equals zero? You all know that, and it made this easy for you. Benny, do you think you know what I mean when I wrote that?

33. Benny

Well like, any number—any—any number, um, plus, uh—

34. Ball

Not plus, minus.

35. Benny

Not—minus, uh, another number—

36. Ball

Not another number—

Benny

x, x—

Ball

The same number. Any number minus the same number—

37. Benny

Would be zero.

1:05:00

38. Ball

Would be zero. Is that true? Does everybody in the class know that?

Students

Yes

Ball

You all knew that any number minus itself would be zero. Right? And that was part of her proof and you—she didn't even have to tell you that because you all assumed it. That was one assumption she could make. Okay? So this is a general way that she could write what she was saying. It's easier than writing all those words. And sometimes in mathematics we can find really easy ways to say very big ideas. Bernadette had a very big idea, and there's an easy way to write her big idea. See how short this idea—this is?

Lin

Um hmm.

1:05:32

Ball

Can somebody try explaining in their own words what that says? Looks kind of complicated but I bet everybody in here could explain it. Who'd like to try explaining what x minus x plus 10 equals 10 means? Could you try, Lin?

39. Lin

Yeah.

40. Ball

What?

41. Lin

‘Cause, x is like—you could pick any number, like, even in the thousands or millions —

42. Ball

Right, and then what?

43. Lin

Take away—you take away—the same number that—that you picked, and then—and then it would be zero ‘cause all—cause any number take away the same number would be—be zero.

44. Ball

Um-hmm. And then? There's some large part that you didn't finish explaining. Here you are at zero and then what?

45. Lin

And then you plus 10, so there's 10.

46. Ball

Okay. I'd like everybody to write down this idea that we talked about in your notebook. And write down the part about that we assumed, that everybody knew, too. Write down that everybody in this class knows that any number minus itself is zero. You can write it in that mathematical way instead of writing all those words.

1:06:35