Li and Tyler have begun to work on the problem. They are using the squares and have made this diagram and table. The teacher begins by asking them questions to see if they have understood the problem and how close they are to making a strategy.

Ms. Nguyen: So tell me what you've found. Can you explain your diagram to me?
Li: First we made a few staircases the way you had and tried to count them up.
Tyler: That middle one is wrong. It should go next. It's later.

Ms. Nguyen: What do you mean?
Li: It's got five steps, but the next one should only have four. He drew it afterwards, see?

Ms. Nguyen: (pointing to the last figure) You mean this one?
Tyler: Yes.

Ms. Nguyen: That last one looks good. Is the one you marked out part of the pattern, though? Tyler, you mentioned it would be later. But how would you get to that top step?

Li: Oh, it doesn't fit. It's adding two, and you only add one.
Ms. Nguyen: Then it isn't in the pattern. What about your table? What are you recording?

Tyler: Well, the bottom is the number of steps, and the top is the blocks you need. Oh, but it's still got 11 in it. That's not right. It should be 10.
Li: That's right. So it's 1, 3, 6, 10.

Ms. Nguyen: So should we try one more case?
Tyler: I can already tell it's going to be 15.

Ms. Nguyen: What's your reasoning about that?
Tyler: It's like you push them up. First one is 1, then you push up 2, and it makes the total 3. Then 3 come up, and you get 6. Two more pushes, and you'll get 15.
Li: What do you mean? You're not pushing anything. You're adding blocks.

Ms. Nguyen: Tyler, can you draw what you mean?
Tyler: Sure. Here.

Tyler adds lines to the drawing to show want he means and numbers the final figure.

Li: That's great. I can definitely see a pattern now.
Tyler: Me, too.

Ms. Nguyen: Can you put it into words?
Li: Each block staircase is one more than the last.

Ms. Nguyen: Is that enough for me to know how to build the staircase? I'm not sure what "one more" or "last" means?
Li: I can say it better. To find the next staircase, you add up all the ones you have, and add the new one.
Tyler: For 3, you add 1 and 2, and for the next one, you add 1, 2, and 3.

Ms. Nguyen: This is good work. What would we have to do for 50 then?
Li: Well, we'd have to know 1 to 49, and then we'd just add 50 to it.
Tyler: That's a lot of work. You might as well build them and count them.

Ms. Nguyen: Well, you've found a pattern and a method, and this is good. But remember what we learned with previous rules: There is sometimes more than one way to make a rule.

  Reflect on the problem