Ep 202: You Could Have Understood Trigonometry (Radian Measure)

April 30, 2024 Pam Harris Episode 202
Ep 202: You Could Have Understood Trigonometry (Radian Measure)
Math is Figure-Out-Able with Pam Harris
Math is Figure-Out-Able with Pam Harris
Ep 202: You Could Have Understood Trigonometry (Radian Measure)
Apr 30, 2024 Episode 202
Pam Harris

All students can learn more math when it is taught as figure-out-able! In this episode Pam and Kim wade into trigonometry and make sense of it as they go.
Talking Points:

Making sense of Pi and angle measure of a circle

• Defining radians
• Understanding the unit circle
• A Problem String using a unit circle
• The importance of modeling
• I Have, You Need with angles

Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC

All students can learn more math when it is taught as figure-out-able! In this episode Pam and Kim wade into trigonometry and make sense of it as they go.
Talking Points:

Making sense of Pi and angle measure of a circle

• Defining radians
• Understanding the unit circle
• A Problem String using a unit circle
• The importance of modeling
• I Have, You Need with angles

Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC

Kim  00:03

Are you notifications off?

Pam  00:05

Yes they are. But I'm ready to start because it's recording. You ready? Okay, here we go. Hey, fellow mathers! Welcome to the podcast where math is Figure-Out-Able! I'm Pam Harris, a former...

Kim  00:17

New. New.

Pam  00:17

Stop. A former mimic or turned mather.

Kim  00:21

And I'm Kim Montague reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:30

We know that algorithms are amazing historic achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:44

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:51

And we invite you to join us to make math more figure-out-able. Kim, that second sentence is really long in this new introduction.

Kim  01:00

It is, we might have to alter it.

Pam  01:01

Yeah, listeners, you can tell we're only on episode 202, and so this is a new introduction that we've talked about. But boy, there's... I might have to pick up that second sentence. I had to take a breath in the middle of that sentence there. Okay.

Kim  01:12

Well, I'm glad you were talking because I coughed a couple of times. Thank you for, Craig.

Pam  01:17

It's allergy season. Oh, (unclear).

Kim  01:19

Apologize.

Pam  01:20

Yeah, listeners, so thanks for joining us while we're both gagging just a slight bit on all the drainage here. If you don't have allergies, you will within three years of moving to Central Texas.

Kim  01:30

Yeah, (unclear).

Pam  01:31

So, welcome, welcome to our delightful. Yeah, alright.

Kim  01:34

Okay, let's get started.

Pam  01:35

There we go.

Kim  01:35

So, I knew that we were going to talk a little bit of high school today, so I actually went looking for a comment. You know, I'm loving reading the ratings and reviews.

Pam  01:36

Oh, my gosh, we love reading.

Kim  01:45

(unclear). Okay, so check this out. You don't even heard this. LoveM314 says, "Awesome ideas. I've taught for 33 years and have many times been frustrated with a lack of numeracy foundation in high school students." Yes. "I Have, You Need is a great place to start." So, anywho. Yeah. Nice Pi reference in the name (unclear).

Pam  02:12

There you go. 314. (unclear).

Kim  02:13

And we agree that numeracy is something that we need to build in all students. And at the high school level, you see the effect of years and years and years of rote memory. I Have, You Need is a super routine for all ages. And we have something to talk about today that's aimed at helping math in high school more figure-out-able. Maybe some new ways to think about I Have, You Need too.

Pam  02:37

Let's do it. Yeah.

Kim  02:38

So, if it's been a minute, and you are not a high school teacher, you might have sweat just a little bit seeing the title of this week's episode. But take a deep breath. We're not going to lead you astray. And you can learn right alongside me. I don't even know what's happening today.

Pam  02:55

Bam. Yeah, alright. So, I planned today, and I think this is going to be a lot of fun.

Kim  02:59

Yeah.

Pam  03:00

One of the funnest, most rewarding things I do is do workshops where I have kindergarten through twelfth grade teachers in the room, and we do something where the first grade teacher goes, "Hey, hey, I just reasoned through that! I just used what I know!" And the high school teacher goes, "Huh. Well, wow. Alright. Like, this is figure-out-able." So, yeah, it's a nice thing. So, here we go. Let's, figure out some high school math.

Kim  03:26

Okay.

Pam  03:26

Kim, do you relate more to snowboarding or skateboarding or maybe even trick bike riding?

