Arithmetic of Complex Numbers Placemat Activity - Algebra 2 + Complex Instruction (CI)

Just because you have an all-groupwork and all-Complex Instruction (CI) format doesn't mean you don't need practice activities too.

Our Algebra 2 kids were getting the concepts of complex numbers and complex conjugates, but were still kinda shaky in terms of fluency in working with them.

Based on ideas I stole borrowed a long time ago from the fabulous Kate Nowak (@k8nowak, http://function-of-time.blogspot.com) and the equally fabulous Rachel Kernodle (@rdkpickle, http://sonatamathematique.wordpress.com ), I proposed a placemat activity to my ever-game Algebra 2 teaching team and they dove right in.

Set-Up
We have typical CI four-person table teams set up in each of our rooms, with each person assigned a specific role based on where they're seated at the table. Our roles are Facilitator, Resource Manager, Recorder/Reporter, and Team Captain, although of course, your mileage may vary. Each role has specific tasks they are expected to perform; for example, only the Resource Manager may call the teacher over for a group check-in or a group question (in our program, teachers only accept and answer group questions).

Each table was given:

• two, double-sided "placemat" sheets for doing work in the center of the table
• a set of problem cards (there are four sets, one for each round of play; to simplify clean-up and organization, I printed each round of cards single-sided on a different color of paper, one set per table group. I've got 7 tables in my room, so I made seven sets of cards. I laminated them and clipped them together, but hey, that's just me)
• the sum to which all four answers for any given round should add up

The sum for each round was written on the whiteboard, though it could have been projected via document camera or Keynote/Powerpoint slide.

Objectives
We had mathematical objectives for the activity as well as CI or norms-based, group work objectives. My students in particular needed reinforcement in group work norms and collaboration. Our objectives were:

     Math Objectives

• achieve greater fluency in the arithmetic of complex numbers (including the distributive property)
• deepen understanding of and fluency with the powers of i
• deepen understanding of and fluency with complex conjugates

     Group Work Objectives

• work in the middle of the table
•  same problem, same time (no one moves on until everyone moves on)
• using table group members as resources

The next time I run this activity, I will definitely give a Participation Quiz because the group work norms are so beautifully reinforced in this activity.

How We Ran It
Recorder/Reporter writes the sum in the central oval of the first side of the placemat. Each group member gets a problem card for round 1 (problem a) and works his or her problem on his or her quadrant of the placemat.

When everybody is finished with their problem, the Facilitator facilitates the addition of all four answers. If they add up to the given sum for that round, the Resource Manager calls the teacher over for a "checkpoint" and the next set of cards for the subsequent round of work.

If their answers don't add up to the given sum, they need to work together through everybody's work on the placemat to diagnose what went wrong and where, as well as how to fix it. Then when they've fixed it, they call the teacher over for a checkpoint and the next set of cards for the subsequent round.

Group Work Benefits — Reinforcing Norms
For my classes, the greatest benefits of this activity came from the fact that it forced students to work in the middle of the table, to use each other as resources, and to talk mathematics. Getting kids to work in the middle of the table is the hardest part of CI, in my view, because it goes against the grain of most of their in-school conditioning. The placemat format makes it nearly impossible NOT to work in the middle of the table. And once they're doing that, it seemed like everything else ran pretty smoothly.

I especially liked the fact that this activity created a context in which students experienced an intellectual need for the using the rules of arithmetic for complex numbers and for the powers of i. It was situationally motivated, but extremely targeted.

Sums for Each Round 
The sums for each round are as follows (if you find an error, please speak up):

• Round 1 (problem a):   26-73i
• Round 2 (problem b):  0
• Round 3 (problem c):  165
• Round 4 (problem d):  2 – 48i

PDF Files for the activity
These are available also on the Math Teacher Wiki on the Algebra 2 page. If you haven't visited the Math Teacher Wiki, you don't know what you're missing.

