7 replies on “Exponentials and Not-Quite Exponentials”

The two parts that the student seems to understand are that the value after 1 year will be lower by 0.35 of the original cost, and that exponentials are involved (somehow). The failure seems to be in figuring out how to translate the concept “decreases by 35% every year” into an adequate function.

I would think that some sort of worksheet example might help: have multiple items depreciate at the same percent, ask students to fill out a table and graph the values, then have some item with value x, and ask for the value after 1, 2, 3 years based on x. Maybe that would help to translate the idea of “decrease by y%” into “multiply by (1-y%)^t”.

Ask the student first to graph what they think the answer should look like, and then to graph the function they created.

Hi, I am not sure where the mistake is in 2a. The answer seems right to me. Maybe I am not thinking straight. Can anyone please elaborate?

After one year, I think that the $19.99 should turn into $12.99. But if I understand this kid’s work correctly, then her formula predicts that it would be $6.99.

Not sure how you got this result. My calculations on this formula give $12.99 after one year of depreciation. This is the formula: V(t) = 19.99 – 19.99*(0.35)^t , V(1) = 19.99 – 19.99 *0.35 = 19.99 – 6.9965 = 12.9935

The student’s formula is equivalent to V(t) = 19.99(1-(0.35)^t)

One form of the correct formula would be V(t) = 19.99(1-0.35)^t

For t=1, they agree, as you calculated. But, for instance, the first is an increasing function with a horizontal asymptote of 19.99 and the second is a decreasing function with a horizontal asymptote.

I like Suevanhattum’s suggestion of graphing since a student with a rudimentary intuitive understanding of depreciation should be able to guess that this quantity should be modeled by a decreasing function.

I’d guess the student vaguely associates “depreciates by” with “decreases by” in her head. If she’s done a lot of the latter problems, she’d jump to subtracting something from an initial quantity, here 19.99. She then guessed what the subtracted quantity should be, assuming it involves an exponential.

Perhaps it is good to encourage her by saying that the two phrasings are different, but she can equate “depreciates by” with “decreases by a factor of” and getting her to come up with 0.65 as the appropriate annual multiplier here.

## 7 replies on “Exponentials and Not-Quite Exponentials”

The two parts that the student seems to understand are that the value after 1 year will be lower by 0.35 of the original cost, and that exponentials are involved (somehow). The failure seems to be in figuring out how to translate the concept “decreases by 35% every year” into an adequate function.

I would think that some sort of worksheet example might help: have multiple items depreciate at the same percent, ask students to fill out a table and graph the values, then have some item with value x, and ask for the value after 1, 2, 3 years based on x. Maybe that would help to translate the idea of “decrease by y%” into “multiply by (1-y%)^t”.

Ask the student first to graph what they think the answer should look like, and then to graph the function they created.

Hi, I am not sure where the mistake is in 2a. The answer seems right to me. Maybe I am not thinking straight. Can anyone please elaborate?

After one year, I think that the $19.99 should turn into $12.99. But if I understand this kid’s work correctly, then her formula predicts that it would be $6.99.

Not sure how you got this result. My calculations on this formula give $12.99 after one year of depreciation. This is the formula: V(t) = 19.99 – 19.99*(0.35)^t , V(1) = 19.99 – 19.99 *0.35 = 19.99 – 6.9965 = 12.9935

The student’s formula is equivalent to V(t) = 19.99(1-(0.35)^t)

One form of the correct formula would be V(t) = 19.99(1-0.35)^t

For t=1, they agree, as you calculated. But, for instance, the first is an increasing function with a horizontal asymptote of 19.99 and the second is a decreasing function with a horizontal asymptote.

I like Suevanhattum’s suggestion of graphing since a student with a rudimentary intuitive understanding of depreciation should be able to guess that this quantity should be modeled by a decreasing function.

I’d guess the student vaguely associates “depreciates by” with “decreases by” in her head. If she’s done a lot of the latter problems, she’d jump to subtracting something from an initial quantity, here 19.99. She then guessed what the subtracted quantity should be, assuming it involves an exponential.

Perhaps it is good to encourage her by saying that the two phrasings are different, but she can equate “depreciates by” with “decreases by a factor of” and getting her to come up with 0.65 as the appropriate annual multiplier here.

I see now. Yes you are right. Thank you!