Wednesday, September 30, 2020
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Geometry PARCC question: equidistant in plane

The following multiple-select question, explained here in hopes of helping geometry students 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 2015 test in geometry, here:

Select from the dropdown menus to correctly complete each sentence.

The set of all points in a plane that are equidistant from a given point is called a {square, sphere, segment, circle}.

The given point is called the {square, sphere, segment, circle}.

[sic]

Answer and references

Correct answers: circle … center. The test as printed for the public release form contains an obvious typographical error, because the correct answer for the second dropdown menu, “center,” is not among the options printed on the test.

PARCC evidence statement(s) tested: G-CO.1:

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Definitions are limited to those in the evidence statement. Plane is also considered an undefined notion.

The evidence statement above references Math Practice 6 in the Common Core:

[MP.6] Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. … By the time they reach high school they have learned to examine claims and make explicit use of definitions.

The question tests students’ understanding of the high school Common Core geometry standard HSG.C.A.2, found under High School: Geometry: Circles (understand and apply theorems about circles), which states that they should be able to “identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.”

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a page on the limited use circles have in algebra, given the inescapable fact that a circle is not a function (it fails the vertical line test).

The Khan Academy, developed by Sal Khan, an engineer who has several videos dealing with circles and their properties, beginning here. The landing page also features links to proofs of various theorems involving circles, which you may be required to know for the PARCC geometry test.

Chapter 10 of the book Geometry for Enjoyment and Challenge by Richard Rhoad et al, all teachers from Illinois, features the following definitions related to circles:

A circle is the set of all points in a plane that are a given distance from a given point in the plane. The given point is the center of the circle, and the given distance is the radius. A segment that joins the center to a point on the circle is also called a radius.

(Richard Rhoad, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge, new edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 1991. The book is used in several geometry classes taught in Illinois high schools.)

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standards it purports to measure and tests students’ understanding of the definition of a circle and its radius. It is considered to have a low cognitive demand.

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper.

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

Challenge

Prove that if two different tangents to the same circle intersect at a common point (outside the circle), the line segments between that point and the two points of tangency have the same length (help here or here).

Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.

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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