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# Common Core Standards: Ratios and proportions

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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.