Sunday, May 9, 2021

Grade 8 PARCC math: solid geometry

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The following two-part 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:

Shot put is a track-and-field event where athletes throw a heavy spherical ball. Shot put balls come in various sizes and materials.

The diameter of a brass shot put ball is 4.2 inches.

Part A

What is the volume of the brass shot put ball? Round your answer to the nearest cubic inch.

Part B

The volume of a stainless steel shot put ball is 45 cubic inches.

What is the approximate diameter of the stainless steel shot put ball? Round your answer to the nearest hundredth of an inch.

Answer and references

Correct answer: Part A: 38 or 39 cubic inches. Part B: 4.41 inches.

Common Core Math Content 8th grade, Geometry

(8.G.C.9) Finding the volume of cones, cylinders, and spheres is new to eighth graders under the Common Core. A formula sheet is provided that contains the formulas, so students aren’t required to memorize the formulas for the test. They are, however, required to know how to use the formulas, and this question, which purports to measure standards in eighth-grade geometry, tests students’ understanding of those standards just fine.

As students “solve real-world and mathematical problems involving volume of cylinders, cones, and spheres,” the Common Core says they should “know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.”

Solution strategy (there are others)

Use the formula for volume and solve for the different variables of a sphere.

From the formula sheet, we find:

V=\frac{4}{3}\pi r^3

Given a diameter in Part A of 4.2 inches, we have a radius of one-half that distance, or 2.1 inches. Plug it in and find V:

V=\frac{4}{3}\pi 2.1^3
V=\pi (1.33\bar{3})(9.261)
V=38.792386\textrm{...}

If you didn’t use the π key on a calculator, you might get a slightly different number, some of which may round to 38 cubic inches. For this reason—because the problem didn’t specify what value to use for π—an answer of either 38 or 39 will be marked correct on this problem, although 38 also results from truncation, not rounding, of the answer above.

Then, for Part B, we are given the volume and must find the diameter. That means we should solve the volume formula for r:

V=\frac{4}{3}\pi r^3
\frac{3}{4}(\frac{V}{\pi})=r^3
\sqrt[3]{\frac{3}{4}(\frac{V}{\pi})}=r

Now all we have to do is plug in the volume and find the radius, then multiply the radius by 2 to find the diameter:

\sqrt[3]{\frac{3}{4}(\frac{45}{\pi})}=r
\sqrt[3]{10.7429586\textrm{...}}=r
2.2065\textrm{...}=r

Multiply that by 2 to find a diameter of 4.413… and round to the nearest hundredth of an inch. Only 4.41 inches is accepted here, according to PARCC’s answer document.

Analysis of this question and online accessibility

Fill-in-the-number questions prevent students from entering words and, in the case of Part A, may even prevent them from typing a decimal point, since an answer is requested as being rounded to the nearest whole number. The question format also prevents guessing, since no options are presented, and the student must enter a number from scratch.

In terms of Part B, an inconsistency in the Common Core standards arises. Under eighth-grade math, expressions and equations (8.EE.A.2), students are required to evaluate:

  1. square roots of small perfect squares
  2. cube roots of small perfect cubes

As a result, the demands of the geometry problem here go above and beyond the Common Core requirements in eighth-grade math. The problem therefore, in Part B, fails to align to the Common Core in eighth grade. The only way to find the diameter of the shot-put is by evaluating the cube root of a number that is not a perfect cube.

The question is accessible for students on any device they may use or on paper. However, because paper test-takers would be required to respond to a multiple-choice format for this question, validity, reliability, and fairness measures may differ among the various delivery modes. Students who don’t know how to find the volume of a sphere might be able to guess the correct answer, from the four options on a paper-based test, whereas students answering the question online won’t have those pre-formatted answers available at test time.

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

Purpose of this series

To help eighth graders and their parents prepare for the PARCC test in math, as administered in at least six states, or to just master content on that test, we provide an analysis of every eighth-grade math problem PARCC released. The series can be found here.

Paul Katulahttps://news.schoolsdo.org
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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