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- [Voiceover] So I'm here with Bill McCall again. I thought what we would talk about today is ratio and proportions. Like always, you know, I always start these conversations, Bill, with just what is your take on how the common core might be different or similar to how people are associating these ideas of ratios and proportions in kind of previous standards? - Well, ratios and proportional relationships and rates I think traditionally have suffered a little bit from a lot of different use of language. People using different words for the same things or the same words for different things. One of the things we tried to do was to just sort of say this is a ratio, it's not the same thing as a fraction. Then, you move from the idea you have for a ratio you have associated rate that goes with the ratio, called unit rate, and then really in grade seven you start looking at what we call proportional relationships which are just situations where you have two varying quantities that are always in the same ratio with each other. - [Voiceover] So let me just back up one second. - I stay away from the word proportion. Sorry, go ahead, yeah. - [Voiceover] Oh, no I just wanted to drill down one thing you said, when you said you know ratios are not the same thing as fractions. I just want to clarify what you are saying there because clearly you can use the fraction representation to represent rations. You could say this is a ratio of two to four, but I am assuming what you are saying is that they are different conceptual ideas. - Yeah, and I think if you have a very strong understanding of ratios, rates, fractions, proportional relationships, then these things sort of meld together in your mind but if you think about the following situation where you have a situation where you are say making a recipe and you have, you know, two cups of one ingredient to four cups of another ingredient, then you'll say the ratio is two to four. If you start adding in more cups, you might double the recipe or you might add one more cup of the first ingredient, two more of the second ingredient. It sounds like I just added something, right? (laughter) Right? I went from two to four to three to six, but if you confuse that with adding two fourths, I'm sorry we went from two to four to one to two, anyway if you confuse that with adding two-fourths and one-half, you're going to get the answer wrong. You have to be a little bit careful about completely melding those two notions together. Two fourths plus one half does not equal three sixths. - [Voiceover] Right. - So, but if you have this idea that like it's all the same thing, that's one of the confusions that can happen there. What is true is that two fourths is the unit rate associated with the ration two to four. Two fourths, the ratio of two to four is the same as the ratio of two fourths to one. If you wanted to say I walk two miles in four hours, it's slow but anyway we are dealing with that ratio, then you could say that's the same as half a mile per hour or two fourths a mile per hour. There is a connection, but you just don't want to say they are the same thing, really. - [Voiceover] Right. - Ratio is really a comparison of two quantities and a fraction is just a single number. - [Voiceover] Right, right, I think that's the key thing at least conceptually, students understand that I guess that idea, single number is a fraction and a ratio is a comparison. - Of two numbers, yeah. - [Voiceover] It's a comparison of-- - Or of two quantities. - [Voiceover] Obviously you can use a lot of the same representations in both, but you have to be careful. - Yeah, I think once people become proficient, they tend to switch back and forth between representations without even really noticing they are doing it. That's fine. That's a sign of proficiency. But, you have to be careful with students that you aren't just confusing them by doing that. - [Voiceover] Yeah, no that's right. I guess the other interesting thing which you started talking about already is kind of the connection, you know, rates can be represented as or ratios can be used to represent rates. You have something per something. - Right. You know, there is a progression in the way people think about that. You might start out by saying "I walk six miles every two hours" and at some point later one, you'll say that's a rate of two miles per hour. You have this notion of a unit called miles per hour. - [Voiceover] Three miles per hour. - There is a progression in between those two ends of the idea. Those are all rates, but they are said in a different way depending on where you are in that progression. - [Voiceover] Right, right. It seems like the common core is going through more work to really make sure people understand this connection. Obviously, the words are obviously related as well. - Yeah, and we are trying to lay out some sort of language and a progression there. - [Voiceover] One thing that I found interesting if we go to the standard right over here, which I think is it makes a ton of sense to me, but it feels like it's different than what I remember learning in school or when it was introduced, is that traditionally percent is kind of introduced as "Hey, here's another way to write a decimal." But it really, one thing that I always mention on Khan Academy, I mean, it literally means per hundred. It really is a rate. That's why it's kind of grouped in with the ratios and rates. - That's right, and the talking used, when we think of how we use percentages, they are used to talk about