Pam  0:00  
Hey, fellow mathers! Welcome to this podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Pam  0:12  
Hello?

Kim  0:14  
I was just thinking are you going to say Pam or Pam Harris? 

Pam  0:16  
Do you remember? 

Kim  0:18  
A 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  0:26  
Because we know that algorithms, while they are amazing historic achievements, they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than we should be using and developing. 

Kim  0:40  
In

Kim  0:40  
this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical

Kim  0:45  
relationships. 

Pam  0:46  
And we invite you to join us to make math more figure out able. Alright, Kimberly, what's up today?

Kim  0:55  
Well, we have the best Journey group ever. Our...

Pam  0:58  
Our live coaching group?

Kim  1:00  
Live coaching group. Yeah, yeah. So, they ask really thoughtful questions. And I'm excited today because we have a question from Brandon that we thought we'd bring to the world. I know that you addressed his question in a Group Coaching session, but I think maybe there's other people in the world who are wondering about the same thing. So, let's do it today. 

Pam  1:19  
Let's do it. 

Kim  1:20  
Brandon said, "I really enjoyed the proportional reasoning workshop and Problem Strings like Sticks of Gum." So, if you are a longtime listener, you know we have some fantastic building powerful numeracy and some high school workshops. So, this is one of them that he took. And you've probably heard of some of our Sticks of Gum Problem Strings. So, he's loving those. And he says, "I've been thinking a lot about helping students see the relationships for y=kx. What would ya'll suggest? I feel like Sticks of Gum and many proportional reasoning strategies have to do with scaling but don't necessarily focus on the relation between x and y."

Pam  1:58  
Yeah, nice. That's a great question, Brandon. And let's dive into that. So, if you can picture a lot of the ratio table Problem Strings that we've done. If I say something like I've got a pack of gum, and it's got 27 sticks, then I might draw a an input output table where I've labeled it packs and sticks and I've got 1 to 27. And then I might ask you several multiplication or division questions based on that. So, I might say, "How many sticks are in 2 packs?" And you would probably double the 27 to get 54. We're not going to actually do this math. I'm just going to talk through a little bit of it. I might say, "What about 10 packs?" And then you might scale times 10 and say, "Well, since there were 27 sticks, you'd have 270 sticks." And then I might say, "How about 9 packs? And then can you use the 10 packs to get find the 9 packs?" Just subtracting a pack. And you might subtract the 27. And I have to think for a second. Is that 243? Yeah. Yeah. So, now I've got some values on the table. And what I've been doing this whole time is scaling. So, you'll see that I have a scaling mark between the 1 pack and the 2 packs times 2. I have a scaling mark between the 1 to the 10 packs times 10. I have scaling marks on both sides of my ratio table to do the 10 minus 1. I don't know if that's a scaling mark, but it's like a bracket to do the 10 minus 1 pack to get the 9 packs. So, we're kind of scaling between values. I could say, "How about 20 packs?" And we could scale from the 2 packs. I could say 99 packs, and we could scale from the 9 to get 90, and then add the 9. Or we could scale and get 100 packs. All of that, those actions, are kind of between the values in the... Oh, maybe I should say within. Ooh, are they within or between? They're they're from packs to packs. And I sort of do the same thing to sticks. 

Kim  3:46  
Mmhm.

Pam  3:46  
So, similarly, if I were to give you something like 2,600 and... I got to think for a second. If I had 100 packs, and I had 2,700 sticks. And then I said 2,673 sticks. And I asked you for how many.

Kim  4:05  
Did you just play

Kim  4:06  
I Have, You Need in your head? 

Pam  4:07  
Yes, I totally

Pam  4:07  
did. 

Kim  4:09  
Okay.

Pam  4:10  
So, again, you could scale kind of from packs to packs and sticks to sticks. Either way, we're sort of scaling between packs and between sticks. What Justin... Justin. Hey, Brandon, you have a new middle name (unclear) Justin. Sorry, Brandon. We like Brandon. He's a thoughtful guy. Brandon's asking about how do I go between packs and sticks? Not just from packs to packs. 1 pack to 10 packs or 1 pack to 100 packs. But how do I go from packs to sticks? How do I help kids sort of write the equation of a line. Well, what if I gave you some random number, and I didn't have anything in my table to use? Do I have some relationship I could use between the number of packs and the number of sticks? 

Kim  4:10  
Mmhm. 

