We did a second go-round of blog posts / error analyses using the same method as last time, but I offered up a different batch of errors.

This time, Alvin, Simon, and Theodore (har, har…) were the students in need of help. Each chipmunk encountered a different, unique problem when attempting to do multiplication (of fractions and decimals) and long division (with two-digit divisors).

Alvin doesn’t change the mixed numbers to improper fractions:

Simon is obsessed with lining up decimal points – as well as the standard algorithm for adding decimals:

Theodore forgets zeros in the quotient:

Feel free to use these errors to drum up discussion in your classroom! Making kids write about the errors is not obligatory – but I think it had a huge impact on retention. Also, I’m seeing my kids get better and better at locating errors, thinking up reasons why students might make errors, and articulating their thought processes. Hurrah!

]]>

This year, my kids have math blogs – which they seem to be digging, which makes me very happy. I’m aiming to have them post about 10 times this year to write about how they are making use of mathematical habits (thanks, Bryan Meyer), among other things.

This week, I had two goals: I wanted my students to do some error analysis pertaining to fraction operations, and I wanted them to do some for-real reflecting about their understanding of fraction operations. Essentially, I hoped to cement student understanding of addition, subtraction, and fraction ‘basics’ before jumping into multiplying and dividing after Thanksgiving break.

I threw together this sheet:

And then this one:

The conversations that ensued were great. (Students discussed in groups, and then we shared out as a class, prior to any writing on that second worksheet.) I loved hearing kids verbalize their conceptual understanding of fraction operations as they remarked upon what Jane, Peter and Sam did correctly and incorrectly. Additionally, my students seemed to take great pride in being able to locate the patterns. (If you haven’t closely analyzed the first sheet: Jane, Peter and Sam each have unique misunderstandings that reoccur in the three problems that they complete.)

I was pretty pleased with the writing prompts, too—which ask kids to think about and propose theories for WHY Jane, Peter, and Sam made these mistakes. (Thanks for the inspiration, Michael Pershan!) So often, when I attempt to get kids analyzing wrong answers, they tell me, “Oh, it was just a careless mistake,” or, “That was a computational error – I need to slow down.” I heard nothing of the sort today, which made me happy happy happy.

Each student will author two blog posts as a result of this activity (alas, each student must choose to neglect either Jane, Peter, or Sam). I’m hoping for good results and good dialogue in the comments.

One improvement I’d like to make, next time, would be to use real student work, in actual student handwriting. If you have other suggestions, I’d love to hear them. I’m hoping to recycle this activity in the future.

PS – Come to think of it, this activity is kind of like an updated, much-improved version of another activity I wrote about here.

]]>

It is very sad, I know.

I want back in – but man, it is so hard to find the time. Also, this break from blogging seems to have revived my intense feelings of fear and intimidation. So many brainy bloggers out there! (mope-mope-mope)

But! I’m going to try again.

I’m going to put forth some updates. Nothing breathtaking, but, you know… I’m feeling good these days. It is nice to be back at work, and I’ve got this feeling that I’m being creative and thoughtful, again. And so, I feel ready to share. (That feeling fizzled a bit in the spring semester, last year – which is part of why I stopped blogging.)

Recently, I did the clock project (which I think came from Coefficients of Determination, originally?) but put a little bit of thought into structuring the activity thoughtfully so my classroom wasn’t mass chaos. (I attempted the clock project last year, too – and things were a bit crazy. So much freedom for the kiddos! Also, so many possible errors when it comes to writing expressions involving the order of operations!) Anyhow, I’d like to write about those improvements here, and also reflect on additional improvements I might make in the future. I dig the project a lot – and am grateful to its inventor!

Also, we did this integers comic strip project that I like pretty well, which some of y’all might be interested in grabbing. I’m telling you here and now that this happened so that maybe I won’t avoid writing about it forever. The project is fun for the kids but could use a bit of sprucing up. Maybe I can grab some ideas from you people?

Integer-nary happened – but this year’s version was new and improved (thanks, Timon!) and I felt pretty good about it. Hoping to use this idea / structure in the future, as I liked how it fostered *thinking* and not automaticity. I’m REALLY pleased (okay, proud!) that I did not at all tell my kids the rules for adding and subtracting integers – not once, ever. (Dude. It was hard to resist.) I made them draw a million number lines and positive / negative chips and they groaned and groaned but then figured stuff out on their own. (Some kiddos are still confused – but when I think about it, I know that this would be the case even if I’d given explicit rules. Unfortunately, I very rarely have 100% of my kids totally on board with what I’m teaching.)

