I gave my three 7th grade honors classes a test last week on Patterns and Unit Rates. With some review problems – just order of operations and basic skills in fractions and decimals. I thought the review problems would help their grades. It turns out over 60% of the students made mistakes in their order of operations problems. Upon further investigation, I realized it was almost all due to errors with integers! Errors in adding or subtracting and multiplying. We didn’t have any division problems.

I started thinking about how we have been teaching integers in 6th and 7th grades – by teaching our students the rules. And making up fun “games” to help them remember the rules. Such as “when a bad thing happens to a bad guy it’s good!” I also thought of the 7th and 8th graders who had asked me after class for some help with integers – they knew that two negatives made a positive – so why were they getting – 17 – 5 wrong when they put 22? or 12? I remember doing this problem with one of the girls:

me: What’s your favorite store?

girl: Albertson’s (i thought she’d say Forever 21 or something like that)

me: Ok, say you bought something for $10 (I write -10 on the board)

me: Now you go to Baskin Robbins and buy ice cream for $6 (it’s by the Albertson’s)

Now the board shows: -10 – 6

me: How much have you spent?

girl: I know you change the minus to a plus because there are two negatives…

me: WAIT! You just SPENT $10 at Albertson’s and $6 at Baskin Robbins! How much have you SPENT?

girl: (looking at board) UMMM

me: Stop looking at the board and JUST THINK ABOUT IT. You’ve spent $10 and $6. How much have you spent?

girl: $16!

me: RIGHT!

We talked more about it, but I was remembering this conversation after I analyzed the tests. Something I didn’t say earlier is I’m the department chair now, so I have a little more clout and I think I can tell the other 7th grade and 6th grade teachers that we need to make some changes. I decided to talk about it at the next department meeting and say “We’ve got to take the time and let our students learn WHY integers work. Our students don’t know WHY anything works, we’ve been so busy trying to teach everything that we’ve developed great ways to help our students. We’ve given them the rules, shortcuts, ‘recipes’ to follow, and we’ve worked really hard doing it. However, what has happened is the students are memorizing how to do the problems, but they can’t tell us why these rules work. Now that we have the CCSS we can slow down a little, and really teach, then reteach if the students don’t understand. And we need to start now. With integers.”

Between the grading and the talking there was a lot of thinking on my part, and talking to a couple of other math teachers, and trying things out on my husband (who is such a good sport, and asks great questions). So by the time I talked to the teachers (about a week) I had put together an Integers Teaching Progression. I had started it with my classes and passed out copies to our teachers. I asked for them to please try it, and give me feedback on what went well and what should be improved, removed, extended, etc.. I don’t think it’s perfect, but it’s better than what we were doing! I’m including the dropbox link: https://www.dropbox.com/s/byz5jj5drrpzmof/Progression%20for%20teaching%20Integers.docx (Updated 10/14/13)

If you’d like to use it and give me comments on how to improve it that would be wonderful. I made worksheets to go with it because one of my teacher friends said she’d like them. So far they seem to be working ok, but it’s a lot of copying!

Ok, that’s it for my first blog post!

I know the troubles of teaching integers well! I agree with your point about making sure that the students know what the integers mean, especially the examples of positives and negatives. I’ve found that the number line strategy helps a lot of my students. Have you tried using integer tiles as well?

HI Sarah!

I haven’t used integer tiles this year, but the teachers teaching the other 7th grade classes (mine are just honors this year) have been using them. I didn’t initially because I didn’t think I’d have a problem! Ha Ha, I shouldn’t have thought that! Thanks for your reminder, I think I’ll use them when I’m doing RTI. (Response to Intervention, twice a week)

I haven’t, and i’ve been thinking about algebra tiles. (I think they’re the same as integer tiles? One side red and the other yellow?) I am trying to have the students relate the numbers to a “real life” story, and tell me what’s going on with money, or football. Algebra tiles are also good for visualization, but I can’t figure out how to write a story with them as well. Thanks for the reply!

I like your progression! I definitely liked the idea of making sure we always tell stories of the problems.

I noticed, though, that your stories about subtraction seemed to involve changing into addition. But that’s not an easy thing to do! When I was working with the Algebra Project, we tried to focus on what subtraction actually means. When looking at the number line, moving is a good analogy for distance. + (-3) means moving 3 to the left. + (+5) means moving 5 to the right.

For subtraction, though, we used “measuring distance” as the analogy. For, say, 8 – 2, you would say “How far is 8 from 2?” The answer is 6, but in particular, it is 6 to the right, or +6. Similarly, if we had (-5) – (-9), that would be “How far is -5 from -9?” It’s 4 to the right, so +4.

HI James,

I don’t really want to change the subtraction to an addition problem, I want the students to rewrite the problem so the largest absolute value is first. This way they can start their story either by “having money” if the first term is positive, or by “lending money” if their first term is a negative. So if the original problem is -3 + 8 I’m saying rewrite as 8 – 3! Then when they use the number line they are starting at 8 and moving 3 left. However, if they are using football in their stories they could say they lost 3 yards on the first play and gained 8 on the second, so they were up 5. To me, it really depends on their stories.

Welcome to blogging! I’m surprised at how poorly my students are doing with integers this year. In a typical year, my 8th graders come in pretty strong and just need a little refresher in September. This year is a different story for some reason.

I looked through your files and the part that jumped out at me the most was the number lines you used for multiplication. With no elementary background, I haven’t seen multiplication displayed that way, but it makes good sense. Thanks for sharing!

HI!

I wasn’t used to number lines either, but Fawn Nguyen showed them to us in a class a couple of years ago and I’ve been thinking about them as a good way to visualize the numbers. I think it makes the students think about what the numbers represent.

Great ideas! My students also have a lot of trouble with integers, and get too focused on the rules. I agree that we need to slow down and teach the concepts, rather than the rules.