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!

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.

Friday Favorites 4

It's the second week of Math Camp... that means I have a little time to post. Yay! Things are still a tad chaotic here - long days, tween drama, field trips, little sleep, etc etc, but I finally have the class I'm teaching all planned out through the end of this week. Phew! Time to write and reflect and observe some great teachers in action.

Also I taught some 13 year old boys how to juggle yesterday. Before I signed up for that duty, I did not consider how many times I would have to say the word "balls." The first time was awkward, but then we naturally took it to a ridiculous extreme. "MALACHI! CONTROL YOUR BALLS!" (Normally I wouldn't post photos of student faces, but this one is on the program's website.)

And now for some fresh faves...

Megan's Easy Way to Start Blogging

Although it's really valuable professional learning for lots of people, keeping up a blog during the school year can be a daunting proposition. An on-ramp can be a 180 blog - just take and publish one photo a day from your classroom. This practice has less overhead in terms of time, but gets you in the habit of noticing things to share. This recent post by Megan Hayes-Golding suggests one way to set this up using Instagram, IFTTT, and Wordpress to make it low friction so that you are more likely to stick with it. If you're unfamiliar with those platforms, don't worry - they are all pretty easy to get started. You could have this up and running in a few hours if you're new to it (a few minutes if you're not). Also, IFTTT works with lots of different services.

Dylan Builds His Intuition

Dylan Kane has been chronicling his growth as an early-career teacher. If you haven't been following along, you should plug into that. I really enjoyed his post about the ways he has to be attentive to avoiding pitfalls of bias and developing intuition that will be productive in his practice, because they paralleled some of the things I realized along the way (although he has articulated them much better).

Meg Encourages MTBoS Users to Make It Work for Them

Much as I love our spirited army of awesome, folks can get a tad dogmatic and judgey from time to time. It can be a turn-off, when you come across some strident prose that makes you feel like you're doing everything wrong. Meg Craig's post speaks to two audiences: seekers of resources and conversations, who are reminded to stick with it and make it work for them. Also sharers of resources and initiators of conversations, who she gently offers ways to phrase your sharing so that it's a bit more inviting and inclusive.

Dan Meyer is going to fix NCTM for Us

Here's how. Thanks, Dan. (Adding some clarification here because I'm afraid this sounded snarky - I'm totally sincere. I'm really excited about the prospect of NCTM taking up the recommendations of the ShadowCon organizers. I think we all of us NCTM members realize that NCTM is not working well for many members and prospective members, and I wholly support these concrete proposals.)

Please Review Our Book

Have you read Playing with Math? Are you going to? (You should! It's so awesome.) It's on Amazon now, and it would be great to get some more reviews. (Since I wrote one of the essays I'm ambivalent about writing one myself.)

Wednesday, July 15, 2015

A Magical Incantation

So this week I'm basically the luckiest girl in the world, because Ben Blum-Smith is on staff at SPMPS, and he observed me teach and then we had a conversation about it. (I know. Be jealous.)

He offered a concrete suggestion enabling student dialog which I want to share. I am pretty good at getting kids to talk to each other about math in pairs or triples...

but I've always struggled with conducting good conversations with the whole group -- getting kids to talk to each other about math in front of everybody. (Aside from the two kids in every class who always raise their hand for everything.)

What we have been doing in this class is having everyone work out solutions to a task on the board. (Classes are small enough (7-11 for my classes), I've partaken of the vertical-non-permanent-surfaces kool aid, and kids at camp are exhausted because it's a three week slumber party, so keeping them on their feet helps with the awakeness.)

When everyone is done-ish, we gather around someone's solution and they walk us through it.

So let's say that a student presents her solution or approach to a task to the whole group. Generally they speak too fast, and gloss over important bits. When they are finished, I have uncovered many, many unproductive questions to ask the rest of the class:

  • Does anyone have any questions for Bianca? (crickets)
  • Miguel, what do you think of Bianca's solution? ("I don't know. It's fine.")
  • Does anyone have anything to add? (more crickets)
  • Did anyone approach it a different way? (Actually I never ask this anymore, because I pick usually two students with different approaches to present.)

But here is the magical incantation that can pick this lock:

Hey, so-and-so, would you explain your understanding of Bianca's solution?

This is a lovely question. I tried it out at every available opportunity today. Interpreting another student's written and verbal solution requires all kinds of nice cognitive work. I imagine that as kids come to expect that they might be asked this question, they're more likely to be more attentive to others' explanations. And, it offers a nonthreatening invitation into the conversation where a student is immediately clear on what she's expected to say.

