**By choice**, I am designing and teaching a class for 8th graders who scored below 50% on their 7th grade Pre-Algebra final exam. It’s tracking. Sort of. I met individually last year with every student who was recommended for this class. Each student currently in the class **chose** to be there. They and their families signed a paper requesting this class (the alternative was a double period of algebra).

And I could talk your ear off about why I believe that creating and teaching this class was an equity decision. And some of you would talk my ear off about why tracking is bad, period. And I would disagree. We’d then have a really heated, informative debate where we’d both push our thinking deeper and become better educators because of it. Sometime soon, over a bottle of wine, the math department at my school will come to my home and we’ll have this very debate. It needs to happen. And as I said, it will make each of us think more reflectively about what we do.

But for now, here’s what I’m excited about: teaching integer addition. Mastering it is critical for success in Algebra 1. All of my Algebra Scholars took Pre-Algebra last year and had extensive experience with integer operations. Nevertheless, on an integer pre-assessment given shortly before I began my unit, only three students could successfully add one digit integers, and only 1 student could successfully add integers with larger addends.

I believe that they have been introduced to far too many models for integer addition, and never been given the time to learn any one model in depth. For this reason, the ONLY model we use in class is a number line, but we have taken a lot of time to have rich activities and conversations about how to use it to model problems involving integers.

I do wonder, though, if using multiple models is useful. I’m curious what others think.

Last year, I realized that when I do math, I love to use colors. It helps to organize my thinking and highlight important discoveries I’ve made while in the middle of a problem. So as of September, pencils are no longer required (though I make them readily available to students and allow them to borrow my supply at any time). Upon entering class, students take these which is a set to share with their neighbor. They are required to use colors when we take notes. They are not allowed to use them for doing their regular work, until I tell them that they should color code. I am trying to preserve the ink for as long as possible. They know that if I find a single dot of this extra-fine German ink on their skin, I will cut off that joint. Ok, not really, but I have been very clear about the purpose of these uber-expensive markers. Having these markers makes students (and teachers) not in my classes quite jealous. And since I teach the lowest ‘track’ of students, I like it that way.

**Phase 1: Comparing Integers.** Using Wu’s definition that a larger number is always to the right of a smaller number on the number line, we placed integers on the number line and compared. To escape bordom, each student had a plastic popcorn box filled with integer tiles from -10 to +10. They chose 3 numbers from their box and compared them on the number line. They then wrote 3 inequalities comparing their numbers. This pushed them to consider 4>-2 as well as -2<4.

Each student made a key at the top of their paper, color coded, using three colors: 1 for positive values, one for negative, and one for zero.

**Phase 2: Adding Integers.** Sticking with the popcorn boxes to generate random pairs, students then chose 2 numbers and added them. Everything was color coded and they asked themselves over and over again, “Will I make it to zero?”

**Phase 3: Discussing discoveries.** Students then took all their addition problems and sorted them onto a mat. I had no intention of telling them rules, but instead wanted them to really be able to ‘see’ in their heads what happens when you add integers. The colors helped to imprint the math in their heads. We had a rich conversation about what the colors showed us on our mats. After analyzing why positive +negative sometimes is positive, sometimes is negative, and sometimes is zero, I felt that well over half of my students had solidified their understanding of when you combine the absolute values (adding) and when you find the difference.

(Apologies for the head tweeking here):

**Phase 4: Tweaking the model, but only slightly.** I realized that asking oneself what the sign of the answer would be was a critical juncture in what should be happening in their heads. Most were really good about asking themselves, “Will I make it to zero?” But I needed a hook, something silly, which tapped into their teen-ager-ish-ness. This whole unit could be summarized by a single question, spoken in my sexyiest possible voice, “Hey Baby…What’s your Sign?” Our final push for mastery of large numbers involved asking each other this question, and stiffling the giggles that ensued. I took away the bulk of the scaffolding (most students had tired of 2 colors on the number line and ironically were begging to simply do their work in pencil). But added one last element for those who needed it: colored dot stickers. A few students still needed a visual reminder to cue them on when to combine the absolute values (add) and when to take the difference. They all had green for positive values and red for negative. 100% of them could now quickly and efficiently respond to the question of their peers, “Hey Baby…What’s your Sign?” And before doing any calculations, they stuck a flourescent orange sticker down with the sign of their answer. It took three weeks to get here, but we made it and I never gave any rules for them to memorize. Success. The real goal, of course, is that they conceptually understand this well enough to use it throughout the year and in the years to come. Stay tuned…Next week’s project is for them to design a series of lessons to teach to the students in the algebra support classes who are still struggling with integers. They can’t wait!

