# How Many? An invitation to #unitchat

Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn’t BelongI showed a way to make geometry playful. Now with How Many? I’m working on a way of making counting playful.

The idea has grown out of the TED-Ed video I did a while back, and the more I play with it, the more I see it in the world around me. My goal is to help parents, teachers, and especially children see it too.

Most counting tasks tell you what to count. Whether it’s Sandra Boynton’s adorable board book Doggies, or Greg Tang’s more sophisticated The Grapes of Math, the authors tell you what to count—or even count it for you.

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn’t specify what to count. Each image has many possibilities.

An example. How many?

Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you’ll see.

It’s important to say what you’re counting, and noticing new things to count will lead to new quantities.

Another example. How many?

A few possibilities: 1, 2, 3, 4, 6, 12, 24, 36. What unit is each counting? Maybe you see fractions, too. 2/3, 4/6, 3/4, 1/12….others? What is the whole for each fraction? The number 3 shows up more than once—there are three unsliced pizzas, and there are also three types of pizza. Are there other numbers that count multiple units?

All of this leads to two specific invitations.

Let me come talk with your students.

(It turns out my schedule filled very quickly, and I’m no longer seeking new classrooms to visit right now—thanks to everyone for your support!)

If you are within an hour of the city of Saint Paul and work with children somewhere in the first through fourth grades, then invite me to come test drive some fun and challenging counting tasks with your students. I have set aside November 17 and 18 and hope to get into a variety of classrooms on those two days. Get in touch through the About/Contact page on this blog.

I’ve been using, and will continue to use and monitor, the hashtag #unitchat, for prompts and discussion of fun and ambiguous counting challenges. Post your thoughts, your own images, the observations of your own children or students, and I’ll do likewise.

How Many? A counting book will be published by Stenhouse late next year.

# Post-Halloween Math Talk

Waiting for the school bus this morning, the two adults and three children discussed last night’s Halloween events.

The neighbor girl, W (9 years old), announced that her brother, E (six years old), had gotten 90 pieces of candy for his trick-or-treating efforts. Griffin (9 years old) announced his haul of 51 pieces.

Me: Did E count each Nerd as one?

Image from Wikipedia

W: Oooo…maybe he did!

P (who is W and E’s father): We were at a house last night that had a bowl with a Take one sign. E went up, then came back and announced that he had taken three.

We told him he had to put two back.

He smiled and said, It’s a package of three!

I love this boy!

I thought for a moment about how various Halloween candies are packaged.

Me:Whoppers?

P: Yeah.

Image courtesy of Free Photo of the Day

I am not proud that I know this sort of thing. But on the rare occasion that my extensive candy knowledge is useful, I am not going to hide it either.

# So what do we learn?

We learn that there is always a follow-up question, and that the follow-up question can bring out fun stories and ideas.

The conversation could have died after E’s 90 and Griffin’s 51 pieces were announced. But I got fun stuff by asking exactly what was being counted.

We have had fun with the question of what counts as one before, when Tabitha and I talked about Eggo mini-wafflesfor example.

# Starting the conversation

North American residents probably don’t need my help here. Your children probably know yesterday’s candy count cold. Ask whether the Nerds (or Whoppers or Smarties or…) count as one piece.

If Halloween isn’t a thing where you are, keep an eye and an ear open for when your children are counting things that are packaged in groups.

# Waffles [Product review]

From time to time, we will be reviewing products here at Talking Math with Your Kids. Sometimes they will be products that are intended to foster mathematics learning, but not always.

Today, we consider one that is not.

We recently bought Kellogg’s Eggo Homestyle Minis waffles.

Tabitha is obsessed with waffles. We typically get the store brand, which as far as I can tell are mathematically uninteresting. But every so often Eggos go on sale, and then it’s game on!

Consider the Minis.

The minis come in sets of 4.

Here is the kind of fun we can have (and, I assure you, that we have had) with this:

• Say, “I’m making frozen waffles this morning. How many do you want?” Leave the unit deliberately unspoken. Child says “one” and is served one mini waffle. Discuss.
• Do the same thing again the next morning.
• Hold up a set of four waffles and ask a young child (say, 2 to 4 years old) how many you have (answer is likely “four”). Then point out to the child that it says there are “10” in the box. Dump them all out and discuss. Key question: What are there ten of?
• Ask a somewhat older child (say 7 to 9) “If there are 10 sets of 4 waffles in the box, how many waffles are there?” Follow up with “How do you know that’s right?”

Finally, this: In addition to (1) waffles, and (2) sets of waffles, there is a third unit to count in that box: servings.

