Project Pentagon

Pentagons are taking over my life.

You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.

My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…

I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.

I consider a pentagon “solved” if I have at least once figured out how it tiles.

I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.

The project is making me think a lot about learning.

For example, tonight I was working on pentagon number 8. I solved it.

2015-12-16 21.19.40

These sets of four can continue to go together in a way I see and can describe.

But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.

2015-12-16 21.20.15

This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?

Another solution uses sixes.

2015-12-16 21.21.30

With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?

I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:

What is (or should be) the relationship between informal outside-of-school math, and school math?

I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:

  • Is this math?
  • Is it school math?
  • Do we value it?
  • Why or why not?

I invite you to join me on this journey.

I’ll keep you posted on the pentagon project.

A new Calculus Activity Builder activity

Let me bring you up to date, in case you have not been following along.

I am on leave from my community college teaching this year, and am working at Desmos remotely from St Paul.

A large chunk of my time involves working on the pedagogy side of Activity Builder, which we released this summer.

Activity Builder lets you build a classroom activity using one of three basic screen types: graph, question, and text with image.

From time to time, I’ll take the opportunity to turn something I’ve done in the classroom before Activity Builder and make an online version. I did that yesterday. (Here is a link if you want to play along as a student—I recommend doing that!)

It’s a simple little calculus activity on the surface. You see a function that is graphed on the coordinate plane, except that parts of the graph are obscured by large black circles.


There are four such graphs, and I ask the same three questions of each one.

  1. Behind which circle(s) must there be roots for this function?
  2. Behind which circles might there be roots?
  3. Behind which circles is it impossible for there to be roots?

After each round of questions, you have the opportunity to move the circles aside to see for yourself whether there are roots.

This is a little routine I developed as a Calculus teacher to spur conversation, and it contrasts with a standard textbook approach, which asserts the importance of three conditions for knowing there are roots:

  • continuity on the interval in question, and
  • a sign change between the interval’s endpoints

In that spirit, you are told in this activity that the first three functions are continuous. You are not told that the last one is.

In a classroom setting, I’ll discuss these examples once students have worked through them. In that discussion, I want to get students to verbalize the following things:

  1. There are sometimes roots where you don’t expect them (Screen 8).
  2. There are sometimes not roots where it looks like there really ought to be.
  3. If the function starts negative and becomes positive, it has a root.
  4. And vice versa. (Screen 4)
  5. AS LONG AS THAT FUNCTION IS CONTINUOUS!!!!! (Screen 16 for crying out loud)

Only after that am I ready to state the Intermediate Value Theorem.

This activity illustrates a curricular principle I sketched out recently, which is that lessons build on students’ experience, and help them to structure that experience mathematically.

This activity creates an experience for students, and then it’s my job to help students structure that in a formal way—through statement of and exploration of the Intermediate Value Theorem.

I’m not a big fan of providing structure for things students haven’t experienced. Typically they see no need for it, and struggle to incorporate these structures into their view of the world. Also, students end up lacking meaningful mental images for representing and triggering the formal structures.

This is theme that plays out in all of my work, by the way. Math On-A-Stick, Oreos, Talking Math with Your Kids….all are predicated on Experience first, structure later.


The tale of Tabitha and the two division problems

Consider these two division problems:

Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?

Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?

These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)

The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.

Do you know which is which?

We talked about this on Twitter today. (Click through for some really outstanding discussion….seriously.)

I have written about the two major types of division problems before, and they are relevant here.

Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.

So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.

Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.

When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.

What’s the moral of the story? Let me know your thoughts in the comments.

A quick plug for Estimation 180

Estimation is more than rounding.

Most of the time we don’t teach this, but it is.

Tabitha (8 years old) had a homework assignment the other night that asked her to imagine she had $100 to spend in a catalog, and to make a list of things she would like to buy from that catalog. She found the latest American Girl catalog and got to work.

There was a table to fill out with three columns.

  1. Description of item
  2. Actual cost of item
  3. Estimate

A couple minutes later she asks, What’s the estimate if it costs five dollars? Should I write $5.01?

She has discerned that estimate means write down a number that is not the exact value.

