Getting Started 
Misconception/Error The student is unable to show the functions in each pair are equivalent. 
Examples of Student Work at this Level The student:
 States that the two functions are onlyÂ equivalent when t = 0.
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 Attempts to evaluate each function for one or more values of t but cannot explain why the functions are equivalent for all values of t.
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 Recognizes whyÂ Â is equivalent toÂ Â but is unable to determine whyÂ Â is equivalent (within rounding) toÂ .
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Questions Eliciting Thinking How are the two functions in the first problem similar? How are they different? What is the relationship between (1  0.1) and 0.9?
Can you use properties of exponents to simplify ? Is Â approximately equal to 0.9? What does this mean about Â and ? 
Instructional Implications 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 a variety of examples. Guide the student to identify the two parameters of an exponential function, the initial amount and the growth/decay factor, as well as the basic form of an exponential function, . Be sure the student understands the difference between the growth/decay factor and the rate of growth or decay. Explain that the growth factor is (1 + r) where r is the rate of growth and the decay factor is (1 â€“ r) where r is the rate of decay.
Show the student that since 1 â€“ 0.1 = 0.9, the two functions are equivalent. Explain the significance of both 0.1 and 0.9 in the context of this problem.
Review properties of exponents and show the student that . Explain that represents the percent of the population that remains after an interval of time that is of t, or one month.
Consequently, 0.9913 represents the percent of the population that remains after one month.
Provide additional opportunities to rewrite exponential expressions in equivalent forms in both realworld and mathematical problems. 
Moving Forward 
Misconception/Error The student cannot explain what the different forms of the functions reveal about the population decline. 
Examples of Student Work at this Level The student demonstrates that the functions are equivalent but is unable to correctly explain what any of the functions reveal about the population decline.
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Questions Eliciting Thinking Can you tell by looking only at the functions that they represent a decline in the population? What part of the function shows this?
What is the significance of the number 0.9? How do you suppose it was calculated? 
Instructional Implications Discuss with the student the differences between the rate of decay and the decay factor. Explain how each is used to reveal different aspects of the population in this problem. Explain that the decay rate of 10% means the population is declining by 10% each year and the decay factor of 90% shows that each year the population is 90% of what it was the prior year.
Review properties of exponents and show the student that . Explain that represents the percent of the population that remains after an interval of time that is of t, or one month. Consequently, 0.9913 represents the percent of the population that remains after one month.
Provide additional opportunities to rewrite exponential expressions in equivalent forms in both realworld and mathematical problems. 
Almost There 
Misconception/Error The student does not recognizeÂ what the function reveals. 
Examples of Student Work at this Level The student shows that both pairs of functions are equivalent. Additionally, the student can distinguish between the rate of decay and decay factor in the first pair of functions and explains what each reveals about the population decline. However, the student is unable to explain what aspect of the population decline is revealed by each of the second pair of functions, Â and .
Â 
Questions Eliciting Thinking How areÂ andÂ Â related?
What does t represent? What do you think is the significance of the factor 12? 
Instructional Implications Review properties of exponents and show the student that . Explain that Â represents the percent of the population that remains after an interval of time that is of t, or one month. Consequently, 0.9913 represents the percent of the population that remains after one month.
Provide additional opportunities to rewrite exponential expressions in equivalent forms in both realworld and mathematical problems.
Consider implementing MFAS task College Costs (ASSE.2.3). 
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 states the two functions are equivalent because (1  0.1) = (0.9). The first form reveals the 10% annual decline in the population and the second form shows that each year the population is 90% of what it was the previous year.
 The student states the two functions are equivalent because Â within rounding. The first form reveals that every month, the population is 99.13% of the previous monthâ€™s population, while the second form shows that each year the population is 90% of what it was the prior year.Â

Questions Eliciting Thinking In an exponential function, how is the rate of growth (or decay) related to the growth (or decay) factor?
What about the form of the function suggested to you that it was related to the monthly change in population?
What is the monthly rate of decline? 
Instructional Implications Challenge the student to rewrite the function so it reveals the 10year rate of decline.
Consider implementing MFAS tasks Jumping Dolphin (ASSE.2.3), Rocket Town (ASSE.2.3) and College Costs (ASSE.2.3). 