## Bad ‘rigorous’ Common-Core-style geometry problem

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.

Published in: on December 3, 2014 at 4:11 pm  Comments (10)
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1. I am teaching Geometry to night-school students trying for credit recovery, so they are high-school age or very close. I was given a “Common Core ” book from one of the few major publishers left. The book is awful. It has the type of problem you have outlined sprinkled throughout every section.
I am a retired math/science teacher (35 years high school level). If a young teacher had to use this book they would be hopelessly confused about what really was expected of the students. I have to look carefully at every problem in an assignment to make sure it makes sense.
I wish I had grabbed some texts when I left teaching.

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• You can look online for used math and science texts. I certainly never saw horrendous problems like this when I was teachng.

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2. Since when did Sandra Stotsky know anything about mathematics, Guy?

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• Probably never.

However, I saw her complaint that geometry shouldn’t be introduced by using transformations, claiming that it had been briefly tried and then abandoned in the USSR at some point. This examplle seems to me to show how such an approach can be made idiotic.

Not taking her as an authority or anything, it seems to me that at some level, seeing ‘congruence’ as defined as having the ability to map identical segments or figures onto each other by some sequence of rigid motions, even as described in this misbegotten problem, isn’t all that bad.

However, in the hands of mathematically and geometrically innumerate peoople like the writers of this particular problem, one can end up with complete and utter absurdity, as we see here.

So, she might have a point. Which is not exactly high praise.

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3. Your objection 3 doesn’t address what was asked in the problem, which was to map segment AB onto segment DE. Your counter-examples did not end up with AB and DE identically placed.

And I’m pretty sure that segment is the perpendicular bisector. Certainly perpendicular, as we’re looking at diagonals of a kite…

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• In response, I would say that “maps segment AB onto segment DE” is unclear as to precisely where points D and E would be located. If they intended this to mean that points A and D would be identical, then they should have said that, AND THE DIAGRAM SHOULD HAVE SHOWED THAT AS WELL. But it doesn’t. So the distance from point F to side DE is a bit less than exactly one half of distance FC’.

In normal parlance, if we say that one segment or figure maps onto another one, it does not necessarily mean that they are located in exactly the same loocation in space, just that there is a one-to-one correspondence between the first object and the second one.

However, if the intention was to say that we have a kite, then why didn’t they just say that and draw that?

Answer: because the people who made up the problem haven’t actually taught geometry, but have been told to produce a lot of rigorous-sounding common-core verbiage, and assigned some low-paid grunt the task of making up a problem, and didn’t check to see if it made any sense.

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• Yes, if the killer problem is the space between the triangles, which there shouldn’t be. (I missed that _that_ was the main objection when I first read this on my phone.)

I agree that there’s a lot of junk at the start of the question which is unnecessary. I think this might have been a neat problem if asked as a reflection: “If triangle ABC is reflected over AB, prove that AB is the perpendicular bisector of CC’.”

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• So we basically agree. Now, why do you suppose that they did NOT ask the very sensible question you just posed?

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• Dunno. Looks like three different problem ideas got together and couldn’t decide which one was going to be in charge. And then whoever made a diagram made a flawed one and nobody caught it in proofreading.

I’m not seeing this as a Common Core problem, though. This just looks to me like an awful setup — you’re right, all the introductory stuff doesn’t need to be there if the student’s not going to be asked to do anything with or about it.

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4. That’s pretty typical for Common Core stuff that I’ve seen. Mind you, the publisher (Houghton Mifflin) has never done anything as lame as these sorts of prolems before.

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