Utah Grade 8 Math

Same side lengths, different order around the triangle

A triangle has vertices A(1, 1), B(4, 1), and C(1, 3). First reflect it across the y-axis. Then, in a separate transformation of the original triangle, translate it two units left. Record corresponding vertices and compare what happens when you follow A to B to C. Explain why side lengths alone do not identify which transformation was used.

Follow the reasoning

Reflection across the y-axis changes (x, y) to (−x, y). Translation two units left changes it to (x − 2, y). Both preserve side lengths: AB remains 3 units and AC remains 2 units, with the same right angle at A. But the traversal A → B → C goes counterclockwise in the original and translated triangles and clockwise in the reflected triangle. The reflection reverses orientation; the translation preserves it. Although A and C happen to reach identical locations under these two transformations, B does not. Checking only one or two conveniently placed vertices would miss the difference. Both images are congruent to the original, so congruence alone does not identify the transformation.

Check the coordinate rule

Reflection across the y-axis changes the x-coordinate’s sign, not the y-coordinate’s sign. Translation changes every x-coordinate by the same amount. A figure’s position, orientation, and side lengths are different properties; specify which one you are comparing. The complete reference includes rotations and dilations as well as these two selected transformations.

Try a rotation

Rotate the original triangle 90° counterclockwise about the origin using (x, y) → (−y, x). The images are A(−1, 1), B(−1, 4), and C(−3, 1). Plot the points on an equally scaled grid and follow A → B → C. The traversal remains counterclockwise, illustrating orientation preserved by a rotation even though the triangle changes position.

Utah reference for this worked activity

The Utah mathematics standards directory identifies January 2016 adoption and lists a revision process that is still underway. The elementary PDF says revised April 2016; the middle-school PDF says revised September 2015–April 2016. These file revision labels do not establish adoption of a new 2026 edition. Use the school’s current learning sequence and consult later department updates.

This selected reference identifies the activity focus. The complete standard and grade curriculum include additional work. These labels do not certify alignment of every independent lesson, worksheet, or quiz in the broader directory.

Choose a next task from the student’s explanation

Use a different set of numbers to check whether the student can explain the same relationship. If the explanation is unclear, return to the model; if the reasoning is sound but a calculation is wrong, work through a shorter calculation and check it against the model. Use the related learning resources alongside the teacher’s assigned goals.

The Utah RISE overview lists mathematics summative assessment in Grades 3–8, including this grade. Follow the school’s current directions and accommodations. Independently published practice has its own timing, tools, question count, and scoring; its result does not establish a state performance level. This original activity is not an official state assessment question. Read Utah assessment guidance.

State sources and the selected complete PDF page checked October 6, 2026. Confirm later guidance and local assignments before planning assessment preparation.