Kim  03:34

Neither except I'm planning to go skiing or snowboarding. I'm going to try that. Ski this year. But I'm going to try snowboarding. So, let's go with that. Maybe I'll give in to the (unclear).

Pam  03:44

Well, so you can picture snowboarding.

Kim  03:46

for sure. I lived in Alaska. I know Oh, yeah, yeah, snowboarding.

Pam  03:48

Well, bam. Okay, so picture any one of those. Skateboarding. Snowboarding.

Kim  03:53

Okay.

Pam  03:53

(unclear) bike riding. And if I said, "Alright, they're on the course, and they just..." See, now I don't even know the lingo. "But they just spun a 360!" What would be happening if they spun a 360?

Kim  04:05

Oh, they turned all the way around.

Pam  04:07

Like, keep going. Like, can you describe that some more? Like, turned all the way around.

Kim  04:10

So, they made a full circle.

Pam  04:12

Full circle. Like, if they're facing one direction, and they spun a 360, they're now facing that same direction. But they went all the way around. Yep. What if they spun a 180? What would that look like?

Kim  04:25

They went halfway around the circle, so they're facing the opposite direction.

Pam  04:28

Bam. Okay, so 180 I'm going one direction. If I spin that 180, I'm now the opposite direction. Like, I'm backwards almost. Kind of. Okay, cool. So, we could use some intuition about these angles that kids have kind of learned. Like, if I were to say, you know, look around right now and find a right angle. What might be something, I don't know, handy that you have Kim that you would like, "That's a right angle." Anything?

Kim  04:52

Oh, on the corner of my paper.

Pam  04:53

Yeah, you got a paper Yeah. In fact, sometimes in geometry we would say, Hey, pull out you're right angle checker." And kids would go, "What?" And you're like, "You know the piece of paper (unclear)." That corner, right? That corner's sort of 90. And how does that corner relate to that 180 that we just spun?

Kim  05:08

So, the corner is 90 degrees, and so it's a fourth of the circle.

Pam  05:14

whole circle.

Kim  05:14

Of the angles, mmhm.

Pam  05:14

Of the Of the 360. And how does it relate to 180.

Kim  05:18

Half.

Pam  05:19

Just like half of that. So, if we spun that 180, and we're now backwards. Like, we've kind of turned backwards. If we only spun 90, we'd kind of just turn a quarter circle. Yeah? Okay, cool. So, let's think for a second about that 360. If we kind of traced the path that we traveled, we could sort of... I think you even said circle, right? We could think about kind of the whole circle. We start here. We spin all the way around. If we talked about walking that circle. Like, if an ant kind of walked around that circle, then that would be the circumference of the circle. We named that distance around the circle. If I were to measure it, you know like, put a piece of yarn or string around the circle, and then I were to pick up that yarn or string, and I would lay it along a ruler, I could measure the circumference of that circle. That would be kind of how far I'd have to walk around. Like, if an ant was walking around that circle, it would be kind of how far that ant would have to walk around, right? We call that the circumference. So, one of the things that I'm not going to develop today is the formula or how you could find the circumference if you were given the radius of a circle. So, I'm just going to tell you today just for fun. So, you could find the circumference of a circle by multiplying 2 Pi times that radius. So, if you knew the radius of the circle, then you could multiply it by 2 Pi. I'm not going to develop that today. We could do that some other time. But so, if I were doing today's lesson, I would expect that students had some experience with 2 Pi times r, times radius, to find the circumference of circle. Are we good? Okay, cool. So, what if... Everybody just picture that circle that I'm kind of talking about. And you're kind of thinking about an ant starts in one place, and he's kind of walking around that circle. And so, depending on how long the radius is, the ant would have a shorter distance to walk or a longer distance or walk. Because like a tiny circle is going to have a short circumference, right? And a big long, big, big huge circle is going to have a long radius. The ant is going to have to walk around that whole circle. And that's a sort of a long circumference. Well, what if I said today the radius of that circle is 1. And sometimes we call 1 a unit, and so I might call this a unit circle just because like the radius is 1.

Kim  07:32

Okay.

Pam  07:32

So, if the radius is one, how far does the ant have to walk?

Kim  07:37

If the radius is 1?

Pam  07:38

Yeah. The radius of the circle is 1. How far?

