Ms. Nguyen stood in front of the class and said the following: "Now that we've spent a few days looking at rules for patterns, I've got a challenge for all of us. Here is a pattern of blocks that are used to build staircases." She made this drawing on the board:

She continued: "I'm not making this with actual blocks, but I think you can see how it would work even by looking at these squares. For the first staircase, I need one block, and for the second I'll need three. For the three-step staircase, it's six."

"The actual staircase I'd like to build is 50 steps high. But how many blocks will I need? Can we apply our skill in looking for patterns and use one to answer to the question, "How many blocks are needed to build a 50-step staircase?"

"Now I'd like you to work in your groups on this. Here is a box of squares, for each group that you can use in your problem solving." As Ms. Nguyen handed out the squares a student asked, "Can we just build the one with 50 steps and count?"

She replied, "Yup, that would work, but for me, I'm sure I'd make a mistake in doing all those addition problems. So this is a case where it may be worth putting a rule to work. But remember that strategy, and after we investigate, we can compare notes."

"Okay, everybody, let's begin. One piece of advice: Please spend a little time getting a feel for this problem and looking at cases. It will help when you move on to finding out whether there is a rule and what it might be. Okay, go to it!"

Now take a moment to think of what student approaches to this problem might be. When you are ready, go to the next page to see some student dialogue.

  Students' approaches