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# Focus on Multiplication and Division as Inverse Operations

During my first few years teaching third grade I didn’t teach multiplication and division as inverse operations. Instead, I taught multiplication and division as separate units. I taught the meaning of multiplication. Then I taught how to multiply by each factor until we completed the multiplication unit. After that, I taught about division as a completely separate unit. At some point we worked on fact families, but this seemed like more of a side note than the heart of the instruction. This means that many students missed how multiplication and division were connected. We know that this connection is important, especially as math problems become more sophisticated. When students understand this relationship, they can be more flexible with their thinking.

Now, I teach multiplication and division much more intertwined. My students quickly realize that if you can multiply you can divide too. My current math series also addresses multiplication and division together, which makes it easier to focus on the fact that multiplication and division are inverse operations as central to students’ understandings. The best is when students starting using multiplication to divide or division to solve for multiplication.

## Introduce Multiplication and Division Separately

I do keep the operations separated at the beginning. We start by focusing on the meaning of multiplication and then on division. We start with modeling multiplication with pictures, on a number line, with repeated addition, and arrays. Click here to read about some of my favorite multiplication strategies.

Once when we have had time to understand multiplication, we start with the meaning of division. Soon after, I ask students if they notice any similarities between the division model and multiplication model. We explore how the models are the same. It is important to emphasize that the method to create the models is different, but that the end model will be the same.

**A misconception students often have is creating a multiplication model for both multiplication and division situations. For instance, if the equation is 8 ÷ 2, students might make 2 groups of 8 in each instead of sharing 8 one at a time in each of the two groups. In class, I really emphasize this difference and we discuss why each model is created differently.

## Fact Families Help The Inverse Operation Connection

After exploring the meaning of multiplication and division with my students, we start working with fact families. We spend a day or two devoted to playing with fact families, looking at the 3 numbers used for each fact family, and discussing patterns.  While there is a special focus on fact families for a day, we will revisit this idea in almost every other multiplication and division lesson.

Now, we start exploring different specific strategies and patterns for each factor.  Immediately following, we work on division with that number as a divisor. Throughout our work, we relate multiplication to division and division to multiplication. This also ties in nicely to solving for unknowns in the middle of different equations.

## Use Symbols for Fact Families

Even with multiplication and division lessons alternating, some students still need extra practice and extra modeling to make the connection. To help aid this, I tie in fact family practice throughout the units. One of my favorite strategies is using symbols instead of numbers to see how different numbers are related, like how we use nonsense reads to check for decoding skills. Using all symbols helps me see if students understand how numbers and equations are related. For example, I could draw that heart multiplied by smile equals tree. If that is true, can you figure out what tree divided by heart is?

Students love coming up with their own pictures and this gives me invaluable information about how flexible my students’ thinking about multiplication and division is. It also makes sure that students haven’t just memorized facts, but that they are thinking about the relationship between operations.

## Practice Multiplication and Division Skills Together!

If you are looking for an engaging multiplication and division activity, check out this interactive activity that has students use multiplication and division equations as their names as they walk around the room and look for an answer who meets certain criteria. With the answers, they will work on solving additional problems, some of which focus on inverse relationships! Just click here to grab the freebie!

I would love to hear about your favorite way to help students develop their understanding of multiplication and division as inverse operations!