# More on complex fractions

Sean objected to my common core objections. He suggested the following context to motivate complex fractions as unit rates, as required by Common Core at grade 7 (and as explicitly proscribed by footnote at grade 6).

John is in a 10 mile walkathon for breast cancer. He looked at his watch when he started walking- it was 7:02. After a half mile, he saw that it was 7:17.

He suggests (and I concur) that students will likely use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate), and then as teachers,

We write (1/2)/(1/4) on the board, and discuss its relationship to 1/2 per 1/4. This may be our major line of disagreement, as I don’t think this a terribly sophisticated jump. Assuming the students have some experience with slope and rate of change, this feels like fair game.

Indeed right here is the point of contention. It’s the transition from:

to:

As I wrote in the comments of my original rant,

[I]f I’m mathematically sophisticated and possess a graduate degree, then moving between (a) 1/2 mile per 1/4 hour and (b) (1/2)/(1/4) miles per hour is a very simple formal move justified by the multiplication algorithm for fractions, I suppose.

But walk into your average seventh grade classroom, make some orange juice from concentrate using the standard recipe (3 cans water: 1 can concentrate), pour precisely 1 cup of juice into a glass and ask, How much of this cup of juice is concentrate and how much is the water I added? My experience is that this is a challenging question for 12 and 13-year olds to make sense of. It’s really important at that grade level. It’s challenging to teach precisely because the relationship is so formally simple. But it doesn’t come naturally to lots of kids.

Answering the orange juice question requires shifting from ratio (cups water: cups juice) to unit rate (water per cup of juice), or from ratio to fraction. That’s hard, and it’s not even complex fractions.

The Common Core writers seem to want to move this from seventh grade to sixth grade. I’m OK with that; I wouldn’t have written it that way myself, but I have no major problem with it.

But they seem to think that we need to ramp it up in seventh grade and that the only way to do this is to do it with complex fractions. From the perspective of a middle school curriculum guy, I question whether it makes sense to do so. I don’t think it’s worth making kids do because it’s an unnatural representation. Furthermore, it’s machinery we don’t need. I don’t see a middle school problem that complex fraction unit rates will solve but conceptually simpler techniques will not. From my perspective as a college teacher, I don’t see it either. College Algebra? Calculus? Neither of these relies on complex fraction unit rates. It is conceptually much simpler to deal with (1/2 mile)/(1/4 hour) by either (1) equivalent fractions (multiply numerator and denominator by 4 to obtain (2 miles)/(1 hour)) or (2) division (divide 1/2 by 1/4, get 2).

I worked for a while with the best route to complex fraction unit rates that Sean suggested:

Assuming the students have some experience with slope and rate of change, this feels like fair game.

This makes sense; think of (1/2)/(1/4) as a slope, rise/run. But this leads me to want to divide. I don’t usually think of slope as 6/3 miles/hour; I divide and say the slope is 2 and the rate is 2 mph. So it doesn’t feel any better when the rise and run are fractions.

Sean concludes:

Obviously this isn’t perfect. But if complex fractions are a necessary component to a middle-school curriculum, where else do they land outside of unit rates and proportions?

Part-whole fractions. Area models. Here’s (1/2)/3:

And here’s 1/(2.5)

I’m all for complex fractions as they arise naturally in Sean’s walkathon problem-and they do arise naturally there. I am not for introducing a way of working with those complex fractions that is unnatural and has no special payoff.

I will gladly consider any and all suggestions for contexts and problems in which the complex-fraction-unit-rate gets me something that equivalence and division do not-and I’ll even accept examples from high school and Calculus. But I’m not optimistic that they’ll arise.

Update 6/7/11: I edited out a confusing statement about the ratios in the orange juice question. My question is about part-whole relationships, the explanation in the following paragraph alluded to part-part relationships. It’s fixed now.

### 27 responses to “More on complex fractions”

1. So very many of the issues I have with math curriculum at all levels can be summarized as follows: “They seem to think that we need to ramp it up in [insert grade or class here] and that the only way to do this is to do it with [insert irrelevant topic here].”

Another related sentiment is “These kids are going to need this skill in X, so we should teach it now,” where X is some course 3 years down the road that 90% or more of the students in question will never take.

2. Christopher

R. Wright, are you trying to get me started on the standard long division algorithm? Well, I won’t take the bait.

Even though it’s supposedly necessary so that students can use it to divide polynomials in algebra. And even though the partial quotients algorithm actually provides better notational and conceptual support for dividing polynomials in algebra. Even if you buy that it’s important to study one algorithm in fourth grade so that you can use it in eleventh grade.

