March 17, 2015

#MTBoS is Exhibiting at NCTM Boston

For full details, hop on over to the Explore MTBoS site, but tldr: There’s going to be a MTBoS booth at NCTM Boston! We could use your help in the following ways:
  • Sometime soon, you can tweet on the hashtag #WhyMTBoS a reason why the MTBoS is great.
  • If you’re attending NCTM Boston, you can sign up to spend time staffing the booth.
  • If there’s a MTBoS project or endeavor that would be great to highlight at the booth, let us know about it!
  • Let us borrow your internet browsing device for NCTM— iPads would be excellent.
And there will be a new Explore MTBoS online excursion after NCTM!

Again, for more details, hop on over to the Explore MTBoS site. Yay!

March 13, 2015

Systems of Equations

My Algebra classes having been playing with solving systems for over a month. A very interrupted month, but a month nonetheless. I decided to go all puzzles at the beginning of the semester to re-engage my classes. And it was awesome. I didn't even realize the puzzles were systems when I first assigned them! Then we did some algebraic manipulation. And we ended with more context. I had no continuity (7 snow days and a week's vacation happened during this unit) so I want to figure out how to run this unit in the future.

We started with Noah's Ark. It was hard. I tried "convince your classmates, then come to me when you have a consensus" but they got mad. So instead I did a few cycles of "work for 10 minutes, then I'll have students share who made progress, then you can work for 10 more minutes" That worked.

Next we did Mimi's shape puzzles (only the first 5 pages because we'll only be solving systems in 2 variables in this course). This went amazingly. They worked at the boards and not only solved them, but explained their work!

Tape diagrams. I like this sheet (except number 5, I meant that to be something trickier than 60*3=men but I don't remember what). It doesn't need to happen in this unit though. It would probably be good to move to September as a way to use diagrams effectively.

We've been solving balance puzzles for months. I thought it would be great to segue into equations/graphing by writing equations from balance puzzles. (See twitter conversation.) Turns out they're too good at solving them to make two variable puzzles challenging enough that they see any need for equations/graphing. It was cool to see how all their equations intersected at the same point when I graphed everything they came up with in Desmos. Next time I'd do this as a whole class activity only. Challenge: write as many equations as you can for this (two shape) balance. Solve it. Check out what happens when you graph all those equations on Desmos. Yay!

Then we spent a while solving algebraic systems: Graphing without context (intro above). Substitution without context (sticky notes intro). Elimination without context (intro with - you can add 5 to both expressions, you can add y to both expressions, if y=5 you can add y to one expression and 5 to the other expression. Okay? Hey look! If I add these two equations ones of my variables disappears. Yay!).

Most kids liked elimination best, which is fine by me. But I did want them to realize that if a problem is set up for substitution, that might go faster. So we did Sam's efficiency rating scale. Then we did a scavenger hunt (solve one system, the solution is on another board, do that problem next) where each problem might be best suited for a different method.

So far I'd only given kids systems with nice solutions. So I had them solve three systems. The point was supposed to be to discover that lines can coincide or be parallel instead of always intersecting. But despite the fact that all three systems included the line y=2x+4, they struggled because they weren't set up nicely for elimination (issues like lining up the = signs arose for the first time). Next time I will have better systems for this task and more challenging systems for previous tasks. Solving three systems takes forever even if they're nice ones... But we finally learned the word coincide and reviewed parallel and practiced those a bit.

We ended with systems in context again, but words rather than puzzles this time. First we practiced interpreting equations with some awesome examples of potential misconceptions (just student pages 1 and 3) then we did the Mathalicious lesson Flicks.

Next time I think I'll do tape diagrams and Noah's ark whenever we need puzzles, not necessarily this unit. Shape puzzles systems is a good starting place. We could even do systems by elimination from there because the method comes up and it is a better transition than solving equations was this year. Then a brief intro to graphing using balance puzzles rather than that whole activity. Hopefully they won't need so much practice graphing so we can use Desmos more. Finally, substitution. With stickies. The context problems don't have to wait until the end but it's not a bad place for them. I think this seems like a reasonable plan (though I have no compelling reason for the order of the methods). It would be awesome to run this unit without all the interruptions. We'll see what crazy weather we end up with next year!

I haven't really been writing much about what I'm doing in Algebra this year since nearly everything I do is from somewhere else in the MTBoS or a cut up Kuta sheet. But maybe it's helpful to see how I put together the pieces to build a lesson? And to remember what resources exist? I'd like to get back to sharing (and recording for myself) so feel free to comment with what you'd like to hear about!

