Friday, May 7, 2021

Grade 8 PARCC math: Pythagorean distance


The following fill-in-the-number question, explained here in hopes of helping eighth-grade students and their parents in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2016 test for grade eight math:

A right rectangular prism is shown:

A right rectangular prism is shown, with a length of 12 inches, a width of 5 inches, and a height of 7 inches. There is a dashed line drawn on the bottom surface from the back left vertex to the front right vertex. Another line, labeled D, is drawn from the front right vertex on the bottom surface to the back left vertex on the top surface of the prism.

To the nearest thousandth of an inch, what is the length of the diagonal, d?

Answer and references

Correct answer: 14.764 or 14.765 (depending on the rounding performed by the student, the computer will accept either answer for this question but no other answer). The exact number is

\sqrt{7^2 + 13^2} = \sqrt{218} \approx 14.76482306\textrm{...}

By publishing both an answer derived from rounding (14.765) and one derived from simple truncation (14.764), PARCC here is saying that when a question says “to the nearest thousandth” of some unit, you won’t get it wrong if you got your answer by truncating, not rounding, the final result.

Common Core Math Content 8th grade, Geometry

(8.G.B.7) Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Understanding of this standard also depends on the associated standards of 8.G.B.6 (explain a proof of the Pythagorean Theorem and its converse) and 8.G.B.8 (apply the Pythagorean Theorem to find the distance between two points in a coordinate system). Students, teachers, and parents should realize that while this example problem only covers B.7, the other standards are fair game as well.

Solution strategy (there are others)

Use the Pythagorean theorem, a2 + b2 = c2, to find, first, the unlabeled diagonal’s length and then the length of diagonal d.

The unlabeled diagonal, drawn along the bottom surface of the prism, is the hypotenuse of a right triangle with legs of 5 and 12. If you’ve worked with right triangles enough, you’ll recognize this as one of the standard, no-decimal side length, right triangles: 5-12-13. If you didn’t recognize that, you can always figure it out using the Pythagorean Theorem:

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

Now that you have that length, you have to see that the diagonal labeled d, which runs from the lower right vertex in front to the upper left vertex in back, is also the hypotenuse of a right triangle. The right angle is the vertex on the bottom in the back left of the prism, and the legs are (a) the height of the prism, or 7, and (b) the unlabeled diagonal, or 13.

a^2 + b^2 = c^2
7^2 + 13^2 = d^2
49 + 169 = d^2
\sqrt{218} = d

Analysis of this question and online accessibility

New to eighth-grade math students under the Common Core is the ability to find diagonal distances on a coordinate plane using the Pythagorean Theorem, according to the North Carolina Department of Public Instruction.

Fill-in-the-number questions like this eliminate guessing as students can’t simply pick the option that looks best without doing the math. They can be delivered easily online and scored without human intervention.

On paper, these problems can be converted to a multiple-choice question, which can be scored without human intervention, or left in a fill-in-the-number presentation, which can’t. This makes the question accessible for students on any device they may use or on paper, although kids who learn information passively may have an easier time with a multiple-choice format. Fortunately that’s not too many students in our schools, and validity, reliability, and fairness measures should not differ significantly among the various delivery modes.

(A minor or irrelevant editorial error remains in the problem as published: The comma after the word diagonal should be omitted—the variable d restricts the meaning of the word “diagonal,” since there are two diagonals shown in the figure. For me, this is another example of either incompetence or sloppiness in the production of this test, for which the taxpayers of Illinois and Maryland have paid tens of millions of dollars.)

No special accommodation challenges can be identified with this question, so the question is considered fair.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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