Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Tuesday, December 11, 2018

Last Post for 250 Miles

Hi everybody,

I have reached the end of my rope with Blogger, and have moved all the contents of this blog to a shiny new location. Any new posts after today will only be posted there. This blog will remain open so that any existing links will continue to work.

Friday, January 26, 2018

Why We Don’t Cross Multiply

(co-authored with Kristin Gray)
“Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.” [from Illustrative Mathematics 6–8 Math, grade 6, unit 2, lesson 12]
We don’t tend to spend much time explaining why we didn’t do things a certain way, preferring to provide a thorough rationale for approaches we did take. But the choice to not spend curricular time on “cross multiplying” as a technique for dealing with equivalent ratios is sometimes contentious and also illuminates the decisions you have to make when writing a curriculum to address a particular set of standards, so let's spend a little time on it here.

First of all, what does “cross multiply” mean? Cross multiplying is sometimes invoked as a technique when solving a problem like “A shade of green paint is made by mixing 2 cups blue and 3 cups yellow. If you have 9 cups yellow, how much blue should you mix with it to make the same shade of green?” The technique is to represent the unknown cups of blue with a letter, let’s say x, write an equation like 2/3 = x/9, and then “cross multiply,” writing 2 * 9 = x * 3 and solving this equation get x = 6. So, 9 cups yellow can be mixed with 6 cups blue to get the same shade of green.

A math curriculum needs to attend to conceptual understanding, procedural fluency, and applications. One implication of attending to conceptual understanding is that you need to help students understand why for every how. So if we were to include cross multiplying as a technique, we would only do it if we developed understanding of why it works.

Take a pair of fractions that are equal but written with different numbers, like 6/9 and 2/3. I can draw a diagram of two rectangles, each representing 1, and partition them into ninths and thirds to show why 2/3 = 6/9:

Now let’s cross multiply. It is also true that 6 * 3 = 2 * 9. Where in my diagram can I see why 6 3’s must be equal to 2 9’s? You can see it, but you have to shift your perspective of what represents “1” (or, said a different way, your perspective of what the fractions represent). It is pretty challenging to explain, based on the meaning of fractions and an understanding of fraction equivalence, why this technique works.

The other way people often justify the move from 2/3 = x/9 to x * 3 = 2 * 9 is by invoking the idea that if you “do the same thing” to each side of an equation, then the equation is still true if the original equation was true. We develop “do the same thing to each side” when the 6.EE conceptual category is studied in its own right in unit 6 and after students understand the vinculum can represent division. However, students work with equivalent ratios and rates in units 2 and 3, so that contexts developed there can be used for learning new content. The standards just say what students should be able to do at the end of the year; a curriculum makes choices about order and emphasis. And decisions have consequences! So the decision to place the study of equivalent ratios earlier in the year means that we didn’t yet have access to “do the same thing to each side.” One could make a reasonable choice to study equation solving earlier in the year and write a different course.

And back up to that problem about the blue and yellow paint. Why did we use 2, 3, 9, and x to write fractions? Why should these fractions be equal in this problem?

It is a jujitsu move to start with a problem that uses only whole numbers and then write a statement equating two fractions. For people who are already intimately familiar with these ideas, it is useful to represent ratios using fractions. But we are introducing this important and new concept in grade 6, here, and students have worked hard to understand in grades 3–5 that fractions are numbers (3.NF.A) and rely on that definition in their study of fractions. The standards define a ratio as a relationship between two quantities (6.RP.A.1) (and an important ratio that is equivalent to a : b is a/b : 1 (6.RP.A.2)). To suddenly assert that a ratio (2 numbers) is a fraction (1 number) runs counter to this definition of ratio and doesn’t build on the understanding of fractions from grades 3–5. To solve a problem about equivalent ratios by jumping to a statement that equates two fractions fuzzes up the definition of a ratio and the understanding of what a fraction is.

