# Milk by the gallon

Milk has been on sale at our local gas station/convenience store. Griffin and I walked up there the other day to buy some milk. Two percent milk for the kids and me; skim for Mommy.

Me: Griff, the milk we just bought was $5.50 for two gallons. How much was each gallon? Griffin (9 years old): With tax included? Or not included? I don’t do tax problems. Note: I weep for the loss of 4% and 5% sales tax rates. They were so easy to compute mentally, and such a nice introduction to the financial world for elementary age children. Minnesota’s sales tax rate is presently $6\frac{5}{8}$%. The city of Saint Paul tacks on another half percentage point. I don’t even bother computing sales tax mentally any more. Me: No worries about taxes. There is no tax on milk. G: OK. Two twenty-five. Er…no that’d be$4.50.

So…

$2.75! Me: How did you do that? G: Well, I thought it would be$2.25, but that’s half for $4.50, so there’s an extra dollar. So I split that dollar in half, which is 50 cents, put that with the$2.25, which is $2.75. Me: Nice. I could see that thinking in your first answer; when you said$2.25. I was curious whether you used that first wrong answer or started over from scratch.

When I thought about it, I did it differently. I thought that half of $5.00 is$2.50, then I need to add half of 50 cents. Same answer, though. \$2.75.

## So what do we learn?

I called in to a Minnesota Public Radio program on math education last week. One of the pervasive questions in such conversations is about how kids are learning to do arithmetic in modern American schools, and it arose in this program.

The thinking Griffin is doing here is lovely, and modern math curriculum is trying to encourage more of it than in the past. He is splitting $5 \frac{1}{2}$ in half, and he is doing it mentally by thinking about the related multiplication facts.

This thinking is not closely related to the standard long division algorithm. One of the big challenges in school curriculum is relating mental math strategies such as Griffin’s to efficient algorithms that are more useful for complicated computations. I have a few resources parents may find helpful over at Sophia.org.

## Starting the conversation

Anytime you find yourself wondering about such things, ask your child to think along with you. I wanted to know whether the gas station price for a gallon of milk was a good one. This required me knowing what the price was for each gallon. Not a hard problem for me, but I had to think for a moment. So then I asked Griffin. Do the same at the grocery store, the convenience store, the hardware store; anyplace where things are priced in groups.

If your kid needs a challenge, ask about gasoline. I paid x for y gallons yesterday. How much per gallon? This one will likely require estimation skills!

# The meanings of division

I was talking with Griffin one day when he was in third grade.

Me: Do you know what $12\div 2$ is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about $26 \div 2$?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade], so it was her magic number. We had 13 pairs of kids.

Me: What about $34 \div 2$?

G: Well, 15 plus 15 is 30…so…19

My notes on the conversation at this point only have (back and forth), which indicates that there was probably some follow-up discussion in which we located and fixed his error. The details are lost to history.

Our conversation continued.

Me: So $12 \div 2$ is 6 because $2 \times 6$ is 12. What is $12 \div 1$?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Me: What is $12 \div \frac{1}{2}$?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and $12 \div 6$ is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

## So what do we learn?

Mathematical ideas have multiple interpretations which people encounter as they live their lives. As we learn more mathematics, we become better at connecting these different ways of thinking about ideas.

In this conversation, Griffin relies on three ways of thinking about division:

1. A division fact is a different way of saying a multiplication fact. ($12 \div\ 2$ is 6 because $6 \times 2$ is 12).
2. Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students).
3. Division tells us how many will be in each group if we make groups that are the same size. (When he was working on $34 \div 2$, Griffin put 15 in each group to start off with.)

We were just talking for fun, not homework or the state test. So I wasn’t worried about his connecting those ways of thinking. I was just curious how he would apply them to some more challenging tasks, such as dividing by 1 or by a fraction.

