## Thursday, September 5, 2019

### Hailstone Series Numbers

We are on day #5 of back to school and I wanted to share a fun lesson with you all today.  We start each year with the Week of Inspirational Math published by You Cubed.  We have been starting the year this way ever since I read Mathematical Mindsets and it has been a great way to jump into a new year and set classrooms up for success.

If you have used Week of Inspirational math in the past, you might know that each year they release a new week.  The week consists of math tasks most of which are low floor high ceiling and videos about having a positive math mindset.  This year, instead of releasing a new week, they re-did the website and it now allows you to pick and choose tasks, videos and other inspirational math options.  I have been working with classroom teachers to take turns picking and choosing old favorites and trying out some new tasks.  Yesterday when I was doing a quick browse on the Week of Inspirational math page, I found a task about the Hailstone series.  This is a series of numbers that I have explored with students in the past but one that I haven't revisited in quite a number of years.  I decided to try it out in my third and fourth grade math class.

If you are not familiar with the Hailstone series, let me give you a quick summary.  You can start with any positive integer (counting number).  If the number is even, you cut it in half (n/2).  If the number is odd, you triple it and add one (3n+1). Mathematicians have found that you can start with any number and follow these steps, you will always end up with 1.  They have not found a number that this doesn't work for yet, but they also haven't proven that it works for every single number.  This means it is kind of an unsolved mathematical mystery which is an angle that my students always seem to find very intriguing.  The reason it is nicknamed the hailstone series is that the numbers go up and down similar to the way hail is formed in a thunder cloud.

I started the class by asking kids what they knew about hail.  The kids were quite surprised by this question and quickly engaged with their classmates on a discussion about what they thought they knew about hail.  After sharing as a group, we watched this quick video clip about how hail is formed.  Because they had already had a chance to activate prior knowledge about hail, they were very interested.

I then spent a few minutes engaging kids in a review of odd and even numbers and did a quick number talk about strategies for cutting numbers in half.  I then explained how the hailstone series worked and we tried it out for a couple of single digit numbers.  Once kids understood how it worked, I had them try a number in pairs.  We had kids coming to the board and adding their ideas to the ones that were already up there.  We came back together after a bit and kids got to share what they noticed.  They were very excited to take what they noticed and try it out on a few more numbers so we made some time for that before wrapping up class for the day.  This is the type of problem that was a great review of important math vocabulary and worked on computational fluency all in the context of exploring a larger mathematical mystery.  It was engaging and motivating for my students, I knew I wanted to repeat it later that day for grades 5 and 6.

In fifth and sixth grades, we had even more fun and excitement because we had to spend less time reviewing things like halving, odd and even and were able to jump into the actual series faster.  They are also much more fluent with computation and were able to generate numbers faster.  They noticed SO MANY mathematical patterns and were making and testing conjectures left and right.  After exploring the numbers to 11 as a group and in pairs, some kids were excited to continue going in order while others wanted to jump around.  I gave them the option of either and they had so much fun playing around with math.  They worked so hard and were so engaged and excited.  At the end of class, I heard one boy say to another, "I am going to work on this all night.  Why don't I facetime you and we can do it together!".  Music to my ears.

This type of problem isn't done after one class period so we put some of our work up on a bulletin board in the hallway where kids can keep exploring this series and adding to our work over the next few weeks during math menu time or even at home!  We listed the series of for the first 15 or so numbers on chart paper, leaving room for kids who were going in order to add to it later.  We also decided to add index cards for those numbers kids did out of sequence.  I can't wait to gather them around this board in a week or two and see what new understandings emerge!  They have some big questions they are still working on answering and for now I am quite content to leave them unanswered and to let them keep exploring!
A big thank you to You Cubed for putting the Week of Inspirational Math out there and working so hard to promote positive mindsets.  Jo Boaler, the creator of You Cubed just release her newest research on mindsets, Limitless Mind!  I think this would make a great book study!  My copy is due to arrive this evening and I am sure it will inspire me to do more with my students.