Canned pumpkin

Fall baking in our house requires canned pumpkin. We were out so I asked for Tabitha’s help at the grocery store, where the pumpkin in on the bottom shelf.

Me: Put four of those bright orange cans of pumpkin in our cart, please.

Tabitha (6 years old): I don’t know if I can carry four.

Me: Do two, then two more.

Photo Nov 17, 10 15 20 AM

T: [With two cans of pumpkin in her hands] I know, because two plus two is four.

Me: Right. You could do three and one, I suppose.

T: OK. Give me one back.

She takes it, picks up two more from the shelf and brings the three cans over to me.

T: I did two and three.

Me: So we have five cans?

T: No! You gave me one back, remember?

Me: So two plus three minus one is four?

T: Yeah.

So what do we learn?

Decomposing numbers is fun.

We tend to think of 2+2 as something to do, and that the answer is 4. But in this case 4 is the thing to do, and 2+2 is one of several possible answers. When we think about different ways to make 4, we are decomposing 4.

Tabitha can keep track of the moves in our complicated decomposition at the end (You gave me one back, remember?) but she does not have practice with the math notation that captures all of these moves (Two plus three minus one is four). That is one of my roles in the conversation.

Starting the conversation

Tabitha gave me the ideal beginning to this conversation—she pointed out that there were too many cans for her to carry. It shouldn’t be difficult to put your own child in such a situation. The grocery store sells lots of things that children can carry a few of, but not a lot of: apples, oranges, cans of soup, etc. Picking up toys at the end of a play session at home or school, or books at the library—all of these are opportunities for you to name the number involved, then suggest a way to decompose it.

M&Ms

Dessert is a good time to get the children’s attention for a little math talk.

A few weeks back, a smallish serving of M&Ms was about to be given to each child, from a large one-pound bag.

In keeping with my assertion that a day should never pass without asking my kids at least one how many? question, I asked Griffin to choose the size of the serving (but unbeknownst to him that this was the purpose.)

Me: Give me a number between 10 and 20.

Griffin (eight years old): What’s the point?

Me: I won’t tell you until you choose.

G: I won’t until I know why.

This is my own doing. I have long told both children that people need to have reasons for asking you to do things, and that satisfying these reasons is more important than following directions blindly. This is an important element of problem-solving and critical thinking. It does have consequences; I understand this.

Me: Tabitha, pick a number between 10 and 20.

Tabitha (five years old): Twelve.

Me: OK. That’s how many M&Ms you each get for dessert.

G: Oh, then I pick 20.

Me: No. The first number I heard. That’s the one I’m using.

G: You should use the biggest.

Me: Nope. The first.

T: Next time, I should choose….thirteen.

This is beautiful, is it not?

I love the realization that things had not worked out for her maximal benefit. I love that she knows some thinking needs to be applied to the situation.

And I love dearly that the result of this thinking is an increase of a single M&M. Griffin comes to her rescue.

G: No, Tabitha! It’s between 10 and 20!

T: Oh. I should choose…nineteen.

So what do we learn?

This was totally devious on my part, and I do not recommend that you behave this way with your children. We do learn, though, that strategic thinking with numbers is something to be learned. The strategy of thinking through the biggest possible number within the given constraints is not obvious to young children. Looking for a bigger number is a prerequisite to thinking hard about the biggest possible number. 

We also learn, of course, that I am a horrible person.

Starting the conversation

Again, I do not encourage you to manipulate your children in this way. Although in my own defense, neither 12 nor 20 M&Ms is such a bad deal for 5- and 8-year olds near bedtime.

The pick-a-number game is fun for lots of things, though. Taking turns (whoever gets closest to the number I wrote down gets the first turn) is a classic example, but you can think up lots of your own. After the picking, talk about the selection. What would have been a better choice, knowing what you know now? What would have been a worse choice? Why did you pick the number you did? Et cetera. Listen to your children’s strategies and share your own.

Multiplication Machine [Product review]

We have in our house a thing called the “Multiplication Machine”. It is a flash-cardy sort of thing. The multiplication facts are written on top of a 9×9 array of spring-loaded buttons. You press one and the button pops up, revealing the product. You can buy such a thing at your nearest teacher supply store. Ours came from Lakeshore Learning.

Talking Math with Your Kids is dedicated to helping parents and other caregivers to identify the mathematical opportunities afforded by everyday life, so we will not discuss here the traditional, intended use of this product (which is drilling and reviewing multiplication facts).

No, I want to talk about this thing as a toy.