Ben mentioned that he didn't really grok the power of this move until well into his classroom experience, and I think I'm kind of in the same boat. I'm sure I've heard of it before, but now I'm in a place where I can really deploy it surgically. Well I mean today I deployed it in kind of carpet bomb fashion. It's like a new toy I can't put down. But I'm going to enjoy the process of integrating it.

Friday, July 3, 2015

Friday Favorites 3

Hey! You thought I forgot, didn't you? DIDN'T YOU?! (Excusable. That would be totally in character.) I just arrived a few days ago at the best mathematical summer thing in the world, the Summer Program in Mathematical Problem Solving, where I'm teaching a course called Math Team Strategies. It's so great, you guys. The staff is the bomb. The campers get here tomorrow. Now for some faves:

Get Your Mathematical Modeling On

...starting here. These are ideas for data students can easily collect, organized by function type. Compiled by Casey Rutherford.

Sean Sweeney's New Video

Who doesn't love a sweet math song? Okay there are people but you have to admit that this is delightful, even if you're not a singing-in-math-class type. Also if you need to catch up on Sean's (and other members of his school's) previous works of art: here's Graph Shop, f(u), and the classic Slope Rider.

Nicora Placa's Math Tasks for Teachers

Nicora breaks down how she chooses or adapts mathematical tasks to use for teacher learning. This one is maybe a bit specialized for folks who work with groups of teachers, but if you are looking for good PD to sign up for as a math teacher, this kind of learning has had a huge impact on my practice.

Google Classroom + Peardeck

If you're at a GAFE school, and you haven't checked out what Google Classroom can do in the past three months or so, you really should get on that. (Especially if you're still using Doctopus. GC is way easier.) And Pear Deck is, of course, the money. And now they're more integrated. Go make some cool shit happen.

(My favorite use of Pear Deck is asking kids to find numbers for (xy) that make an equation true, and then each student plots that point on the Pear Deck slide. The collective points lie along a line, or a circle, or whatever. Connection between multiple representations: made. I used that move on them like a dozen times this year and it never got old. Here's a straightforward post by Jon Orr showing what this can look like.)

Friday, June 19, 2015

Friday Favorites 2

Hey there! Two Fridays in a row! Whaddup! Here are some things that got my attention in a good way this week:

Geoff Krall's Minimal Conditions

Geoff Krall (of PBL Curriculum Map fame) gives an excellent wide-angle view of practices school staff should engage in when they get serious about improving instruction. My favorite thing about this is it seems so do-able. There are things small groups of teachers can start doing with the PD time that's in their control, or if that time isn't yet in their control, suggests some concrete practices to start advocating for.

Allison Krasnow's Virtual Patty Paper

Allison rediscovered a great patty paper book by Michael Serra, and noticed that all of the activities could be recreated on Geogebra. I love this! It demonstrates that ways for students to tinker with ideas -- the important part -- is somewhat independent of choice of technology. Use the patty paper, create a Geogebra version, use both, or give students a choice.

What Collaborating Looks Like

Many of us know that we should be collaborating with building colleagues on the nuts and bolts of planning and instruction, but if you've never done this before, it can be hard to imagine what it looks like. This video series (a collaboration between Teaching Channel, Illustrative Mathematics, and Smarter Balanced) is a really excellent resource including teachers working in elementary, middle, and high school math before, during, and after instruction.

Jackie Ballarini's School's Starting Page

Hey, if you haven't put all the stuff your new teachers need to know in one place, like this, you should! This page was shared during a conversation initiated by Rachel about supporting new teachers, and everybody drooled over it.

Jonathan Claydon is Not Leaving

I really enjoyed reading Jonathan's piece about why he intends to remain a classroom teacher. In this environment it's contrary to so many other articles coming out about folks throwing in the towel, and I think Jonathan shares important sentiments that usually go unarticulated, or at least don't go viral. But should.

Thursday, June 18, 2015

Surprises in Scatter Plots

On Derby Day, in my living room:

"Do you think horse races have gotten faster over time, like people races?"

"We can find out!"

Heads to wikipedia. Does some fancy footwork in drive to convert units of time from M:SS.SS to seconds. Heads to

"Whoa, something weird happened in 1896."

"Is that when they figured out jockeys should be tiny?"

"Oh, look, they made the track shorter."


"Let's only look at times since 1896."

"So, yes, but it's leveling off?"

"Looks that way."


This data could be fun to build out for an activity to get kids using whatever scatterplot-creating tools you want them to use. It's also nice for interpreting plots -- it smacks you in the face that something changed in 1896, and there's a quick and satisfying explanation. Enjoy!