Kim  07:40

It's going all the way around?

Pam  07:41

All the way around. Yep. How you doing there, Kim?

Kim  07:46

Yeah, I don't know.

Pam  07:49

So, okay, what are you thinking? Can you share what's going on your head?

Kim  07:53

I mean, I feel like you're... Did you tell me anything other than it's 360 in a circle?

Pam  07:59

That the circumference we could find by multiplying 2 Pi times the radius.

Kim  08:04

Oh, sorry.

Pam  08:06

That would be another thing. That would be another thing that we...

Kim  08:08

I was like, "Wait, I feel like I don't..." Okay, so 2 Pi.

Pam  08:13

So, why is it 2 Pi?

Kim  08:14

Because 2 Pi times the radius.

Pam  08:18

You're doing great.

Kim  08:18

The raidus is 1.

Pam  08:19

Alright, if the radius is 1. Okay. So, the circumference for if it's a unit circle, if the radius is 1, then the ant has to walk 2 Pi. And Pi is about. Hey, we just said that it's about 3.14. So, 2 Pi would be about? Pi is... How are you doing, Kim?

Kim  08:40

Yeah, yeah. So...

Pam  08:42

Pi is about 3.14. And so,  2 Pi.

Kim  08:45

It's like 6 something. 6.28.

Pam  08:47

6 something or about 6.28. There we go. Okay.

Kim  08:51

Yeah.

Pam  08:51

So, the poor ant would, you know like, be walking for a hot minute. 6.28. Whatever the units were measuring in. About 6.28 units around the circle. Okay, so interesting that there was some work done in mathematics, where people began to realize that there were these periodic things that happened. And one of the ways that they could kind of conceive of modeling these periodic things was to kind of let the skateboarder... Or you're going to go snowboarding. The snowboarder keeps spinning. What do I mean by that? I mean like if you were to start facing one direction, right? And you just said that if you were to spin a 360, then you would spin all the way around. Well, that's kind of the circle. And so, you could picture that. You could be like, "Okay, I'm going to spin all the way around, and now I'm back to where I begin." But what if I keep spinning? Well, I wouldn't call that... I mean, I could call it two 360s. Unless I kind of did it all at once, right? If I literally spun all the way around and kept spinning, then I could say, "Well, I've spun more than 360. And they realized as they started seeing periodic things happening. For example, sound waves. We can model that there are periodic. They do something over and over and over again. The sun rises and sets in a periodic fashion. It rises and it sets. And then, it rises, and then it sets. And then it rises... And temperature. Temperature goes up, and it goes down. And it goes up, and it goes down. And we can actually track temperature over time. And if we were to plot temperature over time, we could see this sort of wavy thing happening. And we call that kind of this periodic behavior. And mathematicians wanted to model that periodic behavior. And they realized that it would be super helpful if they were able to let the angle measure, as they're sort of noticing how many times you're kind of spinning around that circle, that if they let it be the distance that that turtle... Turtle. I don't know why I say turtle. I was saying ant earlier. Whatever is walking around the circle. If we let that ant walk around the circle, we can actually kind of equate the angle measure with how far the ant walked. I don't know if that makes any sense. But in other words, if I said hey, Kim, the ant just walked about 6.28 units. What angle would that be equated to?

Kim  09:43

360.

Pam  09:53

360. So, we could say... Alright, so sort of this 2 Pi. We could kind of say if the ant walked 2 Pi, we could say that was like the angle of 360. Well, what if the ant only... So, if you were... This is the downfall of an audio podcast, ya'll, but I've literally just drew on my paper a coordinate axis. So, I just drew an x-axis and a y-axis, and they're crossing each other, you know like, normal. And then, I'm going to say that the skateboarder or whatever is sort of facing... How do I describe this? On the x-axis, it's kind of facing to the right. It's sort of that, you know like, between the first quadrant and the fourth quadrant. This is hard to describe. So, if you kind of... If the ant is kind of sitting on that first quadrant. And let's say I drew a circle, and the ant is going to walk around that circle. Right now, my ant... This is a unit circle, right? So, it's at the point 1,0. Like, can you find the point 1,0?

Kim  12:11

Mmhm.

Pam  12:12

Okay, so if the 1,0, I'm over 1. I'm not up or down. I'm just like sitting on that axis. The ant is sitting there. And you just said if the ant walked all the way around the circle, then you're saying I would have spun a 360. I just kind of repeated what you said.

