I've been trying to reflect on what I did this year that set my students up for success, and here are some of the things that I think helped.
1. Number lines, number lines, number lines
We used them a lot! One of the very first lessons in our Accentuate the Negative unit is a unit based on the number line. It focuses quite a bit on looking at opposites on the number lines, and comparing which of two numbers are farther from zero.
This year, I decided it was a great day to get kids moving. So instead of doing the lesson out of the book, we did the lesson on a human number line. I set our pieces of construction paper numbered from -5 to 5, and gave each of the kids an index card with a number on it. Some of the numbers were integers, some fractions, and some decimals. Then I called up 3-4 kids at a time to find their "spot" on the number line. Once the kids were on the number line, I started asking the same types of questions that were in the lesson in the book: Who is farther from zero? How do we know that Robbie and Ashley are opposites? Where would Zoie's opposite stand? Whose number is largest and how do you know? How far apart are NiJa and David?
I was so excited at using this as an introductory lesson. I was amazed at how much more engaged the kids were just because I got them out of their seats....they were totally into this lesson. Also, it really got them thinking. On day 2 of the integers unit, I was able to start asking really high-level questions because the kids could see it. They were totally making sense of how far apart -2 and 3.9 are on the number line, and it was exciting to see them making sense of subtraction so early in the unit. I think this also set the stage in student's minds for the number line being a helpful tool that we would rely on during this unit.
2. New way to introduce the chip model
I've always introduced both the chip model and the number line. I really like the chip model and think it is helpful to make sense of things like why taking negatives away makes your answer bigger. But my kids often struggle with the chip model, especially when they are having to take away more than what they have (problems like -3 - -6). This year I changed the way I introduced the chip model. Our team at school uses tickets, so I talked about a new "ticket" system.....where students can get positive tickets, which can be used kind of like money. But now there are also negative tickets that would cause students to owe chores. Then I posed situations to push their understanding. I really tried to get students to see connections....if a student does something good, then a teacher could give them a good ticket or take away a bad ticket. If a student does something they shouldn't, the teacher could give a bad ticket, or take away a good ticket. Students were able to see the connections between adding a negative and subtracting a positive, as they both had the same overall effect.
I also had a card matching activity that I think really helped the kids make connections between addition and subtraction. In the activity, the students had a copy of a number line with a problem on it. They had to match with with an addition problem (example, -4 + -4), a subtraction problem (example, -4 - 4), the answer (-8), and a statement like "starts at -4, decreased 4 in value". I think this really helped kids to see that -4 - 4 and -4 + -4 are really both the same problem, since they are both starting at the same place and moving in the same direction.
3. Lots of short, frequent assessments
I gave quizzes almost every day....they were very short but it really helped me to keep track of what my kids were learning and what they were still struggling with. I used quia or Google forms for most of the short quizzes, to make the grading lots easier.
I also did a couple of days with addition of integers based on this model, and the kids did a great job with it. Right after we talked about adding on the number line, I had a series of short assignments and assessments for kids to work on. This was a great way to differentiate work and push kids to do as much as they could. I had a series of Practice Problems with Exit Slips. I had the kids work on the practice problems and I had the answers posted. When they finished the practice problems, they checked the answers. Then they got the exit slip which I checked myself to see if they were really understanding it. These progressed in difficulty: small integers, larger integers, fractions with common denominators, simple decimals, harder decimals, fractions with different denominators.
I was SHOCKED at how much the kids enjoyed this. One of my least motivated students was on fire during this activity....he was one of two students that finished all of the exit slips and I saw him push himself far beyond what he usually does. They really wanted to work through the levels and worked really hard. I was able to keep track of who I needed to work with in a small group because each exit slip was focused on a specific skill and I knew what they needed. This simple activity was a total win! I'm thinking I will upgrade this activity when I get a chance....make it more like a video game where kids can "LEVEL UP" for each activity, maybe build an avatar or something. But even in its simplest form, it was really helpful.
4. Visualizing the number line
This super simple strategy was really effective. I need to do a LOT more of this in the future. When we got to the point of using integers that wouldn't "fit" on the numbers I had available, I started asking the kids to visualize the number line, and the moves they needed to make. When the kids took the time to actually do it, it really helped....even my kids that were struggling. Hopefully once the kids realize this number line is always with them, they will rely on it even more!
These are all strategies I definitely hope to continue next year. As I have moved into equations with negatives now, I'm still finding that these strategies are paying off. As we talk about how to solve equations, I frequently find myself saying things like "Ok, when we subtract 5, is it rising or falling in value?" or "Picture this on the number line....what direction would we go?". It's been nice because it has made equations feel more connected to our work with integers.