Pam  4:12  
For the particular problem that I just gave you, if I have 1 pack to 27 sticks, I could give you any number of packs and I could say, "Well, if I gave you, I don't know, 19 packs." You could multiply that by 27 and you could get the number of sticks.

Kim  5:08  
Mmhm. 

Pam  5:08  
So, that's like 19 times 27 would give me the number of sticks. And I'm just going to let the number of sticks be y. So, I've got sort of 19 times 27 is y? If I asked you for 37 packs, I could do 37 times 27 and that would give me the number of sticks.

Kim  5:23  
Mmhm. 

Pam  5:23  
If I had... Give me another random two-digit number, Kim. 

Kim  5:26  
82. 

Pam  5:26  
If I had 82 packs, I could multiply that by 27, and I could get the number of sticks. Now, in a ratio table, we would usually go between packs, and we'd find nifty ways of getting to those numbers. But I also could just say to myself, 19 times 27 to get the number of sticks in 19 packs. Or 37 times 27 to get the number of six in 37 packs. Or 82 times 27 to get the number of sticks. So, then I might step back and ask kids, "What are you doing?" And kids might say, "Well..." You know like, what's changing and what's not changing?" And if you looked at my paper right now, I have 19 times 27 equals y. 37 times 27 equals y. 82 times 27 equals y. What if I had any number of packs? And I'm going to say that because then I might write down an n. What if I had n number of packs? Like, any number of packs? Where does that go? Like is that... What's the 27? That's the number of sticks per pack. So, that doesn't change. The y didn't change. What changed every time? If I gave you some random number, where would you put it? Oh, it would be that first number. So, I would have n times 27 equals y. Which is the number of sticks. N or x or whatever. Maybe I'll let it be x. I like to do any and make it n. But if it was x, then now I have x times 27 equals y. And then I might say, "Hey, how do mathematicians typically write x times 27? They typically write that as 27 times x equals y. Or y equals 27 Sticks of Gum per pack times the number of packs. And I've just written x. So, Kim, if you can see what I wrote on my paper, I wrote y equals 27x. But as I wrote that, I said y, the number of sticks, is equal to 27 sticks per pack. And I wrote 27. Times the number of packs. And I wrote x.

Kim  7:09  
Mmhm. So, it sounds like when you're doing those scaling proportional reasoning strings like Brandon was talking about, you can still bring awareness that in all of those proportions, they're all represented by y equals 27x. Like, just bringing awareness to that as you're doing that work with students even in the scaling ones.

Pam  7:29  
Absolutely. Yeah. So, this situation can be represented by this table. 

Kim  7:32  
Mmhm.

Pam  7:33  
But it can also be represented by give me any number of packs, x, multiplied by 27 sticks for gym in a pack. And bam, you'll get y, the number of sticks.

Kim  7:42  
Yeah. 

Pam  7:42  
Yeah. And so, we could bring that out as you're doing the... I would probably do a lot of scaling first.

Kim  7:48  
Yeah.

Pam  7:49  
 In fact, I would use ratio tables and do sticks of gum types of strings with lots of scaling in third, fourth, and fifth grade. And then in sixth grade, when we start talking about non-unit rates, I would do the same kinds of strings but with not 1 to 27, but it would be something like, I don't know, 4 pounds cost $7.87. And so, then I would have the ratio 4 to $7.87. And then I would scale from there. I would do a similar Problem String where I'd say, "Well, what if I bought 8 pounds? What if I bought 10 pounds? What if I..." You know, we would continue scaling with that non-unit rate and give kids experience there. And at the point where I'm ready to start saying, "Hey, what do we call these relationships when all these ratios are equivalent? Oh, we call that a proportional situation." If the ratios are equivalent, then it's a proportional situation. What do graphs of those proportional situations look like? That's where we're graphing the ordered pairs, those x's and y's. Now, I'm going to start to do a lot of work to try to get them to see the relationship between x and y. So, I'm going to keep scaling. We want to keep getting them better at that proportional reasoning. But we're also going to talk about the relationship between x and y.

Kim  9:00  
Mmhm. 

Pam  9:00  
So, one way that I'm going to help students create the relationship between x and y is doing the Problem String and deriving out of that, that relationship. Another way that I can work on that is... Kim, will you sketch a triangle for me? In fact, leave space to sketch a couple triangles. 

Kim  9:20  
Okay.

