Performance assessment: ratios

It’s time for a performance assessment.

This is not multiple choice.

If you have been reading along, you know that I advocate talking math with your kids as the mathematics equivalent of reading with them 20 minutes a day.

Furthermore, you have surely read each of my thousands of words on Common Core’s vision of ratio, rate and unit rate.

And yesterday I proposed a bit of alternate text for the Progression on Ratio and Proportion.

Your assessment task is this.

The task

Imagine you have a 12-year old daughter. She has been learning about ratios and is assigned the task of finding real-world applications of them, as found in the media. She comes across an article that interests her. You strike up a conversation about the excerpt below.

Defend or critique any of the following claims:

(1) The Common Core Progression on Ratio and Proportion will be helpful to you as a parent in discussing the relationship between this passage and her homework.

(2) The distinctions being made (among ratio, rate and unit rate) in the Common Core Progression are useful and meaningful for interpreting this passage.

(3) The discussion proposed in yesterday’s post will be helpful to you as a parent in discussing the relationship between this passage and her homework.

(4) The distinctions being made in yesterday’s post (among ratio, rate and unit rate) are useful and meaningful for interpreting this passage.

The passage

Joe-urban discusses parking at urban grocery stores:

However, David Taulbee, Architectural Manager of Publix, notes that parking at many of their urban stores is full only at peak times, so that sacred parking ratio of five per thousand is called to question, particularly if the store has other parking options nearby like shared, on-street or bicycle parking.

(N.B. That’s five parking spaces per thousand square feet of retail space.)

Your work will be scored on the basis of relevance and the use of evidence. It will not be scored on the extent to which you agree with the scoring committee’s views on these matters.

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Ratcheting back the rhetoric on Common Core

Bill McCallum writes in the comments here at OMT:

 I don’t think that effort [to define terms such “ratio” and “rate” that CCSS leaves undefined] deserves quite the ridicule it is receiving here, but never mind, the criticism will be taken into consideration nonetheless and inform the final draft. I’ll only say that if I had a dollar for every time someone told me the answers to all these questions were obvious, I’d be a rich man. Of course, the “obvious” answers are mutually self-contradictory. This seems to be an area where it is very difficult indeed to find common language, and where emotions run high.

Fair enough. I’m happy to tone things down a bit.

I do need to observe that no one here at OMT (least of all me) has suggested that the answers to the questions at hand are “obvious“. I agree that they are not obvious at all, and I agree that it is very difficult to find common language with respect to these ideas. (Although this last bit is tricky; if we all use the same words but mean different things by them, are we speaking a common language?)

No, my critique is not at all that Common Core has failed to state the obvious definitions.

My critique is that I see no evidence that Common Core-either the Standards or the Progression on rational number-take into account research on how children learn this content, nor do they seem to coincide with everyday uses of these terms. In the era of No Child Left Behind and “evidence-based practice”, I find it troublesome that results of important research work such as that in the Rational Number Project or Cognitively Guided Instruction don’t seem to form a basis for either document.

I find it surprising that there are no research references in the Progression.

In my work with Connected Mathematics, I have many times had teachers ask for definitions of rate, ratio, fraction and rational number. As writers, we have hashed out these ideas many times as well. Answers are not obvious and reasonable people can disagree.

But we sort of agree that answers to these questions ought to be consistent with, and explain relationships to, uses of these terms in mathematics and the world. I don’t understand why this isn’t the starting place for the Progression.

So how about this for a first attempt at the relationships involved here?

A ratio is a multiplicative comparison of two quantities (usually both are non-zero). Conventionally, we use the term “ratio” to apply to part-part comparisons, but this need not be the case.

We can express ratios in several forms. If there are 5 girls for every 3 boys in a certain class, we say that (1) the ratio of girls to boys is 5 to 3, (2) the ratio of girls to boys is 5:3, (3) the ratio of girls to boys is , (4) there are 5 girls for every 3 boys.

