A New Way To Look At Fractions

Math, is a subject that inspires so many feelings in both students and teachers. It’s a subject that some seem to automatically excel in and others struggle. One thing that remains universal is there are varied ways to teach math, which is good because kids learn differently. What makes sense to one child may seem like learning quantum physics to another but a different way of explanation or seeing the way to work a problem or understand a concept can make all the difference! Join Janna as she chats with RightStart Math VP, Kathleen Cotter Clayton about a different way to understand, look at and teach fractions.

ABOUT RIGHTSTART MATH: RightStart™ Mathematics is a complete elementary and middle-school program that uses visualization of quantities, de-emphasizes counting, and provides strategies and games for learning the facts. The primary learning tool is the AL Abacus, a specially designed two-sided abacus that is both kinesthetic and visual. Links mentioned in podcast are here; https://rightstartmath.com/rightstart-fractions-2/

A download pdf is here; https://rightstartmath.com/wp-content/uploads/2022/03/2022-RightStart-Fractions.pdf

RightStart Math is available for purchase at www.rightstartmath.com or www.bookshark.com/rightstart

Janna  00:01: Hi, and welcome to Homeschool Your Way. I'm your host, Janna Koch, and the community manager for BookShark. Once again, I am joined by Kathleen Cotter Clayton and she is from RightStart Math, we are going to be delving into this idea of fractions, how you can make yourself comfortable teaching them, and how your children are going to get excited about learning them. Kathleen, welcome to this episode.

Kathleen  00:25: Thank you so much for having me.

Janna  00:28: Once again, why don't you just give a quick overview of who you are and what RightStart mathematics is? 

Kathleen  00:34: I am Kathleen Cotter Clayton, as you said, and I am the Vice President of RightStart Mathematics. I am in charge of curriculum development. I go to a lot of presentations all around, well I used to say just the US and Canada but I just recently actually, came back from or just recently was in England, and spoke at Cambridge University, ironically, on fractions, which is kind of what led to this podcast today.

Janna  01:05: Well, I know that fractions can be very intimidating for people, so we are going to put them at ease. Let's start off with a hack about fractions, what can we tell our homeschool parents that they can do that can help them start this journey of teaching fractions without any of the negativity that they may have?

Kathleen 01:26: First of all, the number one thing is using a linear perspective with your fractions. So often they come in circles. I want to show you how to do it in a line. And it makes so much sense.

Janna  01:42: I am excited to hear you explain that. Because in my mind, from my school days, fractions are a pie and it's a piece of the pie. And it's a pie or a pizza or something with a circle. So this idea of fractions being on a line is a new concept to me. So let's jump right in. Let's talk about fractions in linear form.

02:05: That sounds wonderful. Well, first of all, when kids are starting to learn fractions, usually they're doing it after they've done addition, subtraction, you know, they know their numbers. They've done addition and subtraction. And now they know that four is bigger than three. I mean, they've been working on this, they can add them, they can subtract them starting to multiply them. And all of a sudden now ¼ is not bigger than ⅓ . Or one, you know, one over four is smaller than one over three. But four is bigger than three. And so it doesn't make any sense. But the interesting thing is, is a fraction itself. ¼ is a number, we write it as a one and a four, but it's actually one number ¼ is a number. And so if the kids can understand that. And so actually it let me back up, if they can understand that, that helps them. So we actually recommend starting to teach fractions in the very beginning. Because if you ask your little one, your kindergarten or your first grader, maybe even your three and four-year-old, do you want a half a sandwich today? Or do you want a fourth of a sandwich? I'm not very hungry, I'm going to have a fourth of a sandwich. And they start to understand that a fourth is an amount. It's a number, it's a quantity, and they can work with it. By the way, as a quick side note, if you notice I said ¼ not a quarter. In math class, use the word ¼ . Not quarter. Because if I say one half, ⅓, one quarter ⅕, one six. It's like I can't get a rhythm as I'm going with this. But if I say ½, ⅓, ¼, ⅕, ⅙, now I have a rhythm. So in math class, call the quarters, fourths are called the fourth, fourth. So you can develop that rhythm. We can understand it, we can see it.

Janna  04:07: It is no wonder that children get very confused when they're being introduced to these concepts. Because when you say quarter, I immediately think of money. Yeah, a fraction.