I’m also in a graduate program that I’m enjoying a lot, which has inspired some good thinking that I’d love to process and share here.

Also going to re-attempt this grand project wherein each kid has his / her personal math blog. Fingers crossed that I see it through to the finish, this year.

And finally, thanks much to Courtney and Jeff for the kind Liebster Award nominations. I’m flattered to be in such good company! Hoping to someday write a post accepting the nomination and also share the love with some other folks.

]]>

**Incident #1:**

A while back, I read a blog post at *Embrace the Drawing Board* about “Integer-nary,” an activity Timon Piccini designed to help students develop their understanding of integer addition.

In case you haven’t read the post, the activity goes like this: One kid draws draws a slip of paper with an equation on it. That kid illustrates (on a mini whiteboard) the appropriate amount of positive and negative integer chips to represent said equation, being sure to cross out any zero pairs (of one positive chip and one negative chip). Kid shares his whiteboard illustration with a buddy (or buddies). Buddy tries to figure out what equation is being represented, and jots that down on his whiteboard. (It’s Pictionary, but with integers. Awesome, right?)

In Timon’s post on this activity, he explains how he has previously observed that students struggle to draw out mathematical models, due in part to their perception that “memorizing mnemonics is true math.” The benefit of the activity, in Timon’s words, is that “it gets students thinking laterally about mathematics, and translating something concrete into something abstract.”

Naturally, I was excited to incorporate Integer-nary. I made a mental note to reserve some class time for it when my unit on integer operations rolled around.

**Incident #2:**

A week or so before my Integer Operations unit, I read an article for a class about Zoltan Paul Dienes (developer of Base Ten Blocks). (I have no idea who the author of this article is. Sorry!) Most notably, I read about (and was fascinated by) Dienes’ “Multiple Embodiment Principle,” which states that:

Mathematical abstractions occur when students recognize structural similarities shared by several related models… It is not enough for students to work with a single model; they must also investigate ‘mappings’ to other models.

And, it occurred to me that I do *not*, in my classroom, embrace the Multiple Embodiment Principle. In the instances when I *do *expose students to multiple models, I have (eek!) told kids that it is okay to stick with the model that works for them. (I’ve noted that some kids prefer number lines, others prefer chips. No biggie, right? Go with what you know—maybe?) I wondered how testing out this Multiple Embodiment stuff might help my kids meaningfully come to grips with the concepts I hope for them to master…

**Ah-Ha!**

And then, I had my “ah ha!” moment—when I thought, “Hey, maybe I can modify Timon’s activity to help my kids think about the mappings that exist between different mathematical models!” It seemed easy enough…

And so, here is the adjustment I made: I required students to draw a number line model (with an accompanying arrow to illustrate the distance ‘jumped’ to the left or to the right of the starting value), rather than simply writing down the equation being represented. My hope was that the process of creating, viewing, and analyzing these two models, concurrently, would spur analysis and investigation of the similarities (and differences) between the models.

After discussing the activity with colleagues, I have recognized that I missed the boat in a lot of ways. In other words, I could have made this activity way more meaningful and worthwhile than I did!

Here are some modifications we thought up, which I’ll definitely try next year:

*What I did**: *Kids began each round by analyzing the numeric equation, first. (For instance, they looked at a slip of paper that said “-5 + 2 = -3.”)

*What I should have done:* At the start of the round, I could have provided slips of paper with number line representations, or even with chips representations! I could have / should have varied the starting point of the activity. This adjustment would have upped the rigor, and inspired kids to more meaningfully make the connections (in multiple directions) between the models.

*What I did:* After the kid who started the round analyzed the equation and then illustrated the chips representation, his partner always wrote out the equation (with numbers)—and *then *provided the number line representation.

*What I should have done:* I think that omitting the ‘write the equation’ step would have forced kids to do more pondering of the models and their similarities.

*What I did: *There was no thoughtful introduction to the activity, nor was there an awesome wrap-up, afterwards.

*What I should have done:* Opening questions—to guide kids’ thinking—might have been valuable. I am also wondering what questions I might have asked after the activity, to help students verbalize the similarities and differences between the models. (I’d love help in brainstorming these questions, as this task overwhelms me a bit.)

What do y’all think? Do you have ways in which you help kids to make these cognitive connections? I want to do more of this! Inspire me, please–as you always do. =)

]]>

The problem is this: My kids kill Expo markers at a staggeringly fast rate.