The 6.RP standards are the start of a long chain of experiences and reasoning that results in students understanding that a linear function is characterized by constant rate of change in grade 8. It goes equivalent ratios and rate → proportional relationships and constant of proportionality → linear functions and constant rate of change.

So the approach that we take in grades 6 and 7 to equivalent ratios and proportional relationships is to 
  • ground understanding in contexts, taking time to develop familiarity with the contexts (mixtures, constant speed, unit price);
  • build a collection of representations of equivalent ratios that are used as tools for understanding and tools for problem solving (discrete diagrams, double number lines, tables of equivalent ratios, graphs in the coordinate plane, equations of the form y = kx);
  • explicitly show how useful a unit rate is for solving problems—which can be thought of as an entry in a table associated with a “1”, or k in the point (1, k) on a graph;
  • work toward using tables efficiently to solve problems (if you think about it, a table with 2 rows and 2 columns looks an awful lot like “set up a proportion”);
  • work toward using equations and graphs to represent situations and solve more sophisticated problems.
I know that lots of people are very comfortable teaching cross multiplying, and this change is challenging. But there’s also nothing inherent to the mathematics that requires the cross multiplying procedure. Our approach fosters conceptual understanding, is aligned to the definitions of ratio and fraction in the standards, and is more extensible to future learning. 

So how would kids using IM 6–8 Math solve the paint problem? I predict they would mostly likely say something like, “Since the recipe needed 3 cups of yellow and I want to use 9 cups of yellow, this is a triple batch of paint. So I’m going to multiply 2 by 3. 6 cups of blue.” You might be thinking, “Oh, but this problem is so easy.” Yeah, but I had to draw fraction diagrams about it before, so I picked easy numbers. To see examples of some more on-grade-level problems, check out 6.2.12 or 6.3.7 or 7.2.6 or 7.4.3.

If you’d like to see more details of how “direct proportion problems without cross multiplying” is handled lesson by lesson, I would recommend starting with the scope and sequence for grade 6 and grade 7. In grade 6, you want to look at units 2 and 3. In grade 7, look at unit 2. To see lesson plans, navigate to the teacher materials for those same units. (You’ll have to create a free account and be logged in.)

Wednesday, January 17, 2018

Low Floor High Ceiling Tasks for Fidgety Adults

I am planning an hour of math for a group of adults that is a daunting mix of math teachers, business professionals with no particular fondness for math, research mathematicians, and other assorted riffraff. First, I was thinking through the feasibility of facilitating us all working on the same problem (eep), but then I just decided to punt and steal John Stevens' idea and set up stations However, when I was looking for a good, single problem to use, I asked Twitter, and got lots of helpful suggestions. I saved them all as a Moment. I'm not entirely sure what that means; hopefully it means I can find them all later. (If you tweeted a response and I missed including it, I apologize; I think I got them all but I don't trust that Twitter showed them all to me when I was looking.) Happy tinkering!

(Choosing a tweeted image as the "cover art" was acting strangely in the embedded moment, so enjoy this picture of Tangie.)

Wednesday, November 22, 2017

Respecting the Intellectual Work of the Grade

A thing that I think we did really well in Illustrative Mathematics 6–8 Math was attend carefully to really deep, important things that adults that already know math can easily overlook. For example, what does an equation mean? What does it mean for a number to be a solution to an equation? What does it mean for two expressions to be equivalent? (This is an example of the crucially important foundational understanding that gets short shrift when we rush kids through middle school math.)

Confusingly, the symbol = can mean a few different things.

  • If we represent the quantities in a word problem with x + 2 = 3x, we might mean, "What value in place of x, if any, makes each side have the same value?" 
  • If we decide to see what happens when x is 4, we might write down x = 4. In this case, the = symbol means, "At this moment, we assign the value 4 to x." 
  • If we represent two quantities that we suspect are equal no matter their value of x with 4 + 6x – 12 = 2(-4 + 3x) and use properties of operations to rewrite each expression until they are identical to each other, = means an entirely different thing: "I think these expressions are equal no matter what I substitute for x, and I would like to know for sure."

