Let’s start with a number story that’s similar to a lot of the stories we see in elementary math:

There are 6 bags of apples. Each bag has 4 apples. How many apples are there in all?

Most students will grab the numbers and try to do something with them: add, multiply, guess. And it’s not their fault. They’ve been taught to look for “what to do” before understanding “what’s happening.”

But what’s actually happening here?

This story isn’t just about apples. It’s about composing equal groups, which is one of just three math main ideas students will see in K–5.

At Structures of Equality (SoE), we teach students to slow down and name the relationship before choosing a model or an operation. We use three consistent visual frameworks—Compare, Parts Equal Total (PET), and Repeated Equal Groups (REG)—to match those three big ideas. 

While the structures all share the same components…

• values
• labels with descriptors,
• a representation of equality)

…they show different kinds of relationships. And understanding which relationship is happening in the story is where problem solving begins.

In this blog, we’ll begin to explore REG.

Why the Structure Matters

Students are often taught that a story like this one is a multiplication word problem. Or maybe a division problem. But those are just operations. What students really need to see is the relationship.

At Structures of Equality, we call this the math main idea. That’s the core relationship the story is describing.

• Composing parts to form a total or decomposing a total into parts
• Composing equal groups to form a total or decomposing a total into equal groups
• Comparing two distinct sets

When students identify the math main idea, they’re not just solving. They’re understanding.

Repeated Equal Groups is the structure we use when the math main idea involves equal groups being put together or taken apart. Think bags of apples, stacks of chairs, teams of kids. The key is that each group has the same value, and we’re either composing or decomposing those groups.

When students start by naming the math main idea, they begin to make sense of the problem before they do anything else.

It’s Not About the Operation

This is a mindset shift for a lot of teachers. When we see a problem like 6 bags of 4 apples, we immediately want to connect it to multiplication. But that’s not where students need to start.

Instead of labeling it as a multiplication problem, we teach students to name the relationship.

We say, “The math main idea of this story is about equal groups being composed to make a total.”

Why? Because students don’t always see multiplication or division right away. But they can notice equal groups if we help them slow down and look for them. And when they do, they start to understand what those operations really mean, not just how to perform them.

This is how we shift from completing a procedure to understanding what that procedure represents.

What REG Looks Like in Classrooms

In early grades, students might model a math story with bags of apples, rows of cookies, or stacks of books. As they move up, they might use drawn arrays, or what we call a fast array, a strategy that helps students model repeated groups more efficiently.

REG, like all structures, needs to be modeled in concrete ways before students can represent it on paper. That’s why we start with objects, cubes, strips, and finally, student-drawn models that make the relationship visible.

I’ve put together a playlist that breaks this all down with classroom examples.

But no matter what tools you use, there are two questions we can always ask: What’s the math main idea? Does the model match the story?

These questions help students build a habit of thinking, not just solving. When the math main idea is clear, students are able to draw visual representations that match the story and that they explicitly know how to connect to that story.

A Closer Look at REG

Let’s go back to this number story:

There are 6 bags of apples. Each bag has 4 apples. How many apples are there in all?

Instead of jumping to multiplication, we teach students to name the relationship first. 

“The math main idea of this story describes composing equal groups to form a total.”

When you first introduce REG, this may be as far as you get. Over time, students will be able to draw visual representations that match the situation as it’s occurring in the story, such as the example below. When they can do this, you’ll know they truly have comprehension.

What If My Curriculum Doesn’t Use This Structure?

If your curriculum doesn’t use the SoE language or tools, that’s okay. This isn’t a program. It’s not a fad. It’s a way of helping students make sense of the stories they’re already seeing.

You can start small.

Instead of jumping to a strategy or computation, ask questions like:

• What kind of relationship is happening in this story?
• What do you notice about the values?
• What’s the math main idea?

Then, model the structure alongside whatever your curriculum suggests. Over time, this habit of thinking helps students make better sense of any problem, regardless of what resource you’re using. When students build the habit of identifying the math main idea, they begin to approach every problem with more confidence and clarity.

Want to go deeper?

This post is just a glimpse into what Structures of Equality is all about. The full book drops Summer 2026. Inside, we take a much deeper dive into each of the three math main ideas and how to teach them with clarity, scaffolds, and student-centered language.

• If it’s before Summer 2026, head to my website to join the interest list.
• If it’s after Summer 2026, head there to grab your copy and start using the tools in your classroom.