Geometric Measurement and Dimension Geometry

Sides and angles

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Tina asks:

Why do kids have such a hard time distinguishing between sides and angles? They are so different in my mind, I don’t even know how to explain the difference.


17 replies on “Sides and angles”

It would be interesting to talk to that student and give them a similar problem (same problem, but slightly different numbers or wording. Was it a mental lapse? Was it that they are memorizing rules and applied the wrong one? This would be a good one to use the Explain Everything Ap on an iPad for the assessment.

This student is very lost and needs more practice. I would try to tackle the concept by asking questions like “Can you create a triangle that has side lengths of 2 inches, 3 inches, and 100 inches? Explain why or why not.” You could also do an activity with this student one-on-one after school using rulers and protractors as manipulatives to make the measurements more real and the concepts more concrete to them.

Although this is not the case for the student mentioned here, in my experience, students typically get these questions wrong because they are not reading carefully.

I think it would be very interesting to ask the student maybe one on one, what the definition of an angle is when it’s not part of a triangle and then what the definition of an angle is when it is part of a triangle. From my experience, students think the angle is defined by the sides so when a question about the sides of a triangle is presented, I’m not surprised if they answer in terms of degrees because they may think you are actually asking about the angles. Terminology and language are extremely important but just because a formal “definition” is given doesn’t mean it is understood. Students tend to create their own definitions and frameworks in their head and the meaning they have made for themselves can be “right” to them, but not answer the question that was asked in the formal mathematical language.

On the other hand, the student could’ve just read “angle” instead of “side” too! 🙂

We did that activity with graph paper rulers and it was awesome. Some kids asked for graph paper or rulers. It’s an issue of getting all the kids to think of the referential task when they see a problem like this one.

Sides are real. You can see them right there. The angle between them is an abstract concept. Not only that, it is one that exists in negative space, the space between objects. People in general tend to lack an understanding of negative space issues, where a thing we identify is really just the name we give to the void between other things.

Carmel’s response is very astute, as she write that “Students tend to create their own definitions and frameworks in their head and the meaning they have made for themselves can be “right” to them, but not answer the question that was asked in the formal mathematical language.” So, you ask them to list the triangle’s side lengths from smallest to greatest. This is a TOUGH problem for the uninitiated. The student MUST understand that the length of a side of a triangle is directly proportional to the size of the angle that is opposite the angle. That is a LOT to understand. Has the student (or, have the students) actually created triangles and then measured the lengths of the sides and then measured the angles opposite the sides and, from those experiments, come up with conjectures about how angles and opposite sides are related. I make the mistake frequently – that a definition or an explanation or a video or a worksheet in and of itself is enough. For some students, yes, but for some, no. This may be one reason for the error – the student does not understand this concept well enough to answer the question correctly. What was done to teach the concept? Were there classwork, exit tickets, homework, do now’s, or other activities that really covered the topic? What did you do when students got the problem wrong? I am just asking….because often these misunderstandings are apparent in the work the students do (if they do the problems at all, or if they are not copying the work of others)and if we can catch the misunderstanding in the lesson or in the work the student’s do, we can help them learn better.

The class did experiments, measured triangles and practiced using a format where they wrote the numbers given in order, then the angle names underneath, then the opposite side underneath that (or sides then angles depending on what was given). Whether or not this student did the experimenting or enough of those practice problems is a different story altogether…

I cannot help but think that part of the problem lies in how we typically identify sides and angles in these diagrams. For those of us who are already comfortable with the relationships, it makes sense to use the upper case/lower case letter distinction where angles are upper case letters and sides are lower case. However, it seems that it would be VERY easy for anyone who is confused about this relationship to be even more so with this terminology. Why don’t we routinely refer to sides with segment terminology instead? Is it simply a matter of looking for a more elegant way to label?

In geometry I only use two letter names for sides (so I was looking for AB, AC, BC as the solution). By PreCalculus I expect students to be able to figure out that “a” is the side opposite from angle “A.” However, we frequently take the shortcut of saying angle A rather than angle CAB which might do a better job of getting students to see the angle as the thing traced out as you move from C to A to B.

I wonder if these are examples of the Pershan Hypothesis, namely that students who are confused and don’t know what to attend to in a problem will map it incorrectly onto a problem they are comfortable with.

For the question, can 3, 5, and 9 be the sides of a triangle, I’ve found students are challenged by that because they want to answer it using arithmetic (non-visual) reasoning. With manipulatives the answer is obvious, but just numbers on a page they think that those numbers should go into a formula and add up to something (or not). The “formula” for the triangle inequality is poorly understood (inequalities! cases! yuck!). So they convert it into a formula they love… A + B + C = 180 (or, in this case, 90?). Three things can make a triangle if you add them up and get a familiar number, so I’ll apply that to these three things. I’m less curious about why the student did that, and more curious about how queasy it made them. If this was a hail mary, I’m pleased… if this was, “duh, I know just what to do here!” then I’m worried.

For putting the sides in order, I agree with other posters that this student seems befuddled by the notation for writing about sides and angles, since they have to use “from C to B and B to A”. I’m not sure if the kiddo is talking about angles or sides, and what he’s justifying by talking about them going in order. I’d like him to talk through the problem with pointing. Either in a mini-interview or using an iPad to record his thinking and pointing. But he may be again using the detail that A > B > C to assume that the side lengths are “in order” too but can’t quite articulate where to start going around the triangle to traverse sides. I also wonder the extent to which he’s attending to the (not-to-scale) diagram.

The Pershan Hypothesis is more relevant than you know. Questions 4 and 5 on this test were angle sum problems so that equation was stuck in her brain.

I can’t say anything valuable in response to the main question that previous commenters haven’t already said (though I would echo @maxmathforum in mentioning that the student is also confused about the sum of the interior angles of a triangle, which is even more evidence that the student is confused about a lot more than just the distinction between sides and angles).

But I did want to make a couple suggestions about how the questions are written, since sometimes tweaking the presentation of the question can help the student stay on track. I would make a distinction between “side” and “side length”, as in “Can 3, 5, and 9 be the side lengths of a triangle?” and “List the sides of triangle ABC in order from shortest to longest.” Besides being more technically correct, this may help the student remember that lengths are associated with sides/line segments and angle measures are associated with angles.

I’m also not keen on the second problem because of the not-to-scale diagram of the triangle. I understand that it is not to scale so that the student has to (effectively) use the Law of Sines instead of visually estimating side lengths. The problem is that that divorces the Law of Sines from physical reality; if using the given angles and the Law of Sines tells me that BC is the longest side, but my eyes tell me that AB is the longest side, it’s going to be hard for me to internalize that the Law of Sines tells me something about the real world. My offhand suggestion for altering the problem would be to tell the student that angle ABC is 60 degrees, etc., without giving them a diagram of the triangle.

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