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!

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.

Monday, January 4, 2016

In Defense of Unsexy

At IM we're writing a sixth grade curriculum, and much of my time is spent writing, reviewing, and begging other people to write and review new grade 6 tasks that really just hit the fundamental stuff. After five-ish years of accepting task submissions, we have some holes. Because nobody wants to write easy questions.

Do you know what teachers have the hardest time finding?

Quality, basic stuff.

It is getting relatively easy to find the rather-complicated application problems, the projects, the whole new grading systems, the elegant warmups, the pinterest-worthy graphic organizers. Many people have invented and shared some very sexy, awesome stuff, and they are changing many teachers' and kids' experience of math. Hooray for that!

But so many teachers aren't helped by sexy stuff. I think one reason is that they don't think the payoff is worth the time investment. Or maybe that changing the whole way they run class is too intimidating. I'd be happy to entertain alternative theories.

Here's a cooking show analogy: in the early-mid 2000's people enjoyed Mario Batali's homemade gnocchi and Bobby Flay's 90-ingredient curries and Alton Brown's coconut cake that takes THREE DAYS (I'm not kidding. Three days.) But you know who I watched every day at 4:30? Rachel Ray. I suffered through her saying "yummo" and "EVOO-that-means-extra-virgin-olive-oil" approximately 19 times per episode, and she taught me how to get a reasonable meal on a plate in 30 minutes and how to chop a damn onion.

This blog grew in popularity (and stays relatively popular even though I neglect it so) not because I invented something big and sexy but because it offered relatively easy swaps for practice worksheets and ugly, fresh-off-the-smartboard rewrites of high school lessons that made the kids do a tiny bit more thinking than usual.

So I'm starting to hear my low-level angst echoed elsewhere and it's bubbling over. I have a request for you if you are a math teacher and you have a blog.

Share your kinda-borderline boring stuff. Your small tweaks that unloaded the right amount of cognitive lifting onto the kiddos. Your rather-basic task or set of tasks that don't seem that exciting, but your kids always seem to readily grasp that topic. Your snippets of classroom dialog where everybody ended up going OHHHH. Your artful arrangement of pieces of instructional units you found lying around. How you took that cool instructional idea you read in that book and figured out how to do it in a congruent triangles lesson.

We need you. Your kinda-lame-but-seems-to-do-the-trick exponent rule investigation is going to make you somebody's superhero. If you share them with me (add a comment on this post, tweet them at me, whatever), I'll re-share them and compile them in new posts. (And probably they will get added to some of those wonderful virtual filing cabinets and wikis.)

Update: Some gems in the comments. And some shared on Twitter. 

Saturday, November 21, 2015

NCTM Nashville Presentation

I had the pleasure of attending the NCTM regional meeting in Nashville this week. I learned some cool stuff that I'm still processing, and I got to do a presentation. In the presentation I tried to explore whether the way I would rewrite and rework lessons when I was a high school teacher can be generalized and communicated to other people. I was, I think, marginally successful.

NCTM is trying this cool pilot where participants can engage with presenters after the conference. So instead of sharing stuff about my presentation here, I'm going to send you over to the presentation page on their site.

Friday, October 2, 2015

Friday Favorites 7

Happy Friday! (It's really Saturday but I'm going to backdate this post and pretend it's Friday. Ha! Technology!) My reading and favoriting has slowed down because I have made the decision to limit my Twitter time, which is exceptionally mature of me, I think. (Using Stay Focusd, which is a chrome plugin that yells at you for not working. It's brilliant.) What I'm mostly doing these days is a zillion math problems, which is pretty fun, actually... You know how when professional chefs see a bag of onions, they get excited because they get to chop a bag of onions? That's how I feel about doing a bunch of math problems. It's a little bit drudgery, but satisfying. Still and all, when something gets a little mentally difficult it can't be too easy to distract myself. Twitter needs to not be an option in those moments.

This is not a favorite because I made it myself, but it's public, so I might as well share it. It's a place to stash mathematically interesting artifacts that I might turn into tasks or assessment questions or lessons. There's nothing worse than needing to write a question in a context and googling for hours. You're welcome, future Kate.

Now here are real favorites:

Capture Recapture with Goldfish

I did this lab in an Algebra 1 class ages ago. It reminds me of that illustration of statistics vs probability: If you know what's in the bag, reach in and grab a handful, and want to predict what's in your hand, that's probability. If you don't know what's in the bag, reach in and grab a handful, and use the handful to predict what's in the bag, that's statistics. It's a good activity, but my first or second year teacher self probably didn't do such a great job with it. Because, obviously, I didn't have Elizabeth and Julie's helpful writeups. I like the way Elizabeth frames how it fits into a bigger Algebra 1 picture. I could also see using it in a stats lab in a way that emphasizes sampling and sample proportions just as easily as a 7th grade-ish solving proportions lab.

