Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim Montague.

Pam Harris  00:11

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 mathematical relationships. We are all about empowering teachers and students. And we answer the question, if not algorithms, then what?

Kim Montague  00:30

Yeah, so the other day, I was working out with a teacher, friend, and she said -  She said, "Hey, you know what you can help me with Kim? I want to get better at thinking about percentages." And you know, that's like my jam. So she said, "It's not something I teach, but I know I could know more than I do." And so since they're my favorite thing, we figured we could have a great ponder together this week. So this week, we're gonna bring out the episode that we're calling Playing with Percentages.

Pam Harris  00:34

You do that a lot. You work out a lot.  Absolutely. And I know the teacher friend that you're talking about. Shout out! And I really appreciate her growth mindset. This adult teacher who doesn't really teach percentages. And she's like, "I bet I can know more about that. Like, while we're working out, let's play with that?" Could we make more math be that way? Let's do it. And, you know, I've seen so many rules and procedures when it comes to percents. And this is one area where we can toss all of those away pretty easily and gain a whole lot of reasoning power, if we'll just reason with students. Help them build their percent reasoning power by asking good questions and helping them make connections.

Kim Montague  01:38

Right. So first of all, we talked a little bit last week about this, but 100% and 0%, kids need to understand what that means to have 100% of something. And while they've never maybe heard the language of percent, it won't mean anything, but more and more with devices, right, they are hearing language and seeing models for percentages. So it's not quite as foreign as it might have been quite a while ago, like our young years.

Pam Harris  02:05

Absolutely. So if you haven't listened to last week's podcast, you might go listen to it, because we talked a little bit more about why kids have experience with percents. We're just going to sort of dive in and reason with percents today. So you just talked about how the kids need to know 100% and 0%. You might find it interesting that the next most important sort of bit is, kids need to understand 10%. Because so much is based on 10% and 1%. So check this out, I started teaching in the state of Utah, my very first teaching gig for two years, I taught in the state of Utah. And it was fascinating to me, when I moved, I didn't really appreciate how well those kids dealt with 10%. And it totally has everything to do with Utah's predominantly a Mormon state, and they tithe. As a part of our church, so I'm a member of the Church of Jesus Christ of Latter Day Saints, and as part of our church, we tithe 10%. And so there's this sort of 10% thing happening all over in the state of Utah. And then when I moved all of a sudden, that was not such a given. We moved to Michigan, and I worked with great kids in the state of Michigan, but just not everybody had as much experience with tithing for their church. And so the 10% thing wasn't happening. And so we had to build it. So we built what does times 10 and divided by 10 look like. And now we can think about 10%. And so it's gonna be really important that you need to establish 10% with experience that you give your kids experience where they divide by 10 and multiply by 10. And they can think about those place value patterns that are happening. And then they could just quickly sort of mess with 10%. So if you're a fourth, fifth, sixth grade teacher, you're going to want to give students experience finding that 10% a lot and then tie it to the work that you're already doing with times 10 and divided by 10, and number shifting, and place value. We'll do some more about place value shifts in an upcoming episode. So stay tuned for that, because that's really important. So now that you've established this sort of 10%, and 1%, check out what you can do. Alright, to do that, we're gonna do a quick Problem String here on the podcast. Alright, you might want to pause a little bit and grab a piece of paper and a pen or pencil to write with. You don't have to, but if it's easier for you to kind of follow what's happening. If you can kind of record your thinking. You also might want to have your finger on the pause button, you might want to think about the question before you hear us give our solutions. It's always better for you to have the relationships in mind in your head before you hear someone else's strategy. So we're going to do some talking out loud about our thinking. But you can go ahead and write down just the problem and the answer at least or record whatever else is helpful for you. Alright, Kim, you ready?

Kim Montague  04:53

I feel a little nervous on the spot.

Pam Harris  04:56

It's all good. Okay, I gotta give you a number and then you're gonna find a certain percentage of it. Alright? 

Kim Montague  05:02

Sure.

Pam Harris  05:02

Everybody ready? Okay, here we go. What is 24% of 88? 

Kim Montague  05:09

Okay, 24% of 88. So I like 25% of 88. And that is going to be - 25% of 88 is 22. So half of 88 is 44, half of that is 25%. So that's 22. But I need 24% of it. So I'm gonna go 1% less. So 1% of 88 is point 88 or 88 hundredths. So it's just 22 minus 88 hundredths and I'm really good with I Have, You Need. And so that's going to be 21.12.

Pam Harris  05:43

Because you sort of found the difference between 88 and 100. Is that .12 or 12 hundredths. And then you just say, subtract it from the twenty-two. That's nice. That's excellent. I had to say that out loud to kind of follow along with what you were doing. Nicely done. 

Kim Montague  05:56

Okay, you're going now. 

Pam Harris  05:58

Alright. All right, back at me, here we go. Okay, good. I got it. I got it.

Kim Montague  06:02

What is 12% of 50? 