Friday, June 12, 2015

Favorites Fridays 1

Hi! Welcome to Favorites Fridays. Instead of just sharing or retweeting on Twitter, which is ephemeral and misses lots of people, I'm going to start collecting my favorite stuff from the mathematical educational Internet from the week here. I've never been one for regular publishing or weekly series-es, but we're going to give this a try. (This may be a dumb time to start this because I'm heading off on vacation next week and I promised my boi-freeeen I'd give Twitter a rest, so I'll skip a week soon but anyway.) I hope you find it useful, but this is also for my personal archival use too. Here goes!

Dandersod's Calculus Projects

Dan Anderson (@dandersod) (does anyone else just think of him in their head as "dandersod?") set a project for his calculus kids, live-tweeted it, and published their reports. You might have mixed emotions about the phrase "calculus projects," but I found these to be super fun, interesting, entertaining reading.

Lani's Memo

This memo focuses on research-based ideas on how to support common planning time so that it has the greatest potential for teacher learning about ambitious mathematics teaching. To that end, we provide a framework for effective conversations about mathematics teaching and learning. We develop the framework by using vignettes that show examples of stronger and weaker teacher collaboration.
"Sometimes, you ask and the internet answers." Lani Horn came through with what Julie, and many teachers are looking for: nuts and bolts direction for teachers hungry for useful professional conversations. We're tired of wasting collaboration time and "PLC time" (a now-meaningless name if there ever was one) on aimless, unhelpful activities that don't have an impact on our practice, and we know there's a better way. This post is going to be a huge help. Bonus: a summary on research about using student performance data.

Tracy Zager's ShadowCon Talk

It will blow your doors off. Tracy is dazzling. Just go watch it. Best use of word clouds in history.

Mike's How to Build a PBL Culture

Mike's PBL is Project Based, but I think this fab collection of activities and recommendations for kicking off a school year would work just as nicely if your PBL is Problem Based.

And that's a wrap! Somebody hold me accountable for doing this next Friday! 

Thursday, May 7, 2015

Pretty Painless Gamification

Today I was at a loss for something fun-ish to review circles in Geometry. I hastily searched my Evernote for "review games" and came across this gem from 2009 from Kim. She called it Ghosts in the Graveyard for Halloween, but since it's springtime I went with a garden theme. I modified the activity slightly.

1. Set up a Smartboard file like so, for six groups to play. The ten objects to populate their gardens are infinitely cloned, and the fences are locked in place so they can't be accidentally moved. (This screenshot shows the "gardens" in the middle of a class.)

2. Students in groups of 3-4. I wrote students' initials (in red) next to their garden.
3. Every student gets a copy of ten problems.
4. When all group members understand a problem, they call me over. I randomly choose one student to explain how they did it.
5. If she can explain their process sufficiently, she can go up to the Smartboard and add the corresponding item to their garden. (If not, I just say okay, I'll be back in a couple minutes.)
6. They were instructed to use the review problems to help them study for the quiz tomorrow, so if they didn't get to all the problems in class, it was okay.

Why I liked it:

  • It did not take forever to set up. I used the review questions I was planning on giving them anyway, and just had to whip up a smartboard page which took all of 5 minutes.
  • You wouldn't think that the state of an illustrated garden on a smartboard file would be very motivating, but they all worked diligently for the entire 30-ish minutes we did this. Thanks, Zynga.
  • I heard lots of good discussion as they made sure all of their group members understood a problem before calling me over.
  • Nobody could slack off and nobody got bored.

Monday, March 2, 2015

Kicking Some Serious Triangle Booty

The children understand that sin, cos, and tan are side ratios. The children! They understand! They are not making ridiculous mistakes, and they can answer deeper understanding questions like, "Explain why sin(11) = cos(79)." I think right triangle trig is a frequent victim of the "First ya do this, then ya do this" treatment -- where kids can solve problems but have no idea what is going on. There's often not a ton of time for it, and it responds well to memorized procedures (in the short term). So, if your Day One of right triangle trig involves defining sine, cosine, and tangent, read on! I have a better way, and it doesn't take any longer.

First, build on what students have already learned about similar triangles. Ideally, this unit immediately follows that one. On Day One, I assign each pair of students an angle. (You guys have 20 degrees. You all have 25. etc etc, all around the room, so each pair of students is responsible for a different angle.) They work through this document (docx pdf), using Geogebra to do the measuring. They write down the length of the side opposite and adjacent their angle, for triangles of five different sizes. It's important that they write down the lengths, divide them with a calculator, and experience surprise and wonder why they are all exactly the same. (Geogebra made this soooo much better and easier than when I did this with rulers and protractors. So much better. In fact, one of my Matt colleagues basically deserves a medal for all the times he's said "Why don't we just do this with Geogebra?" this year.)

