Summary of Article:
The CAS (Computer Algebra System) of the TI-Npsire has made the use of interactive spreadsheets more common and useful in the classroom and in the common core state standards. The article then provides an investigation to be carried out on the calculator. The Devil tells Daniel to put $1,000 in a bank account. The devil will double Daniel's bank account every day. For his commission he will take 10% from the first day, so Daniel will have 1000x2-100= 1900. From then on the Devil will just double his commission each day. The task suggests that students first calculate by hand the first few days of the transaction so that they can make sure they understand what the math of the transaction is, and they can see the benefit of using a spreadsheet. In their calculators students should create three columns: the day (n), the Devil’s due for that day (d), and the balance in Daniel Webster’s account at the end of that day (w). The days should be 0-30, cell B1 is 0, and cell C1 is 1000. Cell B2 is .1*C1, cell C2 is 2*C1-B2, and cell B3 is 2*B1. Then students can let the calculator fill in the rest of each column. Students will see that eventually Daniel will owe the Devil money. When students are looking for why the deal goes so poorly for Daniel, they can look at the graphs of the two columns and the numbers in the table. Students can look at different ratios within the table. Like, the ratio of Daniel's account, current/previous, the ratio of the Devil's commission current/ previous, and the ratio of Daniel's account over the Devil's dues, Daniel/ Devil. The students can predict formulas, like Daniel/ Devil= 20-n for n greater than 0, and the devil= 100*2^(n-1). Students can then explore this situation with different initial parameters. One benefit of the CAS system is that students can replace the initial investment with i, and the following outputs will show how the i affects the result, g for the growth factor, p for the percentage. Students can see that Daniel goes broke when n= 10g. Also, Daniel's balance on day n will be 2/p-n, so Daniel=devil(2/p-n). Which students can use to find Daniel = 2^(n-1)(1000p)(2/p – n). “When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data.” This task helps students, and teachers, understand the mathematics of the transaction, and the value of the technology.
Summary of Class discussion:
We started the class discussion by giving a brief synopsis of the story problem at the beginning of the article. The Devil tells Daniel to put $1,000 in a bank account. The devil will double Daniel's bank account every day. For his commission he will take 10% from the first day, so Daniel will have 1000x2-100= 1900. From then on the Devil will just double his commission each day. We worked on the interactive whiteboard, opened up the TI- Nspire, and had a student set up the initial spreadsheet that the article describes to represent this transaction. We can see that at day 20 Daniel had $0, and from then on he is in the negatives. Dr. Leatham asked us why this sounds like such a good deal, but it doesn't end up working in our favor? What is the math behind it? He also pointed out the situation we have where our bank account amount grows and grows and grows, then suddenly goes to exactly zero, then negative. That seems like a strange "coincidence". We were following what the article told us to do, so next we graphed it. But that's actually where a lot of the discussion came in. A lot of us didn't know how to make a graph from the spreadsheet on our calculators. Dr. Leatham showed us a YouTube video that goes through these steps on a slightly older model of the TI- Nspire. Dr. Leatham pointed out that YouTube has thousands of tutorials, and it can be a very helpful resource if you don't know how to do something. We followed along with the steps shown on the video until we finally had our spreadsheet plotted in a scatter plot on the calculator. We talked about different aspects of the graph, and graphing the points. At first we had our window set at the standard 10 by 10, but we only saw one point. So we had to find new window dimensions to show our points. Dr. Leatham pointed out that this can be a good thing to do with your students because we are using mathematics related not only to the problem, but also to the calculator, and it can be good for students to think about it. We realized that this article is assuming that the reader is familiar enough with the calculator and graphing on it to be able to do this without needing to explain it. It's the math of the spreadsheet that's important to the author, not the processes that are used to figure out the math. We then looked at other things on the calculator. On the calculator in front of the class we made a column for Daniel's amount/ the Devil's amount, and saw that the ratios were steadily decreasing. Some of us also looked at ratios within each set of data. We concluded by talking about why it can be advantageous to do this on the calculator instead of on an excel spreadsheet. Although it is easier to see more of the data on a regular sized spreadsheet on the computer than on the calculator, the calculator has a CAS (computer algebra system) that allows it to do algebra within the spreadsheet. We can put in variables instead of specific numbers, and the calculator will change the data to make an equation using that variable.
Critique:
Usually in our class discussions we talk more about our thoughts about the article, rather than going through the article, but this time we pretty much went through it. At first I didn't like that we didn't talk very much about what we thought, because I figured that most of us should have already gone through the steps of the article, so we knew what happened on the calculator. I wanted to talk about this as an activity in the classroom. But while thinking through this write- up I realized that going through the steps of the article brought up some good things about teaching that might not have been brought up if we had just talked about our thoughts about using it for students. For example, we talked about what we can do as teachers if we get to part of an activity that we can't complete (we didn't know how to graph, so we looked on youtube).
Connections:
The article has a good connection to me and teaching, because it is an example of a task that I can do with students that takes advantage of technology, and incorporates some deep mathematics. The class discussion I think connects more to me because I learned more about getting prepared for lessons. I appreciated learning about more resources (youtube and the calculator) and I appreciated that we encountered a difficulty (graphing from a spreadsheet) and instead of just finding a different task, we worked through it. The task it self also made me realize that I might have technology available to me, but I might not be using it to its full extent, just like we might have never known we can make a graph from a spreadsheet on the calculator.