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!

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.)

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: