Number Strings @pwharris @Kara Imm

Last week, I took a week off from doing any number talks/number strings. I had a visit from the superintendent to work in, and I felt like I was juggling too many other new things. One of my coworkers seems to be able to do number strings intuitively with her kids, based on questions that come up in the moment. For example, when she introduced her students to probability, they asked about a roulette wheel, and so she asked them to consider the chances of getting 00 (which is 1/38). To estimate that percent, she had them first do 1/4 >> 1/40 and notice a relationship, then 1/2 to 1/20, 1/3 to 1/30, and a few others before she had them estimate the 1/38 and then check via a calculator. I was impressed, but I don’t think I’m ready to do it without planning it yet.

I’m trying to figure out how to incorporate some number strings this week for my students. I’m struggling in some ways because I want to be sure that I’m addressing the skills that the kids need in my current unit, but also not giving them strings that require them to make use of strategies from strings we haven’t looked at yet…

In 6th grade, we’ve “finished” our addition and subtraction of integers and we’re in the midst of multiplying and dividing with integers. We’ve done multiplication so far and we’re about to introduce division this week. I’m wondering if there’s a multiplication string that would be appropriate for me to use with them. And I’m also wondering if there’s a problem if I introduce some of the strings with negative factors! Or if I should keep the focus of the multiplication strings on the products themselves, and leave the signs to the day’s lesson.

In 7th grade, we’re working on probability, and there are a few things that have come up that I think could be good for number string work. First, I noticed that while my students have fluency with converting some fractions into percents (more than most kids – not only do they know their 1/4s and 1/2s, they also know their 1/3s, and 1/9s and even their 1/8s!), but the students aren’t fluency with multiple eighths, nor are they fluent with some of the “odder” fractions into decimals/percents yet. I’m wondering if there’s a way to get the students to be able to calculate the percents of 3/8, 5/8 and 7/8 easily (i.e. noticing that 1/8 + 1/4 = 3/8, or that 5/8 = 1/2 + 1/8 and leveraging those skills somehow – or multiplying 12.5 x 3 or x 5).

Second, tomorrow, we’re going to begin proportional reasoning with percents, so I’m wondering if I should do a string where we find “friendly” percents of a number to build a more complex one.

Third, we’ve had to multiply fractions by each other (but mostly “easy” ones), so I don’t know if this is worthy of a whole string right now.

I’m still struggling not only with how to implement these strings, but also how to design and decide on the best string for a given lesson. I know there’s been some back-and-forth about whether the strings should relate to the day’s content or not, and while I’m okay with it not relating directly to that day’s content, I do feel it necessary to relate it to the general topic of the week/unit.

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Visibly Random Groupings

I am a total convert! I read the research this summer and I wasn’t quite convinced, but I rolled it out yesterday and it’s gone fabulously so far! I don’t think I’ve had any kids try to switch cards or complain about their seats. 

I did have four students wind up sitting together two days in a row, but that CAN happen (wow! What are the chances?! 31 cards, 31 students, and two days in a row, 4 of them got the same number as each other! They told me about it, which is why I believe it really happened randomly).

I do think the first month, in September, of sitting alphabetically with name tents so I can learn all of their names FIRST is vital;  I don’t think I could’ve done it otherwise! Calling on students has tested my memory today and yesterday, but the more they’re in different seats, the better I’ll get, I’m sure. 

I think the way I rolled it out has to do with its success so far. At the beginning of the year when we made our name tents, I told the students that one of my goals for us was to be a community of learners together and that I wanted everyone to learn each other’s names. In my sixth grade classes, I did a poll and no one knew more than 5 other kids; in 7 th grade, there was obviously a little bit more familiarity, but no one knew more than 2/3 of the people. Plus, two of my classes, one 6th and one 7th, are “split classes” – kids from different homerooms go to math with me together, but there’s no other class where they have exactly that configuration of people. 

Anyway, I rolled it out yesterday with every class (except my homeroom who I prepped it with on Friday last week). I started by handing the random cards to each student when they walked in and said “find the desk with this card on it” – and the desks are systematically arranged in groups of four and then numbered A-8 (with the four suits in the same relative position). 

Once everyone was seated and starting the warm up/copying down their HW), I paused them and pointed out that we had a new seating arrangement and asked them how I assigned it. We established that it was random, and they saw the cards were shuffled. I told them I would give them a new card and a new seat the next day and the next day and next week. I asked them why they thought I was giving them so many new seats and THEY remembered my goal of learning everyone’s names and working together. I told the kids this would ensure they have lots of opportunities to work with everyone in their classes. I reinforced that their seat would be new each day, though sometimes they might work with the same people or at the same table, that was ok because their seats were only for a day. 

