Name Directions: 1. Start with one 8 ½ by 11 sheet of paper. 2. Hold the paper vertically and fold the paper in half. 3. Unfold the paper. Count the number of smallest rectangles seen. 4. Record your finding in the table below. 5. Return the paper to its folded state, and fold it in half yet again. 6. Again, unfold the paper. Count the number of smallest rectangles seen 7. Continue until you have folded the paper 6 times. Smallest Rectangles 0 1 2 3 4 5 6 7 8 9 10 1 Examine the values in the table for the Rectangles. Do you see a pattern developing? Hint: Can you express the Smallest Rectangles using exponents? Fill in the remainder of the table using the pattern you observe. Let s examine this situation further by preparing a scatter plot with our graphing calculator: 1. Press Y= and be sure this area is empty. 2. Enter the information into the calculator: Press the STAT key, choose #1 Edit. Enter the (1-6) in L1 and the Rectangles in L2. 3. Press 2 nd Y=, choose #1 Plot. Choose the options as shown at the right. Arrow through the choices and hit enter to lock in a choice. 4. Press ZOOM, choose #9 ZoomStat. 5. Prepare a scatter plot on the grid at the right. Please indicate a scale and label the graph.
Analysis of Paper Folding: 1. What is happening to the Smallest Rectangles after each fold? 2. Based upon your observations of the data, write an equation that will determine the Smallest Rectangles based upon the. y = 3. Enter this equation into Y= on your calculator and press GRAPH. Your equation should accurately portray the data plots. Sketch the graph of your equation on top of your scatter plot. 4. Using your equation, determine the Smallest Rectangles that would result from 20 folds. 20 folds = rectangles. 5. The is the independent variable in this experiment and the Smallest Rectangles is the dependent variable. Your equation in problem #2 is a model for exponential growth. An exponential function in general form is written y = a b x. Compare this form with your equation in problem #2. a. In this paper folding experiment, what numerical value does b possess? b. In this experiment, the x in the equation pertains to c. When x = 0, the value of b 0 is. What is the value of a in this experiment? 6. If you examine the areas of the smallest rectangles after each fold, you would be creating a model of exponential decay. a. Describe what is happening to the areas of the smallest rectangles following each fold. b. What is the area of your 8 ½ by 11 sheet of paper?
c. In relation to the area of the original sheet of paper, write a formula for determining the areas of smallest rectangles after each fold. d. Using your formula, complete the table below (round answers to the nearest tenth of a square inch): Area of the Smallest Rectangle (square inches) 0 1 2 3 4 5 6 e. Using your graphing calculator, prepare a scatter plot of the data from the table. Follow the same directions as were used for the first calculator graph, but place the Area of the Smallest Rectangles in L2. Display the scatter plot on the grid at the right. Please indicate a scale and label the graph. f. Will this graph ever reach the x-axis? That is, will the area ever be zero? Explain your answer.
Analysis of Paper Thickness: ( Strange, but true! ) Each time the paper is folded, its thickness is increasing. At 3 folds, the thickness is about as thick as your fingernail. At 10 folds, the thickness is as thick as the width of your hand. If 14 folds were possible, the thickness would be the height of an average adult female. If 20 folds were possible, the thickness would be greater than the length of a football field. If 50 folds were possible, the thickness would be greater than two-thirds of the distance from the Earth to the Sun. Don t believe it? Let s investigate: 1. Does the number of layers of paper in each fold coincide with the Smallest Rectangles? That is, are they the same? 2. Different types of papers have different thicknesses. Suggest a method for measuring the thickness of a sheet of paper. 3. A standard sheet of notebook paper measures approximately 0.004 inches (or 0.1 mm) in thickness. Complete the chart below, using this information in inches. Layers of Layers of Thickness of the Folded Papers Paper Paper (in inches) (exponential form) 0 1 2 0 0.004 1 2 2 1 0.008 2 3 4 5 6 10 14 20 50 x
4. From the chart in question #3, make the following conversions: 14 20 50 Convert to feet: Convert to feet: Convert to miles: Thickness of the Folded Papers 5. It becomes very difficult to fold paper beyond the 6 th fold. In 2005, however, a high school junior named Britney Gallivan folded a sheet of gold foil 12 times, breaking all previous records. Not only did she break the record, she also developed a folding limit equation which determines the least amount of paper length needed to attempt a certain number of folds: n n ( 2 4)( 2 1) L = π t + 6 where L is the minimum possible length of the paper, t is the paper s thickness, and n is the number of folds possible in one direction. In our experiment, the length of our paper was 11 inches (L = 11), the thickness of the paper was 0.004 inches (t =.004) and while we were able to make fold number 6, fold number 7 seemed impossible. Use Britney s formula to determine the needed length of paper, L, if we wished to attempt 7 folds. Be sure to show your substitutions into the formula. Round your answer to the nearest inch needed. 7 folds needs inches minimum of paper