Structures of Equality (SoE) are powerful tools that enhance reading comprehension in math, helping students make sense of number stories. By identifying the math main idea first, students focus on the context of the story and then use models to visually represent what’s happening. The first model we introduce to students is usually the Parts Equal Total (PET) structure. 

If you’re not familiar with the structures, you can start here:

What are Structures of Equality?

Parts Equal Total 101: What is a PET structure?

When visually representing number stories using SoE, the total bar in a PET structure often causes confusion for students. Without a solid understanding of how parts compose a total, sometimes students struggle to understand the relationship between numbers in a math story. This is why it’s important to introduce PET structures gradually and intentionally. Let’s talk about how you can do just that.

Start Simple: Focus on the Parts Bars

One of the most common mistakes when teaching PET structures is introducing the total bar too early. Instead, begin with just the parts bars to keep things simple and focused. This slower approach allows students to build confidence and understanding before tackling more complex relationships.

Start by introducing the context of the math story. For example, let’s look at a story about Josh, who has two groups of marbles: some pink and some black. Begin by asking students questions about the characters, the action taking place, and what they are counting. Think of it as setting the stage for understanding the relationships within the story, or the math main idea.

Using physical objects like pink and black cubes can make abstract concepts tangible for students. By physically representing the parts of the story, they can visualize and manipulate the elements, which helps them connect to the problem. This hands-on approach helps students understand what the parts bars represent. Once they’re comfortable with this hands-on representation, they’re ready to explore the concept further.

Build Context Before Adding the Total Bar

Adding the total bar too soon can overwhelm students and lead to misconceptions. How many times have we seen students latch onto the total as the only important number and completely miss the parts? That’s why it’s key to focus on how the parts come together first.

Using Josh’s marble story as an example, show students how pink and black marbles are separate groups that are combined to form a total group. Reinforce this concept by identifying the units orally. For example: “This pink cube represents one pink marble.” Repeating this helps students connect the pieces of the story before introducing the visual representation, and keeps the focus on the relationship between the parts and the total.

Transition to Visual Representations

Once students are comfortable with the parts, you can transition to visual models. Begin by organizing the parts in a linear fashion and drawing simple representations.

For example, you can draw circles or ovals to represent the marbles. Use colored markers to distinguish between pink and black marbles, mirroring the physical objects you used earlier. Add rectangles around the groups to emphasize their connection. Gradually, students will see how the visual model mirrors the physical objects.

In this model, each black circle represents one black marble. When we modeled the story using cubes, we used one black cube to represent one black marble. This image mirrors the hands-on representation. The consistency helps make the connection more explicit for students.

Here’s a tip: Allow flexibility in where students write values, as long as the total is visually clear. This helps students focus on the relationships in the model instead of worrying about following a strict format.

Introduce the Total Bar

Once students understand the parts, introduce the total bar into the model. Explain that the total bar shows how the parts come together to form the total group. Use guided questions to ensure understanding. For example: “What does this circle represent?”

Encourage students to explain the relationships within the model. For instance, “These three circles represent Josh’s black marbles, and these two circles represent his pink marbles. When we put them together, we can figure out the total.”

Check understanding by asking students to explain what each part of the model represents. This ensures they grasp the relationships before moving forward.

Eventually, you can transition into drawing the PET structure using bars to represent the parts and total.

📹 If you’d like to see this process in action, I walk through what it looks like in “Getting Started with Parts Equal Total.”

Strengthen Understanding Through Retelling

A great way to solidify understanding is to have students retell the math story using only the model. This exercise requires them to think critically about the relationships within the story and the PET structure.

For example, ask students to describe the story of Josh’s marbles by pointing to the model. If they can explain how the parts and total relate, they’ve mastered the concept.

Use retelling exercises to reinforce comprehension and build student confidence. Encourage them to describe the relationships in their own words. Not only does this strengthen understanding but it also helps students internalize these mathematical relationships.

Conclusion

Introducing the total bar in a PET structure doesn’t have to be a source of confusion for students. By starting with the parts bars, building context, and gradually transitioning to visual models, you can help your students develop a clear understanding of how parts compose a total. Retelling the story using the model reinforces these concepts, ensuring long-term comprehension. By taking this gradual approach, you’re setting your students up for success as they begin to encounter more complex problems down the road.