I then give each student an “A#” – an alphabetical number so checking HW will be easier (since we can find them easily on the alphabetical list when they tell us their numbers). 

In my classes, I’ve already noticed some changes: students are talking in partnerships more because it’s a new person each day and some of that is even translating to sharing in the whole class because they feel more confident as they get to know more kids personally. 

Now, tomorrow, my two 7th grade classes will have to sit in the same seats because they were in the midst of debriefing an experiment that they designed in their groups today, and they can’t talk about those experiments with other groups in the same way! But next week, we’ll be back to new seats! 

I can’t wait to spread this practice to everyone! Try it out! 

#NoticeWonder in my classroom

So it was at least three years ago that I first saw the video of Annie Fetter’s Ignite talk about the instructional routine, “What do you notice?  What do you wonder?” (I might have even heard about this routine at the summer institute I did with Lucy West’s company, Metamorphosis). I think it was mid year that I began incorporating it into my class, and I remember being pleasantly surprised at how I heard from students who had been silent in class for much of the year. 

Last year, I began the school year with it, using it so much with my co-teachers that Katie and Kathleen would often tease me about it (though we started using it also in our grown up teacher tasks: what do you notice about these student test scores?). This year, I’ve also been using it right from the get go (in fact, it’s like the second thing in my kids’ notebooks and there are at least three or four N/W charts). I often do it in three stages: draft a list independently, share one or two with a partner and add to your list, share out as a whole class and expand or confirm your list with plus signs for new ideas and checks for ideas you also had. 

Last year, I started to run into a problem with my students that arose yesterday (when I began this post!). My students do “ok” (at worst, sometimes much better) with the noticing, but their wonderings are much less specific and mathematical. Sometimes, I think it might be because I’ve chosen to have them notice/wonder about something that’s insufficiently substantial. Other times, I worry it’s because I haven’t done enough modeling or reflecting on our noticings and wondering about what makes a good one. 
Yesterday, I gave my 7th graders a “probability table” to wonder about. It showed the chances of Nina getting a phone call at her work at an office. 

They noticed things like the probability was written in fractions, they summed to 1, the denominators were multiples of 3, and that she typically got fewer phone calls (they didn’t state that they noticed she got between 0 and 4 phone calls).

They wondered who was calling her, why she had so few calls, where she worked, did she have a good phone plan/what was her rate, Samsung or apple, how long were the calls, what her phone number was. Only the question about how long the phone calls were seemed to me that they could be potentially relayed to the probability, but in the moment, I didn’t make any connections. I just recorded all of their observations and questions. 

In class, after we did this activity, I had my kids answer a few questions about some probabilities and they seemed to do pretty well: but I didn’t make any explicit link between the intro activity we did and how that helped them answer the questions (like P (0 calls) = or what was the number she was most likely to get). 

I’m wondering now how to improve upon my use of notice/wonder. I went to an MfA PD tonight about the instructional routine, Contemplate then Calculate that uses the notice part without the wonder in the beginning of the task and it made me wonder if sometimes, we should just use the notice part. 

The other thing I was thinking about is how awesome it is when we’re able to leverage the wonder part to become the questions we investigate, so it seems like we’re delving into what the kids were asking about (but secretly, it was anticipated and aligns with the objective of the day).

I was also reflecting on the way I began tweaking the routine a bit last year at the end to say, “what could you wonder mathematically about this? ” or what questions could this answer or what mathematical questions could we all about this? I’m wondering now if I should try those again this year. 

The other thing (really big picture for a moment) that I realized at tonight’s PD is that there are apparently LOTS of instructional routines to use in class to teach better: WHY didn’t we learn about any of them when I was in grad school? Did other people? If not, there’s something broken with how we’re training teachers (which I already believe, based on my experience in Teachers College).

Anyway, I guess my closing ideas are this:

How do others use Notice/Wonder?

Does the item you choose affect how well students are able to do this? 

How do you model it and teach kids the routine? 

How do you help students get better and improve at the routine? 

Something New for Next Week (Visibly Random Groups)

So I wanted to begin the year with random groups of students, but I realized very quickly that learning names, handing out name tents, and taking attendance would be quite difficult if I had students in a different seat every day. So I began the year with something that’s “somewhat” random – alphabetical by last name. Students certainly weren’t going to question if I had placed the “smart” kids in one section and the “dumb” in another section or anything.