Kathleen  04:21: Yes. And that's actually funny. I will not tell you how old I was when I discovered. Let me back up. How many quarts or and a gallon? Well, I don't know. I just Google it. I couldn't remember it. I never put it together the court quarter. So wouldn't there be four quarts or four quarters in a gallon? I was flabbergasted when that got put together for me. Like I said, we won't talk about how old I was or how old my children were when I finally put that together. But part of that is our language. So it's not that I wasn't taught right. It was that Our language hinders us. And so you know, I mean, I know, quarters, four quarters and $1. I knew that but for some reason I did not take quarters of money and put it two quarts of gallons, and you know, quarts with liquid? Well, I will admit, I'm 44. And I did not know that I figured that's why we had Google because Google would help us figure out you know, how many isn't this that? Yes. Now I know how many quarts are in a gallon, a quarter? ¼? Of course, it'd be four quarts, and a gallon and one gallon is four ¼. in a gallon. That’s cool. 

 Janna  05:37: Yes. I mean this, this revelation alone is enough to make this conversation incredibly relevant for me. But we have you have so much more than just four quarts to a gallon for us. 

Janna Yeah. So start young in RightStart math, when do children start learning fractions,

Kathleen 05:56:  We actually start in Level A, our kindergarten, for the brand new never been used child, we started right there. Because again, they can look at it, there's so at that age, they're so open and willing to explore and play with things, you know, they understand, Okay, here's one, here's half of it. Here's a fourth of it. Here's an eighth of it, oh, that's just a tiny little amount. And so they can see it. And so they're more open to seeing what fractions are. And so we start at the very beginning. So in levels A, B, and C, they're doing fractions, they're not doing anything, it's really exciting with them. But we're learning them we're becoming comfortable with them, we realize it's another set of numbers, and we can go with it. So by the time we hit level D, which is kind of our third, fourth grade, now we're starting to work with them, we're starting to add them and then level E and F, we're multiplying and dividing them. But we're getting used to them. So we start them out easy. You know, it's kind of like, you know, introducing a child, you know, a young child to a bath or a cold water or something, you you put them in slowly, you know, so that they're comfortable with it. Oh, okay, we can do this. And so for parents, it’s very simple. For parents who have already been teaching math to their children. Is it too late if they're not starting to introduce fractions at the beginning? No, absolutely not. Absolutely not. I will go to conferences, and I will meet adults, and they show them how a fraction chart works. And they're like, oh, that's what fractions are. Yep, got it. And so no, you can start right now today. Now, of course, it's a little bit easier when they're younger, just because they have no wrong ideas that they have to filter through and toss out. But there's no reason why you can't start with a third-grade, or fourth grade, or fifth grader, seventh grade, or eighth grader, adult, 44 year-old, it doesn't matter how old you are, you can start it and learn it and build on it. Fractions are actually so cool. I love fractions. This is actually one of my favorite things to talk about. Because they're just so beautiful. They're so cool. And no, most people are like, she just said fractions and beautiful in the same sentence. They are, they really, really are when they're done right, you can see it. 

Janna  08:17: So give us an example of the beauty that you see when you're talking about fractions. 

Kathleen 08:23: Well, a very simple one is if I take a chart, and actually what I'd like to do, for those who are watching this, I'd like to show my screen because I can show you what the chart is. Those of you that are listening, don't worry, I'm going to go through and explain everything, but it kind of gives me a better way to run through with you. So can we take two seconds? And I'll show you my screen? Yeah, let's do it. Okay. All right. And again, those of you that are watching this, or excuse me that are listening to uh, don't worry about it, you know, we've got it, you know, we'll, I'll explain everything, for those of you that are not able to see it. And I do have a present kind of a presentation of this on our website, that and you don't need to look at it. Now you can look at it later. I'll give it to the very end what that website is. So let's look at a fraction chart. So what am I gonna build a chart, I'm going to start by taking one it can be whatever length you want it to be, whether it's a size of a sheet of paper, or you know, so many inches or centimeters, whatever, just take a length it's one and divide it to you know, take that same one and now take a second one exact same size another one and cut it into equal pieces. And put the name one one half, so one over two, and we'd like them stacked up the one over two, not one slash two. So we can see it one over two. So one half, I'm going to mark that in the center of each One of these strips. So now these two strips, the two, one halfs is the same measurement as the one. Okay, let's take another one and break it into three equal pieces. And I'm going to mark them ⅓, ⅓, ⅓. Take another one the same same length as the one, cut it into four equal pieces, you guys can kind of see where this is going market ¼, ¼, ¼, ¼. Another one tip, break it in five equal pieces. Those are my fifths, create six sevens, eights, ninths, and tenths. Now, once you have them all laid out a lot of work to do it, we have it on our website, we actually have some downloadable ones, that you can just download a go with it. And if you want to make your own, that's pretty cool, too. But once you've got it, you can start to see some of the patterns in there. So if you were to take those numbers that I wrote with a 110, down at the bottom left hand corner, and then you curve your way, all the way up to the one in the top center, and then back down to the numbers down to the 1/10. On the bottom right, you have an arch. And that same arch goes all the way through the chart. And when you look at that you're thinking, okay, that's, that's pretty cool. That's pretty cool, you can see it. Another thing with that you can do with it. And again, if you've got all the pieces, just take the one and put it down at the bottom. And then right above that keeping everything aligned to the left, put the one half. On top of that put the 1/3, the 1/4, the 1/5 1/6, seventh, eighth, ninth, and 10th, which creates another pattern. And so often people don't have any idea that these beautiful patterns exist in fractions. And I just want you guys to see that. And once you start to see that it's easy to start to play with them. So what questions you have so far, Janna, as we're kind of going through looking at this?