Kim  12:26

Yeah.

Pam  12:26

Okay. So, I just wanted to let you know what's on my paper. What's on my paper is a circle that is centered around the origin. And I've got that point 1,0. And I've drawn that circle from that point all the way around. And if I go all the way around, that's either walking a distance of 2 Pi or it's an angle of 360.

Both Pam and Kim  12:47

Yeah.

Pam  12:47

Okay, cool. So, the mathematician said, "You know what? We could just kind of equate those. We're just going to say that the radius is 1. And so instead of saying the ant walks around 2 Pi, we're going to say that the angle measure. That angle measure of going all the way around and ending up back where you were, walking all the way around 2 Pi. We're just going to call them kind of the same. We're going to equate, and we're going to say the angle measure is either 360 or we're going to also say it's 2 Pi. So, we're now kind of talking about an angle measure. Cool. So, what if that ant only walked halfway around the circle? What if it only walked pi around the circle? About 3.14 of those units. What angle was that again?

Kim  13:28

180.

Pam  13:28

That was 180. So, on my paper, I've just drawn like half the circle. I have a circle drawn, but now I've kind of sort of drawn over it as if it's kind of an angle. Like, you know you draw that little angle like arrow thing? It's kind of like... Yeah. So, I've kind of drawn this angle arrow thing over there, and I'm saying. Okay, so, that half circle, or sometimes we call that a straight angle, we could call that 180 degrees like you said, but we could also call that Pi. Because you just said like... How do you know it's Pi by the way? Did I say it was?

Kim  14:00

Yeah, it was You told me (unclear)

Pam  14:00

Did I tell you it was Pi? Well, shoot, I wish I would have done the other. If you've gone that 180 degrees, is there a way that you could reason it was Pi without me telling you it was Pi?

Kim  14:02

halfway around the circle. And if the whole circle is 2 Pi, then 1 Pi is halfway around.

Pam  14:13

Halfway around. Okay, that totally makes sense. What if we only did that 90 degrees? So, now I'm kind of drawing just a little loop that goes sort of up to the y-axis. It's kind of that 90 degree. What would that be in terms of Pi?

Kim  14:28

One-half Pi.

Pam  14:30

Because?

Kim  14:32

Because if a whole Pi is 180, then half Pi would be 90.

Pam  14:35

Nice. And so, I now have on my paper written 2 Pi equals 360 degrees. Pi equals 180 degrees. Pi divided by 2 equals 90 degrees. And I could have written one-half Pi or Pi divided by two, either one, is 90 degrees. Cool. Let's just keep going. What if I wanted to know a fourth of Pi? How many degrees would that be? So, now I'm kind of switching a little bit. I was doing 3s earlier. Now, yeah. Okay, so a fourth of Pi

Kim  15:00

Fourth of Pi is 45 degrees because half of Pi was 90 degrees.

Pam  15:05

Okay, so if half a Pi is 90 degrees, then a fourth of Pi would be half of that. And half of 90 is 45. So, I've just drawn on my paper kind of a 45 degree line from the origin. And I've drawn that little angle symbol. And I've kind of written out to the side that that's a fourth of Pi. I've also written on my paper on the left hand side, I've got 2 Pi equals 360, Pi equals 180, half of Pi equals 90, and a fourth of Pi equals 45. Oh, and each of those numbers had a degree by them. 360 degrees. That little degree symbol. So, fourth of Pi was 45 degrees. Cool. Next question. What about three-fourths of Pi? Or sometimes we call that three Pi fourths. I don't know why we do that. So, I'm going to call it three-fourths pi. What's the angle that's associated with three-fourths Pi? Any ideas?

Kim  15:05

Yeah, if a fourth a Pi was 45 degrees, then 3 times as much would be three-fourths Pi. And that's 135 degrees.

Pam  16:02

And where would that be if I were to... So, let me write that down. 135 degrees. Okay. So, 90. Well, I said 90. That's how I was checking your answer. But you actually multiplied 45 times 3 to get 135. I thought 90 and 45. Either way that would kind of work. Cool.  Where would you, where would I, where would the ant be on that circle? Like, what would the angle be? What would it look like? This is hard to do visually.

Kim  16:17

Yeah.  Yeah. I'm not drawing anything, so give me a second.

