In most classrooms, students are used to seeing word problems that end with “How many in all?” and jumping straight to addition. But without first understanding the story, they’re just guessing. That’s where the Parts Equal Total (PET) structure comes in. Not as an operation shortcut, but as a tool to support comprehension.

Understanding the structure

A PET model can be used to represent stories where the math main ideas describes parts composing a total, or where a total is decomposed into parts. Like all SoE structures, it includes 3 essential components:

• Values
• Labels and descriptors
• A representation of equality

It’s not about keywords or where the unknown is. It’s about how the quantities relate. PET helps students visualize those relationships before they solve. When students draw models without understanding, they might label parts incorrectly, combine unrelated units, or misinterpret the question.

Starting with concrete representations

In early grades, students need time to build this understanding. We start with real objects and real action, connecting stories to what they can see, hold, and act out. We use apples, milk cartons, and sentence strips to represent what’s happening in the story. Only after that do we move to drawings that reflect the story, not just numbers.

Here’s a typical K–2 example:

Liam saw 3 bluebirds in the morning and 5 more in the afternoon. How many bluebirds did he see that day?

The story describes putting parts together and is best represented with a PET structure. Students might start with bluebird cutouts, then use cubes, and eventually move to a visual representation, like the one below:

Consistency across the grades

Across the grades, the PET structure stays consistent. What changes is the complexity of the numbers: larger values, fractions, or decimals—but the structure still helps students make sense of what’s happening.

They learn to draw part bars, label them clearly, and represent equality. The drawings don’t have to be perfect. In fact, we expect to see different orientations, mismatched bar sizes, or color-coded labels. What matters is that students can explain their model and retell the story using it.

Here’s an upper grade example:

There are 74 fifth graders and some fourth graders at the assembly. If there are 129 students total, how many are fourth graders?

Because the math main idea describes parts composing a total, this is still best represented by a PET. The total is known, one part is known, and one part is unknown. What matters is that students understand the relationship and show it in their model, using labels and descriptors.

Listening for reasoning

This kind of reasoning helps students move from “What operation do I use?” to “What’s happening in the story?” That shift is critical. Instead of guessing based on keywords, students learn to ask better questions: Are we putting parts together? Are we figuring out what’s missing? What’s the total?

PET also helps us identify common misunderstandings. If a student adds two values just because they’re in the problem, we can ask, “What does each number represent?” If the model mixes labels like “butterflies” and “caterpillars” but calls the total “insects,” we know to check for unit consistency. These are teachable moments, not errors to fix, but thinking to explore.

Retelling shows real understanding 

One of the most powerful strategies is asking students to retell the story using their model. If they can explain the relationships (what each part means, how they relate to the total, and what’s being counted), we know they’re on the right track. If they can’t, that’s a sign we need to go back to the story.

In the end, PET isn’t about drawing neat boxes. It’s about helping students build flexible, conceptual understanding of how parts relate to a whole. That kind of comprehension supports every number story they’ll encounter and gives them a way in, whether they’re confident readers or still developing those skills.

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📗If you’re reading this before summer 2026 and would like a deeper explanation or PET, join the interest list so you’re the first to know when the Structures of Equality book is available. If it’s after, head to structureofequality.com to find out how to get your copy.