Problematizing Geometry Constructions

I love everything about this. Using a popsicle stick as a straight edge: pro move.

How Parents and Students and Teachers Can Work Better Together a better headline than the clickbaitey one they gave this article. Which is empathetic and treats everyone involved as a professional and a human. Forward anonymously to those parents whose first move is calling the Principal.

Michigan's Teaching and Learning Exploratory

Don't let the boring name fool you - Michigan has done an awesome thing here by posting hours and hours of unedited classroom footage. I learned in the last chapter of Why Don't Students Like School? that looking at video of yourself or someone you know is too scary a place to start, and it's easier to watch and practice constructively critiquing someone you don't know. This resource makes that a whole lot easier.

Thursday, October 1, 2015

Every Bit of This

High schools focus on elementary applications of advanced mathematics whereas most people really make more use of sophisticated applications of elementary mathematics. … Many who master high school mathematics cannot think clearly about percentages or ratios.

Wednesday, September 30, 2015

Exponential Functions and also Area of a Triangle

That title is confusing, right? I know! I just wanted to alert y'all to some tasks that recently went up on Illustrative Mathematics that might address some of your needs, if you are teaching these things.

Exponential Functions: These tasks involve negative exponents in a functional relationship in a context and are aligned with F-LE.
  • Decaying Dice (It's like the penny lab for modeling half-life that kids often do in Earth Science... except with dice.)
  • Predicting the Past (Making sense of negative integers in the domain of a simple exponential growth function.)
  • All Your Base are Belong to Us (Exponential decay and negative exponents, together at last. Bonus points if you get the reference.)
  • DDT-Cay (Interpreting the exponent in a half-life equation.)

Area: These are meant to be used to build understanding as you're working toward a formula for area of a triangle in sixth grade (6-G.1). But they could be useful to reactivate knowledge at the beginning of a study of area in a later Geometry course.
And, hey, it is non-trivial for me to test stuff out with kids these days, so if YOU try them out and you notice stuff or have suggestions, you can comment here or better yet, right on the task on the IM site. (Please let me know if you do that - I don't think I get a notification. And thanks!)

I did draft the initial versions but I can't take credit for these. Tasks published on IM are very much a team effort. Many thanks to Ashli Black who is an ace reviewer and helped me make these a ton better.

Friday, August 28, 2015

Friday Favorites 6

Happy Friday! I am elbows deep in Trello, of all things, but the cat is good company. Here we go...

I took a stab at Activity Builder with an activity that deals with discovering pi and thinking of circumference vs diameter as a proportional relationship. And wow, it's so much better because of their Twitter interaction. I don't know if my favorite part is setting a table to make points draggable only vertically, or their suggestion to share Teacher Notes in a linked google document.

In case you haven't heard, there's also a repository of user-created Desmos activities here. Mileage may vary.

All the Math Talking Points

Are in this shared google folder. If you haven't grokked the magic of Talking Points yet, go read you some cheesemonkey wonders.

OER Curricula and Curricular Outlines

In case I haven't talked your ear off about it yet, I'm of the strong opinion that a school's math department should Decide on a Coherent Curriculum and riff off of that, rather than expecting their teachers to create a curriculum on the fly using random resources they find on the Internet. Some textbook series are good, and there are also decent OER (Open Educational Resource) ones are already out there, and too many people don't know about them.

This Coaching Model

Where your team gets a Teacher Partner - someone who teaches a few classes but also coordinates your collaborative teacher learning. I love this.

I might be a little obsessed with other people's planning documents.

Icebreakers That Won't Make you Cringe

You know what I'm talking about. h/t Lani.

John's Exhaustive Tour of the Good Stuff 

Where do I start? Here.

Wednesday, August 19, 2015

And Then There Was Not Teaching Some More

Waddup, nerds. Just a quick note about what is going on around here, which is that I've joined forces with Illustrative Mathematics to do some very exciting curriculum work. I'll keep y'all posted here as I am able. 

Practically that means it's not a new school year for me, which sucks, because I love the first day of school. There's something so inspiring about a fresh start. And also because there won't be as much to report here. 

But it also means I work at home, which, I'm not going to lie, is pretty boss. I can get all the work done with a cat in my lap and also throw in a load of laundry and also prepare real food for dinner. 
My rig. I realize the television is dominant in this photo, but I haven't actually turned it on yet. It's just extra. That fridge is full of fizzy water.

