Pam Harris  00:02

Hey fellow mathematicians. Welcome to the podcast where math is Figure-Out-Able. I'm Pam.

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

And we're here to suggest that mathematizing is not about mimicking or rote memorizing, but it's about thinking and reasoning, about creating and using mental relationships. We are all about empowering teachers and students. We answer the question, if not algorithms, then what? Alright, Kim, so two weeks ago, we talked about the connection between fractions, decimals, and percents in Episode 19. And then last week, we were playing with percentages in Episode 20. And we had such a ball doing that and we had great response from listeners that we decided to have this week's episode be focused on Playing with Percentages Part 2 continued, because percents are Figure-Out-Able. One of our listeners Kim Axtell sent us this email this week, "Just wanted to share that I totally just found 24% of 88, with my head and without paper or calculator, and pretty quickly, thanks." Great Kim. So super glad to hear from that Kim, about how she learned from our episode last week about Playing with Percentages. Let's play some more.

Kim Montague  01:17

Actually, we also heard from one of your Journey members, Cathy Campbell. And she said, "I don't want to give it away. But as I'm listening to the questions you're asking each other, I hear the second to last question. And I predicted that you were going to make a connection to a previous question and discuss a possible strategy, but you left me hanging, and then you close the podcast with make a comment if you figured out our favorite strategy, and I figured out what you were up to." So that was fun to hear from Cathy. So just like last week, today, we're going to actually do some math. So if you're right now, going on a walk, or you're, you know, on the elliptical, we think you'll be able to hang with us. But if you can grab a piece of paper and something to write with. Maybe just have your finger ready on pause so that you can do some math with us.

Pam Harris  02:04

Alright, so last week, we talked through a bunch of percent problems, but they were actually all the same type. So this week, we're going to work with all three types. So in a percent problem, there are three values involved: there's the total, there's the percentage, and then there's the part the the sort of part of the total that you're finding. So if we're talking about three different kinds of percent problems, it's because one of those parts is missing. So one of the values is missing. And those are the three different kinds of ways that we can ask about percent problems. The first type is all about finding the percent of a number. So if we start with the total, then we find the percent of it. And we talked about these last week. Like for example, finding 12% of 50, or 50% of 12. We did that last week, and 45% of 10 or 10% of 45. So those were some of the problems we did last week. Let's do a couple more like those in case you didn't get a chance to listen last week, or you just want to have more opportunities to build some percent sense. Alright, Kim, so you're ready? You're on first, 

Kim Montague  03:04

Sure. Okay. 

Pam Harris  03:06

Okay. Eighty excuse me, 18% of 20. What is 18% of 20? We want to hear what you're thinking,

Kim Montague  03:14

Okay. Okay, so here's what I'm thinking. If I need to find 18% of a number. Any number, I could find 10% of that number, and 5% of that number and 3% of that number by finding 1% scale up to 3%. But for this one, I'm thinking 18% oh, I can also find 20% and then back up. So that's what I'm going to do. I'm going to find 20% of 20. And what does that 4? 20% of 20 is 4, but then to find 8%, I've got to go back 2% to get to 18. 

Pam Harris  03:49

And I'm actually gonna back up just a minute. How do you know what 20% of 20 is?

Kim Montague  03:53

So 20% of 20 is, that's a fifth, 20% of a fifth. So if I want to think about 20 divided by five, and that's four.

Pam Harris  04:02

Gotcha. Okay, so 20% of 20 is four and how's that going to help you find 18%?

Kim Montague  04:07

So if I know 20%, then I can back up 2% To get to the 18%. And since I already know 20%, then 2% is really nice, because I can just scale down from 20% to 2%.

Pam Harris  04:21

Cuz like divided by 10. Okay, right, right.

Kim Montague  04:23

So so if 20% was four, then 2% is just going to be four tenths, or point four. 

Pam Harris  04:32

Nice.

Kim Montague  04:33

So that's just 3.6 so 18% of 20 is 3.6.

