Posted by: rdkpickle | 08.09.2017

base eight

#tmc17 has come and gone. I traveled from Nashville to Atlanta with Molly, spent five great days at Holy Innocents Episcopal School thinking about math and teaching, and journeyed on to Hilton Head for a post-tmc-vaca with Heather. I began writing this post from the airplane back to California after nearly 3 weeks of travel and am sitting here finishing it from my couch back in my apartment. I’m grateful for a few days before school starts back to read, reflect, and catch up on personal things before school starts back soon.


One of my favorite sessions I got to experience at #tmc17 was Kent Haines‘s session exploring numbers in base eight. Here’s the description:

I was excited about the opportunity to do “math for its own sake,” as I’ve found that these kinds of sessions get my wheels spinning with ideas for my classes, often in surprising ways. I also really enjoy learning from watching other fantastic teachers facilitate a session that is a bit closer to a classroom experience.

Kent’s session didn’t disappoint. We were a small group, anxious to get going by starting discussion even before the session officially began. Kent started with a quick intro (we’re aliens with 8 fingers, base 8 numbers will be green on the whiteboard to make things easier, we’ll still read the numbers aloud using the English words we’re used to so “21” is still pronounced “twenty-one” even though it means “two eights and one one”) and we jumped into things.

First, we counted up to 40, with some quick chats about efficiency. Then we filled out our hundred chart. (Yes, still a hundred chart. Not a sixty-four chart. Hmmmm…) We talked about movement on the hundred chart – how left and right are still subtracting/adding one but up and down movement is now by eights.

Next we moved into doing some operations with our base eight numbers, and this is where things got really fun! My notes are pretty messy here, and I know I’ll fail to capture all of the great insights we had as we were playing… but have a look and I’ll try my best to capture as much as I can below the images.

So first off, it was very cool to be put in a position of “disequilibrium” (as Kent named it) and have to slow down and think about addition and subtraction in a way I just don’t do anymore in our base ten number system. I’ve never taught elementary school and have very little experience with how young learners might think about addition or subtraction (these basic facts being mostly memorization to me at this point), and I definitely haven’t thought about the variety of strategies for two digit addition or subtraction problems outside the standard algorithm. So needless to say I was totally hooked as I started to realize how many different strategies the participants in the session were using to make sense of the problems and the connections between the different ways of thinking through, for example, 55 – 37.

You can see a few of the different strategies folks used on the bottom right-hand side of my work, under the words “some ways”. Luckily, I was sitting at a table with Nicole Hansen (@nleehansen) who has experience as a K-2 math coach and we had a really interesting chat about the different strategies she’s familiar with and how they might translate in base eight. Definitely inspired me to learn more from my elementary teacher friends! One really nice moment from this conversation is kind of cut off in the picture above, but Kent was using the hundred chart (projected on the board) to talk through the subtraction problem 55 – 37, doing so by starting at 55 on the chart, and moving “up” once (subtracting one group of eight), again (another group of eight), a third time (another group of eight), and once more. Then, because that was a subtraction of four groups of eight, he moved to the right to “add one back”. The realization that you only have to add one back to move from subtracting 40 to subtracting 37 actually got an out-loud exclamation from Nicole!

Kent also directed us to notice which problems had the same (apparent) answer in base eight as in base ten, and try to explain why. (Circled on my notes page.)

We then moved on to filling out a multiplication chart. Nicole and I both tried to slow down and focus on making sense of any patterns we noticed, or reason through individual facts by transferring strategies we might use in base ten multiplication. (For example, noticing that multiplying by 4 in base eight feels like multiplying by 5 in base ten. And – what’s up with multiplying by 7? Why does it work like multiplying by 9 in base ten?) There were moments where I felt myself engaging in something similar to “mental math Monday”-type-thinking to compute some of the math facts in this new, unfamiliar landscape and that was pretty cool!

David Petersen (@calcdave) and Jamie Collie (@jcollie44) were also in the session and were inspired by the numbers in the multiplication chart that have the same (apparent) answer as they would in base ten, and decided to hunt for patterns by getting all fancy and color coding those values in Excel. Here’s a zoomed out image from their work (and if you want the file, hit them up on twitter).

I definitely left the session convinced that I wanted to spend a little more time in the alien universe of base eight.

And then a few days later, twitter wowed me even more me by informing me that Fibonacci base exists. COME ON, Y’ALL!! Too much cool! Not enough time!

Definitely a fun hour+ of playing, and a highlight session of my #tmc17 experience. Thanks, Kent! 🙂


  1. Now you can do base sixteen, with 0 1 2 3 4 5 6 7 8 9 A B C D E F
    Look at the coding system for colors in HTML and javascript.
    With the first two digits all the shades of red are covered, with 00 as black, 80 as full red, going to white as FF
    FF is 255 in base 10

  2. I missed the Fibonacci base thing from Edmund. Wow. Mind blown. Now I have a whole new love for him and mathematics.

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