OK, I get it. The Common Core State Standards are about large-scale coherence. Stay focused on the big picture of getting everybody going in the same direction, then tweak things later, blah, blah, blah… I get it.

And yet kids’ education is at stake. And teachers’ jobs in the era of No Child Left Behind and Race to the Top. And the quality of curriculum that has to bend over backwards to align with these standards.

So when I dig into the details in my capacity with *Connected Math*, I get indignant about places where things don’t make sense. Consider the case of ratios at sixth grade:

6.RP.2.Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

I’m OK with this. I’m not thrilled with the “unit rate a/b” part, but it’s not a train wreck. Let’s look ahead to seventh grade, shall we?

7.RP.1.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.

Huh?

The complex fraction (1/2)/(1/4)? Are you kidding me? Just try verbalizing this:

I walked one-half-over-one-fourth miles per hour.

Does anyone ever talk about rates this way? Ever?

No way! The only way to even come close would be to say,

I walked a half a mile in a quarter of an hour.

But then that’s not a unit rate. For some reason Common Core is obsessed with unit rates-strictly defined. If I thought this were a throwaway line, I wouldn’t be worried. But it’s not a throwaway. That sixth grade standard above? It had a footnote:

Expectations for unit rates in this grade are limited to non-complex fractions.

So the Common Core writers didn’t just make this *(1/2)/(1/4) unit rate* nonsense up on their first pass through seventh grade. Oh no-it was important enough to go back and exclude it from sixth grade. And important enough to use up one of only three footnotes in the entire 6-8 math standards. The other two? Here’s the next one:

Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

Are you sensing a theme here?

Right, it’s this other odd obsession with complex fractions (i.e. fractions in the numerator and/or denominator).

And the third footnote:

Function notation is not required in Grade 8.

Phew.

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*UPDATE: Reader Sean steps up to defend this standard in the comments below. I highlight his objections in a later post, and then respond in yet one more post on this topic.

What about, ‘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. ‘

I’m curious about the unit rate.

The description is inelegant, but the standard itself is valuable. It sounds like your problem is with the complex fraction, but I think the comparison between (1/2)/(1/4), half mile/15 minutes, and .5/.25 would be a great discussion about equivalence.

Sometimes I have a hard time understanding what a particular CC standard is asking for, and wonder whether they’re intentionally nebulous. If you, one of the smartest dudes around, is confused/frustrated, how do you suppose states will write their tests? Do you think they’ll continue to use the same questions as always, just rearranged?

Sean, your example is perfect. I agree 100% that kids ought to be able to deal with 1/2 mile in 15 minutes and in getting from there to 2 mph. I am completely groovy with 1/2 mile per 1/4 hour. But

that’s not what is in the standard. And that’s my beef.1/2 mile per 1/4 hour is not a

unit rate. The CCSS authors seem to believe that two things are especially important: (1)unitrates and (2)complexfractions. And that, furthermore, two great things go great together.And if 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 if I’m a kid learning about rates, ratios and fractions? It’s a useless monkey trick with little or no conceptual basis.

Unit rates

areextremely important-both in mathematics and in life. Complex fractions are useful when they arise; 99.5% of the time, it’s with fractional numerators, but I have no beef with fractional denominators either.It’s the unholy arranged marriage of the two that disturbs me.

I guess in my (admittedly hastily written) problem, I was hoping for a flow like this:

1. We discuss what’s happening. Hopefully a few students are curious about either a) how fast he’s going or b) a reasonable approximation for when he’ll finish.

2. We decide on a way to measure how fast he’s going. We discuss measuring by minutes or by hours, and hopefully come to the conclusion that miles per hour serve our purposes best.

3. The students work in pairs. They use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate) in any number of ways. Formal proportional reasoning, informal proportional reasoning, number sense, a graph, a table- whatever is thrown out.

4. We discuss how these strategies are related by placing them side by side.

5. We debate which are the most efficient of these strategies.

6. Now the teacher has a decision to make. Personally, I feel that while it may be unholy and it’s definitely arranged, this is not a terrible transition into a discussion about complex fractions.

7. 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.

8. A number of strategies, again side-by-side, are used to solve this expression. We break it apart visually with manipulatives. We convert both the numerator and the denominator into decimals. We show the algorithm. The students notice that “2” as a solution is the same “2” that they saw before.

9. The teacher states that it may not be the most efficient way to find the unit rate in this particular problem, but that it may be in future ones. We speculate about when.

10. In subsequent days, the teacher can veer towards abstraction with complex fractions. When misconceptions arise, there is the race analogy to give it footing.

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?

Karim, testing is being put together by two consortia of states. This is nightmarishly boring policy stuff, but really important and worth following. Here is a link to a report that came out this week detailing next steps for these two testing consortia.

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I guess I thought about this standard a different way: a frequent style of ratio problem is “How many X are there in Y?” And the standard allows for both X and Y to be fractional: “How many 1/4s are there in 1/2?” I may be wrong about this, but the above question (and several more like it, like “How many 1/3s are there in 7?”) are placed in our Algebra 1, Chapter 1, to try and give students an impression of fraction division and rate problems that doesn’t amount to “invert and multiply”. Students don’t do very well with these problems, because they have a hard time thinking of “1/3” as a unit, and my feeling is the goal of this standard is to correct that issue as much as possible.

I feel CMP does a pretty good job of this with the ribbons activity in B&P3, better than most of what’s out there. But I’ve seen some CMP graduates who don’t have the grasp of fractions and units needed to answer the questions above … some.

I have to say I’m not thrilled with having all students writing and evaluating complex fractions that early; it seems pretty likely to lead to only a procedural understanding of how to solve problems. And the quest for context by using “1/4 hour” as the unit in the sample is a bit of a turd.

While you’re at it, get on that function notation! I’m sick of kids saying y = 3x + 2 is a function, when it’s clearly a relationship between two variables! f(x) = 3x + 2 is a function! 😉

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When the standards in California were written, they were written by Language Arts teachers/professionals. It seems to me, we have the common core written by the same professionals. Math is a language, just not one that the writers seem to have studied beyond their definitions.