Pam  0:01  
Hey fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

Kim  0:11  
And I'm Kim Montague, 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  0:19  
We invite you to consider that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:34  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:41  
We invite you to join us to make math more figure-out-able.

Kim  0:47  
In this episode, we're taking a look back at some of the episodes where we chatted about three important topics in grades 6-8 classrooms. First, we're chatting rational numbers and the five interpretations we need students to experience. Second, we dig into integer subtraction and how to make sense of subtracting a negative, and third, solving proportions. After all, we're talking 6-8, the land of proportional reasoning. Enjoy y'all. Let's get on to the episode.

Pam  1:16  
So, today we're going to start with some important information that, if we kind of have in our heads and our understanding of fractions, decimals, and percents, especially fractions, then it will help as we then start with students, as we do things that are sort of building up for students. So, we're going to talk about the five interpretations of rational numbers. So, what are rational numbers? Rational numbers are things like fractions, ratios, decimals, percents. In fact, a rational number can be expressed as the ratio of two integers. So, for example, one-half is the ratio of 1 to 2.

Kim  1:54  
Mmhm.

Pam  1:54  
Four-fifths is the ratio of 4 to 5. And if I made one of those negative, then we could have the ratio of negative 4 to 5, and so on. So, those are rational numbers. Consider, for a minute, the simultaneity going on in a fraction, things happening simultaneously in a fraction. You have the numerator, a number. You have the denominator, a number, usually a different number. And then you're asked to consider the relationship between those numbers. That's a lot to consider simultaneously. And not only the relationship between those numbers, but then we can consider the relationship between those numbers as a number. So, there's just lots of things going on simultaneously. Susan Lamon, in her book called Teaching Fractions and Ratios for Understanding helps us think about these five interpretations. In fact, that's where we got them from.

Kim  2:47  
Yeah. Yeah, I remember... Pam, I'm going to interrupt you. I remember the first time you ever talked about with me these five interpretations, and it was super important, so I hope everybody's gonna really listen to these five interpretations.

Pam  2:58  
Excellent. Alright, so here they are. So, not in any specific order. The five interpretations of rational numbers are part-whole, measurement, operator, quotient, and ratio. Now, we didn't expect you to get all those down because we're actually going to dive into each of them right now. And so, we thought we'd start today by discussing these five interpretations of rational numbers. Kim, let's start with the most common interpretation in the U.S. That doesn't, by the way, mean it's the best one to start with. 

Kim  3:30  
Yeah.

Pam  3:31  
It just happens to be the most common one. And that is the part-whole way of looking at fractions.

Kim  3:38  
Yeah, it's definitely the most common, right? And in elementary school, it's likely the only way that fractions are taught. 

Pam  3:44  
Yeah, I know.

Kim  3:45  
Right?

Pam  3:46  
Not good, not good.

Kim  3:46  
So, we can picture a typical task in school, in elementary school, right, young elementary school, where there's a pre-drawn object or set of objects, and students are asked to shade in a fractional amount, right? So, like, a rectangle is already cut for them into 5 pieces.

Pam  4:03  
Into 5 equal pieces, right? Already cut into 5 equal. Mmhm.

Kim  4:06  
Mmhm. Everything's done for them. And students are supposed to shade four-fifths. Can you picture that? They might be asked to do the shading when they're told the fractional amount. Or they might be given an image with something pre-cut and shaded in, and might be required to tell the fractional amount not shaded. That's really the only two things that they're asked to do.

Pam  4:28  
So, consider the Development of Mathematical Reasoning, y'all. Remember, there's Counting Strategies come first, and then we build on that to get Additive Thinking, and then we build on that to get Multiplicative Thinking, and build on that to get Proportional Reasoning, and then Functional Reasoning. If a student is given that kind of task. So, let's review the task. They've given a pre-drawn object that is pre-cut into equal sections. 

Kim  4:53  
Mmhm.