If you're reading this you probably realized all of that a long time ago, but none of this is at all obvious to your average 6th grader, or even 11th grader, and 6th grade is where in the common core math standards we are supposed to make a big pivot from arithmetic to algebra.

This is just one task in a 6.EE arc designed to foster deep understanding, but I think it exemplifies the careful approach that we take for the sake of sense making. [student materials] [teacher materials (requires free registration)]

The purpose of this task is to understand that two expressions that are equal for every value of their variable are called equivalent expressions. After the teacher eases students into it, students have a chance to work through the task. The entire point is to contrast x+2 and 3x, which are only equal (the same length) when x is 1, with x+3 and 3+x, which are equal no matter the value you substitute for x. Hey we could have told you x+3 and 3+x are equal no matter the value of x because of the commutative property. I wonder what other properties we can make clever use of. That's the purpose of the next task.

Now, it was hard to choose activities from one lesson to share. This curriculum is multi-faceted and has some super cool stuff. (Akshully, the entirety of unit 6 in grade 6 is one of my favorite things in the world, along with the continuation of focusing on the EE standards in grade 7, unit 6. Check it out.) And this lesson is all steak and no sizzle. But getting the unsexy but necessary bits right is something I'm really proud of.

Monday, July 31, 2017

FAQ: What Can We Change?

We are putting the finishing touches on the Illustrative Mathematics Middle School Curriculum. (For early access to sample units in the pilot, you'll have to share your contact info with us here, but version 1 will be released any day now.) I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. This is the second in a series; here's the first one.

Today's Q can come in many forms: "Do I have to do it this way?" "How much freedom is there to change things?" "Can I still use my favorite activities?"

This is an analogy I learned from someone at Louisiana Department of Education, where they are getting impressive results by incentivizing schools to choose well-aligned curricula. If you were to try and cook a new, complicated recipe, you would probably make it as it's written the first few times you make it. You don't know what all the ingredients are for, you don't know the rationale behind all of the instructions, you don't really understand how it works, yet, before you cook it a few times. Once you start to understand the recipe, though, you can make smart choices to modify it to suit your tastes and needs: substitute green beans for eggplant, leave out the almonds, or take it out of the oven a little earlier, for example.

Just like a dish you want to eat is a cohesive whole, people need to think of a curriculum as a coherent, connected, fairly complicated whole, with dependencies. Standards are one thing—they are a statement of what kids should know at the end. A curriculum makes choices, and choices have consequences. We set up pins in October that we knock down in February. After students have a well-designed opportunity to learn a term, idea, or skill in one unit, we believe that they will be able to remember it in a later unit. This is what you want out of a curriculum. You want kids to be able to make connections between ideas.

The starkest example of this is a question we got from one of our pilot schools: "The word slope just shows up in grade 8, unit 3, as if the kids are already supposed to know what it means. This is terrible! What is going on here?" What was going on was, they skipped units 1 and 2, which were about transformations, thinking transformations were less important, and jumped right to the unit called "linear relations." The end of unit 2 takes a transformational approach to understanding the meaning of slope. (We use dilations to understand what it means for polygons to be similar, learn properties of similar figures, and then use slope triangles (similar right triangles with their hypotenuses lying on the same line) to show why we are allowed to refer to the slope of a line.)

Just like a new recipe, you kind of have to teach a coherent curriculum the way it is written for a couple years before you really understand what is in there. Then, you are in a position to understand what it is safe to substitute or rearrange. 

