A few minutes later I made this.

And then I went scrambling for my notebook.

Yup. Sure enough. is equivalent to the Pythagorean Theorem.

Tagged: kindergarten, triangles ]]>

My first thought was, “Send them all to New York to visit the Museum of Math!” but this was off by a couple orders of magnitude.

As the conversation continued, it became clear that they weren’t seeking advice so much as someone to make it happen. So I said yes.

I am spending three Thursday mornings, and one afternoon, with these fourth graders. Today was day 1.

The theme of the residency is scale. We are playing with small versions of big things and big versions of small things.

A few favorite moments from today:

When asked to share a big version of a small thing, one girl said “Horses”. I pressed her to state her meaning. “If you had a map with stables on it, the horses in those stables would be really small, then when you went to the stables, the *actual* horses would be really big.”

Ladies and gentlemen, I give you the big math idea of *inverse!*

I thought *the horses on the map are small versions of the big real-life horses.* But she was very clear that her experience was *small horses on the map, then see the big ones*. The small-to-big relationship isn’t just the opposite of the big-to-small one; it is its own relationship. These two relationships are inverses—each existing on its own, but with a special connection to each other.

I cooked up a little *Which One Doesn’t Belong? *set in preparation for our work.

*Which One Doesn’t Belong? *never disappoints. (Student/home version and Teacher Guide coming this summer from Stenhouse, by the way!)

We noticed all the things I had hoped for, and more. And then afterwards a girl came up to me to make her case that we weren’t being totally precise about our description of the upper-right image. If—as we claimed—the shape in the upper right is composed of four of the upper-left triangles, then the big triangle wasn’t exactly the same size as the one in the lower left because the triangles have outlines which are not infinitely thin.

I brought in many small laser cut triangles of these seven types:

I gave them time to play with these triangles. One student said she knew what we were going to do with them. So I asked her what that was, and she replied that we were going to see which ones could fit together to make other ones. This was not the plan, but was behavior I was eager to encourage.

She asserted that the pink and the black make the red.

This was a detour worth five minutes, so we took it. Arguments were presented pro and con. The major pro argument was based on the *close enough* principle. Con arguments were of two flavors: (1) put the red underneath and you’ll see some red peeking out from underneath, and (2) the long side on the pink plus black shape is not straight, while it is on the red one.

The main question I wanted to get to—remember that our focus is *scale*—was *Which of the triangles in our set will do what the upper-right shape in our Which One Doesn’t Belong? set does?* Which of our triangles can you make into a larger version?

**Spoiler Alert!**

All triangles do this. But these fourth-graders don’t know that. And because they don’t know that, they got to feel a little thrill of success when they found one that did.

And of course they produced some evidence that the relationship we’re investigating is a challenging one.

This is what we had on the document camera at the end of one of three sessions this morning.

HOLD THE PHONE! LET’S LOOK AT ONE OF THESE CLOSE UP!

Do you see? All the others use four triangles to make the bigger version, and this one can too. But this can scale up to make a bigger version that uses only two of the original!

Of course there is a part of my math-major brain that knows this about isosceles right triangles, but it’s a wonderful wonderful thing to have pop up unexpectedly in the middle of fourth-grade math play.

Overall, a delightful morning of math. We got to only a small fraction of what I’ve got chambered so we’ll pick up where we left off next week. I’m hoping I can get them to build one of these.

Either way, I am thankful for the opportunity to play math with this group of kids. They are creative, enthusiastic, curious, and delightful. Their teachers have been very welcoming and open to the intellectual chaos I began to unleash today.

I chose a set of triangles that would have interesting variety and some discoverable properties.

**Purple**: 3-4-5

**Pink**: Isosceles obtuse

**White**: Isosceles right

**Red**: 30-60-90

**Light blue**: One-eighth of a regular octagon

**Black**: Equilateral

**Dark blue**: One-fifth of a regular pentagon

I also made some yellow obtuse scalene triangles, but they are missing so they didn’t make the trip. Within these classes, these triangles are all congruent. Each class has at least one side that is one inch long.

Tagged: elementary math, residency, scale, similarity, triangles ]]>

You should know that Lakeshore Learning makes a machine for each of the four basic operations: addition, subtraction, multiplication, and division. You should further know that they got the structure of the subtraction and division machines wrong (more on that at end of post). So today, I’ll focus on the addition and multiplication machines.

These have the same format as each other: 9 rows of 9 buttons. Each button labeled on top with *m+n* or *m×n* accordingly, in the *m*th row and *n*th column. The button pops up when you press it; down when you press it again, like a ball-point clicky pen. On the front of the button, visible only when popped up, is the corresponding sum or product. Lakeshore Learning views them as mechanical flashcards and nothing more.

I convert them to Pattern Machines by covering all the numbers with colored vinyl. These are a ton of fun at home and in classrooms—great for patterning, counting, making pixel pictures, etc.

Now I am playing with a Sequence Machine. I have covered all of the tops of the buttons on a Multiplication Machine.

Now you can generate number sequences, without being distracted by the multiplication facts. Above, you see a familiar sequence—the squares.