Pam Harris  06:06

12% of 50. Okay, so remember, we said everything was based on 10%. So I'm going there. So 10% of 50 is five because if I divide 50 by 10, I get five. So 10% of 50 is five, but I'm supposed to get 12%. So 1%. If I know 10% is five, then 1% is a 10th of that. So that's half or point five. So 1% is a half, but I needed two to add on to the 10. So that 2% would be double that half and double half that's easy. That's just one. So I've got 10% is five, 2% is one, so 12% of 50 is six. Whoo!

Kim Montague  06:48

Great. 

Pam Harris  06:49

Did you follow that okay? 

Kim Montague  06:50

Yeah.

Pam Harris  06:51

Yeah, yeah, let me - there's another way that if you don't mind me sharing, so 12% of 100. That's totally easy, right? 12% of 100 is just 12. But I wanted 12% of 50. So 12% of 50 is half of that. And a half of 12 is six. So I could think about it either way. Nice. I kind of like my second way better. Alright, so are you ready for another one? 

Kim Montague  07:13

Yes. 

Pam Harris  07:14

Okay, your next problem is, what is 60% of 75? 60% of 75?

Kim Montague  07:23

Okay, 60%. I'm going to go 50% of 75. And what is 50% of 75? That's half of 70 is 35. And half of seven. I'm sorry, half of five is 2.5. So what is that? 37.5? 

Pam Harris  07:39

Yep. 

Kim Montague  07:40

Okay, and then I'm gonna go 10% of 75. Oh, that's just 7.5. So then I've got 37.5 and 7.5. And what is that? I've got nothing written down. I'm in the air. 37.5 and 7.5. What does that - 45? Okay. So I need to write something down.

Pam Harris  08:08

It ends up that the addition was the hardest part, right? Because you found 50% was 37 and a half, and you had to add the extra 10%. That was - I've lost myself. You had to add the 37.5 and the 7.5. And that ended up being the hardest part of the whole problem. Nicely done. Whoo. Okay. Alright, y'all. We asked you guys to get a paper pencil and then we didn't. Maybe we should have. Alright, what's my next question?

Kim Montague  08:40

Next question. You've got 75% of 60.

Pam Harris  08:44

Oh, you're nice. Thank you. That's a nice one. So I'm thinking about three quarters of 60. So this is one of those times where I'm gonna go from percents to fractions. And I'm gonna think instead of 75% of 60 I think about three quarters of 60. So one quarter of 60 is like 60 divided by four or like half of 60 is 30, and half of 30 is 15. So one quarter of 60 is 15. So three of those quarters, three of those fifteens would be 45. So 75% of 60 is 45. Okay. Alright. Next question. What is, ready? The hardest one so far. What is 50% of 12? 

Kim Montague  09:25

Oh, you're so nice. Okay, 50% of 12. 100% of 12 is 12. So 50% is going to be half that much. 6. Didn't you get a six? Wasn't that one of your answers? 

Pam Harris  09:38

Oh I did get a six. Let's see, my problem was 12% of 50. So they were different problems. Let's go on.

Kim Montague  09:45

All right, you get the last one. Ready? Okay, check it out. You get 45% of 10. 

Pam Harris  09:52

I gave you a nice one... Let's see. 45% of 100 is 45. And if I divide that by 10 to get 45% of 10, then that would be 4.5. Now that's one way to do it. But I'm thinking, let me keep thinking. To get 45% of  10. I could also think about going from 50% of 10. Okay, so like 50% of 10, that's like half of 10. So that's five. But that's 50%, but I need 45% of 10. So I need to get rid of 5%. Well, I already had 50% being five. So 5% would be point five. So now I've got to do five minus point five. And that's another way to get 4.5. Okay, cool.

Kim Montague  10:44

Nice. I love the way that you're thinking about a couple of different ways for your problems, and I'm just happy to get one right now.

Pam Harris  10:50

Well, maybe it's because you're so darn confident. And I'm the one that has to do it a couple different ways to make sure that - and you're the one that taught me to play with different relationships and then choose the one that you like the best, cool. And y'all check it out, percents are figure-out-able. I hope that you're able to follow us on what we just did. So, so cool. We can find percent by using what we know about 10% and 1%, and other handy fractions of numbers to build up. So go figure, go play with percents and have some fun playing around with relationships.

Kim Montague  11:25

So for those of you who are writing down some of the problems and the answers, you might notice one of our very favorite strategies for solving percentages.

Pam Harris  11:33

Hint, hint, hint.

Kim Montague  11:34

Yeah, when you have a minute comment on our podcast with what you noticed today.

Pam Harris  11:39

Alright, so thanks for joining us today on our episode where we were Playing with Percentages. 

Kim Montague  11:45

Hey, don't forget us on MathStratChat on your favorite social media on Wednesday evenings. Thanks so much for your ratings on Apple podcasts. We love reading your comments and we really appreciate you posting them.

Pam Harris  11:56

So if you're interested to learn more math, and you want to help students develop as mathematicians then the Math is Figure-Out-Able Podcast is for you because math is figure-out-able!