The Staircase Problem / Towers / Fancy Staircases

Date | March 2006 | ||||||||||||||||||||||||||||||

Class | Calculus classes | ||||||||||||||||||||||||||||||

Teacher | Ray Weier | ||||||||||||||||||||||||||||||

Notes |
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 “ blocks) very quickly.n2The second part entitled “ The Staircase Problem” uses pictures of staircases that have more and more steps. They are again asked to find a pattern. Most of the groups made a table of values similar to the following:
The third part, “
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: Fancy Stairs” that I might include next time as well. |
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Date | January 2006 | ||||||||||||||||||||||||||||||

Class | 8th grade | ||||||||||||||||||||||||||||||

Teacher | Joyce Cook | ||||||||||||||||||||||||||||||

Notes |
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.) |
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Date | January 2006 | ||||||||||||||||||||||||||||||

Class | calculus class | ||||||||||||||||||||||||||||||

Teacher | Ray Weier | ||||||||||||||||||||||||||||||

Notes |
I decided to try to do one of the “ The objective of the activity was to find all the possible ways to express each number from 1 to 35 as a sum of two or more consecutive counting numbers. They were given a chart to fill in and then were to answer some questions about patterns they discovered while completing the chart. Using these patterns, they were then asked to make predictions as to whether given numbers greater than 35 could be expressed as a sum of 2, 3, 4, or more consecutive counting numbers. I told the students that they had 40 minutes to look at the chart and the follow-up questions and then we would get together during the last 10 minutes of class to discuss the activity. While the students worked on the activity, I tried to walk around the classroom and listen to the discussions that were going on in the individual groups. At first, they had questions about whether they could use the number zero or negative numbers and had to be reminded what a “counting number” was. I was somewhat surprised that a few of the groups started off filling in their charts in a quite disorganized fashion. Some just took a number at random and tried to express it as different sums. It seemed as if it took them a lot longer to complete the chart than I would have expected. Because of the length of time used to fill in the chart, most groups did not have enough time to really do justice to answering the six questions posed in the worksheet. All of the groups eventually came up with a plan that allowed them to get the chart filled in. When we got together as a class during the last ten minutes to discuss any patterns they discovered, both classes made the comment that they could see patterns but that they had a difficult time putting the patterns down on paper as an algebraic expression of some type. They definitely had a hard time abstracting from the computation. One group did mention that they noticed that if they multiplied the middle number in a sequence by the number of numbers in the sequence that that would give them the sum. Overall, I was disappointed in the results of the activity. If I were to do this activity again, I would probably either spend a little more time at the beginning giving them more detailed directions or maybe go through a short, similar type of activity with them first. I would probably also give them the entire period to work on it and then have them write something up and maybe spend the first 10-15 minutes of the next day’s class period discussing their results. |
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Date | Sept. 30, 2005 | ||||||||||||||||||||||||||||||

Class | Algebra 1 | ||||||||||||||||||||||||||||||

Teacher | Becky Bruening | ||||||||||||||||||||||||||||||

Notes | I used The 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. |
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Date | September 2005 | ||||||||||||||||||||||||||||||

Class | 9th grade | ||||||||||||||||||||||||||||||

Teacher | Terry Hagen | ||||||||||||||||||||||||||||||

Notes | T |