Pam  4:53  
And they're asked to shade four-fifths. What kind of thinking would that student probably be doing? So, I'm going to act it out here. I've got this pre-drawn thing. It's cut into equal 5 pieces, and I'm supposed to shade four-fifths. Y'all, I'm going to count the total. Yep, there's 5. It says four-fifths. I'm going to count 4 and shade those 4. And I'm done. That is the only thinking happening. Now, maybe I've given them that pre-drawn figure with it cut into 5 equal pieces and 3 of those pieces already shaded, and now the instruction is what fraction is represented? Again, what kind of thinking is happening? They're going to count the total, write that down in the denominator, count the total that are shaded, write that down in the numerator, and they're done. Like, that's it. They are literally only using counting strategies in order to do those fraction sort of tasks at those young grades. What we haven't done is create some sense of what four-fifths means. We haven't had students like grapple with the idea of what four-fifths are. Or even better, starting with one-fifth to help them think about four 1/5s, or four-fifths. They're just using counting strategies. That's a problem.

Kim  6:09  
Right. Right, so we need to actually help students create a sense of what a fraction means, right? So, in this case, we're talking about four-fifths. What does four-fifths actually mean?

Pam  6:20  
Yeah, so we do need the part-whole interpretation, but we need to do so much more, so it will be richer, and not only the part whole interpretation. In fact, we don't think maybe that you're going to start with... Don't start with the part-whole. So, we need to talk about how to develop this richer, deeper meaning. We're going to do that in an episode coming up. Now, let's talk about... So, Kim. What is 5 minus 3?

Kim  6:47  
Oh, 2.

Pam  6:48  
And I know that's not hard. We're not going to pause very long here. 2. I'm just going to make a quick picture of your thinking, just so we can kind of refer to it after. You didn't have to think about that. But I'm just going to say can we think about 5 minus 3 as 5 subtract 3. We can. But can we also think about it as the difference between 3 and 5.

Kim  7:04  
Mmhm.

Pam  7:04  
We've done some work with that. So, on the board, I would put 3 on a number line. I would put 5 to the right of it. I would draw a little jump above it and put 2. And you're saying the difference between 3 and 5 is 2. Cool. Next problem. What about 3 subtract 5? 

Kim  7:19  
-2.

Pam  7:21  
And how do you know?

Kim  7:26  
For this one, I started at 3 and go back 5.

Pam  7:29  
And if you do that, you kind of skip over the 0.

Kim  7:32  
Yeah.

Pam  7:32  
And you kind of landed on -2. So, I've now written on the board 3 minus 5 is -2. And I'm going to pull out... Oh. But, Kim. See, I think I have to talk about as we go. It won't makes sense if I don't.

Kim  7:42  
Yeah.

Pam  7:43  
So, I'm going to pull out of kids that -2, not 2, you know. And then I'm going to say, "Could you reason about that with elevation?" So, Kim, can you reason about that with elevation? 

Kim  7:52  
So I'm at 3 feet above sea level, and I drop 5 feet, so I'm at -2 feet.

Pam  7:59  
Which is?

Kim  7:59  
Which would be below sea level.

Pam  8:01  
There you go. Okay, cool. And how about temperature? Can you reason that with temperature?

Kim  8:04  
Yeah, it's 3 degrees, and it drops 5, and so it's really cold, -2 degrees.

Pam  8:08  
Brr. And can you do football?

Kim  8:12  
Ugh.

Pam  8:12  
No? No, you don't have to do football. (unclear).

Kim  8:14  
I'm 3 yards ahead of the line of scrimmage, and I get pushed back 5 yards.

Pam  8:20  
Nice. And so, where are you?

Kim  8:21  
I'm 2 yards behind the line of scrimmage, -2.

Pam  8:23  
Bummer, bummer. We're back. That's terrible. And what about... What am I leaving out? 

Kim  8:28  
Money.

Pam  8:29  
Yeah, money.

Kim Montague  8:30  
I have $3.00 bucks. I owe someone $5.00, so I'm in debt to -$2.00

