Why This Big Idea Matters

When we first introduce numbers to students, it’s easy to stay on the surface. Seven means seven things. Five is five objects. But if that’s where the understanding stops, we miss an opportunity to help students build real number sense. One of the most important concepts students need in early math is this: numbers are composed of other numbers.

This idea shows up everywhere in elementary math. When students know that numbers can be broken apart and put back together in different ways, they can think more flexibly and solve problems with greater understanding. This foundational concept is a critical part of both The Fire and Wire Way, my book of daily routines, and the Structures of Equality framework.

Starting with Exploration, Not Equations

When we introduce number composition, we’re not jumping into formal equations. We’re not asking students to memorize number bonds or write expressions right away. We’re helping them explore, observe, and internalize relationships between quantities. This kind of sense making happens long before symbolic math ever shows up.

Let’s look at how we begin to build that foundation with a couple of early activities.

Activity: How Many Bears?

In this activity from The Fire & Wire Way, students explore different groupings of five bears. Sometimes the bears are all in a row. Sometimes they’re grouped as two and three. Each time, we ask: Do we still have five bears?

This simple question helps students develop two big ideas. First, that the total doesn’t change just because the arrangement does. Second, that a number can be made up of smaller parts in different ways. Without writing any equations, students begin to see that five can be made of two and three, or four and one.

This is the beginning of composing and decomposing numbers. It’s also an early entry point into understanding equality, even if we’re not using the equal sign yet.

Activity: Plates and Forks

This activity takes the idea of number composition and adds context. You show students five plates, but only four forks. Some plates have a fork, some don’t. How many have a fork? How many don’t?

Students are now decomposing five into meaningful parts: the number of plates with forks and the number without. And they’re doing it in a real-world context they understand. This gives purpose to the decomposition and helps students talk about their thinking with clarity.

You might hear a student say, “There were five plates. Four got forks. One didn’t.” That’s the beginning of explaining a number relationship. Still no formal equation needed.

Subitizing: The First Glimpse of Structure

Subitizing is often the first time students experience the idea that numbers are composed of other numbers. When students look at a group of dots and instantly say “four,” they’re using perceptual subitizing. When they see two dots and four more and know that makes six, that’s conceptual subitizing.

Conceptual subitizing helps students see parts and wholes at the same time. It’s how they begin to trust the structure of numbers and move beyond counting. And it’s an essential step toward flexible thinking.

From Subitizing to Teen Numbers, Place Value, and Beyond

The idea that numbers are composed of other numbers is foundational for some of the most important concepts students encounter in later grades:

Teen numbers: Students need to see 14 as 10 and 4, yes, but also as 7 and 7 or as 7 and 3 to make 10 with 4 leftover. Teen numbers are some of the first places students work with place value, and we don’t want to limit them to one way of thinking. We want to expand their flexibility.

Place value: It’s not just that 63 is 60 and 3. We want students to understand that 63 can also be 50 and 13, or 40 and 23, or even 30 and 33. The goal is to help students decompose and recompose numbers in ways that make sense for the situation, not to stick to one right answer.

Fractions: Later, students will need to see that 1 whole can be made up of 2 halves or 4 fourths. Or that 3 fourths can be broken into 1 fourth and 2 fourths. If students haven’t developed comfort with number composition in whole numbers, fractions become much harder to grasp.

How This Connects to Structures of Equality

The idea that numbers are composed of other numbers shows up in all three SoE models:

The SoE Overview Poster is available as a free resources on my website: https://structureofequality.com/resources/soe-overview-poster/

Parts Equal Total (PET): When a math story describes parts being composed to form a total or a total being decomposed into parts, it’s represented with a PET model. Understanding composition is essential here. Whether they’re figuring out what’s missing or explaining how two parts make a whole, they’re relying on their understanding of how numbers are built.

Repeated Equal Groups (REG): When a math story describes composing a total from equal groups or decomposing a total to figure out the number of groups or the size of each group, we use a REG model. This is just a more structured version of number composition, but the idea is the same.

Compare: This one might seem different, but it still relies on decomposition. When students use a Compare model, they’re identifying the point where two amounts are equal, and then figuring out the difference. That point of equality is a part of the larger amount. They’re breaking down a number to make sense of the relationship.

Across all three structures, students are building and breaking apart, or composing and decomposing, numbers. They’re making sense of how quantities relate to one another. And they’re doing it through reasoning, not memorization.

How This Supports Standards and Practices

These ideas also align closely with standards that emphasize number relationships and operations. Whether you’re using Common Core or state-specific standards, this work supports Mathematical Practices like MP2 (Reason abstractly and quantitatively) and MP7 (Look for and make use of structure). When students explore how numbers are composed and decomposed, they’re developing the kind of thinking that underpins every major math concept they’ll encounter.

Why This Matters

When students internalize the idea that numbers are composed of other numbers, they:

• Become more confident problem solvers
• Develop mental flexibility that supports future concepts
• Are more prepared for operations, fractions, and algebraic thinking

They don’t just follow steps. They understand what numbers mean and how they work. If you’re working with older students who struggle with operations or fractions, revisiting number composition through these kinds of activities can be a powerful way to fill foundational gaps

And it all starts here: with five bears, five plates, and a willingness to let students play with numbers before writing them down.