I was surprised by how difficult $12 \div \frac{1}{2}$ was for Griffin. Not because it is an easy problem, but because he could have applied his how many of this are in that? idea, or his multiplication facts idea. But he did neither and reinterpreted the task as twelve divided by half-of-twelve.

I was also surprised at the length of the pause he took for $12 \div 1$. It makes sense in retrospect. After all, are you really making groups if it’s just one group? I imagine he had to think that through, rather than the number relationships involved.

## Starting the conversation

When the opportunity presents itself—when you and your child are not under homework stress, not rushing to get out the door or find the dog’s leash; when you happen to be talking about number anyway—ask follow up questions. Even a simple set of division problems got a lot of good thinking out of Griffin. Problems involving 1, 0 and $\frac{1}{2}$ are especially challenging.

Vary the size of the numbers.

Offer a few ideas of your own.

Quit before anybody gets frustrated or bored.

# Days to Christmas: Place value follow up

In yesterday’s post, I told of the Christmas Countdown cube calendar, and of how Tabitha (6 years old) changed my 06 days to 6 days by removing the leading zero. I challenged readers to consider her reasons for this.

Of course I asked her. I showed her the pictures I took and asked why she had taken off the zero. Here are the results of our conversation.

Me: I’m curious about why you took off the zero.

Tabitha (6 years old): Because there aren’t sixty days until Christmas.

A new piece of research has been making the rounds in the media recently. I will write about it at length soon, but for now you just need to know that the common headline is something like, “Young children know more about place value than most people assume they do.” In particular, the research looked carefully at the partial place value knowledge children have, rather than just calling wrong answers wrong.

Helping children develop partial knowledge into better knowledge through exploring and talking is the heart of the work here at Talking Math with Your Kids. Naturally I want to explore Tabitha’s knowledge here.

I get out a piece of paper, a pen and write some numbers, asking her to tell me what they are. I write down her responses verbatim so that I can share them with you.

Twenty-five

## 52

Fifty-two

A reasonable hypothesis for Tabitha saying that “06” meant “sixty” would be that she doesn’t pay attention to the order of the digits in a number. We now know this isn’t true.

More numbers follow.

Sixty

## 06

Sixty

Now things are getting interesting. There seems to be something special about that zero out front. We do some more.

Six hundred

## 006

Six

What? I was sure she was going to say six hundred for this one.

Six

Hmmm….

Zero

## 00

Zero

No hesitation on this one, which surprises me. I thought she might object to two zeroes.

Three

Thirty

## 030

Three

At least this one is consistent with 060.

I am beginning to wonder how she will work with numbers that have zeroes in the middle of them, such as 1002. And I am curious how she will work with numbers that use the same words, but in a different order. I think of 1002 and 2001 as a good example: one thousand-two and two thousand-one. So I build up to those.

Two thousand

## 1000

A thousand

Uh oh. I didn’t expect this. I expected one thousand, not a thousand. I have to change my examples.

Three thousand

## 3002

Three hundred two…er…three thousand two

## 2003

Two hundred three

Beautiful! This is what the researchers were saying—children have partial place value knowledge. What is going on here is this: 3002 looks like 300 with a 2 at the end. This is a very common error, and it represents trying to make sense of a complicated idea.

We do only one more; I can tell she is getting tired.

## 4030

Seven

I cannot tell if she is being silly or serious, trying hard or being clever.

## So what do we learn?

Just as with learning to speak a native (or foreign) language, learning about numbers is not an all-or-nothing proposition. Children have partial knowledge that is sometimes inconsistent. Tabitha self-corrected on 3002, but not on 2003. This conversation supports her thinking about the structure of numbers. She will think about it in the future and be more prepared to pay attention to it because we talked about it. When she wants to know more, she will ask.

## Starting the conversation

Children learn about numbers and language in similar ways—through exposure in their everyday environments. We read to our children to enrich their language environment, and we can expose our children to numbers to enrich their math environment. Waffles, toys and calendars are all examples of everyday objects with mathematical structure that children can play with.