• Complex Numbers Arithmetic – Problem Cards
• Complex Numbers Arithmetic – Placemat Master

And this is why I teach...

It was another crappy Friday in an arithmetic series of crappy Fridays that were running together and threatening to define the limit of my patience for fall trimester as x approaches a mid-sized number that is nowhere near infinity. So I have no idea what possessed me to wake up even earlier than usual to pull together an extra day's practice activity for my right-after-lunch class of rumpled and discouraged algebra students — the ones who believe to their core that California's Algebra 1 requirement is God's own punishment for unremembered karmic crimes they must have committed in previous lifetimes.

But I did it.

The topic was solving and graphing compound inequalities — a skill set that must be mastered in order to have any hope of making sense of and mastering the next topic traditional algebra curricula force-feed to our students: the dreaded topic of absolute value inequalities.

There's really nothing I can say to convince a roomful of skeptical eighth graders that compound inequalities will prove not only useful in business planning (which, after all, is simply algebra writ large across the canvas of the economy) but also amusing and possibly even interesting little puzzles to delight the mind.

To this group of students, they're simply another hoop to be jumped through.

So something in me understood that I needed to reframe the task for them, and to do so using Dan Pink's ideas about intrinsic motivation from his book Drive.

Nothing unlocks the eighth grade mind like an authentic offer of autonomy. As I explained recently to a room of educators at a mindfulness meditation training, middle school students suffer emotionally as much as adults, but they have comparatively little autonomy. A little well-targeted compassion about this can carry you for miles with them, though I usually forget this in the heat of working with them.

For this reason, I like to save practice structures such as Kate Nowak's Solve—Crumple—Toss for a moment when they are desperately needed. I have learned to withhold my Tiny Tykes basketball hoop for moments like this, when students need a little burst of wonder in the math classroom. And so even though I was tired and very crabby about the ever-increasing darkness over these mornings, I pushed myself to pull together a graduated, differentiated set of "solve and graph" practice problems to get this group of students over the hump of their own resistance and into the flow experience of practicing computation and analysis.

And oh, was it worth it, in the end.

The boys who are my most discouraged and resistant learners came alive when they understood that a little athletic silliness was to be their reward for persevering through something they considered too boring to give in to. They suddenly came alive with cries of, "Dr. X— watch this shot!" from halfway across the room. One boy who can rarely be convinced to do the minimum amount of classwork completed every problem I provided, then started tutoring other students in how to graph the solution sets and perform a proper crumpled-paper jump shot.

The girls in the class got into it too, but they seemed more excited about the possibility of using my self-inking date stamp to stamp their score sheets. So I gladly handed over the date stamp to whoever wanted to stamp their own successfully solved and graphed inequalities.

I was far more interested in reviewing their mathematics with them. One of the things I love best about practice structures like this one is that they give me an excuse to engage one on one with discouraged students under a time crunch pressure that adds a different dimension to their motivation. Suddenly they not only want to understand what they have done, but they want to understand it quickly, dammit, so they can move on to another problem, another solution, another graph, another bonus point.

Ultimately, Solve—Crumple—Toss becomes an occasion for conceptual breakthroughs in understanding.

I can't tell you why this happens. I can only tell you that it does happen — often. It makes me feel lighter, more buoyant about teaching them algebra. And it makes them feel happier too.

I wanted to write this down so I could capture it and remember this for a few weeks from now, when it stays darker even longer in the mornings and when I feel crappier and crabbier and more forgetful.

Life on the Number Line - board game for real numbers #made4math

UPDATE: Here is a working link to the zip file: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

Last year I blogged about my work on a Number Sense Boot Camp, so I won't rehash all of that here. This year I want to give the follow-up on how I used it last year, what I learned, and how I'm going to use it this year in Algebra 1.

