Students learn the triangle inequality in geometry and might even recall it correctly 5 minutes after the final exam ("Let's see -- I think it's something like 'two sides of a triangle are more or is it less than the other side'"). Of course, if they can visualize it, they might retain it longer, but, in the end, they should know it as well as they know their basic arithmetic facts. (Uh oh -- bad analogy!).
Here's a fairly straightforward application although many students will answer it incorrectly even if they correctly recall the key geometric fact. Can you see where the traps lie?
A triangle has sides of lengths 12-x, 12 and 12+x. How many integer values of x are possible?
Now for the generalization:
A triangle has sides of lengths a-x, a and a+x, where a is a positive even integer. In terms of a, how many integer values of x are possible?
Comments:
(1) Do you believe we should more heavily emphasize the triangle inequality? The SAT certainly does (not that that is any justification, right?).
(2) What teaching strategies have worked best for you in the classroom when introducing this theorem? How much time is generally devoted to this topic?
Wednesday, August 6, 2008
A Triangle Inequality and Algebra Application
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10 comments:
Hi Dave,
I would probably start with simple questions like:
What is the shortest possible length of a side?
What is the longst possible length of a side?
And then/or before:
How does the triangle look when the sum of two sides
is shorter than the 3rd side?
I believe if presented graphicaly with simple inequality
calculations it should be pretty intuitive and the solution
easy to find (x can be zero too, surprise).
This problem requires facility with negative numbers - good if you're trying to reinforce prior knowledge, but it in this case it increases the cognitive load while introducing a new concept. This is especially true here since it involves visualizing the lengths of the sides and negative distances are more difficult to visualize.
I'd start off with something similar to Jackie's straw lesson with different leading questions.
Florian, Mr. K--
Thanks for the input!
I certainly would not introduce the concept of the triangle inequality using a problem with this many layers.
My problems were just examples of challenges one could give students down the road after they have an understanding of the basic principle and the number/algebra skills to solve it.
Remember, however, that students rarely attempt an algebraic approach to these when they are presented as standardized test questions (which is how I used them). Their natural instinct is to PLUG IN until they reach the maximum value. If they have stronger number skills they will pick up on the 'integer' aspect and recognize the negatives and, of course, zero. These questions were intended as a challenge!
I also wanted to further the dialog about the triangle inequality in the curriculum. Some students may not even know it's called the Triangle Inequality nor might they be aware that this theorem generalizes into n-dimensions or related to several other classical inequalities. May not be appropriate for all, but some should see it!
I start this lesson by asking "How many triangles can you make from a given perimeter?"
I know it's badly worded, it's intentional, I like them to ask questions and refine their understanding of a problem. We remind ourselves what "triangle" and "perimeter" mean, and what "different" means for polygons.
Eventually we get to something like "How many different triangles, with integral sides, can you make from a given perimeter?"
And they're off...I give them straws and rulers, if they ask.
First they figure out they can make 0 triangles of perimeter 2, 1 triangle of perimeter 3, but again 0 triangles of perimeter 4...the skeptical are intrigued...
We generate a table. We generate some form of a "rule" for what sorts of side lengths do and do not make a triangle. It doesn't always look like what I had in mind. It seems to stick with them. This takes ~2 40-min class periods.
Wonderful approach, Kate. Your ideas may help others who are new to this. Obviously you value this important concept and the hands-on demonstration will certainly make a more lasting impression. I really like your idea of asking the open-ended question. Best way to encourage questioning and dialog!
A couple of questions...
Are we talking about hs or middle school students? Having developed these ideas for both middle schoolers and high school students, I know there were similarities in the presentation but also some important differences in the level of abstraction I expected.
I also developed the concept with 4th graders. We collected some branches, some long some short. I remember asking the group if they thought they could make a triangle using one long stick and two very short ones. I held up the sticks and most responded, "No, you can't." One young lady objected, "I can do it." I gave her the sticks and sure enough she made a triangle! Can you guess what she did? She taught me an excellent lesson!
Kate, how do you assess their understanding, retention and application? In the end, what % of the group correctly apply the "rule"?
I am going to guess that the 4th grader only used a short part of the long stick?
I teach high school, these are 9th and 10th graders. My goal for the lesson is some statement of inequality using three letters to represent the side lengths. For example, x-y < z < x+y.
I assess in what is probably a pretty traditional way. My course is organized into units. Each unit involves one or two quizzes and a summative test. I also assign "problem sets" composed of review material that are turned in every week. I don't have the data you're asking for, but my sense of this topic is that my students don't have trouble answering questions involving triangle inequalities after we study it in class.
Thanks, Kate...
Yes, she took the long stick and simply overlapped the shorter pieces! She made it clear to me that I never stated that they must be connected at the ends! That's always the challenge of asking good questions, but, there will always be room for discussion and that's how it should be.
I do like the fact that you included the "other" half of the triangle inequality which is often overlooked (absolute values are needed if order isn't specified). Are you following the text for this development or did you enhance what was in the book (I think I could guess!)?
Sounds to me like your students are very fortunate to have an educator like you.
Thanks again for sharing. Let me know if you ever use any of these 'challenges' with your students and how it plays out...
Dave
Aw, shucks. Thanks.
I noticed I needed the absolute value around x-y right after I hit "Publish". *headsmack*
Regarding clearly stating problems, that's why I go the other way - I intentionally pose terrible, ill-defined questions. I think kids need experience defining what problem they want/need to investigate or solve and what information is necessary, important, or discoverable- that's the way math is going to come at them after they are out of school. And it's much more fun than the semi-paranoid gotcha-prevention mode that is the alternative.
For example in this lesson, I have classes that decide they want to count the degenerate triangles, like {1, 1, 2}, which makes for an interesting conversation and only slightly different outcome.
I use our textbooks to assign practice problems. That's about it. I find them pretty useless regarding conceptual development. I cast my net far and wide looking for lesson ideas. I do very little that is 100% original, I just adopt the best of what I read and see and refine it over time.
It's a nice problem. I could imagine groups who don't need the even constraint.
But I think Kate's is an important problem in developing a general sense of problem solving -- break that expectation of a quick (word choice here, quick, not easy) -- break that expectation of a quick answer every time.
Jonathan
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