As part of my focus on the CRA approach to teaching mathematics, I’d like to spend some time in today’s video talking about the R, which stands for representations. Today, we’re going to investigate two math representations that you need to be using throughout all of elementary school, the number line and the bar model.
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So just a refresher of what the CRA model really is.
So it’s the concrete, the representational and the abstract. If you’ve never heard of this, here’s a link to the video that I did all about the CRA model.
But the idea is, kids need to be using objects, which is that concrete phase. They need to have visuals and models to represent the mathematics that’s happening, and that’s the representational phase. Then the abstract phase is when they are working in just the symbols. Now there’s a time and place where we’re doing one of these more than the other like we’re just in the concrete phase. There are times when we’re just in the abstract phase, but oftentimes it’s nice to be working in all three at the same time, which is what I call the sweet spot.
Well, today we’re going to talk about some of my favorite representations. These are ones that last throughout all of elementary school, and that’s why I feel like they are so powerful. So, if you aren’t using number lines and bar models, I hope that this video will help convince you that you should be.
So let’s start off with a little bit of warning, I guess. This comes from the math practice standards, and it’s practice area number five about use appropriate tools strategically. So, tools often fall in that representational phase, and one of the hard parts is that we are often the ones telling the kids it’s time to use this model, time to use this model.
One of the important points of this standard is that proficient students start to make decisions about when each of these tools might be helpful. We shouldn’t always be telling them, solve this using a number line, solve this using a bar model, or solve this using whatever.
These are representations and tools that we want to help kids see the benefits of, but then kids need to decide when they are going to use these tools and these representations. It shouldn’t always be dictated by us and, if kids prefer one tool or representation over another, that’s okay.
Just because it says, solve using this, as long as the kids can solve it and they’ve got a way that works for them, that is okay. So that’s my little disclaimer before we get started.
Number Line/Number Path
Let’s start off with the number line, and as you can see this is not a number line. But it’s the precursor to a number line. Kids in kindergarten are working with physical objects, and we start to get them to line them up because it becomes a number line later in the grades. So, kindergarten and first grade actually should not be using a number line.
There was wording in the original Common Core Standards that talked about, it specifically called it out, number lines should not be used before second grade. Now there is no wording in a lot of standards that says that, but I want you to check and see because I can almost bet that there is no mention of number lines in your standards until second grade.
Until then, kids should be using actual objects to model, and then we can turn this into a representational phase using what’s known as a number path. Kids still need to see the actual objects, which a number path has those rectangles that kids could still see five things when they are showing five, or whatever it might be.
As they move into number lines, number lines become more abstract. It’s a length view of a model that kids aren’t really seeing five things, it’s the space from zero to five that’s supposed to represent five, and it becomes really abstract for our young kiddos.
But number lines are very important and an essential model to allow kids to be able to visualize some cool mathematics, and eventually, they get to the point where they’re able to work fluently in a number line. The image above shows an example of an open number line because it does not have all of the tick marks that a number line typically has.
I want to just share a few types of problems and have you think about how it makes a difference if we can model this on a number line. I’m not going to go through all the way things that you can model using a number line because it would take forever. I just want to give you a little glimpse into some of the things that are difficult as kids progress up into their mathematics that, if they can visualize these types of tasks on a number line, it can make the mathematics so much easier.
In the grades before they actually get to this actual content, it’s helpful that they are using number lines in lots of different mathematical concepts. Here’s a typical one, where kids have to order fractions from least to greatest. and they will be given crazy fractions. Are we really expecting kids to find common denominators on this to be able to compare them? No.
If you can get kids to visualize where these fractions would fall on a number line, that automatically puts them in order from least to greatest. Now I’m not going to go through and show you all the answers to this.
I just want you to take a second and work on it yourself. Think about how much does that make a difference on this kind of a problem. We’re not jumping to the abstract algorithm of find common denominators and then you can order fractions. Let kids use a representation to help them solve that problem.
Here’s another one that I personally hated in school because it was taught that you have to pull them out, you have to create the equation, solve for the missing part, and it just seemed so abstract to me.
Here’s the problem: I got a dress on sale for 40% off and I paid $36 for the dress. What was the original price? Now there’s a little bit of layering in here. Kids have to realize that that $36 was really 60% of the original price because it was 40% off. So, it’s a little tricky in there.
But, if they can model this on a number line, that, if 100% was the full price of the dress, and what I paid was only 60%, would this help them have an avenue to be able to solve that problem without having to jump to the abstract, create the equation, solve and do the procedure?
It sure helped me. This was a game changer. When I saw finding percentages on a double number line like this, it totally changed my life, seriously. I love doing problems like this now because I can see the relationships that I could never see before. And that’s really what working in the CRA continuum, I guess, is supposed to do for kids. It helps build these relationships and connections that kids often don’t see when they’re just working in the abstract area.
Bar Model
All right, my other favorite is a bar model. Again, kids will start off with using individual objects, but it’s so nice to be able to take those and attach the bar of those objects.
One of the examples in the images is using the bar model to show addition, two parts coming together to make a whole. Some people call this a part-part-whole model, some people call this a strip diagram. There are lots of terminologies that people will call this. I tend to use the bar model.
The other example is showing multiplication using that bar model. The new term, it’s not new anymore, but it was so foreign to me that, when I first read it in Standards, it said a tape diagram. There are lots of different names for this model, but the idea is, you’re drawing the rectangles to represent whatever they were doing in that concrete phase.
You can also do it as they move forward with fractions. The thing that I love about number lines and bar models, the reason I really want you to include them, is they span all the grade levels, and it works for all different types of numbers.
No matter what kind of number sets you’re using, they work. It even works when you start getting into variables. You can have one of those parts in the bar model being represented by an x. It’s a model that will last throughout all of their mathematical careers.
Another type of problem I want to show is this one because, again, we could do all kinds of problems with bar models, but I just want to show you these ones that, I don’t know if you hated them, but I hated them.
It was because I was only taught to do it abstractly. I was supposed to pull these out, figure out how to represent it using variables, and create the equations and solve it. Nobody taught me to draw models, draw a representation of this problem.
Jose is twice as old as his sister Carmela. Carmela is five years older than her brother Silas. Silas is six years old. How old is Carmela and Jose? That seems like a big old jumbled mess of information to our kids. But, if we can pull the pieces out and be able to model this with a representation, it becomes so much clearer of what that information is telling us.
So it might look something like this.
This is one that I drew. Yours might look a little different. Once I put that information into a visual model using those bar models, now I’ve got a place to start. I didn’t even put in the fact that Silas is six. As soon as I put that in, then I’m good to go right here. Before I had that model, these kind of problems were just so tough for me. And they are really tough for our students too.
Alright, well I hope that this video has given you some new ideas of some representations to use and the power of those representations and that you can go back and use those to help your students build their math minds.
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