Structures of Equality (SoE) is a systematic approach that helps students understand what word problems are asking them to do. Although I don’t usually get into problem-solving strategies, I want to focus on helping you bridge the gap between “I know what the problem is asking me to do.” and “How do I solve it?”

Let’s explore the relationship between estimation and comparison in math.

Starting with the math main idea

Word problems that involve comparisons are often the most challenging to comprehend, especially when they involve the concept of ‘less’. But if you have students start by finding the math main idea, it becomes much easier. (This is why I like to call them number stories instead of word problems.)

By focusing on the math main idea, students can begin to break down the relationships in the problem. This process becomes even clearer when we use visual tools to model these relationships.

With the context of the math main idea, they can choose the best model to visually represent the relationships occurring in the story. For example, a story that describes comparing two separate sets, groups, or amounts can be modeled with a Compare structure.

In this article, we’ll focus on the Compare structure and how you can use the model—along with estimation—to  help students make sense of the relationship between the values.

The Compare structure is particularly effective when paired with estimation, allowing students to understand not just what the numbers represent but also how they relate to each other in practical terms.

If you’re new to SoE, learn more about Compare here.

Using the Compare structure and estimation

Let’s start by looking at an example number story.

Hudson has 14 Pokemon cards. Luke has 3. How many more Pokemon cards does Hudson have?

In this Compare structure, there’s a discernible difference between Luke’s cards and Hudson’s cards. When you first start having conversations around estimation with your students, it’s ideal to choose numbers that are far apart or very close together. This allows students to focus more on the mathematical concept and discourages the attempts to solve immediately.

To see this approach in action, let’s explore how estimation can help students understand a simple number story about Pokemon cards.

• Who has more Pokemon cards, Hudson or Luke? (Hudson)

• Use your hands to show me the value of Luke’s cards. (Hold up your hands fairly closely together to represent Luke’s 3 cards.)

Now show me the value of Hudson’s cards. (Move one hand to dramatically show the difference.)

Woah, what did you notice? That’s a big difference. I had to move my hand pretty far. It looks like Hudson has a lot more cards than Luke.

At this point, you want to pose questions focused on estimation and that allow multiple entry points, so all students can actively engage in the conversation. I’d use the questions below to guide conversation. They’re also great prompts for partner talk.

• Point to the model: They have an equal amount of cards up to this point, the line of equality. Then we can see Hudson has a lot more. We have to figure out the difference between Luke’s 3 cards and Hudson’s 14. First, let’s make silly guesses. What’s a number that’s too low for the value of this difference? (Point to the ‘more’.)

By starting with these silly guesses, students can begin to narrow down their understanding of what makes a reasonable estimate, building their number sense step by step.

• Let’s do another silly guess. What’s a number that would be way too big to describe this difference?

• Hmmm, so we said 1 or 2 would be way too small and 20 or 100 would be way too big. What could make sense? About how many more cards would you guess Hudson has?

When students later focus on finding solutions, activities like these help them make sense of their responses. When they draw their Compare structures, you can ask them to consider whether their response makes sense based on where the line of equality falls.

A word of caution: Not all students are developmentally ready at the same time. Some may draw Compare models with bars that don’t accurately represent the quantity of the values. For example, they may draw bars that are labeled 5 and 15 but are similar in size. This is okay. 

Our main goal with the structures is for students to visually represent the math main idea so they can comprehend what’s happening in the problem.

When you have estimation conversations with students, be sure to use models (that you or students have drawn) that are accurate visual representations of the difference.

Extending and modifying the activity

• Repeat the activity with values that are very close together, like 8 and 10. Eventually, as students get more accurate with their estimations, shift to models where the difference between the values is hard to estimate.
• Replace ‘more’ with ‘less’. 
• Use this as a warm-up activity. Context is very important with SoE, but so is helping students understand how to apply the models across standards. You can display a structure out of context and ask students to estimate the unknown values. Then, to bring a comprehension focus back, ask them to create a number story that could match the structure.

These extensions build on the foundational activity, encouraging students to refine their skills and adapt their strategies to more complex or subtle comparisons

Conclusion

Estimation is a powerful tool that not only engages students in meaningful mathematical conversations but also deepens their understanding of relationships in number stories. By combining estimation with visual models like the Compare structure, students can develop critical thinking skills and a stronger sense of mathematical reasoning.

These strategies encourage all learners, regardless of their starting point, to make sense of problems in ways that are both accessible and enriching. As you bring these activities into your classroom, remember that the goal is not perfection but growth—creating a space where students can explore, question, and build their confidence with numbers.