Monday, January 11, 2010

Solve for "X"

“Two plus two is four. Four plus four is eight. Eight plus eight is sixteen. Sixteen plus sixteen is 32 . . .” This is the lunchtime refrain. As the Olders visit and chew, they lob math problems back and forth, reciting what they’ve memorized and often venturing into competitions. The Youngers listen with rapt attention, sometimes repeating what they’ve heard recited so many times before. The Youngers are thrilled when they can answer a math problem posed by an Older, and often appear dejected when they do not know the answers that seem to come so easily to their older classmates. I will often help reframe the problem so that it might make sense to them; “if you had two olives, and I gave you two more, how many would you have?” I might casually hold up fingers for them to count, and the triumphant shout of “Four!! It’s four!!” can be heard from across the street.

Now that we are several months into the school year, the Olders have stepped up their game. Recently, the lunchtime math chatter turned to talk of “x.” One of the boys had mastered the skill of figuring out simple algebraic problems. He quickly explained it to a peer, and they delighted in stumping the other children with new math problems involving the elusive “x.” Some of the interested Youngers looked to me in confusion. X? What in the world could they be talking about? We discussed the fact that x is called a “variable,” a number that we aren’t sure of, a “mystery number.” This, the Youngers could understand. Of course, they wanted to know how to solve this mystery! A younger sibling, C (5 years old), was particularly determined to figure out what “x” was. This was the exchange that took place as we cleaned up from lunch:

Me: Give me two numbers. X plus something equals something else.
C: X plus 19 equals 9.
Me: Well, we need the last number to be the biggest, because we’re saying that the mystery number, x, plus something, equals something else, so the ending number has to be biggest.
C: Ok, x plus 9 equals 19.
Me: That’ll work. So, mystery number, added to 9, will give us 19. (I hold up my hands in front of him to count with my fingers). If we start at 9, how many will we have to count to get to 19? Start with the number that comes after 9, and stop me when we get to 19. What comes after 9?C: Ten.
Me: Right. Keep going. (I hold up a finger for each number he ticks off).
C: 11, 12, 13, 14, 15, 16, 17, 18, 19 . . . STOP!
Me: (holding 10 fingers in front of him) Ok, how many did it take us to get from 9 to 19? (he counts each of my fingers)
C: Ten.
Me: So what plus 9 equals 19?
C: Ten.Me: So x is what?
C: TEN!! (runs over to P and B) It’s 10! X is 10!!

The older boys congratulate him on solving the problem, and C is ecstatic. He tells 4 year old S, “I know what x is! X is 10!” She’s clearly happy to have this information with which to impress her brother. The Olders then give them another problem: X+45=90.

C and S announce in unison, “x is 10!” and the Olders laugh. The Youngers look to me, the air gone from their sails. I break the news:

"X was 10 in the problem we did. But guess what? X is not always 10. X changes when the other numbers change. The good news is, if you know two of the numbers, you can always figure out X. Do you want to try some?” They sure did. We made our way over to the chalkboard as the rest of the students were getting dressed to go outside. The paddock would have to wait for this small group! The Olders wanted to give the Youngers some problems. We started out with smaller numbers, to make things easier.

B: Ok, 5 plus x equals 8. (writes it on chalkboard) So, what you do is, you subtract the same thing from both sides and that will give you x.

S and C look completely confused. I explain the strategy we used to solve the first problem.

Me: We started with the number we knew, then counted how many it took to get to the ‘equals number,’ and that’s how we found x. That might be easier for them to understand than subtraction.

P: Ok, so what do you have to add to 5 to get 8?
Me: Try showing them with something, it will help them see it.
B: (makes tally marks on chalkboard) 6, 7, 8. So what is x?
C looks to me, unsure.
Me: How many did it take to get from 5 to 8? Count them.
C: 1, 2, 3.Me: So x is?
C: Three!!
Me: You just did algebra!
C: (to S) I did algebra! I know it! (turns to me) If I learn all the algebra now, by the time I get to college, I’ll know all the algebra already, and it’ll be easier!

C runs to get dressed for outside, leaving S with me, studying the chalkboard. I ask if she wants to try one, and she enthusiastically says yes.

Me: Let’s try this one. X plus 8 equals 10. (I write it on the board). If we have 8, how many will it take us to get to 10?(S doesn’t seem to know where to begin, so I pull out the basket of chalk pieces). We know we start with 8, so show me 8.(S counts out and lines up 8 pieces of chalk on the rug). So now that we have 8, how many more will we need to get to 10?
S: Two.
Me: Are you sure? Show me. (She takes out two more pieces of chalk, adds them to the lineup).
S: Nine, ten.
Me: So what plus 8 equals 10?
S: Two.
Me: So x is what, in this problem?
S: TWO!!!

She fills in the number 2 on the chalkboard, then runs to the cubbies to D-R-A-G Anne over to see her work. She beams as she shows Anne the equation on the board, and the chalk pieces on the floor that prove her skill. “I did algebra!” she says with a smile, and runs to join C to share the news.

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