However, now we’ve been in those same seats for almost four weeks, and I think the kids are beginning to get to know each other and I’m beginning to get to know them. In my homeroom, I know ALL of their names and faces (because I ALSO have a 15 minute homeroom period where we’ve done A LOT of ice breaker/name games), and in my other classes, I know about 85 – 90% of the kids.

I’d like to start trying out visibly random groups NEXT WEEK, where I plan to tape a playing card to each desk (A – 8 to represent groups 1 – 8, and the four suits to represent the four positions within each group), and then to hand out a second set of cards to students to assign them their seats for the day. I would collect the cards from them at some point (though I’m not exactly sure when), and I would make it clear to them that the seats would only be for the day.

The major logistical thing I’m CONCERNED about is the way in which I check my HW right now. I have students have it out on their desk and I go around, seating chart in hand, and I write their grade next to their name. I can see three ways of coping with checking HW with the random seats:

  1. Collect HW & check it off after class – NO WAY! I wouldn’t be able to keep up with this. I know from previous experience this doesn’t work for me. I will spot check random HW assignments, but I can’t do this daily!
  2. Have a roster with the students names in alphabetical order and look for them to check off their homework. If I do this, then I will definitely assign them “A#s” (alphabetical numbers) which I’ve done in the past but usually at the end of September, so there are no more changes to my roster.
  3. Have a blank seating chart EVERY DAY, and write the students into the seats they’re in and then check off their HW… This seems like a LOT of extra work, even though it would help me know who’s sitting in which seat.

The other concerns I have about this are if my few “squirrely” students wind up sitting together and being off-task.

Another concern I have about this is that I really want to be teaching using group problem solving tasks (so there’s a reason they move around and work with their group mates), but my current school uses a lot of resources from Engage NY, which don’t seem to be very problem-solving-based, so I’m not sure how to implement the necessity of working together.

I’m also deciding about whether or not I want to assign group roles (perhaps based on seat position within the group – facilitator/navigator, resource manager, recorder and reporter are four that I’ve seen before that I liked).

I’m curious about other people’s experiences rolling out the Visibly Random Groupings with their classes, and what has helped make it more successful.

Student Participation & Engagement

Over my years of teaching, I’ve noticed something that many studies have commented on: in many classrooms, there are only a few students who regularly raise their hands to ask and answer questions, while many more are silent observers.

One solution that is often proffered is the idea of cold calling or randomly calling students instead of taking hands. In this way, the idea is that students will be “ready” at any time to share out their ideas. However, I’ve noticed that this causes resentment and anger sometimes on the part of students, as they don’t like being caught off-guard or being called on for an answer they didn’t feel prepared for or sure of.

Another strategy often suggested is giving them wait time or think time or sometimes even writing time or sharing in pairs time. However, I’ve noticed that this often only increases the number of hands marginally, but students who NEVER participate rarely are encouraged to do so by this method.

So this year, I’m deciding to try something new. I let the first few weeks go by like “normal,” observing which students raise their hands regularly and which students are more quiet. Taking note of which classes have more or fewer active participants and if there’s any correlation with the time of day I see them (i.e. more loud, off-task conversations after lunch vs. more heads down and quiet first period).

Now I’ve got several mental lists of students: students who I can “count on” to raise their hands daily (usually multiple times per day), students who will raise their hand when prompted for “students we haven’t heard from lately,” students who only share short answers or only share rarely, and students have not shared at all. Now I’m considering what to do to target those students and encourage them to participate in the verbal conversation of class.

My two ideas that I want to try this year are the following:

During turn-and-talks, hovering near those students and listening for them to share something that I could ask them to share with the whole class OR during work-time on problem solving or answering questions, observing what they’re thinking about, and asking them to share during whole class.

Writing a little post-it note to them telling them I’d like to hear their voice more in class because I think they have valuable things to contribute and either asking them to pick once this week they’re going to raise their hand and I’m going to call on them OR to write me a note about why they haven’t raised their hand in math class at all yet.

I’m also in the process of creating an extensive google form survey to ask my students to reflect on their class participation and how they either feel they do or don’t successfully participate on a daily basis. And in my survey, one thing I was especially proud of doing, was ensuring that when I wrote down the ways you participate, I included both talking in the whole group as well as in partnerships, asking questions, taking notes, listening actively, and solving the problems in their notes. So that allows students to consider a variety of ways they can successfully participate in class, and avoids overvaluing the contribution of the extroverted confident students who raise their hands a lot.

