3 Essential Steps for Teaching Interpreting Remainders Part 1 (EP15)

3 Essential Steps for Teaching Interpreting Remainders Part 1

Hello, everyone, and welcome to another episode of Elementary Math Chat. I cannot believe that we are already at episode 15. It is just flying by. Thank you so much for supporting the podcast and for joining me each week.

Because it is the end of October, I figured a lot of you were probably either getting ready for your division unit, or you are in division right now. So, I decided that this week would be the perfect time to do another content-specific episode, and so we are going to talk all about how to teach interpreting remainders.

This was just one of those lessons for me that every single year I was looking for something new and something better. Don’t you have those lessons where you’re always thinking like, is there a better way a more effective way to teach this lesson? That’s what interpreting remainders was to me.

Thankfully, I was able to make some really small changes and simple adjustments to this lesson that made a really big difference in the end, and so I’m going to share all of that with you today.

Before we jump into the details, I want to talk a little bit about why students struggle so much with interpreting remainders. When you think about this type of problem, they’re always asking about a field trip or about buying cupcakes for a party.

But here’s the reality. How many fourth and fifth graders have ever planned a field trip or gone to the store to buy cupcakes for a party? Probably none, because it’s the teacher who does all of the planning for the field trip, and it’s their parents that probably go to the store and buy the cupcakes.

So, I think the main reason why they struggle with interpreting remainders is that they just don’t have the real-world experience that these questions refer to. This is also why they may not understand why their answer is wrong. They might have the division part of it down, they probably do. But there’s so much more to these types of problems.

And then another reason why they struggle is they don’t understand how the parts of a division problem like the divisor, the dividend, the quotient. They don’t understand how those are related to the story. For example, if they’re doing a word problem and they have to divide 42 by eight, can they tell you what the 42 represents in the story? Can they tell you why they’re dividing by eight?

If they don’t know that they’re not going to understand how to interpret the remainder. I ended up making an activity to help with this just because I saw so many people struggle, and I knew I needed an activity to address this problem. So, I’ll share more about that activity at the end of this episode.

So, let’s start with one of those small changes that I mentioned that made a big difference. The first small change I made was that I decided that I could no longer teach this lesson in one day. I really needed two days. So, that was the first change that I made, and it was so much better because I could slow down a little bit and focus on one or two strategies per day. And I think it was easier for them to take it all in as well.

So, if you are trying to teach this in one day, I highly recommend breaking it down into two separate days. So, let’s talk about what you can do on that first day. 

I think a really effective way to begin this lesson is to have a little conversation around remainders, but don’t actually say the word remainder. Just ask your students to think about situations where something is either leftover or it’s not in an equal group.

They might say that they had leftover pizza from the night before, or they had uneven teams when they played a review game in class. Maybe it was Jeopardy and they were all in teams of four, but then there was one team that was a team of three. Or maybe they remembered from a class party that they had extra favors left over, but it wasn’t enough to give everyone another favor.

This is a great opportunity to point out that things don’t always come in equal groups. We often have things left over, and what we do with those leftovers all depends on the situation, and that’s what this lesson on interpreting remainders is all about. So, this is a great introduction to your lesson because it gets them thinking about the concept of remainders without even knowing the term remainder just yet.

So, let’s move on and talk about the main part of this lesson. My number one tip for you is to follow a CPA approach when you are teaching this lesson. You’ve probably heard of the CPA approach. It’s an acronym, and the C stands for concrete, P stands for pictorial, and A stands for abstract. You might remember back in episode two, I mentioned the CPA approach, and it is just a really effective way to teach a lesson, especially one as challenging as interpreting remainders.

Essentially, what the CPA approach does is, it has students start with a concrete stage, and this is where they use some sort of manipulative or hands-on material to model the problem. And then in the pictorial stage, they draw a picture, or they draw a model to represent the problem. So, they move away from modeling it to drawing it in that pictorial stage.

All of this then prepares them to solve it in an abstract way, where they just write in equation to solve, and that is the goal. I mean, we want them to be able to solve it in an abstract way. But we’ve got to take them through that CPA approach in order to get there.

So, I’m going to walk you through a few different examples using the CPA approach. Today, I’m going to focus on the concrete and the pictorial phase, since that’s what I taught on that first day. And then next week, I’ll take you through the abstract phase, which is what I did on that second day.