# Days until Christmas

Each day, Griffin (9 years old) has taken great joy in setting the Countdown to Christmas cube calendar the kids recently received from my father and stepmother.

On Thursday, I was doing my end-of-semester grading from home and noticed after he had left that he had neglected to set it on his way to catch the bus to school. So I did.

My wife got Tabitha out the door later in the morning while I was at work in my basement office.

Later, I noticed that Tabitha (6 years old) had modified it.

Here I will pause to give readers a day or so to consider why. What might a 6-year-old have in mind that would cause her to remove the zero from in front of the six?

I asked her about it later on and we had a lovely conversation. I’ll report on that soon.

# Multiplication and rectangles

I want to suggest a lovely post by somebody else.

It is written by a math teacher who converses with his niece (who is 7 years old) about rectangles and multiplication. As an example, the rectangle below shows that 6×3 is 18. Or is it that 3×6 is 18? That becomes the focus of part of the conversation.

The girls’ parents look on as the discussion unfolds.

At one point, the math teacher stops the mother who is trying to intervene to help the child see that 4×3 is the same as 3×4. And this leads to the lovely sentence in the blog post:

I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.”

What he means by this is that children can learn from thinking about math, even if they don’t end up with the right answer, and even if they do not experience the full story (here, that multiplication is commutative, which means AxB=BxA for all possible numbers).

Another fabulous math teacher, Fawn Nguyen, told me, “I dare say that it’s not obvious to teachers also.”

Finally, non-math teacher parents may be interested to learn that—consistent with Fawn’s observation—a regular piece of feedback I get from math teachers on my writing here is how impressed they are by my ability to not worry about Tabitha and Griffin getting right answers.

# Talking Math in Minnesota Parent

We are delighted to share a short feature in this month’s Minnesota Parent magazine featuring the “Real Dad” behind Talking Math with Your Kids.

Share widely, please, and do give Minnesota Parent a bit of extra web traffic, won’t you?

Griffin and Tabitha thank you!

# Short notes

I passed a lovely morning with some math teacher friends at the Museum of Math in New York City today. Sadly, I did not have Griffin or Tabitha along. We would have had a ball.

I have three observations…

One. There is lots for kids to do there. It is truly a bonanza of mathematical conversation starters. I left wanting to be hired on as “Docent for Talking Math with Your Kids”. This is because the math does not smack you in the face. Instead, the math in the exhibits tends to surface in the process of playing, experimenting and paying very careful attention to what is going on.

In short, you want to talk about these exhibits and you want to linger.

Two. Many of the exhibits are designed in ways that allow kids to play and to do math beyond the intended activity. The exhibit below, for example, kept two little boys (each 5 or 6 years old) busy for 20 minutes although the original activity was way beyond them.

The first five minutes, they were sharing each of the shapes equally, making sure each of the two boys had the same number of squares, and the same number of triangles, et cetera. The next 10 minutes, they spent arranging the shapes on the screen, marveling at the things this provoked on the screen beneath.

It was lovely.

It could have been a bummer, though. If someone had insisted on doing the intended activity, these kids could not have done it. If the shapes had been electronic instead of concrete, the kids would not have been able to play with them in such creative ways.

Well done, Museum of Math!

Three. On the subway train back to my hotel, I noticed a little girl—probably 4 years old—holding up four fingers as she sat between her mother and brother.

I moved to be in listening range.

It turns out, she had been told they would be on the train for five stops. She had been putting up one additional finger as the train left each stop.

Sister (4 years old): [She is holding up four fingers as the train enters the station] This is our stop!

Brother (8 years old): No. The sign says there’s one more stop.

Sister: But you said five stops.

Mom: One more, sweetie.

The train stops. People get off and on. The doors close. The train starts up again.

The girl keeps exactly four fingers up the whole time.

# 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.

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).

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.

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.

We developed its opposite—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.

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.

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?