Is there a name for when someone thinks they are really bad at something (for example math) and they have learned not to trust their own intuition at all so they make really wild errors by second guessing themselves?
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@futurebird
Physically rotated phone screen so that base of triangle was at x=0 & was 80% certain I was looking at an isosceles if base angles were actually, exactly identical.Re pendulum & isosceles, maybe being too literal but pendulum angle:
1. Could also represent an equilateral triangle.
2. Decreases to zero over time & is no longer an isosceles.I just realized that my default cognitive spatial map is x, y & z coordinates. So pendulum model example would not be intuitive heuristic.
@dahukanna
1 and 2 are totally correct.An equilateral triangle is kind of isosceles triangle. One that works in three ways.
A line is a degenerate isosceles triangle with two right angles and a zero angle at the vertex OR two zero angels at the base and a 180 angle at the vertex. Probably better to call it a line.
Both of these are not always called isosceles but they are the natural extremes.
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@futurebird
Physically rotated phone screen so that base of triangle was at x=0 & was 80% certain I was looking at an isosceles if base angles were actually, exactly identical.Re pendulum & isosceles, maybe being too literal but pendulum angle:
1. Could also represent an equilateral triangle.
2. Decreases to zero over time & is no longer an isosceles.I just realized that my default cognitive spatial map is x, y & z coordinates. So pendulum model example would not be intuitive heuristic.
@futurebird
I interpreted “pendulum could swing all the way around it would make a circle.” On z-axis, not y-axis -
@futurebird
Skimming through is showing me that clearly I'm in the minority of your sample, but nonetheless:The three letter structure is what I grew up with, (but also/and so) was the easiest way to teach one of my kids. Line segments were intuitive to her, meaning the three letter system gave her all the angles on a figure without 'extra' labels. "Just follow the lines".
As a tangent to the thread, kudos once again to the effort you expend on your students.
The three letters eg ∠ABC where B is the vertex are also what I learned and I hated the numbers when I first saw them. But, from experience I find they just work better. My little grumble about "but numbers are for measuring" and also "that's not how I learned it" are something I've gotten past.
I go with what helps the most students get it. Hence I don't use greek letters with ninth graders. Learning the new symbols at the same time was too much for some.
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The three letters eg ∠ABC where B is the vertex are also what I learned and I hated the numbers when I first saw them. But, from experience I find they just work better. My little grumble about "but numbers are for measuring" and also "that's not how I learned it" are something I've gotten past.
I go with what helps the most students get it. Hence I don't use greek letters with ninth graders. Learning the new symbols at the same time was too much for some.
Also different populations of students have different needs. So, maybe I would switch if I thought it would work better with a new group.
I make a big deal about introducing the Greek letters in the spring when we start trig. This gives them time to learn how to write them.
So they do learn them by the end of the year? New symbols are a big deal and deserve space. Don't just spring them on people. I do find the numbers a little "janky" from a pure maths lens.
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@futurebird
I interpreted “pendulum could swing all the way around it would make a circle.” On z-axis, not y-axisWell then you are making triangles in a different plane or triangular prisms maybe.
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Also different populations of students have different needs. So, maybe I would switch if I thought it would work better with a new group.
I make a big deal about introducing the Greek letters in the spring when we start trig. This gives them time to learn how to write them.
So they do learn them by the end of the year? New symbols are a big deal and deserve space. Don't just spring them on people. I do find the numbers a little "janky" from a pure maths lens.
@futurebird
Solid thinking across the board, it sounds like you're the kind of teacher we'd all like to have.I'm curious, has your class size ever affected your decision to introduce 'one more thing', (read: Greek letters in this example)
A 15:1 classroom is a different environment than a 30:1.
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@futurebird
Solid thinking across the board, it sounds like you're the kind of teacher we'd all like to have.I'm curious, has your class size ever affected your decision to introduce 'one more thing', (read: Greek letters in this example)
A 15:1 classroom is a different environment than a 30:1.
Absolutely. If I had 30 students I would not do the compass work. And that would be a big loss, but I would not be able to go around the room and help enough of them to hold it correctly, and keeping that many compasses sharp and ready to go is too much work.
