This one is a little bit hard to read, so I’ve transcribed the student’s writing below the image.



What’s going on, and how could you help?

  • mathtchr

    The student is combining theorems. There needs to be emphasis placed on the fact that ITT says the two angles IN ONE TRIANGLE are congruent to each other. Illustrations could be made to show that this problem is not true using a compass and straightedge.

  • timteachesmath

    Give the kid two compasses or four twizzlers to represent what we know about the two triangles. Better yet, give him or her two twizzlers, and I’ll keep two; we can verify they’re all the same length, then I turn my back and make my two into a triangle and he or she makes his two into a triangle and we see if we made the same triangle. Once he’s disappointed they’re not congruent, I ask what if there ARE any things equal to eachother, if not the two triangles.

  • Carolann

    I don’t know what a twizzler is, but I imagine it must be some kind of stick-like object, which makes timteachesmath’s idea a good strategy. Part of this question seems to be to test how well students read the diagram, so some diagram-reading practice might be helpful. I always teach that diagrams can’t be trusted unless they are specifically labelled as accurate scale drawings.

  • pfordne

    One thing I like/dislike about this problem is the fact that students use their eyes to come to a conclusion; “the two triangles look to be the same, so the angles must be the same.” They need these opportunities to challenge their misconceptions. They don’t consider that there could be other ways to draw a second triangle. So in a way, they were misled by having the drawing provided.

  • pfordne has a good point. Could the class handle the question if it were asked without a diagram? On the SAT, if a provided image *might* be different, it is ALWAYS labeled “Image not necessarily drawn to scale.” Sometimes fixed images still carry that label to keep students alert, but variable images are always labeled. Since the ETS never makes any moves without a research base, I suspect there’s statistical evidence indicating that test takers are unnecessarily swayed by unlabeled variable figures.

    Another suggestion would be giving students lots of time to work with GeoGebra, Geometer’s Sketch Pad, or Cabri Geometry/TI Nspire to give them dynamic experience playing with triangles and determining when given triangle information is unique. I think this is parallel to what timteachesmath and Carolann are offering.

    From a high level, each triangle has only 2 given pieces of information. As a minimum of 3 pieces of information are required to uniquely define any triangle, this problem could have been dismissed from the outset. The fact that the given triangles are isosceles is utterly irrelevant.