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# 1,000 is half of 2,000

Alignments to Content Standards: F-BF.A.1.b F-IF.B.6 F-BF.A.1

A new internet organization called Illuminated Manifolds (IM) is trying to generate 2,000 insightful math problems spanning the K–12 curriculum in a timeframe of 48 months.

For integers $t$ with $0\leq t\leq 32$, let $N(t)$ denote the total number of problems IM has generated by time $t$, measured in months since the beginning of September 2011.  Suppose we are given the following information about $N(t)$.

 t N(t) t N(t) t N(t) 0 0 12 514 24 879 2 5 14 590 26 901 4 42 16 666 28 922 6 163 18 744 30 943 8 400 20 822 32 1000 10 438 22 868

1. Let $R(t)$ denote the remaining number of problems the organization still has to construct at time $t$. Find an algebraic relation between $N(t)$ and $R(t)$ -- how does this relation reveal itself upon graphing the two functions?
2. Let $B(t)$ denote the number of problems produced in the $t$-th month. Using the information from the table, explain how you can approximate $B(t)$ for $0 \leq t \leq 32.$  Were there any extraordinarily productive spurts in IM's timeline?
3. At any time $t$, let $S(t)$ denote the average number of problems that IM has to write per month to meet its goal by the deadline. Find an algebraic expression for $S$ in terms of $N$.  Find some interesting values of $S(t)$.
4. Using all of the data collected so far, decide whether or not you think IM will achieve is goal, and provide a justification for your prediction. Discuss what additional data you would like to have to better inform your prediction.

## IM Commentary

This real-life modeling task could serve as a summative exercise in which many aspects of students' knowledge of functions are put to work. Students must:

• Parse and interpret data in terms of a context.
• Plot and visualize data points.
• Understand algebraic relationships between functions.
• Understand how these relationships manifest themselves graphically, including the effect of transformations of graphs (e.g., that $R(t)=2000-N(t)$).
• Draw conclusions based on the results of their investigation.

As in the list above, opportunities to model the standards for mathematical practice are abundant, ranging from modeling with mathematics, to using tools (to, say, plot data points), to expressing regularity in repeated reasoning as they generalize from sample computations of $S(t)$ to a general formula.  For example, students will need to attend to precision as they formulate a definition of $B$ (for example, deciding between $B(t)\approx N(t+1)-N(t)$ or $B(t)\approx N(t)-N(t-1)$), making sure to attend to the domain of $B$, etc.  It would be worth encouraging discussion between students on this front.

Similarly, to emphasize modeling, teachers should encourage students to be thoughtful and creative in answering the last part.  Students might conclude that since $S(32)$ is much larger than $R(32)$, that the team is unlikely to succeed in meeting its deadline, or they might argue that the comparatively astronomical value of $B(6)$, representing the work done in March 2012, gives credence to the team's potential to accomplish heroic feats.

## Solution

(a)  Since $R(t)$ is the number of the 2000 problems remaining to produce after producing $N(t)$ of them, we have $R(t)+N(t)=2000$, or $R(t)=2000-N(t).$  There are several things we can deduce about the graphs from this relation.  For one, we see the intuitively obvious relationship that as $N(t)$ increases, $R(t)$ decreases.  Plotting the data points, as seen below, also aids in revealing a property:

It appears that the two graphs have a reflectional symmetry at the height of 1000.  This can be algebraically justified by re-writing the equation $R(t)=2000-N(t)$ as $R(t)-1000=1000-N(t).$  This new form of the equation says for any time $t$, the amount $R(t)$ is above (or below) 1000 is exactly the same as the amount $N(t)$ is below (or above) 1000.

