Tag Archives: Tabitha

How tall is the hill? [summer project]

Our house in St Paul sits on top of an odd hill; higher than others around it. Historical reasons for this are murky but it makes the place easy for guests to find. One of my least favorite tasks in all of my domestic life is mowing the hill.

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For a while now, the precise height of this hill has been the subject of family speculation. One recent lazy summer afternoon, Griffin (8 years old), Tabitha (6 years old) and I found ourselves hanging out on the hill with not much to do.

Me: How tall do you two think the hill is?

Tabitha (6 years old): Five feet.

Griffin (8 years old): I don’t know.

Me: How about this: Which do you think is taller; me or the hill?

T: The hill.

Me: Wait. I’m six feet tall. How can the hill be 5 feet tall AND taller than me?

G: You’re six feet, one inch.

Me: Right. Even so…

T: Oh. I don’t know how tall the hill is, but I think it’s taller than you.

Me: Why?

T: Lie down.

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T: See?

Me: Yeah, but just because it’s longer than me doesn’t mean it’s taller than me.

Tabitha seems puzzled by this distinction. Griffin is standing on the sidewalk at my feet.

Me: Look at Griffy’s eyes. Is he looking up or down at my eyes right now?

T: I can’t really tell.

I stand up, right next to Griffy, who cranes his neck back to look me in the eye.

Me: Now?

T: Ha!

I lie back down on the hill.

Me: So how come there’s a difference?

T: You’re lying down now, so that’s not really how tall you are.

Me: So how can we decide whether I am taller, or the hill is?

Nothing much occurs for the next minute or so. We are distracted by butterflies, the edible nature of clover flowers and other wonders of Minnesota’s too-short summers.

Me: Hey! Let’s try this. Tabitha, you go to the top of the hill.

She does, and she stands there, looking down on me with a self-satisfied smile on her face.

Me: OK. So you plus the hill are taller than I am. What about just the hill?

T: I don’t know.

Me: Lie down.

She does, although it takes a few tries to achieve the desired position by which she can look at me from roughly the level of the top of the hill.

Me: Are you looking up or down at me?

T: I can’t tell.

Griffin takes his turn at the top of the hill. He, too, is unsure.

Me: So how can we be sure?

T: You know, Daddy, I don’t really need to know this.

Me: You’re right. You don’t. Nor do I, really. But I have always been curious how tall the hill is. Aren’t you?

G: We could measure a step, then use the number of steps to figure out how tall it is.

I obtain a tape measure.

We determine that each step is 7 inches tall. We notice that the bottom step is shorter than the rest and measure it at 5 inches. Griffin laboriously counts the steps, finding that there are eight of them, plus the smaller one.

G: So what is that altogether?

Me: What? You can do this.

G: Do you know whether you are taller than the hill?

Me: Actually, yes I do, even though I don’t know exactly how tall the hill is.

G: If I figure it out, will tell me whether I’m right?

Me: Yes.

G: [Far too quickly for me to be convinced he has run any computations at all] OK. The hill is taller.

Me: How do you know?

G: Hey! You said you would tell me!

Me: That’s part of doing the math!

G: OK.

A long, thoughtful pause ensues.

G: Eight eights is 64, plus 5 is 69. So you are taller.

Me: But you need eight sevens, which is 56.

G: Oh. Right. Plus 5.

Me: Yes…?

G: Tell me.

Child, please.

Me: Seriously? You can do 56 plus 5.

G: 61.

Me: Yes, and I’m 73 inches tall.

Tabitha, despite her protestations about not needing to know, has been paying attention all along.

T: You’re taller than the hill?

Me: Yes. See? I told you it was interesting.

G: You knew you were taller?

Me: Yes. But I didn’t realize it was by a foot. I thought it would be only by a few inches.

G: How did you know?

Me: Because I look down—only slightly—but I look down at the top of the hill.

In a few days, we will return to the topic of the State Fair Giant Slide and see whether these techniques generalize in my children’s minds.

Incommensurate Cheez-Its

There are now BIG Cheez-Its (U.S. only, it appears). The package claims that they are “Twice the size!” of regular Cheez-Its.

