Getting Started 
Misconception/Error The student is unable to find key features of the graph and cannot graph the function correctly. 
Examples of Student Work at this Level The student:
 Is unable to graph the function.
 Attempts to make a table of values and only partially graphs the function.
 Draws the graph of an increasing function.
 Incorrectly calculates values of N.Â

Questions Eliciting Thinking What type of function describes the relationship between time and the number of fir trees?
In general, what does an exponential function look like?
Is this function increasing or decreasing?
What is the yintercept of the graph?
Are there any xintercepts?
Can you create a table of values for this function? 
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 of the given function. Then, guide the student in making a table of values for t = 0 to t = 5. Assist the student in choosing appropriate scales for the axes and then ask the student to graph the function and interpret what happens to the number of fir trees as time progresses.
Provide additional opportunities for the student to graph exponential functions, both growth and decay, in context. Have the student make a table of values, choose appropriate scales for the axes, and graph the function. 
Moving Forward 
Misconception/Error The student does not completely label or scale the axes or does not scale the axes appropriately. 
Examples of Student Work at this Level The student:
 Does not label or scale either the x or yaxis.Â
 Does not use the same scale throughout the entire axis.
 Uses a scale too small or too large.

Questions Eliciting Thinking What are you graphing on the xaxis? How did you choose the scale?
What are you graphing on the yaxis? What is the largest value of N that you need to graph?
Did you make a table of values for this function? Does the scale on each axis allow you to graph the coordinates you calculated? 
Instructional Implications Guide the student to consider key features of the graph (e.g., initial value and whether it is increasing or decreasing) and the units of the independent and dependent variable when determining how to scale the axes. Remind the student that any positive real number raised to the power of zero equals one. Suggest choosing values of t that are multiples of five such as 0, 5, 10, 15, and 20 which will make the computations easier. Encourage the student to revise the scaling on his or her graph and regraph the function. Remind the student that the two axes can be scaled differently, as long as the same scaling is used throughout each axis. 
Almost There 
Misconception/Error The student graphs the function but extends it beyond a reasonable domain. 
Examples of Student Work at this Level The student graphs the function but:
 Shows it extending through the second quadrant which includes values outside of the domain of the function given its context.
 Shows it crossing the xaxis at some point whose coordinates were not actually calculated.Â

Questions Eliciting Thinking Look at the second quadrant of your graph. What do these values mean in the context of this situation? Are these values of t reasonable?
Will this graph actually cross the xaxis? 
Instructional Implications Guide the student to consider the context of the function in determining a reasonable domain. Encourage the student to evaluate the graph as a model of the number of fir trees that survive over the passage of time and choose values of t that make sense.
Have the student examine other functions and their graphs given in the context of realworld problems and ask the student to describe reasonable domains. 
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 labels the xaxis â€śNumber of Years since 2010â€ť and scales it by increments of five. The student labels the yaxis â€śNumber of Fir Treesâ€ť and indicates the location of values 2000, 4000, 6000, 8000. The student includes a table values [e.g., (0, 8000), (5, 4000), (10, 2000), (15, 1000), (20, 500)], plots the points on the graph, and draws a smooth exponentially decreasing curve through the points. 
Questions Eliciting Thinking What pattern did you notice regarding the values in your table? Why did this happen?
Could this pattern continue for another 20 years?
Why did you choose to evaluate N every five years? 
Instructional Implications Provide additional practice graphing nonlinear functions that describe the relationship between two quantities. Ask the student to consider an appropriate domain given the context. Challenge the student to interpret key features of the graph in the context of the problem. 