No, the markers are not drying out. (I’m sure of it.) Rather, the mushy marker tips are getting squashed by my overzealous, oblivious children.

Enter: The Expo Graveyard! (Click either photo for a closer look at some high-quality sixth grade humor.)

The graveyard is not a solution to my problem – not at all – but it has certainly brought a touch of whimsy to my classroom, and to the front of my ugly grey storage cabinet. (All posters created by students, obviously.) Additionally, my kids and I both react a little bit more appropriately, now, when another Expo bites the dust. Rather than shrugging and pretending they are not to blame, kids take ownership and sometimes put on a bit of a show: “Noooooo! Not another death! It’s tragic!” (Okay, maybe that’s not appropriate behavior…) Rather than lecturing my kids AGAIN about how they really must be more careful (ew, lecturing), I ceremonially tape the marker up with the others and put on a mournful face.

Unexpected bonus: Since the establishing of the Expo Graveyard, two students have surprised me with “gift” packages of brand new Expo markers. I suspect that, with this new awareness of the tragic Expocide taking place almost daily in my classroom, they feel compelled to make amends. Like I said – not a solution, but I’ll take it. (I also won’t feel bad about requiring kids to bring in their own markers, once I’ve run out. And – next year, Expos are definitely going on the school supplies list.)

Lest you write off this post as *entirely* silly and not math-related: Expo markers are the perfect addition to Andrew Stadel’s list of inefficiently designed products. They are in dire need of a “well-thought out modification/enhancement” (his words), and thus could provide excellent fodder for some creative problem-solving activity. So there.

]]>

First we analyzed several situations where, when we multiplied a dividend and a divisor by the same power of ten, the quotient remained unchanged. For instance, we looked at this string of simple division problems – 8,000/4000 = 2; 800/400 = 2; 80/40 = 2; 8/4 = 2; .8/.4 = 2. I challenged kids to decipher what was going on, and to articulate their understanding of the pattern. (My kids still freak out a little when they are analyzing and not directly applying an algorithm. Must work on that.)

Next, I passed out this worksheet.

Here’s how it works – even though I bet you can figure it out, yourself. First, kids rewrite the problem so that it is both easier to solve and produces the correct (same) answer. (Their goal is to multiply the divisor by a power of ten – to turn it into a whole number – and then adjust the dividend in the same way.) Next, students use a calculator to check that they’ve altered the problem correctly. (Do the two problems produce the same solution?) The last step – and, really, the least important one – is to put away the calculator and solve the problem manually, ensuring that the answers match up, yet again.

The focus, as you can see, is on **setting up the problem** – which is really where my students have struggled, in the past. The set-up is also where the interesting, more challenging math is taking place. (A lot of my students weirdly enjoy long dividing – and do it almost thoughtlessly, on auto-pilot.)

The real reason I like this worksheet is because it is a preemptive strike against one particular error that I’ve seen far too often. In the past, I’ve noted that many students – in their efforts to change a messy problem into a simpler one – multiply the divisor and the dividend by *different* powers of ten. (So, *2.8467 / 0.3* becomes *28,467 / 3*, rather than *28.467 / 3*. Eek!) Previously, I would commend these students for turning icky decimals into nice pretty whole numbers – and then point out that they’d altered the problem *too much*: it was unrecognizable and had a different (incorrect) answer.

This worksheet helps students to arrive at this discovery on their own. Via either trial and error or consultation with another student, students can revise the problem until the old and the new produce the same solution. Additionally, if kids are having a hard time with the actual long division, there is plenty of time for me to circulate and help kids, one-on-one.

BTW: This worksheet was much more successful when I did a few of the problems with the class, before setting kids free. (There are many steps, and if kids don’t get the purpose of those steps, the worksheet feels endless.) Additionally, when demonstrating how to do the worksheet, I deliberately accepted (invited?) incorrect student answers, so that I could model the process of self-correction that I wanted my kids to adopt.

And, of course, if you have any thoughts on how to make this activity better still, please share!

]]>

Why the change of heart? Happily (unsurprisingly?), after pinpointing and itemizing my gripes about this unit, I feel much more optimistic and ready to make a change. Now that I’ve moved from moping to actually determining *what* in particular was not working—a necessary step that I’d pathetically avoided, previously—it’s time to get to work. (And… *Yay, blogging!*)

Anyhow, my first attempt to be less lame was inspired by @misscalcul8, who has written a lot lately about her aspirations to *be less talkative*. (She does not need to work on reducing lameness, like I do.) In addition to providing a million examples of how she’s doing this, Elissa writes that, “A good teacher creates opportunities to learn but doesn’t necessarily lead them.” I read this, and I thought, *YES!!!!* And so, Elissa was in the corner of my brain as I attempted to find a better way to teach, reteach, and review (simultaneously!) decimal addition and subtraction.