This was my breakthrough unit last year with my students. It anchored our entire Chapter 2 - Real Numbers unit and really solidified both conceptual understanding and procedural fluency in working with real numbers, the real number line, operations on real numbers, and both talking and writing about working with real numbers. We named it Life on the Number Line.

Here's how the actual gameboards, cards, and blank worksheets looks in action (sans students):

I sure hope I didn't make a bonehead mistake in my example problem!

The most effective thing about this activity was that it compressed a great deal of different dimensions of learning into the same activity, requiring learners to work simultaneously with the same material in multiple dimensions. So for example, they had to think about positive and negative numbers directionally in addition to using them computationally. They had to translate from words into math and then calculate (and sometimes reason) their way to a conclusion. They had to represent ideas in visual, verbal, and oral ways. And they had to check their own work to confirm whether or not they could move on, as no external answer key was provided.

Since they played Life on the Number Line for multiple days in groups of three or four players comprising a team who were "competing" in our class standings, learners felt that the game gave them an enormous amount of practice in a very short amount of time. Students also said afterwards that they had liked this activity because it helped them feel very confident about working with the number line and with negative numbers in different contexts.

I also introduced the idea of working toward extra credit as a form of "self-investment" with this game. For each team that completed and checked some large number of problems, I allowed them to earn five extra-credit points that they could "bank" toward the upcoming chapter test. Everyone had to work every problem, and I collected worksheets each day to confirm the work done and the class standings.

What I loved about this idea was that students won either way — either they had the security blanket of knowing they could screw up a test question without it signifying the end of the world, or they got so much practice during class activities that they didn't end up actually needing the five extra credit points!

Students reported that they felt this system gave them an added incentive to find their own intrinsic motivation in playing the game at each new level because it gave them feelings of autonomy, mastery, and purpose in their practice work.

The game boards were beautifully laminated by our fabulous office aide but do not have to be mounted or laminated. The generic/blank worksheets gave students (and me) a clear way of tracking and analyzing their work. And the game cards progressed each day to present a new set of tasks and challenges.

All of these materials are now also posted on the Math Teacher Wiki.

Let me know how these work for you!

UPDATE 10/27/2016: Here is a working link to a zip file of all the components for this: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

WEEK 1: 'Words into Math' Block Game | #made4math

In keeping with my Week 1 emphasis in Algebra 1 on activating prior knowledge of how to translate words into mathematical expressions, equations, or inequalities (or at least gelling some of it back into place), I've also created a "Block" game for practicing 'Words into Math' in my Algebra 1 classes. There are two levels of game cards that correspond to Lessons 1.3 and 1.4 in McDougall Littell Algebra 1 California edition (for those of you playing along at home).

This is a variation on Maria Anderson's wonderful, tic-tac-toe-style "blocking games" (Antiderivative Block, Factor Pair Block, and Exponent Block — using her generic gameboard, rules, and my own game cards for each of these first three games of hers on her web site).

The game can be played in any number of ways — either competitive or collaborative. Students can compete against each other — tic-tac-toe style — to get four of their counters in a row. Or they can simply take turns choosing the problem and working on solving each problem on the whole board.

I've created two levels of "Words into Math Block": Level 1 (purple problem cards) and Level 2 (green problem cards). I use Maria's generic PDF gameboard and print or copy them on colored cardstock or paper. I have learned the hard way to give each level its own color ID as soon as I create the game cards so I can easily recreate the card sets later whenever I need to.

I allow students to use whatever resources they need to during practice activities, so I expect to see those nifty Troublesome Phrase Translator slider sleeves flying during these two days. :-)

All of my materials, plus the photo above (in case you need a model) are on the Math Teacher Wiki.

Students really love these block games! I have a bunch of different "counters" that they can use as their game board markers: little stars (Woodsies from Michael's), circles, and hearts, colorful foam planet/star clusters, and various kinds of beans.

I'm hoping to get my students to be less flummoxed by mathematical language by giving them practice in using it early and often. Enjoy!

