The Sierpinski Triangle A Conjecturing and Communication Task This task will have you explore patterns in the famous fractal called the Sierpinski triangle. Fractal: A fractal is a geometric shape with the property of self-similarity. That is, each object is made up of smaller objects that have the same shape. A fractal is formed by continuously repeating the same operation on each successive figure. Some famous fractals include the Koch Snowflake, Sierpinski triangle, and the Jurassic Park fractal. In this task, we will learn to construct the Sierpinski triangle and try to answer some questions related to the patterns found in the triangle. You will need triangular grid paper to complete this task. Step 1: Draw an equilateral triangle on the triangular grid paper with sides of 2 triangle units each. Connect the midpoints of each side. How many equilateral triangles do you now have? Shade in the centre triangle. Step 2: Draw an equilateral triangle on the triangular grid paper with sides of 4 triangle units each. Connect the midpoints of each side. How many equilateral triangles do you now have? Shade in the centre triangles as before. Step 3: Draw an equilateral triangle on the triangular grid paper with sides of 8 triangle units each. Follow the same procedure as before making sure to follow the shading pattern. You will have 1 large, 3 medium and 9 small triangles shaded. Step 4: Using a computer (or otherwise) follow the steps below to produce the next two Sierpinski triangles. Using Microsoft word to construct the fractal Sierpinski’s triangle 1. Start with an equilateral triangle 2. Shrink to half size 3. Make 2 more copies 4. Place triangles together (with vertices touching) in quadrants 2, 3 and 4 of an imaginary Cartesian Plane 5. Repeat steps 2 through 4, five more times 6. Shade appropriately using the fill tool. Questions to guide your explanations and conclusions: 1. Look at the triangle you made in step 1. What fraction of the triangle did you NOT shade? 2. What fraction of the triangle in Step 2 is NOT shaded? 3. What fraction of the triangle in Step 3 is NOT shaded? 4. Do you see a pattern here? Use the pattern to predict the fraction of the triangle you would NOT shade in the Step 4 triangles. Confirm your predictions and explain. 5. Develop a formula so that you could calculate the fraction of the area which is NOT shaded for any step. 6. Use your formula to predict the fraction of the triangle that is not shaded in the 8th step. 7. Write the fractions in the above questions in order from least to greatest. Write a statement about how their order connects to the shading in process. 8. Find one other interesting pattern in the fractal called the Sierpinski triangle. Write a paragraph describing this pattern.
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