I LOVE this! For many reasons, first that you are driven by understanding not memorization. I know that most teachers in the blog/twitter community are doing this but sometimes I find that the activities I set up are really memorization wrapped in a prettier design. But then I see units like this and I love it. This is a huge misconception for many of my students as well even though they are in 11th and 12th grade, I can say that 90% of my students DO NOT know how to add and multiply integers correctly – it’s a guessing game for them unless they get to use their favorite thing ever (the calculator).

The second reason I love this is the color. This summer when I taught graphing piecewise functions I used an eleborate chart and color coding (adapted from http://squarerootofnegativeoneteachmath.blogspot.com/2011/05/color-coding-for-sketching-piecewise.html and mixed with http://mathteachermambo.blogspot.com/2008/09/piecewise-function-success.html) The color coding changed the entire process of piecewise functions for the students. Almost every student understood it because they were able to make clear distinctions, see differences, notice patterns, create lists/steps, and ultimately they too were begging to use pencil.

Thank you for the share – I am having a lot of difficulty with this one class period and maybe I will throw the book aside for a minute as one commenter suggested and do a “Hey baby whats your sign” mini unit with these students to bring morale and math-confidence back up.

Great linear thinking (pun intended). Very logical, direct, hands on, thorough, fun and most of all, needed. I’m stealing ur follow up part. I thought we decided on whiskey?

Great ideas! I use activities with a number line and a deck of cards, face cards removed, to teach addition of integers; red cards negative and black cards positive. I also teach the business terms profit and loss. I think the color coding really helps them understand the concept of adding integers and never really teach them the “rules.” PLEASE share how you are going to tackle subtraction. I end up teaching the rule “never subract, add the opposite” just because I haven’t found a good model to explain subtracting negatives. Thanks for taking the time to share.

@Molly: Glad you found this helpful. Over the past few years I have slowly begun using colors more as a visual aid when I’m giving notes and this year I threw in the towel, got packs of beautiful markers, and now insist that they use colors when we’re learning a new topic where I feel color coding ideas will help the visual learners. I have gotten a lot of positive feedback on this from my students. It’s disturbing (but not entirely surprising) to hear that so many 12th graders struggle with these same topics. My experience is that teaching rules has short term success, but not much is remembered after 4-6 weeks if there’s limited conceptual understanding behind the learning of the rules,

@ Mac…whiskey it is…now we just have to choose the bar.

@Diane. Honestly, I just explain that the definition of subtraction is adding the opposite. We look at how these are the same idea on a number line, I find that students who REALLY understand integer addition seem to intuitively understand this idea of subtraction being adding the opposite. One thing which helps is having students circle the terms: 4 + 8 – 3 (as a simple numeric expression). And then talk with them about how we’re combining +4 with +8 with -3 and they seem to have an ah-ha moment that 8-3 is the same as adding +8 with -3. But I’d certainly love to hear other approaches to this.

I love your comparing integers activity! That would probably be a great way to start the year with my ninth grade Algebra I students.

I really like this approach as well because I think it shows clearly the rules work the way they do.

I’ve been teaching with an approach the book we’re using (impact mathematics) has and though this approach is less connected to the actual meaning of negative numbers, I love it because it’s easy and it works for addition and subtraction. They introduce adding and subtracting negative numbers with a “walking the plank” game where a sadistic pirate captain places a poor soul in the middle of the plank (at zero). Towards the boat is positive and towards the sharks is negative. Then the captain rolls a die- one to tell the poor victim which direction to face (towards the boat or towards the water) and one to tell the victim to move forwards or backwards a certain number of spaces. The kids LOVE this game because they get to take turns being the captain and the victim and they play several times- sometimes making it safely to the boat, sometimes falling into the water and dying.

We extend this to negative number addition and subtraction by letting the first number be where the victim starts. The operation tells them which direction they’re facing, then the second number tells them to move that many spaces in the direction they’re facing if it’s a positive or to walk backwards if it’s a negative. So for example, 5+-7 would start you at 5 on the number line, you face the in the positive direction, but then you walk backwards 7 places and end up at -2 on the number line. Once the kids have internalized these “walking” rules, all the other “rules” don’t seem like rules, just common sense (like the fact that adding a negative is the same as subtracting a positive and subtracting a negative is just adding a positive.) There’s also no need to teach them the idea of when to add the absolute value or when to take the difference. They just start working out how to do it in their head without any guidance at all.

I also like it because it lends itself so well to games (we make giant number lines and do long strings of additions and subtractions while students walk along and compete to get to the answer first.)

I don’t like it in that is seems contrived. I’d rather teach them via math logic, rather than through a made up game, but I have to say, so far it has worked beautifully. The students who have learned negative numbers this way with me in pre-algebra have had no difficulties with them as they advanced through our math program. The first group I taught this to I’ve also taught algebra 1, geometry and algebra 2 to as well and I’ve never had to review negative numbers with them.

It’s the only way I’ve taught negative numbers though and I’d like to see if other methods work just as well and aren’t founded on such a contrived premise. I would love to give these activities a try.

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