Question **42** of the Spring 2003, Grade 9, Texas Assessment of Knowledge and Skills (TAKS) released Mathematics test reads as follows:

hour, the relationship between the distance

traveled, *d*, and the time traveled, *t*, is

described by the function *d* = 50*t*. Which

statement is true?

traveled.

traveled.

distance traveled.

traveled.

Out of the 303553 students who took the test, the response percentages for answers **F**, **G**, **H**, **J**, and no answer were, respectively, 20%, 46%, 18%, 16%, and 0%.

Answers **F** and **G** are both correct answers. The Texas Education Agency gave credit for answer **G** but made the mistake of denying credit for answer **F**. The TEA wrongly decreased the scores of those roughly (20%)(303553) = 60711 students who chose answer **F**, wrongly decreased the average raw score expected from random guessing from 13.00 to 12.75 out of 52 questions, and, with respect to this decrease in the average raw score expected from random guessing, wrongly increased by 0.25 the average raw scores of those roughly (80%)(303553) = 242842 students who did not choose answer **F**. The remedy for this mistake is to also give credit for answer **F**.

Answer **F** is correct because, "For a car traveling at a speed of 50 miles per hour", the statement "The time traveled depends on the distance traveled" is true. A car traveling at a speed of 50 miles per hour is also traveling at 1/50 of an hour per mile.

The relationship between distance traveled, *d*, and time traveled, *t*, is the same no matter whether it is expressed as *d* = 50*t*, 50*t* = *d*, *d*/50 = *t*, *t* = *d*/50, or in some other equivalent way. The functions *d* = 50*t* and *t* = *d*/50 are inverses of each other. If *d* = 50*t* is true, then *t* = *d*/50 is true; if *t* = *d*/50 is true, then *d* = 50*t* is true.

The truth of the statement "The distance traveled depends on the time traveled" is not enhanced by the fact that *d* appears as the only quantity on the left hand side of the given equation *d* = 50*t*.* Similarly, the truth of the statement "The time traveled depends on the distance traveled" is not diminished by the fact that *t* appears on the right hand side of the equation, or that *t* is multiplied by 50, or by the fact that the equivalent equation *t* = *d*/50 is not explicitly given.

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Inclusion of appropriate units in the given equation gives the equation *d* = (50 miles/hour) *t*.