 Janna  12:13: Well, I'm amazed by the arches within that chart, I just am in awe of how math really does have beauty and symmetry. And it makes sense, for so long when I was a child. And even as I moved into adulthood, I felt like math was a language I just didn't understand. But seeing it laid out this way, I can now start to see some of the patterns that I don't think I recognized before. 

Kathleen 12:47: Well without the linear view. And that's why it's so important without the linear view, if you're just going with fraction circles, that it's not there. It's not there, it's like it's you can't see it. It's like asking somebody who's, you know, always in a dark house with all the windows pulled. And they only have just the light on in the room, but they can't see outside. And you tell them really outside there are leaves and they're turning colors that were green. And now they're orange and yellow and red, and it's beautiful and the air smells crisp and the kid inside is going I have no idea what she's talking about. That's what we need to do is we need to pull fractions out of that dark room with that dim light and bring them out to where the joy is and, and the happiness and it's just fractions are just so cool. They're just like an idiot, but they're so cool that I want people to have that.

 Janna  13:46: And I want to get there with you, Kathleen, I do so help me understand because right now especially I'm looking at fraction stairs and how they stack on each other and it's to me very reminiscent of Egyptian pyramids and you know, the Incan pyramids. So help me go from these stairs to some practical ideas of how fractions are going to make a difference in my world. 

Kathleen 14:18: Well, it's even like we were saying about the quarts, a gallon to a quart or with money. Or when somebody says, half past a time, you know, it's half past 11 It's half past 12 a half K Well, it's halfway around, or a quarter past or a quarter to.  As a child. I knew how to translate that, but I didn't realize it meant a fourth of the way. I didn't look I just tried to memorize what I was told because I was taught very traditionally. But I memorized right what I was told, but I did not understand enough to bring those connections back in. And that's where we want our kids is to bring those connections in. Because if your kids only know what you've taught them, no matter how much you teach them, that's a failure, in my opinion, because they need to know more than what we know, I want my kids to be smarter than me, because I don't know everything. And I know a lot. And I'm smart, but I don't know everything. I have a daughter that works, in a retirement home, she's a life enrichment director, and the things that she comes up with are just amazing. So does she know more than me in that area? Absolutely. I have a son who's a welder. I don't know anything about welding other than it makes stuff stick together. But he knows all this stuff about welding? And how do you weld with aluminum? And how do you lay a bead? And how do you do this? And how does the arc go here, and we're, you know, reflect off and about. I don't know that stuff. So it's kind of the same thing here. I want my kids, I'm going to teach them the best I can. But I want them to understand it. So they can go above and beyond and create new things, whether it be with fractions, or fractions as part of whatever their next layer is that they're doing.

Janna  19:22: I’m going to venture with a five.

Kathleen 19:25: Five and now I actually like kids to say five fits. Make a whole because five doesn't make a whole. Five-fifths make a whole. So how many sixths make a whole? Six-Sixths, make six. How many eighths make a whole? Eight-eighths make a whole right? So that teacher needed to know that but anyhow so how many let's go ahead let's say how many tenths make a whole we're gonna jump. How many tenths make a whole?

Janna  19:52: 10 tenths, make one whole?

Kathleen 19:55: Yep, how many twelfths to make a whole?

Janna  19:58: 12 Twelfths make one whole.

Kathleen  19:59: And that’s not even on my chart, but see you already saw the pattern. So how many fifteenths?

