Students spend their whole lives working with positive numbers. Then one day we’re like, “Hey, guess what guys? Some numbers are negative. Surprise!” We then wonder why they have such a hard time acclimatizing to this newfound knowledge. How can we break this sad cycle? I find that for many students, if you want them to really understand what they’re doing with adding positive and negative integers, you have to do a lot of background building and even some unteaching from what they’re used to.
I like to teach using interactive notebooks. In our interactive notebooks we use I Can statements to keep track of objectives. Under each topic we have a list of objectives and we check them off as we learn. You can download a FREE set of them here for many 7th grade objectives. In my class, we refer back to these I Can Statements everyday and see how we are progressing (read more about I Can Statements in INBs here).
What I Want My Students to be Able to Do with Additive Inverse
Having this list as road map really helps students understand what they’re learning. You can download your own print and go copy of these I Can statements (including a blank template) here:
Additive Inverse on a Number Line
For adding and subtracting integers we start with understanding additive inverse on a number line. I think this gives students a great visual for what happens with negative numbers when they interact with positive numbers. I really want my students to be able to explain why inverses add up to zero. It may seem like a simple concept, but it’s the backbone of adding and subtracting integers.
Real Life Examples of Additive Inverse
After working with additive inverse numbers on a number line, I want students to be able to give examples of real life situations of additive inverse. If they can tie this concept to something that they know about in real life they are more likely to remember it. Here are some real life examples:
- Ground Level
- Football Drive
You can even have students make a drawing of how this work in their interactive notebook. For example, Johnny owes Dwayne $20. So, he has -$20. Then, he mows the lawn and earns $20. How much money does he have now? Well, assuming he pays his debts, he now has $0 and no longer owes any money.
I want students to choose and articulate an example that makes the most sense to them. Then, they can illustrate it in their notebooks and have it as a reference later on.
Explaining Additive Inverse
Ultimately, we want students to be able to explain what additive inverse is and how it works. After practicing with number lines and real life examples then they’ll have to explain it in an abstract manner. In other words, students need to explain using mathematical terms what additive inverse is. When they *get it*, students will explain it in their own words, without parroting what they heard from the teacher or what’s written in their notes.
This is something that students should be able to do after all the other things we’ve done. They’ll need a couple of opportunities to practice explaining additive inverse on separate occasions. One way I have students explain what they’re learning is in the reflection section at the bottom of the I Can Statements/objectives page. This is a great chance for them to show what they know at the end of this unit.
Again, you can download your own print and go copy of these I Can statements (including a blank template) here:
Understanding absolute value is the basis for understanding negative numbers. I use a number line and arrows that show distance from zero to model absolute value. We briefly talk about the notation for absolute value. It’s something that shows up later in the year when we work with mean absolute deviation. To help students understand and really see what absolute value is, I use a discovery lab activity that you can read all about here.
Show the Distance Between Two Numbers
To practice the second objective and help students understand the distance between two numbers, we use a number line and a SmartPal dry erase sleeve. Students can get many opportunities to show distance between two numbers in a concrete way. Then we practice explaining what’s going on using math terminology. Students want to talk about it without a lot of math terms, so I emphasize that they need to use the following terms to explain what’s happening:
- Absolute value
It’s a little bit more of a struggle, but students get the hang of it. When they’re able to make their own explanation we move on to adding and subtracting integers.
Adding and Subtracting Integers with Automaticity
Once students have explained the idea of distance between positive and negative integers, we do a lot of practice with adding and subtracting integers. My goal is for students to be able to do this automatically. Some students get there super fast and others take longer. The more they have a picture of it in their head of what it is they’re doing, the faster they to learn it.
We take notes in our interactive notebooks to help us remember the rule. I try to make it as simple as possible because they’re working toward not needing the rule at all. If the rule is too wordy then they’ll never get fast at it. Personally, when I add and subtract integers in my head I don’t think of a rule with words. I picture what is happening and instaneously know if I have to add or subtract and if they answer will be positive or negative. That’s what I want my students to be able to do. With the right techniques, the right amount of practice, and a lot of direct feedback students can add integers with automaticity.
Some Practice Tips
Students need a lot of practice adding and subtracting integers over time. This skill is a crucial prerequisite skills for so many other concepts. If you move on too quickly or don’t come back to this topic your students will suffer when they get to other standards. For 12+ activities and resources that will get your students the practice adding and subtracting integers that they need, check out this blog post, “12 Engaging Ways to Practice Adding Integers.”
I like to use the following task cards as a way of reinforcing the skills. Often we use them when we’re learning integers and then we come back to them later. If enough time has passed, then kids don’t remember the answers and they can actually do the same activity again. I have three sets of task cards that I use for this topic. One set practices with additive inverse. The other one focuses just on adding and subtracting integers. The last one is a task card game.
Also, I like to use this adding and subtracting integer practice pocket. It’s something that you can make and use at the beginning of the year. Then, you can bring it out from time to time. It works great as a sponge activity and students get concrete reinforcement.
Making a Plan for Teaching Integers…
In my opinion, the most important take away from this blog post is that you need a plan. Your plan won’t necessarily look exactly like mine, but you do need to have a plan. This plan starts with sketching out the objectives, and you can use the I Can statements above, or modify them as needed. Then, it’s easier to see the path to get students from here to there. I hope that the sequence above helps you in your own classroom, and that your students learn to love working with positive and negative integers.
Thank you so much for reading. Until next time!
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