Likewise teaching them to sharpen the lead on the compass is too much of a class time-sink. (although it's a very cool skill to have, so it make me sad)
I bring only working compasses with sharp lead to class.
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@Bumblefish @3TomatoesShort @EverydayMoggie
OK pizza good to know.
Anyway. Connecting the center of a circle to two points like that is a great way to create an angle. When you copy an angle you are just cutting two pizza slices that are the same.
If the pizzas are the same size, and the distance between the points on the circle is the same. The angle at the top (center) is the same.
@futurebird @3TomatoesShort @EverydayMoggie But why use a compass to do that? Why not a ruler through the middle?
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Using color to identify angles means you can't use it to show which ones are the same. Which is a great use for color if you don't have color blind students. (I don't at the moment but it's always in the back of my mind. )
@futurebird @jetlagjen ps this is the system of arcs I was talking about, to identify which angles are the same size. It works pretty well until you have a more complicated shape with many angles. Is there a name for this system?
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@futurebird @3TomatoesShort @EverydayMoggie But why use a compass to do that? Why not a ruler through the middle?
@Bumblefish @3TomatoesShort @EverydayMoggie
Rulers are less precise than a compass. You could use a ruler if you wanted. But it will only be as precise as the markings you have made on the ruler.
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@Bumblefish @3TomatoesShort @EverydayMoggie
Rulers are less precise than a compass. You could use a ruler if you wanted. But it will only be as precise as the markings you have made on the ruler.
@futurebird @3TomatoesShort @EverydayMoggie Not measuring anything, just using it to get a straight line. Could use any straight edge.
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@futurebird @3TomatoesShort @EverydayMoggie But why use a compass to do that? Why not a ruler through the middle?
@Bumblefish @3TomatoesShort @EverydayMoggie
This is a very deep mathematical question in a way. Why do we do geometric constructions with a "straight edge" and compass and not a ruler and compass?
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@futurebird @jetlagjen ps this is the system of arcs I was talking about, to identify which angles are the same size. It works pretty well until you have a more complicated shape with many angles. Is there a name for this system?
We use this system, but asking which angles are the same when they are marked like this isn't testing the concept I'm getting at.
It's called "decoration" or "tic marks" and the little marks on the sides of the original triangle I posted are a part of the same system.
I would expect a student to mark the angles like this based on the way the sides are marked. My struggling student would make the wrong two the same.
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@futurebird @3TomatoesShort @EverydayMoggie Not measuring anything, just using it to get a straight line. Could use any straight edge.
@Bumblefish @3TomatoesShort @EverydayMoggie
But how would you get the angle at the center exactly the same with just ruler?
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@Bumblefish @3TomatoesShort @EverydayMoggie
But how would you get the angle at the center exactly the same with just ruler?
@futurebird @3TomatoesShort @EverydayMoggie Wherever the cross happens is the peak of the triangle.
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@futurebird @3TomatoesShort @EverydayMoggie Wherever the cross happens is the peak of the triangle.
@Bumblefish @3TomatoesShort @EverydayMoggie
The peak is the angle at the center, right? I don't see a "cross"
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@Bumblefish @3TomatoesShort @EverydayMoggie
The peak is the angle at the center, right? I don't see a "cross"
@futurebird @3TomatoesShort @EverydayMoggie That’s because you haven’t drawn one and for the life of me I don’t know why. You need the cross to cut the pizza.
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@futurebird @3TomatoesShort @EverydayMoggie That’s because you haven’t drawn one and for the life of me I don’t know why. You need the cross to cut the pizza.
@Bumblefish @3TomatoesShort @EverydayMoggie
OK great. I still don't see how to cut two slices that are the same size from the identical circles without measuring with the ruler or using the compass again.
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@Bumblefish @3TomatoesShort @EverydayMoggie
OK great. I still don't see how to cut two slices that are the same size from the identical circles without measuring with the ruler or using the compass again.
@futurebird @3TomatoesShort @EverydayMoggie The opposite one will always be the same size.
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@futurebird @3TomatoesShort @EverydayMoggie The opposite one will always be the same size.
@Bumblefish @3TomatoesShort @EverydayMoggie
Absolutely.
But if you want to copy an angle to a new location that won't help.