Writing my name tent messages with my students for a week gave me an insight into their minds, and I’m already referring back to our messages inside for ideas of ways to engage them in class – I’m even considering putting the post-it note messages INSIDE for them to discover! (As an aside, I haven’t decided what to do with the name tents yet. So far, I’ve been collecting them every day and redistributing them to the students – and by collecting, I mean my student teacher has been doing this, and some of the students have stopped setting up their names tents, in part, I assume, because many of them feel like everyone already knows their name by now – even though that may or may not be true. I’m wondering how long I continue using the name tents, if they go all year or what… A few of my students also asked me if we were going to get NEW name tents to continue writing messages to each other, and I told them no, but I was thinking about having us do it AGAIN in January (or perhaps sooner, if it seems necessary).

Number Talks (@PWHarris & @SaraVanDerWerf)

When I first heard about number talks, I was SUPER excited about them. I learned about them from a great PD with Kara Imm at MfA, and I was blown away by some of the more efficient strategies that they were able to introduce me to. I’ve always used SOME patterns and “shortcuts” but I’ve never used all of them before. Over the summer, I finally read the three books by Pamela Weber Harris, Building Powerful Numeracy, and I was totally on board. I even bought Making number talks matter, and read that whole book, which suggested beginning with the dot patterns and then moving into arithmetic. I read blogs about it, including the numberstrings.com blog and Sara Van Der Werf’s blog on it. I was thrilled when at the NYC mini-TMC, I was able to attend a session about number talks vs. number strings, and to participate in a few.

Now, I was told I should aim to do a number talk AT LEAST 3 – 4 times per week, and during the first two weeks, I should try to do one every day. I’m only in the fourth(ish) week of school, and I’m finding myself struggling with both of these already. First, 43 minute periods make it next to impossible to schedule a consistent routine daily, especially one that takes 10 – 15 minutes to do well – especially since I can’t currently envision how to do it as a “Warm-up” (unless I have photocopies of the handouts from Pam’s book and I use those as the independent warm-up and we discuss the strategies afterward – but I wanted students to have a bit of experience with them before working on the handouts).

The second thing (and this is one that I find hard to admit, which is why I’m pushing myself to make it public) – I have a lot of difficulty when I try to train myself in a new routine or activity, and I often need a lot of practice before I get good at leading it. For example, the first few times I led a number talk, I forgot to ask students what they thought the answer was, and I jumped right into the HOW or WHY they thought about the problem. I also found that for some of the student explanations, I can’t visualize how to represent their method on an open number line (we’re doing subtraction): for example, how do you represent “I switched 3 – 8 to 8-3, but then I made it negative because it’s the opposite.”

Another thing I’m struggling with is actually making it through more than two problems in one ten minute chunk OR keeping the kids’ stamina up to solve more than two problems in a string. We finally did a string the other day that had 6 problems in it, and by the last two, I saw many of my students’ body language (in 6th grade) indicating to me that their attention was spent, and they weren’t participating fully.

When I think about this longer string, I noticed in my students’ body language that they weren’t all listening to each other share out (as we only shared one strategy per problem), and there didn’t seem to be any accountability to paying attention or participating past putting your thumb up at the appropriate time so we could discuss. I don’t like to cold call students, especially in the beginning of the year when we’re still building relationships and getting comfortable with each other, but I notice I’m often hearing from the same 5 students, even with wait time (especially for explanations of their strategies or how they thought about the problems – some of the “quieter” kids are more willing to share their numerical answers to the problems, but only if they feel confident in their accuracy, it seems).

One other thing on my mind: I wish there was a way to gather my students in a circle or move my desks out of the way, because I feel like the students are TOO far away from my when we’re doing the number talks. I want them CLOSER to me so no one can “hide” because their desks. I’ve made our routine very specific to include pencils are down and knees are rotated to face the board and I wait, but the last two times I’ve done it, some students have been reticent to put their pencil down and focus OR to turn their bodies and I’m not 100% sure why.

So let’s talk about my sequence of introduction to number talks:

I began on the second week of school (first week only had two days with kids!). I introduced them to it with the quick images dot talk, and we did 6 dots and 10 dots and 11 dots on three of the four days that week. I ran out of time the fourth day. I did those with ALL of my classes, both 6th and 7th grade, but I noticed the 6th graders had more ways of seeing the diagrams than my seventh graders did.

