One of the challenges in writing modeling problems is that they are inherently messy. If every aspect is cut and dried then you’re not really modeling. The problem should call for making some estimations and making some assumptions. That, however, makes writing a standardized rubric really hard. One student might make different assumptions that hadn’t been expected by the problem writer. Let’s take, for example, the PARCC sample problem called Popcorn Inventory.
The student is given a lot of data in a table (most of which is unnecessary to solve the problem, but again that is part of modeling) that can be used to determine what happened over the last 4 days as well as some assumptions in word form to help us determine what was likely to happen on the previous (or next) 3 days. The problem seems to indicate that last week is typical of what to expect this week. The student is asked to determine a reasonable estimate of how many cups of popcorn seed to purchase in order to handle the popcorn needs for the upcoming week.
The rubric seems to expect a student to use the table to determine the average weekday consumption rate of the various size popcorn containers. It then expects one to use that and the given information to extrapolate the number of containers of each that would be consumed on the weekend (Fri-Sat). Then one would use a given conversion rate to convert the consumed popcorn to an amount of popcorn seed. One would also scale the 4 given weekdays usage of popcorn seed to get how much would be used over the 5 weekdays (ie account for Sunday) and finally throw in some extra popcorn seed to bring us up from our dangerously low level of seed into the desired comfortable range.
This sample solution (provided in the rubric) assumes that on the weekend no popcorn is popped unless it goes out in a box, but the data indicates that at least during the week quite a bit of extra popcorn seed is used. How would graders handle quite different solutions with alternate (probably more reasonable) assumptions? Stay with me past the jump for my solution when I did the problem and some other alternates.
I first figured that about 320 cups of seed were used over 4 weekdays. We are told that weekends (Fri-Sat) are busier. For example we sell twice as many small and medium boxes. So my initial model assumes we can treat weekends as 2 weekdays. This leads to 9 weekdays in our week. So scaling by 9/4 we would need 720 cups of seed plus another 80 cups since we’re currently too low. But we are also told that we sell 200 to 300 large boxes per day on the weekend compared with the 180 we sold over 4 weekdays. Our model would have predicted the same 180 over the 2 weekend days. In other words, instead of selling twice as many large boxes as we did for small and medium, we expect to sell more than 5 times as many. That’s quite the discrepancy and I would wonder what the reason for it was. In any event, there are many different reasonable assumptions that could be made at this point leading to vastly different estimates. I chose to just try to account for the seed needed for the extra 220 to 420 large boxes sold over the weekend. Figuring 5 cups of seed makes enough popcorn for 6 large boxes, I settle on a convenient estimate of 330 extra large boxes and hence 55 x 5 or 275 extra cups of seed. So I settle on a total estimate of 1075 cups of seed and essentially 50 more cups over Friday and Saturday than that estimated by the rubric solution. Would I get full credit for this solution? It doesn’t seem so. The rubric says students should recognize how many small and medium boxes are sold per day and that they MUST first estimate how many cups of popcorn are sold on Friday on Saturday. I did neither. The rubric specifically awards 1 point for “an adequate estimation strategy for two sizes of boxes for both days.” I neglected to do so. In fact I completely ignored all the data in the table concerning small and medium boxes.
The rubric also awards 1 point for accurate use of popcorn seed to popcorn. I happened to do that, but what if somebody made an assumption that didn’t require them to do so. For example, after discovering that our model was way off on the number of large boxes I might have chosen another way to reconcile the discrepancy. I might argue that the since we sell twice as many small and medium boxes, that it is reasonable to assume we should also be selling twice as many large boxes. So maybe the inventory indicates an atypically slow week and shouldn’t be used as a reliable indicator of typical daily boxes sold, but only as an indicator of boxes sold to seed actually used (which is more relevant than the given ratio of seed used to popped seed). How much credit would that strategy get if well argued?
Or what if a student determined that the ratio of large boxes to medium boxes to small boxes sold was roughly 3:1:2 (slightly overestimating small boxes). The ratio of popped corn between the boxes is roughly 5:4:3 (again slightly overestimating small boxes) and thus the ratio of popped corn used is roughly 15:4:6 or 3:2 in terms of cups used in large boxes to smaller boxes. So if we scale the number of large boxes sold by 5 to 6 and the number of smaller boxes sold by 2 maybe it’s reasonable to assume a weighted average of about 4 times as much popcorn needed on a Friday or Saturday compared to the weekdays. Thus we can consider a week as having the equivalent of 13 weekdays and scale our 320 cups of seed by 13/4 for a total of 1040 cups of seed needed next week plus an extra 80 cups or so because we’re currently low for a total of about 1120. This is probably the most reasonable estimate so far and it did not make use of the 1/3 cup of seeds to 8 cups of popped corn ratio at all. What would this solution score?