1. Exponential Growth

2. An Old Problem about Doubling One penny on the first square, two on the next, four on the next, and so on… How tall is the stack on the last square?

3. Developing a Meaningful Formula Write an algebraic expression for the number of pennies on the n th square without using exponents. Hint: What is the meaning of an exponent? This is repeated multiplication by the factor B.

4. Developing a Meaningful Formula Write down three separate expressions for the number of pennies on the 8th square: using repeated multiplication using exponential notation using the actual number in decimal form

5. Developing a Meaningful Formula Write a formula for the number of pennies on the n th square. Describe the meaning of this formula in words.

6. Repeated Multiplication by other Factors Pick a number of squares (other than 1). You will explore the pattern of growth in the number of pennies as you repeatedly advance this number of squares. Determine the factor by which the number of pennies increases each time you do this. Test this for several intervals.

7. Average Rate For several intervals of your group’s chosen size, find the average rate of change and compare it to the number of pennies on one of the squares (choosing the square in a consistent manner). Make a table of your results. What do you notice?

8. Average Rate Individual Reflection: What is the meaning of this average rate in this situation? Groups: Check with each other to see if you understand the meaning of this average rate correctly.

9. What is the average rate of change of the number of pennies per square between the 3 rd and 5 th squares? p=4 p=8 p=16 p=4 p=10 p=16

10. Average Rate in Three Contexts Cars position - time Bottles height - volume Pennies # pennies - # squares Auxiliary Situation Constant Rate Same change in both quantities A 2 nd car taking another trip A 2 nd bottleA 2 nd stacking of pennies At a constant speed With constant width (a cylinder) Increasing by the same number of pennies each square Covering the same distance in the same time With the same change in volume and change in height Over the same number of squares and same change in pennies

11. Number Line Analysis of Rate Complete a number line analysis of to describe the constant rate of change of pennies per square between the 3rd and 5th squares Describe the meaning of the equal partitioning in terms of the context of pennies on the chess board.

12. Generalizing Look at the formula again. What would change if we tripled the number of pennies each square? What would change if we started with 7 pennies on the first square and progressed from there? What number of pennies would correspond to the zero th square? We can rewrite the formula as

13. Summary LinearExponential Additive changesMultiplicative changes Linear coefficient multiplication = repeated addition Base exponentiation = repeated multiplication Initial amount addedInitial amount multiplied Constant rateRate follows the same exponential pattern as amount Rate proportional to the amount