Nights of camping

The following conversation took place in the run-up to our annual summer camping trip recently.

Rachel has no interest in camping, so this ritual is all mine. I started the little ones young with a one-night trip within an hour from home so that we could come home if it’s a total disaster. As they have aged and we have developed our routines, we have gone further afield, exploring wide-ranging Minnesota state parks for two-night stays. We added a weekend fall trip, too.

Last summer, the kids began to ask why “we only go for two nights”.

Ladies and gentlemen, when the kids ask that question, you know you’re doing it right.

So this summer we are expanding to three nights. Tabitha was thinking about that change the other day.

I am straightening some things on the front porch, sweeping and tidying. Not thinking about anything in particular.

Tabitha (7 years old): If we’re going for three nights, is that 2 days and 2 half-days?

Me: Yes.

A few seconds pass.

I realize that I have an opportunity here.

Me: How did you think about that?

T: Every night is a day, except the last one, when we go home.

Me: What if we went for a whole week’s worth of nights? What if we went camping for 7 nights?

T: Easy. Six days.

Me: And?

T: Two half-days.

Me: OK. Ready for a hard one?

T: Yeah!

Me: There are 365 days in a year. So what if we went camping for 365 nights?

T: [slowly] Three…hundred…sixty…four!

Me: Nice!

T: I can even do 400.

Me: You mean 400 nights of camping? You know how many days that would be?

T: Yeah.

Me: All right. Tell me.

She does.

Later, she is in the shower. I am not-so-closely supervising nearby. I get an idea.

Me: Tabitha, what if we wanted seven days of camping?

T: How many nights?

Me: Right.

T: Eight. Am I right?

Me: I can’t trick you at all, can I?

T: Ask me another!

Again, a sign that things are going well. Contrast with her claim a couple years back, “Sometimes I don’t want to tell you about numbers because it’s just going to turn into a big Daddy math talk!”

I have to think hard to dig up something that will be more challenging for her.

Me: You want a hard one? A really hard one?

T: Yes!

Me: Last year, we went camping twice. Altogether, we camped 4 nights. How many days did we have?

T: Three…five…

It turns out that Griffin is lingering in hallway outside the bathroom. He chimes in.

Griffin (9 years old): Four.

Me: Two days, and four half-days.

G: Right. That’s four.

Me: But she’s thinking about it as four half-days, since they aren’t attached to each other. I can see an argument either way.

This summer’s trip was to Lake of the Woods in the far northern reaches of Minnesota.

Griffin posing with an oversized walleye statue in Baudette, MN

So what do we learn?

It may surprise some readers that I have filed this conversation under Algebra.

Like many of the other algebra posts, we are not using x or y, or making graphs or solving for variables. Instead we are thinking about a relationship, and about what that relationship looks like for a wide variety of numbers.

The relationship we are working with here is a simple one: one less. Tabitha had noticed that the number of full days we camp is one less than the number of nights we camp. She had even generalized the idea—notice that she didn’t count the days individually. She said, “Every night is a day, except the last one.” This answer doesn’t depend on any particular number of days; it works for all numbers of days.

What I did in this conversation was help her to apply this idea. By asking her about a wide range of numbers of days, she got to feel the power of her generalization. That is algebra.

The other important part here was continuing the conversation while she showered. Thinking in reverse is an important mathematical skill. We had started with how many days do we get with a certain number of nights? I moved us to how many nights do we need for a certain number of days? The fancy math word for the relationship between these two questions is inverse.

Starting the conversation

Camping trips, vacations, trips to grandma’s house…these are all opportunities to have the conversation we had. If your child doesn’t ask about it, you can ask your child. We are going to grandma’s house for three nights—how many days will you have to play with your cousins while we’re there?

More generally, there are two Talking Math with Your Kids moves I want to emphasize.

  1. It took me a moment to notice that Tabitha had offered me an opening for conversation. I was thinking about something else at the time. When I noticed it, I put those other thoughts aside to talk, ask and listen. That part of the conversation took probably 2 minutes. We can all spare 2 minutes to get our kids’ minds working. We just need to notice the opportunities.
  2. I followed up later on. Following up is good for two reasons: It lets you and your child examine an idea more deeply, and it helps cement memory of the conversation. We remember something we revisit multiple times better than something we only think about once.

The equal sign

It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems.

But I am tired. Wiped out and exhausted.

So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?)

This plan worked beautifully for about five minutes.

She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook.

Eight plus four equals box plus five

Me: What goes in the box?

Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is…

Hey! This doesn’t make any sense!

Me: Why not?

T: 8 plus 4 is something, then plus 5?

Me: What does the equal sign mean?

T: Is. Like 2 plus 2 is 4.

Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1?

T: No!

I let it go and we move on with our evening.