Pam Harris  04:37

Because you went back 2% from 20% and four subtract four tenths is three and six tenths. Nice. All right, cool. Totally, totally. Figure-Out-Able. Excellent. And that's what mathematicians do. Right? So you just heard Kim have one attack where she could have found 10%, 5% and 3%. And that's a way to get 18% but for this particular problem, she's like, "Actually, I'm going to go ahead and find 20% and backup 2%." And mathematicians do that. They think, they decide on a course of action, they rethink and then they might choose which one like feels the best to work with that day. Very cool. 

Kim Montague  05:14

Right. Okay, you ready? For one? 

Pam Harris  05:16

Yes. 

Kim Montague  05:16

Okay. Alright. What if you were asked 20% of 18? 

Pam Harris  05:22

20% of 18. Okay, so my plan of attack is going to be to find 10% of 18. And then double it to get 20%. 

Kim Montague  05:31

Okay 

Pam Harris  05:31

so 10% of 18 is one and eight tenths or 1.8. 

Kim Montague  05:36

Okay. 

Pam Harris  05:37

10% of 18 is 1.8. But I want 20%. So 20% is double 10%. So I need to double that one and eight tenths, or that 1.8. And so double 18 I know is 36. So double one and eight tenths would be three and six tenths or 3.6. 

Kim Montague  05:54

Nice. 

Pam Harris  05:54

Yeah? Okay.

Kim Montague  05:55

Well done. So the first type of percent problem is finding a percentage of a number like 18% of 20, or 20% of 18. So Pam, if that's the first type of percentage problem, let's talk about another one.

Pam Harris  06:08

Okay, so we could also think about, the first type is finding percents of a number. We could also find the percentage. So if we're given the total, and we're given the part then we could find out what the percentage is. Or if we're giving the part and the percentage, then we could find the total. So let's work first on the part where I'm going to ask you for the percentage. We don't know the percent at this time. So like, if I were to say, we're gonna start with 20. And I said, hey, 20 is what percent of 20?

Kim Montague  06:41

Let me wrap my head around for just a second. 20 is what percent of 20. That's just gonna be 100%.

Pam Harris  06:47

Yeah, cuz 20 is 100% of itself. Right? 

Kim Montague  06:50

Right.

Pam Harris  06:50

Okay, good. Very good. Okay. 10 is what percent of 20?

Kim Montague  06:56

10 is just half of 20. So that's going to be 50%. 

Pam Harris  07:01

So 10 is 50% of 20. Very nice. Okay, good. What about five is what percent of 20?

Kim Montague  07:09

5 is what percent of 20. So five is going to be 25% of 20. So 10. Right? 10 was 50%. So five is just half of that. So that's going to be 25% of 20.

Pam Harris  07:22

Cool. And another way to think of that as five is a quarter of 20.

Kim Montague  07:25

Right. 

Pam Harris  07:25

20 divided by four is five and a quarter. 25%. Okay, cool. 

Kim Montague  07:29

Okay. 

Pam Harris  07:30

We're still focusing on 20 here.

Kim Montague  07:32

Okay. 

Pam Harris  07:32

15 is what percent of 20?

Kim Montague  07:36

Nice. So five was a quarter of 20. So 15 is going to be three quarters of 20 or 75%.

Pam Harris  07:45

Nice. Okay, cool. So 15 is three quarters of 20 or 75%. So, I'm back in fractions. We want to say percents today. Okay, cool. Alright. So staying with that whole 20, that total of 20. One, the number one is what percent of 20?

Kim Montague  08:04

We just said 10 was 50% of 20. So I'm gonna scale down. So if 10 was 50%, then one is just going to be 5% of 20.

Pam Harris  08:16

Interesting. You got any other ways of thinking about that? 

Kim Montague  08:18

Yeah, actually, I was thinking about, I was just talking about coins with my son the other day. And so I know that there are 20 nickels. And there's a relationship for me about -

Pam Harris  08:28

In a $1.

Kim Montague  08:29

- in a $1. I could also think about if we said five is 25%, then you could scale down by five. 

Pam Harris  08:38

Oh, nice. 

Kim Montague  08:39

So five is 25%, then one would be 25% divided by five. So that's 5%. 