Pam  9:21  
Okay, so you got a triangle and one side is a. And it's kind of long. Draw it kind of a long, skinny triangle

Kim  9:27  
Okay. 

Pam  9:28  
Long, skinny triangle.

Kim  9:29  
Made a guess. Yep. 

Pam  9:29  
Yeah, so long, skinny

Pam  9:31  
triangle. 

Kim  9:31  
Yep. 

Pam  9:33  
Not the longest side, but the next longest side. Okay, I don't know just draw an equaliteral. Don't draw a isosceles triangle. Three different lengths. 

Kim  9:45  
Okay.

Pam  9:45  
The middle length is a. That's a long.

Kim  9:48  
Okay.

Pam  9:48  
Okay, the shortest length is 8 long. Okay. Okay. Now, I want you to draw a triangle that looks very much the same shape that you just drew, but it's smaller. 

Kim  9:58  
Okay. 

Pam  9:58  
Now, I want... Do you have your smaller one? 

Kim  10:02  
Yep. 

Pam  10:03  
And do they look the same shape? 

Kim  10:05  
Yep. 

Pam  10:05  
So if I grab that little one and I stretched it on a computer, so it stretched in all the directions, it would look like the bigger one? 

Kim  10:12  
Yep. 

Pam  10:12  
Okay, cool. That middle length side is 12. 

Kim  10:16  
Okay.

Pam  10:17  
And the smallest length is 4.

Kim  10:20  
And I don't know the other lengths? 

Pam  10:22  
No, but what you do

Pam  10:23  
know is the two triangles are similar. 

Kim  10:25  
Okay. 

Pam  10:26  
So, that's kind of important. Now, we haven't really defined similar triangles. But it's same shape. 

Kim  10:31  
Mmhm.

Pam  10:31  
But the side lengths have been scaled. The side links are in proportion.

Kim  10:34  
Mmhm. 

Pam  10:34  
So, the ratio should be equivalent. So, do you have a gut instinct on this one? What do you want to?

Kim  10:39  
Like, to find a? 

Pam  10:40  
Yeah, to find a. 

Kim  10:41  
Yeah. Yeah. I want to double 12. 

Pam  10:43  
Why would you do that? 

Kim  10:45  
Because the side lengths that are 4 and 8, the 8 is double the 4. 

Pam  10:49  
Nice. 

Kim  10:50  
And so like a is double of 12. 

Pam  10:53  
I'm

Pam  10:53  
going to slow you down. (unclear).

Kim  10:54  
Sorry! 

Pam  10:54  
No, no, no. It's okay. I just... I thought, "Do I interrupt?" And then I started to, and I was like, "Don't interrupt! Let her talk!" So, you said the 8 is double the 4. So, on the two triangles, I've now drawn a scaling arrow from that 4 to the 8. So, that's kind of between the two triangles.

Kim  11:06  
Mmhm. 

Pam  11:06  
Right, because the 4 is on one triangle. The 8 is on the other triangle. So, between the two triangles, I've drawn this arrow. And you said it was double, so then I put times 2 kind of above that long scaling arrow. 

Kim  11:19  
Yep. 

Pam  11:19  
And then you said, "Well, if that's times 2, and a corresponds to 12..." I'm putting words in your mouth. But then you also doubled that one? 

Kim  11:28  
Yeah.

Pam  11:29  
Yeah. So, now I have the same kind of scaling arrow between the 12 and a, and I'm writing times 2 above it. So, you're like, "A has to be double 12." And you said it's 24. Yeah.

Kim  11:38  
Yeah. 

Pam  11:38  
Cool. So, I would then bring out to students. Well, like, make sure that makes sense to everybody. And we would call that a between strategy.

Kim  11:47  
Mmhm. 

Pam  11:48  
Because you looked between the two triangles.

Kim  11:51  
Mmhm. 

Pam  11:51  
And said, "Okay, I know they're similar, so they've got to be the side lengths are in proportion. So, 4 to 8 has to be the same scale as 12 to a."

Kim  11:59  
Mmhm. 

Pam  11:59  
And that was times 2. Cool. Then I would change colors, and I would say, "Did anybody stay within the second triangle, the little triangle? Did anybody look between the 12 and 4?" Because 12, remember, is the middle, the length of the other triangle. It's the one that corresponds with a.

Kim  12:17  
Mmhm. 

