Thinking Tasks My Students Love: "Open MIddle"

Building Thinking Classrooms in Mathematics fans, I've rekindled an old flame. 

Back in my mimicking classroom days, I made some beautiful music with the collection of "challenging math problems with solving" at Open Middle.  It had a perfect place in my instructional setup back then.  I could always count on it to give the kids enough of a challenge to keep their engagement for quite some time and to provide a satisfying payoff for the ones who were able to find the highest level solutions.

We were quite the pair.

In time, however, we grew apart.  I changed grade levels.  I changed schools.  I spent two years Building A Thinking Classroom in Mathematics. 

I moved on.

For whatever reason, when I overhauled my mathematics philosophy, I didn't bring Open Middle with me.  I fell into new habits.  I stubbornly insisted on planning the d*#$ thinking tasks myself.  It wasn't that Open Middle changed or failed to fit in my sparkling new Thinking Classroom, I just kind of.... forgot about it.

In the Spring of 2026, however, I am happy to announce that we're back together again, educating and delighting students as reliably as ever.

The satisfaction of conquering a "problem worth solving" from Open Middle.
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What Are the Problems At Open Middle LIke?

The collection of problems at Open Middle are self-differentiating challenges that can accomplish anything from giving students a motivating way to practice skills to offering them a high-level, puzzle-like challenge to tackle.  Here's a 7th grade example that is representative of how the tasks look and work:
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So simple to understand, so manageable to start, and so very deceptively complex to solve.  Kids, I find, can't wait to get started on these when presented, and then they get really, really invested in maxing them out.

Here's one I found to be particularly clever and instructive for my 3rd graders:
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Here's a high school geometry example:
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They look so modest that, if you've never tried one, it is hard to explain just how drawn into these challenges the kids get.  Even better, the payoff when they solve one can be everything that makes being a teacher rewarding and worthwhile:
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Why Do Open Middle CHallenges make Good Tasks in A Thinking Classroom?

Let me count the ways.

First, they're literally designed to fit Liljedahl's description of thinking, which is "what you do when you don't know what to do." Inevitably, the first time I give the kids an Open Middle problem each year, a whole bunch of them even say "I don't know how to do that," expecting me to teach them how rather than saying "oh good - that means you have something worth thinking about!"

The problems have low floor, a high ceiling, and an "Open Middle," meaning there is no direct route to a solution.  Perfect for real thinking.  There isn't anything to mimic.

Second, they have a low floor.  They don't look all that challenging.  All I have to do is fill in some boxes?  I can do that!  The kids don't hesitate to get started because they aren't intimidated by the complexity.  The magic is in the mild, I've been known to say, and these challenges look mild enough to reliably create that magic.

The problems are simple, they're just not easy.

Third, they are oh-so-easy to thin slice. 

Some examples from recent experience.

Example #1 (3rd grade)  I want to start with an Open Middle Challenge that I had to thin-slice downward​.  This third grade example I showed above was too tough for the first group of kids I gave it to.  
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We entered the "frustration zone" pretty quickly.  All I had to do to get them out of the frustration zone was to move the 291 for an easier entry slice.

As you can see, once I got the entry point right, we were all smiles and confidence. 

The next slice was to put the problem back as it was originally intended, but I didn't limit them to using the digits only one time each (this is my go-to way to lower the entry point with most Open Middle challenges - remove the "once each" restriction).  That one went quick, and then the last slice was the original challenge, and they were ready to wrestle with the tricky regrouping needed to get that middle 9.  It still wasn't easy, but the thin-slicing made them confident enough to persevere until they got it.
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​Example #2 (5th grade) Here's an example that I had to thin-slice upward for my 5th grade class.
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I absolutely loved this challenge, and so did the kids.  I expected this one to be too hard, but it was just right as is.  The kids didn't hesitate to start, and nobody was able to solve it easily.

But quite a few groups were able to solve it in under ten minutes, leaving me in need of some follow-up slices.  On the fly, I cut it down to two fractions, but made the sum 2 for the next slice.
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Don't you just love it when you can literally see them thinking as hard as they can?

It was great.  I had kids telling me it was impossible since fractions have to be less than 1.  Then when they thought of improper fractions, they got a little too attached and wanted both fractions to be improper.  Plenty of struggle, and they were plenty willing to do it.  

A perfect slice.

The next slice was a sum of 3.  The last slice was "how high can we make the sum before it gets impossible?"

Example 3 (3rd grade)  This was one that I was sure was going to be too hard.

I sliced this one down from the get-go with the "feel free to repeat digits" tactic, which really helped.  It also allowed the kids to reinforce the fact that multiplication and division are inverses, which is no small feat for 3rd graders.  Its pretty abstract.  Once they got that, I set them on the tougher challenge with the "no repeats" constraint.  