But that’s not what estimation is about at all. Estimation is about finding a number that makes sense, and not worrying about whether it’s the exact value or not.

The image below seems to be going nuts on the Internet today (despite my exhortations to the contrary! Oh, Internet! When will you learn to listen to me?)


“Is this reasonable?” is a great estimation question. Rounding is one way to answer the question. But if a kid can quickly find a number that makes sense and it happens to be a precise number, then we probably haven’t asked a good estimation question. Rather than mark it wrong because the kid didn’t round, we should ask this kid a more challenging question next time.

What does a good estimation question look like? What would be more challenging?

I’m glad you asked.

Estimation 180. Thinking of a number that makes sense is much more interesting when you have to bring your knowledge of the world to bear.


Is 75 inches a reasonable answer for the difference between the father’s height and the son’s? Is 75 centimeters reasonable?

Oh! And here’s the first in a nice sequence about numbers of pages.

Kindergarten questions

I am spending a bit of time a couple days a week in kindergarten this year. It was part of the now-changed sabbatical plan, but important to me to follow through on.

Today was my first day. It was awesome.

The young ones are working on patterning. AB patterns, AAB patterns, ABB patterns and ABC patterns. I’ll leave the curricular questions for when I know more. Today I’ll take these activities at face value, which is to say: this is the mathematics these children were working on today.

The children were instructed to use square tiles to make an ABC pattern. If you haven’t spent time studying curricular approaches to patterning in early elementary, this means that they were to use the tiles to make something such as this:

Screen Shot 2015-09-17 at 7.12.35 PM

Color is the only variable attribute of the tiles the children were using.

I had several interesting conversations about this task with children today. The one I want to report is the following.

I made this pattern:

Screen Shot 2015-09-17 at 7.17.32 PM

I asked the girl sitting next to me whether I had made an ABC pattern.

Girl: Yes.

Me: How do you know?

Girl: [blank look; long pause]

Me: Show me how you know it’s an ABC pattern.

She carefully points to each tile, saying one letter per tile in the following way.

Screen Shot 2015-09-17 at 7.20.53 PM

She pauses.

Girl: And you need another C.

Learning is messy—beautifully messy.

I left today with two big questions on my mind; each relating to this exchange, and to others not documented here.

  1. What would these children have done if asked—prior to instruction—to make a pattern with their tiles?
  2. How does this kind of patterning work interact with learning to count?

I hope my readers will see that these are not questions I expect to be answered in the comments. I hope you will see that these are big and important questions worthy of wondering about for days, weeks, and beyond. I hope you’ll join me in wondering about these questions, and the consequences of potential answers to them.

I argued a while back that learning is having new questions to ask. I hope you’ll join me on my learning journey.

New year, new job

If you’ve been following along (and honestly, I cannot imagine how anyone could possibly have time to do so!), you are under the impression that I’m on sabbatical leave this year.

There has been a change of plans.

I’ve taken an unpaid leave from my college and am spending the bulk of my professional time on curriculum development work at Desmos as a (nearly) full-time teaching faculty member.


The job actually involves almost no sitting on small children.

Our team is growing (Do you know any awesome designers? Send them our way, please!) We have a lot of great stuff in the pipeline. I’m delighted and grateful for the opportunity to work with this amazing team. It’s well known that I’ve been working with them on an extracurricular basis for some time now (>1.5 years), and this has made the transition super smooth.

I am especially fortunate to be able to set aside a portion of my professional life for ongoing projects that are outside the scope and focus of Desmos (although they are certainly consistent with the overall Des-mission of more and better mathematics for all learners!) I’ll spend a couple mornings a week in a kindergarten most of this school year, for example, and Which One Doesn’t Belong is still slated for a 2016 release from Stenhouse. (We still need to sort out Math On-A-Stick for next summer, but that’s a year away.)

Find what you love. Do more of that. [#tmc15]

Here is the text of the keynote I planned to give at Twitter Math Camp. Actual product may have varied substantially in content (but not in spirit) from the typed original.

If you’ve never seen me talk in a large group you’ll have to imagine the energy, cadence and passion you see in my ShadowCon talk brought to this longer form.