SOLVE - CRUMPLE - TOSS in Algebra 1: hommage à Kate Nowak

Kate Nowak creates some of the most innovative and engaging practice activities anywhere -- especially for those skill/concept areas that are more like scales and arpeggios than like discovery/inquiry lessons. Some skills, like basic math facts, simply need to be practiced. This is true not because students need to be worn down but rather because it takes the mind and body time and first-hand experience to process these as matters of technique. It takes time to get used to the new realities they represent.

Nowhere is this more true than in tinkering with the multiple different forms and components of linear equations in Algebra 1. No sooner have students gotten the hang of finding the intercepts of a line than they're asked to find the slope. They figure out how to find the slope and the y-intercept, and they're given the slope and a non-intercept point. They figure out how to crawl toward slope-intercept form, but fall on their faces when asked to convert to standard form. Standard form, point-slope form, slope-intercept form, two points and no slope, it's a lot of abstraction to juggle. Mastery is part vocabulary work, part detective work, part scales and arpeggios, and part alchemy of different forms. It's a lot to take in.

Enter Kate's Solve - Crumple - Toss activity. I have loved this practice structure since the day I first read about it, but I have struggled with the fact that the most engaging part of the activity destroys the paper trail/evidence. This was less important with high school students, but it is really important with middle schoolers, I find, because they are so much more literal.

For today's linear equation-palooza in class, I created a basic "score sheet" for each student and I numbered each of the quarter sheets on which I glued blocks of problems (4-6 problems per mini-sheet). I also differentiated them from "Basic" level (Basic-1 through -4, Level 2-1 through -4, etc), so that students could choose their own levels. Students were also invited to work in pairs or groups of three because I find it encourages mathematical language use and increases risk-taking. It also seems to be more fun.

After the "Solve" part of the activity, students brought their solved mini-sheets to me to be checked. If they completed the problems correctly, they got a stamp on their score sheet and proceeded to the back of the classroom where I'd set up the Tiny Tykes basketball hoop over the recycling bin. There they completed the "Crumple" and "Toss" stages, awarding themselves a bonus point on the honor system if they made the shot. Then they returned to the buffet table of problems and chose a new mini-sheet.

Because my middle school students like to bank extra credit points toward a test wherever they can, I like to attach these to practice activities such as this one or Dan Meyer's math basketball. Being more literal and concrete than high school students, middle schoolers seem to find great comfort in the idea that they can earn extra credit points ahead of time in case they implode on a quiz or test. What they don't seem to realize -- or maybe they do realize and they just aren't bothered by it -- is that if they participate in the process, they win no matter what. Either they strengthen their skill/concept muscles and perform better and more confidently on the test; or they feel more confident and less pressured because they have banked a few extra-credit points for a rainy day; or both.

It was fun to hear my previously less-engaged students infused with a rush of sudden, unanticipated motivation to tease apart a tangled ball of yarn they have previously been unmotivated -- or uncurious -- to unravel. And something about the arbitrary time pressure of trying to complete as many problem sheets as possible in a short period was also fun for them. I'm feeling a little ambivalent about not having found the secret ingredient of intrinsic motivation in this required blob of material. But I am grateful that, once again, an unexpected game structure generated what the late Gillian Hatch called "an unreasonable amount of practice."

The last word goes to the one student who put it best: "The crumpling is definitely the most satisfying part."

Beautiful, fluid chaos -- or what "learning in flow" actually looks like to the trained eye

One of the things I like best about my new school are my colleagues. In fact, I don't believe I could stand to teach English (instead of math) if I did not happen to have my particular grade-level team of amazing, insightful, reflective, open-minded collaborative English-teacher colleagues.

