How many do you see? How do you see them?

This summer, as Gwyneth and I were packing up Othello, I started playing with different arrangements of discs – mostly arrays – and asked her “How many?” I remembered the following arrangement, taken from AIMS’ Cookie Combos activity.

3^2 + 4 * 4

“Sixteen plus nine, so nineteen plus six… twenty, twenty-five,” she said. (I don’t think that she actually said “twenty” aloud. That came after my clarifying question: “Wait. Huh?”)

There’s a lot happening in Gwyneth’s bridging through twenty strategy – partitioning of quantities, place value, commutative property, breaking apart to make (a multiple of) ten. All within a three count, standard algorithm be damned.

This invented strategy discussion was a happy accident. The goal of this problem when we pose it to teachers is to see different ways to visualize the group and represent these using expressions. It’s about valuing different methods; the solution – counting 25 cookies – is easy enough.

How many do you see? How do you see them? How many different ways can you find?


Some popular solutions:

7 + 2 * 5 + 2 * 3 + 2 * 1
3 * 5 + 2 * 4 + 2
4 * 5 + 5

If you look just right, you can see two arrays:

4 * 4 + 3 * 3

A creative solution that involves counting what’s not there:

7^2 – 4 * 6

And moving what is:


If you plan on using these images with your students, I recommend displaying the photo with just white discs. This leaves the problem open. Two colours were used above to illustrate various visualizations. This can steer student thinking. (See how the use of colour is intended to be helpful here.) If students miss one of the visualizations above, display that photo and ask for the expression (or vice versa).

[TMWYKS] Rainbow Loom

Christopher Danielson brought you #tmwyk, or talking math with your kids. I bring you #tmwyks, or talking math with your kid sister.

It happens to every parent, I think: the kid says something and nobody has to ask “Where’d she hear that?” Maybe it’s the kid’s choice of words. Or maybe it’s the tone, pitch, or rhythm that gives you away. Rare is it for me that these are proud parenting moments.

A recent exception:

Gwyneth (9 years old): What patterns do you see?

Rainbow Loom

Keira (6 years old): Red, white, yellow, red, white, yellow, red, white, yellow.

Gwyneth: Great! Can you find another pattern?

[TMWYK] Aero Bubble Bar

Recently, Nestlé launched the new AERO bubble bar throughout Canada and the UK.

For the benefit of the American readership:


From the press release:

As well as offering a unique bar design, guaranteed to stand out from the crowd, AERO’s innovation isn’t just for show. The new design sees the bar divided into ten easily snappable ‘bubbles’, making it less messy to eat and more portionable. What’s more, each of the ten ‘bubbles’ are designed to melt more easily in the mouth, maximising the taste of AERO’s signature bubbly chocolate.

I brought one home a couple weeks ago. I put the bar’s portionability to the test.


I snapped off two bubbles each for Keira (5), Gwyneth (8), and Marnie (N/A). Plus, two for me. (Missed math teacher opportunity, I know.) Two pieces were left over. “How much more should we each get?” I asked.

“Half,” Keira answered. She told me to make two cuts: two becomes four, or n(Keira’s family). For shits and giggles, we played with different cuts. What I learned from Keira:

the halves and the halve nots

“Or two-quarters,” Gwyneth piped up.

“Huh?” I returned, caught off-guard. “Tell me more,” I recovered. Gwyneth told me to cut each of the two bubbles into four quarters, giving us eight quarters. Eight pieces can be shared equally between four people. Each of us should get two pieces, or two-quarters.

Gwyneth’s strategy–divide each piece into fourths rather than make four pieces in all like her sister–surprised me. It’s a strategy that makes sense to her: dividing each piece into fourths means she’ll be able to form four equal groups. It’s a strategy that’s flexible: I don’t think she’ll be fazed by a curveball, like an additional bubble or family member.

Symbolically, we have:


The result is trivial; her thinking is not.

For more math talk with kids, please follow Christopher Danielson’s new blog.

Pentagons and Poodles

Gwyneth: Dad, is this a pentagon prism?


Me: It is! Pentagonal.

Gwyneth: Look, Dad! There’s a pentagon inside the pentagon.


