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This kind is near the boundary: “Oh, I’ve seen triangles with those angles before, this side will be sqrt(3) times that side” is a kind of pattern to recognize, but its realm of application is narrow enough that it might not be worth looking at as hard as we do, when there are other much more heavily re-usable patterns like the Pythagorean Theorem or the idea of using symmetry to understand a diagram.Kuta Software Properties Of Parallelograms Answer Key Click to find a set of quadrilaterals worksheet with special right site to any set of parallelograms, calculus in a in a helpful resource that focus on I’d characterize the “good” kind of efficiency more as recognizing regularity, generalizing, and identifying patterns. On the other hand you do want to generalize and abstract (“abstract” here used in the computer science sense of finding the commonalities in a bunch of things and rethinking them as a single thing with a variable parameter, for instance), so those types of efficiencies are vital. Often if you want the efficient method, you’d pick up a computer instead! Humans can understand, calculators can calculate. I think in many cases (as hodge suggests) there’s no reason ever to go to the efficient method. I think your conjecture in part (1) is correct, that a lot of this comes from rushing to “efficient” methods before students are ready for them. Some students are so good at this that it can be difficult to distinguish from actually understanding what’s going on, until at some point in their mathematical career they get to the point that they simply can’t keep so many ad hoc rules in their minds at once and the precarious structure collapses. I call the “throw stuff into the blender” method “coping strategies” - you’re lost, you have no idea what’s going on, so you do something desperate to try to survive.
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It should also help make sure that the hypotenuse is actually the longest side! And cut down a lot on the memorization, too.Īs you described in (2) above, I’m glad that the students came up with the idea of the impossibility of these numbers and showed the value of estimating in checking the work. I like the suggestion of drawing the rest of the square or the equilateral triangle to make at least some of the side relationships visible. (Further evidence that we-myself and my students included-pay too little attention to the hypotheses of our theorems, and then wonder why our (mis)use of the conclusions leads to wonky answers.) If you don’t truly know what’s going on with the Pythagorean relationship, why not toss some angles in and press liquify? After all, using that a^2+b^2=c^2 nonsense has yielded the right answer in the past. And so on.īonus thought: What looks like an outrageously misguided approach to someone “in the know” (whether a teacher or another student), might seem as reasonable to the student who is struggling as any other “throw-some-values-and-an-equation/formula/algorithm-into-a-blender” approach. For example, in the first image (isosceles right triangle with hypotenuse of 1), the legs (which must be smaller than the hypotenuse) cannot possible by approximately 1.41 in length. One approach students used in that followup conversation was the impossibility of many of the responses given the decimal approximations of the side lengths in radical form.
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But they made for great classroom conversation at the start of class the next day. They are signs of medium to massive misunderstanding. (2) Needless to say, the responses captured in these four images were more than mildly disappointing. So here’s my question: Did these students perform so terribly on this four-question exit slip because I was pushing them to be efficient before they were ready to be efficient? And if that’s the case, how do we know as teachers when our push for efficiency is actually detrimental for students? How do we know when an individual (or small group, or entire class) is ready to move from slow-and-steady-and-meaningful to generalized and more efficient? Should the push ever come from the teacher, or only from the student?
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Many students made meaningful observations/connections, some did not (as evidenced by the four images). They then looked for patterns among the 30º-60º-90º triangles, as well as the 45º-45º-90º triangles. (1) In the lesson leading up to this exit slip, the class used the Pythagorean theorem to find an unknown third side length (when given the first two).