I'm not saying that teaching math is one big continuous picnic of sparkly rainbows, unicorns, and effortless class periods of absorptive learning, but in English Language Arts -- particularly at the middle school level -- you have to teach some of the most thought-numbing, soul-dissolving parts of the curriculum ever to torture the human mind. Spelling. Grammar. Vocabulary. I throw up in my mouth a little bit each time I have to schedule time for them on a new weekly assignments calendar. But they're a part of the curriculum that's mandated, and so they have to be taught.  Some things you just have to pinch your nose and swallow as quickly as possible.

On the other hand, or possibly as my reward for fulfilling these less satisfying obligations of my curriculum each week, I get to work with kids on one of the deepest and dearest endeavors of my heart. I get to teach them writing.

Now, one of the things I have learned in my many decades on this planet is that writing -- and learning how to write -- is MESSY. Learning to write a first draft is more about learning how to tolerate the waves of revulsion that come over you as you confront the your own feelings of inadequacy at what you put down on the page than it is about about learning how to structure a proper academic paragraph. In fact, I'm convinced that I could teach a goat to structure a proper academic paragraph. What takes genuine human maturity and emotional/psychological courage is learning to get a first draft down on paper.

For that, you have to gain the willingness to produce what writer Anne Lamott calls "a shitty first draft."

Since I'm not permitted to use that kind of language in a public school classroom with middle schoolers, I use the methods I first learned from, and later taught with, celebrated writing teacher Natalie Goldberg. Her system of "writing practice" emphasizes "separating the creator from the editor," and basically involves a small number of inviolable rules for producing your first draft of an idea. These are:

• keep your hand moving
• don't cross out
• don't worry about spelling, punctuation, grammar, or other rules
• lose control
• don't think
• go for the jugular
• follow and trust where your mind takes you
• give yourself permission to write the worst poop in America / on Planet Earth / in The Milky Way Galaxy

So for first drafts in my classroom, this is how we practice.

We close the door and the windows (so we don't bother anybody trying to do more traditional learning), I make earplugs available to anybody who needs to block out noise in order to think, and I tell my students to let it rip.

As soon as they finish a draft, they can come find me or a peer-editing partner and they or we look at what they have done. First we do so aloud but with no comment until the draft has gotten a first hearing. Natalie says, you can't know what you've written until after you've written it, so first we give it a hearing, then we go to town giving it a quick edit. We look for what our rubric tells us to look for. Then they go back to their desk (or to the floor, or to the back table -- wherever they want to be writing) and they bang out another draft.

The beauty of this system is that it gives them a lot of practice compressed into a very short space of time. Everyone can get a fast, free, immediate edit from a published working writer without time for judgment, shame, or a sense of disgrace to take hold. The kids have very quickly grasped how to use writing practice to harness the flow state, get their juices flowing, and not become too attached to what they've put down on paper. It gives them a wonderful experience of the feedback loop in writing and it gives them immersive time in the flow state that has rapidly improved everyone's basic writing skills noticeably and quickly.

The, um, downside of this system is that while we are having one of these in-class writing workshops, my classroom looks like a chaotic free-for-all. Or so I thought until the other day.

See, when we're doing this, even though I am not actually writing, I fall into the flow state too. I get absorbed in reading, writing, listening, editing, coaching, and cheerleading and I completely lose track of time. In a good way.

But the other day, one of my English colleagues and I were scheduled to trade classes halfway through the period to teach each other's classes part of a jigsaw lesson we were doing. I knew we were doing so, and I had everything ready and prepared, I just lost track of time. So when she arrived in my classroom to trade, she got treated to the sight of me on my knees next to somebody's desk giving one quick edit after another, kids reading aloud and giving each other peer edits, one kid sitting under the phone table next to the flag because he concentrates best down there with earplugs in using an Algebra textbook as his lap desk, and other kids writing (with earplugs in) either alone or together, but in what I imagined to look like total unfettered chaos.

Fortunately, this teacher is both an enlightened person and a reflective practitioner in her teaching, so she was absorbed for a few minutes just watching what was going on, taking it all in and finding it extremely effective and engaging.