Janna  20:04: Make a whole? 15 fifteenths?

Kathleen  20:07: And how many hundreds make a whole?

Janna  20:10: 100 Hundredths make one whole.

Kathleen  20:14: And ooh, doesn't it sound just a wee bit like decimals? Anyhow, moving right along. That's another topic. But see how they're all linked. And if you have this layer, okay, we've got the foundation now. Now I know how that works, then we can start to do some comparing, again, we still haven't added yet. We're still comparing what's more, three-fourths, or four-fifths, three-fourths, or four-fifths. So again, those of you that are listening, just think what three-fourths would be, do you think four-fifths is going to be less or more than three-fourths? It's actually going to be more because you think about it, it's a whole, not including 1/4. And it's a whole not including 1/5 1/5 is smaller. So four-fifths must be bigger. It's kind of a different way of looking at it, but you can kind of move it around and make it fit what the kids need.

Janna  21:18: Let's do and again the pattern of because I was like, Okay, well, if you start getting further down, is five-fifths greater than four-fifths, right? So because I could see the pattern, I could easily answer that question where the first question without the pattern I was like, Yeah, nervous for a minute.

 21:39: idea. Yep. go even farther down, and we say what is more seven-eighths, or eight-ninths. And again, in my head, I can see the chart. I mean, even without anything in front of me, I can see the chart. So I know my answer. Eight-ninths is going to be bigger. But again, working my way through it, because I understand how the chart is laid out. I'm going to say okay, well, here's my whole. And I've got, I'm going to subtract 1/9 of the eight-ninths, okay, that's this big, and I'm gonna subtract the ⅛  off the seven-eighths, you're off one to get seven-eighths. Well, I know 1/8 is going to be bigger, so I'm subtracting more. So therefore the other number is subtracted less. Therefore eight-ninths is going to be bigger. It's another way of thinking instead of going, Oh, what's the rule? Oh, my head hurts. Gosh, I hate fractions. You can start to see it and work with it. And like it was said before. So often we look at fraction circles. And I've got them shown here. Think about a circle where I've got the fifths marked, I've got the six marks. What's more, four-fifths or Five? Six?

 Janna  22:52 It would be very hard to Yeah, it would be very hard to visualize the answer right from a circle.

Kathleen 22:59: Exactly. And when I look at this, I'm just thinking, Oh, it's circles. Maybe I should is it pie? Can I eat it? Is it pizza? I can eat that too. You know, but it doesn't help me. I can't see it. It's not easy. It's easier to use the fraction chart. So continuing with this again, before we get into addition and subtraction, because that's a very good question. Let's continue to build on how the numbers go together. So I'm going to create a partial chart because you have, you know, at home if you've done this yourself, you've created your pieces. So I'm going to take my one and a half. I'm gonna lay them right below the one and then my Forte's so I'm gonna lay them below. So I've got my four-fourths, and then my eights and how many eighths I need. eight eighths. So I've got this lined up. Now, in your mind. Take this, you've got this build, you got your one-halves, fourths, and eights. If you can try to erase the horizontal lines that go across and get rid of your numbers. What does it look like? And something on a ruler, you're looking at a ruler exactly I have ever had people almost fall off their chairs when they saw that a fraction chart ends up looking like a ruler. It absolutely floors people. So again, we're still in the comparison we're comparing we're still trying to learn how fractions work. How do they link to each other? How can I maneuver them? I'm still just playing with fractions before I start doing anything to them. I'm just finding out how they're related to each other. So actually, the way that we have the kids practice is we play a game. Do you guys remember the Game of War? You lay down two cards, you turn over the face card and whoever has the higher one takes that you try to get the whole deck from your buddy What if we do these with fraction cards, I'm going to use the same thing with the one halves, fourths, and eighths. Because those are important. We use those a lot. We don't use thirds of an inch very often, but we use the ones half sports and Ace a lot. So I'm going to take a deck of cards. These are fraction cards, flip them over. One person said, Janna, you can have the three-eighths. I have 1/4. Who is more, three-eighths, or 1/4? 