I came up with a totally-not-attractive, handwritten worksheet. (I’m seriously ashamed that it is so sloppy. There was, alas, no time to prettify. It served its function – and that’s what matters most!) I passed it out, and explained to kids that I had culled the errors (in the left-hand column) from previous students. (That was a white lie.) I then told them it was their task to analyze the mistakes and write a *super*-helpful description to the confused student who made those mistakes. That confused student was counting on them!

Here’s one student’s work. (Hopefully you can tell the difference between kid handwriting and my handwriting…)

The idea of helping a struggling student was quite motivating for my kids, and gave my classroom a certain spirit that honestly surprised me. Here’s what I mean: Kids were not just analyzing errors, but rather exhibiting their wisdom and generosity – and taking pride in doing so. They made efforts to present their ideas and their work more thoughtfully, thoroughly, and neatly than usual. (Until I saw the awesome work that kids were producing, it had not occurred to me what perfect training this activity provides for my error analysis sheet, which is a compulsory step for students who want to reassess in my class. It’s almost like I planned it! Except, no, I’m not so clever.)

There was also (yep) a handful of kids who behaved as though the mystery students’ errors were CRAZY, implausible, and beneath them. That, too, was actually sort of pleasant and amusing to watch. (I probably should have discouraged this sort of response. The kids responsible were being so goofy and having such a blast that I just shrugged and let them keep working.)

Oh! I also explained to my kids that, in each group of four, I wanted to see all students on the same problem, at all times. I distributed mini whiteboards, so kids could teach one another improvisatory lessons, on the spot, in case anyone in their group could not locate the error or figure out the correct solution. (Speaking of which: I’ve been doing SO MUCH MORE group work this year, that I’m thinking it might be wise for me to nab Amy Gruen’s lovely idea, on helping students understand what it means to be a good coach.) I saw a lot of interactions that I liked, but some kids could to stand be way more supportive and helpful to their table-mates. I want to encourage that.

Anyhow – in case you hadn’t drawn these conclusions on your own, I liked this activity a lot! Here’s the rundown of how it met my students’ needs.

- Kids who might have simply gone through the motions, in the past, were forced to write a little bit about why various algorithms work.
- Kids who would have likely fallen into the usual traps were (a) spared the trouble (someone already made that mistake! No need to do it again!), and (b) challenged to articulate why various errors are tempting, but must be avoided.
- Kids who would likely have been utterly lost received mini-lessons from peers. (Also, I could spot those kiddos and work with them fairly easily, given that I was NOT in front of the class blah-blah-blahing. Oh – in case it wasn’t clear: There was
*no*blah-blah-blahing. I just let kids dive in.) - Kids received training and practice in analyzing errors, which is an all-around useful skill – one that I’ve always wished they knew how to do more effectively!

If I have a bit of time to make a less ugly version of this sheet that you can use, I promise I will update this post and add it.

]]>

And so, I want to respond to a prompt that I was a bit nervous about addressing when it was offered up a few weeks back. Today, I want to talk about my *least* favorite unit to teach – my unit on decimal operations.

Although I’m starting to think up some solutions to the problems that I itemize, below, I’ve definitely got a long way to go. I would *love* to know if any other folks have struggled with this unit (or another unit) in the same ways that I am struggling.

Without further ado, here are the reasons why I am less-than-enthralled about teaching sixth graders decimal operation:

- Kids always perform at a variety of ability levels – of course. That we’re not ‘all on the same page’ feels true in an especially pronounced way during this unit, and I’m pretty sure I’m not just imagining it. For instance – some kids solidly mastered decimal operations in fifth grade. Others need to sort out minor confusion, due to lack of practice over the summer. A few kiddos never had a clue.
- Moments of discovery (you know, the moments when class is totally fun and exciting and you love your job the most?) are less frequent, largely because this skill is simply
*not*brand new for anyone. - Kids who decidedly
*don’t*fully understand the skill are convinced that they do, because they’ve seen it before. They zone out, rush through their work, and make lots of careless mistakes. - Kids use a whole bunch of different algorithms, and a wide variety of terminology, which they’ve learned from a handful of different feeder schools. (This is fantastic, but it leaves me feeling stumped – a bit paralyzed, really – when it comes to leading whole-class discussions and working example problems.)
- Algorithms are at the heart of decimal operations. Yes, conceptual understanding is crucial (duh), but let’s face it: When kids are successful at adding, subtracting, multiplying and dividing decimals, they are
*meticulously*following a whole bunch of rules.