Nope. I won’t take the bait.

I will not carry on about the bogus argument that the standard long division algorithm is important because it leads to polynomial long division which then leads to synthetic division. The last of which I have not once met a student who retained it for more than a month afterwards. And which is not really even that closely connected to the original long division algorithm it supposedly springs from anyway. (If it were, wouldn’t there be a version of synthetic division that we could perform on whole numbers? But there is not.) And by the way, why do we need synthetic division? So that we can evaluate a polynomial for a given value of x? In the 21st century? Really?

Nope. I am not going to get started. Wouldn’t be productive.

Nor will I discuss the supposed superiority of invert and multiply over the common denominator algorithm for fraction division. And I absolutely will not discuss my successful effort to teach an entire Calculus I course through exclusive use of common denominators when division arose without once inverting and multiplying. I won’t go there.

I will not take your bait, sir.

3. Chris,

Man. Poor Common Core Standard 7.RP.1. A painful way for any standard to go down. Appreciate this very much. Every CCS deserves this kind of chisel treatment.

Two quotes, two questions:

1. ‘Furthermore, it’s machinery we don’t need.’

Curious as to what else- outside of the long division and fraction-division algorithms- you feel falls under this umbrella. More specifically, if you could eliminate one skill/concept/idea commonly taught in an algebra 1 course, what would it be and why?

2. ‘This makes sense; think of (1/2)/(1/4) as a slope, rise/run. But this leads me to want to divide. I don’t usually think of slope as 6/3 miles/hour; I divide and say the slope is 2 and the rate is 2 mph.’

Fair enough, with one comment. Say my students have to graph something like y = 2x + 4. Almost all of them will use one of three strategies:
a. a quick graph with the slope and the intercept.
b. a table of points.
c. finding both intercepts (a stray and admirable few)

It is the first group that I’m concerned with. After developing the notion that ‘yeah, if it’s in slope-intercept form, you can literally graph this thing in four seconds,’ students slowly and inevitably disconnect a little bit.

To combat this, I attempt to create crisis. I’ll ask one of the students from group A how to do it, and they’ll reply, “Put a point on (0,4) and go up 2, over 1.” I’ll reply that I don’t feel like going up 2, I feel like going up 8. I’ll ask if I’m allowed to do this. Someone will say inevitably say no. Someone will say yes, but that I have to go to the right 4. A spirited debate will hopefully ensue. We’ll go through the song and dance, and hopefully a more refined understanding of a) slope and b) equivalence is established.

If things are flowing, I might ask two more questions- What if I felt like going down 12 instead of up 2? And secondly, what if I decided to up 1/2 of a unit? And how could I use these equivalent slopes to re-write my original equation of y = 2x + 4 if we felt so inclined?

That we would just simplify the (1/2)/(1/4) from your original quote is certain. That it’s not valuable as a learning tool (as stated) I’m not so sure.

4. Christopher

Sean:

Man. Poor Common Core Standard 7.RP.1. A painful way for any standard to go down. Appreciate this very much. Every CCS deserves this kind of chisel treatment.

I’ve got my eye on you, 5.NF.5, first bullet. Although that may only be due to the extremely poor quality of the writing. Not sure. I haven’t been able to fully parse that sentence yet.

At the same time, every standard should be so lucky as to have such a staunch defender as yourself.

Curious as to what else- outside of the long division and fraction-division algorithms- you feel falls under this umbrella. More specifically, if you could eliminate one skill/concept/idea commonly taught in an algebra 1 course, what would it be and why?

What, three topics aren’t enough? You want me to cut more? Are you authorized to give me this kind of power?

The next one on the chopping block is easy.

FOIL.

Here CCSS and I are in complete agreement.

I’ll get to posting this in full sometime soon. I did an historical textbook search a couple years ago. Mid 1950’s and earlier? No FOIL. About 1960, I found an annotated teacher’s edition of an algebra text in which FOIL appears as a teaching tip for students struggling with multiplying binomials. Today’s mainstream published texts? It’s a standard. From useful but optional mnemonic device to essential topic; such is the decline of American curriculum.

To combat this, I attempt to create crisis. I’ll ask one of the students from group A how to do it, and they’ll reply, “Put a point on (0,4) and go up 2, over 1.” I’ll reply that I don’t feel like going up 2, I feel like going up 8. I’ll ask if I’m allowed to do this. Someone will say inevitably say no. Someone will say yes, but that I have to go to the right 4. A spirited debate will hopefully ensue. We’ll go through the song and dance, and hopefully a more refined understanding of a) slope and b) equivalence is established.

