Formative Assessment 75

Assessment Problem: 

Suppose that turtle and rabbit are racing to a pole and back.  Suppose turtle travels at 30 ft./sec. on the way there and 50 ft./sec. on the way back.  How fast would rabbit have to travel [at constant speed] to tie with turtle.

Common Responses: 

I think that rabbit should travel at 40 ft/sec, because it is half way in between.

Mathematical Issues: 

It is common for students to think of speed as a distance. Many students see speed in terms of chunks of distance (or speed-lengths): a speed of 60 feet per second is understood as a chunk of 60 feet that can be traveled in 1 second. Although this is not incorrect, it only allows children to reason in one direction: given a speed and an amount of time, the student can add up “speed-lengths” to find the distance. Given this thinking, it is difficult for students to find the average speed of an object. In this case, the student partitions the distance to arrive at an incorrect solution. A more robust understanding of speed includes the understanding that there is a direct proportional relationship between distance and speed. Partitioning the total time to travel a given distance implies a proportional partition of the distance traveled, and vice versa.

Analysis of Change
Rates of change
Learning Progressions: 
Proportional Reasoning
Common Core Standards: 
7.RP: Ratios and Proportional Relationships
Research References: 
Thompson, A. G., & Thompson, P. W. (1996). Talking about Rates Conceptually, Part II: Mathematical Knowledge for Teaching. Journal for Research in Mathematics Education, 27(1), 2-24.