I don't know how well the gravity of this will come through, but the way they solved it was to set the multiplication side at 1 x 5 and try to figure out what to put on the other side.  They decided they wanted to divide by 4, so they kept "filling" groups of 4 until there 5 in each group​, then seeing if the result was one that didn't force them to reuse any digits.  They were leveraging the relationship between multiplication and division in a way that exceeded my wildest dreams.  It was absolutely extraordinary. 

​For the groups for whom that went well, "can you find another solution?" was a great follow up slice.  Repeating the task once uncovering the logic behind it is still no easy task.  It was a real joy to watch them find ways to make what they learned repeatable.  One of the kids was so proud when she found a second solution that she went home and figured out increasingly outrageous solutions without my even asking!
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She was so interested in the problem that she kept working on it at home until she had a method for producing increasingly outrageous correct solutions.  I mean, c'mon.​

In addition to using the Open Middle challenges as tasks themselves and thin-slicing within them, many of them would also simply serve as a great final slice to an existing thin-sliced task.  For example, this is the seminal example of thin-slicing that Liljedahl provides in the original Building Thinking Classrooms in Mathematics​ book.

It seems to me that this challenge from Open Middle would provide an excellent culminating slice to this existing task:
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​What kinds of Thinking Tasks are Open Middle Challenges?

I wrote a post some time back called The Six Ways I Use The Two Types of Tasks.  Liljedahl categorizes tasks for Building Thinking Classrooms in Mathematics as either curricular or non-curricular, but I've found that those two types of tasks "manifest" in six different ways in my practice:

1. Culture-setting non-curricular tasks
2. Problem-solving curricular tasks
3. Figure-it-out thin-sliced curricular tasks
4. Introduce the minimum, thin-slice to the maximum curricular tasks
5. Non-curricular tasks preceding direct instruction
6. Test day non-curricular tasks

For the most part, I use the challenges at Open Middle as core curricular tasks (#3) themselves, or as a "spicy" final slice to an existing task on the same topic.  These are the tasks I use most often and that are most critical to get just right.  I'm grateful to have Open Middle as such a reliable source to help build these, because as you may have read, I am notoriously resistant to using curricular tasks that I don't create myself.  

​My kids this semester, however, have grown to really love these challenges, and I could see myself using them as #5 and #6 in the future, too.  When I'm just trying to get the kids brains activated ahead of something else, I think these would be as good as any tasks for that purpose provided that my future students like them as much as my current ones do.
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This isn't even class time.  They asked me for "one of those problems with the boxes" when they were waiting on the rest of their class to finish something.  I gave them just about the hardest one I could find, and they dove right in.
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How Can I Get STarted Using Open Middle Tasks?

Lucky us, the Open Middle team has their full collection of tasks

• available for free
• organized by grade level
• sub-organized by Common Core strand

on their website.  Amazing!  There is also a primer on the philosophy behind the challenges and some ideas for using the tasks on the site.

To really make the most of Open Middle, however, I'd recommend the book that Kaplinsky published by the same name.  It provides a bunch of great, high level information like:

• when Open Middle challenges are best deployed
• how to get the kids ready for them
• what to do if they end up too hard or too easy (a Building Thinking Classrooms core practice!)
• how to conduct a consolidation conversation (another BTC core practice!)
• how to make your own Open Middle style tasks (which I particularly enjoy doing!)

Kaplinsky and the Open Middle team know far more about math and math instruction than just how to make these challenges, and that really shines through in their book.  It is a terrific window into a great mathematical mind.  It offers far more value than the sticker price, and it also provides a great way to share the philosophy with other teachers Thinking Classroom or otherwise.
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A Treasure Chest Of Thinking Tasks!

I have found Open Middle to be a treasure chest of tasks for my Thinking Classroom.  I know that folks are always on the lookout for engaging tasks, and Kaplinsky and the team at Open Middle have provided me with literally hundreds!  My thanks to the team for the site and for the book, both of which I've really enjoyed reading myself as someone who loves math. I hope you enjoy them as well!
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Want to read more right now?  You're in luck - this is my 113th post!  Two posts similar to this one that you might enjoy are:
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​Thinking Task recommendation: "Math Games With Bad Drawings"

These posts were linked in the one you just read.
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Three Tips For Thin-Sliced Curricular Tasks

Thinking Classrooms are Lifelines for the Inattentive

Thin-Slicing:
The Magic Is In The Mild

Are thinking Classrooms Out Of Step WIth Cognitive Science?

Six Different Ways That I Use The Three Types Of Thinking Tasks

Or, you can browse past posts by category:

• ​Building A Thinking Classroom
• Leaning Into The Science and Engineering Practices
• Classroom Practices
• Classroom Stories
• Ideas and Opinions
• Pandemic-Related Issues

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