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Sunday, October 23, 2011

Triangle Centers

I'm attempting to incorporate triangle centers with constructions. locus, and parts of coordinate geometry, because they all go together anyway. This is one of the things I love about Geometry - there are many topics, but they all relate to each other.

Some of my questions are exact replicas of those found in Exeter Math Book 2. I'll attach all the documents I used, and I know they are imperfect. But this is the basic progression:

Incenter
Where is a point that is equidistant from all sides? Conjecture that there is such a point on Nspire.



Figure out how to construct it by pouring salt on triangles.



Circumcenter
After doing some problems to review that there is such a thing as the Pythagorean Theorem...Where is a point that is equidistant from all vertices? Use coordinate geometry to see that there is such a locus of points on the coordinate grid.


Centroid
After a detour into deriving the midpoint formula... Where is a point that is the center of gravity in a triangle? Use area and coordinate geometry to dissect a triangle into two equal areas.



Demonstrate center of gravity by demonstrating that the triangle will balance on that point.



Where is the intersection of the altitudes? This point is not interesting, so get through it as quickly as possible. But take the opportunity to teach what an altitude is.

Then there's an Nspire document with a summative review sheet so that they can keep straight the four different, confusing kinds of triangle centers.

Next we are going to do the famous locus scavenger hunt, after a day of basic locus notes, which will allow me to basically skip the whole pointless locus unit altogether.

And then I think I am going to have the children make their own locus-based scavenger hunt for the towns of Fayetteville and Manlius, for the fun, and for the backwards learning. Though I don't happen to have those documents prepared yet.

12 comments:

  1. Are you going to mention the Euler Line?

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  2. Meh. I don't know. I keep thinking about it, and then thinking, "enh." They'll be like, "oh." But they can't prove it and it won't be that interesting to all but a handful.

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  3. we use geogebra.org for constructions, since it is free & web based

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  4. GeoGebra is great for this, however, I think I read somewhere that it calculates the center of gravity incorrectly on some polygons.

    You have some nice activities there. My students are 18+, so I usually make this into a "sit two by a computer and figure it out"-activity. In the end I provide my version with the five triangle centra in it: Link

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  5. The orthocenter IS interesting! If you have three points and the orthocenter of the three points, any one of the these points is the orthocenter of the other!

    There is also Fagnano's Theorem! Given an acute-angled triangle, find the triangle with the smallest perimeter whose vertices lie on the edges of the given triangle. Turns out this triangle is formed by the feet of the altitudes, and the orthocenter of the given triangle is the incenter of this one!

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  6. I have been thinking about your activity...after we learned the centers and their properties and how to construct them, I would turn these problems around. I would give my students a segment, tell them it was a median, and have them construct the triangle around it. They might need to learn some of the other tools, though, but it was always a good puzzle.

    I would do these kinds of construction puzzles for any of the centers. If this point is the incenter, then construct the triangle around it. If this circle is the circumcircle, then construct the triangle around it (yeah...easy...but they never see it). If this point is a centroid, then construct the triangle around it.

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  7. I solved this exceptionally interesting question concerning orthocentres and circumcentres using the vectors perspective:

    http://www.whitegroupmaths.com/2011/03/sixth-term-examination-papers-step.html (see Q9)

    Just sharing. Hope it is useful. Peace.

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  8. Hi kate, can you explain how that experiment works with the salt. I see the picture but can't quite tell.

    Mike

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    Replies
    1. You just pour on as much salt as it will hold. The salt piles up into a little pyramid with edges on the angle bisectors and the vertex at the incenter

      Delete
  9. Thanks for your response! On another note, i have been using your grading scheme for a few years now and was wondering if you use an online gradebook. I have been using a pencil/paper approach and have been tallying at the end of the quarter to do averages. Thanks again.

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  10. I'm glad someone's using it. :) I've had to modify mine quite a bit for the culture of my new school.

    At my old school, we were using mygradebook.com. Now, I'm using a combo of paper/pencil and excel.

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  11. ok...thanks...just checked and they are going out of business! I would use excel but when I think I would have to manually double the second score on the second time they assess the skill. Unless there is some fancy formula to give excel. Thanks again. mike

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