In the third week of school (my first full week: last week!), I began my unit on integers with my 6th graders, and I began doing subtraction number talks with them. I originally planned to use those same number talks with my 7th graders, but I ran out of time the first day, and then every day after, I didn’t quite see how subtraction strings were close enough related to the current unit of study (probability and statistics), so I was unsure about taking 1/3 of class to discuss mental math skills that aren’t directly relevant to the work we’re doing RIGHT NOW. I considered doing different number talks with them, but I wasn’t sure what – I feel like fractions/decimals/percents would be most useful to them right now in their current unit, but I’m not sure exactly how to write strings for that. One thing we’ve done a bunch of so far in class is talk about the eighths as percents and the sixteenths as well. I’m wondering if there are strings I could design that would help them think about those percents more easily; I know the books don’t recommend starting there, but with my 7th graders (who seem so disaffected, honestly!), I’m worried about subtraction being seen as “too easy” by them.

At least in sixth grade, even though we’re thinking about subtraction (and we often all have only one answer), it appears much more related to our current unit of study. We looked at the comparison of the difference and removal ideas about subtraction (although I noticed that because of the types of problems I’ve given students from Kent Haine’s balloon and sandbags game, most of them are using only the removal strategy – I just thought of how I need to tweak it to encourage them to use the distance idea: “How far did the balloon move if it started at # and wound up at #?” I think I’d love to try that in a number talk on Thursday (after our quiz tomorrow on subtracting with negatives, but before we move into multiplying with negatives!). We also did a few problems from Pam Harris’ integer strings, and I’d like to incorporate more of those, because that’s where I saw some real sense-making beginning to happen, as they began to use the previous number statement to reason about the sign of the new one or to think about whether the balloon went up or down relative to the first number (i.e. doing 3 – 8 and then doing -3 – 8). During this third week, we did a number talk three of the five days of school.

This week, we’ve done NO number talks so far. I felt a bit rushed in trying to get my kids ready for their quizzes and keeping to a department timeline of when we should be moving into the next topic, so I decided to skip them (not a decision I’m necessarily proud of or happy about, but there it is). I know I don’t have time tomorrow to do it, but I want to be sure to try to do some on Thursday and Friday.

Anyway, I’m feeling a bit lost with my number talks right now. In 6th grade, I think I just need to get in the habit of making three days of the week “number talk day” and NO MATTER WHAT, we end with number talks that day, using the last ten to fifteen minutes (OR we begin with them??). In 7th grade, I want to think about fractions, decimals, and percents – I’ve noticed that MOST of the students are pretty comfortable with the basics, so I don’t think I need to go back to basics, but I don’t know that they have familiarity with any types of representations of their thinking – I think most of it has just been mental math done quickly and without analysis/thinking about how they know.

I find it a bit scary: researching a new teaching technique, reading all about it, planning for it, and then when you try it out, it doesn’t go exactly the way you wanted it to, and you’re not sure how to modify or fix it. My colleagues at my current school are NOT using number talks: at least not formally. Two of my colleagues don’t use them at all, but one of my colleagues instinctively builds them in as issues arise in class. For example, the other day, she told me her class was discussing the probability of getting 00 in roulette (based on a student question), and they were looking at the fraction 1/38 and considering how to turn it into a percent. She led them through a set of problems that she came up with on the spot to help them see how if they knew 1/4, they could easily figure out 1/40 or if they knew 1/3, they could easily figure out 1/30. I think that type of problem (relationships between fractions) might be a good place to begin with my seventh graders.

I feel like I’m not ready to completely give up on number talks for a few reasons, even though I feel like I’m having a rocky start to them:

  1. I think that the number sense and flexibility my students will gain from being exposed to different strategies and discussing them will be infinitely valuable to them.
  2. I think that different students may participate in number talks than in the “rest” of class, and I want to keep this activity that allows different groups of students to share out.
  3. I think that these strings can highlight strategies that the students may never have thought of before and I want to give them as many tools as possible to create versatile problem solvers.
  4. I think that the number strings will help them become fluent in the language of math calculations, if we do them enough times.

Anyway, I’m hoping to begin a conversation with other people who are trying or who have tried number strings/number talks, because I’d like advice or feedback about where to go next or how to troubleshoot the problems I feel like I’m running into so far.

As an aside: I’ve taken a bunch of pictures of my board after doing my number talks, and if I can figure out how to sync my photos on my phone with my blog, I will! Maybe this weekend….

