As reported in the Houston Chronicle the Texas Educational Authority yesterday announced a revision of the scores of the 10th grade TAKS exam that was given this spring, because of an error in one of the questions.

Readers with some knowledge of high school trigonometry may find it interesting to see the problem. The question is reproduced in the Houston Chronicle, or one can see the TEA original (look at question 8). The question shows a drawing of a regular octagon, indicating the inscribed radius as being 4.0cm and the circumscribed radius as being 4.6cm. The question is what is the perimeter of the octagon to the nearest cm. The choices are 41cm, 36cm, 27cm, and 18cm.

The data are contradictory: an octagon with inscribed radius 4.0cm has circumscribed radius about 4.33cm. Taking the 4.0cm and 4.6cm at face value a student might reason that the perimeter of the octagon is somewhere between 2*pi*4.0cm and 2*pi*4.6cm, and this leads to the answer 27cm in the multiple choice format. Or the student could apply trigonometry and obtain perimeter 26.5cm by starting from the given inscribed radius or 28.2cm by starting from the given circumscribed radius. A fourth approach is to use Pythagoras's theorem on a right triangle that has hypothenuse 4.6cm and one right side 4.0cm; then one finds that the circumference of the octagon must be 36.3cm. That (or rather, 36cm) was the intended answer.

According to the TEA press release, "item eight on the 10th grade math test could have been read in such a way that the question had more than one correct answer". That is putting a very kind spin on their blunder - there is in fact no reading of the question under which it has just one correct answer. It is amazing that the TEA would have this test composed and reviewed by people that fail to recognize that one cannot arbitrarily specify both the inscribed and the circumscribed radius of a regular polygon. According to the TEA press release: "Each test item goes through a rigorous review process that includes a field test of the items and two separate review sessions by professional educators who have subject-area and grade-level expertise and who are recommended by their district." The TEA didn't mean that as an explanation, but for me the "professional educators" part goes a long way just the same.

[**Addendum**, Aug 09. Please see the figure accompanying
question 8 in the exam. The line segments that I described as inner
and outer radii are not, in fact, identified as such in the figure or in
the question. They meet at a point that certainly appears to be the
center of the octagon, but that is not labelled either. There
*is*, therefore, a reading of the question under which it
has a single correct answer. Under that reading the given data are
all correct, the special point is not meant to be the center of the
octagon, and the figure is simply distorted in what happens to be
a highly misleading way.]

Posted by Bas Braams at August 7, 2003 02:14 PM

It is of some peripheral interest to revisit that TAKS question under the interpretation that the given radii are measured quantities, subject to measurement error. Now, a correct treatment of the problem would say that we have two indirect measurements of the perimeter: 26.51cm and 28.17cm. We average and round, and arrive at a perimeter of 27.3cm with estimated error (+/-)1.2cm. The alternative treatment used in the TEA answer (based on Pythagoras's theorem) involves subtraction of two nearby measured numbers, resulting in a large error bar. No one with any training in data handling in the physical sciences would use that approach.

Posted by Bas Braams at August 8, 2003 04:44 AM