Explain your thinking using words, numbers, and/or pictures.
Mr. Vann asks for our thoughts on the latest math journal quandary.
“What about kids who flunked?” Dan shouts out. “You told us nobody moved away, but you didn’t say anything about flunkers.”
Mr. Vann nods. “I am proud to announce that all of last year’s fourth graders are currently enrolled at Highbridge Middle School. Whether the same will be said of this year’s fifth graders, we will have to wait and see.” That shuts Dan up for the rest of class.
Mr. Vann says we don’t ever have to hand our math journals in. They’re a private place to record our thoughts and processes. But he does walk around a lot and look over our shoulders while we discuss the answers.
I don’t know how Mr. Vann knew I was thinking about this particular math journal “quandary,” but I liked drawing the answer. I had extra time to work on it in the cafeteria before class.
Even though it doesn’t make a lot of sense, I think the hardest part of middle school is lunch. There are dozens of starving aliens trying to locate nourishment and find an acceptable place to ingest it.
I thought Ella and I were so lucky to have the same lunch period, but the way things work out, Pod One is usually halfway finished before Pod Two even makes it to the start of the line.
By the time I got through the line today, Ella was already cleaning up her tray. Plus, she was sitting with a bunch of girls from Pod One who I don’t even know, and there were no empty seats at the table.
“You can have my seat, Mika,” Ella said, standing up. All the other girls stood up, too.
“Tomorrow we should wear blue pants and green shirts,” one of them said.
“And purple hair ties!” another one suggested.
“We have a new club,” the first girl explained. “The Onesies!”
“Because we’re in Pod One,” Ella said with a little shrug.
They left to go bus their trays and head outside. I pretended like it was taking all of my concentration just to open my milk carton. Then I took out my math journal and worked on my aliens a little longer.
We’ve moved on to time in math, even though time is covered at the end of chapter four.
“When a topic becomes relevant,” Mr. Vann explains, “that is obviously the best time to explore it.” Everyone has been coming to math late, so Mr. Vann has decided that time is a relevant topic.
It’s not our fault. Math comes right after lunch, and when you’ve finished eating, you can go outside. They don’t call it recess because this is middle school, after all. But most kids wolf down their food and then head outside to play Frisbee or soccer or just hang out.
And you can’t blame people for not wanting to come back inside for class. It’s still “warm as toast,” as my mom likes to say. The sky is that shade of blue that seems clearer in the fall, and the light slants at interesting angles.
Today, the clouds look like cotton balls that somebody pulled apart and tossed up into the air, like something Monet would paint. Take a look at his Village Street painting. That’s the kind of sky I mean. All wispy and soft and exactly right.
Monet is probably my favorite artist, and I know for a fact that he spent a fair amount of his time “drawing.” Over the summer, Mom and I went to a museum in Utica. They had a traveling exhibit of all the French Impressionists.
My favorite painting there wasn’t by Monet, though. It was by an artist I’d never heard of before—Morisot. Peasant Girl Among Tulips.
I spent a long time just staring at that girl’s face. She was sitting in a bunch of flowers with her hands clasped together, looking like she was waiting for something, or maybe somebody. The colors of the flowers matched the colors in her dress, and her face was the same shade of peach as the tulips.
“What do you think she’s thinking about?” Mom asked me.
I didn’t say anything. I didn’t really have an answer.
“Maybe about what middle school is going to be like?” she asked next.
I knew Mom was just trying to “check in,” as she says, but I had to give her an extra-exaggerated eye roll for that one.
“I’m pretty sure she doesn’t go to middle school,” I said.
Math Journal Entry #3
Think about something you experienced recently. Estimate when the activity began and when it ended. Then figure the elapsed time spent on the activity to the nearest quarter hour.
Explain your thinking using words, numbers, and/or pictures.
Dan P. says that we can easily solve our coming-to-math-late problem by having class outside. That way, kids can just stay outside after lunch and they won’t be late for math.
Mr. Vann surprises everybody by saying, “Makes perfect sense.”
We all follow Mr. Vann through the side doors and sit in a big circle on the grass. It’s funny to see the classroom from the outside. It looks so empty.
Mr. Vann holds up a stack of paper plates. “One to a customer,” he says in a silly voice, like he’s hawking games at a county fair. Then he asks Dee Dee to pass them out.
Dee Dee was in my discussion group for the estimation unit. She’s also in my science class, and she is really into science. She always wears these T-shirts with funny science stuff on them. Today, her shirt says: Don’t trust atoms. They make up everything.
Dee Dee tosses me my plate like a Frisbee. Then she gives me a thumbs-up.
Mr. Vann takes a thick black marker out of his pocket. He draws a point in the middle of his paper plate. Then he draws two arrows pointing directly up. One is shorter than the other.
“What is the time?” he asks.
Dan looks at his watch and yells out, “One-eighteen.”
I’m not about to raise my hand and point out that coming outside in order to save time has actually made us start class a full eleven minutes late. It’s fun sitting crisscross-applesauce on the grass. It’s warm as toast, and I like feeling the sun on my face.