Me: Cool. Hey Keira! What did you make? A puppy?


Keira: It’s a POODLE, Dad! And it’s got a SQUARE body!


What happened to five?

At 10:00 pm Saturday I returned home from #NCTMDenver. My daughters Gwyneth (8) and Keira (5) were glued to me for the next two and a half hours. Mostly playing with the Zometool kit I picked up at the exhibit hall, filling me in on the past five days.

In September, Gwyneth was concerned about the precise use of language. She’s still at it, researching dog breeds on the internet. Hasn’t stopped. Saturday night/Sunday morning, she wanted me to see this:

pedigree select-a-dog

Remember, her little sister is five.

“What should I click, Dad?” she asked. I was just about to reply “Doesn’t matter, just pick one” before I stopped myself. Instead, I told her to pick the best wrong answer. I was just curious, not trying to prepare my daughter for future success on bubble tests. “Six to eleven,” she quickly answered. Her confidence surprised me. “Nah, gotta be under 4,” I said.

With some prompting (needling?) she presented three arguments. First, Gwyneth reasoned that since Keira was “five and a bit” her sister was closer to six than four. She argued that it’s less than a year until her sixth birthday and it’s been over a year since her fourth birthday.

Second, she reasoned that “five and a bit” was more than five, the halfway point between four and six.

She gets it. Kids get it. They get that 37 is closer to 40 than 30. They get that 7.3 is less than halfway between 7 and 8. They get it until we ask them to memorize things like “Five and above? Give it a shove.”

Third, Gwyneth argued that since she is eight and her sister is five, the best answer is the one that includes the two of them. A stretch to connect this to measures of central tendency?

I’m not sure if Gwyneth enjoys finding these things for her dad or if she thinks it’s getting her one step closer to this:


A fun conversation, either way.

Furry Logic

This is my favourite photo. It’s of my youngest daughter, Keira, and my dog, Skye, while on a walk this past spring. I said “Say cheese” and they did. Both of them. Unknown to me at the time, it would be the last picture I would take of Skye. In the next two weeks, Skye’s health rapidly deteriorated to the point where my wife and I had to make the difficult decision. It has been tough on all of us. I still catch myself holding the gate open for her behind me as I go between the front and back yard.

This has been particularly hard for my 7-year-old daughter, Gwyneth. She understands why we are not getting another dog. But that hasn’t stopped her from researching dog breeds on the internet. Non-stop. If you ever meet my daughter, she’ll ask you questions like “D’you know that Labrador retrievers have webbed feet for swimming?”, “D’you know that pugs have a hard time breathing because of their flat faces?” and “D’you know that poodles are hypoallergenic?” Think Jonathan Lipnicki in Jerry Maguire. She’s that kid. And I love her for it.

But this is my math blog…

The other day Gwyneth came to me to tell me she wasn’t happy about what she had read on Here it is:

Did you catch what was troubling my daughter? Here’s two more:

Here’s our conversation, as I remember it¹:

Gwyneth: They say Golden Retrievers are the smartest. And they say Papillons are the smartest. But they also say Poodles are the smartest. Shelties too!

Me: So, what’s the problem?

Gwyneth: They can’t all be the smartest.

Me: So, what should it say?

Gwyneth: One of the smartest. Not the smartest.

The smartest means:

Golden Retrievers > Papillons
Papillons > Golden Retrievers

Not okay with Gwyneth.

One of the smartest means:

Golden Retrievers ≥ Papillons
Papillons ≥ Golden Retrievers

She’s cool with that.

At the same time as this conversation, Dr. Keith Devlin was writing about the use of language in the case against Lance Armstrong:

Though the layperson typically thinks of mathematicians as being focused on numbers, that is actually not the case. That false view is a consequence of the mathematics taught in high school. Only at university are you likely to encounter the mathematics done by the professionals. High among our real areas of expertise are logical reasoning, rigorous proof, and the precise use of language.

Maybe it’s because her dad is bothered by things like “increased student scores by 50%” when they mean “increased the number of students passing by 50%” that my daughter is concerned about the precise use of language. And I love her for it.

¹The Department of Giving Credit Where Credit is Due asks you to check out Christopher Danielson’s talking math with your kids posts.