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Lesson Plan: Geometric Sequences Objective: -Students will become familiar with geometric sequences through investigations with fractals. Prerequisites: -students will be familiar with sequence notation -students will have had some experience with fractals -students will have at least average skills with Geometer Sketchpad. Introduction: Consider the following sequences, what relationship do you observe between successive terms of these sequences? 3n 1,3,9,27,81,243,..., ,... 3 2,4, 8,16, 32,64,...,( 2) n ,... n 1 1 1 1 1 , , , ,..., ,... 2 4 8 16 2 Sequences with this property are called geometric sequences. The ratio between consecutive terms is called the common ratio. Review of Fractals : 1. What is a fractal? 2. Using the National Archive of Manipulatives students will review the properties of fractals Investigation of Geometric Sequences: Question: Is there a shape that has an infinite perimeter, and an area of zero? To answer this question, students will do investigation 1 and 2, found at the end of the lesson plan? -Students will find the common ratio of a sequence. -Students will find the nth term of a sequence. -Students will determine if sequence is geometric. -Students will investigate sequence as n approaches infinity. The Sum of Geometric Sequence Let S n a1 a1r a1r 2 ... a1r n 2 a1r n 1 and rS n a1r a1r 2 a1r 3 ... a1r n 1 a1r n by finding the difference we have: S n rS n a1 a1r n so S n (1 r ) a1 (1 r ) and n 1 rn S n a1 r 1. 1 r Investigation 3: The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the triangles are shaded (See figure in GSP applet). If this process is repeated 9 times, determine the total area of the shaded region. What if the process is repeated an infinite number of times, what is the total area of the shaded region? -Students will use all information they have learned from investigations 1 & 2, and in addition will need to sum their sequence in investigation 3. -Students will also investigate the sum of an infinite geometric sequence in investigation 3. Investigation 1 1. Construct triangle ABC, with midpoint D, E, F. B F E A D C 2. Construct the interior of triangle ABC. B F E A D C 3. Iterate the construction of triangle ABC to triangle AFD, and then to triangles FBE, and DEC (Hint: Use Add New Map in structure menu to iterate all triangles at once, and choose Final Iteration Only from the display menu). Once this is done, hide the original interior. Your image should look like the following after 1iteration: B F E A D C 4. Use the + and – keys to explore other iterations of the Sierpinski Gasket. 5. Suppose a stage 0 gasket (a single triangle) has area 1. What would the shaded area of stage 1 (1 iteration) be? Justify. 6. Fill in the rest of the table: Stage 0 1 2 3 4 Area 1 7. As the stages increase, what happens to the shaded area? 8. Does the area at each of the stages give us a geometric sequence? If so what is the common ratio, and what would the area of a stage n gasket be? 9. Using Geometer Sketchpad, define the coordinate system and plot the function between A(n) (area of shaded region at stage n) and n. What do you notice about A(n) as n gets really large? What would the area of the gasket be at stage infinity? 10. Suppose a stage 0 gasket has a perimeter of 3, fill in the following table and determine what the perimeter will be at stage infinity? Stage 0 1 2 3 4 n Area 3 Investigation 2 1. Using the GSP file Geometric Sequences. Click on the page marked square gasket. 2. Find the geometric sequence for its area using each of the stages, and write an expression for the nth term. Suppose the area of the square is 1 at stage 0. 3. Find the geometric sequence for its perimeter using each of the stages, and write an expression for the nth term. Suppose the perimeter of the square is 4 at stage 0. 4. At stage infinity, can you draw the same conclusions as you did for the triangle? 5. What does the triangle and square gasket sequence for area have in common that would make it behave the way it did at stage infinity? What about the sequence for perimeter? Using these observations, make a conjecture about the behavior of geometric sequences at stage infinity? Be ready to defend this conjecture? Investigation 3 The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the triangles are shaded (See GSP page extension). If this process is repeated 9 more times, determine the total area of the shaded region. What is the area of the shaded region at stage infinity? 1. Open the GSP file Geometric Sequences, and open the page labeled extension. 2. If I create a sequence of total shaded area for each stage, is it geometric? Why or why not? 3. Fill in the following table: Stage 1 2 3 4 n Area of new shaded triangles 3. Find the total sum of shaded region at stage 9 using the sum of a finite 1 rn geometric sequence formula: Sn a1 r 1 . Verify your 1 r result by adding the first 9 terms. 4. What is the area at stage infinity? Verify this with an algebraic model?