On this day, they just do opposite/adjacent ratio, share the ratio for their angle in a shared spreadsheet, and then everyone has access to the shared spreadsheet (an opp/adj-only trig table) to solve some problems (in that same document). The thing is, they are figuring out how to use what they have learned to solve the problems; they're not just repeating a procedure that was demonstrated. This took one 45-minute period, including checking Chromebooks out and in. I collect the sheets and look for students who had a strategy for #11 (how to solve when the variable is in the bottom of the ratio) so they can share their strategy next class.

Next two classes, I provide them with a table of all three ratios (to tape in their notebook) for angles of 5-degree increments, and they work their way through this page with appropriate help. For example, in the first set of problems, I just had them label the sides first. Then choose the ratio for all the problems, then solve for an answer. This particular document is not terribly pretty, because I had limited time to put it together. In every class, someone wanted to know why they couldn't use like hyp/adj if that equation was easier to solve. For those that asked, I pointed out that we couldn't look up hyp/adj in the table, BUT, they could use the other angle in the triangle. (And yes I'm aware they could use 1-over the ratio in the table, but that seemed like an overly complicated strategy to suggest.) I gave them a few find-sides and find-angles problems (limited to the angles in their table) to practice for homework. They did not all get to the back, but the kids to catch on/work quicker had something to do after the basic problems.

Today I spilled the beans that these ratios have special names, and we could look them up in our calculator. We mostly spent the period getting used to looking stuff up in the calculator including some hot Plickr action, and working on these problems which they are finishing for homework. I told them they only had to do one "Explain why," but they had to complete all the rest.

Tomorrow on our block day, we are going to go outside and figure out the heights of some really tall things (docx pdf). There are lots of "measuring tall things" activities out there, but I heavily adapted this document, so thanks to Christopher Conrad for posting it.

Friday, January 23, 2015

On Making Them Figure Something Out

Often when I don't really know a great way to teach something, I end up defaulting to Making Them Do Something and Making Them Notice Something, and then finally Making Them Practice Something. I think lots of people do, and it's not bad. It's loads better then Telling Them Something, and it's certainly better than Children with Nothing To Do. But lots of times, the learning that comes out of MTDS and MTNS doesn't really stick that great. They can maybe do an exit ticket, but ask them a question that relies on The Thing in a week, and you just get a bunch of blank stares. So in my planning, I'm taking as a signpost Making Them Figure Something Out or MTFSO. I think the best existing curricula depend on MTFSO, and it must be nice to be working with one of those.

So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (question 50 of 50 in this document).

You can find all kinds of MTDS/MTNS lessons about the discriminant. Here and here, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.

For this particular lesson, they first solved four equations using the quadratic formula: x2 - 10x + 9 =0, x2 - 6x + 9 = 0, x2 - 7x + 9 = 0, and x2 - 4x + 9 = 0. When we debriefed their solutions, we spent time describing the types of solutions, but we did not belabor the point about why the roots came out each way.

Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:
A1)  Come up with a new, original quadratic equation whose roots are real and irrational. Demonstrate that your equation works by using the quadratic formula to solve it.
A2)  Come up with a new, original quadratic equation whose roots are real, rational, and unequal. Demonstrate that your equation works by using the quadratic formula to solve it. 
B1)  Come up with a new, original quadratic equation whose roots are imaginary. Demonstrate that your equation works by using the quadratic formula to solve it.
B2)  Come up with a new, original quadratic equation whose roots are real, rational, and equal. Demonstrate that your equation works by using the quadratic formula to solve it.
(In the first class to do this, I gave out the tasks haphazardly. But from that experience, learned that "rational" and "equal" are harder to find than "irrational" and "imaginary." So I adjusted accordingly for the second class. Each set above consists of one of the easier ones, and then one of the harder ones.)

I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.

After groups had a chance to figure stuff out and explain how they did it (it all came down to paying attention to what kind of number was under the radical, of course), we summarized with some notes:

And that is that. A general understanding of what was going on persisted to the next day... We'll see how Monday goes.

Sunday, January 11, 2015

SSA ASA and All the Rest

I finally, Finally, FINALLY have a plan I like for introducing triangle congruence theorems.

For a few years there (2? 3?) I had been trying to make a go of Triangles a la Fettucine as described in this MT article. And I just couldn't work it! The principle of the thing is sound, and it always started out okay, but quickly got tedious and the kids would both lose interest and not get the point. Many students just would stubbornly not get the memo that you had to use the entire length of the colored-in sides, but you could use any length of the uncolored sides. The mechanics of the thing got in the way of seeing the larger picture.

