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, November 19, 2014

Graphles to Graphles

New game! My Algebra 2 students struggle with stating the domain and range for reasons including: trouble understanding and writing inequalities, and a lack of comfort with the coordinate plane. We spent a day on looking at graphs and identifying their domain and range. We learned to deploy our wonderful domain meters and range meters that I learned about from Sam. But for maybe 25% of the students, the cluebird was stubbornly refusing to land.

So, I thought asking the question backward might be a good way to attack it. Instead of here's the graph, what's the D and R? ask, here's a D and/or R, draw a graph. I mean, I know this is pretty standard fare. The thing is, I didn't want to do examples and a worksheet, or hold-up-your-whiteboard so I could somehow assess 22 graphs in a split second. It seemed like there should be a better way.

So I did what I do, which is ask on Twitter. And I got lots of helpful ideas, but this was the one that I latched onto and ran with:
The end result is, I'd argue, more like Apples to Apples than Charades (hence the title).

To prep: Make game cards. I printed each page (docx pdf) on a different color card stock. Student play in groups of 4-ish, so plan accordingly. I printed 6 sets. (John suggested having students submit constraints, but, for this crew, I decided to unload that part and create cards with the constraints.) You'll also need a mini-whiteboard, marker, and eraser for each student. Check your dry erase markers, because nothing kills a math game buzz like a weaksauce marker. (I'll admit to a minor teacher temper tantrum where I uttered (okay, yelled) the words "I'M NOT THE MARKER FAIRY! I DON'T POOP MARKERS!" Teacher of the year, right here, folks.) Also, you'll need some kind of token that players can collect when they win a turn. I use these plastic counting chips that I use for everything, but anything would work, candy, whatever.

Doing a demo round with a few kids playing and everyone watching will pay off, in the more-kids-will-know-what-is-up sense.

Here's how the game plays:
  • Someone is the referee.
  • To begin the turn, the referee turns over two (or one, or three) different-colored cards, and reads them out loud. (I feel the reading aloud is important practice for interpreting inequalities.) You could do, like, first round is one card, second round is two cards, third round is three cards. Whatever suits your needs.
  • The other players have one minute to sketch a graph meeting the constraints on the cards. The referee is responsible for timing one minute.
  • The players hold up their mini-whiteboards so the referee can see. 
  • The referee disqualifies any graphs that don't match the cards, and explains why. Other players should police this, too.
  • Of the remaining graphs, the referee picks his favorite. This player wins a token.
  • The turn is over, and the player to the referee's right becomes the new referee.
  • Repeat.
It was great! Here are things I liked about it:
  • 100% participation 100% of the time. At no point should anyone be kicking back.
  • Nowhere to hide. There were a couple kids who had to come to me and say, "Miss Nowak, I really don't know what's going on." which I don't think they'd be compelled to do if we were just doing some practice problems.
  • Good conversations. Especially reasons for why graphs were disqualified. "You need an arrow there! The domain goes to infinity!" That sort of thing.
  • Students were necessarily creating and evaluating. Take that, Bloom
  • Built-in review of what makes a graph a function vs not a function.
  • My chronic doodlers had a venue to express themselves. Especially if the graph didn't have to be a function.
  • Authentic game play. You could use your knowledge of what a referee liked to curry favor.
Here are some action shots. Let me know if you try it, and how it goes!

Thursday, November 13, 2014

We Got a Problem

We spent practically the whole period (35-ish minutes) on one problem today. This one, that Justin wrote about recently, that he found on Five Triangles:

Since we just spent a few days naming pairs of angles made by parallel lines and proving what's congruent and what's supplementary, I was provisionally hoping we'd get, at the end, 10 or so minutes for students to present various solutions to the class. That did not happen in any of my three class periods. Because I didn't have the heart to interrupt them. At the 10-minutes-left mark, too many were still making passionate arguments to their small groups about why they thought their solution worked.

Here's what we did: The big whiteboards were on the tables. Groups of 3 or 4. I stated the problem while showing this diagram. Made a big deal of starting with a regular old piece of copier paper and making a single fold. 3-5 minutes silent, individual think time. No class made it to 5 minutes without a buzz starting. I wrote a time on the board and said, by this time, everyone in your group needs to be prepared to present a solution to the class. Also, the answer is not as important as the reasoning that got you there. If you are saying something like "this angle has this many degrees," you have to explain the reason why that must be true.

