The Staircase Problem / Towers / Fancy Staircases
The Staircase Problem -Towers (“Algebraic Strategies” activities)
The activity actually has three main parts to it. I had students work in pairs on each activity for about 5- 10 minutes and then we discussed each part as a group. The first part is entitled “Growing Squares” and uses table tops made out of square blocks. The first table top has one block, the second table top has four blocks, the third table top has nine blocks, and so on. Students were all able to come up with the pattern (nth table top has n2 blocks) very quickly.
The third part, “Towers”, was more challenging. This used three-dimensional shapes. They again made tables like the following:
Creating a formula was challenging for them. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. They then tried to use the previous formula from the staircases here in this problem as well. Two of the groups concluded the formula for the nth tower as: 2n^2 - n. During the last few minutes of the class period we worked together as a class to see how this formula could be derived.Overall, I was pleased with this activity and will probably try it again. There is also another activity entitled “Fancy Stairs” that I might include next time as well.
It was a short class, so students had about 20 minutes to work on it. Almost all could figure out the number of bricks in a row when they knew the actual row number. Only about half of them could describe a rule to figure out the number of bricks in a row for any number. No one used variables to describe it (even though we have done a lot of work with variables in this pre-algebra class.)
I decided to try to do one of the “Algebraic Strategies” activities (Sum of Consecutive Numbers) with two of my calculus classes on a Friday afternoon after having taken a chapter test the previous day. I had them work in groups of two in one class and in groups of three students in the other.
|Date||Sept. 30, 2005|
I used The Staircase Problem / Towers / Fancy Staircases from the Algebraic Thinking class in my HOTS class. (HOTS stands for Higher Order Thinking Skills and is a non-mandatory mini math class that we offer opposite band where we play with math topics as well as puzzles and thinking games. I have 5 students in the class this semester, which I divided into 2 groups.
One of the groups immediately saw a pattern in the staircases and computed the answers. While they could describe the rule, they could not put it into an algebraic form. The second group, while having less formal math training, actually attempted to create an algebraic formula. It was cumbersome and ugly – but it worked. They were somewhat frustrated with the what their results looked like after working the whole period on it so I sat down and we made it nicer looking together – but pointed out that it was the same thing that they created.
On the towers they developed strategies to compute the 1, 2, 3, 4, and 10th towers. I told them that once they had these done I had a story to tell them that might help them with the 100th (since they haven’t learned about arithmetic sequences yet) and then related the fable of Gauss and his teacher asking him to add all the numbers of 1 to 100 and how he arrived at the added the sum forward and backwards etc… It was a nice extension and eased some of the arithmetic while still concentrating on the patterns of the towers.
The fancy stairs were very difficult to take to an abstract level, but seem to become easier if you break time into “odd fancies” and “even fancies”.Overall we spent anywhere from about 45 minutes for the fastest (least abstract thinking) group to 90 minutes for the group that really tried to go to the abstract. I did not find any changes that I would make.
Towering numbers. This activity went very good. They did the first one done by using the picture. When we got to the third part to find a rule the faster students had it right away, but were so eager to tell the other students that they didn’t have the chance to think of it on their own. The next time I have them do this activity I will have them work in pairs or in groups of three. I think things went well and I will do towering numbers next year.