Getting Started 
Misconception/Error The student does not recognize the exponential relationship between the variables. 
Examples of Student Work at this Level The student writes a nonexponential equation involving N and t such as:
 N = 13000  3t
 13000  0.03N = t

Questions Eliciting Thinking What does decreasing by 3% per year mean?
Can you use the information given in the problem to determine the number of trees in 2011?
Is the number of trees increasing or decreasing each year? Would your equation predict a declining number of trees over time? 
Instructional Implications Review the basic form of an exponential function and its two parameters, the initial value and the growth/decay factor. Ask the student to identify the initial value and the growth/decay factor in the given problem. Then guide the student in making a table of values for t = 0 to t = 5 such as:
Ask the student to rewrite the expressions in the table representing N using exponents. Then, relate the form of the expressions to the basic form of an exponential function and assist the student in writing the function.
Provide opportunities for the student to explore and investigate exponential functions, both growth and decay, in context. Have the student make a table of values for each example. Then, guide the student to identify the two parameters of each exponential function, the initial amount and the growth/decay factor, and write functions to represent each example. Be sure the student understands the difference between the growth/decay factor and the rate of growth/decay. Remind the student that the growth factor is (1 + r) where r is the rate of growth and the decay factor is (1  r) where r is the rate of decay.
Provide additional opportunities for the student to write exponential functions from verbal descriptions, tables of values, and graphs. 
Making Progress 
Misconception/Error The student makes an error in some component of the equation. 
Examples of Student Work at this Level The student recognizes that the relationship between N and t is exponential but makes an error in representing the initial amount or the decay factor. For example, the student:
 Neglects the initial amount and writes the equation as .
 Writes the rate of decay in place of the decay factor writing the equation as .
 Writes the decay factor as 0.7 or 1.03.
 Writes the exponent as t â€“ 1 rather than t.
 Writes the function correctly but then attempts to simplify it subsequentlyÂ writing it incorrectly.

Questions Eliciting Thinking How many trees were there in 2010? How did you represent this initial amount in your equation?
What does the 3% represent (the percentage of surviving trees or the percentage of trees that were lost)? How should it be represented in your equation? 
Instructional Implications Review the basic form of an exponential function, , and the meaning of the two parameters, a (the initial amount) and b (the growth/decay factor). Remind the student that the growth factor is (1 + r) where r is the rate of growth and the decay factor is (1  r) where r is the rate of decay. Ask the student to revise his or her equation. HaveÂ the student evaluate the function for t = 0 to determine if the exponent should be written as t or t  1.
Provide additional opportunities for the student to write exponential functions from verbal descriptions, tables of values, and graphs. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly writes the equation asÂ orÂ .

Questions Eliciting Thinking What kind of function did you write? What about the wording of the problem indicated that it was exponential?
What would the graph of this function look like?
If you wanted to determine the number of trees in 2015, what value of t would you use in your equation?
Will the formula ever produce a number of trees that is not an integer? 
Instructional Implications Consider introducing the 'Rule of 72' (72 divided by the rate of growth will predict the approximate number of years for the initial amount to double). Â Ask the student to determine if this would apply to exponential decay. Since 72/3 = 24, have the student determine the number of trees remaining after 24 years and ask the student to compare this number to half the initial amount.
Ask the student to write a function involving N and t that takes into account the following: In an effort to partially offset the loss of trees, the forestry service decides to plant 200 trees each year. 