Updated June 18, 2025

You survived the wait! I’ve been teasing for the last 3 weeks, building up to the unveiling of Structures of Equality. Last week, you got a sneak peek. Today, you get a full introduction.

A Quick Recap

I created Structures of Equality when I became frustrated with common word problem solving approaches, and their many pitfalls.

One of the most effective strategies I found was bar models. But even they had limitations. I realized it was possible for a student to create a bar model and not fully grasp the context of a story problem. I knew the missing piece was a reading comprehension tool that would address this issue.

The Creation of Structures of Equality (SoE)

In addition to bar models, another approach with many merits is Cognitively Guided Instruction (CGI). The barrier for me was that there were so many problem types, I sometimes found it confusing to pick a model. I knew I needed to present students with a variety of problem types, but identifying each one by name (as suggested in CGI) was cumbersome and felt unnecessary.

So I took the best of both, along with a few other elements, and created Structures of Equality. I don’t often call things groundbreaking, but I truly think they are.

So…what are the structures and what makes them so special?

Structures of Equality (SoE) are graphic organizers that help students map out a math story. The 3 different SoEs are:

• Parts Equal Total
• Compare
• Repeated Equal Groups

A SoE is made of three crucial elements. They MUST:

• Contain values (numbers)
• Have labels (units of measure, the things you are counting…)
• Represent equality

When a student includes all three, it indicates comprehension.

These are the only models you need to diagram any problem type in the elementary Common Core standards.

Can my students do these independently?

Yes! After explicit instruction and consistent use, students will internalize the structures. They will have a reading comprehension tool to confidently work through any word problem.

If your students are not yet able to decode words on their own, you’ll still need to provide that support to them.

How do I get started?

You’ll want to approach teaching the structures systematically. Initially, your students will need support to feel comfortable. Over time, instead of frustration, they will be excited to solve problems that previously challenged them. 

There are a variety of free resources available on my website and YouTube channel to help you get started.

Conclusion

By combining bar models and Cognitively Guided Instruction (CGI), I was able to create something truly revolutionary. With these graphic organizers – Parts Equal Total, Compare, and Repeated Equal Groups – your students will have the tools and confidence to work their way through any word problem.