The Skyscraper Problem
Lesson 1 of 18
Objective: SWBAT find the partial sum of an arithmetic sequence.
Launch and Explore
One of the purposes of today's lesson is to get students thinking about what they learned about sums of sequences last year in Algebra 2. I start this lesson by telling students that they must put all textbooks away and they are not to look at their phones. I find that students will look in the book, use the partial sum formula of an arithmetic sequence, but have no concept of why the formula works. (You and your students will have to be persistent for this task to be worthwhile; I talk about this in the attached video.)
I give students the task worksheet and have them read through the problem. Next I tell them that they may or may not remember a formula to figure out the total cost, but I want them to solve this problem in a way that would be accessible to a student who never took Algebra 2 and does not know any formulas. It is definitely a challenge for students to approach this problem from a conceptual level without using a formula, but it is definitely worth it.
The problem does not need much introduction (I found this task in an issue of Mathematics Teacher from NCTM). When I give students the task worksheet I let them go at it and work for about 25-30 minutes with their table groups. Here are some strategies that I used for groups who were struggling:
1. For a group that did not know how to begin: I asked them what it would cost to wash just the first floor windows. Then I would ask for the cost for the second floor windows. Then I would ask about the third, fourth, and then the 50th floor window. This helps students to see the need for a shortcut to get from the first to 50th floor.
2. For a group that does not know how to add the terms of the sequence together: I ask them to work on a simpler problem. We could easily find out that 1 + 2 + 3 + 4 + ... + 9 + 10 = 55, so how could we devise a faster way to get to that solution?
After giving students a sufficient amount of time to work on the skyscraper problem, I bring the class together and have a discussion about it. It usually helps to start with the hint of using a simpler problem and talking about 1 + 2 + 3 + 4 + 5 + ... + 9 + 10 and how we can get the answer of 55 without manually adding every number. It is difficult not to give in and just provide the method of adding the first and last term and multiplying it by the number of pairs, but the students need to see it for themselves.
If lots of time is passing and students are still not getting it, you may want to write 55 as 5(11) and see if that helps them out. My students eventually realize that there are five pairs of numbers that all add up to 11, so we could use that shortcut. My students named this the Rainbow Method because it resembles a rainbow when the pairs are connected with a curve. Then, I had students try this method with the skyscraper problem to see if they could get the cost for the 50-story building.
One of my students related the sum of the sequence to finding the area of a trapezoid. This diagram shows how the student represented the cost of each floor as a segment and then they can be arranged to form the shape of a trapezoid. The sum of the bases of the trapezoid would represent the sum of the first and last term of the sequence and the height of the trapezoid corresponds to the number of terms. I thought this was a nice connection to Geometry and it provides a visual element to the sum.
Using the Rainbow Method will be slightly different for the 103 story building. Since there are an odd number of floors, there will be 51.5 pairs. We discussed why this is and made sense of the fact that the middle term is half of the sum of the pairs of numbers. Again, I used a simpler problem of 2 + 4 + 6 + 8 + 10 to think about having an odd number of terms.
Once we review the questions from the task, I want to summarize the process for finding the partial sum of the arithmetic sequence. For the hotel with 50 floors we multiplied 25 by 1083 to get the total sum. For 1 + 2 + 3 + ... + 9 + 10 we multiplied 5 by 11 to get our answer. When I asked students to summarize this, one student said we multiply the number of pairs by the sum of each pair. Another student said that we multiply the number of terms divided by two by the sum of the first and last term.
After this summary, I ask students if they are familiar with what we worked with and we review some of the associated vocabulary. The terms I want them to bring up after today's lesson are given in the list below.
- arithmetic sequence
- arithmetic series
- common difference
- partial sum