In this new lesson, I deliberately separated the triangle-creating phase from the seeing the larger picture phase.

Before the lesson, I gave students the first page of the pre-assessment from this Shell Center Formative Assessment Task. (Thanks someone on Twitter who suggested that.) I did not use the lesson itself, because I felt my students weren't at the point of being able to understand what it was asking. The pre-assessment was compared to what they could do afterward, of course, but also to kind of get their juices flowing about angles and side lengths and what congruent means. They worked on the pre-assessment for about 15 minutes.

Phase 1 was constructing some triangles out of construction paper based on various given information using straight edge, protractor, and occasionally a compass. Here is the instruction page. (Thanks again, Twitter, for some helpful feedback making it better before it went live to the children.) Students were in groups of 3 or 4, and the group was responsible for creating the nine triangles (There are ten on the sheet, but the last one is impossible.) For thoroughness, I'd love if every student had to create all the triangles, but I was afraid 1) that would take way way too long and 2) many students would tire of it midway through and check out. I think my instincts were right on both counts. The more difficult constructions were marked with a *, which I told the students, which allowed for some self-moderated differentiation (by less-confident students quickly claiming responsibility for non-* triangles).

Before they started I ran a quick protractor clinic, and they were off. The groups worked on creating triangles for 30-40 minutes. Beforehand, I created a set of reference triangles out of cardstock, so I could quickly assess their products for accuracy when they were done. I just kept the correct ones and discarded the inaccurate ones (where an angle was not measured correctly, for example.) (Instead of making them re-do incorrect ones, but I think that would be a fine thing to do if you can swing the time.) The groups that finished first, I handed off the cardstock triangles and put them in charge of assessing other groups' work. By the time triangle construction was complete in all three of my Geometry sections, I had 10+ correctly-made copies of each triangle.

I suppose you could try to recreate this experience with dynamic geometry software somehow, but I'm dubious that there's a replacement for creating physical triangles with your hands. Students directly encountered having choices for how to finish making the triangle (when they were only given the lengths of two sides, for example) vs being locked into only one possible triangle (when they were given ASA, for example). In the future, I'd like to be more deliberate in assigning each student one of each.

After triangle construction, we ran through a quick lesson on notation and naming conventions for congruent polygons. Then I put up a kind of standard-looking congruent triangle proof, where enough information was given (or able to be inferred) to show that all three pairs of sides and all three pairs of angles were congruent. We wrote out a 9-step proof that proved the triangles congruent by SASASA. "Wasn't that a pain?" I said. "Yes," they said. "Wouldn't it be nice if we could know triangles were congruent to each other with a smaller set of information?" I said. "Yes," they said.

after school gluestick par-taaaayyyy
In the meantime (the lesson covered by this post spanned five days, FYI) I had glued all the triangle A's to a poster, all the triangle B's to a poster, etc. Before they looked at them, I had them predict (using the original instruction sheet) whether they thought all the copies of each triangle had to be congruent. Will all the triangle A's be congruent? Will all the triangle B's be congruent? etc. Reminders that for congruence, it's okay if you have to reflect or rotate one to make it look exactly like the other. 

Justin Lanier and Pershan? possibly others? brought up the issue on Twitter that triangle-uniqueness (will this given information only allow you to make one triangle?) is a cognitively different thing from triangle congruence (can I be sure these two triangles are identical?). That was a slippery thing that always poked at the edges of my thoughts this time in the school year, but I'd never thought to explicitly address it. I think this lesson does a nice job of bridging those two related understandings. The triangle-constructing compels one to think whether there are choices to be made in what this triangle looks like... or is it unique? But putting all the triangles together on a poster highlights the question of do all these triangles have to be identical, using certain given information? Pretty seamlessly, I think, and without having to dwell on it.

Using a recording sheet (created by my colleague, Matt), they did a gallery walk, recording whether, in fact, all the triangles were congruent, which parts were the given information, and drawing a sketch. (He provided the triangle outlines in the rightmost column -- I took those off.) Then we had a quick discussion and they made their first foray into identifying which theorem applied based on given information (the back of that sheet).

And that was that! On their quiz Friday (usually we have a half-period quiz on Fridays), I asked another question very similar to one of the pre-assessment questions, and every single student showed growth in their ability to explain why the given information was not enough to guarantee unique triangles.

(I'm not sure if this lesson is useful in Common Core land -- over there, you're supposed to link congruence to rigid transformations. Which I do here in Virginia, informally, but it doesn't rise to the level of students performing transformational proofs.)