Then she came over, tapped me on the shoulder, and reminded me that we needed to switch classrooms and finish the jigsaw activity.

At lunch later, she told me how much she had enjoyed getting the chance to watch our process unnoticed because it gave her a totally different vision of what an in-class writing workshop could be.

That made me feel both grateful and relieved, since I had felt certain that doing what I was doing was the straightest path to getting myself fired (except for the part about my kids' writing improving more and faster than many other teachers' classes).

Anyway, it was really interesting to have been observed while I myself was in the flow state.

Renegade Math Teacher Brings SBG into the English Classroom: Film at 11

SBG works so well in my Algebra classroom I've been looking for ways it could transfer into my eighth-grade English classroom -- especially with regard to writing. One of the hardest things about teaching persuasive or expository writing is that each "skill" is composed of multiple, discrete sub-skills, each of which could itself be broken down further into tinier and tinier (i.e., more refined) sub-skills. In many ways, it's an M.C. Escher-like process -- a mise en abîme of nested skills.

BACKGROUND

Our school and district use our own combination of two methods that work for us, adapted through our own collaborative practice, reflection, and research over many years to fit our district goals and population. As a starting point and foundation, we use the Jane Schaffer method for teaching the actual composition process and a slightly modified version of 6 Traits program for assessing the finished product (or the work-in-progress). 

What I like about the 6 Traits assessment system is that it has a strong SBG orientation. It is a rubric-based system that assesses idea development, organization, voice, word choice, sentence fluency, and conventions -- the main basic categories that young writers need to master to produce both competent and coherent arguments, paragraphs, and essays. In our district, we have whittled it down to a four-point scale, which gives most middle schoolers a fighting chance of making sense of their scores.

What I like less about the 6 Traits assessment system is the complexity and denseness of its rubric. As Edward Tufte, the infographics pioneer might say, its information density approaches near-total opacity.

While I appreciate that its authors are trying to be comprehensive, my experience is that, for a middle school or high school student trying to juggle the many skills that come into play in writing a persuasive paragraph, it's just too damned complicated.

MY IDEA

One of the things I've noticed early on in this school year is that even our strongest writing students tend to have only a tenuous grasp of what makes an effective topic sentence. And having taught literature at the university level, I have seen how this confusion tends to persist and worsen over time.

So my goal was to come up with an activity that integrated two tools I've found useful in SBG in the math classroom: (1) a clear, simple, compelling four-point rubric for judging the effectiveness of a topic sentence, and (2) an activity to give students practice in judging a wide range of topic sentences, along with practice in using the rubric as a basic for analyzing, debating, and justifying their assessments of each one.

With that in mind, I created the two tools which are attached here: a Topic Sentence Rubric and a "Judging Topic Sentences" activity for use in pairs or small groups. The "Judging Topic Sentences" activity sheet includes twenty topic sentences I wrote based on a recent writing prompt for that staple of the eighth-grade English curriculum, "Flowers for Algernon." The writing prompt (which was deliberately broadly written) asked the student to compose a persuasive paragraph regarding the author's message in the story about cruelty toward people with mental disabilities. 

I gave them 30 minutes in class to work together on the assessing activity before we came back together as a whole class to discuss and give closure to the process.

RESULTS

What was fascinating as I circulated among the groups was how quickly everybody grabbed hold of the idea of using criteria from the rubric as the basis of their judgments. Suddenly I was hearing arguments about how, yes, a certain claim was definitely true and supportable but was basically pretty trivial! I was also hearing students argue that another example made an "original and juicy claim," but that it was awfully long and wordy and could easily be improved with better word choice and sentence construction.

When we came back together as a class, I asked for examples of the worst topic sentence on the list and the best. The discussion was productive in that it brought students to an understanding that an "OK" topic sentence could kick off a really great paragraph if the writer used all the tools at his or her disposal. It also made them realize that a truly outstanding topic sentence could launch a truly mediocre paragraph if it was followed by weak use of evidence from the text and lame or badly written analysis and interpretation.

