Predicting the Past

Task

Leo leaves his house one morning and notices a small plant growing in the yard that he has never noticed before. Out of curiosity, he grabs a ruler and measures it; the plant is 3cm tall. Exactly one week later, Leo notices the plant has grown quite a bit so he measures it again; now it is 9cm tall. And one week after that, it measures 27cm.

Leo knows that it is unlikely that the plant will continue to triple in height each week indefinitely, but he starts to wonder about the height of the plant before he started to measure it, and how he could model its growth mathematically. Suppose that the plant follows a rule, “triples in height each week.”

1. Read the information contained in the table to understand what Leo has written so far, and then complete the table. Write any heights that are less than 1 cm as fractions.

   Week	-4	-3	-2	-1	0	1	2	3	4	5	w
   Height (cm)	Â	Â	Â	Â	Â	3	9	27	Â	Â	Â
   Height Expression	Â	Â	Â	Â	Â	Â	\(3^2\)	\(3^3\)	Â	Â	Â

   1. Express the height of the plant, h, as a function of the week it was measured, w.
   2. Explain in words the meaning of h(0). 
   3. Use your function to find the height of the plant on week -4. Write this value as a fraction. Does the result of the function agree with what you wrote in the table?
2. Suppose that a different plant also demonstrates exponential growth, which means it grows by a constant factor, but the following height measurements are taken instead. (Blank cells are provided as convenient workspace, but you don’t necessarily have to write anything in them.)

   Week	Â	Â	Â	1	2	3	Â	w
   Height (cm)	Â	Â	Â	10	20	40	Â	Â
   Height Expression	Â	Â	Â	Â	Â	Â	Â	Â

   1. Write a function that expresses the height of this new plant, g, of this plant as a function of the week it is measured, w. 
   2. Use your function to determine the height of the plant, in centimeters, on week -4. Write this value as a fraction. 

IM Commentary

The purpose of this instructional task is to provide an opportunity for students to use and interpret the meaning of a negative exponent in a functional relationship. In grade 8, students understand negative exponents in terms of how they interact operationally with other expressions, for example, since \(2^5 \times 2^{-4}=2^1\), that must mean that \(2^{-4}\) is equivalent to \(1 \over 2^4\). When students learn about exponential functions in high school, they can express regularity in repeated reasoning (MP.8) and create a function to describe an exponential relationship. They can interpret and evaluate the function in order to understand the meaning of a negative exponent in a context. 

For all parts of this question, the teacher should use discretion about how much freedom and how much scaffolding to give students as they work. Depending on work done previously, a teacher might want to reactivate students' understanding of the meaning of negative exponents from eighth grade with a quick review. Another good idea might be to stop the class after 5 minutes or so of working on part "a." Some students might get off on the wrong foot by writing "0" for the height on day 0, or by writing negative values for heights instead of working out a consistent pattern of change. Also, students may be inclined to express values less than one as decimals, especially if they are allowed calculators. Expressing these values as fractions should be encouraged (as stated in the problem), because fractions make the meaning of the negative exponents much more clear. For example, \(3^{-2}={{1}\over9}\) vs. \(3^{-2}=0.\bar1\). Students should be encouraged to notice when numbers they have written "break the pattern," and given opportunities to discuss with each other how they can find numbers that follow a consistent pattern. Before moving on to part b, the teacher should make sure through a whole-class discussion or assessment-for-learning that everyone understands how the patterns in part a work (both row-wise and column-wise).

As the problem statement explains, it is not necessary to fill in the table for part b, but it might help to write everything out. The correct answers to both i and ii in part b are elements in the table, when the table is completed correctly. The jump from part a to part b is non-trivial, since part b involves an initial value other than 1. For example, students may initially write \(g(w)=10^w\) as their function for part b. They should be encouraged to try inputs of w = 2 and w = 3 and observe that the outputs don't match the heights given in the table. Teachers should think carefully through questions they might pose to support students here. This will sound different for different classes, depending on what the class has done previously, but will probably depend on appealing to consistency as students work in the table in part b. Example questioning might sound like:

"In part a, what was the height in week 0? [1 cm] Okay so we started with 1cm and kept tripling it. What is the height in week 0 for the new plant? [5 cm] Okay so we're going to start with a 5 and keep doubling it. How can we write that mathematically? What would make sense for the "Height Expression" row?"

Solution

1.  

   Week	-4	-3	-2	-1	0	1	2	3	4	5	w
   Height (cm)	\(1\over81\)	\(1\over27\)	\(1\over9\)	\(1\over3\)	1	3	9	27	81	243	\(3^w\)
   Height Expression	\(3^{-4}\)	\(3^{-3}\)	\(3^{-2}\)	\(3^{-1}\)	\(3^0\)	\(3^1\)	\(3^2\)	\(3^3\)	\(3^4\)	\(3^5\)	\(3^w\)

   1. \(h(w)=3^w\)
   2. h(0) is the height of the plant the week before Leo noticed it.
   3. \(h(-4)=3^{-4}={1\over{3^{4}}}={1\over81}\)

2. Week	-2	-1	0	1	2	3	4	w
   Height (cm)	5/4	5/2	5	10	20	40	80	Â
   Height Expression	\(5(2^{-2})\)	\(5(2^{-1})\)	\(5(2^0)\)	\(5(2^1)\)	\(5(2^2)\)	\(5(2^3)\)	\(5(2^4)\)	\(5(2^w)\)

   1. \(g(w)=5(2^w)\)
   2. \(g(-4)=5(2^{-4})=5{1\over{2^4}}={5\over16} cm\)