October 27, 2020
Episode 19

Math is Figure-Out-Able with Pam Harris

Ep 19: Fractions, Decimals, Percents

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Math is Figure-Out-Able with Pam Harris

Ep 19: Fractions, Decimals, Percents

Oct 27, 2020
Episode 19

Pam and Kim explore what it means to be flexible with fractions, decimals, and percents. They discuss how we can become more flexible as we deal with rational numbers, and how we can be judicious problem solvers.

Talking Points

- Why kids of today are more familiar with percentages
- Why flexibility matters
- How to become more flexible using different representations of rational numbers

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Pam and Kim explore what it means to be flexible with fractions, decimals, and percents. They discuss how we can become more flexible as we deal with rational numbers, and how we can be judicious problem solvers.

Talking Points

- Why kids of today are more familiar with percentages
- Why flexibility matters
- How to become more flexible using different representations of rational numbers

Pam Harris:

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

Kim Montague:And I'm Kim.

Pam Harris: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. And we answer the question. If not algorithms, then what? Alright, in today's episode, we are going to do some math, we like to have an episode every once in a while where some mathematics happens. So today, you might want to have a paper and pencil handy if you'd like to record things, if it's easier for you to think about stuff. If you can write something down. You might have your finger on the pause button, just in case you want to think about something before we kind of move on.

Kim Montague:Right. So in today's episode, we want to talk about one of our favorite mathematics topics, fractions, decimals, and percents. Specifically, the connections between them. So Pam, we both find it interesting that some people really like one of these more than the other right, they find them more relatable or easier to work with and favor them in some way. Like, for instance, I really like dealing in percentages, and I have -

Pam Harris:You are the percent queen.

Kim Montague:But I have a feeling that my husband probably prefers fractions, because of all the building that he does.

Pam Harris:Totally. And I'm actually a bit more comfortable with decimals, I sort of like to hang in decimal land, and I'll deal with fractions and percents if I kind of have to. Yeah. So we had a discussion on our membership site Journey the other day, and it seemed like one of the teachers that we were talking to is really more comfortable in fractions. It's so interesting that we kind of had these comfort zones.

Kim Montague:Right. And either way, whatever your comfort zone is, we want to suggest that we don't want to leave anyone in just one of those forms, because the power is in the connection between them.

Pam Harris:Yeah, so today's episode is all about the

connections between these three:fractions, decimals and percentages. So what do we mean by that? Well, like all math, we want them to be Figure-Out-Able.

Kim Montague:So here's an interesting tidbit. For a lot of places, one of the early mentions of fractions is pretty young, but it really lands heavy in like, third, fourth, fifth grade. And the first mention of decimals is writing money amounts, younger, like second grade, and it hits heavy with computation in fourth and fifth grade. But percentages are not mentioned mostly until about sixth grade.

Pam Harris:Yeah, so it's kind of interesting, right? There's a mathematical reason for that, to really understand percentages, percentages are all about ratios. It's all about the ratio out of 100. And so it's really this relative amount. And we usually wait until six/seventh grade to do a lot with ratios and proportional reasoning. And so the curriculum sort of usually waits that long to deal with percentages, but we both feel - you and I Kim - we feel like we can use the real world experiences that kids have with percents to do stuff way earlier. So Kim, talk about some of those.

Kim Montague:Sure. So right, we live in a digital world. And so we have children who have percent bars on devices everywhere, right? So they're, they're playing games, and they have download percentages. They have cell phone batteries that showed decreasing battery life.

Pam Harris:Totally it's like a picture of a battery, right? Filled in how much battery life they have left.

Kim Montague:There's progress bars and games and on e-books.

Pam Harris:Games! Nbody is using games these days. Nobody plays a game. No electronic games going on. And ebooks? Uh huh.

Kim Montague:Yeah. So and just online in general, and computers, they're seeing these pictures of batteries with percentages right next to them. And so if nothing more, students are understanding 100% as the whole or the unit and zero percent as the beginning.

Pam Harris:Sure, when they look at those sort of percent bars, representing any of those things you just talked about, they can tell Oh, if I'm like closer to the zero, then nothing's downloaded different closer to the hundred I'm almost done. And they're really getting this, this kind of at heart feel for what's happening.. They have this sort of gut level instinct for percents. And it's when they're really young.

