Do your students struggle to understand and remember the distance formula? Boy, have I been there. In this blog post I’m sharing with you a powerful way to teacher the distance formula that really pushes students to struggle productively and ultimately learn the distance formula at a deeper level. When students get to the place of struggle, and then you give them just a little support, then they understand and remember the distance formula more than ever before.
I have a lot of sad stories related to distance formula and me. I shared all about how my own short-cut blew up in my face in this blog post.
Shortcuts sometimes seem like a great idea, but sometimes, even when they seem to go right, they go so wrong. When I first had to teach the distance formula, to be honest I wasn’t sure what it was. I knew my kids would struggle because they have a hard time memorizing and of course they struggle with adding and subtracting integers. I created a shortcut where they put the two coordinate points on top of each other and subtract to find a and b in the Pythagorean Theorem.
What went wrong?
It was a momentary fix and they could pass a test. But it didn’t last. Students couldn’t solve these problems a few weeks later, let alone when they saw the distance formula again in high school. Then, this past week I was in a colleague’s classroom with my coaching duties and I saw her using this shortcut. I was shocked to see that someone who had been teaching for so long had gotten a hold of my shortcut and was using it. If I could go back in time I would make sure that shortcut never saw the light of day. This discovery lab that I’m sharing with you today took the place of that shortcut, and man, what a difference it has made.
My colleague had found herself in a place of desperation. So, she used a shortcut. It wasn’t just any shortcut, it was my shortcut. This made me feel terrible. I wish she would have used the discovery lab that I used because it made the rest of the distance formula unit so easy for my kids. This taught me an invaluable lesson about making sure kids get a conceptual background and not just getting them to pass a test.
Anyways, I would like to share this discovery lab with you because if you do it right, it will change how your students understand the distance formula. To read more about discovery labs in general, check out this post.
Step 1-Building background
When you do this discovery lab activity with your students, they shouldn’t already be familiar with the concept. It works best as an introduction lesson, used before notes. I always include some kind of background building in a discovery lab. First, this is included because it literally builds on their previous learning and kick starts their brains. Second, it can serve as a means of seeing what they know about a prerequisite skill. In this case the prerequisite skill is the Pythagorean Theorem.
To get warmed up, students looked at an example of the Pythagorean Theorem and they had to explain it without any words. They had to use symbols, numbers, formulas, and arrows to explain what was going on in the Pythagorean Theorem without using any words. This was a great way to see their understanding of the Pythagorean Theorem, which of course would be essential to understanding the distance formula.
Students came up with different ways of showing it. Some put little tally marks in all of the little boxes. Others wrote the formula and then drew arrows to the different parts of the diagram. A couple struggled with the directions and got stuck on the idea of explaining. They thought it just required a simple answer, so they wrote out the formula and left the rest of the page blank.
All in all, it got them warmed up and thinking about the Pythagorean Theorem. It also showed me that they knew the theorem. This type of response gave me the green light to move on to the next part of the activity.
Step 2-Observations
The set-up
In the observations portion of the discovery lab, the fun really begins. Now, students have to see something and try to figure it out. They have guided questions and as the teacher you can step in with small pieces of information if the struggle has stopped being productive for too long. In this discovery lab there is a lot of struggle and a lot of frustration, especially for the kids that normally understand everything quickly.
This discovery lab starts off with them having to look at a right triangle and find the lengths of legs two different ways. First, they can count them. Then, they have to find the lengths a different way. This is when I saw a lot of staring and fidgeting. For this to be successful, you have to monitor the class and keep redirecting them to the page. I had them work with partners and I could see exhaustion on some of their faces. They got distracted because it was “too hard” to figure out. I let them struggle for a bit. 
Moving things along
When the struggle was no longer productive I started to give students little hints like asking them if they had tried adding or subtracting any numbers. Then, the lights started to go on in their heads. They saw that they could subtract the y-coordinates to find the length of the vertical line. I told them to write down their process. Then, they kind of didn’t know what to do again. I asked them if they tried the same thing with the x-coordinates. Their minds started racing and they were pleasantly surprised to find out it worked with the x-coordinates, as well.
They showed each other what was happening and showed them on the board a few times. At this point, just about everyone got it. There was still one main hold out. She usually gets everything before the others, so she dug in her heels that she didn’t get it this time. So, of course, she decided it was pretty much impossible to get. I rolled with it because I knew she would eventually let her guard down and get it once her ego recovered.
Step 3- Synthesizing their new knowledge and the Pythagorean Theorem
This next part was the best part. I asked students how they could find the distance of the hypotenuse with their new found knowledge and the Pythagorean Theorem. Back to the struggle we went. I let them struggle and try to figure things out. One group with a little push got it and then another. They talked to each other and I gave small nudges in the right direction. Finally, all groups except for one could see how you can find a and b by subtracting and then use the Pythagorean Theorem to solve for c. If there was a movie there would have been some swelling music at this point 🙂 .
We did the bonus section with the question, “What is the distance between points a and b.” I showed them what the distance formula looks like. They felt pretty confident and acted like they couldn’t believe how easy it was to solve. My little teacher heart got pumping because they got it– without a short-cut. I could hardly believe it, especially because I work with students who are not confident in math. I think this is one of the harder things we do, but they thought it was easy!
Step 4- Writing conclusions from this discovery lab
Every discovery lab includes a conclusion section. Students need to synthesize what they learn, and writing their conclusions helps them do just that. From their conclusions I could see their thinking and understand their misconceptions. In addition, students put a lot of effort into this part of the lesson, usually. When they don’t, then I try to figure out why they’re resisting.
For example, several of my students didn’t write a question they still have. They didn’t want to write a question, so instead they wrote things like “nothing” as a response. So, I had them write their question on a post-it and then showed them on the doc cam.
Next Steps- Taking Notes in the Interactive Notebook for Distance Formula
I don’t know how I would survive without interactive notebooks. They provide such a great way to organize graphic organizers and kids can always refer back to them. For the distance formula, I used a foldable graphic organizer that shows an example of how to find the distance on a graph, and how to find it from just the coordinate points. It’s pretty simple and straightforward, but it provides a great place for students to refer back to and organize everything we just discovered in one place. We went through the notes together as a class.
At this point my students felt pretty confident. Many of them even completed the example ahead of me. This was the last thing we did before we started practicing. To read about the activities and resources we used to practice the distance formula, check them out in “9 Ways to Make Teaching the Distance Formula Awesome.”
Try it Out
This discovery lab was a great experience for my kids and I believe it will work for your students. My students became so confident on this topic that some of them were doing the calculations in their heads. It was amazing and I’d love to share that kind of success with you and your students.
Want to try this discovery lab out in your own class? You can find all the materials, ready to just print and go, here.
If you want to use the same foldable notes that we used, you can get it here.
Until next time, thanks for reading!