Pam  12:17  
So, Kim, if I direct your attention there, what are you thinking

Pam  12:21  
about? 

Kim  12:21  
Yeah, so I see that the short side, which is 4, is a third of the 12. So, 4 times 3 would be 12. Which means that 8 times three would be a.

Pam  12:33  
Nice. So, as you were talking. Inside the triangle. Or not inside. But like on the triangle. At the triangle. The smaller triangle. Between the side length of 4 and the side length of 12, I have a red arrow, and I've said times 3. And then between the 8 to the a. Oh, that's terrible that I have an 8 and an a. Maybe I should have made that x. From 8 to a, then it would have to have the same scale factor times 3. And sure enough, there's our 24 that we found before. So, in two ways, we're really clear that a is 24. And that time you used a relationship within the two triangles.

Kim  12:57  
Mmhm. 

Pam  13:00  
So I kind of... I have two different colors. So, I have a between strategy. It's the scale factor between the sides of the two triangles. And I have... In red, I've written a scale factor within the sides of a triangle, and then we've used that same scale factor in the other triangle.

Kim  13:27  
Mmhm. 

Pam  13:27  
So, you might be like, "Pam, what does this have to do with y=kx?" Well, in a big way, we have a proportion, and we want to get kids acknowledging, realizing, constructing the idea that there is a between relationship and a within relationship. 

Kim  13:45  
Yeah.

Pam  13:45  
And sometimes one might be easier to use than the other.

Kim  13:48  
Mmhm.

Pam  13:49  
And so, we want kids to look at both of them.

Kim  13:52  
Mmhm. Oh,

Kim  13:52  
can I tattle on myself? 

Pam  13:54  
Oh, go. Yes. 

Kim  13:55  
So, somebody, asked a question in the teacher Facebook group, you know, that we love. And good questions. Join us there.

Pam  14:02  
Math is Figure-Out-Able teacher Facebook group, Yep, mmhm.

Kim  14:04  
Yeah. So, somebody asked a question, and I was like... You know, I've said, I'm lazy sometimes when I read things. And I only looked... I can't remember which one. I think I only looked between. 

Pam  14:04  
Okay.

Kim  14:06  
And I was like gave some like answer about, you know, being an estimate or whatever because the between was kind of like funky. And then somebody else, you know, shot back like, "Oh, well, if you look within." And I was like, "Dang it!" I totally didn't even bother. So...

Pam  14:31  
So, are you saying... You're saying that the relationship you used, the numbers were funky enough that you just estimate? You just estimated.

Pam  14:40  
(unclear) times 10 and a 1/3 or something. Between. So, you estimated, and you're like, "If it's about 10 and a 1/3, then about 10 a 1/3, the other one, would be this. It's about. It's close enough. Whatever."

Pam  14:49  
Like, all these people are like, "How about this way? This way? This way?" Oh, man..

Kim  14:49  
Actually.

Pam  14:50  
You got me. 

Pam  14:50  
So, that

Pam  14:50  
would be a moment in class where we get the "ooh" factor, right? We could go, "Oh, wish I would have looked

Pam  14:50  
at that one. Yeah. 

Kim  14:50  
I think that's what i said, "Oh, shoot! (unclear)."

Kim  14:51  
Yeah. 

Pam  14:51  
"I want my brain to do that next time."

Kim  14:51  
Yeah.

Pam  14:51  
"Alright, next time." So, that's a good moment that we want to engender in class.

Kim  14:51  
Sure. 

Pam  14:51  
We want kids to go "Right, right. I can look within, but I can also look between." Cool. So, a way that that could look for this particular problem that we just did, I could write a to 8. So, like, I'm going to write kind of a fraction a "over" 8. And I'm only saying "over" because  it's an audio podcast. And so, picture the fraction a/8 equals 12/4.

Pam  15:29  
Because...

Kim  15:30  
a/8 equals 12/4?

Pam  15:31  
Is that right? 

Kim  15:32  
4/12? 

Pam  15:34  
a/8, 12/4. A, the letter a to 8.

Kim  15:39  
Sorry.

Kim  15:40  
I wrote mine backwards. (unclear).

Pam  15:42  
I shouldn't have used a and 8. That was dumb. 

Kim  15:44  
That's okay.

Pam  15:45  
So, a to 8 is equivalent to 12 to 4. 

Kim  15:48  
Yes.