Janna  25:30: The three-eighths I win

Kathleen 25:32: The three eighths. So you win. So she takes those cards. All right, next one. Okay, Janna has five eighth's. I have three-fourths, whose is bigger. Five eighths, or three-fourths? It would be mine. Go me. And so we were playing this game. My son and I were playing this game one time. And I think actually he had seven eighths and I had three fours. And he said, Mama, I won. And I went, Oh, he said, he said, Mama I want he said I beat you by an eighth. And I went what?!. And I looked at the chart, sure enough, he beat me on that hand by an eighth. Now at that point, we were, I was not teaching him how to add and subtract fractions. I was just teaching them how to compare them. So he can become familiar with it. But he had already taken that next step, going beyond what I had taught him. And was applying it to a new situation. Yes, that is good parenting. That is good teaching. We actually have an app for people who want to have their kids play because, you know, sometimes you just have to have some screen time. You know, if you're driving and rather than having your kid play some silly game, you can have a maybe do a game we've got, we've got a very cleverly called fraction war. And you can buy the, on Apple and Android. And the beginner one just does as we talked about the one, one-half, fourths, and eighths, then there's an easy version that adds in thirds and sixths, and medium adds in fifths and tenths. And the hard puts in percents. So there's a way to practice this as you're going through and learning it.

 Janna  27:20: So now let me ask you this really quick as you are, just as you explained the app that you had. So do you start talking percentages after you feel like you have a solid foundation with the fraction chart?

Kathleen  27:36: Yes, and what a lot of kids miss myself included, I now get it. But what does percent mean? It means percent cent means like a century or one 100 Or Centurion so 100. So it means per 100. So 50% is 50 100s, which is the same thing as one-half,  25% 25 per 100 is the same thing as 1/4. So in our curriculum, we start talking percent, I want to say in about level D or E, which would be kind of our third fourth fifth-grade level because we need to make sure before we throw in percents, which is built on fractions, you have to have fractions laid down, solid, clean, total, and then percent per 100. Then it makes it very simple. And then again, from there, percents to decimals, they just link back and forth per 100. Hundreds. Wouldn't that be point five zero? Well, it's just to 25% 25-100s. So that you would write that numbers point two, five 2500s.

 Janna  28:58: You make it sound so simple, is Could it really be? It really be?

Kathleen  29:04: It is that's what's so cool about it. But the hesitation that you're or kind of the not hesitation but kind of the amazement that you're having is normal because you I assume we're taught a lot like I was were just this was what it is. Don't give me any backtalk. This is what you have to do. This is why it works. Move on. I don't remember learning fractions as a child. I'm sure I did. But I have no recollections of them. I remember learning multiplication division. I have no recollections of fractions. And I could do one but I never really did them very well until I came into this linear way of looking at the fractions.

 Janna  29:50: Well, and when I think fractions, and again, this is my old school way of thinking, I just think okay, well that's a division problem. I never saw it really as I mean, I guess we I knew it was part of a whole. But I immediately see it as a division problem. And I'm starting to see that maybe that's where I went wrong in some of my thinking,

Kathleen  30:14: well, and then there's another way to look at fractions. That's a good point. Because if you look at fractions as the one half as one divided by two, it's one it's a whole one divided into two equal parts, you know, and two-thirds is two divided into three equal pieces. It's another way of looking at it. But to me, that's such a hard way to look at it. I don't I don't like it that way it doesn't. It doesn't sit well, in my head. I like it better as one half is a number, two one half is the same thing as one, which by the way, is actually adding fractions. But yeah, logical

 Janna  30:55: It’s true. Because even as you said that it's very, it's not clean. I don't see a pattern I get no, really, I get really confused very quickly if I'm looking at it through the lens of division, as opposed to this linear way.

Kathleen  31:16: Yep. Exactly. Exactly. And this way, just, I don't know, it's just simpler. It's just it to me. It's just so pleasantly basic. It's just, it's just happy. It's joyous. And basic.

 Janna  31:29: I'm not sure. Yeah, I'm not sure most parents would say fractions are pleasantly basic.

Kathleen  31:34: Yeah. But they are when they're taught right when they're presented right. 

Janna  31:39: Why do you think math has been overly complicated? Over the years?

Kathleen  31:48: I think it's because people don't understand it. So if I don't understand it, then the rule to manage it is the following. Let me get my 48-page book out, or 4000-page book out, because I don't know how to do it. But I've got a thing on page 93. It says this is what I do. Because I don't know how to manage it. So I really don't know what it is. So I think that's why it's become that way people don't understand the basics, which is why I spent so much time on the bases. Because if you have the basics you can build up. But if you don't have the basics, yeah, you can put your walls up if you're trying to build a house and put a roof on but it's not going to hold if you have no foundation, you have to have the foundation. Math needs to be taught, so we understand it. And so often it's taught so that you have to memorize stuff. But in fact, it should be the other way around 95% should be understood. And only 5% memorized.

