Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW

SEASONS GREETINGS!

While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...

The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.

The extension at the bottom goes beyond what might be expected from the beginning of this exploration.

Math educators can adapt this for Algebra 1 through AP Calculus students...

THE PROBLEMS

What are the number(s) described in the following?

1. A number equals its reciprocal.

2. A number equals 25% of its reciprocal.

3.  A number equals twice its reciprocal.

4.  A number equals the opposite of its reciprocal.

5.  A number equals k times it's reciprocal. Restrictions on k? Cases?

Answers:
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined

OVERVIEW and much more...

• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:

(*) New = (Old + k/Old)/2

Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]

Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!

A Recursively Defined Sequence to Challenge Your Algebra Students

In continued tribute to Dr. Mandelbrot, here is a challenge problem for your Algebra 2 students which develops the ideas of iteration and recursively-defined sequences while providing technical skill practice.  From my own experience, even some of the strongest will trip over the details so don't be surprised if you get many different answers for the 5th term in part (c) below! We all know that current texts do not provide enough mechanical practice and this becomes more evident as our top students move into the advanced classes.


THE CHALLENGE

A sequence is defined as follows. Each term after the first is two less than three times the preceding term.

(a)  If the first term is 2, determine the 2nd through 5th terms.

(b) If the first term is 1, determine the 100th term. Explain.

(c) If the first term is x, determine simplified expressions in terms of x for the 2nd through 5th terms.  To help you verify your answers, the 5th term is 81x - 80. Show all steps clearly.  Compare your results with others in your group and resolve any discrepancies.

(d) Write a general expression for the nth term if the 1st term is x. It should work for all terms including the first! Explain your method. Proving your formula works for all n is optional.
Answer:  3^(n-1)x - (3^(n-1) - 1)
NOTE:  Students who have learned the formula for the nth term of a geometric sequence should recognize the first term in this answer! Help them to make the connection...

(e)  Extension:  Change the recursive relationship to: Each term after the first is three less than twice the preceding term.  Redo part (d) for this new sequence. The pattern is more challenging!
Ans:  2^(n-1)x - 3(2^(n-1) - 1)
NOTE: For the more advanced students, have them prove their "formula" by induction.

Final Comment: In what form do you think this kind of question would appear on the SATs and, yes, this topic is tested and has appeared!


"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught from year to year, It's got to be drummed In your dear little ear. You've got to be carefully taught." --from South Pacific

Squeezing Circles Into the Corner: An Infinite Sequence Investigation in Geometry

Another summer diversion from geometry...

The number of variations for tangent circles is endless and this is one of my all-time favorites. Math contests and SATs seem to have a preference for circles inscribed in squares or tangent circle problems and this one is along those lines. However, the real payoff comes from developing recursive thinking leading to an infinite geometric sequence and its sum! Students will be asked to intuitively "guess" the value of this infinite sum and to then verify their conjecture. Proving it requires nothing more than the classic formula for the sum of an infinite geometric series but, at the outset, this problem is eminently suitable for your geometry classes. Don't hesitate to use it in your "regular" classes. Questions that are deemed appropriate only for honors classes are often suitable for most students if the groundwork is laid (background, examples, etc.) and hints are given strategically.

PART I In the diagram above the larger circle has radius 1, the two circles are tangent to each other and to the two perpendicular segments (you can think of the larger circle being inscribed in a square if you wish).

(a) Make a conjecture from the diagram without computing: The ratio of the radius of the smaller circle to the larger is approximately
(A) 0.05 (B) 0.15 (C) 0.25 (D) 0.35 (E) 0.5
Note: This part may be omitted.

(b) Show that the radius of the smaller circle is exactly (√2 - 1)² = 3 - 2√2
How was your conjecture?
Note: Your decision about giving them the result like this. Obviously if they see part (b) on a worksheet, their estimate in part (a) will be pretty good! My intent was to focus on the method. Of course, feel free to rephrase this.

PART II
Of course we will not stop at 2 circles! Squeeze a third circle into the corner between the 2nd circle and the right angle. Determine its radius by using the result from part (a). [The key here is to think ratios!]

PART III
If we label the radius of the largest circle R₁, the radius of the 2nd circle R₂, the radius of the 3rd circle R₃, etc., we can now define an infinite sequence of these radii.
(a) Find a formula for the nth term of this sequence, n = 1,2,3,..

(b) What is the mathematical terminology for this type of sequence?

(c) Think intuitively here: From the diagram, what should be the "sum" of the original radius R₁ = 1 and the diameters of the remaining infinite collection of circles. [Another formulation: As n-->∞, this sum approaches what number?]

(d) Using the formula for the sum of an infinite geometric series, verify your conjecture in (c).

Comments:

• As always, feel free to use this with your students and revise as you see fit. However, pls use the attribution in the Creative Commons License as indicated in the sidebar.
• Finding the radius of the 2nd circle is a challenge by itself and the problem could stop there. The extensions can be assigned as a long-term project or for those wishing to do extra credit. I always liked having additional challenges for the students who were capable of going further, although relating this problem to geometric sequences or series is of importance. Of course, I am well aware of time constraints faced by the instructor.
• Your thoughts...
  

How Recursion is Tested on the SATs and much more...

[Wonderful discussion going on in the comments dealing with math terminology and the underlying concepts embedded here. As always, my post is just the appetizer. The main course is to be found in the comments section. If you enjoy this discussion you may also want to visit some of my recent posts touching on Singapore Math and a review of an important book by Alec Klein on teaching gifted children.]

The following is similar to a question that appeared on the PSAT a couple of years ago. The College Board has been testing recursively-defined sequences for some time. Students take the test, then it's 'out of sight, out of mind'. Educators may want to explore the ideas behind these problems in much greater depth. Recursive sequences are an important topic and are now included in many sets of standards. Should this be viewed as exclusively a high school topic?

Consider the sequence 11,6,5,... defined as follows:
Each term after the second is the nonnegative difference of the two preceding terms.
What numbered term has a value of zero?

Things to ponder beyond the answer...
(1) Is this question appropriate for middle schoolers?
(2) Would the terminology nonnegative difference be problematic for some students?
Note: This is the actual wording from the exam. Would you have reworded this using absolute values?
(3) If an educator wants students to develop these ideas in a more extended investigation, how much preliminary groundwork needs to be laid if any? It's important for our students to understand that mathematical researchers often start with specific examples like this without knowing the general relationship. They analyze many specific examples looking for patterns just like students can be trained to do.
(4) How many particular examples would students need before they can begin to formulate an hypothesis? Should the first two terms be restricted to positive integers (or even integers for that matter!)? Should the instructor suggest different starting values or allow the students to explore? As always, the issue is the role of the instructor while students are exploring. Teachers inexperienced with facilitating these kinds of investigations can benefit from observing those who have been doing this for awhile. It ain't obvious and it sure ain't easy!
(5) Would most students conjecture that the sequence will eventually become 'stable' at some point if the initial two terms are integers? Would middle schoolers be able to explain why this would happen?
(6) Do you think some students would be able to formulate a general theory for this type of sequence? Develop a formula for the number of terms required for the terms to reach zero? How many students in middle school or high school would suggest starting with decimals or even irrational numbers like pi? Is there ever an end to these investigations?
Well there is an end to this post! [Notice that the original question seems less significant now.]