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SUPER-IMPORTANT INSIGHT FROM CONVO WITH JOHN GOLDEN (08/26/14):
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Please feel free to add as many crazy, fiendishly ambiguous Geometry Talking Points as you like here. On my prep, I will create and publicize other spreadsheets for other course levels!!! - @cheesemonkeysf 08/26/14 7:07 a.m.
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Talking PointQuestionsResponses & GuidanceAlways - Sometimes - Never
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There are as many points in a 2 inch line segment as there are in a line.
Is this question getting into the territory of comparative infinities? (ES)
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A three-legged stool cannot be made rocky.
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A four-legged table can [always] be made stable by adding to one leg.OK, just... help! (ES)I think the answer is yes. Three points define a plane (these are the bottoms of the table legs that hit the floor). If you change the height of a fourth leg, you can get it to also lie on the plane (SS)Sometimes - Three non-linear points define a plane... perhaps thes four legged table is unstable becuase the legs are all linear. The addition of the fifith leg in the same line would not create stability. The stipulation might be that the unstable four-legged table starts with at least two sets of non-linear points. Adding a fifth point that doesn't exist on either line would likely create stability. (LP)
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Points take up no space.This is all predicated on how students think about the words "take up"... If it's area/volume, the answer is that a point has 0 area/volume... But I can see students going back and forth, because how can something which has no area/volume exist? So do points not exist if we say "no"? (I don't think there is a right answer since it feels like an ill defined statement, but that's a good thing for this, methinks) (SS)
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Lines take up no space.(see above -- SS)
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The edge of a circle takes up no space.Help! (ES)(see above -- SS)
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Lines exist in the real world.Help! (ES)This is philosophical. If you ask "can you create a physical line that exists in the real world" the answer is no. You can approximately draw lines. But one might argue that mathematics is a theoretical construct in the real world, so we can come up with the idea of lines, and define them, and in that way they exist in the real world. At least that's how I see this. (SS)
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If you keep on adding sides to a shape, you keep on adding area.Help! (ES)No. You can definitely take a hexagon, for example, and remove one of the sides, and in that missing space put two line segments "facing into the interior" of the hexagon... You've removed a side, added two more segments, and reduced the area. (SS)
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If you keep on adding perimeter to shape, you keep on adding area.Help! (ES)No. See above. You've added perimeter but reduced area. (SS)
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Everything has both perimeter and area."Everything"? (ES)Maybe "every object" would be an improvement? I wrote this question, I've had fun conversations with kids about whether 3D objects have perimeter and area. (MP)
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A person is made up of points and lines.
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A circle is made up of a line.Help! (ES)Poorly worded question. I (maybe) retract. The possibility that I was going for was that a circle is just a line bent into a circle. (MP)
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Two points make a lineIs this getting at the distinction between "defining a line" and "making a line"? In other words, is this TP setting up the ambiguity of two points that are NOT connected into a line? (ES)I suspect this is getting at how we define line... a straight "line segment" or a "curve"... Can we "bend" a linesegment to get a curvey "line piece" and still call it a line? (SS) I wrote this question, and I was aiming for Elizabeth's ambiguity. (MP)
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Interesting! I was actually thinking about whether the two points were NEXT TO each other to "form" a line, as opposed to naming two points on a line as the means of "defining" that line. This is why the ambiguities are so rich. (ES)
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A point and a line have the same area.
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Lines have a perimeter.
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You can chop a line into two lines.
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You can chop a point into two points.
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You can chop a square into two squares.
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If you're flying from New York to Moscow the straight line (great circle) route is over the north pole. Going across western Europe is actually curving way out of the way.
