Pizza!

As many of you know, the third and fourth grade classes order pizza each Friday. Children throughout the lower school put in their order; runners from the 3-4s pick up the orders and the money, determine the number of pizzas to buy, and hand-deliver it when it arrives.

Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.

How does pizza relate to mathematics? Glad you asked. Let us count the ways...

1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?

And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!

Some of the proceeds, up close and personal:



This young man is clearly enjoying himself.
Think Scrooge McDuck.



Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:



One of the "Grand Totalers," making bundles of ten for easier counting:



And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):

Getting to School

If you're like most of us, you've always wanted to know how kindergarteners get to school. Lucky you! Now's your chance to find out--because the kindergarten recently put together a graph showing that information. Voila--!



(You can click on the picture to enlarge it.)

The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.

There are the reading-the-graph questions, of course:

*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?

And the interpreting-the-graph questions too:

*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?

But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.

n (or maybe n+1) Flies on the Wall

The 3-4 classes generally begin the year with work on number sense, number patterns, place value, and number in general. This year we're starting off with some projects involving functions and some simple algebraic ideas.

Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):

Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--

Student: 38!

Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--

Students: 48.

That's right. Okay, let's make a table and try it using some other numbers for n:

n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55

Looks good. Okay, what patterns do you see? How does n change when you add 10?

Students: The ones digit stays the same.

Yeah? Always, or only most of the time?

Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.

That's right. Can you think of a number n where the ones digit would change after you add 10?

Students: several suggestions, all of them withdrawn upon further reflection.

...Why doesn't it change?

Student: The number 10 has 0 in the ones column, so it doesn't change the ones.

Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]



What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?

Students, less hesitantly than before: Always.

How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.

Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.

n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88

Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!

Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!

By how much? Let's say it as a rule.

Students, cautiously: It goes up by one in the tens column and one in the ones column.

Always, or just sometimes?

n-2 or n-3 students, where n is the total population of the class: Always.

Two or three students: Sometimes.

Why sometimes?

2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...

2 or 3 more students: Ohhh!

Let's extend the table--

59 59 + 11 70
69 69 + 11 80

n/2 students: It goes up two in the tens!

The other n/2 students: And it goes down to 0 in the ones.

Okay, let;s make the rule. Help me out here:

[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]

Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??

[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...]

Learning from the DVD...Player

Teachers of today can choose from a wide array of technologies to spice up their lessons and increase students' understanding. There's Powerpoint, of course, and calculators, smart boards and video cameras, wikis and spellcheckers, voice-to-text programs and DVDs, Excel spreadsheets and Activote systems, GPSes and, um, electric pencil sharpeners; the list goes on.

Most of these educational technologies get plenty of respect within the educational world. (Well, maybe not the pencil sharpeners.) Whole conferences are organized around these technologies and how they can help teachers do a better job of preparing students for the 21st century [Q: At what point will we start saying "preparing students for the 22nd century"?]. BUT there is one technology that is sadly overlooked. It is the Rodney Dangerfield of the educational technology world. I refer, of course, to the lowly DVD player. Not the DVD; the player.



"How did you know so quickly that 8 + 8 was 16?" I asked a first grade girl earlier this week. (If this question sounds familiar, it's probably because you read the previous entry in this blog.)

"Well," she said, "we have this DVD player at home and it has arrows. And if you want to speed through the movie it says 2, 4, 8, 16, 32, and then it goes back to 2 again. And I know that 2+2 is 4 and that 4+4 is 8, so 8+8 must be 16, and I guess that 16+16 would be 32. But then the pattern stops because it goes back to 2 and 32+32 is...something, but it isn't 2."

What can I say? Clearly, we should as a nation reduce our spending on old-boring-and-ineffective technologies such as computers, projectors, smart boards, and digital cameras, and load up classrooms instead with DVD players. Who's with me?

--Actually, this is a really good example of a child not only noticing but using math in everyday life. No one taught her that 8 + 8 was 16. She was struck by a sequence of numbers that appeared in her environment, and spent time and energy deciphering the pattern--learning, and evidently mastering, the fact that 8+8=16 along the way. This is the kind of thinking we always want to see in our students. As our report form puts it, one of our goals for children is that they "recognize and construct mathematics in daily life." It's lovely to see such a clear example.

That's All She (w)Rote

You have 8 cubes, I say.

The child, a first grader, nods happily. He's just counted them, accurately, and showed me how you could split them up so that we each had the same number (4 apiece, if you were curious), and answered several other questions about them as well.

What if you had 8 cubes and I had 8 cubes too? I ask. How many would we have in all?

This isn't necessarily an easy question for six-year-olds, and they vary in their approaches--also in the speed with which they answer. 28, says the boy, just as automatically as you please. There's no lack of confidence here. (Not a lot of accuracy, either, but hey, it's still early in the year.)

28? I ask, just to make sure.

28, he says. No. I mean, um, 34. Yeah, 34.

34, I repeat, resisting the temptation to ask, Regis-style, whether this is his final answer. And how did you know?

Oh, I didn't know, he says with a grin. I guessed.

Okay, I say, and go on to do a few more activities with him. I wrap up with a nice open-ended question: What else do you know about math that you'd like to tell me?

Well, he says eagerly, one thing I know is that 8 plus 8 is 16...

SMACK! goes my hand (metaphorically at least) against the side of my head.

This little anecdote nicely illustrates the difference between knowing a fact and KNOWING it. This boy knew that 8+8 was 16, but he didn't KNOW it--that is, while he could repeat it, he couldn't use that information in a real-life context. His verbal knowledge isn't yet supported by his conceptual understanding.

There's nothing wrong with learning some kinds of things by rote. Indeed, sometimes it's necessary. It's just that you have to be careful with kids and not automatically assume they KNOW everything they know....if you know (KNOW?) what I mean!

Spiiiiiiiiders

So there I am in the Pre-K, "just visiting" as they say on the Monopoly board, and the children are doing watercolors, and one child brings over her picture to show me.

"I painted a tabantula," she explains, her eyes as wide as a four-year-old's can get. Wider, if possible.

"A tabantula, huh?" I say. "Sounds mighty scary."

"It has eight legs," she says, and proceeds to count them, which she does very well (us Math Guys notice these kinds of things), and lo and behold, guess what, there ARE eight.

"Well, that's a good thing," I say, "because taban, I mean, taRANtulas are spiders, and spiders are supposed to have eight legs. Good for you for knowing that. I guess you're an expert on spiders."

She ignores this comment as the typical babbling of the Adult and points instead at a swirl of red paint. "That's the tabantula's head," she explains. "Do you want to know what that red is for?"

"Tell me," I say.

She leans in very close, stretches up, finds my ear, and stage whispers "IT'S BLOOD."

Can't wait for Halloween!

The whole nine yards; or, Dressed to the nines

If I'm timing this one correctly (and I'm probably not as the margin for error is not exactly huge), this post will be time-stamped

09 [month]
09 [day]
09 [year]
09 [hour]
09 [minute]

or

09/09/09 at 09:09

The nines have it!

[Edited to add: Ooh! Missed by ONE MINUTE. Oh well...I tried.]