Kim Montague:Yeah, and they sure know when to plug in, right?

Pam Harris:Absolutely. Like they know when they're about ready to like, hit it. Oh, I can go now and you know, whatever the thing is, then, yeah, I need to go recharge right or I'm ready to go conquer the next whatever in the game, or oh, no, I better do something because my health is low. But all those things they really are gaining this gut level sense of what percents mean. So we also see percents in things like sales and tips, and of course, all those digital places that we just talked about. As students recognized Something like a 50% download or 50% progress in a game or 50% Health Points, that's all one half, right? That's the fraction one half. And they're sort of seeing that percent bar be - if you'd see my hand right now I'm like waving like like you can kind of see I've got the whole percent bar and then I'm landing in the halfway mark. And when students see that sort of 50% part and written right next to the bar often is that 50%, then we can sort of rename that right there, then over their shoulders, we go, Oh, you're halfway through. And we can use that sense of relative size to our advantage. And then go ahead and get formal with percents in sixth grade. But I think we can do a whole lot of informal stuff earlier.

Kim Montague:Right. If we can tap into that idea, we can say things like, oh, you're at 50, out of 100. 50%. We can help connect what they see on the bar to what they're experiencing and use that sense that they're developing. It makes me think of 'just in time' vocabulary, where you see students experience progress bars, and you can drop in the language kind of over their shoulder, like you just said.

Pam Harris:Yeah, so instead of just in case vocabulary, like frontloading, let's do all this stuff with percents and be real formal about it. 'Just in time', as it happens, you can just drop that language in and give students different ways to sort of talk about what they're seeing. All right, so let's talk about how to reason from a fraction to a decimal. So sometimes students learn this very formal procedure, that I have this fraction, what I must now do is do this long division thing in order to get the decimal. We want to, uh, definitely, definitely at the beginning, we want to help students really think about the connections between fractions and decimals. So let's start with an important benchmark fraction, like one half, and then we could use money to talk about the decimals. So like the fraction one half is 50 cents. And so how do we write 50 cents we write it point five or point five zero. And so that's a really natural way to serve, start connecting the fraction one half half of the dollar 2.5 in that decimal form. So Kim, talk to us about how we can use that to then reason further about fractions like one quarter and one eighth.

Kim Montague:Yes. So if you know one half is point five, or point five, zero, then you can know the connection between half and one fourths. So one fourth is just half of a half. So that's going to be one quarter point two, five or 25 hundredths.

Pam Harris:Because we're talking about half of that point five half of that 50 cents is going to be that 25 cents.

Kim Montague:And so one eighth, you can just think about the relationship to a quarter, one eighth is half of a quarter, right? So half of a fourth, which is .125, or 12 and a half hundredths.

Pam Harris:Woah, you did that really fast. If you've never thought about half of a quarter half of 25 cents, then you might have kids actually have to think about half of 25 cents. They could do maybe half of 20 cents, and that's 10 cents, and then half of that five cents is two and a half cents, right? So they could sort of add that 10 cents and that two and a half cents to get that 12 and a half that you were just talking about. You might also have kids think about half of 24 cents. Oh, that's just 12 cents. Now they only have one cent leftover, and well, what's half of that one cent? Sure enough, that's just half a cent. So that's another way of sort of thinking about that 12 and a half. So we want kids to really think about half of 25 and work with that and mess with that a little bit. And it'll become you know, pretty, pretty reasonable to do. In fact, I have a story about eighths. So my neighbor, Russ, I don't know if I - have we talked about Russ on the podcast?

Kim Montague:I think so, yah.

