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:

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.





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