Tabitha was hard at work pressing buttons on the Multiplication Machine the other evening. When I peeked in on her, I saw a scene that looked an awful lot like the one below.

patterns.3

She was playing with the arrangement of up and down buttons, not with the multiplication facts written on them. Patterns are tons of fun. So I went with it.

We developed the up, up, up, down pattern. We went across each row from left to right, top row to bottom row, as you would read a book.

Up Up Up Down, etc.

We developed its opposite—Down, down, down, up.

Down down down up.

We developed the Up, up, up, up, down pattern. This proved much more difficult for Tabitha, as she could not subitize the four ups. She counted them on her fingers, which she also needed for pressing buttons. She worked it out, though.

Up up up up down.

Before executing this last one, we noticed the right-to-left diagonals we had gotten from the Up, up, up, down pattern and predicted what we thought would happen when three ups became four. She correctly predicted the left-to-right diagonals, but I do not know why she predicted this.

We have not yet investigated the down-down-up pattern together, but I suspect she will get a kick out of it.

Down down up.

There are many more cool patterns to play with here. A few ideas that I am sure we’ll explore in the coming weeks:

  • What will happen with lots of different combinations of ups and downs?
  • What if we do columns instead of rows?
  • Are there any patterns where you cannot tell whether the person did rows or columns?
  • What if we follow a path back and forth across the rows, instead of starting at the left-hand side of each row?
  • What if go right-to-left? Or bottom-to-top?

The biggest number

I do not recall the beginning of this conversation, but I do recall that we were eating pizza at the dinner table when Tabitha anticipated my turn in the dicussion.

Tabitha (6 years old): I know what you’re going to say, Daddy. “Counting never ends.”

Me: I suppose that sounds like something I would say, yes.

T: What’s the biggest number, though? Googolplex?

Quick tutorial. A “googol”—spelled that way—refers to this number: 10^{100}, or “a one followed by a hundred zeroes”.

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

It is, of course, a very big number. Far too big to be practical in any meaningful sense. The very idea of such a large number having a name is fascinating to children. Most children (in my experience) encounter one googol in their social interactions with other children. The googol does not appear in the Common Core State Standards.

A “googolplex” is 10^{10^{100}}, or 10^{googol} or “a one followed by a googol zeroes”. You cannot write this number out in standard form.

You may Google googol for lots of interesting characterizations of how extremely silly this very large number is.

For example, you will not live for one googol seconds.

Indeed, the universe has not existed for one googol seconds (not even by the greatest estimates of its age—not even close).

You get the idea.

Me: Well, like you said I would say, counting never ends, so no googolplex is not the biggest number.

T: If you counted by 10,000 could you ever get to googolplex in your life?

Me: No.

T: If you counted by 11,000?

Me: No.

T: 12,000? 13,000?

Me: No. Even if you counted by googol, you couldn’t get to googolplex in your lifetime.

T: Well, what if you counted by googolplex?

Me: Well sure. It would the start of your count, wouldn’t it?

She decides to demonstrate this (Side note, we have been counting by various numbers of late).

T: Googolplex.

She smiles broadly, congratulating herself for successfully counting to what she has perceived to be the largest number.

We discuss further the existence of a largest number. Then Tabitha makes a claim that takes us in a different direction.

T: Eventually, numbers just go back to the beginning.

Me: So if you keep counting, you get to zero?

T: No.

Me: One?

T: No, Daddy! Don’t you remember there are numbers before zero?

So what do we learn?

Big numbers are fun. Boy howdy are big numbers fun. Children love to talk about the biggest number, and whether one exists. There is all kinds of lovely thinking going on when they ask these kinds of questions.

Talking about big numbers often leads to talking about infinity. If there is no biggest number, it is because numbers go on forever. The only thing Tabitha has experience with that goes on forever is a loop. She drew on that loop metaphor in imagining that numbers go back to the beginning eventually.

Starting the conversation

Listen for the biggest number talk. It often surfaces when children are comparing their athletic prowess (I can jump 2 sidewalk squares! I can jump 100 sidewalk squares! Pretty soon, someone is claiming to be able to jump googol or infinity sidewalk squares.)

When it surfaces, support it. Play and explore with your child. Answer questions. Ask questions. Talk about it and have fun. Look stuff up together when the questions go past your own knowledge. Shoot me a question here at Talking Math with Your Kids if I can answer any of those for you.