(b)  Since $B(t)$ is the number of problems produced in the $t$-th month, we have the algebraic relation $B(t)=N(t)-N(t-1)$ for $t\geq 1$.  Unfortunately, this does not help in computing values of $B(t)$, since we never have data for $N(t)$ for two consecutive months.  To make use of the data that we do have, we could approximate $B(t)$ by taking the amount of items generated between $t-2$ and $t$ (instead of $t-1$ and $t$) and dividing by 2, leading us to the approximation

$$B(t)\approx \frac{N(t)-N(t-2)}{2},$$

representing the average rate of change over that time interval.  Other reasonable approximations might include $B(t)\approx \frac{N(t+2)-N(t)}{2}$ or $B(t)\approx \frac{N(t+2)-N(t-2)}{4},$ representing average rates of change of $B(t)$ over the intervals $[t,t+2]$ and $[t-2,t+2]$ respectively.  We'll use the approximation centered above for computing values.  For example, we get

$$B(20)\approx \frac{N(20)-N(18)}{2}=\frac{822-744}{2}= 39 \text{ problems}.$$

The most extreme value of $B(t)$ appears to occur around March 2012 ($t=8$), when we find $B(6)\approx 118.5.$

(c)  We note that $S(t)$ is the number of items remaining to complete (out of the 2000), divided by the number of months left to complete them.  Since at time $t$, there are $2000-N(t)$ items left to complete, and $48-t$ months remaining, we arrive at

$$S(t)=\frac{2000-N(t)}{48-t}.$$

Some notable values of $S(t)$ would be $S(0)$, the average production rate needed from the start, $S(32)$, the average production rate needed at the end of the recorded data, and times when it appears that $S(t)$ is exceptionally large, small, or changing quickly.  For convenience we include below all the calculable values of $S(t)$, in tasks per month, rounded to two decimal places.

 $t$ $S(t)$ $t$ $S(t)$ $t$ $S(t)$ 0 41.67 12 41.28 24 46.71 2 43.37 14 41.47 26 49.95 4 44.50 16 41.69 28 53.90 6 43.74 18 41.87 30 58.72 8 40.00 20 42.07 32 62.5 10 41.11 22 43.54

Not obvious from the original data is the extent to which $S(t)$ is almost always increasing as a function of time.  Note the exception of $t=8$, the time following the 2-month span of exceptional productivity as noted in the previous part of the solution.  Note also that IM has produced 1000 problems in 32 months, averaging $31.25$ problems per month. So it is worth remarking that IM will have to work $\frac{62.5}{31.25} = 2$ times as quickly as it did in its first 32 months in order to achieve its goals.

(d)  There is no right answer to this question, but lots of relevant information that could be used in drawing a conclusion.  As noted in part (c), the track record seems to show that IM needs to increase its production significantly in order to meet its quota, potentially justifying an answer of "No." On the other hand, IM has also shown the ability to give drastic production increases as noted in part (b), in which the value of approximately 118.5 for $B(t)$ achieved during March 2012  seems to be more than sufficient to cover the value of $R(2)=62.5$ demanded for the last 16 months until the deadline. Perhaps the largest question that remains to be answered is whether or not that level of productivity can be reached again -- other information that might be useful in this regard is the extent to which IM has developed a national community of problem-producers large enough and capable enough to complete the task. It would also help to have a corporate jet.

#### Bill says:

over 3 years

In the answer to part (b), if I apply the formula B(t) = N(t+1)-N(t) with t=0, then I get B(0) = N(1)-N(0). This is the number of problems produced in the first month. The stem for (b) says "let B(t) be the number of problems produced in month t." So you seem to be saying that the first month is "month 0." I wonder if some students might find this confusing. Maybe it would be better to say "let B(t) be the number problems produced during the t-th month" and then use the formula B(t) = N(t) - N(t-1). I also think there a potential confusion in the definition of N(t), because it just says t is measured in months since September, and students might be confused as to whether t=1 means the end of September or the end of October, or something else. So maybe you could say "let N(t) be the number of tasks produced t months after the beginning of September" or something like that.

#### Cam says:

over 3 years

Dear Bill McCallum,

I agree entirely, and I have attempted to incorporate your suggestions into the task. Thank you for your support of Illustrative Mathematics.

#### Kristin says:

over 3 years

I kind of liked the other version better. The ambiguity in the problem statement about the meaning of t=0 could have provided an excellent opportunity for students to debate the best interpretation of that and to choose one, thus engaging students in the "formulate" part of the modeling cycle. But this version is fine as well.

#### Cam says:

over 3 years

Dear Kristin Umland,

I agree entirely, and have updated the commentary to address your recommendations concerning implementation. Thank you for your support of Illustrative Mathematics.