Question **8** of the Spring 2003, Grade 10, Texas Assessment of Knowledge and Skills (TAKS) released Mathematics test reads as follows:

centimeter of the regular octagon drawn

below?

[To view the drawing, click here]

Out of the 246816 students who took the test, the response percentages for answers **F**, **G**, **H**, **J**, and no answer were, respectively, 8.4%, 52.2%, 16.0%, 22.9%, and 0.5%.*

One angle is specified to be a right angle. It is part of a right triangle. The height of this right triangle is specified to be 4 cm. The hypotenuse of this right triangle is specified to be 4.6 cm. The base of this right triangle is related to the hypotenuse and height by the Pythagorean Theorem (which is given on a beginning page of the test),

base = [hypotenuse^{2} - height^{2}]^{1/2}
= [(4.6 cm)^{2} - (4 cm)^{2}]^{1/2} = 2.2715... cm.

The angle on the left side of the dotted line that is supplementary to the right angle is also a right angle and this second right angle is part of a second right triangle. The height of this second right triangle is identical to the height of the first right triangle. The hypotenuse of this second right triangle is specified to be 4.6 cm, which equals the hypotenuse of the first right triangle. The base of this second right triangle is, therefore, equal to the base of the first right triangle. The bottom side of the regular octagon equals the sum of the bases of the two right triangles,

side = 2(base) = 2 [hypotenuse^{2} - height^{2}]^{1/2}
= 2 [(4.6 cm)^{2} - (4 cm)^{2}]^{1/2} = 4.5431... cm.

The perimeter of the regular octagon is 8 times one of its sides,

perimeter = 8(side) = 16 [hypotenuse^{2} - height^{2}]^{1/2}
= 16 [(4.6 cm)^{2} - (4 cm)^{2}]^{1/2} = 36.3450... cm.

This shows that answer **G** is the correct answer and that answers **F**, **H**, and **J** are incorrect.

The Texas Education Agency gave credit for the correct answer **G** but made the mistakes of also giving credit for the incorrect answers, **F**, **H**, and **J**, and for no answer. The TEA wrongly increased the scores of those roughly (47.8%)(246816) = 117978 students who chose answers **F**, **H**, and **J**, and no answer, wrongly increased the average raw score expected from random guessing from 13.75 to 14.50 out of 56 questions, and, with respect to the average raw score expected from random guessing, wrongly decreased the scores of those roughly (52.2%)(246816) = 128838 students who chose answer **G**. The remedy is to remove credit for answers **F**, **H**, and **J**, and for no answer.

The TEA issued these statements of 4 August 2003: TAKS Released Test Items.

The TEA states that question **8** has "more than one possible correct answer." This TEA statement is false; answer **G** is the only correct answer.

The TEA states that "If the item is solved in this manner, the perimeter is 36 cm (answer choice G)." The clause "If the item is solved in this manner" is irrelevant; the perimeter is 36 cm (to the nearest centimeter) and answer **G** is the only correct answer no matter in what manner question **8** is solved.

The TEA states that "However, if a student used one of the 45-degree angles at the center of the octagon and trigonometry to solve the problem, he or she may have chosen answer choice H (27 cm) or determined that there was no correct answer." This TEA statement is based on the incorrect assumption that the apex of the triangle is located at the center of the octagon; the TEA knows that this assumption is contradicted by the specified geometries and lengths.** Any student who made the incorrect assumption had the opportunity, using nothing more advanced than simple geometry and the Pythagorean Theorem, to find that the assumption was contradicted by the specified geometries and lengths. Any student who "determined that there was no correct answer" had the opportunity to check for, and find, his or her incorrect assumption(s).

The TEA states that "Because there was more than one possible answer to this problem, we have determined that students did not have a fair opportunity to demonstrate their understanding of this mathematical skill." The TEA claim that "there was more than one possible answer to this problem", by which the TEA means that there was more than one correct answer to the problem, is false. The TEA claim that "students did not have a fair opportunity to demonstrate their understanding of this mathematical skill" is false; the solution requires nothing more advanced than simple geometry and the Pythagorean Theorem, which students were expected to know.