Counting by fives

The time I spend doing the dishes is frequently productive. Tabitha (6 years old) or Griffin (9 years old) will often linger nearby playing and talking. Sometimes they talk to me. Sometimes they talk about math, as on a recent evening.

Tabitha (6 years old): 5, 10, 15, 20, 25, 30…

She continues until 170, at which point she becomes bored.

T: Dad! I can count by hundreds. Want to hear?

Me: Yes. Yes I do.

T: One hundred, two hundred, …

She continues to nine hundred, when she pauses. I wait a few beats to see what will happen. When nothing does, I ask.

Me: What comes next?

T: Ten hundred.

Then…

T: Hey Dad, when do you get to one thousand?

Me: Awesome. Ten hundred is one thousand. That’s the word for ten hundreds.

T: What comes next?

Me: One thousand one hundred.

T: One thousand two hundred, one thousand three hundred,…

She continues. She reaches one thousand nine hundred.

Long pause.

T: Two thousand. [She smiles broadly.]

Later that evening, she is doing her weekly homework, which involves counting by fives. It is not clear from the directions whether she is to start over for each kind of five, or whether she is to continue counting all the way down the page.

count.by.fives

Tabitha decides to continue the counting, and this seems the more appropriate challenge for her.

She gets to 95.

T: How do you write one hundred?

Me: A one and two zeroes.

T: How do you write one hundred five?

Me: You tell me. You try, and I’ll tell you whether it’s right.

T: One-zero-five.

Me: How did you know that?

T: Replace the zero with a five. How do you write one hundred ten?

Me: What do you think?

T: One-one-zero.

So What Do We Learn?

The first activity was about number language. Tabitha was following the patterns in number language and puzzling over the places where the patterns break down. One hundred, two hundred, three hundred…how does one thousand pop out of this sequence?

The second activity was about writing numbers (or numeration). Tabitha had no trouble counting through one hundred five, but she wasn’t sure how to write it.

A common error is to write this: 1005. The thinking goes like this: One hundred is 100, so I’ll write that, then put a 5 at the end.

Starting the conversation

Any time a child asks if you want to hear her count, the correct answer is yes. Then listen. Listen for the challenging bits (the teens are difficult, as is anything following a ‘9’—such as 19 to 20, or 49 to 50). Be ready to help if they get stuck but also to let them think it through and try for themselves. Then talk about how they knew to do what they did. Talk about their thinking.

Post-Halloween Math Talk

File this under Talking about talking math with with your kids.

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.

whoppers

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.

Doll years

Out of the blue on our recent camping trip, Tabitha had an announcement for me.

Tabitha (6 years old): I am 12 in doll years and Griffy is 16 in doll years.

Her brother Griffin is 9.

T: So how old are you in doll years, Daddy?

Me: Well, how do doll years work?

Photo Oct 12, 1 34 29 PM

T: Well, I’m 12 and Griffy’s 16.

Me: Is it twice as old? Then I would be two times as old, so nearly 86.

My birthday is coming up next week. This has been a point of discussion around the house recently.

T: No! It’s 6 times!

Me: You’d be 36 then.

T: No. I am 12 in doll years.

Me: Oh! Six years older not 6 times as old!

T: Yeah.

Me: Then Griffy is 15, not 16. And I would be almost 49.

So What Do We Learn?

Children build lovely and complicated imaginary worlds. For a long time, Griffin and Tabitha would play “creatures” together. Whole societies of stuffed animals, dolls and plastic figurines rose and fell. These societies had celebrations and tragedies. There was Creature Christmas that could take place at any time of year. Also a Creature State Fair. Et cetera.

Combine this parallel creature/doll universe with learning about the passage of time and pretty soon doll years are going to pop up.

Griffin and I talked about tortoise years and dog years a while back. At the time, Griffin was 8. He was comparing life spans of tortoises to those of humans, as we do with dogs to generate the 7 dog years per year comparison that is commonly known.

Tabitha is firmly grounded in comparing by counting and addition, as is appropriate for a 6 year old. Somewhere between third and sixth grade, children transition from always comparing by addition and subtraction to being able to compare by multiplying and dividing. This difference is what Tabitha and I are discussing in this conversation. She says Six times but means Six more.

Starting the Conversation

Listen for the comparisons your children make. Here, Tabitha compared ages. But heights, dollar amounts, number of Tootsie Rolls in a candy dish, et cetera; all of these are possible comparisons that children will naturally make. Ask a follow-up question. How do you know? is a good place to start. What if? is a lovely follow up. For example, What if there were a newborn baby in our family; how old would it be in doll years?