Pam  15:48  
Okay, cool. So, I could have that proportion is two fraction looking things equal to each other. We know they're equal to each other because we started the problem saying the two triangles are similar.

Pam  15:58  
Right. And that's part of what we would want to help kids to develop is that therefore the sides are in proportion.

Kim  16:04  
Mmhm. 

Pam  16:04  
So, we would want kids to be able to look at that proportion, a to 8 equals 12 to 4, and say to themselves... Now, I don't know. Looking at it that way, you might instantly go, "Gee, 12 divided by 3 is 4, so what divided by 3 is 8? Ah, it's got to be 24." Like, that's just screaming at me. But a kid also might say, "Yeah, but I can get from that 8 to 4, the two denominators 8 to 4, by like half of that. Well, so then I've got to take half of something to get 12. Bam, that's got to be 24." That be another way of thinking about it.

Kim  16:34  
Mmhm. 

Pam  16:35  
So, with this one, I gave you particular numbers that happen to go either way. Often, t's kind of like the one you got caught on the Facebook problem. 

Kim  16:43  
Yeah.

Pam  16:44  
That one way is is slicker than the other.

Kim  16:46  
Sure. 

Pam  16:46  
And it might depend on the day where one relationship pings for you better than the other relationship. We want to encourage kids to look between both. Just like...nicely done Brandon...we don't only want kids to scale in ratio tables. We also want them to develop the relationship between x and y.

Kim  17:03  
Mmhm.

Pam  17:03  
We want both of those.

Kim  17:04  
Do you ever do a Problem String with a series of these triangles where like first one might be like this where it's pretty simple relationships, but then you like go through the string, and you end up where like one would be better than the other? 

Kim  17:17  
Totally.

Pam and Kim  17:18  
Yeah. 

Pam  17:18  
Yeah. I'll also do just flat out worksheets like that where I'll give them a couple, you know, three or four triangles where the first one is pretty accessible, that you could do either way. And then there's one that there's an obvious one way. And then there's one where there's an obvious the other way. And then there's one where, ooh, neither of them are great, and you're going to have to work for both of them. 

Kim  17:40  
Yeah, yeah.

Pam  17:41  
Yeah. And that would be a fine time for some independent thought. You know, kids are, after we've done a lot of work, sure. I think I wonder sometimes if people hear me and go, "So, is there ever homework? Like, independent work?" Yes.

Kim  17:54  
Mmhm. 

Pam  17:54  
There is definitely independent work. And that is an example of something that I would definitely have kids do and talk about. 

Kim  18:01  
Yeah. 

Pam  18:01  
And then I would want to have them share, "Hey, what did you choose on the first one? Yeah, everybody did the within. Hey, what did you on the second one? Yeah, everybody did the between. What did you do on the third one? Hmm." And then it's not who's right for which one they chose, but it's like to build the different strategies that are coming out of the two different ways you do it. 

Kim  18:18  
Mmhm. 

Pam  18:19  
The two different ways you would solve. And then same on the harder one. You know like, let's share strategies. Now, I might, on a given day, only do one of those. We might just choose like the third problem where everybody had... Maybe not the hardest one. Maybe the hardest one was just to kind of keep kids going and really try something. But let's actually share the strategies on the one I know everybody sunk their teeth into well. 

Kim  18:40  
Sure.

Pam  18:40  
That would be a good one to kind of share out. Okay, so there's one way, Brandon, and everybody else, that we can help kids develop both within and between relationships. So, let's do another one. Kim. 

Kim  18:58  
Yeah.

Pam  18:59  
Quick Problem String. So, draw a ratio table, vertical ratio table. And I wouldn't make a kid do that, but I'm going to have you do it because you can't see what I'm drawing. And I'm going to say that we're starting with the ratio of 4 to 6. So, I've got a vertical ratio table. On the left side is 4 and on the right side is 6. Do we have the same thing? 

Kim  19:02  
Yep. 

Pam  19:02  
Cool. Alright, then I'm going to tell you that the next thing I'm going to write down is, on the right hand side, I'm going to write an 18. And I'm going to ask you what corresponds to 18? If it was 4 to 6, it's blank to 18. What would that be? 

Kim  19:19  
12.

Pam  19:20  
Because?

Kim  19:21  
Because 6 times 3 is 18, so I'm going to scale 4 times 3 to get 12.

Pam  19:26  
Nice. What if I then said, actually, I mean 3 on the right hand side. So, 4 to 6, but now I've got blank to 3.

