In last week’s video, we talked about three ways to help kids who are struggling with subtraction facts. One of the ways was to help them build the connection between addition and subtraction. But that is easier said than done. 

I’m Christina Tondevold, the Recovering Traditionalist, and today we’re going to take a look at Building The Connection Between Addition & Subtraction and Multiplication & Division in our quest to build our math minds so we can build the math minds of our students.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)

Last week’s video about How To Make Subtraction NOT So Hard

Susan O’Connell’s books Introduction to Problem Solving (pick the grade band appropriate to what you teach)

BrainingCamp Virtual Manipulatives

Now, building the connections between the operations is often referred to as inverse operations. The inverse of addition is subtraction. The inverse of subtraction is addition.  The same goes for multiplication and division. One of those big “aha moments” mathematically for me, was the moment that I realized I never have to subtract again. I can use addition to solve any subtraction problem. 

Now, sometimes it may be easier to just subtract, but the understanding that I can use addition to help me subtract was a huge game changer for me. I know that seems like a big overstatement but it really wasn’t. It really did change the way that I thought about mathematics and about the operations. Unfortunately, that moment didn’t come until I was an adult, but it really did help me to be able to do math mentally. 

I always struggled when I needed to do subtraction or division mentally. I needed paper and pencil to write it down and really go through the steps and algorithms that I had been taught to be able to figure out the answers. But once I learned that I could use my understanding of addition to solve subtraction, or my understanding of multiplication to help me solve division, I was finally able to do those operations mentally. 

Now I do multi-digit subtraction and division using those inverse operations but it all really starts when kids are young and we’re helping them see those connections between the operations using just small numbers. But no matter what grade level you teach, if your kiddos have not built this understanding of inverse operations, we really do need to do that for them. 

There’s three things that I feel like you can do to help them build that. 

#1: Focus on Key Concepts NOT Keywords

We need to help kids understand key concepts not key words. I know I was that teacher. I had the key words poster up there. I taught my kids to use keywords to help them solve story problems. But really it’s not about key words. It’s about helping kids develop key concepts. 

This image was a big part of what helped me better understand these inverse operations; when I really understood the key concepts.  

Once I understood these key concepts, it really did better help me understand how subtraction is related to addition, how division is related to multiplication, and even how addition is related to multiplication, and subtraction is related to division. It really does help me see how all four of them are connected. 

#2: Use Contextual Situations

When you put math in a contextual situation for kids, it really does help them better understand these key concepts. If I just have 3 + 4, it doesn’t really tell me anything about 3 + 4. That’s why we felt like we needed those keywords. But if we put it into a contextual situation, otherwise known as story problems, then kids actually have a context wrapped around those problems. So kids can solve problems when they’re in a context that they can’t do when it’s just numbers on a piece of paper, because they can actually imagine the situation and they can act it out oftentimes. 

So put the mathematics into story problems and then have the kids really delve into those key concepts, have them compare and contrast different types of situations. That’s what really helps them focus in on key concepts, and see the connection between the key concept of multiplication and the key concept of division. So here’s an example:

Yes, we want kids to be able to solve story problems but when we’re showing these two together, it’s more about doing a “same, but different” task with them.  We’re showing these problems and we’re asking “what’s the same, what’s different? What does that mean? What’s the key concept we’re doing in each?” And you can even model the problem alongside it. 

I’m showing an example with multiplication and division, but the same thing holds true for addition and subtraction. Show a problem that is addition and then use the exact same context, exact same numbers but turn it into a subtraction-based problem. Then talk about how the modeling of each is similar but different, how the actual problem is similar but different, and help kids build the connections between those.

#3: Purposeful Practice

Kids do need practice. They need practice with seeing the connections between these inverse operations, but it’s not just fact family practice. I have seen so many lessons, textbooks, and worksheets that are out there on the internet just of fact families. 

It isn’t enough just having kids see that 4 + 3 is in a fact family with 7 – 3. We want them to talk about in lots of ways like, what’s the same, what’s different? What connections can we build here? So doing a number string is really, really powerful to do this. 

I am picking these problems on purpose so that we can have a discussion about it. It’s not just having them solve to find all the answers. The power comes when we have the discussion about how these are the same and how they are different. 

So if I start out with 4 + 3, and the kids we talk about that it’s 7. The next problem is 7 – 3. I don’t want them to just give me the answer. I want to have a discussion about how these are the same as that last problem we just did. How is it different? How does 4 + 3 help you know what 7 – 3 is? You can move into 7 – 4, and you might even put 3 + 4 and you can talk about commutative property here. 

You can create these numbers strings in whatever fashion you want. But the idea behind the creation of this number string is yes, we are doing fact families, tt’s purposeful practice, but we are focusing on the discussion and the connections we’re helping kids build as they do it. 

It doesn’t have to be just the abstract version I have here. I made some images using the website brainingcamp.com. I love them for their virtual manipulatives. 

You could do all kinds of things to be able to animate these visuals. But the point is, it doesn’t have to be just the abstract version of the fact families. It can also be showing the connection between addition and subtraction, or multiplication and division using the models that kids are being exposed to. We want them to be able to have that purposeful conversation about the connections between these operations and that can come by doing the problems, but it can also come by the visuals that we help kids see. 

So when we are helping kids understand the key concepts, we’re helping them see connections between those contextual story problems, and we give them purposeful practice that builds the relationships and connections…We are helping them build an understanding of the inverse operations that is going to progress and develop into them actually understanding algebra. 

How many of us really understood algebra? The cool part about algebra is it’s a lot of just understanding inverse operations. If we can start that at a very early age to help our kids understand and become fluent with the operations, and the by-product is that they’re going to understand algebra even better, to me that’s a win-win!!

But reminder, it isn’t just about having kids work on their fact families. It really is about building that understanding of the operations and the connections between the operations and putting those types of experiences into place in your classroom that help the kiddos do that. 

I hope that this video has helped you build your math mind so you can go build the math minds of your students. Have a great day!

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