Pam Harris  08:46

Ah, very cool. Very cool. I like the nickel. And that's really cool. One out of the 20 nickels would be five, like five cents or 5%. Cool. Alright. So how about two? Two is what percent of 20?

Kim Montague  08:59

Oh, good. I'm glad you just asked me about 1% or one. One is 5%. So then two would just be double that 10%.

Pam Harris  09:07

Cool. You got any other ways of thinking about that?

Kim Montague  09:09

Two, what else have you asked me? Oh, yeah, so 100% is 20. You stopped the 20. So scale down a 10th of that. So a 10th of 100% would be 10%.

Pam Harris  09:21

Nice, very cool. Now, we use some numbers here that are pretty reasonable, pretty Figure-Out-Able. But we can get pickier. We could get into some gnarly or uglier numbers, but probably not on a podcast. Like we would have to, I would want to use a percent bar where I would put the information down that we know and then I would chunk things and write other things I can figure out. And I can get really into some pretty gnarly solving problems either the first step that we just did or that we did today or the finding the percentage either way, if we wanted to get a little pickier. But we would just want to use relationships like we just did. And we would just have to, you know, keep going and get a little bit more precise. Alright, cool.

Kim Montague  10:02

Can I ask you some? 

Pam Harris  10:03

Yep, I'm ready. 

Kim Montague  10:05

Alright.

Pam Harris  10:05

Okay.

Kim Montague  10:06

Alright. So let's work with the number 12. You gave me 20. We'll do 12. 

Pam Harris  10:11

Okay.

Kim Montague  10:11

So six is what percent of 12?

Pam Harris  10:16

Alright, so six is half the 12. So it's 50%. So 6 is 50% of 12.

Kim Montague  10:20

Yep. Good job. What about three? Three is what percent of 12?

Pam Harris  10:25

Okay, so three is a fourth of 12. That's one way to think of it. So that's 25%. But it's also half of the six we just had, and six was 50%. So three is half of six. So half of 50% is 25%. So three is 25% of 12.

Kim Montague  10:41

Excellent. Alright, let me go with 1.5. 1.5 is what percent of 12?

Pam Harris  10:47

Whoa.

Kim Montague  10:48

Sorry about that. 

Pam Harris  10:50

Big meanie. Okay, hang on. So if three was 25%, and one and a half is half of three, then I want half of 25%. And that's 12 and a half percent. Okay. So one and a half, 1.5 is 12 and a half percent of 12. Whoo.

Kim Montague  11:09

Nice. Okay, one more, one more. Let's go with 4.5. 4.5 is what percent of 12?

Pam Harris  11:18

Kim! Okay.

Kim Montague  11:19

You can do it. 

Pam Harris  11:20

Okay. So we already know that three was a quarter of 12. And one and a half was 12. And it's almost like I feel like I'm dealing with eights here. Because that 12 and a half percent is like one eighth. Anyway, it's just sort of feeling like it. So once I add the three and the one and a half to get four and a half, I'm thinking about 25% and 12 and a half percent. That's 37 and a half percent. Yeah, I'm definitely thinking in eighths here. That's interesting. So 4.5 is 37 and a half percent of 12. Or it's also three eighths of 12. If you want to go that direction.

Kim Montague  11:52

Ah, well done. Alright, let's talk about this third type of percent problem.

Pam Harris  11:59

Right. Okay. So the third type, the one we haven't talked about yet is where the total is unknown. So we know the part. We know the percentage, but we don't know the total and these might be the most challenging for me.

Kim Montague  12:11

Okay, give me an example. Let's see if I can make it.

Pam Harris  12:16

Alright, so let's use 20. Again, right? What if I say 20 is 100% of what number? We don't know the total. 20 is the part, it's 100% of what number?

Kim Montague  12:29

Okay, that one's good. 20 is 100% of 20. 

Pam Harris  12:32

Because just itself, right? 20 is 100% of itself. Okay, cool. You thought these are going to be hard. 20 is 50% of what number?

Kim Montague  12:41

20 is 50% of 40. So 20 is just half of 40. I got it. Okay.

Pam Harris  12:48

Okay, cool. So 20 - Yep. Good. That works. Okay. 20 is 10% of what number?