Kim  19:35  
Then I've got 2 on the left side because 3 is half of the 6.

Pam  19:40  
Ah, so you go for that beginning one? 

Kim  19:41  
Yeah. So, 2 is half of 4. 

Pam  19:43  
Could you also go from 18? 

Kim  19:44  
Sure. 

Pam  19:44  
Either way. 18 divided by 6 is 3. 12 divided by 6 is 2. Cool. So, so far we have 4 to 6, 12 to 18, 2 to 3.

Kim  19:52  
Mmhm. 

Pam  19:52  
You know, Kim... No, I'm going to mention that in a minute. Hang on. Write that down. Okay, I'll mention that in a second. The last one I'm going to give you. Again, on the right hand side, I'm going to say 21. And I'm curious what would go on the left hand side.

Kim  20:07  
You're saying 21? 

Pam  20:07  
Correct. 

Kim  20:07  
On the right side. Then I'm going to go 14 on the left side. 

Pam  20:09  
Why? 

Kim  20:09  
Because I'm going to add the corresponding 12 to 18 and 2 to 3.

Pam  20:15  
Ah, okay. So, 18 and 3 gave you 21.

Kim  20:18  
Mmhm. 

Pam  20:18  
So, then you're going to add the 12 and 2 to get 14. Cool. "And then, class, did anybody do it another? Is there any other relationships that you might?

Kim  20:28  
To find 21?

Pam  20:28  
Yeah. It's okay if not. You know, I always

Pam  20:32  
ask for more.

Kim  20:32  
Can I tell you? I just guessed that you were going to ask 9. 

Pam  20:44  
Oh. 

Kim  20:44  
I don't know why I was trying to anticipate your thinking, so let me erase that. Okay, 21. 21, 21. 

Pam  20:44  
That's awesome. 

Kim  20:44  
I could scale from

Kim  20:44  
3. 

Pam  20:44  
So, really, you weren't kind of actually paying attention. You were doing your own thing? 

Kim  20:44  
I mean, kind of. 

Pam  20:44  
Should I take points off for that? 

Kim  20:45  
Probably. I'm not very compliant. 

Pam  20:45  
That's the last thing I'm going to take points off for. 

Kim  20:51  
Tongue in cheek. Okay. 

Pam  20:57  
You're playing with numbers. I

Pam  20:58  
love it. 

Kim  20:58  
Yeah. 3 to 21 is times 7. 

Pam  21:01  
So,

Pam  21:01  
you could also go from the 3 and multiply by 7. Cool. Alright, so so far we have 4 to 6, 12 to 18, 2 to 3, 14 to 21. I'm going to lift a couple values out of that ratio table and kind of say it's as if I gave you 4 to 6. 

Kim  21:14  
Yep. 

Pam  21:15  
And I asked you x to 21.

Kim  21:18  
Mmhm. 

Pam  21:18  
And this time you chose to go from the 6 to 21.

Kim  21:23  
Mmhm.

Pam  21:24  
But not in one fell swoop. Like, we did some things to get to 21.

Kim  21:28  
Mmhm. 

Pam  21:29  
And we did a couple different things. But, I don't know, can you? Do you have? If I were to say, "Put some words to what you did to get from 6 to 21," do you got

Pam  21:38  
any words?

Kim  21:39  
I did times 3 of them. And then half. So, like basically from 6 to 21 is times 3.5.

Kim  21:46  
Is that what you're asking?

Pam  21:48  
I don't know. I'm not sure. Is that what? Oh, because that's the half. I got you. Yeah. Yeah, because you got like 18. And then 18 and 3, and that's 3. And a half. Got it. That took me a second. Whoo. Nice. And then we could say then the 4 times 3.5, then also gives you the x. Which we found was 14.

Kim  22:07  
Yeah.

Pam  22:08  
Cool. New problem. Well, actually, first. Sorry, before we do that. So, I'm just going to suggest that I gave you 4 to 6, asked you then what would be the corresponding x to 21. You found the x by scaling from the 6 to 21 and dragging the 4 along with you. 

Kim  22:23  
Yeah. 

Pam  22:24  
Okay. Next problem. This time, I'm going to start with, completely unrelated, another vertical ratio table, 6 to 21. So, 6 on the left. 21 on the right. And this time, I'm going to give you numbers on the left.

Kim  22:38  
Okay. 