Things quickly get more complicated, though. If you read each of the sequences below from left to right, ask yourself *What comes next?*, *What would be the 10th or 23rd term in the sequence?* and *What is the general relationship between the term number and its value (i.e. What is the nth term?*)

There are many more sequences to be made on these; many more questions to explore.

I’m working on a follow up question about the relationship between the sequence generated by the same button patterns on Sequence Machines built from an Addition Machine, and from properly redesigned Subtraction and Division Machines.

*What is a properly designed Subtraction Machine? *you ask? I’m glad you did! Such a thing would have *m*–*n* or *m*÷*n* on the button in row *m*, column *n*. This is not how Lakeshore Learning makes them. They make them like this:

]]>

Congratulations! [Child] successfully completed a group of DreamBox Learning lessons.

Are you familiar with the concept of compensation? This is a strategy that can be used to make addition problems “friendlier”. To use it, just subtract a “bit” from one number and add that same “bit” to the other to create two new numbers that are easier to add mentally.

[Child] learned this strategy by completing a series of lessons using the special DreamBox tool, Compensation BucketsTM. For example, when shown the problem 29 + 64, [Child] turned it into 30 + 63. Towards the end of this unit, [Child] was adding 3-digit numbers with sums up to 200!

On the Run: Friendly NumbersTake turns supplying two numbers to add. The other player has to make the two numbers into a “friendlier” equivalent expression. For example if you say 38 + 27, [Child] might say 40 + 25, or 35 + 30.

Remember, your encouragement to play two or more times per week for at least 15 minutes each time will help [Child], because knowledge is built most effectively when concepts are presented regularly. You can always check [Child]’s latest academic progress on parent dashboard.

Best regards,

The Teachers at DreamBox Learning

I’ll let my friend say what he found so great here. I agree with every word.

I just wanted to share this Dreambox support email I got today as [Child] was working. This is perhaps a more impressive feature than the software itself. Not only does it provide a very simple explanation of what skill she is learning, it provides the jargon needed to have a conversation with her: “Bits”. As a middle school math teacher that was always a roadblock for a lot of parents: the language parents didn’t use themselves when they were in math.

Compare against the report I receive on a weekly basis from XtraMath.

You should know that DreamBox is a for-profit corporation that charges for its services, while XtraMath is a non-profit that provides its services for free while soliciting donations to continue the work.

But here’s the thing: XtraMath is not a free version of DreamBox.

Math fact practice with an emphasis on speed is not a version of conceptual development. It’s a totally different mission based on completely different understandings about what it means to learn mathematics.

DreamBox is trying to help my friend support his daughter’s mathematical development. There is an actionable message: *Play with this idea that relates to something she’s been working on*.

XtraMath is giving parents “the information they need to know how well their children know their math facts, and the progress they are making toward mastery.” But what are parents to do with that information? How do they support their children in moving forward?

Ultimately, this is a question about data in ed-tech. I’m not here to single out XtraMath; it’s just the case that’s in front of me each and every week. There are lots of ed-tech products out there trying to do what XtraMath is doing.

When kids do things on computers, it is easy to collect data about what they do. It’s easy to turn that into a report or a dashboard.

It is much more difficult to determine what it all means. But data without meaning isn’t useful.

For instance, what does Tabitha’s report above *mean*? What am I to understand about Tabitha’s mathematical knowledge based on this report? This is the major representation her school offers of her progress in third-grade mathematics. What am I to make of it? More to the point, what is a less mathematically knowledgeable parent to make of it?

But what is a school to do?

I say provide more meaning, less data.

Don’t include *free* as a primary criterion for adopting materials.

Lay off the speed-based fact practice and find ways get kids talking, thinking, and doing.

And what is an ed-tech company to do?

Invest its limited resources in crafting materials consistent with research on children’s learning and on parent communication.

]]>A while back I wondered what it would be like to decompose a triangle and play with its parts. So I cut up a triangle and got busy.

My play started simply.

Things quickly got more complicated, with symmetry, patterns, and tilings.

I saw happening in myself what I kept telling people I saw in children at Math On-A-Stick last summer. The longer I persisted, the richer the ideas I had. These are in a bowl on the dining room table (along with my favorite pentagon and some materials for Tabitha’s decimal study), available for play whenever we like.

You may not have a laser cutter, but you certainly have access to a compass, cardstock, and scissors. I recommend getting down to business so you can play with my favorite quadrilateral.

]]>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.

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.

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.

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.

Tagged: learning, pentagons, school math, solutions, tmwyk ]]>

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.

- Behind which circle(s) must there be roots for this function?
- Behind which circles
*might*there be roots? - 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:

- There are sometimes roots where you don’t expect them (Screen 8).
- There are sometimes
*not*roots where it looks like there really ought to be. - If the function starts negative and becomes positive, it has a root.
- And vice versa. (Screen 4)
- 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*.

Tagged: activity builder, calculus, desmos, intermediate value theorem ]]>

**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.

Tagged: cookies, division, Tabitha ]]>

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.

- Description of item
- Actual cost of item
- 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.

Tagged: 8 years old, ccss, common core, estimation, Tabitha, talking math with your kids ]]>

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:

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:

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.

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.

- What would these children have done if asked—prior to instruction—to make a pattern with their tiles?
- 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.

Tagged: kindergarten, learning, pattern, questions ]]>