OK. Quick survey. Two questions.
Ignore all motivational and tangentially related conceptual issues. Accept for now that two people are using ratios to compare amounts of money. Jim has $300; Christopher has $250.
question 1
Jim writes a ratio this way:
The ratio of the amount of money Jim has to the amount Christopher has is 300:250.
Christopher writes it this way:
The ratio of the amount of money Jim has to the amount Christopher has is $300:$250.
The question: Is one of these more correct than the other?
Question 2
Jim and Christopher notice that all of their money is in 10-dollar bills. Therefore, it makes sense to write their ratios in a different form.
Jim writes the new ratio this way:
The ratio of the amount of money Jim has to the amount Christopher has is 30:25.
Christopher writes it this way:
The ratio of the amount of money Jim has to the amount Christopher has is $30:$25.
Same question: Is one of these more correct than the other?
Bonus questions
What is the general principle at play here, in your view? And to what extent does that principle matter?
Overthinking my teaching 
Here’s a ramble.
I think all four are fine, but I actually like keeping the money. Context matters a lot, and how the math does or does not relate to the context matters. For example, given that Chris and Jim knew the amounts were $300 and $250, writing them next to each other with a colon in between doesn’t help them compare all that much. So I want to say that the second two ratios has the potential to tell them something new that they didn’t know before: For example, they might be able to say
(1) For every $30 I have, you only have $25. This is a new insight.
(2) The second thing one could say is Jim having $300 dollars when Chris only has $250 dollars is the same as Jim having $30 dollars when Chris only has $25 dollars. –> Without more clarification, this doesn’t make much sense to me.
Taking away the dollars completely, whether 300:250 or 30:25 might lead someone to think different things, such as dividing to get a percentage and interpreting the problem that way.
I’m guess what I’m getting at is that each one has different affordances for think about the context, and part of that context is the purpose–why are they trying to compare.
I believe the ratios without the dollar signs are more correct. Ratio implies comparison of two amounts of the same thing. I wouldn’t use 15 in. : 12 in. for a ratio of length to width even if those were the respective measurements. I would state “the ratio of the length to the width is 5:4.” Units aren’t necessary when comparing items measured in the same unit. I do believe the distinction matters because once you compare different types of things you are more accurately using a “rate” such as mi/gal or words/minute.
However, I do agree with BWFrank about the context being important. Once it has been established though, the units aren’t necessary.
I tend to agree with MikeB – we know the context and so the units are irrelevant. It’s like a situation I find myself in quite often -when I am deciding whether to call a bet in a poker game, I calculate pot odds – it’s a strict ratio and no units are needed – doesn’t matter whether the bet is 10¢ or $10, the odds are the same and my decision about whether to call or not doesn’t factor in the units – just the ratio.
Mathematically, I’d ditch the dollar signs. They’d “cancel” (hmph) one another out in the fraction, anyway, and 300:250 seems to emphasize the relationship of the numbers themselves. On the other hand, I’m a sucker for units. They give math meaning, and I’d probably still end up keeping the signs, math purity notwithstanding.
In terms of the 30:25, that’s a bit tougher. I’ll echo @bwfrank, and will add a slight caveat. One could argue that, if you keep the dollar sign, $300:$250 vs. $30:$25 are very different, insofar as they feel different. After all, a student might look at the second ratio and say, “I’d rather have $25, since I’d only need $5 more to catch up.”
I’m getting away from the proportion, I know, and am ignoring Chris’ statement that 30:25 referred to $10 bills. For me, this suggests two things: that we either remove the dollar signs in the second ratio; or that we stress the for every mantra.
Care to expand on your hmph, Karim?
For me the most important question is which helps us communicate the idea we’re trying to convey better? This is a question that I would pose to students and let them decide which way they think is better. A question such as this one is a great opportunity to let students take ownership of their mathematical decisions. Because does it really matter that much what they decide to do?
I’ve always hated colon notation for ratios. They are fractions, after all, and we already have a symbol for that. I like all the presentations EXCEPT $30:$25. If you are going to use units, then the numbers associated with the units should be correct for those units. 30 bills:25 bills is ok, but $30:$25 is not an accurate pair of measurements.
Incidentally, if the students are going to do science, they should get in the habit of always starting with all the units expressed. They can cancel units later, once they are “simplifying”.
Not surprisingly, I suppose, we seem to have a split here. When we view this through our math glasses we want to lose the units. When we view through the lens of science, we want to retain them. Karim and gasstation both picked up on an important subtlety; that the units have changed in the second set of ratios.
Karim:
This is a really interesting point-the for every business. My inner middle-schooler objects to using for every when the ratio represents the exact quantities in the situation. If Jim has $300 and I have $250, then why would we go to the trouble of saying for every $300 of Jim’s, I have $250. More plausible is for every $30 of Jim’s, I have $25. But as Karim and gasstation point out; that’s not the ratio I’m referring to in Question 2.
In any case, it will surprise no one that this is more than an academic exercise. Common Core requires ratio at sixth grade and I’m working on some text in Connected Math that introduces it. So while I’m with The Space between the Numbers (may I call you TSBTN?):
I’m asking for a different purpose. My classroom/my kids? We’re going to talk it out. Writing text that’s going to go into many, many classrooms and be the place where the conversation begins? That’s a different situation altogether.
My inclination is to lose the units here (sorry, scientists) but instead to be really careful about the language that surrounds the ratio. I like Jim’s ratios better in the examples in the post. They read more cleanly to me and put the focus on the numerical relationship. But I’ll be having this debate with my colleagues later this week for sure.
My apologies to gasstation for the colon notation. Love it or hate it, we’ve gotta introduce it. And one of the most challenging tasks I’ve taken on in this project is writing text everyone can live with that discusses the relationship between fractions and ratios. If we want to be precise about answering the question What is a fraction? we really do need Modern Algebra-a fraction is an equivalence class. That is, of course, not helpful at sixth grade. Finding a mathematically acceptable, conceptually justifiable description of a fraction and how it relates to/differs from a ratio-for sixth graders-is challenging.
As a quick window into the world of why this matters, consider what happens if Jim and Christopher each get another $100. If we write the original ratio as a fraction: 300/250 and the ratio of the new money as a fraction: 100/100, then we add the numerators and the denominators to get the new ratio: 400/350. And it feels to a middle-school kid an awful lot like we just added those fractions.
Thanks to all for your participation; thoughful responses all around and really useful stuff.
Seems like the moral of all this is “ratio notation stinks”.
Ratios don’t have units, that’s part of what makes them ratios. So, in my opinion, 300:250 and 30:25 are fine (also 3/10:1/4, heh), while $300:$250 is not. Writing it out as a fraction would allow you to include the units as (300 dollars) / (250 dollars) or whatever it is. Ratios can have a name after the ratio, such as “4:3 aspect ratio”, that can help with the meaning of what it says.
The sad part to me is that this hurts students’ understanding of fractions as a single number; 4/3 is a number while 4:3 is a ratio. The other sad part is that some high school curricula (and state standards, but not Common Core) teach “odds” as ratios, when “odds” are almost exclusively used in gambling contexts. “Odds” should just disappear from math education completely in my opinion.
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