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