Here is a problem put out by Houghton-Mifflin and labeled ‘test prep’ that I came across as a worksheet for a 9th grade Geometry student I’m currently helping here in Washington, DC.
How many things can you find wrong with this question?
If you have a hard time reading the question, it asks, “Why is segment DE the perpendicular bisector of segment FC?” (FC wasn’t drawn by the makers of the worksheet; that was added by the student right before I took the photo with my phone.)
Here is what I find objectionable:
(1) Segment DE is in fact NOT the perpendicular bisector of segment FC. It might be perpendicular, if EB’A’D is a rectangle, which is nowhere given in the problem, but even if the skinny space between the two triangles is a rectangle, DE does not intersect segment FC’ at its midpoint.
(2) We should not be asking students to prove things that are not true.
(3) The first two paragraphs of verbiage* are unnecessary, but if we take them at face value, you could end up with either of these two diagrams instead:
or else the following:
In neither case is it true that DE is even perpendicular to undrawn segment CF, even though I definitely used ‘rigid motions’ to transform triangle ABC into triangle DEF.
(4) In geometry classes, an apostrophe after a capital letter means something: that a point is the result of some sort of transformation performed on the point that lacks the apostrophe. In this problem, points A, B, and C should not be written as A’, B’, and C’.
(5) Here is a much more straightforward problem: You are given the congruences shown in the diagram. Explain why (or “Prove that”) segment HJ is the perpendicular bisector of segment IK.
Maybe this is what Sandra Stotsky was complaining about when she dissented from approving the Common Core math standards, objecting to making transformations (such as rigid motions or dilations) the core of geometry .
*For those of you unfamiliar with the term, a rigid motion means a rotation (turn), a reflection (flip), or a translation (slide) — they normally change the location of a figure without changing its size, area, or angles. And, yeah, I taught geometry to 8th and 9th graders for many years, so I know a little bit of what I’m talking about.