I'm around, on the Internet, of course, and I want to keep doing the Friday Favorite thing. You are welcome to yell at me when I slack off.

Friday, July 24, 2015

Friday Favorites 5

Happy TMC, everybody! I know all the TMC-ers are busy TMC-ing right now, but it's Friday! Here we go...

John's MTBOS search engine

What a good idea. I don't know how I missed this.

Tracy's Proof Games

Here at camp there's a "Games and Strategies" class running this week, and kids keep running up to staff proposing games like "We start at zero, take turns adding 1, 2, or 3, first one to 19 loses." These are kind of addicting, is what I'm saying, and motivate and "Support Generalizing, Conjecturing, Strategy, and Proof-Like Reasoning," as the title suggests. And here are a zillion of them in one document!

I Am Not Tom Brady

Just putting this out as a public service announcement that schools that pull shit like this exist, so you can walk away quickly if you get a whiff of it in an interview. h/t Lani for the share.

Cathy's Write-up of a 17-Armed Spiral

Here's some recreational math for you, in the spirit of math camp.

Shelli's Teacher Binder

Back in the days of student-ing, my life was all about my paper organizer. I had very specific requirements and shopped and shopped until I found it. These days I'm a more scattered leaving-digital-detritus-in-my-wake kind of organizer, but this makes me think maybe it's not too late. 

Look How Pretty

The #mathphoto15 Flickr stream. 

Friday, July 17, 2015

Summer Problem-Solving Course

This summer I have the privilege of teaching a problem solving class to mathematically-inclined rising eighth graders. The course is called Math Team Strategies because a big goal is to get kids more ready for contests like MATHCOUNTS and the AMC contests. But we are also looking to highlight problem solving strategies that are broadly useful, whether kids decide to participate in contests or not.

I'm going to make this post pretty nuts and bolts just the facts ma'am - it's the nitty gritty details for people who want the ideas.

I lovingly plucked from the work of, and want to give tons of credit to:


Eight days, two hours a day, one focus strategy per day. On the final day, instead of a new strategy, students experience a somewhat-complete MATHCOUNTS contest.

The Strategies

(Most of these are chapter titles in Crossing the River with Dogs - but that book has many, many more chapters. It's awesome. You should check it out.)

The Lesson Flow

For each day, I selected problems that lent themselves to that day's strategy. Some problems are from Crossing the River, some are from old MATHCOUNTS contests, and some I made up. Additionally, we developed a few mathematical shortcuts over the course of a few days, like counting permutations with repetition and the length of a diagonal of a square. I cut the problems up onto slips, so students would only have one problem at a time. (For a longer course, or perhaps for older students, I'd probably elect to use Crossing the River as a text.)

All the students worked on the same problem at the same time, standing at chalkboards. I had anywhere from 6 to 12 students in a class, so this was manageable. I also had a TA who was a math-major undergrad. Nirvana. Before I left home I grabbed a handful of fridge magnets, thinking they might be useful for something, and we used them so students could stick the current problem to the chalkboard.

The Posters

The intention was for the whole class to go over each problem before everyone started the next one. (See this post about group discussions.) Of course, some students took longer and needed support. When I am helping, I tend to make the same suggestions and ask the same questions over and over. This poster was for students to refer to if both the TA and I were busy when they got stuck.

Also, of course, some students finished more quickly than the group. I also tend to always make the same suggestions when students say they are "done," so I made this poster for them, too.

The Self-Assessment

Before we went over each problem, I asked the students to turn in their problem slip with their name and a rating of the problem from 1 through 4. I did compile this data in a spreadsheet, but I'm not sure what to do with it. But I thought the self-assessment couldn't hurt.

The Resources

Will be here until someone holding a copyright yells at me to take them down. Or maybe this is fair use. I dunno. I hope it's good advertising for the publications cited above. Some of the problems turned out to be too easy, and I'll be changing them if I'm back next year. Some were too hard, but I thought it was okay to give kids at most one problem a day that was a big stretch for them. When that happened, I invited the TA to share their solution.

And That's about That

This was a really rewarding course. The kids loved it, I loved it, we all just had a grand old time talking about math for two hours a day! It was refreshing to not feel pressure to cover content at a breakneck speed, or sell kids on math (these kids already like math), or have to assign grades. (This morning when we did a sample MATHCOUNTS Sprint, a girl asked "Does this count? Oh, wait. We don't have grades." And she worked hard on it anyway.)

Questions, feel free to throw them in the comments.