Then I started listening and circulating. There was almost 100% engagement, and I have to think it's due, to a high degree, to the problem itself. This problem just felt do-able, but not obvious, to every learner in the room -- the sweet spot.

When a group would start crowing, in their 9th grade way, that "MISS NOWAK. WE GOT IT," I refused to confirm or deny that their answer was right, and asked a randomly selected group member (everyone was supposed to be able to explain the solution) to walk me through the reasoning. I played the role of highly annoying and dense skeptic. "Wait, how did you know that angle was 90?" "Because it's a RIGHT ANGLE." "Wait, how do you know it's a right angle?" "... ... ...BECAUSE A SHEET OF PAPER IS A RECTANGLE." "Oohhh, right." And then, when they got to a part that was not justified (an assumed bisector, an assumed isosceles triangle, trying to use two sets of parallel lines to leap to a conclusion about congruent angles), I wasn't shy about saying I wasn't convinced. In their groups, they had already harvested the low hanging reasoning fruit. I figured my experienced eye was valuable for training a spotlight on flaws in their arguments. And they responded well, in a back-to-the-drawing-board kind of way.

I'm in the habit of slagging myself on here, but I'm going to take a moment and describe a few times I witnessed and celebrated some great, inspired ideas with at least one learner today:
  • You extended that line to make it intersect another line!
  • You marked those lines as parallel with some arrows! And those other ones too!
  • You made an estimate of a reasonable answer!
  • You grabbed a piece of scrap copier paper and made a physical model to look at and play with!
  • You suggested your group start over and draw a clearer diagram, so more people would know what you were talking about!
The Value
This seemed evident: the importance of being able to articulate how you know things are true. I think that was one of the purposes of proof that Pershan hit on last summer... knowing why a right answer is right. In a world (of school Geometry) where if I'm careless, I'm too often asking kids to "prove" things that they think are already obvious, I want to make as much room as possible for problems like this where something is not obvious and needs justification.

I'm not entirely comfortable with leaving 72 kids hanging about what the correct solution was, and why. What I want is for this to keep bugging them, and for them to make little doodles and sketches of it in their spare time, and for them to not be able to leave it alone. I did not want to ruin anyone's fun. At the same time, I think many kids could have benefited and learned from seeing a few different correct arguments for why the angle had to measure 140 degrees. This is still an open question for me -- what's the best way to handle kids/groups that don't arrive at the correct answer? Do you let the question hang, or do you interrupt everybody so groups who made it down a valid path can have time to show what they did? I'd love to pick it up tomorrow, but I'm going to be out (whaddup, #NCTMRichmond!) so we'll have to see if anyone has any memory of what happened today on Monday.

They should have sent a poet. I should have done this on a block day.

Monday, November 10, 2014

DDT, Y'all

Today in Geometry we tried Dance, Dance Transversal as popularized by Jessica and Julie. The kids dug it, and nailed an exit ticket identifying names of pairs of angles. I followed Julie's plan pretty closely. I loved that the kids were up and moving around for a good 20 minutes of class. (Was anyone else traumatized by that Grant Wiggins article? I'm very on the lookout for ways to make kids move.)

I just want to add one more resource to the arsenal: a powerpoint I made to show while playing. The slides auto-play the different moves. There are some initial slides that demonstrate where to put your feet for each cue. Then, the first two game slides are timed with a 1.5 second delay and worked well with Problem, and the second two game slides are timed with a 1 second delay and worked well with Dynamite.

Here's video. In case you're wondering, I did also play along, every period. Because their dancing did not have enough FLAVOR, and I had to demonstrate. Try to ignore the one kid doing some kind of demented hopscotch:

Saturday, November 8, 2014

Gallery Walk for Noticing Features of Inverse Functions

I put a call out on Twitter last week for good things for inverse functions. I got a few helpful responses but nothing that was really the thing. So here's what I made.

The day before, we had worked with inverse functions as doing and undoing equations. I started with ciphers. Students walked in and the board said, IQQF OQTPKPI, DGCWVKHWN DTCKPU! with no explanation from me. I just greeted them and took attendance and acted nonchalant. One kid sidles up and goes all sotto voce, "Miss Nowak, does the first part say Good Morning?" Since the good morning part was a pretty easy crack using context, after a minute or two someone notices that all the letters are shifted over by two, and can't keep from blurting it out, and we're off.