Pam  8:35  
Cool. Alright, so then I'm going to say, hey, I'm a little curious because we just said on the first problem, 5 minus 3, that we could think about that as the difference between 2 numbers. But every scenario you just told told me about, you were kind of falling. The temperature was falling 5 degrees. You were going down below sea level. It was removing. You were going in debt, like removing money from you. What did I leave out? Debt? Elevation? Falling? Football, you were being pushed back 5 yards. All of those were kind of a subtraction removal context. Could we look at the difference between 3 and 5? Could we let the subtraction? Because right now I'm pointing at the subtraction symbol 3 minus 5. I'm pointing at that subtraction in between the numbers, and I'm saying if that means difference, that's what we've said. We said with whole numbers it can be difference. Then I should also be able to look at the difference between 3 and 5. Ooh, ooh, but the difference, the distance is 2. But you just told me the answer was -2. Ah, crumb. I guess we can't use distance to think about subtraction. Or can we? So, I would have... For both problems. Right now, I only have one number line on the board. We talked through those other contexts when you were removing 3 minus 5. But I've really only got the one number line, and that is from 3 to 5 is this distance of 2. So, I'm going to say could we use distance? Could we find out how far apart the numbers are, but then think about removal to say, yeah, they're two apart, but is it going to be positive 2 or -2? The distance is 2, but from 3 feet you fell down 5 feet. Where are you? You're 2 feet below sea level. From 3 degrees, you fell. So, we can actually use distance to find out the magnitude of the answer, the absolute value of the answer. But then we have to think. We have to use removal to decide is it going to be positive or negative. I wonder if that will work. Let's try another problem. So, Kim, what if I said 7 subtract -2?

Kim  10:39  
I just drew a number line, and I put -2. And then I also put 7, and so I...

Pam  10:45  
Where are they?  Can you orient us, because nobody can see?

Kim  10:49  
Yeah, so -2 would be to the left of 0. I didn't have a 0, but I'll put it right now. So, -2 is to the left of 0. And then 7 is to the right of 0. So, the space between those is 9 units, 9 whatever.

Pam  11:06  
Okay. I've got that -2 to 0. I've got the 0 to 7. You add those together, you got this. So, 7 and -2 are 9 apart. Okay, okay.

Kim  11:15  
Yeah.

Pam  11:16  
Mmhm.

Kim  11:16  
And so, the way I think about it is I asked myself is it going to be positive or negative by thinking about the context? So, if I'm at $7.00... No, I don't actually do. I don't. I...

Pam  11:32  
Yeah, this is tricky.

Kim  11:33  
I think I think about if I remove a number that is smaller or larger. Like, am I removing something that is less than what I had? Or am I removing something more than what I had?

Pam  11:44  
Well, and let's pause for just a second and look at the first two problems that we did. Because we had 5 minus 3 was 2. And 3 minus 5 was -2. So, you just kind of said bigger, smaller numbers. And this is exactly what we're going to do with kids. We're going to kind of dive in. How did you know the distance between 5 and 3 and 3 and 5 was 2. What was the size of numbers involved? Your whole life up until we started bringing in integers. Like, when you were subtracting in grade two.. Grade two. I've been talking to Canadians lately. Can you tell? In second grade. Where was the big number, where was the small number in most of the problems you were ever subtracting?

Kim  12:23  
You were always subtracting the smaller number.

Pam  12:25  
Yeah, so from a big number, you were subtracting something smaller than it. Then you would get a positive answer, right? Like, you had money and you subtracted just a little bit of money, you still had money. If you were above sea level and you fell down just a little bit, you were still above sea level. If the temperature was whatever, and it dropped just a tiny bit, like you're not close to 0, so just drop less than getting you to 0, then you're still above 0. So, all of those contexts help us think about if from the first number you just subtract something smaller than it, bam, the answer is positive. But what happened in that 3 minus 5? How does that?

Kim  13:05  
You're subtracting the bigger number now.

Pam  13:08  
Yeah, so from 3, you're subtracting something bigger than it. Oh, from $3.00 bucks, I'm spending $5.00. I'm in debt. From 3 feet above sea level, I'm falling down 5. I'm 2 feet below sea level. From 3... let's see, what's the context?  Sea level. Temperature. From 3 degrees, the temperature dropped 5 degrees. Brr, I'm below 2 degrees. All of those contexts help me think about from a number, if I subtract something bigger than it, the answer's got to be negative. So, what does that mean for 7 and -2. From 7?

Kim  13:47  
Oh, sorry. You're asking me?

Pam  13:48  
Yeah, sorry.

Kim  13:50  
So, I think about those as from 7.

Pam  13:52  
Mmhm.

Kim  13:53  
I'm subtracting a number smaller than it, so then my answer is going to be positive.

Pam  14:00  
And that is the clincher of comparing those numbers. From 7, are you subtracting? So, as I've written on my paper, I have 7 subtract. And I wrote in parentheses -2. So, from 7 you're subtracting something smaller than 7. So, you actually have to get kind of in your head. How are 7 and -2 related? -2 smaller. They're 9 apart. Since it's smaller, I'm removing something smaller. That's like what I've done since second grade. Bam, the answer is gonna be positive. Cool. Well, so then what if I just turned that problem around on you and said same numbers, -2 subtract 7?