Pam  16:30

Ah, because you're probably watching what I was drawing. Which...

Kim  16:34

I'm not watching. I'm not listening to you.

Pam  16:35

I know, but that's what would have happened in class, right? I wouldn't have expected students to be drawing anything either. I just would have been drawing it on the board. But yeah, so.

Kim  16:44

So, okay. So, I just drew a circle, and I really quickly put 90 degrees is half of Pi. And I put, Pi is 180. So, it's going to be between those. So, it's going to be between... Yeah. between half Pi and 1 Pi.

Pam  17:02

And between 90 degrees and 180 degrees. So, it's kind of that 45 degree angle right there in the second quadrant. Could be a way to talk about that. Cool. If I was in class, I might say to anybody, "Did anybody use the Pi?" And like Pi would be too much, right? Because three-fourths. We were finding three-fourths of Pi. I might ask. Could you think, Kim about like using Pi to find three,-fourths? Because you used a fourth of Pi, and you scaled that up times 3. Brilliant, brilliant. Love that, love that. But I'm wondering if you could use Pi also to get to three-fourths of Pi?

Kim  17:36

Yeah, I could do Pi was 180. But (unclear) go back a forth because I only need three-fourths of Pi.

Pam  17:42

Nice.

Kim  17:42

So, back 45 degrees.

Pam  17:44

From that 180. And that was also 135. Cool, nice. Alright, good. Let's get even a little bit weirder. What about seven-fourths of Pi. Where would that be and how many degrees? You could do either. In either order. Degrees first, where it is?

Kim  17:59

I'm going to go back from the 360 because 2 Pi would be eight-fourths Pi.

Pam  18:05

Nice.

Kim  18:06

And that's 360. So, I'm going to go back and forth, which we said was 45 degrees. So, that's 315 degrees.

Pam  18:17

So, did you think about... When you said 45, you took a deep breath. 45 degrees. You just subtracted that from the 360. 360 minus 45 is a 315. Cool. And where? I'm sorry. Did you already say where it was?

Kim  18:27

I did not. It's between. So, around my circle. Like on a clock, the 9:00 would be 270 degrees.

Pam  18:38

The 9:00?

Kim  18:39

Like, if I've drawn a circle. I didn't draw it on a coordinate grid, so I drew a circle on my paper.

Pam  18:44

Okay.

Kim  18:45

And 6:00 would be 180. I'm just trying to give a visual from where I am. 9:00 would be 270. Yeah. So, I'm like between 9:00 and 12:00.

Pam  18:56

So, you know what? Assuming that you could see my paper, what I didn't tell you was I need your angles to all start at 3:00. How weird is that?

Kim  19:06

Oh, okay. My 360s at the top. Okay.

Pam  19:09

Yeah. Do you mind redrawing?

Kim  19:11

No wonder you were confused.

Pam  19:12

I mean, I followed you. But yeah. So, standard position is that the angles all start... If we're looking at a clock, they start at 3:00. That's where sort of a 0 angle would be.

Kim  19:23

I'm going to rotate.

Pam  19:25

Yeah, if you don't mind. So, let me actually walk everybody listening because sorry, everyone. Let me. I should have probably started there. If the ant starts at 3:00. That's the 0 angle, and he starts walking, then the 45 degrees would literally kind of be 45 up from that. And so, it's going to be... Golly, what is that? Pointing at the 1...

Both Pam and Kim  19:45

1 and a 1/2

Pam  19:46

o'clock?

Kim  19:47

Yeah.

Pam  19:47

In between the 1 and 2.

Kim  19:48

We call that 1:30.

Pam  19:51

The 1 and a 1/2 o'clock. The 1:30. That's funny.  And then, if the ant had gone to a half a Pi, then that would be up at the 12.

Kim  20:01

Yeah.

Pam  20:01

And then, if the ant keeps going at the 180 degrees, or Pi, that would be over at the 9:00.

Kim  20:08

Mmhm.

Pam  20:09

Okay, cool. So, where would you be now if you were at seven-fourths of Pi?

Kim  20:14

I just rotated my paper. So, it would be at... So, if 360 was at the 3:00.

Pam  20:20

Yep.

Kim  20:21

Then 315 would be... Gosh, how do I say this? At the 1 and a 1/2 o'clock.