Thanks to the Twitter Math Camp organizing committee for inviting me to talk, and to the whole community for supporting my work over the years. It means a lot; I hope to return the favor many times over.

Lisa Henry’s introduction:

Christopher Danielson teaches and writes in Minnesota. You may know him through his documentation of his children’s mathematical antics on Talking Math with Your Kids, through his exhortations not to share bad “Common Core” homework assignments on Facebook, or through his shapes book Which One Doesn’t Belong?

Of course you may not know him at all. But if you do, you’ll recognize that he holds little sacred besides the responsibility we take on in this profession to foster the growth of young minds.

He is currently working on a teacher guide for Which One Doesn’t Belong? to be published by Stenhouse in the spring. He encourages each and every one of you to promote the heck out of Common Core Math for Parents for Dummies. Most of his time this summer is devoted to bringing Math On-A-Stick—a new event to support children and caregivers in informal math activities—to the Minnesota State Fair, and he continues to work with Desmos on developing online networked classroom activities such as Polygraph and Function Carnival.

This is a very American talk about teaching. From what I’ve learned about teaching in other countries with robust educational systems—Singapore, Finland, Japan, Germany, and so on—the U.S. is unique in its tradition of sink-or-swim for teachers.

We equip new teachers with a modest set of tools and experiences, and we say Do the best you can with what you’ve got!

At the policy level, we understand that this is a disaster. But in the American fashion we try to legislate and standardize our way to improvement. We issue pacing guides and measure fidelity to adopted texts. And of course we measure teacher quality by testing students in an effort to standardize learning.

In this sense, the message of my talk today is a very American one. My message is this: Find what you love. Do more of that.


Viewed one way, this is advice to teachers trying to survive and to serve their students well in the era of NCLB and high-stakes testing. (There are probably several such folks in our midst today.)

And that will be a valuable takeaway, but it’s not the heart of my talk. The heart of my talk is more forward-looking and hopeful.

Sending minimally prepared teachers into the field, leaving them to figure it out on their own, and then evaluating whether they have—these things we do well in this country. If you believe that quality comes from sorting out the bad apples, then we’ve built a good machine for this, and the major impediment to improving it is the unions. I assume that this is familiar rhetoric.

What we don’t do well is orient and induct teachers to a community of professionals. We don’t structure our communities to draw on the diverse strengths and passions of its members. This is something that I understand those other nations I mentioned do much better than we do.


For me, the group assembled here, together with the ones who would be here if they could, and many more of the teaching professionals we interact with—whether regularly or sporadically—for me, this group is a community. A community that grows and changes in response to the contributions of its members. It’s not a community that agrees on everything—no community can while remaining honest, open and vibrant. Instead, disagreements offer healthy opportunities for the community and its individual members to grow.

So the hopeful vision in my message (Find what you love. Do more of that.) is that in identifying where your heart is in this profession, you can strengthen your voice and focus your efforts as you contribute to and help shape this community. What our larger American educational system does poorly—foster a professional community that grows and responds to the diverse strengths of its members—the MTBoS does quite well.

So I put two big questions in front of you:

What do you love?

How can you incorporate more of that in what you do? (In your classroom and in your community)

When I ask, What do you love? I don’t want to hear that you love…

Rectangles or stats or 3-Act Lessons or Spirals or Technology or Groups.

I want you to dig deeper. Those things embody what you really love. Whatever you are truly passionate about is bigger than these things. If you can say why you love rectangles or stats or whatever, you’ll be closer to the kind of thing I have in mind.

So now I’ll tell you what I love and how that—in the context of what I have to do—helps to guide my teaching and my contributions to our community.

This is so nerdy. I really hope this is a safe space for this.

I love ambiguity.

The spaces between the certainties are much more interesting to me than the certainties themselves. Ambiguity can provoke wonder, surprise, reflection and clarification.

I’ll share with you how I incorporate this love in the work I do in the classroom and in our community. Let’s start with a video.

This video lacks ambiguity.

Children are smarter than this.