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A line has an infinitely small thickness.Q: Doesn't a line have absolutely NO thickness? (ES)Yes, but don't you think that this will generate a good discussion about the difference between "infinitely small" and "no" thickness? And what really "thickness" means? (SS)
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Yes, that's why I LOVE the statement! I just wasn't sure if *I* was understanding the situation correctly. (ES)
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A trapezoid has at least one pair of parallel sides.The debate on this will be over whether a trap with two parallel sides is still a trap (I would hope) (SS)
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There is no such thing as a perfect square (circle?) in the real world.This is true. Geometry is an idealization (reading the beginning of Lockhart's Measurement is a powerful reminder of this). -- Oh! Good point! (ES)
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A line is uniquely defined by four points.This is tricky. It hinges on how students understand the word "uniquely." Yes, a line is uniquely defined by four points. It is also uniquely defined by three or two points. :)
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Aha! So the important thing here is that a line is not ONLY uniquely defined by four points! Very fiendish! (ES)
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A circle includes both the perimeter and the inside.Q: I thought the circle includes only the circumference, but we can find the area "intercepted" by the circle. Is this an incorrect or limited understanding? (ES)It does only mean the circumference. But I think this question will generate a solid discussion about how important having a common definition (that we all agree upon) is... Because I think a good number of kids coming into geometry will think of a "filled in dot" as a circle. (SS)
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Oh, phew! Good! I was afraid I did not understand the underlying situation. (ES)
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Another Q: could this same distinction apply to triangles, squares, etc.? (ES)
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The sides of a polygon can meet at places other than vertices.There are contradictory definitions. Some define polygons as having sides that don't intersect with each other... others have two classes of polygons (simple and complex... where complex can intersect)... importance of language.
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We live in a three-dimensional world.Who knows?!?! Fun for kids who like to think of time as another dimension, or who have read about string theory or cosmology. (SS)
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Being, myself, a zero-dimensional object, I hesitated to weigh in on this controversy. ;) (ES)
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The paper this is printed on is a plane.Nope. It has a thickness, no matter how small. (SS)
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You cannot have a triangle with more than 180 degrees.They might all agree on this one. Then show them a 90-90-90 triangle on a balloon. Huzzah! I suppose this might not be a good talking point as kids won't know this (SS)
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OK, but sometimes the important thing is that WE are having fun. :) (ES)
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The interior of a sphere and the sphere itself form the "inside" of the sphere; everything else is "outside" the sphere.Q: OK, this one is completely confounding to me. Can someone explain? (ES)Screw it. I was trying to articulate a confounding statement about "inside" and "outside" -- and was trying to get at the notion that perhaps the sphere itself isn't inside or outside (e.g. is the rubber of a balloon "inside" the ballon, "outside" the ballon, "neither", or "both"). But I couldn't articulate the talking point well. Sorry. (SS)
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Don't be sorry! That's why I *LOVED* this one. I just wanted to be able to facilitate the after-party conversation demanding an answer. :) (ES)
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A zero dimensional object does not exist.And yet... here I am. ;) (ES)Existence. Ha. Fun. Also they might not realize a point is a zero-dimensional object, so this could be good (SS)
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Yes! Yes! Yes! This is a brilliant distinction! (ES)
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The perimeter of a circle is a one-dimensional object.Technically, it is a one dimensional object that lives in a two-dimensional world. But I think it's a confusing enough to cause a nice debate (SS)
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Oh, nice! I love that it's confusing. (ES)
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Two points can exist at the same location in space.Q: Um... help? Couldn't two distinct points "coincide" at the same location in space? (ES)I think a single location in space is the same thing as a point. So I think the answer is no. You can call them all the different names you want, but they are always the same "point." I think Jurgensen talks about this early on in the book, but I could just be making this up. (SS)
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But Euclid says that a point is that which has no part, i.e., it has nothing but location. In addition, two distinct lines can "coincide" in the plane or in space. So...? (ES)
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One can measure the length of a line segment with perfect precision.With a ruler, no. Given the endpoints on a coordinate plane, yes. Ha!
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#TEAMDESCARTES :) (ES)
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A polygon with a finite area can have an infinite perimeter.Koch snowflake... But can you call an infinitely many sided figure a polygon? I just think it might be fun to introduce something they think is impossible but is... but I don't know if it's a good talking point or not because I can't really "envision" a good discussion from 9th graders about this (SS)
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You can have a triangle with side lenghts 3, 2, and 7.False. I think some students might say "of course you can have a triangle with ANY side lengths" (if you do this at the beginning of the year) while others will say "I can't picture it". So it could be a good talking point, possibly. (SS)
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A point is a circle with radius zero.
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