Spiraling the Hundred Chart and Beyond

Now that I have a little more time on my hands, I decided to write up some of my spiral explorations so that others can more easily play along or use them in the classroom.

Spiral Instructions


Spiral Templates – 100 1 sheet

Spiral Templates – 100-4sheet

Spiral Template 1 to 196

Spiral Template 1 to 1089

Google Document Files:  Instructions   Templates


  • Because my school is not 1:1, I printed these out and had the kids use markers or colored pencils.
  • I have them start with the spiraled hundred chart (printed 4 to a page) to have them play around with patterns at first. This is also a good opportunity for them to test conjectures.
  • Connecting the numbers in the pattern adds another element of visual interest.
  • Students looking for an extra challenge can try a variety of quadratic patterns.  I usually have them put a function into Desmos or a graphing calculator and use the table.

Here are the pentagonal numbers done on the large spiral:Pentagonal.jpg

Spiraling Math on a Stick

I’m sitting on an airplane, about to embark on a nine day professional development bonanza in Baltimore, Maryland.  Right before we were given the airplane-mode directive I get this message from Christopher Danielson:


Now, quick background:  I adore Christopher, everything he creates, his wife, his kids, and pretty much all that he touches.  I’m being hyperbolistic (which is totally a real word) but if you made a word cloud of my blog posts over the last year, it would be borderline embarrassing how large the words Christopher and Danielson would be.  S0, to make a long story a little bit longer, I was out-of-my-mind excited about this opportunity and wanted to text everyone I knew.   The hawk-like gaze of the flight attendant made that a difficult pursuit.  Anyway, it was a long flight.

Fast forward through a handful of prototypes, test runs with the resident 5-year old mathematician of the household, 250 colored card stock copies, and a pep-talk from the Professor himself, I was ready to hit the Minnesota State Fair to be the visiting mathematician/artist at Math on a Stick.

I settled on using 1-100 at the recommendation of Christopher because many children would be familiar with a hundreds chart either from home or school.  I could easily help kids made the transition from that familiar chart to this “different” way of arranging the same 100 numbers.  Making that comparison was something I would not have thought of, and it ended up being incredibly important.


Admittedly, I was a tad nervous that my idea would flop.  That all of the fun I’ve had with this spiral wouldn’t translate to a wider, younger, more playful audience.  All of that fear melted away when that first adorable young face sat down, and grabbed a marker.

One of my favorites of the day was the girl who figured out that if she colored the multiples of 10 and then connected them, they turned into triangles.  Always triangles.  She tried again with multiples of 5, with the same result.  The glow in her eyes was precious.

Simply stated, this was one of the most fulfilling mathematics experiences I’ve had ever.  What Christopher Danielson has created with Math-on-a-Stick represents the paradigm shift in the way adults view mathematics through the playful exploration of children.


Spiraling Out of Control

I wanted to collect all of my spiral art on my blog.

Here is the document I used (spiralled numbers 1 – 1024)


Here’s the google doc: https://drive.google.com/file/d/0B6QZoh9vKRK4Qm5DVkowTnRlYXM/view?usp=sharing

And here is what I’ve created so far.  Enjoy!

Link to Google Photos Album: https://goo.gl/photos/CzAQU8Q4AhRoBMBLA

0601161833e 0601161833d 0601161833c 0601161833b 0601161833a 0601161833 0601161833(1) 0601161832a 0601161832 0601161831g 0601161831f 0601161831e 0601161831d 0601161831c 0601161831b 0601161831a 0601161831 0601161829c 0601161829b 0601161829a 0601161829 0601161828b 0601161828a 0601161828 0601161827e 0601161827d 0601161827b 0601161827b(1) 0601161827a 0601161827 0601161826g 0601161826f 0601161826e 0601161826d 0601161826c 0601161826b 0601161826a 0601161826 0601161825g 0601161825f 0601161825e 0601161825e(1)

Next up:  Hexagon grid spirals.


