Working with positive and negative integers is something that can throw students off at first. One of my favorite ways to introduce new topics is with a discovery lab. Why? Well, these activities help students face their misconceptions head on. Discovery labs are an ideal way to start a new topic because it helps students build their own conceptual model right from the get-go. Then, when you start the teaching and modeling, they have a place in their brain to put that new information. Today I’ll share with you how my students get introduced to adding positive and negative integers through a discovery lab approach.

**Positive Versus Negative Integers**

Through all of my experiences teaching about adding integers there is something that’s always true. Students have a very strong background in positive integers. That’s great! The flip side of that is that you have to work hard to break that schema so that they’ll understand negative integers. I’ve even found that many teachers have the same rigidity when it comes to adding and subtracting integers. Getting past this strong built-in understanding can take a while. It feels like you’re breaking down a wall and then rebuilding it.

I find that using number lines as much as possible is a great cornerstone for the building of the conceptual understanding when working with integers. The additive inverse discovery lab that I’ll walk you through here does a great job of giving students a concrete view of what is happening when they are adding and subtracting integers. The best part is that they have to write rules, in their own words, to explain what is happening.

**Using a number line work mat**

I start the **discovery lab** with a workmat that has a number line on it (see the pic below). We take a few minutes to just establish how to use the number line. I model a couple of examples and have the students follow along. You can do this without getting into the math yet. The important part here is just to show them how to use the arrows and the number line itself.

Students will be given positive and negative integers. They’ll build the expression on the number line by starting at zero, placing the arrow that represents the first number, and then placing the second number. That’s the first step of the discovery lab. Before moving on, be sure students understand that 1) they will always start at zero, 2) each arrow has a different length to represent a number, and 3) when a number is positive, the arrow points to the right; when a number is negative, the arrow points to the left. Once they’re clear on these rules, they’re ready to move on to the challenges.

**Explaining the Challenge**

The main premise of a discovery lab is that students, through trials, will draw their own conclusions and make meaning of the math they’re exploring. There’s a fine balance between students struggling and not getting too discouraged. It helps to frame the discovery lab as a challenge. Remind students that they’re trying to figure something out that you haven’t even studied yet. This will help students to put their guard down.

For this discovery lab, explain to students that they are trying to figure out the rules related to adding and subtracting positive and negative whole numbers, or as we are going to call them from now on, INTEGERS. They’ll model a few different examples of expressions that follow a rule. Then, they’ll have to write a rule about that case. Again, it’s important to focus on the fact that it’s a challenge. **Trying** is the most important part of what they’re doing.

**Complete the Trial**

When my students get going with this discovery lab, I have each partnership or group work through the first trial together. I give each student their own mat and have all students do the modeling. This partnership part is most important for the discussion that happens. In the first trial I want them to conclude that when a positive number and its inverse are put together they make zero. Some students may need help with how to start writing the rule. So, I give them sentence starters like these:

- When you…then…
- Every time…then…

In your classroom, after the first trial you might want to let students share what they wrote for their rule. Give examples of how to write it. I wouldn’t do this for every trial or some students will wait to write anything until you have people share (sneaky, sneaky!). Students are very used to waiting for the answer and then writing it down. A discovery lab is for building background and not for notes to refer to later. Keep in mind that you’ll give notes later on in the unit. The discovery lab has a different purpose all together.

**Continue with the Trials**

Let students work at their own pace as they complete all 6 trials. Students will move from working with additive inverse, like 3 and -3 to working with positive and negative integers with different values. For each trial, students build expressions, write observations, and then write a rule that they believe applies to the expressions they examined. There are a wide range of how students will word their rules and it is important to let them use their own words in writing their observations and their rules.

Also, when I did this with my students I was amazed at some of the conclusions they came up with and how close it was to how I would explain it. Here are some examples of things they said when we were completing this discovery lab:

- “A positive number and its negative always make zero.”
- “When the negative number is bigger than the positive number then the answer will be negative.”
- “If all the numbers are negative then you add them together and the answer has to be negative because all of the numbers are going to the left.

Students are very capable of drawing these types of conclusions. Often we don’t give them a chance because we’re honestly in a rush and we don’t think we have enough time. Or, we’re afraid they’re going to fail and we want to prevent any mistakes. But letting students play around with the math, letting kids try on their own, they consistently surprise me with their thinking. As far as the time goes, I think that letting kids discovery and make their own conclusions is the most valuable activity you can do with your time. You will turn your students into thinkers instead of regurgitators. And by the time I get into the “teaching,” things move much, much faster than before.

**Next steps…**

After completing this discovery lab it’s time to give some notes. This gives students a place to refer back to through the rest of the unit. There are many ways to help students remember what’s happening when they add and subtract integers. I find it helps to give students a few different ways to remember what they’ve learned, and then let them choose what works for them.

The best way to boil the rules down is, “Same signs Add, Different signs Subtract, Always keep the sign of the bigger number.” It’s short and sweet. If they can get that phrase nailed down, then they can always simplify these expressions. The more they practice the more fluent they get. To learn more about the sequence of objectives for teaching this topic, check out this blog post.

**Let Your Students Discover…**

If you want to get a print and go version of this discovery lab to try with your students, you can find **it here**. Or, you can follow the steps laid out above to start your students off with exploring and inquiring about integers. The challenge that I would give you is to let your students use discovery to understand additive inverse and adding positive and negative integers. I promise they’ll be more likely to remember this concept when they are the ones discovering it. I’d love to hear all about your experience, so please feel free to drop a comment below.

Thanks so much for reading. To read more how I break down and teach additive inverse and adding integers, I shared all about it in the post, “How to Make Adding Integers Stick Forever.” If you want to check out other ways to get students the practice they need, check out this post, “12 Engaging Ways to Practice Adding Integers”.

For more examples of discovery learning in action, check out how I used discovery labs to teach my students about area and circumference, simple probability, compound probability, and absolute value, among other topics. Thank you again. Until next time!

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