My fellow eighth-grade English teachers used this activity in their own way over the next few days and found it to be very helpful in getting students to think about what makes a strong and effective topic sentence.

So now it looks as though it will become a regular part of our writing curriculum. 

Another triumph for SBG!

Number Sense Boot Camp - Request for Feedback and Input

Maybe all of your Algebra 1 students showed up on Day 1 every year with a solid and fluent grasp of basic number sense, but mine sure didn't... and it scared the crap out of me. And then afterwards it haunted me, ALL   YEAR    LONG . . .

The stuff they didn't get was just mind-boggling to me:

• subtracting
• adding a negative number
• the basic concepts of the real number line
• fractions
• measuring
• counting
• basic ops with fractions
• absolute value (any related topic)

I mean, this is basic citizenship numeracy stuff, on the same order as basic literacy.

So since this does seem to be a general condition I am likely to encounter anywhere I am likely to teach, I decided to develop a "Number Sense Boot Camp" unit I could use to start the year off with, diagnose critical number sense deficits, use as an occasion for teaching basic classroom routines, give students a chance to dust off (or remediate) their basic arithmetic skills, and basically give us all a fighting chance of getting to some introductory algebra work.

Another thing that worked this year was stealing adopting game-like practice structures, such as those advocated by Kate Nowak in New York state and by the late Gillian Hatch in the U.K. As Gillian Hatch said, a game can provide "an intriguing context" as well as "an unreasonable amount of practice" in vocabulary, reasoning, procedural skills, generalizing, justifying, and representation than they might otherwise be inclined to do. As Hatch said, it also seems able to lead students "to work above their normal levels." As anyone who has tried any of Kate's practice structures can attest, there is something about introducing this playful element that really gets students to dive in.

IDEA #1
One thing I did this past year that worked for many individual students was to do some specific work with the real number line. I made a printable number line and gave each person their own number line (downloadable from Box.net folder) and a plastic game piece to use with it as a calculating device.

Since the rudiments and rules of board games have such wide currency in our culture, most students found this a helpful physical metaphor that gave them both conceptual understanding and procedural access to basic counting, addition, and subtraction experience that had eluded them in their previous nine to eleven years of schooling.

These had the added benefit of conferring prestige upon those who had shown up for extra help and received their very own set (though I gladly handed them out to anybody who requested one).


IDEA #2

Emboldened by my initial success, I realized could expand the idea of a number line "board game" to use as a basic structure for practice – both in using the number line and in many other basic number sense activities.

It even dawned on me that this could be made extensible by having different kinds of "task cards," depending on whether a player has landed on an even number, on an odd number, or on the origin (a decent justification for considering even- and odd-ness of negative numbers here ; go argue over there if you have a problem with this).


Players move by rolling one regular die and one six-sided pluses-and-minuses die (+ and –) (kids seem to need grounding in the positive and negative as moving forward and backward idea). Kids earn "points" in the form of game money, which could carry over and be used to purchase certain kinds of privileges (such as a "free parking" pass for a day when they don't have their homework to turn in).
Your thoughts?

UPDATE: 
Here are links to the different game boards, along with descriptions of each.
Basic Printable Number Line For Use With a Game Piece:
http://www.box.net/shared/eyy4nvhbtn5xx1qdc9j2

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 1-up version (for use with your basic at-home printer):
http://www.box.net/shared/nv0sdz65hy5p3hv8ix1x

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 3-up version (prints a 24" x 24" poster at FedEx Kinko's--costs about 2 dollars):
http://www.box.net/shared/d4uly1arl88lm3qu9vsm

I made Game Card files using Apple's Pages software (for Mac OS X) and MathType equation editor. You can use these as templates or make your own:
http://www.box.net/shared/s6ha4ol1o6tk0xltp1y1
http://www.box.net/shared/7or8g5klub7jiymshq0f
http://www.box.net/shared/5vq6cmpcd9f9lq1qhud1

Here is a link to the folder itself if you'd like to share and upload your own documents or samples:
http://www.box.net/shared/ftzkun7cvi5vxgvanvh5

Please share any experience or insights you have with them. Enjoy!