“Willing suspension of disbelief,” Mr. Vann replies. “Please give me the time represented by my theatrical stage prop, not the real time.”
A boy named Malcolm raises his hand. “Twelve o’clock,” he says.
“Twelve a.m. or p.m.?” Mr. Vann says with one eyebrow raised. It’s the look of someone asking a trick question.
“Beats me,” Malcolm says with a shrug.
“Correct!” Mr. Vann shouts. Then he starts howling like a wolf or a coyote. “The moon, the moon,” he calls, holding up another paper plate to play the part.
Mr. Vann can be really goofy, but the funny thing is, because he’s the one goofing off, none of the kids do.
“Twelve o’clock a.m.,” Malcolm says.
“Correct again!” Mr. Vann yells.
Then he takes out a pair of teacher scissors and starts cutting his plate into different sections to show us how fifteen minutes is the same as one-quarter of a plate, and thirty minutes is the same as one-half of a plate—if you remember that the plate is a clock, that is.
Chelsea tentatively raises her hand. “Um,” she starts. “Is this actually part of the fifth-grade curriculum? I mean, we made plate-clocks in second grade.”
“Oh, we’re not making plate-clocks,” Mr. Vann says with a chuckle. “Although that is an excellent idea.” He takes a scrap of paper out of his pocket, jots a note on it, and puts it back. “We are visually representing fractional parts of a whole using the two-dimensional form of a circle.”
He pauses because he realizes we’re all pretty confused. “We’re gonna cut the plates up!” he says enthusiastically.
Of course, nobody thought to bring scissors when we came outside, so the cutting part is a definite challenge. Kids start folding and ripping their plates, or just coloring in the different sections. I take out my colored pencils.
First, I divide my plate in half and then in half again. I color each quarter a different shade of blue. Then I divide each quarter into three equal parts (each representing five minutes, of course).
I leave the two sections on the outside plain blue. But in the center sections, I use my eraser to make clouds. It’s like taking the color away to leave the white. It gets all smudged and blurry, but I like the way it comes out, kind of like a Monet sky.
“That is so cool,” Dee Dee says, pointing down at my plate.
“Thanks,” I say.
This is more like art than what we did in art today, which was walking through the field to collect rocks and acorns. That was more like science. But in science, we wrote poems about the circulatory system, which was more like language arts. If I’d had music today, we probably would have played a good game of soccer.
* * *
—
When I get home from school, I take my oil pastels and one of my blank books, and I go outside to draw for a while. I’m trying to get the color of the leaves. They’re not all red and orange and yellow yet, but they’ve changed a little bit, like someone started turning down the dimmer on their light.
I barely get started when Mom calls me inside. I know I’m not doing my homework “first things first.” (Mom always says it’s important to do your must-do’s before your may-do’s.) But usually if I’m drawing, she doesn’t care that much.
“Mika!” I hear again. I gather up my stuff and head inside.
“I have a doctor’s appointment,” she tells me.
“What?” I say. “Now?”
I am totally confused. I should explain that my mom makes a to-do list every day. She has a pad of paper with a magnet on the back of it that sticks to the fridge. It says To-Do Today in swirly letters at the top. Below that, she writes the date, and then she makes a list of everything that’s “on tap” for the day.
To be exact, she makes two lists—one for her and one for me. But my list only ever has the same three things on it: 1) Make bed. 2) Go to school. 3) Do homework.
If my mom had an appointment, it would have been on the fridge, and I would have known about it.
“This was not on the list today,” I say, shaking my finger and making a face like I’m the parent and she is in really big trouble.
“They squeezed me in last minute,” she explains.
It seems like there should be an easy comeback to that, but I can’t think of one at the moment. I open the fridge and look around. Maybe I’ll grab a snack for the road.
“Mika, fridge,” Mom says. Mom gets on my case for holding the refrigerator door open. “One of the largest avoidable utility expenses,” she likes to remind me.
I close the door. “Why do you have to go to the doctor, anyway?” I ask.
“Oh, I just noticed a spot on my leg that looks a little strange,” she says.
“You always look a little strange,” I say, making a silly face.
Usually Mom would make her own silly face back, and follow it up with: “I know you’re strange, but what am I?” But she doesn’t say anything.
“Why can’t I stay home?” I ask.
One problem with it being “just the girls” is that wherever Mom goes, I have to go, too. “Everybody else’s parents let them stay home alone,” I try.
Mom delivers her expected “Good thing I’m not everybody else’s parent.” But when she says it, something weird happens. Her voice cracks. I’m startled out of my whining.
“Please, Mika,” Mom says quietly. “Let’s just go, okay?”
I suddenly get this weird feeling at the back of my neck, like someone is squeezing it even though no one is. I drop my drawing stuff on the table, grab my backpack, and head out to the car.
Math Journal Entry #4: A Math Lib
“Yankees all-star pitcher Daniel Pimental started pitching the American League Championship’s seventh and deciding game at one-oh-seven p.m.,” Dan reads out loud from his math journal. “He finished pitching his perfect game at three-fifty-two p.m. He spent two hours and forty-five minutes clinching the American League Championship.” Dan ends his Math Lib with a fist pump. Mr. Vann gives him polite applause.
