What Are Open Middle Math Problems?

What is an Open Middle problem?

“Open Middle” might sound like a strange name for a type of math problem, but an Open Middle task refers to a very specific type of math task that encourages deeper thinking and creative problem-solving. Open Middle problems have:

• A “closed beginning,” meaning that students all start with the same initial problem.
• A “closed end,” meaning that students typically end with the same answer.
• An “open middle,” meaning that there are multiple ways to approach and ultimately solve the problem.

Open Middle problems generally require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. Unlike many traditional problems where students follow a straightforward procedure, Open Middle tasks require students to explore different strategies, test their ideas, and refine their approaches, making math feel more engaging and interactive.

Why use Open Middle math problems?

Open Middle math problems help students develop mathematical reasoning, critical thinking, and perseverance because the path to the solution is usually not obvious. These problems also promote rich math classroom discussions, as students compare strategies and justify their thinking. By engaging in Open Middle problems, students strengthen their conceptual understanding and improve their ability to approach unfamiliar problems with confidence. Plus, they provide teachers with valuable insights into student thinking beyond just correct or incorrect answers.

What is the difference between open-ended and Open Middle problems?

When all students get the same solution to a math problem but used many different strategies for solving it, discussing the strategies can lead to wonderful conversations. That math discussion is more challenging to facilitate if students used different strategies and got different answers.

When a problem has a closed end (so it’s not open-ended), it means that students arrived at the same answer. And when the middle is also open, then they have the potential to use many creative strategies. Compare this with a closed-middled problem where it tells you a specific strategy to use, like the standard algorithm.

Personally, I prefer problems that have open middles and closed ends. These problems all have the same answer and allow for great discussions around the strategies students used. When the problems have open ends, it can be challenging to have discussions because you can’t readily compare how different approaches ended with the same result.

To be clear, I am not saying that open-ended problems are bad and open-middled problems are good. What I am saying is that each have their own purpose and using Open Middle problems that encourage a variety of problem-solving strategies is worth including.

What are some strategies for teaching Open Middle math problems?

Open Middle problems can lead to wonderful classroom experiences, but they take a little more planning to achieve better results. Some tips to consider include:

• Do the problem ahead of time, ideally with colleagues! When has a lesson ever gone exactly like you hoped where every student gets it right? The reality is that these problems are challenging, and students will correctly and incorrectly solve them. If you get the problem right on your first try, you’re not ready to handle all the incorrect strategies students might encounter. So, I highly recommend that you try to anticipate the most common correct and incorrect strategies students will use. This will help you to prepare with teaching the lesson and allow you to assist students more easily.
• Use an Open Middle worksheet. I have templated worksheets that are freely available on openmiddle.com (along with hundreds of problems) and encourage students to have a growth mindset about the problems. There are spaces for the first, second, third, fourth, fifth, and sixth attempts! How often do students encounter math problems where they are told up front that they are unlikely to get it right on their first few tries and that this is totally normal and ok? Using these templates will encourage students to grow instead of shutting down.
• Initially, give students Open Middle questions that are below their grade level. If students have never seen an Open Middle style problem, they may be confused by the boxes or restrictions like using specific digits at most one time each. I recommend getting them familiar with the structure first. Then, when they are more comfortable with the format, you can move to grade-level content.

Where can I find Open Middle math resources?

If you’re looking to incorporate Open Middle problems into your teaching, here are some great resources:

• Every unit in HMH’s Into AGA (for Algebra 1, Geometry, and Algebra 2) series includes dozens of Open Middle problems. It’s an easy way to incorporate them into your curriculum.
• The openmiddle.com website also has dozens of problems at each grade level for you to browse through. A few sample problems for elementary students are provided below.
• I have written a book called Open Middle Math: Problems That Unlock Student Thinking, Grades 6–12 that goes deeply into all aspects of using Open Middle problems.
• Follow me on social media at @robertkaplinsky or at @openmiddlemath.

The views expressed in this article are those of the author and do not necessarily represent those of HMH.

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Engage every student in Grades 9–12 with Into Algebra 1, Geometry, and Algebra 2 (AGA), our core math curriculum that supports learners in developing conceptual understanding and procedural fluency.