Kim Montague  12:55

20 is 10% of 200. I can just scale up times 10. So, yeah, 20 is 10% of 200.

Pam Harris  13:07

Because if you've got, if you know 20 is 10% of a number, that number has got to be much bigger than 20. Right? 

Kim Montague  13:13

Right. 

Pam Harris  13:13

Like you said, it's gonna have to be 10 times bigger because we're talking at 10%. Cool.

Kim Montague  13:17

Yep. 

Pam Harris  13:18

So 20 is 10% of 200. 20 is 5% of what number?

Kim Montague  13:24

Ooh, it's gonna be pretty sizable number. So 20 is 5%. So I want to go from 5% to 100%. So I'm going to scale up times 20. So 20 is times 20 - 400. 20 is 5% of 400.

Pam Harris  13:41

Because 5% of 400. You can think about that. That's 20. Yeah, cool. 

Kim Montague  13:45

Yeah. 

Pam Harris  13:45

Cool. See, these are too bad, right?

Kim Montague  13:47

Not bad. Alright. Can you go?

Pam Harris  13:48

Can you describe what was going on in your head a little bit with these a little differently?

Kim Montague  13:52

Um, yeah, so I actually thought about that last problem two ways. So I had already said that 20 is 50% of 40. So now, there's a relationship between 50% and 5%. So I have a 10th of the percent. So I need 10 times greater of a number.

Pam Harris  14:17

That blows my mind just a little bit. So if 20 is 50% of 40. Then you're thinking 20 is 5% of a much bigger number, right? How much bigger? 10% Bigger, right? So 10% times 40 is 400. Interesting. Okay, cool. Cool.

Kim Montague  14:32

Okay, you want to go? 

Pam Harris  14:33

Alright, bring it on. Okay.

Kim Montague  14:34

I'm gonna ask you the same questions, but I'm gonna give you a crunchier number. Okay?

Pam Harris  14:37

Of course. 

Kim Montague  14:39

Alright. So let's do, we did 12 last time. Let's do 12 again. 

Pam Harris  14:42

Alright. 12 is my friend.

Kim Montague  14:44

12 is 100% of what? 

Pam Harris  14:45

That's easy. 12 is 100% of 12.

Kim Montague  14:47

Okay. You can do this 12 is 50% of what?

Pam Harris  14:51

So 12 is 50% of something. It's half of something. So 12 is half of 24. So 12 is 50% of 24.

Kim Montague  14:58

Okay. 12 is is 10% of what?

Pam Harris  15:03

Okay, if 12 is 10% of a number, then the numbers are big. And it's going to be 10 times bigger, right? And so 12 times 10 is 120. So 12 is 10% of 120. Because now I'm gonna think in the other direction. If I have 120. And I want 10% of it. Yeah, that's 12. Okay. Yep.

Kim Montague  15:22

Alright, what is 12 is 5% of what?

Pam Harris  15:27

Okay, so this is gonna be a bigger number. Because if 12 is 5%, then 12 is 5%. It's 5% of this big number, right? Because it's just 12, little tiny 12 is just 5%, five percent is like. So I could do a couple things. I'm actually tempted to say if 12 is 5%, then I know that double that, 24 is 10%. And so if 24 is 10%, then I can multiply that by 10. And so that's 240. So 12 is 5% of 240. 

Kim Montague  16:03

Nice. 

Pam Harris  16:04

Yeah, that's one way. And then I'm gonna try your strategy because this is not as natural for me. I know that five percent is like a nickel and there's 20 nickels and $1. So I could, if 12 is 5%, then I could scale the 12 up by 20. So I could do 12 times 20. And since 12 times two is 24, 12 times 20 would be 240. So 12 is 5% of 240. Right. That's a strategy. Okay. Bam. 

Kim Montague  16:27

Okay, I'm actually give you one more.

Pam Harris  16:29

Thanks a lot. 

Kim Montague  16:30

Uh huh. 

Pam Harris  16:31

No, I like it. I like it. Playing is good.

Kim Montague  16:33

It is good. 12 is 150% of what?

