Category Archives: Problems (math)

My favorite quadrilateral

I am a known fan of triangles and hexagons. I have also been having quite a bit of fun with laser cutters at a local maker space.

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.

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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.

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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.

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.

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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.

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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.

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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.

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.

Fruit snacks

Kellogg’s has issued Froot Loops fruit snacks in the shape of digits. (Side note: Cheez-Its need to get on board with this! There have been Scrabble tile Cheez-Its for years. We want numbers, operations and relational symbols!)

Naturally I bought some.

Tabitha (8 years old) asked—as she does in these scenarios which occur with great frequency—Are you just buying that because it’s mathy?

Yes, sweetie. Yes I am.

But how to put them to use?

After many rejected ideas, here’s my favorite.

The task

Here are the contents of one pack.


That’s 5, 2, 9, 1, 3, 2, 4, 3, 9. Their sum is 38.

I’m setting the over/under on the sum of the next pack at 41. Do you want the over or the under? Why?

Play along with your questions and answers in the comments.

I’ll open the pack on Wednesday, May 27.

Wiggins question #7

Question 7

Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

All measures of center hide variation.

This is what makes them useful, and it is what makes them problematic.

Using the mean makes zeroes a problem in grading. Wildly divergent values (such as a zero in a gradebook) will greatly affect the mean. It is hard to argue that 2 A’s and a zero is the same as consistent D work. Yet this is how the mean plays out.

But going too far down this road will only lead to critiques of the whole system of grading students at all. I find that system to be indefensible and counterproductive. I have made my peace with it, and I try to do as little harm as possible with the responsibility I have to assign grades in my work.

All of which is to say, it is not using the mean that leads to a poor measure of achievement. It is mistaking quantitative measures for accurate ones that leads to a poor measure of achievement.

Wiggins question #5

Question 5

“Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea is.

In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.

You see, any rich enough mathematical idea has multiple meanings. What is subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it takeaway? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an equivalence class resulting from the equivalence relation,

\frac{a}{b}=\frac{c}{d} if and only if a\cdot d=b\cdot c

Is that what a fraction is?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. Hours times miles per hour gives miles. Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: A\times B means A groups of B (I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). A, in this interpretation, is expressed in one unit. B is the unit rate (things per group). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Image from The New York Times.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”


Why do we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—one-to-one correspondence. One number word for each object, and one object for each number word.

The answer is that mom really did not count the corner meatball twice. The first time, she counted the meatball to establish that each row has 4 meatballs. The second time, she counted the rows. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or What does this have to do with mathematics?)

Twister (on sister site, Talking Math with Your Kids)


Wiggins question #4

Question 4

Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002

One sixtieth is the biggest of these, as the others are between one and two one-thousandths (or between one one-thousandth and one five-hundredth).

Then, in descending order, we have .002, .00156, .0015 and .001.

It is common that students will order the decimals this way:

.00156, .0015, .002, .001

I cannot predict where a person who does this will place one sixtieth.

Nonetheless, when we read these decimals as point zero zero one five six, we encourage students to ignore place value, and we encourage the misapplication of whole-number rules to the right of the decimal point.

We really do need to use place value language for decimals in classrooms. One-hundred-fifty-six one-hundred-thousandths.

Much, much more about this and related ideas in the Triangleman Decimal Institute posts from last fall. Short version: learning decimals is WAY more complicated than most people have any reason to imagine.