The Fractal Tree
Lesson 2 of 18
Objective: SWBAT describe sequences recursively and explicitly.
Yesterday we used an arithmetic sequence to represent the cost of washing all of the windows of a skyscraper. Today we will use difference sequences to represent characteristics of a fractal tree. Many of my students are not familiar with fractals, so I give them a quick overview of them. Here are the things I want them to know about them:
- Unlike most topics in high school math, fractal geometry is a very new branch of mathematics because of its reliance on computers to generate visual representations of complicated processes.
- Things you study in geometry (lines, triangles, circles) never occur in real life. Nothing exists that is perfectly linear, triangular, or circular. Object in fractal geometry are jagged, rough, and imperfect like things that actually occur in real life.
- Fractals are created using an iteration process. You designate a process and then repeat it over and over again. Theoretically the process can continue on forever. I go over the iterations for the Koch snowflake (shown below) as a demonstration. I draw it on the board and manually draw the first three iterations. When I begin to draw the fourth iteration students usually start laughing because they get the idea of how long it would actually take me to draw it - they understand how the fractal is getting exponentially more complicated.
- Fractals have self-similarity. If you "zoom in" to one portion of the fractal, it will look the same as the shape as a whole.
This is a great task because it is accessible to every single student regardless of their ability. Students can usually figure out the recursive pattern of each column easily, and then fact that they cannot easily figure out the explicit formula is perplexing. I find that my students are extra motivated to find a formula that they know so much about but cannot quite get. The last column of the table is probably the most challenging and I could feel the urgency in the room – my students were walking around the room and bouncing ideas off of other members of the class.
I will give students about 20 – 25 minutes to get as much of the worksheet completed as they can. Here are the teacher moves that I will do during this time that they are working:
- Encourage students to check the formulas for the nth step by plugging in values that they already know to see if they still work.
- If I notice that two students in a group have different formulas, I will ask if they are algebraically equivalent.
- If a student cannot find many of the formulas, then they may be working with decimals. It is much easier to notice the pattern if the number are in fraction form, so I will encourage them to write them that way if they are not making progress.
I begin our class discussion by talking about the first column. I ask a student for the formula that they got for the nth stage of the pattern. Most students write it as 3n-1 but I will ask if other forms are possible. A student may have it written as 3n/3 or 1•3n-1. The last difference may seem trivial to students, but if I write the sequence 7, 21, 63, … and ask what the formula for the nth term is, they will quickly be able to know that it is 7•3n-1. The great thing about that form is that you can see the starting value and the common ratio.
This formula has n – 1 again, just like yesterday with the skyscraper problem. I revisit this and make sure that a student can tell me that for the 50th term, for example, we multiply by 3 forty-nine times, not fifty. So the number of times we add or multiply is one less than the term number.
The last column will usually be challenging for most students. I ask them why and usually a student will say because the sequence is not arithmetic or geometric. We have not used these terms extensively, so we will discuss their meanings. When we go over the answer, I explain that it might be helpful to look at the numerator and denominator separately, and find a pattern for each.
Finally, I emphasize that it is interesting that for the last column, students could have found the next term in the sequence over and over without much difficulty, but figuring out the formula was tough. Working recursively and explicitly is very different and finding an explicit formula can be very challenging even when you know the recursive pattern. In this video I explain how I differentiate the terms recursive and explicit with my students.
This is a good stopping point for the lesson. I explain that we have worked with arithmetic and geometric sequences and ask for an example from the last two days. Then I say that each sequence can be written as a recursive or explicit formula. We will be spending the next few days looking at these concepts in depth.