Saturday, July 29, 2017

Your Opinion of #MTBoS Has More to Do with You Than It Does with #MTBoS

"Someone's opinion of you has way more to do with them than it does with you." I have a smart mouth and also get upset when other people are upset with me, so I've likely heard this aphorism more than the average person. It's been floating into my head lately, not because I think someone is upset with me (for once) but because of thunderstorms on Twitter over use of a hashtag. I'd like to propose that what someone thinks of MTBoS (Math Twitter Blogosphere) has more to do with them than it does with MTBoS. Consider:

  • a mid-career math teacher who checks out Twitter, finds a hashtag he doesn't understand and conversations under that hashtag he doesn't understand
  • an organizer who has poured immeasurable energy into welcoming first-time attendees to TMC under the banner of MTBoS
  • a popular blogger and speaker who wants his ideas to have a broad and lasting impact on the way mathematics is taught, and has evidence that #MTBoS is a barrier to interested people accessing those ideas
  • an early-career math teacher who figured out what #MTBoS means by asking someone or google and periodically checks out the hashtag for inspiration
  • an early adopter of blogging and twitter who found many friends for life in MTBoS who make up a part of her support network and social circle
  • a math teacher who discovers #MTBoS, tries asking a question on twitter with that hashtag, and gets no response
  • a math teacher who had good results with resources found through MTBoS, but doesn't feel like a member of the club because she doesn't want to start a blog
Here is me anticipating people getting upset and trying to head that off: I'm not trying to characterize any of these as selfishly motivated. All of these archetypes exist only because they want what is best for their students, all of humanity, or both. Also, all of these people's feelings are legitimate, because of course they are, because they are having them, and I'm not suggesting otherwise. Finally, if none of these describe you, I'm sorry and you still matter. This isn't an exhaustive list, it's my musings over breakfast.

My prediction is that #MTBoS isn't going anywhere anytime soon. At least until the current crop of organizers of all things MTBoS retire, or as long as they remain good at generating energy among newcomers. 

My other prediction is that other hashtags will grow and fade in popularity. Easier to interpret hashtags are appealing because there is a lower barrier to entry, but they also tend to get diluted by spammy marketers, and then people stop paying attention to them. One possible explanation for the longevity and strength of #MTBoS as a hashtag is that it's a bit of a secret handshake.

Here is one idea I have: when you use MTBoS not as a hashtag, but in longer form (on a blog post or while speaking), always follow it with "Math Twitter Blogosphere." The way Rachel Ray always said "E-V-O-O extra virgin olive oil." Clue the noobs in. It's a kindness.

I'm looking forward to meeting and learning from new people on whatever hashtag we come up with and maintaining my enthusiasm for MTBoS and all we have accomplished and all of the good work yet to come.

Friday, June 9, 2017

FAQ: So When Do I Teach?

We are putting the finishing touches on the Illustrative Mathematics Middle School Curriculum. (For early access to sample units in the pilot, you'll have to share your contact info with us here, but we're looking at mid-July for the release of version 1.)

We're often in the position of talking to teachers who have heard about the materials and are evaluating them, or whose district has adopted them and they are just learning about them. I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. I am thinking to hash some of the Q's out in blog form, first. So theoretically this one in the first in a series. If you want to fight with me on anything I have to say, please speak up!

Imagine this scenario: you demonstrate a problem-based activity with a group of teachers. You let them know that this is a grade 6 task where students have already learned to use double number lines and tables to represent a set of equivalent ratios. By this point, students are also familiar with recipe contexts; they know that an equivalent ratio of a recipe tastes the same. Here is the task:

Lin and Noah each have their own recipe for making sparkling orange juice.
  • Lin mixes 3 liters of orange juice with 4 liters of soda water.
  • Noah mixes 4 liters of orange juice with 5 liters of soda water.
How do the two mixtures compare in taste? Explain your reasoning.

The task is launched with a notice and wonder, they start happily working away, and you monitor what they are doing. You invite a few of them to make their reasoning visible to everyone, deliberately selecting them to share in a way that highlights a particular nuance you want to make sure everyone will understand, making mathematical connections between their approaches. (If you're savvy, you'll recognize this structure as Smith and Stein's 5 Practices, though my short description here isn't really doing it justice.) After conducting this discussion, many voices have contributed. Earlier in the day, you did another activity that loosely followed this same structure. You think, hey, I've done a pretty good job demonstrating the basics of how a problem-based classroom is meant to operate.