Integers from @KentHaines

On Monday, September 26th, I rolled out my introduction to integers with my 6th grade classes. I’m making heavy use of the resources that Kent Haines has shared on his google drive, here: https://drive.google.com/drive/u/0/folders/0B2-Wy7iXL83zU0NSU3d1LVNzc0E

I’m going to give an overview of what I did each day of my week, and then delve more deeply into my reflection on it.

Monday, I had a double with one class and a single with the other. I began by doing an entry slip on a post-it note, where I asked the students to write down everything they thought they knew about integers/negative numbers so far, and then to rate how well they felt they knew it in three categories (strangers, say hi in the hallway, and besties). Most kids were in say hi in the hallway, with some in both of the other two categories. I read over the responses after class, and I found it interesting which details certain students chose to include: some students just said they knew they were numbers below 0, while others said they knew how to do operations (sometimes even giving specific rules they remembered).

After they placed their post-it notes (in my double, we briefly discussed what patterns they noticed and what surprised them about where the post-its were), we switched gears. My double spent the first period finishing up our discussion of the growing pattern from WIM (@Jo Boaler’s WIM year 1, day 5), and we put their work under the document camera and came to the “rule” or shortcut that they could add one to the stage number and square it to get the total squares. We introduced some new routines (giving a student two snaps when they come to the board and restating what other students said by sharing with our partners what we heard, and asking the presenter questions). In my other class, we had managed to do that the previous week (since they have their double on Thursday, not Mondays), so by the end of the first period, both classes were now in the same place.

The next thing we did was a number talk. The previous week, I had done at least two dot pictures with each class (my 7th graders also! 3 pictures in two of my classes that had doubles on Thursdays), and so they were used to the norms of showing their thumbs while they think. I asked them to think about two subtraction problems this time. I was originally going to do a number string from Pamela Weber Harris’s book to introduce the idea that subtraction could be thought of as distance between two numbers or as removal, but as they started sharing out their strategies, they had used such a variety of strategies and were so eager to share all of them, that I decided to focus just on those first two problems and elicit as many ideas as possible, thinking about this initial subtraction number talk as more of a formative assessment tool, to see how students were currently thinking about subtraction.

I showed them a picture of a hot air balloon and asked them what they knew about them. We pulled out some important information – in one class, we discussed the idea that helium makes some balloons go up, and we also discussed the idea that hot air rises, so that makes the balloons go up. They also said they knew that sandbags could add weight and make the balloon go down, but releasing them could make the balloon lighter and go up. I introduced the context for the game, and we began discussing the rules for how it would be played. Then, we talked about starting at 0 and how we could change our cargo as long as it didn’t change our height. I showed them two starting cargo loads, and asked them what their heights should be (both 0), and the students told their partners where they thought the balloons should start and why (most came up with the idea that there were an equal number of balloons and sand bags, so it stayed at 0). I showed them two different cards and asked them how we should change our cargo, and we shared out how that would change the balloon heights. In the double, we had more time than the single, so they had an opportunity to play in partnerships for a few minutes, whereas the single didn’t have enough time to play on Monday. I knew we’d need to play the game again in both classes on Tuesday.

Monday’s exit ticket asked students to tell me the cargo for a balloon to be at a height of 3 or 5 (depending on which class). Almost every single student came up with multiple correct loads! I had less than three students in both classes combined who had difficulty with it. Some of them noticed patterns and one of them even articulated that the pattern was as long as there were three more balloons than sandbags, you could load many combinations on there!

Tuesday, we began with an entry slip, analyzing student mistakes. We looked at the handout (also from Kent Haines) about making the height of 3, and we did a thumb vote so all students had a chance to participate and share their ideas about whether the balloons correctly modeled three or not. Everyone did really well on this, and most students seemed to get it completely correct, with good reasoning and answers.

Because I wanted to ensure that all students had enough time to play the balloons and sandbag game, we skipped our number talk I had intended for Tuesday. It takes the kids so long to get settled (just taking out supplies, copying the homework into their planners, putting away tonight’s HW and taking out tonight’s HW takes 5-7 minutes – and then they finally begin on the warm-up, and that can take another 5 minutes of work time!).

I had to pause the first class during the game play to review how they could do EXACTLY what  a card said by changing their cargo without changing their height (i.e. “Adding zero pairs to their balloon”), but in the second class, I anticipated this confusion, so we discussed it BEFORE beginning to play the game in partnerships. I then gave each class about 10-15 minutes of play time before telling them to choose the three worst cards and three best cards with their partner.