Children can handle ambiguity, which is what I have found so appealing about the Which One Doesn’t Belong framework. (Megan Franke and Terry Wyberg mentions)

In case you aren’t familiar with it, I’ll give you the rap I give kids.


Let me tell you what children say in response to this richer set of shapes. 


[do that]

As they talk about these things, I summarize, paraphrase, probe, review and restate. By the time we’re done, we have a list of properties of shapes. Which of these properties are they supposed to use in deciding which one doesn’t belong? Which of these properties are important? Which ones matter? It depends.

Here’s another set of shapes. 


Sometimes ambiguity comes from studying new objects for which we don’t have a repertoire of vocabulary.

Which properties are important? Which ones matter?

We don’t know yet when we’re looking at a new class of objects. But when we do agree that a property is important, we can name it.

So all that vocabulary which is associated with geometry—that vocabulary isn’t important on its own. Instead it points to important properties, or to properties which somebody has deemed important.

(A non-math example: at some point, it became clear that this growing collection of math teachers online needed naming. The community’s historians are still working on the lineage of the phrase mathtwitterblogosphere and the abbreviation mtbos, The point is that the name exists because there was a thing that needed naming)

Here is what kids do with the ambiguity of the spirals.

[say what that is]

One last WODB example.


The shape in the lower right is the one that provokes discussion. (get to vertices)

Returning to the theme of what you have to do…

In my capacity as a College Algebra teacher, I have to teach rational functions.

Lots of rules and vocabulary and certainty are associated with standard textbook treatments of rational functions. There are asymptotes (vertical, horizontal, oblique…), rules to go along with locating these. There are zeroes and intercepts and symmetries and on and on…

So I played Polygraph: Rationals with my students.


The design of Polygraph is that it puts you in the position of needing to describe to somebody else what you see before you have a shared vocabulary for it—as with WODB. But now importance has an implicit definition. A property is important if it’s useful for distinguishing between functions, and for helping your partner to do that too.

At the beginning of the game ,we shuffle the functions to suggest to you that location in the grid isn’t important. But maybe you think it is. So you ask about it. And you’re likely to get burned.

Orientation, by the way, is important in this context. The difference between 1/x and -1/x is one of orientation.


So orientation doesn’t matter in plane geometry, but it does matter in coordinate algebra.

And the graph on the left is a function, but the one on the right isn’t.


Orientation matters.

But these are both squares.


(Tangentially related…How far can you turn a parabola and have it still be a function?)


I’ll finish with an example from the community. Meg Craig cares about kindness and empathy. She wrote about this on her blog recently. She urged us all to remember that We as teachers are all trying, to the best of our ability, to have students reach the best of their ability.  If you haven’t read this post, you need to.


That kind of thing has a lasting impact on our community. Just the other day, I got worked up about fences. 


There is so much wrong here. Unambiguously wrong, and I called it out. Then I was thinking later that day. OK. That was angry Triangleman. What could kinder gentler Triangleman do?

Well, what do I do in my classroom? What do I do at home with my children? In both of these contexts, I educate patiently. I accept people for where they are and I try to help them see new perspectives; to think differently about things than they do right now.

What does that look like on Twitter? I don’t know. It really is such a great medium for ranting. But I do know that the reason I’m asking myself this is that MathyMeg brought her strengths and passions to our community.


I told you to find what you love, and to do more of that. I told you that I love ambiguity. Maybe you’re the opposite of me. Maybe what you love is certainty in mathematics. How can you help your students appreciate and understand the unique nature of mathematical truth (different from all other disciplines)?

Maybe your heart is with the beauty of geometric forms, or the rhythmic regularity of patterning. Maybe you love how statistics can inform us as we strive to make equitable decisions in an unjust world.

Truth, beauty, regularity, fairness…

Each of these is a more important grounding for your classroom perspective than a pacing guide or textbook sequencing. But none of them is antithetical to these either. What you love can be found in what you have to do.

So my message to you is simple. Name what you love—be explicit about what makes your mathematical heart sing; what resonates in the depth of your teacher soul—and look for it in every corner of your professional life. Share with your students and share it with us, your colleagues and your community.