I had one of them explain how the encoding was done with this example. Then, they wrote secret messages using their own shift n cipher, traded, decoded. I babbled a little bit about Caesar and Enigma (I really want to show them this Numberphile video, thanks for the tip Mike Lawler). The encoding and trading and decoding only took about ten minutes, we went through one from beginning to end: what was your message, how did you encode it, how did you decode it. The alphabet was written on the board along with a counting number under each letter, the idea being that if your encoding added 5, the decoding would be subtract 5.

We spent the rest of that day couching inverses in terms of equation rules. x + 5 and x - 5 is fine and pretty obvious, but what about more complicated rules. Kids had mini whiteboards, I'd throw a function on the board and they'd try to write the inverse. Each time, they wrote down operations done in the original function, inverse operations in reverse order, then do that to an x. So for example if the given function is 3x2 - 5, they write down "square, multiply by 3, subtract 5" and then write down "add five, divide by 3, square root." Plop down an x and do those things to it. The biggest hurdles were order of operations (so they might write down "multiply by 3, square, subtract 5"). Also, always undoing the whole of what came before. So in this example, they'd be likely to write x + 5/3 instead of (+ 5)/3. But we just kept going and honestly, they didn't want to stop until they were getting them right. (I just got an idea about how to make this part better. Compute like f(3) (or something) and run the result through their inverse to see if a 3 comes out. (Instead of just you're right or wrong because I say so.) Have to figure out how to make that manageable.)

The next day, I wanted them to notice all the nice things that are true for functions and their inverses: the symmetry over y = x, that the inputs and outputs trade places, that f-1(f(x)) = x. So, each student got one of these cards. They figured out the inverse of that function using the technique from the day before. There was another student in the room with the inverse of their function, so they had to get up, talk to people, and then sit with their partner.

Each pair of students got one of these (the first page). They tacked their cards to the paper, completed the tables, graphed each function in a different color, and computed f-1(f(0)) and f-1(f(1)). They needed various levels of support interpreting instructions, but it helped to have them working in pairs on the same piece of paper - there was a natural reason for them to talk to each other to figure it out. My colleague Lois is teaching the same course, and got a coach to come in for one of her sections, which was a great move.

As the mini-posters were completed, they were hung up around the room. I said, hey, you all had different functions and now they're up there with their inverses. There are some neat things that are always true about a function and its inverse. Walk around and look at them all, and write down at least two things you notice. If you look at page 2 of this same document, the first question has space for them to write down observations.

They sat back down, and they shared their noticings with the class. I had Desmos up on the projector with some pre-loaded functions, so we had a concrete thing to point to as they were sharing.

Then they got to work on the rest of that page 2, which is lifted directly from lesson 6 of this eMathInstruction textbook (thanks Sam for pointing me to this resource). Some of them were able to just do those problems, some needed help restating the given information and what the problems were asking.

So there you have it. I especially liked this lesson for the social, discussion, get-up-and-move-around aspects. These Algebra 2 classes have not responded positively to problem-posing when they haven't been "shown how" to do a problem first, but, we have been successful with lessons like this where we break the questions into clear chunks while still requiring that they do some thinking and figuring out. It's a bit of a tightrope walk but that's how you get down a tightrope, right? One tiny step at a time?

Friday, November 7, 2014

Fire Up Blogging Machine

Yo. This year is hard. New building, blah blah. I'll pause a minute for no one to be surprised.

I feel, very often, like I'm not that good at this. I know that everyone does sometimes. I know, I know. I think it has very much to do with attending to formative assessment every day. (Every. Damn. Day.) Measurement: making it hard to lie to yourself since... measuring was invented.

The children are charming and testy and pathetic and confident and devious and brave, all in the same day, all in the same 45 minutes. There are 70 ninth graders that move through my room, and the thing is that a ninth grader is like the weather in Buffalo in April -- if you don't like it (or if you do), just wait five minutes.

I have lessons that I want to write up for this blog, the problem being my artifacts (documents, pictures, student work, etc) are all over the place. The file system on the school network is unreliable, so teachers all use either Dropbox or Drive or flash drives to store and/or keep a backup of everything (even though Dropbox isn't installed at school -- it's web interface only, 2005-style). Colleagues have been very generous sharing (bewildering Virginia-standards-based) materials with me. So all the stuff I've modified or created and used is on... the school file system and Dropbox and Drive and a flash drive. That puts just enough of an annoying-barrier in the way of assembling blog posts. I have got to get my computer file organizing act together.

So, those are some lame excuses for the radio silence. More coming. I'll figure this out.