# Tag Archives: ratio

A few weeks back, bedtime was spiraling out of control. The kids share a room; they would be wound up at bedtime and the transition to sleep was not happening smoothly. We had a big, big problem on our hands.

We solved the problem with 10-minute reading time. The kids have to be in their beds. We dim the lights. We set a timer for 10 minutes. It has to be quiet during that time. Then we turn out the lights, do the picture (remember the elephants?) and sleep comes more easily.

Complete transformation. It is awesome.

Tabitha, the other night, wanted to color. We talked and agreed that she could do it sometimes. As is the nature of 5-year olds, she soon wanted to know the limits.

The following conversation took place on Wednesday night this week.

Tabitha (five years old): I know I can’t color every night but can I tonight?

Me: Yes.

T: Then read, then color the next night?

Me: I don’t know. I think reading twice before the next color is better.

To be clear. It was not my intention to get into a math conversation at this point. I just wanted her to go to bed.

No, this move on my part was truly about literacy, not math. I don’t want 10-minute reading time to turn into 10-minute coloring time. I really, really like the idea that books will become part of my kids’ independent bedtime routines.

But Tabitha loves to know the rules she’s playing by. And when those rules are based on numbers, they’re going to lead to math every time.

Me: Right. That sounds like a good ratio.

Whoa.

Couple things.

First of all, I used the term ratio with absolutely no expectation that she would process it, and I am quite sure that she did not. I have long been an advocate of using good vocabulary with my children—there is no shame in not knowing the meaning of a word, but also no sheltering them from the fact that these words exist. This is at least partly the source of their substantial vocabularies. But I do not believe she knows the word ratio.

Secondly, Tabitha’s reformulation of the 2:1 ratio as 4:2 blew me away. It nearly slipped past me without notice. I was focused on getting them to bed; we were in the truly final phase of that process. I had pretty much tuned her out.

But when I looked at her, I could see she was expecting a reply. She needed to know whether she could get two coloring nights in a row by doing four reading nights in a row.

So I replayed her question in my mind, counting the reads.

Me: Yes. That would be fine. You may do that.

## 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.

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.

## 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.

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 $\frac{5}{3}$, (4) there are 5 girls for every 3 boys.

The fraction notation $\frac{5}{3}$ is problematic in early ratio instruction because children may confuse it to mean that $\frac{5}{3}$ 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.

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 $\frac{5}{3}$ girls per boy enrolling at the school, or $\frac{5}{3}$ girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are $\frac{3}{5}$ 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: $\frac{7}{4}$ lawns per hour and $\frac{4}{7}$ 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.

I always appreciate when my local Cub Foods does the math for me. It saves me a lot of time.

But what if I want 5? Or 17?

## Quick survey on mathematical correctness

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?

## 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.

## 50/50

### The setup

The Walker Art Center in Minneapolis has an exhibit titled “50/50”. The premise is that half of the exhibit was selected by public vote online and the other half of the exhibit was selected by museum curators in response to the public selections.

### The questions

In the photo below, we see the public half of the exhibit on the left. The curators’ side on the right is masked in black. In fact, you can only see about 2/3 of the public’s side; the remainder is past the left-hand edge of the photograph.

(1) How many works are in the exhibit?

(2) How many did the public choose?

(3) How many did the experts choose?