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# PARCC Algebra 2: Modeling a delivery route

The following constructed-response question, explained here in hopes of helping algebra 2 students and their parents in Maryland prepare for the PARCC test near the end of this school year, appears on the released version of the PARCC Algebra 2 sample items released following the 2015 test.

The arrangement of a distribution center and four stores to which it delivers is shown on the grid. Each unit on the grid represents 5 miles. The grid lines represent the roads.

The distribution center operators will use a single vehicle and must decide between a large truck and a small van. They will base their decision on this information.

Large Truck

• fuel efficiency: 9 miles per gallon
• delivers to all stores in one round-trip
• uses the shortest route to go to Stores A through D and then back to the distribution center

We can, of course, determine the exact point at which it will become more expensive to use the small van and skip the $12 fixed fees but buy more gas. The two graphs intersect at the point where $8\frac{8}{9}x = 6\frac{2}{3} x + 12$ Remember, in our model, x is the price of gas in dollars per gallon. $8\frac{8}{9}x - 6\frac{6}{9}x = 12$ $2\frac{2}{9}x = 12$ $x = 5\frac{2}{5} = 5.40$ So, a little less than my eyeball estimate of$5.50, so I conclude my answer makes sense. If gas is more expensive than that, it will be better for the company to use the large truck to make the deliveries and pay the extra $12 fee for docking, because the truck won’t have to return to the distribution center each time and guzzle all that extra gas. ## Rounding differences Note that the price for gas I found in part B differs from that shown on the PARCC scoring guidelines for this problem. Instead of keeping exact numbers, PARCC rounded to the nearest tenth in the models used for part A. That gave them an answer of$5.45 in part B, which is just as correct as mine, given that PARCC rounded.

They did all the right computations in their work and justification and executed them without error. Rounding is not an error, but it resulted in a difference in the exact number of 5 cents.

A lot was being made a few years ago about how “Common Core math” seemed to say several different answers would be considered correct, as long as they were justified properly, given the correct mathematics. That didn’t sit very well with some parents, who also thought everything we teach third graders through high school students has to meet the exacting standards of a graduate engineering student. This problem is a prime example of where an answer of $5.45 can be just as acceptable, and earn the student just as many points, as$5.40.

On a multiple-choice test, the two different answers would never be available to students, and on a constructed response question like this one on the PARCC test, the student gets most of the points on this problem for the modeling and the justification, not for the correct numerical value for the price of gas.