Then you get the question, maybe timid but very curious, "So, when do I teach?"

So here is a response that I'm turning over.
Can you say a little more about what it looks like when you teach, as it looks in your mind, here? Okay, it sounds like synonyms for what you are describing might be telling or explaining. Is that fair? Okay. It's expected that you'll do some telling and explaining when using our stuff as it's meant to be used. The difference is in the timing. Let's think about what we did in the sparkling orange juice activity. You had a chance to work on a task, a few people shared their approaches, and then we made some observations about their approaches. What do you think the mathematical learning goal of that activity was? 
"Well, I remember seeing two sets of equivalent ratios represented with a double number line and with a table, and then so-and-so explained how she computed how much orange juice for 1 liter of soda water for both mixtures. It seemed like the point was that when you want to know which mixture tastes stronger, you need to create equivalent ratios so that one of the quantities is the same for each mixture. For example if orange juice to soda water is expressed as $15:20$ and $16:20$, you know that the second recipe tastes stronger." 
Okay cool. Do you think you got out of that activity what was intended? Does that mean you learned something? Does that mean teaching happened? 
There's still telling and explaining. Mathematical playtime is awesome, but a problem-based classroom is not just about mathematical playtime. We have clear learning goals for the course, each instructional unit, each lesson, and each activity. 
The way it's different than you might be used to is when the explaining happens. Perhaps you are used to first explaining something, and then kids do some work on the thing you just explained. In problem-based instruction, this is reversed. Kids have a chance to try and figure some stuff out first, you see what they come up with, and then after they've had a chance to get good and familiar with the context, the question being asked, the constraints, and they at least make some progress. . . then you take steps to make sure the relevant learning goals are made visible. Sometimes this part looks like explaining or telling.
I'd suggest that teaching is a really broad and complex set of skills and behaviors, and telling or explaining is just one of them, and that telling or explaining isn't the only way to help kids understand something. In fact, does that approach work well for every student? How much do your students remember of what you explained the next day, or the next week? If you're completely satisfied with how things are going, awesome, but I bet you're here because either you or someone in your school endeavored to look for ways of conducting a math class that might work better for more kids, so that things made sense to them and the learning stuck around. 

Did I miss anything to address this particular question? (Please note that this is one vignette from two days of learning, and we spend time on a whole bunch of other things as well.) Does any of that come across badly? I want to acknowledge the person's completely understandable discomfort but also not shy away from asserting that teaching and learning happen in a problem-based classroom, and that we did it this way because we think better teaching and learning happen.

Friday, March 3, 2017

Anyone Want to Classroom Test Something? (grade 7)

Hi! We are field testing all of our new materials in pilot schools, but I have one activity where the first draft was unworkable, and we have to come up with something totally new, and since the pilot schools are past this point I can't throw another version back to them. So...Internet... want to try something out for me? This is working toward the CCSS standard 7.EE.B.4a, so it's for seventh graders or students working on grade 7 material. The assumption is that they already have some strategies for reasoning about and solving equations of the form p(x+q)=r and px+q=r but that throwing negative numbers into the mix is relatively new.

Mainly what I am worried about here is that question 2 will go awry and students will go overboard and way far away from equation types they know about. And I also don't know whether that would be a good thing that students and teachers can just roll with, or if it's going to present challenges that are too much for too many people.

So, if (and only if) this fits in with your plans, please try it out and let me know how it goes! Thanks in advance!

Okay here's the task:

1. Here are some equations that all have the same solution. Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question.

x = -2
x - 3 = -5
-5 = x - 3
500 = -100(x - 3)
500 = (x - 3) ᐧ -100
500 = -100x + 300

2. Keep your work secret from your partner. Start with the equation -5 = x. Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation.