In the past, I’ve ended up teaching boring lessons on one or two algorithms, peppering my explanations with lame disclaimers. (For instance: “Yes, you’ve learned this before—we’re just keeping it fresh,” and, “I know you don’t do it this way; that’s just fine. See if you can learn a second method.”) I am not at all happy with this approach. It sucks, basically.

I need to either think up some terrifically amazing way of differentiating (ensuring that all students are suitably challenged), *or* develop some brilliant method of having kids share their knowledge and coach their peers through the process of mastering these skills. Does anyone know of any structures for making this happen? (It’s not just collaboration that I’d want to happen, but rather, one kid would actually teach the skill to another kid.)

*Seriously. What do you do in situations such as these!?*

PS: Worth mentioning: I really struggled to nail down what I wanted to say for each of those bullet points, up there. I felt a sheepishly embarrassed to recognize (mid blog-post, too – eek!) that I’ve clearly spent an insufficient amount of time scrutinizing this situation. Ah well. I wrote the blog post, so up it goes! (I suppose there’s a lesson, there, too…)

PPS: I *was* going to share one recently attempted, halfway decent method of overcoming these struggles, but I think I’ll wait on that and see if y’all have anything better for me. (You’re just DYING of suspense, I bet. Sorry!) Also – I’m still in the middle of this unit; it’s not too late for me to change things up.

]]>

I was elated about this problem for a multitude of reasons. I knew that it would: (1) serve as the topic of my third New Blogger Initiation Post, (2) extend and up the rigor of my Week #1 whiteboarding activity, and (3) welcome kids back to school with a bang, after their unexpected week-long hurrication. (During this electronics/internet-free break from school, I had the pleasure of reading Driscoll’s book, in its entirety. I highly recommend it as a source of challenging, worthwhile problems; it also has gotten me thinking about how I might help smooth my students’ transition from arithmetic to algebra.)

Anyhow, take a look at the problem, first, which I slightly revised from its original form to increase the clarity for my little people. See if you’re as excited as I was, okay?

Eight adults and two children need to cross a river. A small boat is available that can hold one adult or two children. (In other words, there are three possibilities: 1 adult in the boat; 1 child in the boat; or 2 children in the boat.) Everyone can row the boat. How many one-way trips does it take for all of them to cross the river?

Bonus: Can you describe how to work it out for 2 children and

anynumber of adults? How does your rule work if you have 100 adults?Finished early? Think about how to visually represent your solution.

In case you’re not in the mood to play around with this puzzle, here is one version of the solution. (Or, way more fun option: Scroll down to check out my kids’ solutions! The level of clarity varies from whiteboard to whiteboard, but all boards contain accurate information.)

And now that you’re familiar with this lovely little problem and its answer, I’m sure you’re *dying* to hear about how my kids attacked it.

First: I gave everyone a solid five minutes to make sense of the problem, by reading, re-reading, and note-taking. The room was silent, and most kids (judging by facial expressions) were pretty perplexed.

Next: I let students join heads for a few minutes. What I noticed, here, was that only a very small handful of kids had a clear idea about how to begin breaking down and solving the problem. Those kids – the ones who “got it” – helped their peers wrangle their way through the facts and make sense of the information provided.

Now, the fun part! After frustration had just begun to swell and some kids were feeling a little stumped, I gave one envelope to each table – each of which contained the pieces that you see, here.

I didn’t need to explain what the pieces were. (In fact, two students were working on their own concrete, movable models before I distributed my envelopes: so great!) As soon as envelopes were opened, students knew exactly what to do. They were no longer stumped, but eager to use the simple, multi-colored, cut-out shapes to *tell the story *of the river-crossing.

After ten minutes of playing with these low-tech, low-glam models, all of my students noticed that for each adult, the boat made four trips across the river. (“Why are you making these poor kids do all the work!?” one observant student inquired. “This is child abuse! The adults are hardly doing anything!”) Alternatively: Students independently discovered that they needed to seek out structure within this problem, and then they did just that – without my saying a word.