Pam Harris  16:38

Wow, that is, we haven't done any that are over 100. I guess it's time that we did. 100%. Okay, good. Alright. So 12 is 150%. So this is different than all the other ones we have done in the last little bit. Because now if 12 is 150% of something, it's bigger than the sort of total. It's bigger than the referent that we're referring to. So 12 is like three halves. That's kind of actually how I'm thinking about it. I'm thinking about it, like three halves is 150% is like 50%, 50%, 50%. So it's like three parts of the original. And so if I cut 12 into those three parts, that's like four is the part. But I want two have those parts to get the whole 100%. I don't know if that's making any sense. And so eight. So 12 is 150% of eight. 

Kim Montague  17:27

Nice. 

Pam Harris  17:27

Yes. Yeah. Do you think about that that way?

Kim Montague  17:31

I did. Actually, I was following you because it's exactly what I was thinking about.

Pam Harris  17:35

Okay. Alright. Cool. Because if I've, if 100% of eight is 8, then 150% of eight would be another half of it, which is 12. So yeah. 12 is 150% of 8. So I just kind of went backwards to sort of check, make sure that we're good. Okay, cool. So this is interesting, right? Because traditionally, typically, if you look at a typical textbook or a traditional classroom, we tend to see teachers kind of use only one way of thinking about percent problems. And I typically see two main sort of one ways of thinking about percent problems. One way is that teachers will say, "If it's a percent problem, set up a proportion, put the part over the whole equals the percent over 100, and then cross multiplying, divide, and you can solve for the part that's missing." And so what happens is kids like do the same thing every time. All they have to do is figure out, do I have the part? Do I have the whole? Do I have the percent? Put it in where it goes, and then solve for the missing thing by using kind of Multiplicative Reasoning to cross multiply and divide. Or, hopefully, at least Multiplicative Reasoning or maybe not even that if they're using an algorithm or calculator at that point. But my point is, they sort of do the same thing every time. That's one traditional way. Another traditional way is that they turn the percent problem into an equation. This is a typical algebra kind of solution. Maybe the proportion one might be a typical middle school kind of solution. But the same thing happens because as soon as I turn it into an equation, then I kind of write everything down, like x times the percent equals the total or something like that. And then I kind of solve for the missing part. And I use the equation solving techniques to solve for the missing part. But I'm kind of doing the same thing every time again. I'm putting in what I know. And then I'm sort of solving for what I don't know. What I'm not doing is actually making my brain think and reason about the percent. I'm either translating it into a proportion, or I'm translating it into an equation. And then I'm solving the same way every time. What I'm not doing is really thinking about what what do I know, and how does that relate to, how do these kind of the pieces relate to each other so I could figure out the missing part. And then that actually, like makes my brain be able to handle more and more complex relationships. So it's an interesting, this is a perfect moment for us to think about what we're advocating in math education that's different from traditional. Because traditional says, just do this one way and do it the same way every time. Practice it until it kind of gets this like go to and I've got this. Like people call it sometimes procedural fluency. And that's not so much what we're interested in. Do I want students to be fluent? Absolutely, but I really want them to be fluent in thinking and reasoning and using relationships. I want their heads to get, their brains to get more, to be able to handle more and more sophisticated things. Or, or, by the time we get them in later math, all we can do is say, "Oh, your brain isn't ready to handle all this sophisticated stuff. So we better just give you rules and stuff to mimic and procedures to follow the steps." Because that's all they can handle. We don't want that, right? We want to build brains, we want to empower people to be able to think and reason using what they know, and then they can handle more and more crankier problems. Alright, so let's recap a little bit. There are three types of percent problems that we mentioned today. One is where the part is unknown, one where the percent is unknown, and one where the total's unknown. Every time ask yourself what you know, and then use friendly, friendly relationships to solve for what you've don't know. 

Kim Montague  21:05

So much fun today. Hey, don't forget to join the fun MathStratChat on your favorite social media on Wednesday evenings. Thank you guys so much for the five star ratings on Apple podcasts. We have loved reading comments, and we really appreciate you posting them. 

Pam Harris  21:19

So if you're interested to learn more math, and you want to help students become mathematician where they mathematize their world, then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able.