3. Write the equation you ended up with on a slip of paper, and trade equations with your partner. See if you can figure out what steps they used to transform -5 = x into their equation. When you think you know, check with them to see if you are right.

Saturday, February 25, 2017

Is This Thing On?

Hello, Blogoworld! I'm not sure if anyone is still listening, but if you are, I have a short assignment for you. I'm preparing a talk where I'll show different people's sample work to the same problem. So I'd like to collect a bunch of different responses. Here is the problem:

A sloth can go 50 feet in 7 and a half minutes. How far can it go in an hour and a half?

If you'd like to participate, I need a good photo of your hand-written work. Upload it wherever, and share in the comments of this post. Bonus points for use of representations with more structure than dividing and multiplying. If you have access to a young person, it would be cool to have some samples that are in little kid handwriting. Thank you!

Wednesday, January 20, 2016

What I Think a Rate Is Right Now

Stating a few assumptions before I get into this:

I'm going to explain how I use the word "rate" and the phrase "unit rate" (and also throw around the word "ratio" somewhat recklessly) and it might not match what's in your textbook or how you use the words in your classroom. Some textbooks proclaim that ratios may only involve like units whereas rates use unlike units. In the physical sciences they typically use "rate" to refer to a measurement with respect to time, specifically. All names for things are conventions. I'm not trying to say that you or your textbook or the physics teacher are wrong. Here is a complete list of the arbiters of correctness when it comes to conventions: 
  1. mathematical consistency
  2. people in the act of communicating about the same situation understand each other
  3. you're not setting up a person for massive confusion later on 
The definition of a trapezoid is a good example. Is a trapezoid a quadrilateral with one and only one pair of parallel sides, or is it a quadrilateral with at least one pair of parallel sides? Said another way, is a parallelogram a special type of trapezoid, or is a parallelogram by definition never also a trapezoid? Answer: ¯\_(ツ)_/¯ It depends on a choice made by a person. Textbooks often present definitions like, "This is what the word means!" when they really mean something more like, "This is a choice we made in order to move forward." 

I'm working on a common core aligned math curriculum for sixth grade. So something to understand as a consequence of that: I'm thinking about how to make these ideas make sense to kids in middle school. So I don't want to write a post about mathematically ironclad definitions that would pass muster with research mathematicians; I want to write a post about stuff that it would be wonderful for kids age 11-13 to understand and is also flexible and useful to build on in later studies.

And one last preliminary: sometimes it's important for teachers to understand some nuances and it's not as important for students to understand them at the same level of detail. So, I'm not suggesting that any of this post is appropriate for instructional or assessment purposes with students. For example, an appropriate question for a student might be "In a fruit punch, the ratio of cups of grape juice to cups of soda water is 2:5. How many cups of grape juice for every cup of soda water?" But this question would not be appropriate for sixth graders: "In the ratio 2:5, what is the unit rate?" Because, ew.

Okay so here we go

A tortoise travels 10 inches in 3 minutes. A snail travels 8 inches in 3 minutes. Are they traveling at the same rate? (Assuming they're both traveling at a constant rate.)

No they are not traveling at the same rate, but I hope you didn't need to compute anything to know that. You can tell because they traveled different distances in the same amount of time. In this context, you have their distances traveled, you have the time it took, but then you have this third thing that means something concrete in the context -- how fast they are going. Their rates. We can express the tortoise's rate as 10 inches in 3 minutes or around 3.33 inches per minute or a foot-and-a-quarter every 270 seconds but the real live concrete in-context rate is the concept of how fast (or in this case, slow) he is moving.

(Note for curriculum nerds: at some point you have to make it explicit to students that "are these happening at the same rate?" is structurally the same question as "are these equivalent ratios?" Not super relevant to this discussion but it seems worth mentioning.)

