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Friday, November 30, 2012

My Centroids Lesson Keeps Stalling Out Right Here

and I don't know how to fix it.

The kids and I literally stared at each other for ten minutes over this. I wanted to take them all outside and drown them in the pool. (I know it's not their fault, though, obviously.)

I don't have any other good way to come at this, though. Other options for asking relevant questions seem too ambiguous for this age group. Applications of triangle centroid are thin on the ground, or at least I haven't thought of any yet.



  1. So you're saying drowning them is not an option?

    I love this activity, though. What do you think was their biggest hangup over it?

  2. Their biggest hangup is they still default to procedures they know rather than understanding the problem.

    The first thing most of them did was construct an angle bisector. Why? Because it seemed reasonable that it would divide the area equally, and it's an easy construction. None (none!) of them noticed that it didn't actually work until I pointed out a few problems with that solution. That's when the staring contest began. And the homicidal ideation.

  3. If their hang-up is that the angle bisector didn’t work, maybe it is a perseverance issue. What do you think would happen if you just said something like, “Well, I guess that didn’t work. Why did you think it would?” (Insert meaningful student comments here.) “Hmmm, those are good thoughts. Maybe your approach is correct but you just chose the wrong line. I mean after all, we have learned about angle bisectors, but also perpendicular bisectors, medians, altitudes, and midsegments. Perhaps we should try one of those.”

    If you kids are rewards motivated, maybe offer a reward to the group that finds the right method first, or better yet, to the group that can explain/prove why their method works. I suppose at some point you’ll need to consider whether or not you have enough time to carry something like this all the way to conclusion.

  4. Those are good suggestions. Thank you. And, you are spot on, perseverance is a big part of it. How to help them grow from "I tried something and it didn't work, oh well I must be stupid, I give up" to "I tried something and it didn't work, what else can I try" has been a challenge for me with this group.

    Thing is, they don't know what a median is yet - that's the point of this activity. :) I should probably have said what I ended up doing, which was focusing on the area calculation. What could we change in here, so that we end up with 7.5? That kind of eventually broke the stalemate, but I felt like I had to give too much away.

  5. Maybe showing another graph with 3 triangles would help. Triangle 1: base=3 height=5 and show area. Triangle 2: base=3 height = 5 and show area. Triangle 3: base=6 height=5 and show area. Maybe draw the triangles in such a way that they suggest triangles 1 and 2 are just triangle 3 cut in half. But don't point it out. Maybe that will prime them for the graph you showed. (A side note, maybe change the scale of the 3 triangles where the area of the big triangle is a factor of 2...just so we play with whole numbers.)

  6. I'm going to be annoying and talk about motivation.

    Is there a reason for them to care about the center? I don't mean giant big picture motivation, but just something immediate and tangible.

    When working with MS geometry students, my default mode is to make a thin/weak game that everyone can play on the way in, where knowing who won involves the definition/construction I'm aiming for.

    (and, yes,receives TEH PRIZEZ!)

    Since this is ActivePrompt week, I made these two:

    I probably wouldn't use that the first time through. Last time I taught geometry had a great set of colored dot magnets from IKEA that would stick to the board if you threw them in a gentle arc. I'd use something like that and offer a prize to the CLOSEST to the center. That way there's some extra distance from a simple "do I know the math answer?"

    I'd hope that "who won?" would be a compelling question. If not, I'd publicly choose a "winner" who was clearly off, and let the middle school rage at such gross injustice push us into the exploration.

  7. Off the cuff, non-teacher perspective, not guaranteed to be helpful thoughts:

    - The word dissecting sounds an awful lot like bisecting. Not sure if it matters, but they were obviously anchored on that concept.

    - The first sentence in #4 is a non sequiter for completing its task. It seems more relevant as a hint/reminder when working through #5.

    - The fact that these 3 examples were chosen (acute, right, obtuse) might be a red herring. It almost suggests that there are differences between these three classes of triangles for this problem.

    - The task for #5 is "come up with an efficient method for...". womp womp. Yeah, right after I fill out my TPS reports. Maybe a little better: "There are many ways to divide a triangle into two sections. Can you split each triangle into two sections so that you know the areas are equal?

  8. Andrew I'm normally on board with the "first guess/estimate" approach as a way of motivating the new thing, and use it regularly, but I don't know how to make it workable here. By geometric center do you mean centroid/center of gravity? Or would any of the other three triangle centers be an appropriate response? I am afraid opening that question to this group at this time would introduce an annoying amount of ambiguity, as they already know about incenter, circumcenter, and orthocenter. Unless the question were very specifically worded, like, at what point could you balance a triangle flat? or something. Thoughts?