Pam  22:38  
So, I'm going to say if 6 is to 21, then what's 2? 2 is to?

Kim  22:43  
14. 

Pam  22:44  
2 is to. That's terrible. If 6 to 21, 2 to? 

Kim  22:46  
14.

Pam  22:47  
14. And how did you

Pam  22:48  
do it? 

Kim  22:48  
6... Which one do you give me? 2. 6 divided by 3 is 2. What did you give me? 6 to 21, and then you give me 2. 

Pam  23:00  
Mmhm.

Kim  23:01  
So, divided by 3, divided by 3. Just kidding.

Kim  23:07  
7. 

Pam  23:08  
Okay, so 6 to 21 and 2 to 7. Those are equivalent ratios.

Kim  23:11  
Yeah. 

Pam  23:11  
6 to 21, 2 to 7. Cool. Next problem. I actually want 4 on the left hand side. So, 6, two, now 4. 

Kim  23:11  
Now, you get 14. 

Pam  23:15  
Okay, that's 14. Cool. And you just doubled from there? 

Kim  23:21  
Yeah.

Pam  23:22  
Okay, cool. So, I'm going to lift those values out of the ratio table. So, it's almost like I gave you 6 to 21,

Kim  23:28  
Mmhm. 

Pam  23:28  
And then I said 4... No, then I said... Oh, I got to think. 6 to 21. What did I give you? I gave you 4. 4 to something. 4 to x. And you found that that X was 14.

Kim  23:41  
Mmhm. 

Pam  23:41  
Do you notice...

Kim  23:43  
Yeah. 

Pam  23:44  
What do you notice?

Kim  23:45  
It's funny because towards the end of the other ratio table I was drawing. 

Pam  23:50  
Yeah.

Kim  23:51  
I noticed the 4 to 6. And I was like, "Dang it, that's only two-thirds of..."

Pam  23:56  
Oh. 

Kim  23:54  
So, here it's 4 is two-thirds of 6. So, I need the number that is two-thirds of 21.

Pam  23:56  
4 is two-thirds of 6. And I gave you 6 to 21, and then I asked for 4.

Kim  24:08  
Yeah.

Pam  24:09  
So, did you just find three 1/2s of 4? No, that's not right. What?

Kim  24:14  
I said that 4 was two-thirds of 6.

Pam  24:18  
Oh, so you need two-thirds of 21. I followed you now.

Kim  24:21  
Yeah.

Pam  24:22  
Whew. Good luck, listeners. You might have to write some of that down in order to follow some of that. So, part of my point here is it's actually the same proportion. 

Kim  24:30  
Yeah.

Pam  24:31  
(unclear) problems. 

Kim  24:31  
I see that now.

Pam  24:32  
In the one problem, we started with 4 to 6, and I said, "So, then what corresponds to 21?" In the second problem, we started with 6 to 21, and I said, "What corresponds to 4?" So, notice, in both problems, I gave you 4, 6, and 21 and asked for the fourth value. In the one, we started kind of with the 6 and scaled to the 21 and dragged the 4 along with it. In the other, we started with the 6 and scaled... No, wait. And scaled it to 4 and dragged the 21 along with it. 

Kim  25:04  
Yeah. That's so fun.

Pam  25:05  
It is kind of fun.

Kim  25:06  
It is really fun. 

Pam  25:07  
So, that's an example of a Problem String that I could do with kids to go, "Hey, when I'm given..." Now, very clearly, I'm writing next to each of these vertical ratio tables the proportion because I want them to be able to see, "Oh, it is the same proportion. Golly, I could look within and between when I'm given a proportion, and I could choose which way I'd like to go." And, Kim, you taught me this. If I start going a way and I don't like it, I'm going try the other way. 

Kim  25:30  
Mmhm.

Pam  25:30  
And I might like that one worse, and so I might go back and finish the first one.

Kim  25:33  
Mmhm. 

Pam  25:34  
But we have options. And part of mathematical behavior is seeing which way you kind of want to go today. What relationships are sparking for you? And kind of (unclear). So, there's...

Kim  25:45  
And good

Kim  25:45  
heavens. It's like creative, and it's interesting, and it's capitalizing on relationships, and it's just so much better than bat and ball.

Kim  25:54  
Honestly. 