Dee Dee reads next. “Dee Dee started looking through her Orion SpaceProbe Altazimuth Reflector Telescope promptly after school at three-oh-two p.m. She finished looking through her Orion SpaceProbe Altazimuth Reflector Telescope directly before bedtime at twelve-forty-eight a.m. She spent nine hours and forty-six minutes looking through her Orion SpaceProbe Altazimuth Reflector Telescope.”
“Very inspiring,” Mr. Vann says. “By the way, what kind of telescope do you have?” He winks.
Dee Dee just winks back at him.
“Mika?” Mr. Vann says next. “Care to share? Because remember, to share is to care.”
“No, thanks,” I mumble.
Maybe if I had done this math journal exploration before my mom’s doctor appointment instead of after, I would have written something different. I wonder if Mr. Vann ever thinks about time that way—how things can turn out differently depending on when you do them.
If I had done my homework first things first (like I was supposed to), I might have written something funny like Dan or clever like Dee Dee. I might have written: Mika started tap-dancing at three-forty-five p.m. She finished high-kicking at five-forty-six p.m. She spent two hours and one minute perfecting her “Shuffle Off to Buffalo” routine.
That probably would have gotten a good laugh or maybe even a few bonus points. Mr. Vann is always giving random bonus points for interesting answers, and random demerits for hasty answers. “Fools rush in where angels fear to tread,” he likes to remind us.
Instead, I wrote about waiting at the doctor’s office because it felt like I was waiting there forever—one hundred and one minutes, to be exact. At least I had plenty of time to work on my drawing.
Mom said she just wanted the doctor to take a quick look, but they must have taken more than that because when we got home, she had a big bandage taped to the back of her knee.
“We’ve spent enough time on time,” Mr. Vann announces. “Since it’s all relative anyway, we’ve spent relatively enough time on it. So we’re moving ahead to the beginning of chapter four, which, as we are coming from the end of chapter four, is a bit like going back to the future, so in a sense, it seems that we are still contemplating time after all.”
Like many things in fifth grade at the middle school, Mr. Vann often makes very little sense. In any case, learning about time wasn’t really helping with our coming-to-math-late problem. If anything, it just gave kids funnier excuses.
“I started opening my locker at one-oh-two,” Dee Dee says as she hurries in the door. “I couldn’t get my locker open until one-oh-nine. I spent seven minutes of elapsed time trying to open my locker. And that’s why I’m late.”
“I entered the boys’ bathroom at one-oh-one,” Dan begins.
“I suspect,” Mr. Vann interrupts just in time, “that what we are having is not a time problem after all, but a space problem.” He looks at Dee Dee knowingly. “That is why we’re moving back to the beginning of chapter four: measurement.”
Everybody flips backward to the correct page in the textbook. Chelsea, who is never late to class, looks particularly confused.
“If our classroom were closer to the cafeteria”—Mr. Vann rubs his chin—“then we might have less difficulty arriving punctually after lunch. Perhaps we should hold math in the janitor’s closet directly across from the cafeteria.”
I can see Chelsea fussing with the top button of her sweater. She’s probably worried about how we’re all going to fit into the closet, where she’s going to sit, and what we’re going to use for a board.
Mr. Vann opens the top drawer of his desk and takes out a bunch of wooden rulers. He spreads them like a fan. “Seri
ous thinkers need serious tools,” he says very seriously.
I’m not sure how a wooden ruler is all that serious, but it is better than the cardboard ruler that you punch out of the back of the textbook. Those get all bent and fuzzy.
Mr. Vann starts passing out the rulers. “One to a customer!” everybody shouts.
“Now, let’s imagine,” Mr. Vann says as he walks around the room, “our dear Dan is catching for the Boston Red Sox.” Dan laughs because he’s the biggest Yankees fan ever. Plus, he’s a pitcher.
Mr. Vann keeps talking and passing out rulers. “And let’s say Dan wants to measure the distance from third plate to home base.”
“It’s third base to home plate,” Dan says with a sigh.
“That’s what I said,” Mr. Vann says, even though we all know that is not what he said. “What unit of measure would be most appropriate for Dan to use?”
A few kids raise their hands.
“No,” Mr. Vann interrupts before he even calls on anyone. “Better scenario. Let’s say Dan is dancing with the Joffrey Ballet.”
“No way,” Dan mutters. But then Chelsea glares at him. I guess Chelsea is into ballet. Good for her.
“Dan needs to measure the distance across the stage to make sure he has ample space to complete his world-famous grand jeté. What unit of measure would be most appropriate?” Mr. Vann sits down at his desk and starts opening drawers like he’s searching for something.
After a moment, he looks up at us. “Well, go on already,” he says as if we are all misbehaving by staying in our seats. “Time to start measuring.” He spreads his arms, pointing to the classroom like it’s a prize on a game show.
“And please remember to attend to precision, dear thinkers. If nothing else, measurement is a way for us to get a handle on things. Measurement lets us know exactly where we stand.”
Solving for M Page 2