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Traffic Jam


Alignments to Content Standards: N-Q.A

Task

Last Sunday an accident caused a traffic jam 12 miles long on a straight stretch of a two lane freeway. How many vehicles do you think were in the traffic jam? Explain your thinking and show all calculations.

IM Commentary

This task, while involving relatively simple arithmetic, codes to all three standards in this cluster, and also offers students a good opportunity to practice modeling (MP4), since they must attempt to make reasonable assumptions about the average length of vehicles in the traffic jam and the space between vehicles. Teachers can encourage students to compare their solutions with other students.

In addition to giving practice with working with units (N-Q.1), the task gives students an opportunity to engage with N-Q.2 by working with a new unit appropriate to the question, namely the notion of the length occupied by a vehicle in a traffic jam, which consists of its length plus the gap to the next vehicle.

The task also provides students the opportunity to attend to the appropriate level of accuracy when reporting quantities (N-Q.3). Given the uncertainties in assumptions, it does not make sense to report the number of vehicles to any greater accuracy than 100s.

Finally, teachers should be conscious of possible interpretations of the problem, and limitations of the model. For example, students may not know the standard convention that a "two lane freeway" has two lanes total, as opposed to two lanes in each direction. The problem is also ambiguous as to whether the traffic jam affects both direction of traffic, or just one (as assumed in the solution). Teachers can decide whether to make these conditions explicit to students, or just note that students who adopt different conventions may have answers that are, e.g., larger by a factor of 2 or 4. For a limitation of the model, students might correctly note that the length of a traffic jam is not a constant, but rather a function of times that increases and decreases based on many complicated factors. Teachers may have to help students who start on this (reasonable, but difficult) path make enough simplifying assumptions to reduce it to a solvable interpretation.

Solution

The solution depends upon the estimate of the length occupied by a vehicle in the traffic jam. Students should be given wide latitude on how they determine this length as long as they explain their thinking clearly and the estimates are reasonable, from cited references. For example: According to answers.com, the average mid-size sedan is about $13.5$ ft long and the average large pick-up truck is about $16.4$ ft long.

If all of the vehicles in the traffic jam were average mid-size sedans, and there was no gap between them, there would be approximately $\frac {\mbox{5,280 ft/mile}\times\mbox{12 miles}}{\mbox{13.5 ft}} \approx 4,700$ cars in the jam.

If all the vehicles were large trucks, and there was no gap between them, there would be approximately $\frac{\mbox{5,280 ft/mile}\times \mbox{12 mi}}{\mbox{16.4 ft}} \approx 3,900$ trucks. As in the commentary, note that if different conventions are made about how many lanes are involved in the context, this might affect this estimate by a factor of 2 or 4.

A more reasonable assumption would be that there are both trucks and cars, but fewer trucks than cars. For example, if there are $1500$ trucks, they would occupy $1500 \times 16.4 = 24,600$ ft, leaving $38,760$ ft for the sedans. This would amount to $38,760/13.5 \approx 2900$ for a total of $4,400$ vehicles in the traffic jam.

Making allowance for the gap between vehicles, and using a created unit of "length occupied by a vehicle in a traffic jam", would decrease all these estimates.