rates. We talk about the interest rate on a loan, and it's quoted as a percentage. - [Voiceover] Yep, yep, no that's, I think, really, really interesting. This is also where kind of unit conversion comes in. It's not kind of called out as a separate unit conversion thing. It's just a continuation of rates. You have 5,280 feet per mile, you have 1,000 meters per kilometer. These are rates. - Right, right. One of the things that sometimes in traditional curriculum that they make a distinction between situations where you are comparing two quantities with the same units and then when you are comparing two quantities with different units and like you use the word rate for one of them or ratio for the other, I forget the details, we.... There doesn't really seem to be any mathematical or scientific rationale for making that distinction. We don't make that distinction. - [Voiceover] Right. Then as we go into-- That was all essentially sixth grade, is most of what we have just been talking about. Then seventh grade, you are starting to kind of I would say, you are starting to manipulate them. Doing a little more complex arithmetic with them, and starting to kind of treat them a little and starting to solve I would say more algebraic problems with proportions really. - So, there's actually a fairly big shift that comes in seventh grade which is you start to worry about proportional relationships which are situations... When you first start to compare a ratio, you might just have a single pair of numbers. Going back to that recipe example. On the other hand, you might also have lots of equivalent ratios, because you might want to make different quantities of that recipe. You might make a table of equivalent ratios. Even when you are looking at that table of equivalent ratios, there's just each row is a ratio. There's a sort of shift in your mindset that happens when you start looking at those columns and you say "Okay, this column represents "cups of flour, and this column represents cups of sugar." Those are varying quantities and there is a relationship between those two quantities. Then you are talking about proportional relationship between the number of cups of flour and the number of cups of sugar. The unit rate is what gives you that relationship. Number of cups of sugar is one half the number of cups of flour. So, these ideas begin to get very tightly interwoven. There is actually quite a lot of thinking through to do with these tables of equivalent ratios. Yeah, so you can see, as what you are drawing there is that it has this property that if you multiply by two on one side, then you also multiply by two on the other side. There's an additive pattern there too. If you add two on the left, you add four on the right. Then there is the horizontal pattern where you go across and say that two times what's on the left is on the right. All of those patterns require sort of thinking through and developing, but then they all get summarized by the idea of a proportional relationship. You eventually just represent algebraically with an equation Y equals two X, Y being the number of cups of flour and X being the number of cups of sugar. You don't have to use Y and X, but there is a sort of movement from the idea of ratios to proportional relationships, eventually to the idea of a function, really, in eighth grade. There is that progression. There's quite a lot of detail there and stuff to figure out and opportunities to get confused if you're not careful about your language and about the progression. - [Voiceover] Right, right. We see here, it's this first standard right over here compare unit rates associated with ratios and fractions, including ratios of lengths, and you're actually learning to manipulate a half a mile and a fourth mile. This is really kind of taking what you did in sixth grade to the next level. - It's taking it from whole numbers-- In sixth grade the ratios appear as whole numbers and in seventh grade the ratios are pairs of fractions. - [Voiceover] And then the second standard that we are talking right over here... The second standard, this is where it's really what you are talking about. You are really starting to appreciate... You're really starting to not only appreciate these patterns that we saw over here but you can represent them by equations and you're also starting to graph them so you're also going to start seeing-- - Yes, right. - [Voiceover] You know, so if this is sugar, sugar, this is flour, and if you were to graph it you get essentially a graph like that and it's really helping you to start to think about things like linear functions. - Right. - [Voiceover] Then this last standard seems kind of a little bit of a capstone. Solve multi-step ratio problems, you know, interest, tax, mark ups, mark downs, gratuities. - Yeah. - [Voiceover] It's kind of like putting it all together. - That's not only pulling together all the work you've done with ratios and proportional relationships, it's also pulling all the work you've done with number and fractions, and getting proficient with working with computations with fractions. That's actually a standard that pulls together a huge amount of stuff that's come before even starting all the way in kindergarten, and really sets kids up for going on to high school with a solid problem-solving ability, with the beginnings of algebra, with the idea that mathematics is useful. That's a real sort of mobile standard. - [Voiceover] Right, right, I definitely see that. Awesome, well this was very helpful. Thank you! - Okay, great. - [Voiceover] Alright. - See ya.