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Exponential Growth Name: __________________ Worksheet Question: What relationship exists between the number of rectangles formed and number of folds when a piece of paper is folded repeatedly? Exploration: 1. Fold a rectangular piece of paper in half. The fold divides the paper into two regions, each of which has half the area of the paper. Q1. Fold the folded paper into half again. Into how many regions has the original paper been folded? What fraction of the paper’s area does each region have? 2. Continue to fold the paper until it is no longer possible to make another fold. After each fold, record in a table as shown below: Number of Folds 0 1 2 3 4 5 6 Number of rectangles 1 3. Make a scatter plot of the data in the table on your fathom document. Q2. Identify the variables for this relationship and describe the domain and range for this situation. Q3. Describe how the number of rectangles changes as the number of folds increase. Q4. Express symbolically the relationship between the variables. Q4. If your paper had 128 rectangles, how many folds would you have made? Explain your answer. 1 Q5. Describe the graph of your data. Chapter 10: Inverse Variations, Exponential Functions, and Other Functions Q6. What is the relationship between the original number of rectangles and the number of rectangles after one fold? Q7. What is the relationship between the number of rectangles after one fold and the number of rectangles after two folds? Q8. What do you think the relationship will be between the number of rectangles after two folds and the number of rectangles after 3 folds? Q9. What pattern do you notice in the number of rectangles as the number of folds increase? Q10. What would you have to do to the original number of rectangles to get the second number of rectangles? Q11. How do the exponents relate to the values in your table? Q12. Suppose you could continue to fold the paper and extend your table to include 10 folds. How many rectangles would there be? Make a slider and rename the variable as ‘a’. Add the function, anumber of folds and adjust the slider until you have found the value that fits the data points. Plot the function you have determined in Q4 by selecting the graph and entering the function. Q13. Does the function fit the data points? Plot the residual plot and verify if the function you have plotted is the best model. Explain the model. 2 Extension: Q14. Describe how the situation would be different if the paper had been folded into thirds each time instead of halves. Q15. Suppose that you began with a square sheet of paper that was 2 feet on a side. What is the relationship between the number of folds and the area of the rectangle after each fold? Draw the table to show the relationship. Q16. Explain how the graph of the function you found in Q15, would be different from the graph of the original function, A = 2n. 3