Pam  25:54  
Yeah, so bat and ball is a term that we heard somebody describe cross, multiply and divide because you kind of draw the big loop is the bat. We're not... Don't memorize that. I'll just maybe say something because it's on my mind. I was in a presentation recently. A very kind, good... Let's see. Good humored presenter said it's okay if you do the tricks after you are well grounded in reasoning. 

Kim  26:19  
Yeah. 

Pam  26:19  
And I'm going to respectfully disagree. That I want to stay well grounded in reasoning, and I don't want to introduce tricks because as soon as I introduce tricks, then we just get a whole bunch of tricks. And yeah, they were grounded in reasoning, but that we're not going to hang on to the tricks, and we're just going to have more and more tricks. And kids will be able to get away with using less sophisticated reasoning. And what happens then is, sure, some kids will hang on to tricks and get answers. But what they won't do is continue to solve problems, building their brain. 

Kim  26:50  
Yeah.

Pam  26:50  
That's kind of a subtlety. Like, you might be saying, "No, they're grounded, so from now on out, they can use tricks because they know what the question is asking. They can use a trick. We're done." But I'm suggesting we are missing out on the possibility of all the time that they are now using that trick, they could have been exercising their brain to get better at the kind of reasoning those problems are calling for.

Kim  27:13  
Mmhm. 

Pam  27:13  
And that's kind of an important loss.

Kim  27:15  
Yeah, it's a little short sighted to share these tricks with kids because it might give them... You know, if kids can use the tricks, it might give them answer today. Might give them an answer in your year. But it's not building them for future mathematics, and that's really unfortunate.

Pam  27:35  
It's helping them get answers in the moment.

Kim  27:38  
Mmhm. 

Pam  27:38  
And maybe in the future, but without building. Yeah, that's well said, yeah. So, Kim, the last one I'm going to mention is kind of similar to what I did when we talked about the Sticks of Gum. But now, it's not a Sticks of Gum Problem String. Now, it's a Rich Task. And in the Rich Task, I'm going to lob out something that kids can think about. So, I've given them enough information. Like, I might have said to them, "Hey, for this particular problem, you've got 12 ounces of frozen yogurt, and you put it in, and that frozen yogurt costs $0.40 an ounce, and you've put it in a waffle bowl that costs $0.50. Can you figure out the cost of that frozen yogurt in that waffle bowl?" And the kids will figure that out. And that won't take them long. But then I will then give them a problem where they don't know the number of ounces. And we've done this in a clever way in my Building Powerful Linear Functions workshop, where we've shown them the waffle bowl on a scale, but we've stuck the cup of spoons in front of the scale, so you can't see the ounces. And we'll say, "Well, hey, for this one that's in a in a dipped waffle bowl. So, that's like $0.50 for the... Or $0.60 for the waffle bowl." So, now we're going to add that waffle bowl, $0.60 plus some ounces. "Miss, I don't know how many ounces it is." I'm like, "I know. What are you going to do? Is there some way you could represent all of the possibilities? And then kids usually look at you like, "What do you mean?" And I'm like, "Well, what if it was one ounce?" And they're like, "Okay, so that's $0.60 for the bowl plus that $0.30 an ounce for the yogurt without toppings times 1. Well, what if it was 5?" And they're like, "Okay, that's $0.60 an ounce plus $0.30 an ounce times 5 of those." Well, what if it was 18 ounces? I'm hungry! We're getting lots of frozen yogurt? That's $0.60 an ounce plus $0.30. And if you can see what I'm writing down, I'm writing those in a row, so that the $0.60  stays the same. The $0.30 an ounce stays the same. What's changing? It's those ounces. Because I don't know, right? So, how could I represent all of the possibilities? Well, that's going to be the variable. So, I'm going to end up with an expression that's like $0.60 plus $0.30  times x or n the number of ounces. And that's another way where I'm going to help students build towards the relationship between x and y, between the input and the output. So, I think we need both. I think we need good Rich Tasks to get kids to do that, and we also need Problem Strings to really build their proportional reasoning. And together, we've got kids reasoning proportionally. It's a beautiful thing. 

Kim  29:56  
Mmhm.

Pam  29:56  
Yeah, so check out Building Powerful Linear Functions, ya'll, if you haven't. If registration is not open right now, get on the wait list. It might be my best work. Which I'm pretty proud of a lot of what we do, so that's...

Kim  30:08  
Yeah.

Pam  30:08  
That's saying something. Alright, Kim, thanks. That was fun. Thanks 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!