Side note: This problem is so beautiful because it is nearly impossible for kids to find the solution without noticing that certain events are repeated over and over and over within their narrative. That there are *eight* adults (rather than, say, three or four) forces even the most ridiculously stubborn of kids to pause and think, “Huh, haven’t we repeated those steps a whole bunch of times?” (I’m talking about those students intent on plodding through, step by step by step, and steering clear of anything remotely resembling pattern or structure or a shortcut. You know the type, eh?)

Next, I gave students time to plan their visual/pictorial models. After plenty of lively discussion, here’s what they came up with. (Click on any of the collages, below to take a closer look – or zoom in much, much more by visiting the collection of photos, here.)

Many students used some variation of the models that were the clearest and most compelling way back when we solved the *first* river-crossing puzzle. In other words, they opted to recycle previous work and ideas, rather than considering the unique variables facing them in the present moment. Alas: These groups’ models (though lovely and correct) did not quite capture how the solution involved cycling through the same four steps eight times in a row.

The other groups—the ones that developed novel, often circular models—ended up with pictorial representations that better (IMHO) captured the essence of this problem and its solution. Regardless, I’m pleased by the variety of approaches and the abundance of creativity – and also at the notion that my kids are developing more nuanced ideas about clear mathematical communication. (Students did, of course, have a chance to look at and critique one another’s boards, at the end of the activity.)

The simplest ‘model’ of the solution, which we explored at the end of class, was of course an algebraic equation: T = 4a + 1, where T = total trips and a = the number of adults. That my kids got to develop and make use of an algebraic equation was just a little added bonus – icing on the cake! – although I can imagine using this problem as a jumping-off point for all sorts of fun algebra-related analysis. (Driscoll also suggests altering the number of children to really make things interesting.)

]]>

During my first year teaching, I taught my kids to find the greatest common factor and least common multiple by making these long, awful lists of factors and multiples. (Uh huh: My approach was somewhat like this. Yes, that’s a link to Khan Academy.) That was the only approach that my kids learned from me. No alternative methods or shortcuts – that was it. (Man, I am outing myself, tonight!)

Now, it is not news to you that prime factorization is an often more efficient path to finding both GCF and LCM. I’m still, however, totally psyched about it (several years after learning this), so, forgive me if I wax poetic for a bit – and share with you some of the reasons why I dig teaching kids to find GCF and LCM with prime factorization! (Also: Via a very informal analysis, I’ve gathered that this stuff gets glossed over, often – so this is not purely an expression of my love. I’m hoping maybe some other folks will get on board, too?)

- It is an approach that is highly appealing to visual learners. (Let’s face it: Factor trees are pretty!)
- It is an alternative – another option! – preferred by many to the drudgery of listing. (It also offers me the chance to reiterate this big idea that any number of paths can lead to a good solution.)
- It builds a solid understanding of our number system, and of prime numbers as building blocks.
- It is not
*always*better or more efficient. (I like this, because it encourages my kids to assess the problem and consider, “which is the best course of action?” before jumping in.) - It requires a much deeper, more nuanced understanding of both numbers (in general), and GCF and LCM (specifically).
- It is way more fun and interesting than listing, listing, listing.
- When kids get comfortable finding GCF and LCM with prime factorization, they also become pros at reducing fractions using prime factorization. (That’s another thing I never considered teaching, back when I knew even less than I know now.)

Okay, okay. I’ll stop. But before I go, I want to share one more thing! Take a look, first, then I’ll describe, okay? (Click to enlarge.)

Last year, we were working on GCF and LCM when I broke out the big white boards for the *very* first time. (I didn’t have a blog, so I didn’t ask kids to refrain from writing their names; you’ll have to pardon the big black blotches.) Each group was to find the GCF and LCM of a different pair of large-ish numbers, and share their solution (I’ve highlighted them in yellow) with the class.

Here’s the fascinating thing: The boards you see above all feature the same pair of numbers (252 and 120) – but only one board offers an entirely correct solution. (Hint: It’s not the super-organized, hyper-neat whiteboard. Sorry, my darlings.) Although all of the errors are fairly minor – no need to sound the alarms – it is fun to explore the teeny, tiny details (and omissions) that tripped up my kids. Also: What could I do to prevent this type of student error in the future?

PS: On these boards, no one made the common mistake of multiplying ALL of the prime factors together, to find LCM. I have definitely seen that one, before, and the result is a freaking ginormous solution.

PPS: I’m thinking that this could be a great skill to use for this Mistake Game that I keep hearing so much about. Still wary about trying that one out with my still-very-young crowd.

]]>