Today was our last day of school before a loooong spring break – we do not return until March 14. We were asked not to have any assessments today as some students have term finals tomorrow. So, I wanted to find a flexible problem that all 3 of my courses could wrestle with today. I teach AP Calculus BC, Discrete Math, and Geometry so this was a bit of a challenge. I found a lovely problem here :
I was so delighted by how my students engaged with this problem today. A little background first. My BC kids are on the verge of learning about power series so a series/sequence question is right up their alley right now. We have been talking convergence and divergence tests. I also had some competition problems in my back pocket because I knew this would not take them very long. My Geometry class just finished a chapter on similarity and we have spent the past two days playing with Taxicab Geometry. A blog post on that adventure is coming tonight or tomorrow night. My Discrete kids just finished their winter term where we studied patterns (numeric and visual) as well as some theory about voting and ballot strategies and they, too, have played with Taxicab Geometry this week.
I want to share a few of the insightful comments that some of the students made about this list of sequences. I prompted each class with one question first: Why does they say that these are related sequences?
In all of my classes students first focused on the rules for each arithmetic sequence and made observations about the pattern of differences moving from 2 to 4 to 8. In one of my BC classes a student instantly said ‘Each first term is 2^n and then you add 2^(n+1)’ Amazingly fast pattern recognition, but more than I hoped for right out of the gate. Most of his peers were taken aback and seemed happy to focus on smaller pieces. In each Discrete class and in my Geometry class I had students noticing the doubling pattern from one sequence to the next. Only my Calc students used recursive language technically, but all classes had students recognizing that pattern. It is interesting on reflection to see how formula driven (or is that formula comfortable) my Calculus students are compared to the other classes.
I tried to get a series of ‘what do you notice?’ comments going and the following popped up in every class;
- The first sequence is the only one with odds
- They are all arithmetic series (either by description or by use of the formal language)
- The difference in each sequence is increasing by an twice as much as the difference from the previous sequence
- The first term is a power of 2 (my Geometry kids needed prompting to remember about the 0 power)
- All the sequences other than the first have only even numbers
After gathering a series of observations about the sequences, we directed our attention to the charge of finding where the number 1000 might be hiding. Luckily no one wanted to list all the terms of a sequence until 1000 arrived or was passed by. So the following suggestions came my way;
- Subtract the first term from 1000 and divide by the common difference to see if 1000 is on the list
- Divide 1000 by 2 repeatedly until we arrive at a term that is more manageable and more clearly on one of the lists
- 1000 is 10^3 so we need to find 5^3 since 10 = 5 * 2 and we know that 2s are built up row by row
I was really pleased by the focus on 1000 being built up by factors of 5 and 2. One of the discrete classes built up to 1000 while the other kept dividing by 2 to bring it down to the 125 necessary. Once we were focused on 125 it was clear in all classes that the first sequence was the only place that 125 could live. My first class of the day is one of my AP Calculus BC classes and after realizing where the 1000 is there was no discussion of whether that 1000 could appear anywhere else. In my second class, one of my Discrete Math classes, they focused on the plural in the question and wondered whether there might be multiple landing spots for the 1000. We counted out 1 – 20 together on the lists and noticed that no number was repeated. We were pretty confident that this pattern would hold. In my second AP Calculus class – the one where a student generated a formula right away – he stepped up and showed a terrific proof that this had to be a unique solution. Writing each term as 2^n + (2^(n+1))*k where k represents some multiple of the number of differences in the sequence. By setting this equal to 1000 and factoring out a 2^n he made the argument that 1000 needed to be written as a product of a power of 2 and an odd number of the form 1 + 2k. Listing factors of 1000 it was pretty clear that only 8 * 125 satisfied the conditions of the problem.
Finally, my Geometry kiddos had the opportunity to dig into the problem and I was pretty darned pleased, I must say. It was the last period of the last day of school before a two week spring break. They are the youngest of all my students and they are the least experienced mathematically. What I saw today was real evidence that these students have been growing as problem solvers. They are more patient and persistent than they were in the fall and they are more willing to make guesses out loud than they were when we started the year together. I am so happy about the conversation we had. Other than the concern about whether 1000 exists in any of the other lists, they were able to nail all of the important pieces of this problem.
I discovered the problem at about 5:45 this morning and I could not be more pleased about the conversations I had with my students today.