@3TomatoesShort @EverydayMoggie @futurebird I was going to suggest paper modeling. I can’t calculate anything to save my life, but give me a concrete thing I can shape with my hands and things start to make sense. You mentioned compasses. Nobody has reliably convinced me they can do anything except make circles. The whole thing where you use them to make angles is deep magic that only wizards can perform.

@futurebird @3TomatoesShort @EverydayMoggie However if you had asked me to cut it like a pizza we’d have been fine.

@Bumblefish @3TomatoesShort @EverydayMoggie @futurebird
Using a compass to draw an isosceles triangle is a step too far for me.
I’d start with physical paper, then model different scales in a digital tool and then translate that to angle degrees + line proportions.
I’ve repeated this process enough that I can “tear” a flattened 3-D cube out of a sheet of paper in capital T or lower-case t configurations.

@pthane

I'm trying to understand this.

If I flip a board "l-r" will the result be different than "r-l" ?

The way I'm picturing this the answer would be "no" right?

@futurebird
Yeah l-r or r-l is the same but many kids would flip top-bottom or rotate.

@dahukanna @Bumblefish @3TomatoesShort @EverydayMoggie

That sounds like so much work to me.

If you want someone with two parts that are the same length that start at the same point they have to be on a circle. No need to measure anything.

One of the harder things to accept as a teacher is that some kids will just never like math. That's fine though. I can cope.

What everyone *can* like (at least) is getting the right answer and knowing it's right. Knowing that yourself not needing anyone to tell you.

So I'm hoping we can at least get to that stage.

Right now I think she thinks I'm just making up geometry as I go and it's basically Calvinball.

@futurebird I‘ve no cognitive intuition for geometry, never been able to tell left from right, am a visual learner & can imagine an Apple+smell it+hear crunch when I bite into it, …
So validating a flat planer isosceles triangle needs deliberate cognitive effort, else i’m guessing based on “way wind blows” or my mood.

As a digital Designer, developed systems that enable expressing visual scene by literally, mentally mathematically raytracing scene. POV ray was “heaven”- camera, lights, action!

@jetlagjen

We use the numbers consistently because I find using three letters is more confusing for most students.

eg ∠ABC = ∠BCA

vs. ∠3 = ∠2

Which do you find less confusing?

We could also do ∠B = ∠C but naming angles by single letter is a bad habit since there can be multiple angles at a vertex.

@futurebird My experience as a tutor and a teaching assistant in math-heavy subjects was that, while it's not everyone, there are a lot of kids who basically get the material but who need someone to occasionally encourage them that they're on the right track when they seize up and second-guess themselves.

The second-guessing can be useful in more advanced study if it's targeted appropriately. If I didn't know for sure that that diagram was supposed to represent Euclidean geometry, I *couldn't* actually say that the congruent angles followed from the symmetry of the triangle without further assumptions. (Maybe the metric space it's in doesn't share that symmetry!) Then, that student's suspicion of intuition would be warranted. It's just misfiring in situations where it's not.

@futurebird @jetlagjen I like the system where you have one arc in one angle then two arcs, etc., and an equal number of arcs for angles that are the same size.

But I guess if you are asking the student which angles have the same size, you can't use arcs in the problem.

Using letters to represent the line segments is the standard? I agree, naming angles by the combination of letters was confusing.

What if you use the dot of color to identify each angle? I guess that wouldn't photocopy well..

@dahukanna

That's interesting. I *also* have no sense of left or right. "you injured your left leg which one still hurts?"

But, this has always been a secret advantage in mathematics since seeing patterns backwards happens as fast as forwards.

I don't mind doing measurements but here is another shot at avoiding it:

An isosceles triangle is a pendulum. The distances are the same because it's the same string. If a pendulum could swing all the way around it would make a circle.

@jetlagjen

We use the numbers consistently because I find using three letters is more confusing for most students.

eg ∠ABC = ∠BCA

vs. ∠3 = ∠2

Which do you find less confusing?

We could also do ∠B = ∠C but naming angles by single letter is a bad habit since there can be multiple angles at a vertex.

@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
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
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.

@jetlagjen

@phpete @jetlagjen

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.

@futurebird
I interpreted “pendulum could swing all the way around it would make a circle.” On z-axis, not y-axis

@phpete @jetlagjen

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.

@jetlagjen