At one store, 2 pounds of M&M's cost $14. At a different store, 2 pounds of M&M's cost $16.95. Which is a better deal? Did you have to compute anything to know that? No, you can compare the good-deal-ness without computing the cost of 1 pound or how many pounds you can get for $1. The rate is a third thing going on here capturing how-good-is-this-deal that could be expressed in different ways, one of which is a unit price.

Those examples were different types of quantities (distance and time, weight and cost) but we can talk about rates with same quantities like volume and volume. 

Kate mixes 2 oz of gin with 5 oz of tonic water. Ashli mixes 3 oz of gin with 7 oz of tonic water. Are they sipping the same beverage? This is not so easy because none of the quantities match up. So now we need to find how many ounces of gin for one ounce of tonic water for each beverage right? We could, for sure, but we don't have to. We just have to compare equivalent ratios for the same amounts in the different concoctions:

If I used 3 oz of gin to mix a beverage that tastes the same as my original drink, I would need 7.5 oz of tonic water. Ashli only mixed her 3 oz of gin with 7 oz of tonic water, so Ashli's was a bit stronger than mine. Here, that third hidden thing going on is the potency of the beverage, and I'm still asserting that it's a rate, and I still haven't figured out how many of anything per one of anything.
So, let's sum up what we have so far: in any set of equivalent ratios that represents a context, there is a third thing that characterizes something meaningful about those two things happening at the same time. It could be land speed, how much of a good deal you are getting, beverage strength, the tempo of a song (number of beats to number of minutes), how crowded my neighborhood feels (number of people to square miles)... This third thing hidden within a set of equivalent ratios is a concept I'm calling a rate.

But then, it's often convenient to refer to the special equivalent ratio that is something-paired-with-a-one: "how many of these for every one of those?" It is convenient for at least two reasons (and probably more). First, it helps you solve equivalent ratio problems pretty quickly. For example, I know that I get a certain lovely shade of orange acrylic paint if I mix 3 teaspoons of yellow paint with 2 teaspoons of red paint, but I want to make the same shade and I need alot of it so I want to use up the 9 teaspoons of red paint I have on hand. How much yellow paint should I mix it with? I might approach that problem in any number of ways, but a good way is to reason that 2:3 is equivalent to 1:1.5, so to solve 9:? I just need to multiply 1.5 by 9. (This explanation would be clearer if I drew you a ratio table or another double number line but I am getting tired and it's almost cocktail hour.) It's convenient to use a word to name the 1.5, and "unit rate" is as good a name as any. I like how the "unit" part reinforces that it has something to do with 1. A question kids should be able to answer as part of their process is, "What does the 1.5 mean in this context?" and they should be able to say "there are 1.5 teaspoons of yellow paint for every 1 teaspoon of red paint."

Second, it's a way to express that third thing in a set of equivalent ratios with just a single value which can be algorithmetized (like if you want to tell a computer how to do it.) In the gin-and-tonic example above, we could have computed that Kate's drink had 2/5 oz gin for every ounce of tonic water, and Ashli's drink had 3/7 ounce of gin per ounce of tonic water, and since 3/7 is greater than 2/5, Ashli's was stronger.

Then actually later in seventh grade I could write an equation for the relationship that is my recipe, g = 2/5 t, where t is volume of tonic water and g is volume of gin, and 2/5 is re-named the constant of proportionality for the set of all gins and tonics of that particular strength, and I could graph this equation and an equation representing Ashli's recipe and note that the line representing her recipe is steeper, but we're really getting ahead of ourselves here.

Okay, this was a long post, but we're almost done. I believe that my interpretation is supported by the CCSS standards and the RP progression document, although I also believe that those documents also allow you to conclude that rate only means "how many of these for every one of those" (because the only examples they give for "rate" are quantities per 1). But if you're going to use rate to mean how much of this for every one of that, I think you need to come up with another word for that third-thing physical quantity that I am calling rate.

Alright. Comments are on. Come at me, nerds.