We had a debrief at the end of class as to why they thought certain cards were good or bad. I was pleased to see that they saw that removing sandbags and adding balloons were both “good” while adding sandbags and removing balloons were both “bad.” I ended with an exit ticket, where I asked students to place four balloons on the number line based on their cargo. Almost every student in both classes got all four of the balloons in the correct location. The exceptions were a few students who had clearly miscounted how many chips there were (my projection screen sucks, so I wasn’t too surprised).

On Wednesday, we began with a warm-up where students had to brainstorm how to move a balloon that was 3 ft above the cliff to 5 feet above the cliff. We discussed how you could EITHER remove 2 sandbags or add 2 balloons and concluded that it didn’t matter which way you did it, because they were equivalent. On this day, we also began to explicitly match the game to our numbers: balloons represented positive numbers while sandbags would be represented by negative numbers; putting new cargo on was the same as addition and removing cargo was the same as subtraction. We also established that removing balloons was equivalent to adding sandbags (both would make you go down) and adding balloons was equivalent to adding sandbags (both would make you go up).

Then we went into the number talk I had originally planned for Tuesday. This time, I gave them two more subtraction problems (one at a time) and asked students to give me thumbs up when they thought they had an answer and a strategy. We shared out strategies, and I highlighted the removal strategy and the distance/difference strategy. We discussed how they looked (difference seems to add on whereas removal uses subtraction), where the answers were (on the number line vs. the total jump), and when you would choose to use one vs the other (when the numbers are close together vs. far apart).

I then modeled with the students how to use the open number line to represent a balloon story. In one class, that was all we had time for, before I sent them to work on the open number lines for HW.

On Thursday, we began connecting number sentences with our experiences drawing on the open number line. The entry slip used the example from the previous warm-up (at 3, get to 5), and showed how to write a number sentence for the two main ways (i.e. 3 – (-2) and 3 + 2), and then asked students to explain how to change the cargo and write a number sentence to match a picture showing a balloon at 1 above that wanted to get to 4 below. The most common mistake I saw students make was thinking about only 3 balloons or sandbags instead of four (i.e. not thinking enough about the “Extras” as they cross the 0). I don’t think there’s quite enough understanding yet of the symmetrical nature of the number line.

We then went into checking the HW, where I first asked students to try writing a number sentence to match each open number line diagram they had drawn for the various situations (i.e. start here, change cargo like this, end here). Then we compared their number sentences with the answer key, and gave them some immediate feedback.

After the HW check, we went into a number talk, to solidify our understanding of the two main ways to picture subtraction – removal and distance/difference. We discussed when to use each, where our answers appeared, and how to recognize which one had been done. We then copied those notes into our notebook, so we would have the two methods for reference. In the class that had a single, we transitioned right into our exit ticket, where I gave the students two scenarios: one in which they had to add sandbags to a negative number and go more negative, and one in which they had to remove sandbags from a negative close to zero and go into the positives (as well as going up). Students had to both represent their thinking with an open number line, and write a number sentence to match the situation. In both classes, most students were able to correctly find the first answer, but some represented it as a subtraction sentence (instead of adding a negative, as the scenario would dictate), and in the second one, some of the students were able to correctly find the answer and represent it correctly, but more students got confused and added a negative instead of subtracting a negative (or adding a positive). I realized that my students would benefit from analyzing the mistakes that were made, so I created an entry slip for Friday where the students would view two common mistakes and have to explain what the mistakes were and why the student might’ve thought they were right.

In the class that had the double on Thursday, we spent the second half of the double practicing how to translate between the number sentences and the stories. After we went over the homework, I asked them who felt like they needed more practice starting with the story and representing it in a number sentence and who wanted to move on, and almost everyone voted to move on (I think I need to do more votes with them closing their eyes so they’re not unduly influenced by each other!), but it wound up being okay. We determined what question each number in the number sentence answered (i.e. “Where do you start? How do you change the cargo? and Where do you end?”), and then I gave them a chance to work on their own. I had them use a new routine to check their work (new to them; one I’ve used many times before). I find it to be better than always going over the answers as a whole class for the entire handout – I have them read their answers in groups, going around with one person reading answer 1, then the next person reading number 2, and so on. The listeners check off or circle their answers – checks meaning they agree and circles meaning they disagree. At the END, after reading ALL of the answers, they then discuss any they disagree about. Finally, they write on a post-it note any they disagree about, so I can track any patterns.We also might share as a whole class about one or two of those problems if there’s still a lot of confusion. After they did this activity, they did the same exit ticket as the other class.

