Here is the link to the Carnival of Math Edition X.
The following is the first in a series of investigations in recursive sequences and functions for middle school and secondary students. This apparently advanced topic is accessible to prealgebra students at an introductory level. The first few parts of the investigation below are appropriate for the younger students. The remaining parts require more algebraic facility and reasoning. The problem in the title of this post doesn't begin until more than halfway down the page (after some background is developed). Do not skip the background below since it's referred to frequently in the activity. My personal experience is that this topic is highly engaging to students. Considering the connection between recursion and fractals, this topic is certainly part of most standards-based curricula. The terminology of recursion (recursively-defined sequences, recursive description, recursive function, recurrence relations, etc.) is quite confusing at first. Many confuse these ideas with iteration, a general term for describing repetitive algorithms.
Finally, from a pedagogical point of view, please note how the Rule of Four is implemented in the activity below: We start with a verbal description of the rule of formation of a sequence (in natural language), followed by a concrete numerical representation of the terms, followed by symbolic representation. One could also depict the terms graphically on a number line or in the coordinate plane if the function model is used for the sequence.
I should probably save this for the new school year but it's hard for me to suppress ideas when they begin to crystallize. I've been thinking for some time about how we can introduce recursive functions in prealgebra through advanced algebra and beyond. I enjoy taking sophisticated ideas and reducing them to basic principles, then developing lessons that explore the topic in some depth. Moreover, this particular topic reveals the interconnectedness of mathematics in a particularly elegant and beautiful way.
Background (Needed for the Investigation Below!)
Consider the sequence 1,2,4,8,...
Elementary students can generally guess the most likely value for the next term, 16. They also are expected to identify the 'rule' of forming the 'next' term, namely doubling or multiplying by 2. This is an important stage in their development of algebraic reasoning - abstraction or generalization. In addition, they should begin to recognize that the terms of the sequence can be described generally as powers of 2, even though a formal introduction to exponents normally begins in 7th grade.
Middle school students should progress to the function table format of a sequence:
Elementary and middle school students should be able to verbalize in natural language that 'you double the terms'. As educators, we need to lead them to a more formal relationship by a line of Socratic questioning like: "Double what? To get what?" Students should then be able to express the idea that each term is twice the previous term. We can ask, "Which term doesn't follow that rule?"
To symbolically describe this sequence, we can write:
a1 = 1
an+1 = 2 ⋅ an, n = 1,2,3,...
This is known as a recursive description of the sequence. Try it - replace n by 1,2, and 3 and see if it produces the terms above.
[Note: Later on, in more advanced algebra, students should be able to express this as a recursive function: f(1) = 1; f(n+1) = 2f(n), n = 1,2,3,...]
The closed or general form requires a knowledge of exponents but is accessible to 7th graders
an = 2n-1, n = 1,2,3,... Try it!
(If you're questioning my sanity (you wouldn't be the first!) about introducing such sophisticated mathematics to general 7th graders, well, I do have a legitimate basis for this curricular decision - more later...).
Powers lend themselves naturally to a recursive description and this is why I begin with the above example. Recursive thinking develops when we ask questions like:
If we know what 25 is, how would we obtain 26?
To deepen this understanding further:
If we know what 298 is, how would we obtain 2100?
Does the exponent key on a calculator help students see these relationships? Not really! The calculator is useful to demonstrate powers and exponents but not for this discussion. Later on, the graphing calculator can be used to enter recursively-defined functions (after they've learned the ideas!).
If you're very familiar with recursively defined sequences and functions, you've probably left this page already! However, the idea of an operation or function being defined in terms of itself is a beautiful and very important notion in mathematics. This type of thinking was necessary for Mandelbrot to develop the notion of fractals, which defines a process in which each stage is defined in terms of the preceding stage or stages - that is recursive thinking!
Ok, by now you're wondering what happened to the title of this blog!
TAKE ANY NUMBER, ADD THREE, DIVIDE (OR MULTIPLY) THE RESULT BY -1. NOW REPEAT THIS SEQUENCE OF OPERATIONS ON THE RESULT YOU OBTAINED.
1. Start with the number 6 and follow the instructions above. Repeat this 2 more times. List the first 4 terms of the sequence obtained. Write a brief description of what you observe about this sequence.
2. This time start with a different integer. Again, list the first 4 terms of the sequence obtained and your observations.
3. By now you've concluded that the sequence obtained will alternate in the form a,b,a,b,...
Which one of the original operations (add 3, multiply result by -1, etc.) do you believe is causing the sequence to repeat like this?
The remaining parts require algebra background.
4. If the first term is x, verify algebraically that the sequence will alternate.
5. You've now determined that the sequence appears to be repeating but not constant like a,a,a,a,... For what value of x, the first term, will the sequence be constant, i.e., all terms will have the same value?
6. Write a recursive definition (refer to how we did this for powers) for our sequence whose first term is x. We'll start you off:
a1 = ____
an = __________, n = _______
The algebraic formulation of the recursive description in #6 was fairly straightforward, since it is just a symbolic representation of the verbal "Take any number, add three, then divide the result by -1." As useful as this may be, we often a need a general formula for the nth term as a function of n, rather than in terms of preceding terms of the sequence. That last sentence was fairly complex, so here's an illustration:
Let's assume the first term is 6. Then the nth term can be described as:
an = 6 if n = 1,3,5,7,... (i.e., n is odd)
an = -9 if n = 2,4,6,... (i.e., n is even).
This would allow us to find any particular term, say the 100th term, without knowing the values of preceding terms. Such a description is known as a general description or the closed form of the sequence. Such a formula is often very hard to determine, whereas the recursive form is easier to formulate. In our problem, the general formula for the nth term had to be given in two cases or piecewise as mathematicians term it.
It is possible to give a single formula for all of the terms of our sequence as a function of n:
an = -7.5(-1)n - 1.5, n = 1,2,3,...
Verify this formula for our sequence above: 6,-9,6,-9,6,-9,...
8. Change the original problem to:
Take any number, add 2 to it and multiply the result by -1. Repeat.
(a) Starting with an original value of 6 (as the first term), list the first 5 terms of the sequence.
(b) Write a recursive description for this sequence.
(c) Write a piecewise formula for the nth term as in the background example above.
(d) Write a single formula (closed form) for the nth term in terms of n for this sequence.
9. (More Challenging)
A sequence is defined verbally by:
Take any number, add k to it and multiply the result by -1.
(a) If the first term is x, write a recursive description for this sequence.
(b) Write a piecewise formula for the nth term.
(c) (Super Challenge) Write a single formula (closed form) for the nth term in terms of n for this sequence.
Wednesday, June 20, 2007
Take any number, Add Three, Divide the Result by -1. Now Repeat this! Recursive Sequences and Functions Part I: Grades 7-12
Here is the link to the Carnival of Math Edition X.