The Staircase Problem / Towers / Fancy Staircases

Date March 2006
Class Calculus classes
Teacher Ray Weier
Notes

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 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:

 Staircase # # of blocks 1 1 2 3 3 6 4 10 5 15 6 21 7 28
Again all were quick to notice the consecutive numbers in the right column differed this time by 2, and then 3, and then 4, and then 5, and so on ….  When I asked them for a formula for the nth staircase, most groups came up with the formula

The third part, “Towers”, was more challenging.  This used three-dimensional shapes.  They again made tables like the following:

 Tower # # of blocks 1 1 2 6 3 15 4 28 5 45 6 66

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.
Date January 2006
Class 8th grade
Teacher Joyce Cook
Notes

Towering Numbers

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.)
About ¼ of the students could figure out how to find the total number of bricks in a tower when they knew how many rows there were.  And only 2 or 3 of those could describe the rule in words.  Again, no one used variables.
I went over the problem the next class day and we talked about using variables.  I showed them how to use variables for this particular problem.  Hopefully, some will be able to on the next exercise.

Date January 2006
Class calculus class
Teacher Ray Weier
Notes

Towering Numbers

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.
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.

I thought that these students should have done a better job expressing their patterns as algebraic functions and feel that we probably need to try to do more of this type of activity to develop their skills at building rules.
Date Sept. 30, 2005
Class Algebra 1
Teacher Becky Bruening
Notes

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.
Date September 2005
Class 9th grade
Teacher Terry Hagen
Notes

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.