Formative Assessment 74

Assessment Problem: 

If I get 8 miles per gallon out of my car during the first leg of a journey but for some reason only get 4 miles per gallon during the second leg, what is my car’s performance over the entire trip?

Common Responses: 

Many students will add together the values given in the problem, 8 + 4 = 12, and conclude the rate is 12.
A more sophisticated response recognizes that two variables are involved.  A student finds that the first leg has 8 miles for 1 gallon, the second leg has 4 miles for 1 gallon, so together you have 4 + 8 = 12 miles and 1 + 1 = 2 gallons, so the rate is 12 miles per 2 gallons, or 6 miles per gallon.  However, this second method is also incorrect, for it assumes that the trip is exactly 8 miles for the first leg and 4 for the second leg.  The number of miles traveled in each leg of the trip is not given in the problem, and so the problem is not solvable.

Mathematical Issues: 

When working with rates problems, students frequently have trouble recognizing that a rate is a numerical description of a relationship between two varying quantities. In this example, it is common for students to confuse a rate of consumption with the actual quantity of consumption. Namely, if the car consumed gas at a rate of 8 miles per gallon, that does not mean that the car traveled only 8 miles.

Analysis of Change
Rates of change
Common Core Standards: 
6.RP Ratios and Proportional Relationships
Research References: 
Thompson, P. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-234). Albany, NY: SUNY Press.