On seeing this claim, I thought for sure that we were gonna have a We mean four times, but say twice sort of a situation on our hands. So I bought some.

And then I asked Tabitha (6 years old) and Griffin (8 years old) what they thought. I started with Tabitha when Griffin wasn’t around so I could get her pure thoughts.

She put one cracker on top of the other and proclaimed, “No”.

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I wanted to know the source of that. I thought she might be making the classic linear v. area error (i.e. interpreting twice to mean twice the side length). So I asked.

She pointed to the uncovered part of the BIG Cheez-It and argued that this didn’t constitute another full regular Cheez-It. Score one point for argumentation, but minus one for spatial visualization.

A few minutes later, it was Griffin’s turn. He ran like a chipmunk with his two crackers into the dining room. Experiment over, right?

Nope.

He was in search of paper and a pen. He carefully traced each cracker, cut out the uncovered part of the BIG one and attempted to partition and reassemble this remainder on top of a tracing of the regular cracker, which it did not completely cover.

Sadly the cut outs are lost forever.

Sadly the cut outs are lost forever.

His conclusion: BIG Cheez-Its are almost but not quite twice the size of the regular Cheez-Its.

Volume perhaps?

Addendum 1

If the crackers are twice as big, but the mass of one serving is constant, and if one serving of regular Cheez-Its consists of 27 crackers, how many crackers should be in one serving of BIG Cheez-Its?

There are 14.

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Addendum 2

If the area of a BIG Cheez-It is about twice the area of a regular Cheez-It (as Griffin confirmed), then the side lengths should be in a ratio of approx. 7:5 (a reasonable estimate of the square root of 2).

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addendum 3

Notice the progression in the children’s strategies. The six-year old worked with the crackers. The eight-year old worked with representations of the crackers. Similar conclusions were reached; the child who worked with representations could manipulate those representations in order to achieve a greater degree of accuracy, and to investigate hypotheses that the child working concretely could not.

Neither child used tools to calculate areas.

Armholes (6-year old topology)

We were packing for a trip recently. I have developed a system for getting the kids packed. It is beautiful. Here’s how it works:

  1. Send kids to basement to get suitcases.
  2. Keep suitcases on first floor.
  3. Send kids upstairs to get one type of item at a time. E.g. Three pairs of underpants. Then three pairs of socks. Et cetera.
  4. Kids throw each type of item in the suitcase.
  5. Repeat steps 3 and 4 as often as necessary.
  6. Done.

Seriously. It’s awesome.

I made an observation with Tabitha partway through.

Me: Isn’t it strange how a pair of socks is two socks, but a pair of underpants is only one thing?

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Tabitha (six years old): Yeah. It should “a pair plus one” because there are three holes.

Me: Wow. I hadn’t thought of that. So how many holes does a shirt have?

T: Three….No four!

Me: How do you figure?

T: The one you put your head through, the arms, and the head hole.

If you are like me, you may be a bit behind the curve on her language here. “The one you put your head through” is the one that ends up at your waist once your shirt is on. I had to think about this for a moment.

A few days later, I was curious to probe her thinking a bit further. She was getting dressed (a process which is always slow, and occasionally very frustrating for the parents):

Me: Do you remember how you said a pair of underpants has three holes and a shirt has four?

T: Ha! Yeah!

Me: I was thinking about that and wondering whether there are any kinds of clothing that have one hole or two holes.

T: Socks have one hole!

Me: Oh. Nice. Sometimes Daddy’s socks have two holes, though.

T: Yeah. When they’re broken.

By this time, she finally has the underpants on and her pants are being slowly pulled on.

Me: Wait. You need socks!

She goes to her dresser and proceeds to sort through the very messy sock drawer.

T: There are no matches.

I find what appears to be two socks balled up together.

T: No! Those aren’t socks! Those are for putting over tights to keep your legs warm.

We look at each other.

Big smile.

TThose have two holes!

Zero=half revisited

A few weeks back, Tabitha asked Why are zero and half the same? I was curious to know whether that conversation had affected her thinking in any way. So I asked.

Me: Tabitha, do you still think zero and half are the same? Or have you not thought about that in a while?

