Every teacher has a topic that takes them time to figure out. The distance formula is one of mine- it’s taken me a few years to really figure out. I have wanted to make sure that my students understand it and not just that they can use it. This year, as I was planning for this topic with my I can statements I started to see it more clearly. As a result, my teaching of it was a lot better. I’d like to share with you how I used these I can statements to chunk the learning for students and keep them accountable.. Plus, I’ll share with you some really great lesson ideas that will help students really see and understand the distance formula.

## What I want my students to be able to do with the distance formula

In our district we have to teach students to find the distance between two points on a graph as well as when given two coordinate points. We make sure they are super proficient at the Pythagorean Theorem before we start. Therefore, they can do the one on the graph easy peasy. They just count the distance for the legs and then do the Pythagorean Theorem. The distance formula on the other hand is the Wicked Witch of the West. I was afraid of it at first, and it was definitely not something I remembered doing in school. When this happens, and we are pressed for time, and I am working with a group of students that struggle, I might look for short-cuts. So, a couple of years ago I came up with a really good one for distance formula and it got a lot of kids to pass a test.

Fast forward 3 years. This crazy thing happened to me. My district had an emergency at the high and they shifted me up to the high school for a month to teach Geometry. Guess who was in my geometry class? And guess what topic we learning while I was there? If you guessed all those kids I taught a short-cut to, and we were studying the distance formula, then you would be right. My own short-cut came back to haunt me because it turns out they didn’t remember the distance formula. At all.

## The I can statements for distance formula

This year when I started teaching the distance formula, I approached it using I can statements. This made a huge difference in how I chunked the learning for kids. Here are the I can statements that I had my students use and refer to in their interactive notebooks:

- I can derive the distance formula through an example.
- I know the distance formula.
- I can find the distance between two points on a coordinate graph.
- I can find the distance between two coordinate points.

So, we put a copy of these statements in our interactive notebooks and we reviewed them on a daily basis. It was a great opportunity for students to see where they were at with what they were learning. I love seeing them get excited when they realize that they are good at something now that they just started learning about a few days earlier. This is a straightforward, logical order to teach this topic.

## I can derive the distance formula through an example.

I’ll admit that having students derive the distance formula is a little advanced, but many of my students lived up to the challenge. The couple of kids that didn’t put forth much effort and gave up quickly eventually rode the coattails of others. But in the end, they all had a strong hand on the formula. I had them learn the derivation of the distance formula through a discovery lab. It was probably the most grueling of the discovery labs. It started off with a lot of blank stares from my students when I asked them to look for certain patterns.

In the end students could explain that they were finding the legs a and b and plugging them into the Pythagorean Theorem. It was a miraculous moment, and I only gave them a little bit of guidance along the way.

## I know the distance formula

In the past I just used tricks to get students to remember the distance formula. This year, though, the majority of my students could remember the formula without any tricks thanks to the discovery lab. For the students who were still having problems I used this example from Mrs. E Teaches Math. It is a face with eyes (parentheses), pupils (subtraction signs), a nose (plus sign), eyelashes (power of 2), and hair (the radical). This was a great reinforcer of the formula and it was a fun way to help students remember.

**I can find the distance between two coordinate points**

Finding distances between coordinate points can be difficult even if you have started by building a foundation of conceptual understanding. If they don’t have this foundation, then you will probably see you best students get it and most of the rest just get overwhelmed. Because we spent the time up front to play around with the distance formula and build understanding though discovery, my student already knew and understood the distance formula before we even tried to solve one problem. After the discovery lab and foldable notes we did a series of I Do, We Do, You Do activities. I like this format because it gradually releases the control to the students.

We completed 5 problems together. I released control on each progressive problem. On some of the middle problems I gave a head start to the kids who were comfortable and ready to go. Then, the less confident students did another one with me. This was very successful. I knew it was successful when I heard comments like, “When do we get to take the test? I’m gonna ace it!” or, “This is so easy, why didn’t we learn this in 3rd grade?” These kids struggle, so comments like this reassure me that this set of I can statements combined with the right activities really does work.

**I can find the distance between two points on a graph**

Oddly enough just giving students two points on a graph and assuming they will see that they could draw a triangle to solve doesn’t work. Often times when students learn something in isolation they have a hard time generalizing it to other situations. This is a great opportunity to get students to see of they can figure out how they could find the distance between the two points based on what they’ve learned so far. Some kids will see it, and some will not.

Of course, once I modeled one problem they all saw it rather quickly. They counted the distance of the legs, which gave them the a and the b for the formula for Pythagorean Theorem. It’s important for students to see examples with a variety of distances during their practice. Besides that, this is a pretty straightforward concept.

**Common Misconceptions**

Why do students struggle with the distance formula? First, there are many details that need to be remembered for the formula itself. If you don’t pay attention to detail, it messes the whole thing up. Usually, when students make this mistake they can figure out what they did wrong just by rechecking their own work. These mistakes are usually from doing too much in their head. They might forget to square the a and b, for example.

Second, integers strike again. If students lack proficiency in adding and subtracting integers then they will struggle with getting an accurate answer. We work on integers almost on a daily basis. By the time in the year when we do this topic, students have usually gained proficiency with integers. This is something you have to be thinking about all year long because it is a part of so many concepts.

Finally, some students almost refuse to write down all of the steps and along the way they forget to square something or square root at the end. Requiring students to show work and not just final answers can combat this misconception. This includes making sure that they subtract from the same x as the y. If they flip them around they will also end up at an incorrect answer.

**Take-aways and getting the show on the road**

Chunking the learning like this and being very clear about objectives is essential. Student canalso keep track of their learning as you move from one I can statement to another. My biggest take-away from this unit was that students are much more likely to remember the formula if they are the ones who discovered it. You can read more about this distance formula discovery learning activity here.

If you are interested chunking the learning in a similar fashion, please use or modify these I can statements. Also, be sure to check out more ideas for teaching the distance formula and all our resources for distance formula here. Thanks so much for reading. Until next time!

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