Some wonderful people on twitter have made these into pretty neat computerized visuals.  Here are some of my favorites:

John Golden:


GHS Mathematics Department:

Dan Anderson:



Math on a Stick – Encore Edition

First, thank you from the bottom of my math-loving heart to Christopher Danielson for allowing me the privilege to be one of Math on a Stick’s visiting mathematicians and spend the day talking math with all of the kids.  It never ceases to amaze me that when you bring in something mathematically simple and open ended how much creativity and wonder kids will bring to it.  I mentioned this last year and the year before, and it is worth repeating today:  We need to get out of the kids’ way.

The school year is about to start, and the standards will dominate our conversations as teachers.  I don’t want to dismiss the importance of common national standards as a foundation to ensure that each and every student has access to important mathematical concepts.  But, we secondary math teachers have a reputation as self-proclaimed masters of content knowledge which can be important.   Still, I notice, we spend an awful lot of time making sure kids can expand a fourth degree binomial and not nearly enough time listening to the children make sense of ideas and letting them create and explore mathematically.  Kids who can manipulate algebraic expressions fluently can do just that.  (Perhaps they could use it to manipulate other algebraic expressions. Such joy.) On the other hand, students given opportunities to play with math have a chance to develop a deep understanding and love for mathematics.  For my own child, I’d rather have an ounce of the former and seven tons of the latter.

Thank you Desmos for sponsoring the day (and the awesome shirt).  Thanks to the Math Forum, Annie Fetter, Sara Vanderwerf, Ellen Delaney, the Minnesota State Fair Foundation, and all of the amazing people that have helped create this special corner of the fair where math isn’t scary or anxiety-inducing.  There are no tests on math facts or multiplication charts to memorize.  There are no lectures, nothing to practice.  And it’s the highlight of the Minnesota State Fair for me every year.

Highlights from a Hard Year

There really is no two ways about it: this school year has been hard. Really hard. I taught AP stats for the first time, it was my first year as a 6-12 math specialist, and our district piloted new curriculum. I was challenged as a teacher and to be completely honest, as an overall human being over the last 9 months like I couldn’t have predicted. I pride myself in finding the silver lining, always. This isn’t a skill I was born with; it’s something I practice. And this school year had plenty of silver even though it took some polishing to make it shine. But I believe that I have the ability to create happiness from within so I found 10 great things from this year.

 Here are my top ten highlights from the 2016-2017 school year:

1. As part of our math professional development, I got to share Principles to Actions with 30 math teachers from across the district. 

2. We used M & Ms a LOT in AP Stats. And that was delicious for everyone.

3. I shared this amazing book with tons of teachers with young kids at home. 

4. I found a coherent learning sequence for  non-AP stats that is both challenging and engaging.

5. I got to present at NCTM on Stats and Social Justice with this amazing​ human.

6.  I subbed in a theater class and learned a ton about another curricular area and experienced the joy of watching kids be creative.

7. I introduced my algebra class to my spirals.

8. I got this lovely note on teacher appreciation day.

9. I learned so much about the intensity required to teach an AP course with fidelity. 

10. I learned to accept change as a natural part of teaching, even if it means letting go of something you don’t want to release

There are 10 school days left for me this year. I know they will be filled with joyful learning, exciting transitions, and some sad goodbyes. But the pool is calling my name for the summer. But I’m eager to dive in to next year’s adventures. 

Afterward, Kindness.

Teachers had a uniquely difficult job today:  Support students who feel unsafe about the future of our country.  Do so with grace and humility while simultaneously dealing with our own feelings of uneasiness and hopelessness.

Based on what I hear my students say on a daily basis, it seems as though their views on the presidential race fell into one of four categories:  1.  Trump 2. Hillary 3.  Neither  4.  Indifferent.  This isn’t surprising for high school students in the far north suburbs but it seems important to note that my students’ views fall all over the political spectrum.

I began each of my classes today as I do at the beginning of each trimester:  You have my respect because you are a human.  My number one job is to keep you safe, and the most important thing that happens in this classroom is kindness.  

This isn’t anything new to them and it’s the culture we create in my classroom.

Then I pointed to this on my wall:

A gaping gash in the blue faux concrete. Many of them have never noticed it before but now their eyes are fixed on it.

