Grade 8 PARCC math: geometric transformations

The following two-part (drag and drop, multiple-choice) 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:

ΔPQR is transformed to the image ΔP′Q′R′.

Part A

Describe the single transformation that maps ΔPQR onto its image ΔP′Q′R′. Determine whether ΔPQR maintains its shape as a result of the transformation.

Drag and drop the correct words and phrases into the boxes.

The transformation is a __(blue)__ __(black)__.

ΔPQR is __(red)__ to the image ΔP′Q′R′.

The color-coded words and phrases:

BLUE: dilation, rotation, reflection, translation
BLACK: by a factor of ½, by a factor of 2, 90 degrees about vertex R, 180 degrees about vertex R, to the right, across the line, to the left
RED: not congruent, congruent

Part B

A translation is performed on ΔP′Q′R′ to create image ΔP′′Q′′R′′. How does line segment P′′Q′′ compare to line segment P′Q′?

A. Line segment P′′Q′′ is longer than line segment P′Q′.
B. Line segment P′′Q′′ is shorter than line segment P′Q′.
C. Line segment P′′Q′′ is congruent to line segment P′Q′.
D. There is not enough information to compare the two line segments.

Answer and references

Correct answer: Part A: The transformation is a reflection across the line. ΔPQR is congruent to the image ΔP′Q′R′. Part B: (C) … congruent to …

Common Core Math Content 8th grade, Geometry

(8.G.A) Understand congruence and similarity using physical models, transparencies, or geometry software.

(8.G.A.1) Verify experimentally the properties of rotations, reflections, and translations: (A) Lines are taken to lines, and line segments to line segments of the same length. (B) Angles are taken to angles of the same measure. (C) Parallel lines are taken to parallel lines.

(8.G.A.2) Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

(8.G.A.3) Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

(8.G.A.4) Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Solution strategy (there are others)

Recognize that the triangle is flipped over the line, representing a reflection, which is a rigid transformation and results in an image that is congruent to the original figure.

A tool from the Khan Academy to perform reflections
Transformation worksheet from a school district in Arizona
Transformations are new in eighth grade under the Common Core, according to the North Carolina Department of Public Instruction.

Analysis of this question and online accessibility

Translations, reflections, and rotations are examples of rigid motions, which are, intuitively, rules of moving points in the plane in such a way that preserves distance.

Some topics learned much earlier, including what students learned about angle and line measurements in fourth grade, is put to good use in the eighth-grade geometry standards now in the Common Core. And in some states, including Maryland, the ability to recognize and even perform any of the three rigid transformations was expected in third and fourth grade. The Common Core introduces these rigid transformations first in eighth grade under the umbrella of congruence of geometric figures.

The Common Core adds dilation, which results in a similar image. The dilated image will only be congruent to the original if the dilation factor is 1. As a result of the choice of dilation, a student’s answer in Part B is in jeopardy if he identifies the transformation incorrectly as a dilation by a factor of anything other than 1 (a factor of 1 is not one of the options).

As a result of the shift in the transformation skill to eighth grade, the question assesses students’ knowledge of the Common Core standard it purports to measure. Note that even many secondary school math teachers are reluctant to teach rigid transformations in geometry, according to a recent study conducted at the University of California, San Bernardino.

Multiple-choice questions like Part B can be delivered easily online and the drag and drop has an easy conversion to a multiple-choice format for paper-based tests, which makes this question accessible for students on any device they may use or on paper. Validity, reliability, and fairness measures should not differ significantly among the various delivery modes.

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