In recent years, mean absolute deviation and variability concepts have been added to 7th grade math. It seems like a huge jump from their current understanding to these concepts, but we as teachers always figure out ways to make the content accessible to students. The first time I taught mean absolute deviation I just told the kids that absolute value was the distance a number is from zero. It seems like a simple concept and in my rush to get to the meat of the topic, I left it at that. Well, needless to say, the majority of my students don’t retain information just because I say it a couple of times. I know, I know, teaching lesson learned (again!). Let me share with you how teaching absolute value through discovery made teaching the whole topic of mean absolute deviation and absolute value in general so much easier.

**Making my teaching better**

This year, when I knew that mean absolute deviation was coming up again soon, I thought about how to get students to own absolute value. It’s not too hard or difficult, but they just needed somewhere to put it in their brains. The more I thought about it, the more I realized that absolute value was very intuitive and that it would be great for a discovery lab. If students discovered that numbers and their inverses have the same distance to zero, then we could use that knowledge to build toward mean absolute deviation.

If you’re not familiar with discovery labs, you can read more about how I got started with them here. As a quick refresher, a discovery lab is a 3 step process where students figure things out before they are given formal instruction.

The 3 steps are:

- Building students’ background
- Making observations and performing trials
- Writing a rule and/or drawing conclusions

When I thought about absolute value, I realized that students needed to explore absolute value with different types of rational numbers. The discovery lab could work for positive and negative fractions, decimals, and whole numbers.

**Building background for absolute value**

Brain science tells us how important building background is when students learn new concepts. Students will remember things better if they have a schema built in their brain that they can attach it too. For this discovery lab, the building background part of the lesson is about distances between numbers. I asked students to visually represent the distances between numbers on a number line.

Students first marked the two numbers on a number line and then showed the distance between them. This should not cause any problems for students. If they can’t do this, then the rest of the discovery lab will be difficult. My students completed it with ease and this let me know that they were ready to move on to the next step.

**Observations about numbers and their inverse**

In this part of the discovery lab, students were given a rational number and its inverse and they had to make observations. From past experience, I learned that you need to give them a little direction when you ask kids to make observations. If you don’t give them a focus, they’ll come up with a lot of observations that aren’t mathematical. They will notice things like, “The number line has arrows on it,” or, “3 is next to 4.” I have one particular student who is excellent at finding these types of observations!

The focus of this observation is about distance. I told my students to look at the numbers and make observations related to distance. This definitely helped them make appropriate and keen observations. I’ve included some student work to give you idea of what they might write. Keep in mind that during the observation portion of the discovery lab, we emphasize that there are no right or wrong answers. This helps students to feel comfortable making observations.

Most partnerships came up with some iteration of how the inverses were the same distance from zero as each other. I had them find someone who was not their partner and share their observations. This gave them a chance to see someone else’s thoughts on the experience.

**Writing a rule about absolute value**

Before they wrote their rule about absolute value I gave them the term absolute value. They then wrote their own rule. I encouraged them to use the word inverse and absolute value as keywords in their rules.

I love reading their rules and seeing how they express what they have just observed. It can be tempting to swoop in and try to make sure they’re writing the rule just as you would write it. I would discourage you from making it seem like there is one way to write the rule. This gives students the opportunity to express their understand.

**Conclusions and other thoughts**

There are so many ways to get students to reflect on their learning. I like to find different ways for them to reflect and give them enough time to think through their responses. They need a little bit of time and they need some encouragement.

The questions we used for the discovery lab conclusions were:

- How will you remember what you learned today?
- Why do you think we used a number line in this discovery lab?
- How is this related to adding and subtracting positive and negative numbers?

You’ll notice that these questions have lots of possible responses. Asking open ended questions like these, I get lots of different answers and really get to see how deep their understanding is.

**Tying this into mean absolute deviation**

I used this discovery lab before we got to mean absolute deviation. Then when we were doing MAD we could refer back to this experience. Moving into MAD, students use the mean as the number they are finding the absolute value from, instead of zero. Making this connection with students between their new and deeper understanding of absolute value and MAD worked so nicely. I really didn’t have any students struggle with this aspect of MAD. They remembered the discovery lab and even referred back to it unprompted when we were doing MAD.

Overall, we spent a day on absolute value, and then completed 3 absolute value mazes on the 3 days following the discovery lab to reinforce their learning. Then, we spent two days practicing mean. When we got to putting it all together, finding the mean absolute deviation was actually pretty easy. Only a couple of students struggled and the vast majority were proficient with it rather quickly. It worked so much better than last year when I just jumped right into the grade level concept.

**In your classroom… **

Discovery lessons can seem scary at first. It can be a big shift from what we are used to as traditional teachers. But I think everyone should give them a shot. Don’t worry, you can give notes and have students practice after the discovery lab. It just gives the students background and they remember the things you’re teaching better if they had to come up with some of the information instead of just trying to memorize it.

Take the plunge and try this absolute value discovery lab in your class. It’s one of the shorter and easier of the discovery labs. Please share your experiences in the comments below and let us know about your discovery experience.

If you want to use the print and go version of this discovery lab, it’s available for download in our TpT store here. Thanks for reading. Until next time!

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