This is where my “This is a safe zone” sign used to hang, I say.  When I painted the pillar, I had to remove it, and after sticking it back on, the adhesive wore out and it fell into the garbage without my knowledge.  But a wall with a sign doesn’t make this room any more safe. We create a culture of kindness and respect and use mathematics as a catalyst. THAT is what makes this a safe zone. 

We then had great conversations in both classes:  Spirals in college algebra, sampling bias in AP stats. But none of that matters if students don’t feel safe. As a teacher, I’ll do that every day as no part of my job is more important.  I’ll continue to fight for access to high level mathematics for all of our students and promote kindness in all of my classes. No one gets to vote on that but me.


On Which I Pat Myself on the Back

Warning:  Some of you are going to find this really dumb.  You’ve probably been doing this with your STEM-fueled 5th graders for years and are wondering why a 35 year old secondary math teacher is so excited about.  Four words: This. Isn’t, About. You.

So in April, I went with two great #MTBoS friends, Julie Wright and Danielle Reycer to the Exploratorium in San Francisco after NCTM.  To be honest, I was awed by the place, but my motion sickness unexpectedly overcame me, and I spent most of my time at the exhibits that didn’t require visual attention.  But in the back, there was a “maker-space” (or whatever you edu-folks call it) where they were making Scribbling Machines.  I took a look at it and thought “I want to do that this summer.  Maria will get a kick out of it.”

Step one:  get a 1.5 – 3.0 volt motor.  [Crap. Something with wires and electricity.  I can’t do wires and electricity.  Cuz I’m math, not science.

Tip:  Don’t purchase a new one.  Re-purpose one from a child’s toy.  [Sweet!  Look out, loud spinning, jump contraption from Hell’s fifth circle, I’m taking your motor!]

What I thought was the motor was not, but when I attached the battery, it became magnetic, which was neat.  I loosened about 8 more tiny screws and finally extracted the actual motor.  Now the game was on.  I was like a mad scientist, tongue to the side, laser focused on getting this scribble machine to function.  No, darling spouse, I won’t tell you what I’m doing, but you can see when I’m done.  Now go away.  

Here are a couple of videos of the Scribble Machine in action after a lot of adjusting and reconfiguring.  Yes, I know, I listen to great music:  Listen to the Music Radio on Google Play Music.

Here are the instructions if you are interested in making your own Scribble Machine.  Now if I could just get it to spiral…

You might be thinking, Ok, Beagle.  What’s the point?  You made a thing and now you want us to be excited for you?  Well, no.  Yes, I’m excited I repurposed a motor from Lucifer’s Leaping  Musical Spin Toy of Satan.  But confession:  I was one of those women who was convinced that robot, electrical, and computerized toys were designed with boys in mind.  These Lego robots weren’t exactly screaming my name:


But this girl looks genuinely excited, right?


Until I saw something I wanted to make, I was convinced I couldn’t.  My daughter is growing up in a country where for the first time a woman is running for president and a woman will adorn the $20 bill.  But what’s more important than those major accomplishments for me and my daughter is for her to see her mom try stuff and fail.  And try again, and fail again.  And then try one more time, and fail repeatedly until I have a working Scribble Machine that doesn’t do much but prove that I wanted to, so I did, tripping over myself along the way.  Maria wants to learn to ride a bike this summer.  And while I won’t be the one out there helping her, I want her to believe that she can and she will.

I see my own mother do this with her sewing.  I’ve seen her tinker and toil over stitches and fabrics and techniques until she creates something so unique, beautiful and truly one-of-a-kind. Like this (oops, stained) Doc McStuffin’s jacket:


Note:  Maria’s analysis of the Scribble Machine is still pending.  Will update with reactions and artwork.


Beagle Blockade

Last weekend we took the beagles to the dog park.  Although they are small, and somewhat unassuming, Herbie seems to be able to hold his own when it comes to keeping up with the bigger dogs.  Here’s a 15 second snippet of the adventure:

Needless to say, the beagles were thirsty by the end of the trail.  Of  course the park was equipped with water at dog level, but for some reason, Herbie refused to partake.  This little hound had just ran his tail off, howling all the way but would not drink any of the water overflowing the canine-sized bowl available to him.

You can bring a beagle to water, but you can't make him drink.