Pam  20:28

Well, not 1 and a 1/2, right? Wouldn't it be 4 and a 1/2? Is it the 4:30?  I mean, it's both.

Kim  20:31

No, if my 3... I'm going back to 315 degrees. Oh, that was your 0. You didn't call that 360 at the 3:00. Oh, true. That's true. So, then 315 would be between what a normal clock would be 12:00 and 3:00.

Pam  20:53

So, you're going back.

Kim  20:55

Right from 360 to get to seven-fourths Pi?

Pam  20:59

Which direction was your ant walking as you were walking around the circle?

Kim  21:03

From the 3:00 towards the 12:00.

Pam  21:06

Okay. So, then when you got to 360, if you keep going up towards the 2:00 and the 1:00, then you're adding degrees. But from 360, you got to subtract Pi fourths to get to seven-fourths of Pi, right?

Kim  21:24

Yeah, I think I'm going to need to see your visual because 0 and 360 to me is at the 3:00.

Pam  21:30

Yep. Yeah, agreed.

Kim  21:33

But if that's 360, which is 2 pi, and I want seven-fourths Pi.

Pam  21:38

Do you want less than 2 Pi or more than 2 Pi?

Kim  21:40

I want less than 2 Pi.

Pam  21:41

So, if we were together, I would be taking my my pencil, and I would start at that 3:00, and it would open up... Golly, how do you describe that? Counterclockwise.

Kim  21:54

Which way does it open up?

Pam  21:56

Counterclockwise. So, it goes from the 3:00...

Kim  21:58

Ah, okay, okay, okay. See, yeah, that's good to know. It opens up counterclockwise, so then it's going to be below the 360 on a clock. So, it's going to be like more like 4. Between 4 and 5.

Pam  22:09

(unclear). Yeah, for seven Pi fourths. Yeah.

Kim  22:11

Okay.

Pam  22:11

Oh, wow. So lots of visual things I was assuming. So, you had your aunt walking sort of down and clockwise?

Kim  22:18

Yeah. my ant was walking clockwise this whole time.

Pam  22:21

Gotcha. Yeah, the ant has to walk counterclockwise. That's standard position. The angle opens counterclockwise. Weird, right?

Kim  22:29

Yeah.

Pam  22:29

Super weird. Whew! Okay, I don't even know if we should do one more. But let's just try one more.

Kim  22:34

Okay.

Pam  22:35

Where would a sixth of Pi? What would be the degree measure and where would a sixth? Again, it's starting at that 0 or... Yeah. Sorry, 1, 0. Starting over there at the 3:00. And it's opening counterclockwise.

Kim  22:50

Yeah. Okay. So I have to think about this. So, I know the degrees. That works for me. So, if Pi is...

Pam  22:50

Can you tell us?

Kim  22:51

Pi is 180.

Pam  22:52

Oh, sorry. Mmhm, mmhm.

Kim  22:56

If Pi is 180, then a sixth of Pi is 30 degrees because it's just a sixth of 180.

Pam  23:08

Okay, and 180 divided by sixth is 30. So, alright, so Pi sixth or one-sixth of Pi is 30 degrees. Okay.

Kim  23:15

So, if my ant is walking counterclockwise.

Pam  23:19

Yep.

Kim  23:20

And I'm at 180. That's Pi.

Pam  23:25

Why are you at 180? Oh, okay.

Kim  23:26

Because that's where I started.

Pam  23:28

Why?

Kim  23:29

Because I used Pi to help me figure out a sixth of Pi.

Pam  23:33

Oh, okay. Okay. Thank you.

Kim  23:35

So, I want a sixth of the distance between Pi and 2 Pi. No?

Pam  23:43

Well, I don't know. That feels like it would be more than Pi, and you only want a sixth of Pi.

Kim  23:49

Oh, because again, I'm clockwise, so it's going... So, I only want a sixth of the way. Good gravy.

Pam  23:54

(unclear).

Kim  23:54

So, I'm at the 1:00.

Pam  23:57

Because?

Kim  23:58

Because it's a sixth of the way between 12:00 and 6:00 on the clock.

Pam  24:03

12:00 and 6:00.

Kim  24:05

There are 6 numbers between 12:00 and 6:00.

Pam  24:10

Ah, okay.

Kim  24:10

If I'm dividing that Pi by 6.