So, let’s go back and talk about the concrete stage. Again, this is where you’re going to begin by acting these problems out with some sort of manipulative. One year just for fun, because I always taught this lesson around Halloween, I went to the store and bought the M&Ms, the ones that come in the little small party pack, and that’s what we used as our manipulative. They were super excited about that. So, if you want them to get excited about math, M&Ms will definitely do it.

I think maybe 30 M&Ms came in the bag, so it was perfect. It was just enough. And of course, the first question they always have has nothing to do with math. It’s can we eat them when we’re finished? And I always say, Classroom A gets to eat them and Classroom B does not. It always circles back to Classroom A and Classroom B. You might remember that reference from episode one.

Now if you want to do candy, you could also use the small bags of Skittles. But if you don’t want to use candy, which I totally get, try using the dried pinto beans that you get at the grocery store. These worked really well because number one, they were small and they were easy to move around and put in groups. But number two, a lot of them come in a bag. I think I maybe grabbed two bags for my two classes.

And you know, I never would have considered them to be a manipulative, and I honestly can’t remember how I thought of it. It was probably Pinterest or something. But I tried them, and they worked well. So, that’s what I stuck with.

Like I said, they do come in a pretty large bag. So, before I used them, I divided them up into quart-sized Ziploc bags, and they shared these with their shoulder partner. I also gave them sticky notes to represent the number of groups as we modeled. And then I have them model all of this on a dry erase board.

So, we took our materials, and we modeled each problem by using the sticky notes to represent maybe the number of boxes, or packages, or vans, whatever the problem was about. And the pinto beans represented the total number of muffins, or people, or books that needed to be divided up. And of course, whatever beans were left over represented the remainder.

So, we modeled the division problem, and then we went back to the story to determine if we needed to add another sticky note to include all of the pinto beans. Or, did we not need the remainder and we could just drop it in our answer? And then we also discussed the possibility that the remainder was the answer. Those were the three ways that we interpreted the remainder in fourth grade.

Now if you want to add a little engagement to this lesson, you can surprise them by putting their names in these word problems. I mean, they can’t wait to tell you when it’s their cousin’s name or their friend’s name that’s in a word problem. So, having their own name is super exciting for them. Just a little bonus tip there.

All right, here’s an example that you can use with your class on interpreting remainders. Emma has 34 books she wants to donate to a used bookstore. She has small boxes, and each box holds five books. How many boxes does Emma need to hold all of her books?

So, in this case, the pinto beans represent the books, and the sticky notes represent the boxes. So, they’ll need to count out 34 pinto beans to represent the total number of books and then they’ll need to organize their pinto beans into groups of five. And remember that the sticky notes represent the boxes. So, they’re going to take their groups of five pinto beans, and they’re going to put them on top of a sticky note.

Make sure as you do this that you are constantly reinforcing what these manipulatives represent. These pinto beans represent the books. So, when they put them in groups, they’re not just putting pinto beans in groups, they’re putting books in groups. When they get their sticky notes out, they’re not just sticky notes, they are representing the boxes in this problem.

They have to know how these numbers connect to the word problem. Otherwise, they are really going to struggle to understand this concept.

When they’re finished modeling, they should have six sticky notes with five pinto beans on each note, and then they’ll have four left over. Some will see right away that they need to add another sticky note to store all of her books. But then others may not understand why the answer isn’t just six remainder four.

Now, I told you earlier that I made small changes to this lesson that made a big difference, and one of those changes was to write an answer statement. And I actually encourage you to do this before you even model the problem. So, going back to our example, the question is how many boxes does Emma need to store all of her books? So, the answer statement could be Emma needs _____ boxes to hold all of her books.

This is such an effective way to really help them see why answering something like six remainder four would not make sense in the story. Just read the answer statement and put that in there. It does not make sense. So, that will help them see that she needs one more box. She needs seven boxes.

So, I believe answer statements are a must when you’re teaching interpreting remainders, and that was the second change that I made to this lesson that made a really big difference.

Now with your manipulatives, make sure you model one example for each way to interpret the remainder, and then move on to the next phase, which is the pictorial phase. Remember in this phase, they are not going to model it with manipulatives. They are going to draw a picture to represent the division problem.

I do want to point out something that is super important and something that you need to consider before you start teaching this lesson. In third grade, they are used to drawing the circles first and then putting the dots in. So, let me give you an example.

Let’s say they are modeling 12 divided by four. They would draw the four circles first, and then they would put one in each circle, and they would count all the way until they got to 12. So, they would end up with three in each group, and that is the correct answer. So, that works fine for third grade.

