer and/or processes in finding solutions to the extension problems or student created extensions), peer or self-evaluation, evaluations (quizzes and/or tests) to replicate the lab results. Several forms of assessment should be utilized when lessons are inquiry/discovery based. Bonus: For interested students, the winning design would be placed on t-shirts, brought from home, using an iron-on print. Page 2 Project AMP Dr. Antonio Quesada – Director, Project AMP PAPER FOLDING INVESTIGATION FOLDING A FRACTAL – CREATING THE DRAGON CURVE Team Members: ____________________ ____________________ File Name: ____________________ Introduction: The first dragon curve fractals were constructed by folding paper in a prescribed manner. Each added crease doubles the number of folds and complexity. When the paper is opened, the outline of the dragon curve fractal appears. As a result of getting the outline of the dragon curve, students will have a visual and a preview of the dragon curve they will create while completing the lab, using Cabri. Directions: Follow the procedure below to create a fractal pattern with at least three repeats, or iterations, using a sheet of paper (3 inch by 5 inch recommended). The sheet of paper represents level 0, or the generator level. Procedure: 1. Create level 1 by folding the paper in half. The folded paper reveals the outline for the basic seed: two edges. 2. Create level 2 by folding the paper in half again. This fold creates level 2 because each line in level 1 is divided in half. How many edges are in the outline of the dragon curve now, if you were to look at it opened up from the side? 3. Create level 3 by folding the paper in half again. This fold creates level 3 because each line in level 2 is divided in half. How many edges are in the outline of the dragon curve now if you were to look at it, opened up from the side? 4. Unfold the paper. Look at the distinctive dragon curve emerging. Carefully trace the outline of the dragon curve below. 5. Extension: If you were to fold the paper one more time, how many edges are in the outline of the dragon curve now, if you were to look at it, opened up from the side? By folding the paper a fourth time, you would have created level 4. Can you draw the outline of this fourth level of the dragon curve? Give it your best attempt. In theory, you could continue until it is impossible to continue folding the paper. Page 3 Project AMP Dr. Antonio Quesada – Director, Project AMP Fractals Rock: Dynamic Geometry Lab – Creating the Dragon Curve Team Members: ____________________ ____________________ File Name: ____________________ Lab I: Rotating Objects about Vertices. Setup: Open Cabri Geometry II. Set it so that it fills the screen. Choose “Show Attributes” under the Options menu. Click on the color option (top button on the side). The color palette should appear on the screen. While holding the mouse button down, move the cursor to the middle of the screen and release the button. The color palette should now appear on the screen in its own window. Move the palette to the right side of the screen. [You can add the other option tools to the palette window by clicking on them – however, the other tools are not needed for this task] 1. Rotating a line segment around an end point. a. Show the axis. Using the Numeric Edit Tool create a measure of 4.00 centimeters (assign the units using CTRL-U). Move this number (using the Pointer Tool) below the palette. Using the Numeric Edit Tool create a measure of 25 degrees. Move this number below the 4.00 centimeter number. b. Using the Measurement Transfer Tool create a point located 4 cm from the origin along the x- axis. Click on the origin (when it indicates “this point”), then click on the number 4.00, then move the dashed line that appears until it lies on the x-axis and click to put the point there. Using the Segment Tool create a segment between the point at the origin and the point at the 4 cm mark on the x-axis. Change the color of this segment to Green. c. Now using the Rotation Tool rotate the segment 25 degrees about the point at the origin. Click on the segment (“this segment”), click on the point at the origin (“around this point”), and click on the 25 degree number (“using this number”). A new line appears. Change the color of that line to purple. Describe this line compared to the original line. d. Using the Numeric Edit Tool, click on the 25 degree number. Using the up and down arrows at the side, change the number. What happens to the line as the numbers change? Page 4 Project AMP Dr. Antonio Quesada – Director, Project AMP e. Where is the line when the number reads 165 degrees? f. Set the number back to 25 degrees. Using the Angle Tool measure the angle between the two lines. Using the Numeric Edit Tool, change the number to 125 degrees. What happens to the measure of the angle? g. Change the number back to 25 degrees. Grab the point at the 4 cm mark on the axis (the end point of the original segment) and move the first segment to the 4 cm mark on the y-axis. What happened to the measure of the angle? h. Now change the number to 80 degrees. What happened to the measure of the angle and where is the rotated line relative to the original line? i. How would you define the rotation of a line that is rotated 120 degrees about a point? 2. Rotating a polygon about a vertex. a. Clear the screen (Select All from the Edit Menu and press the delete key). Put a point in the middle of the screen using the Point Tool (2nd button). Using the Triangle Tool, create a triangle with one vertex at the point you created and the other vertices wherever you like. Keep the triangle fairly small. Label each vertex (A, B, and C) using the Label Tool (10th button). Color the perimeter of this triangle green. b. Using the Numeric Edit Tool create a 30 degree number and put it below the palette. Then, using the Rotate Tool rotate the triangle 30 degrees about vertex A. Color the perimeter of this triangle purple and label the two new vertices D and E where D is the vertex above the AB side of the green triangle and E is the vertex above the AC side of the green triangle.
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