The fraction notation  is problematic in early ratio instruction because children may confuse it to mean that  of the students are girls. Children are accustomed to fraction notation being reserved for part-whole relationships; for this reason the notation should be saved for later instruction.

The term rate suggests change. We tend to talk about a “ratio” in static situations where the values remain constant but a “rate” in a situation where the quantities are changing. In the girls and boys situation above, it would be correct to say that there is a rate of 5 girls for every 3 boys, but this feels awkward. If students were enrolling in a school and there were 5 girls enrolling for every 3 boys, the term “rate” is a more natural fit.

A unit rate is a rate where one of the quantities being compared is 1 unit. If we enroll five girls for every three boys, this is not a unit rate. We could say that there are  girls per boy enrolling at the school, or  girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are  boys per girl.

What counts as a unit varies. When computing a “unit rate” for buying pop, we could compute the cost per ounce, the cost per can, the cost per six-pack or the cost per case of 24. Which of these is considered a unit rate depends on our choice of unit.

To summarize the discussion, ratios and rates are different mainly in connotation. Each expresses a multiplicative relationship between two numbers (as opposed to an additive relationship, for which we use the term “difference”). Unit rates are important forms of rates because of their intimate connections to algebraic and calculus ideas such as slope and rate of change.

It’s just a first stab at discussing these terms in this context, but is consistent with common usage and focuses the discussion on the main idea that is important at this level-rates and ratios are about multiplication relationships, which are the heart of proportional reasoning.

Going back to the lawn example that has been the focus of discussion here as well as over at the Common Core Tools website, this would suggest that “7 lawns in 4 hours” is a rate (there is change involved, and it’s not a part-to-part relationship), and that there are two unit rates: lawns per hour and  hours per lawn.

Again, I am not claiming that these relationships are obvious. But if a couple of important goals for the Progressions work are (1) clarity and (2) usefulness for teachers, professional development and curriculum development, I think my proposal above is an improvement over the present document.

What is a rate? Common Core revisited

A commenter (not me) asks over on the CCSS Progressions blog:

Are rate and unit rate interchangable? Or should a teacher define them for a middle school students as… Rate: a quantity derived from the ratio of two quantities that describes how many units of the first quantity corresponds to one unit of the second quantity. Unit rate: the numerical part of a rate (e.g. For the rate 8 feet per second, the unit rate is 8.) If these are correct, I would then ask for clarity on the phrase “at that rate” in this example from 6.RP.3b. “For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?” Does “at that rate” here really mean “at the rate implied by the ratio of 7 hours to 4 lawns”? You aren’t suggesting that “7 hours to mow 4 lawns” is a rate? The rate, which you ask for in the last question, is “7/4 hours per lawn”?

The answer to this last question is going to be “yes”.

Whether it matches the meaning of these terms in real life or not, the answer will be “yes”.

Whether it matches the grammatical structure of the English language, in which unit would be seen to modify rate, the answer will be “yes”.

A unit rate, in the Looking Glass world of Common Core is not a kind of rate; it’s a different thing altogether. A rate is a numerical/linguistic construction. A unit rate is a number. Each is associated with a ratio.

But why?

The best sense I can make of this is that CCSS wants these terms to be precisely enough defined to admit a sort of mathematical clarity. No such definitions previously existed. So CCSS made them up.

Objections to my Common Core objections

I want to give a little breathing room to a thoughtful debate going on in the comments. I know it requires an extra click to get to those comments so maybe we’ll get the attention of a few more readers by dedicating a post to it. Additionally, if Common Core spurs thoughtful debate about relationships between math content and students’ minds, we’ll all be smarter for it.

Sean objects to my Common Core unit rate rant and offers the following classroom scenario:

A problem

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.

the lesson

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.

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?

Really thoughtful stuff. Much appreciated. I have more to say but I’ll hang back and let Sean’s ideas simmer for a few days.

Huh? Making sense of Common Core

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.

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.