It turns out that 1 serving is 3 sets of 4 waffles. How awesome is this?

You can ask an older child to predict what the number of “Servings per Container” will be on the Nutrition Facts label. I would have gotten it wrong. I would have applied too much mathematics to the problem and said $3\frac{1}{3}$. You can see the “correct” answer below.

You didn’t think I had something to say about the waffles as food, did you? I’m sure they are everything one would expect of Eggo waffles. You probably already know whether you consider that a good thing. Tabitha likes them.

P.S. My own father turns 70 today. He certainly supported my own mathematical development growing up. Thanks, Dad! And Happy Birthday.

# Things that come in pairs

Talking math with your kids takes two forms: (1) seizing opportunities the kids initiate, and (2) creating opportunities where there might previously have been none.

This is story of creating an opportunity—first with my four-year old, then with my seven-year old.

I was doing the dishes one morning while Tabitha (who was four, nearly five) drew in the other room. She came in for help.

Tabitha: (4 years old) Where are the scissors?

Me: I don’t know. How many pairs of scissors do we have anyway? We have a lot, but we can never find them. Why is that?

T: I just need one pair of scissors.

Me: Isn’t it weird that scissors come in pairs?

T: Yeah.

Me: What else comes in pairs?

T: Pants do. And shoes.

Me: Oooh. Good. What else?

T: Legs. And ears. And noses [giggles].

Me: Noses?!? Noses don’t come in pairs, silly!

T: Eyes do. And glasses.

Me: Nice! Eyes. You know, it’s not just people who have eyes that come in pairs. Fish do, too.

T: Of course!

Me: How many pairs of eyes are in our aquarium?

T: Seven.

Me: So how many eyes is that?

T: Nine?

[Griffin (who was seven years old) wandered in from the living room.]

Me: Griff, how many pairs of eyes are in our aquarium?

G: Seven.

Me: So how many eyes is that?

G: Fourteen.

Me: How did you know that so fast?

G: Seven plus seven is fourteen.

Me: Right. But that’s two sevens. Don’t we need seven twos?

G: Yeah, but it’s the same answer either way.

# So What Do We Learn?

This conversation was about groups and units. If kids are going to understand place value, they’ll need to be able to think about different units. Sometimes a unit is a thing (an eye); sometimes a unit is a group (a pair of eyes). We counted both.

When Griffin finds two sevens instead of seven twos, he using the commutative property of multiplication. Sometimes called the rearrangement property, it is a fancy way of saying that when you multiply, it doesn’t matter which number comes first. 7×2=2×7. Following up on Griffin’s strategy requires more math knowledge than many parents are comfortable with. That’s OK. That’s why Talking Math with Your Kids is here. Shoot us a note with your questions.

But if you are comfortable with the idea (and/or the vocabulary) of commutative, you could follow up with Does this always work? or a discussion of what things in the world are commutative and what things are not.

Real-world example? In the bathroom (forgive me), Wipe, Flush, Wash is not commutative. Do those in a different order and get different results. But Use the potty, Brush your teeth, Put on your pajamas? Totally commutative; the order we do them in does not matter.

# Starting the Conversation

Give kids practice counting both groups and individual things. Just help them notice that some things usually do come in groups. Eggs, bicycle wheels, grapes; these are all things that usually come in groups. Ask whether the groups are always (or almost always) the same size.

Notice with your children that your grocery store (probably) has this package of eggs, and wonder aloud how many eggs are in two-and-a-half dozen.

Correcting Tabitha when she added 7 and 2 to get 9? That wasn’t nearly so important. She was four at the time; multiplication facts typically become important around eight or nine years old.

# The unit is the thing that you count

Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out of the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

# So What Do We Learn?

What makes this more than just a funny story is that Griffin and Tabitha are clearly counting different things. They are talking about different units.

When we make cookies, everyone agrees on the unit; we know what one cookie is.

But brownies are different. Tabitha seems to think that a brownie is the thing that comes out of the oven. Griffin seems to think that a brownie is what you eat in one serving.

One brownie according to Tabitha.

One brownie, according to Griffin

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.

Fun follow-up question: Does the thing in this video count as one brownie?

# Starting the Conversation

Anytime there are things in groups—or things being cut—is a good time to talk about units.

Grocery stores usually have express lanes where you have to have Ten items or fewerAsk your child whether someone with a dozen eggs could use that lane. What about someone with 12 apples in a bag? What if the apples are loose?

When your child asks for two slices of pizza, take one slice, cut it down the middle, smile wryly and ask whether that’s OK.

In all of these cases, the central question is What counts as one? Play with that question.

Also, watch that video together. It’s a ton of fun.