Nothing to argue with here, and nothing with your logical extension to going up 1/2. In going through this with kids, you have masterfully negotiated some important ideas about slope as a ratio.

How does the part go when you ask students to write down the walking rate over that interval of 1/4 hour as a unit rate, though? If you’re telling me that, after all of that, when you ask, What was my rate in miles per hour over this 1/2 mile? even one kid is going to say (1/2)/(1/4) miles per hour, then I need to visit your classroom to see what’s going on there.

My point isn’t that it’s impossible. It’s that this is not a useful enough idea to turn into a standard. When it’s a standard, it gets tested. When it gets tested, it gets drilled. It’s just not worth our time.

The slope exercise you posit is absolutely worth our time. (1/2)/(1/4) miles/hour is not.

5. Hey, speak for yourself, we dumped FOIL in our high school books 😉 I guess that means we’re not mainstream!

There’s actually a recent speech by Common Core author Phil Daro where he specifically says FOIL should die instantly, while he argues in general about the need for fewer topics taught deeply and less of a focus on “coverage”.

6. Amen on FOIL. The next thing to die needs to be my dear aunt Sally. I can’t tell you how many misunderstandings I have seen because of her.

On synthetic substitution: I don’t believe it comes from the original long division algorithm at all – it is just the generalization of factoring a polynomial into nested form. Its only tie to long division is that you wind up getting the same result.

Also agreed on common denominators for fraction division. I bring it up every chance I get with my high schoolers.

7. Student attachment to FOIL can be disquieting. After teaching it, you’re assured that it will come up as a possible solution strategy to almost anything. How do I graph this? FOIL. Is there a way to find the circumference? FOIL. Is there a different way we could’ve solved this? FOIL.

I wish there were a way to quantify how much more damaging it is than helpful. It must go.

I guess I’m more curious about bigger ideas. For example, there seems to be a shift in some curricula towards a more focused approach to linear functions and less emphasis on quadratics and exponentials in grade 8, and even in Algebra 1. In CMP 3, which I think is designed for eighth graders, there’s quite a bit on all three, and some really rich stuff about the comparing different rates of growth.

If I’m not mistaken, CMP 3 also has expectations that students can go from equation –> graph and from graph –> equation with quadratics and exponentials. This is more rigorous than CCS suggests for Grade 8. For non-linears, all that’s expected is a ‘qualitative description based on a graph’ (8.F.5). What do you think? Considering there’s so much confusion in later grades re: functions, I wonder what the best path is.

Lastly, thanks for the introduction into the absurd world of 5.NF.5, where the words factor, multiplication, scaling, resizing, factor (again), and product were used in the same sentence.

8. Christopher

Aaron:

I don’t believe it comes from the original long division algorithm at all – it is just the generalization of factoring a polynomial into nested form. Its only tie to long division is that you wind up getting the same result.

Hmmm…I’ll have to check a few sources next week to make sure I have correctly represented the argument. I don’t think I’m making up the synthetic-division-as-shortcut-for-polynomial-long-division argument, but if so I’ll retract. Thanks for keeping the rant honest.

Bowen:

Hey, speak for yourself, we dumped FOIL in our high school books. I guess that means we’re not mainstream!

Is this news to you, Bowen? Is it bad news?

More for you, Bowen, in a new post this week.

Sean:

If I’m not mistaken, CMP 3 also has expectations that students can go from equation –> graph and from graph –> equation with quadratics and exponentials. This is more rigorous than CCS suggests for Grade 8… What do you think? Considering there’s so much confusion in later grades re: functions, I wonder what the best path is.

First, some notational clarification. You refer to Eighth grade CMP2 here. “Connected Mathematics 2” is the official title of the current published curriculum, it being the result of a major NSF-funded revision of the original materials. CMP3 refers to the next, ostensibly smaller, revision in light of Common Core (I’m working with the original authors on this revision, hence the Common Core rantings at this site-want more info on CMP3? I maintain that blog too-it’s new).