Later on, though, after putting on jammies but before toothbrushing, I follow up.

Me: Tabitha, I want to ask you a follow up math question.

T: OK.

Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1?

T: Yes! Of course! Easy!

Me: Can I let you in on a little secret?

T: A secret secret? Or not really a secret?

Me: Not really a secret. But something you might not know.

T: [rolls eyes] OK.

Me: The equal sign means “is the same as”.

T: Of course! I know that!

Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1.

T: Oh.

So what do we learn?

This is kind of a big deal.

We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!)

But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as.

Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right?

Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do.

The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right.

You can follow Tabitha’s and my adventures in equality in the coming weeks.

Starting the conversation

If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun!

Postscript

Coincidentally, a fourth-grade teacher wrote up his class’s explorations in equality today. If you’re interested in what this can look like in school (easily adaptable for homeschool), head on over.

Bedtime

In the thick of the holiday season last month, Griffin (9) turned his attention to the mythical midnight on New Year’s Eve.

Griffin (9 years old): Can we stay up until midnight on New Year’s Eve?

midnight

It is a nice touch that he includes his sister in this request. They may argue but they are a team.

Me: No.

G: We can when we’re 12.

At one point a year or two ago, he forced me to commit to the age at which he will be allowed to stay up to welcome the new year. The future me may be angry about the commitment I made, but for now it is paying dividends.

Me: Yes.

G: When I’m 12, I’ll get to stay up a lot later every night, since my bedtime is a half hour later every year. I’ll be able to stay up until….10.

Me: We’ll see about that.

Griffin restates his rule. I am pretty sure that I never endorsed this rule, but he is committed to it. I figure that I may as well exploit it.

Me: By your rule, what will your bedtime be when you’re 18?

G: Twelve to eighteen….so…three more hours…One o’clock!

Me: I should have asked about an odd number. How about when you’re 21?

G: Two-thirty in the morning.

Me: By your logic, what would have been your bedtime when you were 1?

G: Well, at 8 it was 8, so…4 hours…Four o’clock!

Me: I disagree. I don’t think it should be 4 hours from 8 to 1.

Griffin defends his answer.

G: It’s eight years, which is 4 hours.

Me: I don’t think 1 is 8 years younger than 8.

G: Oh. Yeah. Four thirty.

Then when I was 3, it would have been six o’clock.

He pauses thoughtfully.

G: Did I go to bed at six when I was three?

Me: I honestly don’t know. We make sure your bedtime matches your need for sleep.

So what do we learn?

This conversation exemplifies an important Talking Math with Your Kids principle—use whatever interests your children as an opportunity to talk about math. Another related principle is to use the conversation you don’t want to be having as a launching point for math talk.

I assure you that I am a relatively normal parent; I dread discussions of bedtimes. There is so much opportunity for whining, wheedling and comparisons to classmates. I understand that these classmates are mythical, but my children assure me that they all stay up later.

The key moment in this conversation was when I grabbed the bull (as it were) by the horns and asked him to apply his rule. What will your bedtime be when you are 18? That question got us talking about some good math, and it turned a potential power struggle into a fun conversation.

Using a rule to make a prediction is an important aspect of algebra. Griffin’s half hour later for every year older rule is a wonderful example of rates. I asked him to predict both forwards (what will be your bedtime when you are 18?) and backwards (what was your bedtime when you were 1?)

I missed two opportunities here.

It is a good idea to ask questions that make kids think in the opposite direction. So I should have asked something like How old will you be when your bedtime is midnight? This would force Griffin to think bedtime-to-age when he has been thinking age-to-bedtime.

The other missed opportunity is to play with the silliness of extending this rule too far. I could have asked, How old will you be when your bedtime is 8:00 p.m. again? If his bedtime keeps getting later, it’s going to come back around. Could he work out that this would require 48 years? Or would he reject the question as silly and put a limit on his rule? Either way, it’s a productive math talk to have.

Starting the conversation

Anytime your child wants to enter a negotiation, there is an opportunity to turn it into a math talk. How many M&Ms can I have for dessert? How many pumpkins can we buy? When can I stay up later? How many friends can I have at my sleepover? All of these and more are opportunities to ask what if questions involving rates, predictions, past and future quantities.

So don’t dread the negotiations. Take advantage of them!

More patterns on the multiplication machine

When we left off last week, I had challenged Tabitha to find a pattern on the multiplication machine so that there would be the same number of buttons up as down. This challenge followed up on her sophisticated argument that her down-up-down-up pattern yielded more downs than ups.

ups.and.downs

There are 81 buttons, so the task of evening out the ups and downs is not possible.

But Tabitha is 6 years old. She knows little about even and odd numbers. Searching for a way to share 81 things equally (between up and down in this case) is a good way to get her thinking about the idea.

