I think, in this case, I might get the student to directly measure the two angles to see that they aren’t the same, and then ask them to explain why that must also be true mathematically.

I think that’s a good call, John. I want to add something that might just be explicating what it means to “explain why that must also be true mathematically.” Part of the problem here is that the student is collapsing a bunch of quadrilaterals into one category. Meaning, the student isn’t just seeing this picture and thinking that it should be symmetrical. He also knows that there are 4-sided shapes that ARE symmetrical. So part of the work that has to be done with this student is creating separate conceptual categories for the shapes that have symmetry and the ones that don’t. The most effective thing would be for the student to do that conceptual work.

Also the use of patty paper activities in class where the students see visually where the symmetry line truly is can be beneficial. Along with the emphasis on details . . . the congruency marks are important and need to be taken into account. Also, assignments need to have kites drawn in different planes (vertical alignment vs horizontal vs slanted/diagonal). I really like this site . . . great idea!

It’s also possible the kid just rushed through it, and will figure things out in a few seconds upon seeing the graded paper. As always, not enough context.

I’m a firm believer that there are very few sloppy mistakes. Every mistake that seems sloppy is a combination of at least two things (1) distraction by some part of the problem that is taking up too much brain power b/c it’s confusing and (2) a plausible alternative. It’s true that (1) is the cause of a lot of mistakes, but nobody just writes down random answers when they’re confused. They write down plausible alternatives, and we can help this student in two ways. First, by making that alternative implausible. Second, by giving the student more problems that help make this problem more and more automatic. (Which is NOT an argument for kill and drill. There are better ways to help make a procedure more automatic.)

I have a few thoughts: 1. Often they are following a set of patterns, a skill of how to find the angle. They aren’t seeing symmetry so much as following the process they were told to use in similar problems. They have to think about this conceptually rather than as a process.2. They need practice with other shapes that are similar but not symmetrical. I’ve found that a quick study of optical illusions helps them see the need to trust measurements rather than what they first see (though not as helpful here). 3. Students need to be taught logic and rationality. They know symmetry. They know that those two aren’t symmetrical. So, somehow they aren’t checking it logically. Fewer problems and more reflection are both a part of the solution. Note: I was “that kid” who made those mistakes all the time because I was slow at computation and I would rush through problems like this so that I would have time for longer, multi-step equations.

Maybe a way to get people to stop seeing false symmetry is to ask them whether there is or is not a particualar symmetry.

Have the student start with a scalene triangle. Make sure they know that all three angles have different measurements, and I would suggest coloring them in using three different colors. Then (using patty paper or what have you) have them reflect the triangle over the longest side. If they understand that the image is congruent to the original triangle, they can color the new angles. Across from each other will be two angles of the same color (the largest angle in the original triangle and its image), and two angles of *different* colors. The different-colored angles are not congruent.

If they were to do the same thing with an isosceles triangle, color the base angles the same color (of course) and reflect over the base, the student should see a rhombus, both of whose opposite angle pairs are the same color (and therefore congruent).

## 8 replies on “False Symmetry”

I think, in this case, I might get the student to directly measure the two angles to see that they aren’t the same, and then ask them to explain why that must also be true mathematically.

I think that’s a good call, John. I want to add something that might just be explicating what it means to “explain why that must also be true mathematically.” Part of the problem here is that the student is collapsing a bunch of quadrilaterals into one category. Meaning, the student isn’t just seeing this picture and thinking that it should be symmetrical. He also knows that there are 4-sided shapes that ARE symmetrical. So part of the work that has to be done with this student is creating separate conceptual categories for the shapes that have symmetry and the ones that don’t. The most effective thing would be for the student to do that conceptual work.

Also the use of patty paper activities in class where the students see visually where the symmetry line truly is can be beneficial. Along with the emphasis on details . . . the congruency marks are important and need to be taken into account. Also, assignments need to have kites drawn in different planes (vertical alignment vs horizontal vs slanted/diagonal). I really like this site . . . great idea!

It’s also possible the kid just rushed through it, and will figure things out in a few seconds upon seeing the graded paper. As always, not enough context.

I’m a firm believer that there are very few sloppy mistakes. Every mistake that seems sloppy is a combination of at least two things (1) distraction by some part of the problem that is taking up too much brain power b/c it’s confusing and (2) a plausible alternative. It’s true that (1) is the cause of a lot of mistakes, but nobody just writes down random answers when they’re confused. They write down plausible alternatives, and we can help this student in two ways. First, by making that alternative implausible. Second, by giving the student more problems that help make this problem more and more automatic. (Which is NOT an argument for kill and drill. There are better ways to help make a procedure more automatic.)

I have a few thoughts: 1. Often they are following a set of patterns, a skill of how to find the angle. They aren’t seeing symmetry so much as following the process they were told to use in similar problems. They have to think about this conceptually rather than as a process.2. They need practice with other shapes that are similar but not symmetrical. I’ve found that a quick study of optical illusions helps them see the need to trust measurements rather than what they first see (though not as helpful here). 3. Students need to be taught logic and rationality. They know symmetry. They know that those two aren’t symmetrical. So, somehow they aren’t checking it logically. Fewer problems and more reflection are both a part of the solution. Note: I was “that kid” who made those mistakes all the time because I was slow at computation and I would rush through problems like this so that I would have time for longer, multi-step equations.

Maybe a way to get people to stop seeing false symmetry is to ask them whether there is or is not a particualar symmetry.

Have the student start with a scalene triangle. Make sure they know that all three angles have different measurements, and I would suggest coloring them in using three different colors. Then (using patty paper or what have you) have them reflect the triangle over the longest side. If they understand that the image is congruent to the original triangle, they can color the new angles. Across from each other will be two angles of the same color (the largest angle in the original triangle and its image), and two angles of *different* colors. The different-colored angles are not congruent.

If they were to do the same thing with an isosceles triangle, color the base angles the same color (of course) and reflect over the base, the student should see a rhombus, both of whose opposite angle pairs are the same color (and therefore congruent).