Kim  14:37  
Yeah.

Pam  14:37  
So, same difference, right? Same distance between the numbers? Okay, but.

Kim  14:41  
But this time, I'm starting at -2, and I'm subtracting something larger than what I have, so it's going to be negative. What did I say? -9.

Pam  14:51  
-9. The distance is 9, but we're removing something bigger than we started with, bam, the answer has to be negative. So this is not trivial... 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.

Kim  15:10  
Mmhm.

Pam  15:10  
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 eight 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  15:56  
Mmhm.

Pam  15:56  
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  16:33  
Okay.

Pam  16:34  
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  16:41  
Make a guess, yep.

Pam  16:43  
Yeah, so long, skinny triangle.

Kim  16:44  
Yep.

Pam  16:48  
Not the longest side, but the next longest side.

Kim  16:52  
Okay.

Pam  16:52  
I don't know, just draw an equilateral. Don't draw isosceles triangle. So, three different lengths.

Kim  16:57  
Okay.

Pam  16:58  
The middle length is A. That's A long.

Kim  17:01  
Okay.

Pam  17:01  
Okay, the shortest length is 8 long.

Kim  17:04  
Okay.

Pam  17:05  
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  17:12  
Okay.

Pam  17:14  
Now, I want... Do you have your smaller one?

Kim  17:16  
Yep.

Pam  17:17  
And do they look the same shape? 

Kim  17:18  
Yep. 

Pam  17:19  
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  17:25  
Yep. 

Pam  17:26  
Okay, cool. That middle length side is 12. And the smallest length is 4.

Kim  17:33  
And I don't know the other lengths?

Pam  17:35  
No. But what you do know is the two triangles are similar.

Kim  17:39  
Okay.

Pam  17:40  
So, that's kind of important. Now, we haven't really defined similar triangles, but it's same shape.

Kim  17:44  
Mmhm.

Pam  17:44  
But the side lengths have been scaled. The side lengths are in proportion.

Kim  17:48  
Mmhm.

Pam  17:48  
So, the ratio should be equivalent. So, do you have a gut instinct on this one? What do you want to...

Kim  17:52  
Like to find A?

Pam  17:54  
Yeah, to find A.

Kim  17:54  
Yeah. Yeah, I want to double 12.

Pam  17:57  
Why would you do that?

Kim  17:58  
Because the side lengths that are 4 and 8, the 8 is double the 4.

Pam  18:03  
Nice.

Kim  18:03  
And so, like A is double of 12.

Pam  18:06  
I'm going to slow you down. 

Kim  18:07  
Sorry!

Pam  18:07  
No, no, no. It's okay. I just 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 2 triangles, I've now drawn a scaling arrow from that 4 to the 8, so that's kind of between the 2 triangles.

Kim  18:22  
Mmhm. 

Pam  18:22  
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  18:32  
Yep.

Pam  18:32  
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  18:41  
Yeah.

Pam  18:42  
Yeah, so that 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  18:51  
Yeah.

Pam  18:52  
Cool, so I would then bring out to students, like make sure that makes sense to everybody, and we would call that a between strategy.

Kim  19:01  
Mmhm.

Pam  19:01  
Because you looked between the two triangles.

Kim  19:04  
Mmhm.

Pam  19:04  
And said, "Okay, I know they're similar, so they got to be the side lengths from proportion. So, 4 to 8 has to be the same scale as 12 to A.

Kim  19:13  
Mmhm.

Pam  19:13  
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 triangles.

Kim  19:28  
Yeah.

Pam  19:28  
Its the one that corresponds with A. So, Kim, if I direct your attention there, what are you thinking about?

Kim  19:34  
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 3 would be A.

Pam  19:47  
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. 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'll use that same scale factor in the other triangle. 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  20:58  
Yeah.

Pam  20:58  
And sometimes, one might be easier to use than the other.

Kim  21:02  
Mmhm.

Pam  21:02  
And so we want kids to look at both of them.

Kim  21:09  
Alright, y'all, we tackled some key topics that you 6-8 teachers tackle every day. Moving students from multiplicative thinking to proportional reasoning is at the heart of your work, and we are thrilled that you tune in each week to learn more about how to make math figure-out-able for your students. Keep up the great work.