You can bring a beagle to water, but you can’t make him drink.

I made the joke on Facebook, “You can bring a beagle to water but you can’t make him drink.”  We use this phrase often as high school teachers (with “horses” obviously), but the sentiment is the same.  “I can’t teach a student who isn’t willing to learn.”  Now before we get defensive, I want to break this down a little.  I’m guilty as any of buying into this mantra at one time or another.

One of the best things to happen in my district is the adoption of a PLC model which requires that we not only reflect on how we teach but on what kids actually learned.  Simply bringing the horse (or beagle) to water is not sufficient (nor was it ever), but we now have a district-wide culture that supports students who reject the water the first time.  In my beagle example, Herbie was afraid of the giant yellow pump that was pouring water into the bowl.  When he got home, he was still thirsty and proceeded to slurp down the entire contents of his regular bowl on the floor in the kitchen.  I owe my dog another opportunity to drink water because, well, he’s a dog.  High school students, no matter how much we desire the contrary, are not adults.  They are children.  And sometimes they will reject learning the first time, and the second time (and sometimes repeatedly just to spite us).  But as educators, we owe them continuous opportunities for learning.

Well, it’s the first weekend of summer, and I’m coloring spirals and making beagle metaphors on my blog.  The next three months promise to be pretty fun.


Tales from the Nerdery

I think math is neat.   I like to play with math.  And sometimes I come up with something that supports my theory that math is neat.

First, I took the numbers 1 – 100 and spiralled them around some regular graph paper.  I wondered what would happen if I colored in the multiples of 4.  I examined my work and thought, “Huh.  Well that’s neat.”


Of course then I needed to spiral even more numbers and test out everything I could think of.  Multiples of 5, 6, 7 and so on.  Linear patterns, quadratic patterns, prime num…nope, nevermind.  I don’t do prime numbers.

Anyway, it’s that time of the school year where stress relief is necessary so I have been playing with these spirals for over a week. Today I came up with something worth sharing on my blog.  I took the positive y-values of y = x^2, y = x^2 + x, y = x^2 + 2x and so on, and colored those squares.  Then I put the images together and made a gif. Obviously, holding the camera at a steady angle is not a skill I have mastered.  But I still think it’s pretty darn neat.


The Power of a Network

I joined twitter in 2008 and started tweeting more actively in 2009.  Thanks to my attempt at Justin Aion’s Twordle experiment, I was reminded that my first tweets were annoyed snark toward the women of the View.  They never responded.  Shocking.

Five years later, my twitter usage has evolved into something that has helped transform my teaching.  Attending Twitter Math Camp last week provided some proverbial icing on the cake.   Glenn Waddell reiterated that Twitter Math Camp is 150 teachers who all believe they can change the world.  It’s hard to capture the magnitude of this incredible event and hard to explain in words how much positive impact these “friends in our phones” can have on the actual students in our classes.   I thought perhaps a picture could capture it. Over the last few days, I attempted to capture the essence of this twitter network.  I wanted to visually represent the inter-connectedness and strength of a group of math educators who feel that by interacting in person for four days in the summer, they’ll have the power to make their students’ world better.



#TMC14 Twitter interactions from May to July. (in a sine wave)


Twitter mentions in a spiral

Random Interconnections

Random Interconnections




This last graphic is the most powerful to me.  These are the twitter interactions among the TMC14 participants SINCE the event.  Have you ever gone to professional development where you kept interacting with so many people from the conference?  Me neither.  There’s a ton more that I want to do with this network software, but I’ve poured over it for days, and I wanted to share what I had so far.

There were 36 teachers at TMC12, 110 at TMC13 and 150 at TMC14.  I know that these networks spread far beyond the attendees in Jenks, Oklahoma last weekend.  But a strong foundation has been built. It’s an unspoken commitment to one another that says, “when standards-based grading (or interactive notebooks, or problem-based instruction, or group communication) isn’t going as well as you’d hoped, I’ll be there to get you back on track.”  It’s a network of teacher’s across the country that come together over mathematics, but truly bond over their inherent desire to help all students succeed.  And it’s open to anyone who has the desire to be one of the connecting threads.