  9. When an activity has a "trick" bit (that is, their gut reaction answer won't work) I usually hang together as a whole class through that part before I set them loose. Somehow it's less frustrating that way; sure-it-will-work but doesn't with their first stab working on their own can be overly discouraging.

  10. Hi Will! Those were very helpful, actually, thank you. I love me a nice non-teacher perspective.

    I hear you on dissect vs bisect, and how that word choice may have led them to try the angle bisector. Nice observation.

    The first sentence in #4 is intended to remind them that by base and height, I mean the base and height they are accustomed to from area calculations. And also set them up for 5, yes.

    I will have to think about the utility of drawing right, acute, and obtuse. The point is more that the method turns out to be the /same/ for all three. Even if it seems at first like one might need a different method for each.

    I do like your wording for 5 better. I get a little carried away with economy of words sometimes.

    Lots to think about! Thanks.

  11. Jumping in here with something that might seem comical, but also might help. The answer here is: SPLICE IT! It's like an Active Prompt as mentioned above, but it's hidden in an APP game. An app that I've recently downloaded because of one of the people I follow on twitter kind of recommended it. Some ex-pat living in Argentina.

    Kate, how about playing a round or twelve of Splice It!? Aim for 50% and 50%, and while you're at it, play a few non triangle episodes.

    Si? Que te parese che?

  12. It is very much like Slice It, Nico. :-) I don't mean to sound like I'm poo poo ing every idea, but having a class all play Slice It would be logistically difficult. I know there are ways to show my ipad through the projector, I just don't know if it's worth the hassle for this.

  13. center of mass concept ..... it is the balance point for a triangular pinwheel..easiest rotation

  14. If you're looking for motivation about how/why a triangle might need to be split into two equal areas, why not go to the dessert table? I remember arguing with my brother about the last piece of pie and how it could be cut equally- we usually ended up going right down the middle, but there are lots of other ways to slice it and still get equal areas for each person. This assumes that the crust has the same appeal as the center, which is debatable, but it has real world context if that's something you're looking for.

  15. I had students map three locations on Google maps like school, the mall, and a friend's house and then print them off and find the 3 triangle centers with a ruler and protractor. They then had to decide where they should build a house and why and also which was equidistant from all 3 locations.

  16. I have a theory that students are thrown when we ask for half of something, when the formula for "something" already has a factor of 1/2.

    So by asking for half the area you want the students to visualize that they need (1/2)b [or I suppose (1/2)h] but they look at the formula and see there's already a factor of 1/2 [(1/2)bh] and they have trouble seeing past that.

    I notice it in physics when I ask about half the kinetic energy or something similar.

  17. @Kate, I can understand why you think you are giving too much away with the approach you ended up taking, but I think you can give the students perspective that will allow them to generalize it in the future. In this situation, the general problem would be something like "How do you use a single line to divide a triangle into two sub-triangles of equal area?". One might try some lines and notice that in order to get two sub-triangles the line must pass through a vertex, which narrows down the question. Then one might try a few sample triangles such as the ones you used, and figure out how to get triangles with a specific sub-area given a known area of the larger triangle.

    I think the concern about motivation is important, but we should also keep in mind that problem-solving does not come naturally to most people, and that teaching kids how to problem-solve is as important as giving them motivation to solve a problem.

    @Chris: There's a classic puzzle involving how to divide a square cake into n slices of equal area and equal crust length.

  18. Different approach:

    Draw a triangle on the board, with a flat base. Have the students calculate the area as a class, so they remember the formula.

    Now draw a line from the top corner to the base. Make sure it's *not* a median. Ask them - which of these two new triangles is bigger?

    They should be able to tell you which is bigger, and figure out that it's because the base is longer.

    At this point, rub out the line and tell them - in their own books or on personal whiteboards - to figure out to split the triangle so the areas are exactly equal.

    I think with this set-up, they'll be much better equipped to come up with the answer.

  19. I see that centroid is being equated here with splitting the triangle into equal areas. Did you know that this isn't always true for other sorts of shapes?

  20. Scott - cool observation.

    Hao - yes - that's the sort of thing I wish I knew how to foster the students' coming up with, not just me showing them.

    Alex - that approach makes the solution rather obvious, no? I agree it would work. Just not comfortable with how little they'd have to struggle for resolution.

    Sue - I'm not sure off the top of my head the definition of or method for finding centroids for n-gons with n>3. I'm sure I knew it at one time and could work it out again. Center of gravity was a big deal in at least two engineering courses I sat through 15 years ago. :)

  21. Jason - yes, I agree with you now. In my final section with this problem set, we dispensed with the angle bisector guess as a group before turning them loose. I just didn't anticipate that response ahead of time! Frustrating. You'd think I'd know better by now.