March 8, 2015

Cookies in the Cookie Jar (#tmwyk)

Hi! Still here. Still have an awesome 13 year old here. Last week I bought cookies and dumped them in the cookie jar before J got home. When she noticed them, we had this conversation:

J: How many cookies are in the jar?
What do you think?
J: 50
I see 16 on this side. There's like 3 sections so that would be 48. But I feel like there are more cookies in the middle.
J: 61
I happen to know that they came in three rows, so it's probably a multiple of three.
J: No fair! How do you know that?
I put the cookies in the jar. There were three rows in the package.
J: I get to change my answer then...
Why do you want to change it?
J: 61 isn't a multiple of 3. So... 66.
Okay. Let's see what the package says.
J: (looks at snapchat on her phone while I retrieve the package from the trash) Ha! He said 11. There's 11 right there!
(I figure out that she sent a photo of the cookie jar to her friend and asked him to guess too.) Tell him to try again! So the package says the serving size is two and there are about 27 servings per package.
J: Doesn't it just say how many there are? (Grabs package) It has the weight... How are we supposed to know?
You can do this. So if one serving is two cookies and there are 27 servings, how many cookies is that?
J: I don't know.
One serving is two cookies. How many cookies in two servings?
J: 4
Three servings?
J: 6
Want me to keep going until 27?
J: No... Isn't there a problem we could do?
J: What is it?
You tell me!
J: I dunno
Well, how did you get 6 for three servings?
J: I added two... So I keep adding two... Until 27... So we do 27 times two?
Great idea! (She starts opening the calculator app.) What's twenty times two? (She types it in her calculator anyway.)
J: Wait, what's your guess?
(I realized later she meant my estimate for cookies in the jar, not my guess for 27*2.) 54
J: I was way off. I thought it was like twice as many.
No, you were pretty close. Twice as many would be 108. So it says about 54, does that work? Is it a multiple of three?
J: No. (Looking at snapchat again.) Ha. Now he guessed four. He's not even trying!
(I explained too low and too high guesses for Estimation 180 but her friend didn't want to talk math with us.)
J: Wait but what was your guess?
Oh. Um. When I counted the 16 on one side I thought there were more than 48 so I guess I thought it would be over 50.

This conversation was less about me probing J's thoughts on estimating and more about me modeling mine. I'm curious how she got 50 as an initial guess (surprisingly close!) or why she picked 66 as her multiple of 3 when 60 and 63 were closer to her last guess. But I knew that we were going to figure out the number of cookies by serving size and I wanted to keep her interested through to that part of the conversation. I was fascinated by her phrasing "Isn't there a problem we could do?" She didn't think of "How many cookies?" as a problem. She didn't think of "one serving is two cookies and there are 27 servings" as a problem either. She only thinks of "27*2=__" as a problem.

January 14, 2015

Nix the Tricks: Second Edition

It's finally officially here!!

I've been hard at work on the long overdue update. While the brunt of the writing got done in November as I'd hoped, there were images to design plus editing, editing and more editing that dragged on. But I think the final product is worth the wait (on your part) and the effort (on mine).

The second edition includes:

  • 29 new tricks.
  • Updates to several tricks from the first edition.
  • A new chapter organization method that's more balanced and  aligned with strands of topics.
  • A conclusion, including testaments from teachers nixing tricks in their own classrooms!

The new book is available in a variety of formats. The regular size pdf and tablet/kindle friendly pdf are available for free download. (I updated both today for some minor typos if you've been paying attention and already downloaded one.) The paperback is now 100 pages! You can order it directly from CreateSpace. It will be on Amazon soon (they need a few days to process). This edition is twice as long as the first, but not twice as expensive! (You're welcome.)

I can't possibly begin to thank everyone who has helped me get here. Michael Fenton did an impressive amount of editing so he gets a personal shout out - Thanks Michael! But the biggest thank you has to go to every single person who so much as mentioned this project to someone else. I never imagined how quickly or how widely word would spread. There were over 9,000 downloads of the first edition directly from the website (which launched just over a year ago). I cannot begin to imagine how that number extrapolates to how many people have read the book - between people emailing pdfs, printing copies and downloading from other sites that have posted the pdf.* It amazes me every time someone outside the MTBoS tells me that they've heard of the book. And as I've done no advertising, all of that is you.

Thank you all. Good luck nixing tricks and I hope to hear from you soon with successes or struggles (or typos, though I can't bear to typeset this thing another time this month). I will be at NCTM Boston and TMC and I'm always around on twitter.

*Which is in accordance with the copyright so long as the title and copyright pages are included and the download is free. More sharing is always better, but given the choice I'd rather people provide a link to so visitors have access to other resources including updates!

January 9, 2015

Concept Maps

The semester is coming to a close so I assigned a concept map to my PreCalc students. We have been studying trigonometry so their task was to take all the concepts of trigonometry and make connections.

I started by giving them three minutes to independently list the topics they have studied this semester. They were encouraged to do it by memory first, and then to flip through their notes to see what they missed. At the end of three minutes of silence I collected all of their ideas on the board. I purposefully spread them out and grouped like ideas together. Then I showed them this:

This map is unrelated to anything we've studied this year, but it shows what I expect from a concept map - both topics and connections. Since we did this on the last day of class before break they had the option of completing their map in class or building a fractal with me and completing their map for homework.

When we got back from break I got some awesome maps!

I didn't get a complete photo,
but he included a concept graveyard!!

But those are the only completed maps I received. And while my classes are small this year, I don't have a class of three. So I took all of the topics they included (as well as the students who handed me admittedly incomplete assignments) and made a table of concepts. The students who still needed to complete the assignment then had the option to use the existing concepts (I printed the table and they were responsible for cutting) and organize them into a map. So far I have one of those, and it looks equally awesome:

It is encouraging to see students making connections that I didn't specifically highlight. And having the whole semester on one page has to help students get some perspective as they head into studying for midterms. I was hoping to do a gallery walk so students could see connections they may have missed, but the lack of follow through in this class was rather prohibitive. I think I made the best of the situation though, and there might just be time for one next week if a lot of students surprise me on Monday!