Patterns, Patterns Everywhere!

This week's #MTBoS mission is to write about a favorite open ended problem that I use in my math class.  I have thought about his idea for a few days and while I really don't have one favorite problem, I decided that I would write about a favorite type of problem....finding and expressing patterns.

One thing I love to do is give my middle school students a seemingly simple pattern, like this,

and then ask a variety of questions.  Sometimes, I let my students ask the questions instead.

I might begin with the obvious and build in complexity:

• How many squares in the next figure? the 10th?
• What is the area of the next figure? the 10th?
• What is the perimeter of the next figure? the 10th?
• If each unit square is 2 feet long (or 5 feet or 1/2 foot) what is the area/ perimeter of the next figure?

By the time I have exhausted the possiblities for whichever line of questioning I focused on I am ready to lay the big question on them...

• What if I asked you to find the 100th figure?

Since the main strategy for most of my students involves counting, they usually have to pick their chins off the floor at this point because they do not want to continue their data tables all the way to 100 or 1000 or 752 or whatever crazy big number I decide to throw at them.  So, it's back to the drawing board to design a new strategy and look for new relationships in the data.  

It usually doesn't take very long for a few to start seeing the big picture and trying to explain to others in their groups what kinds of patterns they discovered.  Eventually, we end up with a few different relationships that all come out to the same answer and, depending on which grade level I have at the time, I either lead them to an appropriate equation or ask them to translate their relationships into equations that anyone can use to find any of the figures.

I am embarassed to say that the first time I tried this "What about the 100th?" was kind of by accident.  I needed an extension for a group of quick finishers.  I knew I didn't want to find something else for them to do, and I was trying to create a culture of persistent problem solving, so the question just kind of popped into my head and I ran with it.  Not only were they able to generalize a model, but they had fun doing it.

Now, I try to give my students a similar type of problem pretty often.  We don't always spend a lot of class time on it, but they are practicing a pretty important skill while having fun.

And now, for a little extra...

I know we are only supposed to write about one of the prompts this week, but I really want to take a moment to share what I love about my classroom, too.  I teach at a K-8 charter school with one class per grade level...our max enrollment is 180 and in the middle school we have no more than 60 students. 

Because of our smallness, I have the unique experience of teaching my students for their entire middle school careers.  That's right, folks.  When I get those little guys as 6th graders, I get to keep them until they go to high school!  That means that I really get to know my students as math learners and as people.  I get to watch them grow (many times from Kindergarten all the way through) into independent thinkers who are ready to tackle the world.

I am so proud of our little school that really feels more like a family than a school.  I get to leave my house each day and go to a place that feels more like home than work and I get to change kids attitudes toward math while I'm there.  

I know it made this post too long, but I just had to share that extra little tidbit!

Thanks for visiting!