  22. If I am asked to split the right triangle into two pieces, my eye is drawn to the lower right vertex - not what you want. Also, bisecting the angle comes pretty close to working. When I am asked to split the acutte triangle into two pieces, my eye is drawn to the "top" vertex - which is what you want.

    Maybe provide the first triangle (acute) where bisecting the angle clearly won't work. For example: (0,0) (1,3) & (10,0).

    As a follow up ask them to draw the right & obtuse triangles with same base & heights.

  23. Hm, the kids eyes were drawn right to that right angle, they split it into 2 45''s - looked close enough. They picked the angle in each triangle to bisect that looked the most promising for generating equal areas. But yes, maybe a more specific triangle where no bisected angle would look reasonable is called for here.

  24. Did they generally pick the biggest angle to bisect?

  25. Yes but I dont think that's it. They picked the one that looked closest to the vertex of an almost isosceles triangle.

  26. Can you make your height 4 or 8 instead of 5? Some things will happen then:

    - the triangles will be less isosceles-ish
    - as a consequence the angle bisector will maybe be more obviously not the right answer
    - now from any vertex I can easily eyeball (on my graph paper) where I could draw a line to the opposite side so that my new base is 3 or my new height is 2 (for h=4) or 4 (for h=8) (not that finding 2.5 was really hard but making it evenly divisible by 2 might help)

    One more idea: Reword Q5 so that they have to come up with a method for easily cutting the triangles "in half" ... it might force a discussion like "Do you mean half the AREA?"

  27. Yep yep yep. Already changed it to 8. You know for next time. :)

  28. For me, centroid as center of mass is the most interesting aspect to explore. And there's an fun way to find it for planar figures.

    Take a physical triangle (cardboard, foamboard, etc); affix it to a board with a single pushpin, placed anywhere, and let it freely hang; drop a plum line from the point of suspension and trace that that line on the triangle. Repeat, using a different starting point.

    The two lines intersect at the centroid of the figure. Compare, discuss, prove, explore with new figures.

    A good way to decorate the room is to then suspend all the objects from their centroids, as they should balance.

  29. Following up on Patrick's idea, once your students know how to find the centroid, you might have them make triangles out of thick cardboard (the material can't be too lightweight) and then find the centroid on their triangle. Make a nice fat dot at the centroid, and then have them draw a couple of concentric circles around it, something like a bull's-eye. On the other side of the triangle, have them pick a point that is clearly *not* the centroid and do the same thing.

    If you toss one of these triangles (with some spin on it) in a nice arc, say across the front of your classroom, what the students will see will vary depending on which side you have facing them. If they are looking at the non-centroid side (I would do this first), they'll just see a blurry triangle spinning around as it crosses the room. Try it again with the centroid side facing them and they will see the bull's-eye follow a nice smooth parabola and hardly notice what the triangle is doing. The difference is pretty striking. (It also works really nicely with crazier shapes, like a guitar or a whale or something, but that's somewhat off-topic I dare say.)

    You could start with this instead, as sort of a magic trick. "I can make the bull's-eye stay in focus when I toss this shape across the room, or I can make it blurry." Amaze them with your ability to make it happen the opposite way from what they predict every time, until they figure out that you're showing them different sides. (This works best if the "wrong" bull's-eye is not tremendously far from the centroid, of course, so the students hopefully won't immediately catch that the two sides are different. Now, I don't know how that leads your students to figuring out that the center point of the side that works is the intersection of the medians, so this method may not be useful for your purposes.

  30. Similar to Alex's approach, I'd start off with an already drawn triangle and ask them how to exactly double that triangle's area by adding on another adjacent triangle. (You will need to work on the phrasing a bit.) Essentially, get them to do the task backwards, and then give them a new triangle (acute, obtuse, or right) and ask them now to find out how to divide up the area equally.

  31. Maybe the question isn't very good. How about applying the concept of equal area to solve problems instead?

  32. Way too late to this party, but:

    Put a point on the base of the triangle that isn't the midpoint, then draw the two triangles formed. Ask "Which of these two triangles has more area?"

    Then move the point and do it again. Ideally put it somewhere that looks suspicious. If you want, you could use an angle bisector -- my personal favorite is the angle bisector to the "24" side of a 7-24-25 triangle, which is 5.25 units along the 24 side. (Sorry, this needs a picture, and sorry that I have a favorite angle bisector.)

    Do it one more time, this time with a new triangle where you label only the base and not the height, hoping kids may notice that it only depends on the bases chosen.

    Then ask "where can we put the point so that both areas are equal?" and it should work.

    No homicides!

  33. Hello! I'm new to this EduBlogoTwitterSphereUniverse thing. (How long do I get to play that card?)

    I just wanted to say "hi" and that I really like this investigation.

    A question: What might be some contexts to embed this problem into? Baking and splitting triangular shaped food in half is all I can think of.


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