But if they use this same process when they get to remainders in fourth grade, they may accidentally put the remainder in another group. And they may not realize that it’s left over. So, to prevent this from happening when you’re modeling, have them think of the divisor as the number in each group.

Let me give you an example of what I mean. Let’s say the problem is 12 divided by five. Instead of having them draw five circles and divide the 12 dots among the circles, have them make the groups of five dots first and then circle the groups. So, this way they’ll end up with two groups of five and then two left over.

Modeling this way is more appropriate anyway for fourth graders because when they get to long division, they think along these lines. You know, how many groups of four are in 34? Or how many groups of nine are in 87? Plus, it also aligns with thinking in terms of multiples, which is also how they think when they learn long division.

One more thing I want to mention in this pictorial phase is when they modeled the division problem, I had them label their leftover pieces with an R. That way they didn’t accidentally circle them and make them another group. They knew that they were separate, and they knew that they were the remainder.

So, let’s do one more example together, and I’ll take you through how I had them model the problem in this pictorial stage. Sarah has 29 pictures to place in a photo album. Each page holds six photos. How many photos are on the last page?

First, we’ll start off by writing that answer statement. So, for this one, you could write there are _____ photos on the last page. So, you’re just changing the question and rewriting it as a statement. And then next you move on to drawing a picture. Remember, you want them to draw the six dots first, and then circle each group until they can’t make another group of six. So, they’ll end up with four groups of six and then one final group of five to label as the remainder.

Now remember, they might be tempted to write the answer as four remainder five, and if they do this, have them read it within the answer statement so they can see that it just doesn’t make sense. Sarah doesn’t have four remainder five photos on the last page. She has five photos on the last page.

So, that takes us to the end of my first day. Again, we did the concrete examples, we did the pictorial examples, and then I saved the abstract examples for day two. Now, I did continue practicing the pictorial stage when they came to my small group, and I will put a link in the show notes to the activity I used. You could use it in your whole group lesson, but my lesson was already long enough, and so I just saved this for small groups. 

All right, well, I’m gonna go ahead and end this episode here. Next week, I’ll tell you a little bit more about that small group activity, and I’ll also share what I did on day two of interpreting remainders in that abstract phase. So, that means it is time for today’s teaching tip of the week.

Halloween is right around the corner, and you are probably looking for some fun ways to incorporate a little spookiness into your lessons, and I have the perfect small group activity to help you do this.

There are actually Halloween-themed ping pong balls that have these scary-looking eyeballs on one side, and then the other side is blank, and that’s where you’re gonna write in some math problems. So, take a Sharpie, write math problems on the back for students to solve, and then put them inside a bucket. Maybe you have a Halloween-themed bucket that you can use, but it can be any bucket really.

Just go ahead and put them in the bucket, and then they’ll have to dig their hands in there. Like they’re putting their hand and a bucket of eyeballs to grab the questions that they’re going to solve.

The only other thing you’ll need is something for them to put the eyeballs in. You could do a little basket or a little cup. Otherwise, they’re gonna roll off the table. So, just be prepared for that.

As far as what skill to pair this with, it really depends on what you’re doing this time of year. But I was always teaching division. So, I wrote a mix of two-digit by one-digit division problems all the way up to four-digit by one-digit. And what’s nice is the eyeballs come in three different colors. So, the red ones I wrote two-digit problems on. The yellow ones I wrote three-digit problems, and the blue ones I wrote four-digit problems. So, the colors represented the level of difficulty.

Now you can also have them estimate quotients. So, if you write the division problems on them, they can estimate it and then they can solve it. This works for both multiplication and division, as well as decimals. You can throw those in there as well if you teach fifth grade.

But if you feel like there’s not enough room for that, you can also just label them with whole numbers, and then they can list the factors of the number that’s on there. Or you can write a decimal and they can compare the decimals or order the decimals. Once you start thinking about it, you can come up with a lot of different ways to use these.

For the materials, you might be able to find these at Dollar Tree, but I’m sure they’re probably out of stock right now at this point, but Amazon also has them. So, I’ll have that link in the show notes if you want to grab a set of these, and I’ll also include a picture of the bucket that I used and how I put that all together if you need a good visual.

I even gave them a piece of gum when they were finished, and the gum also had an eyeball on it. So, it was super fun. Kind of disgusting if you think about it, but they loved it, and I’ll have that link in the show notes as well.

All right, friends, that is all for today’s episode. Have a great week, and I will see you next Tuesday for day two of interpreting remainders.