AND FUTHERMORE:
Julia (@jreulbach on Twitter who blogs at ispeakmath.wordpress.com) has started a Number Sense Boot Camp page on the Math Teacher Wiki where you can share and find other Number Sense Boot Camp ideas and activities. Available at http://msmathwiki.pbworks.com/w/page/42105826/Number-Sense-Boot-Camp .

UPDATE - 14-Sep-11:

It's only been one day since I introduced the tournament of "Life on the Number Line" but I am already excited about how well this is working out. It is exposing ALL kinds of misconceptions and misunderstandings about adding a negative and about interpreting negative and positive as movement along the number line. Students are playing individually as a "team," and the team with the highest number of correctly worked problems will win 10 free points (2 problems using the 5-point rubric for each person) on next Friday's unit test.

     Since they are surfacing all kinds of misunderstandings about + and - movement on the number line, this is leading to vast amounts of mathematical conversation to get it figured out. So basically, they are teaching each other about adding negatives and subtracting negatives and interpreting that as movement along the number line. 

     I can see that each day it will make sense to give some daily "notes" at the start of class on clearing up common misconceptions I've seen the previous day in students' work so they can solidify their conceptual understanding as well as their procedural fluency a little more each day.

     Best moment yesterday: a girl looked up at me beaming and said, "This is way more fun than doing math!"

     I said, "Good!" but I was thinking, "You have no idea how much math you are actually doing!" :-)


ANOTHER UPDATE:
Here are the game cards to use on the first day: http://msmathwiki.pbworks.com/w/file/45547360/1st%20batch%20of%20game%20cards.pdf


And here is a generic worksheet (front and back) you can print out and give to the kids to use as their template:
http://msmathwiki.pbworks.com/w/file/45547628/generic%20worksheet%20for%20Life%20on%20the%20Number%20Line.pdf

If you have only a ton of basic 1-6 6-sided dice, use Post-Its to make two (2) plus-and-minus dice for students to use with one (1) regular numbered die. This is a good task to give to a student helper. ;-)

FINAL UPDATE:

Four final things:

Thing #1
This unit confirmed me for that kids really do need active, multi-day practice in "living life on the number line" to gain a sense of positives and negatives as directions WHILE AT THE SAME TIME they are developing a sense of positives and negatives as additive quantities. It's not enough for us to just wave the idea of life on the number line at students. It doesn't make sense to them. They really needed experience alternating between (a) positives and negatives as indications of directional movement and (b) positives and negatives as additive or subtractive quantities in the process of deepening their additive reasoning skills.

Thing #2
Right before we started, I had the bright idea to give every group TWO +/- dice and ONE six-sided number die. If you don't mind my saying so, this ended up being a master stroke because it forced students to think about rolling (–)(–)(3) and rolling (–)(+)(3) and every possible combination thereof. This one thing alone might have done the most to deepen their sense of additive reasoning and of +/– as directions of movement.

Thing #3

Here's a link to a zip file that contains ALL of the game cards I created for this unit (on the math teacher's wiki): Game Cards- ALL

For all those who have asked and those who are thinking of asking, I'll say that my school uses the California edition of the McDougal Littell Algebra 1 textbook (by Larson, Boswell, Kanold, and Stiff). For this reason, the game cards are targeted at each of the lessons in Chapter 2. However they are not tied to that textbook and could easily be used with any curriculum or textbook (just sayin').

Thing #4

I'll have to take a photo of the final game boards our instructional aide mounted and laminated for us. They are a true work of art!