Pam Harris:Everyone should know Russ. Russ is a brilliant neighbor because he can fix anything. So Russ has also had several different employment positions over the years. He likes to do different things. He's a very talented guy. At one point as we've known him over the years, he was a bank loan agent, or he was working at the bank, and he wanted to become the loan agent or something, something like that. And he hollered at me, and he said, Hey, can you help me understand something? And I was like, Sure, yeah, no problem, because he helps us all the time, right? I'm definitely going to help him with whatever he asks. And he said, so as I'm about to take this test, they keep throwing out these numbers. I don't understand where they're coming from. I was like, Well, what kind of numbers he goes out, like 37.5 and 87.5. And all these, like, I don't know where these numbers are coming from. And I just kind of smiled. I said, Oh, I bet they're talking about like loan rates. And he's like, yes, we're talking about loan rates. And I said, so are they are they also throwing numbers around like, 3 and an eighth or four and seven eighths. At that point loan rates were a little bit higher. And he goes, Yeah, yeah. So on the one hand, they talk about these eighths and fourths for for loan percentages. But then they also talk about these decimals and I just can't even tell what's going on. And so I just said, I kind of walked him through what you just did. I said, Well, like Russ, what's 50%? He said, Oh, that's easy. That's point five. That's a half. And so I said, Okay, well, what about the quarter? 25. Well, then what about an eighth? He's like, Oh, that's like, Oh1 12 and a half! That's where the 12 and a half is coming from? And I was like, yeah, so then what would it be if I if we're talking about seven, eight, so he goes, Well, that's just 12 and a half back from 100. That's just 87 - That's where! And he just like was on fire when he realized that I could just think about seven eighths as just one eighth back from 100. And what s 100, minus that 12 and a hal . Well, sure enough that's just eighty seven and a half. And I now I'm doing that kind of qui kly. But I've messed with eighth a lot now. And so I kind f have some numbers at my finger ips to sort of think about the elationship between the eights as a fraction and the d cimals and the percentages tha are those sort of equivalent . Cool. So he's a fraction gu in a huge way, Russ was more omfortable dealing with fractio s, because he's also does a lot with measurement, which I think is true for your husb nd. Right? They're both carpe ters, they measure a lot. So t ey're really comfortabl with frac

Kim Montague:Right, but if you only live in one of them, like I'm just a decimals person, or I'm just a fraction person, you're going to run into situations where thinking about a different form might be more helpful. Right. So -

Pam Harris:Yeah, absolutely.

Kim Montague:So I prefer percents. But there might be a time where I'm - or decimals. But if I'm thinking about one third, it's totally nicer to think about fractions, because the decimals are kind of gnarly with the repeating point 333333. Right?

Pam Harris:Totally.

Kim Montague:So 40%, really nice. I'd rather think about 40% of something, then four tenths or two fifths of it.

Pam Harris:And I've totally heard you like, you'll be looking at a fraction problem and you'll just zing right over to percents because it's so cool. And that's really nice. You were just mentioning 40%. That's like sort of two fifths. So we think about fifths, one fifth is one 20th of anything. And, and four fifths, the denominator is a factor of 100. So those are really nice, because we can sort of use a lot of relationships back and forth, because of that factor of 100. Part. So like one fifth is just 100 divided by five. So that's why it's connected to sort of 20%. And really brilliant, but maybe would you really go to decimals if I asked you sort of like two fifths of five?

Kim Montague:Oh, no, probably not then right? There are times when it makes more sense to stick in a particular area. I generally land in percents, but there are times where fractions make sense, right?

Pam Harris:Let's walk on just really quick. So if I'm thinking about one fifth of five, one fifth of five anythings would just be one, right? So then what are two of those one fifths would just be two. So two fifths of five is just literally two. You probably wouldn't go, "What's 40% of five?", you would just think about a fifth of five and then scale that up to get two-fifths of five. Yeah. But like you said, there are some cases where we would probably hang in one or the other. Like, for example, you mentioned one third. So let's talk about one third.

Kim Montague:Yeah, I might want to think about a third infractions rather than decimals or percents. Because it can be a little messy. I mean, it can be done, it just gets a little bit messier.