Pam  24:12

Okay, okay. But we had (unclear)

Kim  24:14

I don't know how to describe

Pam  24:15

A third of

Kim  24:15

it other than giving you numbers for location between them. I guess I could have said it's a third of the way between 2 Pi and between 0 and... Oh, mine's backwards now. 0 and 90. So, it's a third of the way between 0 and 90, which would be closer to 0 than it is the middle between the two.

Pam  24:16

the... Where's your 0 again?

Kim  24:22

My 0 is at the 3:00 on a clock.

Pam  24:34

Okay. And it's opening counterclockwise from there, right?

Kim  24:47

Yep. So, I'm only going 30 degrees.

Pam  24:49

Up from that, from the x-axis.

Kim  24:52

Oh, so that's on 11:00. Good gravy. No, no. See, I'm trying to figure out the clock and the thing at the same time because I've never seen a unit circle.

Pam  25:01

Yeah, yeah.

Kim  25:02

And I don't know which way I'm supposed to go versus.

Pam  25:06

Yeah. So, maybe do this for me. Put your pencil. So, draw the axis.

Kim  25:12

Yep.

Pam  25:13

Okay, now draw circle.

Kim  25:15

Yep.

Pam  25:16

And our starting place is going to be at the 3:00 or the point 1,0.

Kim  25:21

Yep.

Pam  25:22

Okay, I want you to put your pencil so that the point is at the origin and it's laying on the x axis.

Kim  25:29

What point? The 1,0?

Pam  25:31

Sorry, the point of your pencil.

Kim  25:33

Okay.

Pam  25:33

Haha, that wasn't very good. The front of your pencil. The front of your pencil is on the origin, and the rest of your pencils like going to the right.

Kim  25:44

Okay.

Pam  25:45

Yeah, I don't know if... So, the way mine looks, I have the circle, and then I have a pencil laying flat on it horizontally. And it's going from the origin to the right. So, it's horizontally. So, if I'm going to describe angles, they're all going to start here. And now, I'm going to rotate that pencil counterclockwise, leaving the front of my pencil at the origin.

Kim  26:09

Yeah, so I'm going to rotate it 30 degrees.

Pam  26:12

Okay.

Kim  26:12

So, if I were looking at a clock, and the 360, the x-axis that you had me start at.

Pam  26:23

Mmhm.

Kim  26:24

Then, if that's 3:00, then I'm going to be at 2:00.

Pam  26:29

That's 30 degrees.

Kim  26:30

Yes.

Pam  26:30

Of the 2:00.

Kim  26:31

Yes.

Pam  26:31

Because?

Kim  26:33

Oh, was I saying 1:00?

Pam  26:35

Yeah.

Kim  26:35

Ugh. Okay, yeah. 2:00. I meant back one number from 3:00.

Pam  26:39

Oh, back one number. That makes a lot of sense.

Kim  26:42

Sorry.

Pam  26:42

Yeah, yeah. And I kind of like when you were saying like it makes sense that a number, a movement of a number, is 30 degrees because you got 12 of those numbers for the whole 360 degrees. So, each each move into that number is 30 degrees.

Kim  26:58

Yep.

Pam  26:59

Awesome. So, you sort of. It didn't move very much, right? The 30 degree angles not very much off of that. It's just kind of 30 degrees. It's one number movement kind of off of that. That will be 30. Alright.

Kim  27:08

Yeah.

Pam  27:09

Kim, we can do it. I only have one more problem. I promise we're done. eleven-sixths of Pi.

Kim  27:14

Mmhm.

Pam  27:16

But again, your pencil has to start on that horizontal 3, pointing at the 3:00.

Kim  27:24

Okay.

Pam  27:24

Okay.

Kim  27:25

So, twelve-sixths is 2 Pi.

Pam  27:28

Yeah.

Kim  27:29

So, that's 360 degrees.

Pam  27:30

All the way around.

Kim  27:32

Yep. I need one-sixth less. And that was 30 degrees less. So, that's going to be 330 degrees.

Pam  27:42

Nice.

Kim  27:45

So, it's going to be 30 degrees less than all the way around the circle. Cool.

Pam  27:51

Cool. And we just decided that 30 degrees was kind of that movement between numbers if we're on a clock, so are we down there at the 4:00?

Kim  27:59

Yeah, yeah, yeah. It's one beyond.... Um... Yes because it's going the other direction. See, I'm having a hard time with the counterclockwise, I think.