Now to your question. Yes, CMP asks much more of eighth graders than does Common Core w/r/t algebra. I’ll have to write more on this. But the short answer is that the authors wanted to stand behind CMP2 as covering an Algebra I curriculum (a stand which had been unnecessary when writing the original materials in the early 1990’s). In the last 15 years, the rhetoric of Algebra for all eighth graders has been steadily ratcheting up. And then Common Core doesn’t do much with algebra at eighth grade. It remains to be seen whether districts will be willing to let up on the algebra requirements at eighth grade and just do the Common Core work, or whether eight grade algebra will continue to be a de facto standard. We haven’t gotten far enough into the CMP3 revision to make any decisions on this point.

My own gut feeling is that I’d like to see the curriculum let up a bit on the algebra to allow some breathing room for other topics at eighth grade. I’d like to see eighth graders do the two units you allude to: Growing, Growing, Growing (exponentials) and Frogs, Fleas and Painted Cubes (quadratics). But I’d like to lay off the factoring a bit. And then I’d like them to do the Say It with Symbols unit, which does a nice job of summing up the algebra work across the curriculum. And I’d lay off the factoring there, too.

As for the intro to 5.NF.5? My pleasure.

• To clarify on synthetic substitution: no argument that it is used as a shortcut for polynomial long division. I just don’t believe it is in any way derived from the long division algorithm. It’s just fancy factoring.

9. I’m coming very late to this discussion (and I haven’t read the whole thread), but it popped up on my blog, so I thought I’d give an author’s perspective. Here’s the standard (sans formatting):

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.

Chris advocates computing the unit rate using either equivalence or division of fractions, and I completely agree, and I think this interpretation is clearly consistent with the standard. If I understand correctly, the objection is to having kids explicitly deal with the concept “(1/2)/(1/4) miles per hour”, as opposed to simply having them compute the rate by dividing 1/4 into 1/2, and get 2 miles per hour. Note that the example does indeed indicate 2 miles per hour as the final result. In fact, I’m having a hard time seeing why the example doesn’t suggest exactly the computation Chris suggests. Still, his fears of crazy assessment writers taking this to an extreme are not without basis. Maybe the progression document about ratios and proportional relationships should say something about this.

10. Christopher

Thanks for stopping by, Bill. It means a lot.

You write:

If I understand correctly, the objection is to having kids explicitly deal with the concept “(1/2)/(1/4) miles per hour”, as opposed to simply having them compute the rate by dividing 1/4 into 1/2, and get 2 miles per hour.

Indeed, that is the objection. The standard seems really clear that (1/2)/(1/4) is to be treated as a unit rate, not just as a rate. If the intention of the standard is to allow thinking about this complex fraction as (1/2 mile)/(1/4 mile), then I have no beef.

Your comment, Bill, seems to suggest that I’m worrying about something that wasn’t really intended to be a big deal in the CCSS. But I disagree. My reading of both the standards and the progressions is that:

1. rate and unit rate are treated as equivalent to each other, and

2.complex fractional unit rates are important, not just a tossed-off mention (this reading is supported by the footnote at grade 6-they’re important enough to exclude them from 6th grade)

I disagree on both of these counts. I think of (1/2 mile)/(1/4 mile) as a rate, and I think it’s reasonable to suggest that it is to middle school students. And I don’t think these same kids are ready to deal with the distinction between this and (1/2)/(1/4) miles per hour. But the standard in question is calling on us to teach this distinction (and to write it into curriculum).

11. There’s no question that the “For example …” part of this standard could have been written more clearly. I can only say that your interpretation 2 is an unintended consequence. Complex fractions are indeed introduced in Grade 7 (see footnote to 7.NS.3), and students are intended to divide fractions by fractions to compute unit rates in Grade 7 (7.RP.1), and the “For example … ” text here does indeed combine the two, but this was not intended to push the notion of a “complex fraction unit rate” to the fore. The footnote in Grade 6 was meant to signal that, although students learn to divide fractions by fractions in Grade 6, they are not expected to do that in the context of rate problems. That restriction is then lifted in 7.RP.1. I would also note that in general the italicized “For example …” text of a standard is just that, an example. The main body of the standard starts out “Compute unit rates associated with ratios of fractions … “, which supports the division-of-fractions interpretation of the “For example ..” text in my previous comment.

But you should look again at the progression, because I don’t think it does support your interpretation 2, and in fact it tries to clear it up. In particular, the rate language in the progression is pretty consistently of the form “x units for every 1 unit” rather than “x units/unit”. If that language were applied to the “For example …” text in this standard, it would read something like” “For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as (1/2)/(1/4) = 2 miles for every hour.” If I understand your critique, this would be better, no?

As for your interpretation 1, I don’t see how you get it from the standards, and certainly not from the progression. But maybe we can save that discussion for another time. It’s Thanksgiving day, after all.