 Janna  32:53: And if we could understand this concept for homeschooling our children or even as adults looking to career change, or add quality of life, anytime I do something when I understand it, it excites me, right? Like, like, I get why I'm doing it, I know what the result is going to be. It takes all of that frustration and guesswork out of it. And I think, unfortunately, I'll speak for myself. So much of math was guesswork. It was like a roll of the dice. I hope I got that right. I think maybe I did that correctly. But as you said, maybe my formula was off, what was the formula, how to memorize how to do that, as opposed to understanding it. And then being able to practice it with confidence, knowing that I'm not going to get 100%, right. But when you're doing something and 95% of the time, you're hoping that it comes out the right way, that is no way to execute any type of activity.

Kathleen  33:56: And that's very true. When you think about, like a world-class violinist, do they make mistakes? Of course, they do. But they catch them quicker than we can hear it. And that's what a good mathematician does. Do they make mistakes? Absolutely. But they will catch them quickly. So I might do a calculation just recently. I don't know if I was telling you on the previous podcast or somebody else. I'm a quilt designer. And I was designing something I was trying to figure out, you know how many something I needed to cut. And I came up with a really weird number. And I thought that can't be right. And so I calculated again, and I had done the first time wrong. But I knew enough with my first calculation that was not the right answer. It didn't feel right. But how often in math, do we start with something, and as you said, we go through the algorithm we go through the process who doesn't move on? I have no idea of my answers are even remotely right because they don't even know what I'm expecting them to be. And that's, I think a piece part of the understanding layer that's missed. One thing I want to point out is, it's harder to teach for understanding. Because if I can just tell my kids, here's the rule, you know, you put on your socks, then you put on your shoes, here's the rule. But if they understand, if I put my shoes on, then my socks will one, I'm gonna get blisters, and two of them gonna get my socks, all messy, and three, Mom's gonna be really mad at me. But if I can understand why I put them on in the order, I put them on socks then shoes. I don't have to have I mean, I may have to explain that took a lot more time to explain why you don't put your shoes and your socks on. But they'll never do it wrong once they know the reason why that was wrong. Why do I put them on in this order? So it takes longer to teach for understanding than it does just to give them the rules. But as we know, sometimes the rules, if you don't understand why the rules are important, you have to keep telling the kids the rules over and over and over. And then they forget them. And then it's more things you have to memorize or write down their little book of life. And, oh, this is so exhausting. Instead of finding that joy and happiness on doing it right.

 Janna  36:20: And I would say that, although maybe on the front side teaching to understand could take some more and does take a little bit more time. On the backside of that the time that you invested in the beginning, I don't think you'll have to exhaust yourself as much as you continue on through the process, because they do understand the why.

Kathleen 36:42: Yes, and they can start to make leaps with you, or maybe even ahead of you. And that's what we want them to do.

 Janna  36:50: Yes, I know I do. I know I look at my children, and I want them to be able to do any type of math that's put in front of them, to not break out in a sweat, to not know for sure if they're going to you know, question their entire life choices, when a math problem is presented to them, either in academics or in life. So I appreciate this new way of thinking about math. I have said it again, I will continue to say it had I been taught the RightStart way, I may have very well ended up as an engineer instead of a Communications major.

Kathleen  37:28: But then we wouldn't have your podcast, Janna. 

Janna 37:30: It's so true. Everything happens for a reason. But I do get excited when I see the possibility of doing something differently. And sharing that with other parents so that they don't have to continue these patterns of lifelong, we have to just get through math no matter what and they can start to see the creativity, the patterns, the creativity and how math in and of itself can actually awaken the imagination, which I just think most people that feels like heresy, but it is true. It is true. So Kathleen, thank you so much. Thank you for sharing these concepts.

Kathleen  38:08: Thank you. One quick thing for people who are listening and would like to see more about this, go to our website, https://rightstartmath.com/rightstart-fractions-2/ and then you can see the presentation. And this will make and I've got a lot more information on there. But there are a lot of there are free things, you can download the fraction chart. So that way you got it and you can have something to start jumping off with your kids.

 Janna  38:46: And we will have that link in the description as well for our listeners and viewers. So hopefully we will get people to start understanding that math can be more than just some rote memorization and of rule-following. And so we encourage you guys to look at that website and get more information to help you in your homeschool as you delve into fractions with your children. Kathleen, thank you so much for being here. We appreciate your time. 

Kathleen  Thank you, Janna. 

Janna And thank you, guys, until next time, bye-bye.