Gemini Puzzles- Create a Headache for Order of Operations!

OK, the MTBoS has really influenced the way I approach teaching and assigning problems.  Coming back from Christmas Break, I needed a different way to get students thinking about math again, and found that thanks to Math = Love in the form of Gemini Puzzles.

What are Gemini Puzzles?  In short, they are equality statements that are missing any mathematical symbols. Here is an example:


The thing I love about this is- I put these two on the whiteboard for students to see when they first walk in and instead of asking what we are doing today or talking about what they did last night, they started thinking about what was going on.  The other great thing, I didn’t have to tell them what to do, they started playing around with the numbers to get it done.

When I asked my students what they were doing, there was a common theme:


Which is great until I ask students about the name of the problem, Gemini.  I then get answers about twins and I tell them that’s relevant to the problem.  They are given two equivalence statements, and they have to be twins.  In other words, anything they add to one statement has to be the exact same as what they add to the second.

Then students start working in a stream of consciousness and they have this as an answer:


It doesn’t take too long for a student to say that there’s problem with this.  Students are thinking “one plus one is two, times two is four”, but they are not properly showing that.  Then the talk of Order of Operations hits, and students realize the first statement is only 3.  By throwing in another set of symbols they find the correct solution:


By providing students with two equivalence statements that need the exact same symbols to make them true, you are giving students a reason to think about the order of operations and how they interact.  I have loved listening to student talk when working on these, and I even had students proudly come up to me with different solutions to a problem.  They are amazed that there can be different ways to implement mathematical operations and get the same result (see if you can find which one they discovered!)

Gemini Puzzles are a great activity for the classroom, I know I’ll be using them from now on, I hope you enjoy them as well.

I currently have a range of students from 6th to 12th grade in my classes, with various mathematical skills ranging from 4th to 12th+, and these puzzles have engaged and challenged them all.  It’s a great thing to see.

Open Middle Introductions, Please.

Hello all, you may or may not know that although I do create a lot of Open Middle problems, I do work with a group of 4 other awesome educators who create the Open Middle team.  This blog is an introduction (re reminder) of what Open Middle is, the people behind Open Middle, and if you wish to accept the challenge- what kind of information we need to publish your great problem to the Open Middle site.

What is Open Middle?

Taken directly from the OM site (blatant plagiarism, thanks Robert K.):

Dan Meyer introduced us to the idea of “open middle” problems during his presentation on “Video Games & Making Math More Like Things Students Like” by explaining what makes them unique:

  • they have a “closed beginning” meaning that they all start with the same initial problem.
  • they have a “closed end” meaning that they all end with the same answer.
  • they have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

Open middle problems require a higher depth of knowledge than most problems that assess procedural and conceptual understanding.   They support the Common Core State Standards and  provide students with opportunities for discussing their thinking.

Some additional characteristics of open middle problems include:

  • They often have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method. Example
  • They may involve optimization such that it is easy to get an answer but more challenging to get the best or optimal answer. Example
  • They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it. Example
  • They are generally not as complex as a performance task which may require significant background context to complete. Example

In other words, we are asking kids to solve non-routine problems that may or may not have one single correct solution, but have multiple ways to arrive there.  This promotes rich classroom discussions about the mathematics behind the problem which develops deeper understanding.  Hopefully this is what you are all about in your classroom.

The people behind Open Middle

On to the people who drive Open Middle (well those who organize the website, because we appreciate everyone who uses our site!)  The project was founded with Robert and Nanette.  After flooding Robert with questions and problems for the site, I was invited to join the team.  Open Middle started to gain a following and problem submissions started pouring in, and we welcomed Dan and Zack to the team.  Here’s a rundown of each member (or what I can dig up through a web search).

Robert Kaplinsky (taken from his website  Hi, I’m Robert.  I train mathematics educators who want their students to be better problem solvers.  I help them build the tools needed to get students critically thinking and articulating their reasoning.  I do this while continuing to work full time for a K-12 school district in Southern California as a mathematics teacher specialist.  It helps me stay connected to current issues in education as a typical month involves me teaching students at multiple grade levels, mentoring teachers, and providing professional development.

Nanette Johnson (taken from her blog I am a wife to one amazing husband <3,  mommy of 3 awesome kids…a math teacher to more than 2000 students (over the years…and still counting).  I love teaching, and watching students break down their previous self-imposed beliefs that they “just weren’t good at math”.  I’m concerned about how teachers are expected to teach in ways in which we did not learn and in ways which were not appreciated.  I want to find ways to help teachers go from surviving to thriving.

Bryan Anderson:  Currently I am working at a public school that services a juvenile center in my area.  As such, I teach mixed classrooms where student age could fall anywhere between 10 to 18 years.  I also see a wide range of proficiency levels as most of my students typically have not attended school for at least one year.  A large majority of my current students qualify for special education services, and as a requirement to this I am picking up my special education certification for learning disabilities.

Dan Luevanos (taken from his blog Math Rockstars):  My name is Daniel Luevanos. I live with the intent to rock ‘n’ roll. I believe that we can change the world by challenging ourselves and others to do the unexpected.  I’ve been a secondary math teacher in San Marcos Unified School District since 2009, teaching at the high school and middle school level. Currently, I’m a Secondary Mathematics Teacher on Special Assignment in San Marcos Unified School District.

Zack Miller (taken from his blog A Math Education that Matters):  My name is Zack Miller and I work in math education in the San Francisco Bay Area. Teaching math was my first job out of college, and since then it has developed into my (sometimes fully-consuming) passion. Lately, I’ve been leading the charge re-envisioning the curriculum, assessment, and instruction for my charter network’s math program.  In the time since I started my career, I have earned a degree from the Stanford Teacher Education Program, been a founding teacher at two innovative charter schools, presented at numerous conferences, and became a Math for America Master Teacher Fellowship. I’m eager to share what I’ve learned, and to learn more from the community of folks focused on this important profession. That’s what this blog is all about.

What Information Do I Need to Submit a Problem to OM?

The first step is to click on the link for submitting problems on the Open Middle website, which can be found here.  It will take you to a page that looks like this:


This page gets slightly long, so let’s just cover the essentials

  1. Your name, address and blog (yes, please let us link your blog!)
  2. Any co-contributors
  3. Problem Title- please make this as descriptive to the problem as possible
  4. Directions- We are working on having a consistent wording for these.  Typical Open Middle format follows along these lines: Use the whole numbers 1 through 9, at most one time each, to find the largest ____________.
  5. Image link- please make sure this link is accessible as we have different members preview the problem, edit if needed and uploading.
  6. Hints- what can teachers say that could prompt student thinking about the problem?
  7. Answers- please, please provide at least one correct solution
  8. DOK- please indicate what Depth of Knowledge your problem requires.  If you are unsure, check out this DOK site.
  9. CCSS-M Standard.  Typical formatting for these are not the same as the CCSS site, we record the grade level, mathematical strand, and standard.  for example: CCSS.Math.Content.6.NS.B.3
    Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.                                                                                                                            Would be labeled as 6.NS.3 for the Open Middle website.

Thanks for checking out the Open Middle website, we hope that these problems provide you with an alternate way of providing purposeful practice for students, and those practices will create rich discussions and empower students as learners.

Comparing Numbers- Open Middle Problem

Directions: Using the numbers 1 to 9, each only once, fill in the blanks to make the inequality statements true.  The middle number of each inequality has to be odd.


the symbol < means “is less than”


EDIT: Thanks to Graham @gfletchy for talking this out with me- if symbols seem inappropriate for K, use the following:

___ is more than ___ but less than ___

___ is more than ___ but less than ___

___ is more than ___ but less than ___