Atul, Brenda, and Demetrius are also working together on the problem. Here is what they have done:

Ms. Nguyen: This looks interesting. Tell me about your work.

Brenda: After we made some cases, Atul pointed out that we could make the square and look at what you are taking away.

Atul: It's easy to calculate the square for 50, or for n, it's just n squared. So the rule must be what you take away to make one specific staircase.

Ms. Nguyen: So what have you found?

Demetrius: Well, we're stuck. We've noticed that each square can contain the next smallest staircase. And we think there is an "n squared minus something" pattern.

Atul: But the minus something is the previous staircase, so you haven't gotten away from having to add them all up.

Ms. Nguyen: Brenda, can you paraphrase what Atul's just said, what you've found so far?

Brenda: Sure, we do have an equation. See, here they are building (n • n) - 1, then (n • n) - 3, and (n • n) - 6, but we'd like to find a way to express the - 3, - 6, - 10 as a general rule.

Demetrius: Your drawing makes me think of another idea. What if you worked backwards?

Atul: What do you mean?

Demetrius: Well, when you are actually building the staircase you would put block on block. But in math, you could start at 50 and look at 49. Would that help with a general rule?

Atul: That's interesting. Let's try it.

Brenda: Here's an easy case, 3 to 2. (She points to the bottom figure in the drawing.)

Ms. Nguyen: That's interesting. Has that helped?

Brenda: But you still have to do the subtraction?

Ms. Nguyen: What do you mean?

Demetrius: You have to know the size of the previous case. It's not a general rule.

Atul: But wait. I think I see something when you draw it that way.

Brenda: What?

Atul: Flip around the smaller staircase like so. It's like a pyramid now, but it's still n squared.

Ms. Nguyen: Hmm. That's interesting. Does that hold? Are we really looking at the case for 2?

Demetrius: No. Actually, that was the case for 3, but I think it will work.

Brenda: How do you mean?

Demetrius: Adjust the pyramid a little. Draw it like this. First you have the staircase. Then add n.

Atul: Now you can just make a rectangle by adding n again -- n squared + n. That's cool.

Brenda: But is that the answer? For three, n² is 9 + 3 = 12. But the staircase is only six.

Ms. Nguyen: What do you think? Is this a reasonable answer? Have we gone far enough?

Brenda: Oh, I see. This isn't the answer. The rectangle we've made through this method would build two staircases of that size, but if you divide by 2, you get the answer.

Atul: That's right. So there's a rule that will build any size staircase. Just divide in half -- N squared + n over two.

Ms. Nguyen: That's good work. Can you check it out with a few cases now and also work out the n = 50 and n = 100? Also, I'd like to come back in a bit and ask you to explain how you found this using both the diagrams and the table.

  Reflect on this group of students