On Friday, I had singles with both classes. In both classes, I wound up deciding to skip the number talk because I didn’t have enough time. I only wound up doing 3 number talks instead of the 5 I intended to, but I think we used our time in a meaningful way.

We did our entry slip first, with finding the mistakes. I liked having the students share about the mistakes (and the reasons they think the students might’ve made those mistakes) with their partners; I made it transparent that the mistakes I shared with them were REAL mistakes from their classmates on the exit ticket the day before. I also told them we were going to postpone our quiz that was supposed to be on Monday based on how many errors I saw. Only about 7 – 10 students in each class got all of it correct!

After we went over the mistakes as a whole class, and discussed why they were mistakes (i.e. translation errors vs. removing sandbags being the same as adding balloons and making the balloon go up!), we began the next task.

In the class that had the single the previous day, we introduced how to write the story from the number sentence and represent our thinking on the number line, and they got the same practice handout as the other class did the previous day. In the class that had had the double, we began the handout with the open number sentences – instead of the “answer” always being the one that’s missing, sometimes one of the two numbers being added/subtracted were missing.

In both classes, we did the same exit ticket. I put up five problems on the board (“naked number sentences”) and asked the students to copy them onto their post-it notes and write the answers. I found that ALMOST everyone in both classes got the addition problems correct (whether they were adding two negatives or adding one negative and one positive), but that the subtraction problems threw the kids off still. More than half the class still got these correct, but I realized there were still a significant chunk of students in both classes who were adding the two numbers instead of subtracting when the signs were different (i.e. 4 – (-1) became 3 instead of 5) I realized the “double negative” is what seems to be throwing kids off the most still (expectedly). I was pleasantly surprised that there only seems to be about four kids (two from each class) who were struggling with the addition ideas still.

So this brings me to my debrief for this first week of integers and my planning for this upcoming week:

  1. It seems like an AWFUL lot of prep for the game to have only played it once! I don’t think I realized how rarely we would actually play it, and I wonder if there’s a way to give the kids more of a chance to play it again (or to reuse the context or something at some point). Perhaps we can make it a choice-time activity they could earn some Friday or weird day (like when we have class on a Tuesday with Monday and Wednesday off for Columbus Day & Yom Kippur?).
  2. I liked our ability to refer back to the game over and over, and using only one context in the beginning. I’m wondering how/when I should introduce the students to the idea that there are OTHER contexts out there where negatives will show up. I know I need to because on the state exam (and in high school, etc.), they will be asked to represent situations using negatives that don’t involve hot air balloons and sandbags, and I want them to be prepared for how to think about this too.
  3. I’m impressed with how few students seem to have problems with the addition of negative numbers (whether it’s both negative or whether it’s one negative and one positive), as well as how many students are able to consistently subtract a bigger positive number from a smaller one to get a negative result (i.e. 5 – 8 = -3).
  4. I’m also impressed with how fluently the students are using the idea that adding a positive is the same as subtracting a negative and adding a negative is the same as removing a positive.
  5. I’m concerned about how many students are still confused about subtracting negatives correctly. I don’t know exactly what activity I should do next (this week) to reinforce that concept. I haven’t quite done all of the open number sentences yet from Kent Haines, but I’m going to do those on Monday. I want to make sure my students are fluently able to add and subtract with negatives before we move into multiplication and division (which I’m planning to start on Wednesday/Thursday).
  6. I don’t think I’ve done enough work around emphasizing the symmetrical nature of the number line (with 0 in the middle, the numbers go outward on both sides in the same pattern). I don’t think we’ve done enough work with this idea yet.
  7. I also don’t think I’ve done nearly enough work with comparing the numbers (which balloon is higher?) nor have I done enough work with absolute value (who’s farther from the cliff?), but I plan to incorporate that this week and next.

I think my plan for Monday is going to be to use the open number sentences with the students. In one class, we started those on Friday, and they were starting to make the connection between inverse operations, and I’m wondering how to make that more explicit. I’m also thinking about using the number talks/strings from Pamela Weber Harris’ book, as well as the most recent articles on integers from NCTM’s magazine teaching middle school math.

In general, I think my kids enjoyed the game and using the context of it to talk about numbers, but I definitely need some idea to help differentiate this experience because some of my students are already quite comfortable with the number lines and the operations and I need to challenge them to think more deeply (I just found a few problems on Open Middle that I can give to a few of the kids on Monday to think about).

If I hadn’t already made all of the materials to play the game, I don’t know if I would play the game again next year, but since I’ve already made the supplies, I do think I will keep using the game next year.