Pam Harris:So let's actually get a little messy because at some point, we need to have kids think about one third as a percentage or decimal, what that equivalent is, and you just sort of throw it out as .33, but we want kids to kind of reason through that at least once or twice in their life so that they have like a real feel for how that one third in fraction form relates to that decimal, and percent. So you can have kids think about a third of 100. And the thing about a third of 100, how they're going to break that down one third of something nice like 90, so a third of 90, that's almost 100. So if one third of 90, that's 30, right, third of 90 is 30. So now we have left, if we've got, we're trying to get a third of 100, we already have one third of 90 being 30. Then we sort of have 10 left, what can we break that into Oh, we can take one third of nine, and one third of nine, that's just three. So so far, we have now one third of 99. One third of 99 was that 30 and 3, so one third of 99 is 33. Well what do we have left? We still have one left, right? What's a third of one? Hmm, well, third of one is just a third, right? And so now we sort of have one third of 99 is 33. One third of one is one third. And so we have 33 and one third. And then we can sort of keep going and do some more reasoning. And that's kind of how it turns into this sort of point 333333 thing happening because we're sort of thinking about that 33 and one third, and what's one third was 33, and one third, and it's kind of keeps going. So we sort of reason through that 33 and a third percent. Well, then can we reason about one sixth? Snd we kind of did that fast a little bit earlier where we were talking about the relationship between a half and a fourth. We need kids to be able to think about half of a half. If we have one half that we can think about half of a half to be one fourth. Well, similarly, if we have one third, what's the definition of a half of a third, is that a sixth? Like, if we've cut something into three pieces, one of those pieces is one of the three, that's one third. But if we've cut that those same pieces in half, now we have six pieces, six total pieces. So now we have one out of six. So half of one third is one sixth, we need kids to be able to do that kind of reasoning. So then what would the decimal equivalent for one sixth be? We just talked about how the decimal equivalent for one third would be point 3333, or 33 and a third. So now we can think about half of that, can we get kids thinking about half of 33 and a third? Now kids need to sort of break out how, and they could do it in several different ways. And I'll just think about one of them. If I'm thinking about one half of 33, and a third, could I think about half of 32? That's 16. Okay, cool. So if I'm trying to think about half of 33 and a third, and I know half of 32 is 16. Then I still have one and a third left? Well, half of that one and a third. Let's see, that's kind of four thirds. What's half of four thirds? It's just two thirds, right? So now we're at 16, and two thirds, and bam, we have the equivalent for one sixth, it's 16 and two thirds or point 1666666666. So those are some ways of sort of reasoning through about thinking about the connection between some of our favorite fractions to what their decimal or percentage equivalent might be. And I've said decimal and percent kind of equivalent, we would obviously need to help kids sort of think about getting between decimals to percents. So let's talk about that a little bit. How does that all connect?

Kim Montague:Well, so we can think about a third in decimals and percents. But often, it's so much easier just to think about one third, like one out of three. Like if I asked you for the price of a $30 sweater that has sale price of one third, you wouldn't really want to do anything except for find a third of 30. Right? Absolutely. Yep. So it's a thing moving fluidly between fractions, decimals, and percents makes you a more well rounded mathematician, it was fun to just listen to you go back through those. And frankly, it makes real life mathematics become more manageable, right, you're given a fraction or a decimal that you don't really want to mess with. And if you're fluid between the three forms, you can choose what makes more sense at that time. So if our listeners try to live in one form or another, or have students who do, what can they do to become more flexible?

Pam Harris:So a lot of things. But in just this podcast, we'll just talk about one real quick here. Whenever you can, talk about them interchangeably. When you say one half, write point five. When you write point five, say 50%. When kids are watching progress bars, sort of drop in over them and talk about where they are, I see you're just a little bit done. So that says 10%. That means you still have 90% to go. So teachers make your units less about one representation only and instead focus on the connections between all three forms

Kim Montague:Right I think it's really easy to ask your students early. Do you have a comfort zone? Is there one topic that you feel like makes more sense to your brain and help them get better about exploring the relationships and connections rather than treating them as distinct topics.

Pam Harris:Yeah, sort of lean away from rules and steps to follow and more towards using what they know and how they're related.

Kim Montague:Don't forget to join us on MathStratChat on your favorite social media on Wednesday evenings where we explore problems and when Pam asks a fraction question, you'll see how people use all three forms to answer it. Thanks tons for the five star ratings on Apple podcasts. We love reading your comments and the 'aha's you're having as you're listening.

Pam Harris:So if you're interested to learn more mathematics 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!

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