You may recall that I had shooed Tabitha off to her bath on giving her this challenge. This is where our story picks up.

At the end of the bath, she puts on her jammies and announces…

Tabitha (6 years old): I know how!

She runs into the room to get the machine.

T: Now Daddy, I don’t know if this is going to work, so just keep your ideas to yourself.

This line is awesome, is it not?

I do as I am told.

She produces this:

Photo Nov 20, 9 41 52 PMT: Oh no.

Me: What?

T: These [she points to top and bottom rows] are both up.

She tries again, producing this:

Photo Nov 20, 9 42 20 PMT: Oh no. Still too many up.

At this point she gives me a look which I take to mean that I can have a try. So I go back to her first pattern.

Photo Nov 20, 9 41 52 PMAnd I start to share out the bottom row—half up, half down.

patterns.6Tabitha: But Daddy! That’s not a pattern!

So what do we learn?

The raw beauty of Tabitha’s line, “I don’t know if this is going to work, so just keep your ideas to yourself!” strikes every time I think about this conversation.

Children enjoy investigating their ideas. I have to work very hard to get many of my college students back to a mental place where they trust that they have mathematical ideas worth investigating.

The best thing a parent or teacher can do in this situation is be quiet and let the kid work it out.

Starting the conversation

As all interesting conversations do, this one had a trajectory. We started in one place (making fun patterns), focused our attention on one part of what we were doing (comparing the number of ups and downs) and finished off with a “what if” question (what if ups and downs were equal in number, what would that pattern look like?)

You can practice that with your child. It doesn’t matter whether any particular conversation goes anywhere (many of ours do not), eventually you’ll hit on something interesting to both of you and pretty soon you’ll notice that 10 or 15 minutes have gone by.

And then the next time will be easier. Soon it will be a habit.

 

Patterns on the multiplication machine

Tabitha (6 years old) has been playing with the multiplication machine off and on for a few months now. We have never once used it for learning multiplication facts, but we have had a ton of math-learning fun with it.

Recently, she carefully did an up-down-up-down pattern that she continued along all of the columns.

ups.and.downsWe discussed her patterning a bit before I sent her upstairs for her bath.

I followed behind and, when I knew the machine was out of reach, asked her a question.

Me: You know how some buttons are up and some are down right now? Are there more ups or more downs?

Tabitha (6 years old): There are the same amount.

This was the answer I expected. As we will see, her reason for the answer surprised me.

Me: How do you know?

T: I did two patterns. Up down up down… and Down up down up. Let me show you.

We head downstairs together.

Me: OK. In this one column [I point to the first column on the left], are there more ups or downs?

T: [She counts each type] Downs. But in this one [the next column to the right] there’s more ups.

Me: So how about the whole thing?

T: Same!

Wait.

I’m gonna figure this out.

1,1,2,2,3,3,4,4,5

There’s more downs.

Me: How do you know?

T: Watch. [She points to columns as she counts, one column for each number word] 1, 1, 2, 2, 3, 3, 4, 4, 5.

Me: Right. But what does that mean?

T: One. There’s more downs. Then one. There’s more ups. Then two, and two, like that. But five is more downs.

Me: Wait. One has more downs, then the other one has more ups. So together the 1s have the same number of ups as downs?

T: Yeah.

Me: So then what?

T: Same for the 2s and 3s and 4s. But 5 has more downs.

Me: So there are more downs total.

T: Yeah.

Me: I have a challenge for you. If you can, try to make a pattern that has the same number of ups as downs. But not now. Now you have to get in the bath.

(to be continued)

So what do we learn?

We have to keep an open mind when our kids are telling us what they think.

I knew there were more downs than ups because we started with a down in the upper left. Moving across the row, there is an up for each down. The last down at the end of the row has an up at the beginning of the next row.

Every down has an up.

Except for the bottom right corner. That down has no up to pair with. So there are more downs than ups.

This is not at all how Tabitha saw it. She made groups—each column has an extra up or an extra down. Then she grouped these groups—each pair of columns has equal downs and ups.

By keeping an open mind, I was able to listen to her thinking. This let me ask follow up questions, which helped Tabitha make her own thinking better and more clear. I try as much as possible to have explain your answer be about convincing each other of something. You think they’re the same? Convince me. You think there are more downs? Convince me.

Starting the conversation

Patterns. Notice them. Play with them. Ask about them.

Any repeating pattern will do.

Even if the pattern had been Down down up, down down up (i.e. two downs for every up), we would have had a lovely conversation about which there were more of. The keys to the thinking in this pattern were:

  1. There were too many things for counting to be a convenient solution, and
  2. We started thinking about it when we didn’t have the machine in front of us.

I would have been content if Tabitha had counted 41 downs and 40 ups. But I was very much hoping to push her to use the pattern she had created to reason rather than to count.

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 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.