Pam  28:09

Yeah, yeah. I can see that. And I think, ya'll, this is so important if we were doing this visually...which, obviously, we would like do that in class...then, I would be doing a ton of movement with my arms. I usually like put my arms kind of where I'm telling Kim to put the pencil, and I'm opening up, and it's always going counterclockwise. Starts at that sort of standard position, kind of from the 0 to the 3. And then, instead of the clock hands going the way they normally do, they're going backwards. So, I'm not sure I've ever had anybody use the clock to help describe. But it has been helpful visually to kind of describe where we are. But I wonder if it's also getting in the way of...

Kim  28:10

Yeah.

Pam  28:10

...hands moving the wrong direction. (unclear).

Kim  28:10

Yeah, yeah, yeah. I think that's been part of it for me because I can't see what you're... And this is why modeling is so important.

Pam  28:13

So important!

Kim  28:14

Right?

Pam  28:14

Absolutely. (unclear).

Kim  28:18

The teachers are sometimes just saying all the things. And seeing them definitely would have been helpful.

Pam  28:48

Well, and, Kim, I'm really glad that you were helping me become clear about what you were thinking, so that everybody listening could go, "Oh, okay. Like, Kim's, the clock hands are going like normal clock hands."

Kim  29:14

Yeah.

Pam  29:15

Pam's saying, "Nope. Sorry. With radian measure, the turtles got to walk the other direction." Like, the angles have to open the other way. Yep. Alright, so let's just end this a little bit by saying you might be like, "Why are we doing this thing with Pi, and angles, and all the things?" Well, I'll just mention that we needed a real number measure for angles in order to model periodic functions in a very particular way, so that we could do things with sound waves and temperature swings. But then, even more importantly, we added complex numbers. And bam, we get electricity and computer chips. So, all of that became necessary. We do need a real number measure for angles in order to sort of do tons of things that came after that. Ya'll, maybe the thing that you can take away from this is, you don't need to memorize the unit circle. You may have been in a trig class someday when they were like, "Alright, here's the unit circle. I'm going to give it to you. Memorize all this crazy stuff." Actually, it's all based on relationships. And today, we just did sort of the first part of it talking about how we can find the angles that are in these measures related to Pi, but they are all figure-out-able. Kim, we can actually play I Have, You Need with radian measure. I'll be honest with you, it's really a bit more about fractions. But I could say things like, hey, if my total is 2 pi, and I've got one-third of Pi, what do you need to make 2 Pi? And should we do that? If I've got one-third of Pi, but our total's

Kim  30:45

(unclear).

Pam  30:46

If the total is 2 Pi. The total is 2 Pi, and I've got one-third of Pi. I probably should have done the angles we already did, but.

Kim  30:53

1 and 2/3 Pi.

Pam  30:54

You still need 1 and 2/3 Pi leftover? Sure. (unclear). Oh, go ahead.

Kim  30:57

But also. But also people can use what they... 360 degrees, right? So...

Pam  31:03

Absolutely. Yep.

Kim  31:04

Kids are having some difficulty with the difference between some of the radians.

Pam  31:10

Yes. And middle school teachers when you're doing, fifth grade teachers when you're doing degree measure for angles, you could totally say, "Hey, if I have 360?"

Kim  31:18

Yeah.

Pam  31:18

"If that's the total, and I have 180, what do you need? If I have 90, What do you need?" When you're doing complementary angles, you can say, "Today, our total is 90." Bam. When you're doing supplementary angles, you could say, "Today our total is 180." And you can ask, "Given the one, what's its partner?" So, lots of different things that we can do with I Have, You Need and angle measure. Kind of fun.

Kim  31:39

Yeah. If you got nothing else out of today, then maybe I was able to verbalize some of the challenges that your kids are having as they're trying to follow the conversation. Right? And modeling is super, super important. We love how you were making connections to the podcast with what's happening in your classroom, but we love it even more when you tell us what you love, and share the podcast with others. So, I just checked recently, and there were 204, Pam...

Pam  31:40

Oh, nice.

Kim  31:40

...reviews. Yeah, so...

Pam  31:41

Sweet!

Kim  31:41

...we would love it if you would continue to share it with others and rate and review, so that more people can find it.

Pam  32:13

Yeah, thank you. And thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com And keep spreading the word that Math is Figure-Out-Able!