The TEA states that "Therefore, TEA has decided that all students should be credited with a correct response to item 8 of the Grade 10 mathematics test." The TEA's decision to give credit for the incorrect answers, **F**, **H**, and **J**, and for no answer is due to incompetence. Texans are ill-served by such incompetence.

The TEA states that "When this credit is given, an additional 4,640 Grade 10 students (1.8% of the 246,816 tested statewide) will meet the standard, and an additional 936 students (.3% of the total tested) will achieve commended performance." These students should be informed that they met the standard or achieved commended performance falsely, due to scoring mistakes made by the TEA.

The TEA states that "TEA has requested that Pearson Educational Measurement provide districts with revised Confidential Student Reports and Labels for each Grade 10 student who originally gave a response other than G or who did not respond to item 8 on the Grade 10 mathematics test." This TEA request of Pearson Educational Measurement was inappropriate.

The TEA states that "We regret that item 8 was not a valid question, and we apologize for the inconvenience this will cause campus and district personnel." The TEA claim that "item 8 was not a valid question" is false.

Mark Loewe notified the Texas Education Agency of the scoring mistakes and some of its false claims in a letter of 19 May 2004. The TEA has failed to correct the mistakes and is continuing to propagate the mistakes and false claims without correction. On 15 July 2005, Mark Loewe gave written and oral testimony to the State Board of Education on the topic "TEA failed to correct TAKS mistakes". The SBOE has also failed to correct the scoring mistakes. Texans are ill-served by the TEA and SBOE failures to correct the scoring mistakes.

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* These response percentages are from "Texas educators flunk minimum skills math test", by Nancy T. King, *University Faculty Voice*, September 2003. The Texas Education Agency withheld response percentages for question **8** from the Item Analysis Summary Report. Nancy T. King obtained the response percentages through a Freedom of Information Act request.

** For any regular octagon, the ratio of distance from the center to (the midpoint of) a side divided by distance from the center to a vertex is given by

(center to side)/(center to vertex)
= (2^{1/2} + 2)^{1/2}/2.

This formula is exact and is easy to derive using nothing more advanced than simple geometry and the Pythagorean Theorem. A decimal value for this ratio is 0.9238.... A trigonometric expression for this ratio is cos(22.5^{o}). The fact that this ratio differs from the right triangle's height divided by hypotenuse,

height/hypotenuse = 4 cm/4.6 cm = 0.8695...,

contradicts the assumption that the apex of the triangle is located at the center of the octagon.

For any regular octagon and any right triangle whose base is half the side of the octagon, the distance from the center to a side of the octagon is given in terms of the hypotenuse and height of the triangle by

center to side
= (2^{1/2} + 1) [hypotenuse^{2} - height^{2}]^{1/2}.

This formula is exact and is easy to derive using nothing more advanced than simple geometry and the Pythagorean Theorem. Plugging in the hypotenuse and height specified in question **8** gives

center to side
= (2^{1/2} + 1) [(4.6 cm)^{2} - (4 cm)^{2}]^{1/2}
= 5.484... cm.

The fact that this distance differs from the triangle's height contradicts the assumption that the apex of the triangle is located at the center of the octagon.

For any regular octagon and any right triangle whose base is half the side of the octagon, the distance from the center to a vertex of the octagon is given in terms of the hypotenuse and height of the triangle by

center to vertex
= (4 + 2^{1/2}2)^{1/2} [hypotenuse^{2} - height^{2}]^{1/2}.

This formula is exact and is easy to derive using nothing more advanced than simple geometry and the Pythagorean Theorem. Plugging in the hypotenuse and height specified in question **8** gives

center to vertex
= (4 + 2^{1/2}2)^{1/2} [(4.6 cm)^{2} - (4 cm)^{2}]^{1/2}
= 5.935... cm.

The fact that this distance differs from the triangle's hypotenuse contradicts the assumption that the apex of the triangle is located at the center of the octagon.