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’ve 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 throughout this unit felt 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.
To be completely honest, I was afraid of it at first, and it was definitely not something I remembered doing in school. When this happens, and I’m pressed for time, and 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. It was so good that it got a lot of kids to pass the distance formula test.
Fast forward 3 years and this crazy thing happened to me. My district had an emergency at the high school 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 my former students didn’t remember the distance formula. At all. Yikes!
This painful realization moved me from trying to find a short cut to really helping students understand more deeply how to solve using the distance formula.
The I can statements for distance formula
This year when I started teaching the distance formula, I approached it using I can statements. This has 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 their level of understanding was with everything we were learning. I love seeing students get excited when they realize that they’re good at something now, even though they just started learning about it a few days earlier. These statements provide a straightforward, logical order for teaching this topic.
(You can grab all of my 8th grade math I Can Statements, including these for the distance formula, right here. This download also includes blank templates for any special topics in your classroom.)
#1- 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 handle on the formula.
To walk them through the derivation of the distance formula, I used a discovery lab (read more in this post). It was probably the most grueling of the discovery labs I’ve done with them. The activity started off with a lot of blank stares from my students when I asked them to look for certain patterns.
Despite the rough start, students hit their stride. 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.
#2- I know the distance formula
In the past I relied on 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’s a face with eyes (parentheses), pupils (subtraction signs), a nose (plus sign), eyelashes (power of 2), and hair (the radical). This was a great way to reinforce the formula, and students thought it was a fun way to remember.
#3- I can find the distance between two coordinate points
Finding distances between coordinate points can be difficult, even if you’ve started by building a foundation of conceptual understanding. If they don’t have this foundation, then you’ll probably see your 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?” Many of my students typically struggle, so comments like this reassured me that this set of I can statements, combined with the right activities, really does work.
#4- I can find the distance between two points on a graph
Oddly enough, just giving students two points on a graph and assuming they’ll 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 make a connection. I like to ask them if 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’ll 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. Integers is really something you have to be thinking about all year long because it’s 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 with I Can Statements and being very clear about objectives is essential to having a unit run smoothly. Student can also keep track of their own 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’re 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!