Tabitha (six years old): I think…Half isn’t a number. I mean, it’s made of numbers put together, but it’s not a number.

Me: What is a number?

I love this question. How people answer it can be revealing. I asked a version of it of Griffin when he was in Kindergarten.

T: 4\frac{1}{2} is a number.

Me: Oh? 4\frac{1}{2} is a number, but not one-half?

T: Yeah. But it doesn’t really get used.

Me: What do you mean by that?

T: Well, people say, 1, 2, 3, 4, 5, 6, but not 4\frac{1}{2}.

Me: Oh. So when we count count, we skip over 4\frac{1}{2}?

T: Yeah.

We are both silent for a few moments, thinking.

T: Zero, too. People don’t count starting at zero. They say 1, 2, 3…

Me: Yeah. Isn’t that funny?

T: It should go half, zero, 1, 2, 3…

It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of \frac{1}{2}; that it expresses a number between 0 and 1.

A kindergartener on units [Talking math with your parents]

The following conversation took place in my house the other day. Tabitha (6) had been informed by her mother that she (Tabitha) needed to eat something healthy before eating a chocolate-covered donut. I was—and remain—ignorant of the origins of this donut.

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I came in partway through the conversation.

Rachel: I’m going to cut you a small slice of this apple.

Tabitha (6 years old): Do I have to eat the whole thing?

R: The whole apple? No.

T: No, the whole slice!

R: Yes!

If you are unaware of the fun we have had with units around our house, you may wish to check out our discussion of brownies, and (of course) the following.

Sharing tostadas

It is perhaps not widely known that I love good Mexican food, and that—with the assistance from afar of Rick Bayless—have developed a number of specialties de casa.

Among these specialties is tostadas, which I make starting with corn tortillas. A bit of oil and 10—15 minutes in the oven makes them crispy. We build from there.

The tortillas fit nicely in a 3 by 3 array on my favorite cookie sheet. There are four of us in the family. You can see where this is going, I am sure.

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Griffin served himself a second tostada the other night.

Tabitha (six years old): Griffy’s having another one?!?

Me: Yes. There’s a second one for you, too.

T: How many did you make?

Me: Nine.

T: That’s not a fair number!

Me: What would be a fair number?

T: One where everybody can have the same amount.

Me: Right. But how do you know 9 isn’t a fair number? And what would be one?

T: I don’t know.

Griffin (eight years old): Eight would be. Or 40.

Me: Oh! Forty! Then we could each have 10. Would you like to eat 10 tostadas, Tabitha? But then I would need to buy a second pack of tortillas.

T: [Silent, but her eyes get big and she nods vigorously.]

G: Or 20. Or 12.

The final count is 2 tostadas each for Mommy and Tabitha, and 2\frac{1}{2} tostadas each for Daddy and Griffin. Along the way, I promise Tabitha a taco if she finishes her second tostada and is still hungry. This strikes her as fair.

Zero=half

A propos of nothing the other day, Tabitha asked a strange question.

Tabitha (six years old): Why are zero and half the same?

Me: They aren’t.

T: Like seven is one more than six, but zero and half are the same. They’re both nothing.

Me: One half? if you have half of something, that’s more than nothing.

T: But half, the number, that’s the same as the number zero.

Recall that last fall, she was not convinced that one-half was a number at all.

She now accepts that one-half is a number. But she hasn’t really dealt with the idea that there are numbers between other numbers. She is doing a bit of beautiful kindergarten logic here. Her premise is that there is only one number less than 1, namely 0. She has also accepted that one-half is a number less than 1. Therefore, one-half and zero are the same.

And—rightly—she is suspicious of this conclusion. The logic is sound, but it doesn’t make sense.

I go to work on that first premise.

Me: Oh. I see. Well, one-half—the number—is between zero and one.

I draw this picture, which I feel is certain to be totally unconvincing.

I was writing upside down. Forgive the crummy 2's. Note the complex fraction. Take that, Common Core!

I was writing upside down. Forgive the crummy 2’s. Note the complex fraction. Take that, Common Core!

But then again, we hadn’t talked about one-half being a number since October. That last conversation seems to have been fermenting all this time, so maybe this one will do the same.

To be continued, I am sure.