Math questions that make me groan

Not as often as you might think, but every now and then someone in our office asks me for help on a homework problem for one of their kids. Today I got asked to help out on this question:


In case the picture isn’t clear, you have an isosceles trapezoid with sides of length 24, 3x + 2, x – 4, and x – 4 and you are asked to find x. Plug and chug through the math and you’ll find that x = 14, but there’s a problem. Plugging in x = 14 you find a trapezoid with sides 24, 44, 10, and 10, which only works if the configuration is a straight line (or if you want to save a little face for the problem writers, a degenerate trapezoid).

I don’t like this question at all and I have all kinds of sympathy for any student who is confused by it.

Of course, there have been many examples of other poor test / homework questions. Patrick Honner quickly pointed out a bad trapezoid question on the New York State 5th grade math exam:


Two of my favorite examples of homework / exam problem goof ups are (1) this geometry problem from V. I. Arnold on Tanya Khovanova’s blog:

A Vladimir Arnold problem on Tanya Khovanova’s blog

and (2) the Pyramid puzzle which we talked about at the end of this Family Math project:

Octahedrons Tetrahedrons and the Pyramid Puzzle


But, despite the occasional fun story this type of question annoys me on homework and makes my blood boil on standardized tests. Hopefully today’s example will be the last one I see, but I’m not counting on it 😦

Proof and Problem Solving in Geometry

One of the fun things about home schooling for me is teaching subjects that I’ve never taught (or really thought about in years) for the first time. I’m in a section in our Geometry book about properties of triangles with my older son and it is an absolute blast work through with him.

It could easily be that I just wasn’t totally tuned in to the problem solving / proof process previously, but I feel as though I’m seeing my son pull together a lot of different ideas for the first time. It is so cool to see this process.

Last week we talked about this theorem – if a triangle has two equal medians, the triangle is isosceles. I just love watching the ideas from basic geometry come together here.


Tonight we looked at how you could find the radius of the circumscribed circle for an isosceles triangle. This is more problem solving-related than proof-related but I still really enjoyed watching the ideas come together. The extra connection with algebra here even brings in some math beyond geometry:

[a little post publication edit as I realized that you could use Patrick Honner’s Desmos program about circumscribed circles to explore the 2nd problem, too]

Patrick Honner has a nice Desmos program you can use to explore circumscribed circles. Setting the three points of the triangle in his program to be (0,0), (10,0), and (5,12) you can see (by clicking the “calculations” tab) that the radius of the circumscribed circle is indeed 169/24 = 7 1/24. Fun!

Patrick Honner’s Circumcircle program in Desmos

[end of post publication edit]

Before going through this Geometry book I’m not sure that I really understood why we were studying geometry. In fact, I’m sure that my answer wouldn’t have been that different from “that’s just what you study after algebra.” Now that we are in the middle of the book, though, I’m starting to get a better understanding of how studying geometry helps build your mathematical reasoning skills. It is so fun to watch those skills develop.