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.

#MTBoS30- Day 9

Brute force…


This has really gotten me thinking lately.  I gave my students the following picture:


Yesterday my students and I did a problem adding and subtracting a string of numbers.  We looked at patterns that could be found in it, and even looked at why PEMDAS is misleading for order of operations.  They were finding patterns, multiple strategies and stretching their thinking.  I had hoped that this would translate to this problem- I was wrong.

#1 answer (98% of my students): I counted all of the squares.

I had tick marks, numbers, dots, colors, everything on their sheets showing how they made a 1-1 relationship with each square to find the total.  I was very disheartened, but I didn’t show it.  I asked if anyone had an alternate method, and only received a small handful (of which I will also share).  So I asked those who counted all of the squares: “How many of you still find yourself using your fingers when computing?”  All of them raised their hand.  This is a strong indication of where my students are developmentally with math, and how I need to provide them with opportunities to explore and stretching their thoughts about problems.

So, I then reminded them about yesterday and asked what we looked at with that problem.  They talked about the patterns and multiple representations- they remembered the whole segment!  So I asked, “How can that apply here?”


4+3+4+3+4+3+4 or 4×4 + 3×3


1+3+5+7+5+3+1 or 2×1 + 2×3 + 2×5 +7


15+4+4+1+1 or 15+ 2×1 + 2×4

And then the bomb dropped


5X5 square


A student came up with this and blew the minds of every one of his peers.  I let it sit there for 3 minutes before I even uttered a word.  I could see each of them processing what just happened.  Finally one student’s face lit up and he said “Oh, I totally get it!”

It took a while for them to realize that yesterday was not a “one and done” day for math, that I will expect them to do this every day they are with me.  Tomorrow we will have another pattern and see how they do with that one.

Go out and drop the bomb on your class.

#MTBoS30- Day 8

Putting it all together- that’s what my mind is trying to do.  There are a TON (really, there are) of great resources out there on the web via the #MTBoS.  There are times where it is hard to decide what to use, when to use it and where.  I am lucky enough to be slightly flexible in my curriculum (and also a curse because of the nature of the placement).

So, because I teach in MN and there are no “perfectly pre-packaged curriculum” (really who even wants that truly?), my mind is trying to wrap itself around what would be a good meld of components in the classroom.  Like I mentioned in my previous post, Science Practices in Math, I really believe that restructuring the lesson layout will help not only my students, but others as well.

Currently I have been using parts of David Wees a2i, an online curriculum for Algebra 1 and 2 as well as Geometry.  It has opening activities and I like the exploration of topics.  I also need to work with resources I find great (and my students do too) such as Estimation 180, Open Middle, WOBD, Would you Rather, Visual Patterns.

This summer will be interesting for me…

Insurance Rates- Expected Value

Consider the following insurance quotes:

Comparing Rates.jpg

If you plan on only having once accident in the next 5 years, with a damage total of $2,000, Which policy is better for you?


Is there an amount of damages where neither policy was the better deal? (That after 5 years, you would end up paying the same amount of money)

Create a statement that demonstrates which policy is the best deal over 5 years.

Science Practices in Math


My thinking this year has been stretched a lot, and I’m still trying to decide what was good thinking and what wasn’t.  One thing that is really sticking in my head is the thought of learning methods I experienced in Science- and I’m wondering why I don’t implement them in Math.  I’m not sure how your science classes went but here is the breakdown of mine:


This is very similar to Dan Meyer’s 3 Act Math, and perhaps one of the reasons I connect so well to that model.  Show me something cool like this that will set my brain on fire:

(and of course I wouldn’t show the whole video to begin with)

Now you have me, I’m totally hooked on what is going on in class.


Once again, just like the 3 Act model, ask me about what is going on.

  1. What do you notice?
  2. What do you wonder?
  3. Why do you think that is?

Kids are endless fonts of ideas and knowledge, let their thinking dominate the classroom, not yours.  I typically record every student’s response (yes, every one that is appropriate for school- excluding those that imply language, race, sex, etc.).  Like I have blogged about before, students will hit upon what your learning concept of the day is, why give it to them when they can supply it themselves?


After we had this great classroom discussion about what we saw, we learned about about it.  This is where formal instruction fits within the class.  This is where you implement your practice, whatever form it may take.  The secret here, everything was done in class.  This is NOT the time to assign homework, and if you run out of time that’s awesome!  It means you are sending students out the door with that itch in their brains about what just happened in class today.  Their brains will be kicking those ideas around all night, whether they know it or not.  Typically I see an increase in engagement and clarity of focus from students on the second day.

This is where normal instruction ends.  This is where the great story of Mathematics dies.  We practice, we know it, we show mastery of some degree.  We wait until the end of the week, month, semester or year to show that mastery on some formal assessment.  This is where we need to be like my science class and take it all one step farther…


Take what you just learned about and test it out or apply it.  This could be a different spin on the activity that introduced the topic, or another application of the mathematics all together.

Take Fawn’s Barbie Bungee.  It seems to do this right?  It does, to some degree.  In Act 3, we learn if our calculations are correct, the same with Barbie Bungee.  The thing is, we are still building upon the knowledge we created in that lesson, even though we are justifying our thinking, we aren’t building new knowledge.  What do I mean?  I typically ended Barbie Bungee with this:


This is my bungee jumper now, and there’s a lot more at stake with this one than our Barbies.  This one will create a huge mess in the room if I mess up.  I have never seen students so excited and purposeful about their calculations and mathematics than when I introduce the egg.

This whole process gives my students a whole new appreciation for this:


My newest mission: Don’t stop at practice, push students to reflect, rethink, reapply their mathematical knowledge for different experiences and scenarios.