12. Christopher

Bill:

As for your interpretation 1 [that rates and unit rates are treated synonomously], I don’t see how you get it from the standards, and certainly not from the progression.

This would be the passage that suggests it to me. From the progression p.3 (emphasis added):

For example, the ratio 3 feet for every 2 seconds has the associated rate 3/2 feet for every 1 second; the ratio 3 cups apple juice for every 2 cups grape juice has the associated rate 3/2 cups apple juice for every 1 cup grape juice. In Grades 6 and 7, students describe rates in terms such as “for each 1,” “for each,” and “per.” The unit rate is the numerical part of the rate; the “unit” in “unit rate” is often used to highlight the 1 in “for each 1” or “for every 1.”

• But this is making a distinction between rate and unit rate, right? Still, it’s subtle distinction, and I’m guessing what you want is to call 3 feet for every 2 seconds a rate rather than (or as well as) a ratio, reserving the term unit rate for rates that are expressed per 1 second, whereas we call the latter thing a rate, reserving the term unit rate for the numerical part of the rate, which is later understood as the constant of proportionality in a proportional relationship. Your way is certainly a common way of speaking, and perfectly fine among people with a secure understanding of ratios, rates, and proportional relationships. The rationale for narrowing the use of the term in the progression is to prepare for later understanding of proportional relationships $y = kx$ in which $k$ is a rate. Later, perhaps in high school, proportional relationships become a powerful, if sophisticated, way of looking at many different problems, eliminating some of the mystery from “setting up proportions.” So that’s where this is all heading. But this was certainly a knotty issue even for the authors of the progression, not all of whom agreed. At some point we decided it was time to make the draft public and see what others thought. I did mention in my post that this draft was rougher than others, and you are seeing that. I don’t know how this will shake out eventually, but thanks for digging into the details and giving us your thoughts.

13. Christopher

So 3:2 is a ratio, 1.5:1 is a rate (and also a ratio) and 1.5 is a unit rate. Do I have it right now?

If so, then I think you want k to be a unit rate (not a rate) in y=kx.

14. Christopher

Oh and by the way…YOU CAN DO TEX IN WORDPRESS?!?!

Nice.

15. Christopher

Bill:

“For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as (1/2)/(1/4) = 2 miles for every hour.” If I understand your critique, this would be better, no?

I would be much happier with this, yes. But why not just:

If a person walks 1/2 mile in 1/4 hour, compute the unit rate as 2 miles per hour.

What do we lose by not specifying the exact notation that is used here? I’d be perfectly happy with someone computing (1/2):(1/4) and scaling by a factor of 4 to get 2:1. But your proposal above is better than the current text for sure.

16. Answering all these at once:

1. Almost, but 1.5:1 is still just a ratio. However, 1.5 miles for every 1 hour is a rate, and yes, 1.5 is the associated unit rate.

2. Yes, you enclose the TeX code in dollar signs as usual, but put the word latex after the first dollar sign. It’s not perfect, sometimes looks dropped below the line, but it can do pretty cool things. You can also do displayed equations by using double dollar signs I think, let me check:
$$\int_{-\infty}^{\infty} e^{-x^2} \, dx.$$

3. You proposal is fine too.

17. OK, I was wrong about the displayed equations. I think you just enclose them in a center tag and use \displaystyle:

$\displaystyle \int_{-\infty}^{\infty} e^{-x^2}\,dx.$

18. And yes, you are right that $k$ is a unit rate, although I think eventually, in high school, we just start using the word rate, because we start to assume that all rates are expressed unit rates (e.g. (1/2)/(1/4) m/hr (joke)), and expect the whole per unit thing to become internalized into a student’s conception of rate. Part of the problem that is causing this discussion is exactly when to make all these transitions.

19. Pete

Two great things that have come out of the comments. First off, In NYS, the 8th grade assessment one year literally said, “Use FOIL to solve,” to which I was thinking, “I don’t teach FOIL. I teach distribution!” Secondly, in regards to slope and rate of change and per, I insist to my students that when they hear the word per, that means that some type of division has been going on. I have some teachers that will battle with me because the students have to multiply to get their answer. This is true because they have to multiply to get the unit analysis to work out and have the terms cancel! It’s so frustrating to have another teacher say, “I had these students that said per means divide but they have to multiply to get the distance.” To which I say, “Yes, they had to take the miles PER hour by dividing and then multiply by the hours to get the distance.” Just my \$0.02.

20. Alex Otto